Discussion:
Abstract Algebra Broken
(too old to reply)
Tim Golden BandTech.com
2020-08-31 12:50:10 UTC
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.

https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation

These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.

After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.

To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.

Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.

We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
Mike Terry
2020-08-31 15:10:13 UTC
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
[snip rest of ridiculous rant]

This all reads like you are serious, but you clearly have never studied
abstract algebra, and know next to nothing about it - certainly not
enough to criticise even a basic construction which you don't
understand. Or is it some kind of subtle joke? (If so I miss the point.)

Regards,
Mike.
Tim Golden BandTech.com
2020-08-31 15:40:24 UTC
Post by Mike Terry
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
[snip rest of ridiculous rant]
This all reads like you are serious, but you clearly have never studied
abstract algebra, and know next to nothing about it - certainly not
enough to criticise even a basic construction which you don't
understand. Or is it some kind of subtle joke? (If so I miss the point.)
Regards,
Mike.
Hi Mike.
a1 X
is not a binary operation. agree?
Mike Terry
2020-08-31 22:09:42 UTC
Post by Tim Golden BandTech.com
Post by Mike Terry
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
[snip rest of ridiculous rant]
This all reads like you are serious, but you clearly have never studied
abstract algebra, and know next to nothing about it - certainly not
enough to criticise even a basic construction which you don't
understand. Or is it some kind of subtle joke? (If so I miss the point.)
Regards,
Mike.
Hi Mike.
a1 X
is not a binary operation. agree?
Assuming a1 is an element of the underlying ring...

By a1 X, do you mean:

a) the polynomial a1 X
[i.e. whose X coefficient is a1, and all other coefficients zero]
b) the product of the polynomial a1
[i.e. whose constant coefficient is a1, and all others zero]
and the polynomial X
[i.e. whose X coefficient is 1, and all others zero]
c) something else?
[e.g. maybe X represents multiplication or something!]

Mike.
Lalo Torres
2020-09-01 01:51:09 UTC
" but this operation has not been defined "

Bring to the table just one known or not-known example where this is not defined

For example in square matrices :

a₀·I + a₁·X + a₂·X² + ... + aₙ₋₁·Xⁿ⁻¹ + aₙ·Xⁿ = O

I : https://en.wikipedia.org/wiki/Identity_matrix
O : https://en.wikipedia.org/wiki/Zero_matrix
· : https://en.wikipedia.org/wiki/Scalar_multiplication#Scalar_multiplication_of_matrices
https://en.wikipedia.org/wiki/Matrix_multiplication#Powers_of_a_matrix

" They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers "

tropical geometry is a semiring, but just in case :

Tropical Polynomials by Bryant Mathews (semiring)
https://circles.math.ucla.edu/circles/lib/data/Handout-2121-1848.pdf
Tropical algebra by Amanda Ellis
https://www.math.utah.edu/mathcircle/notes/MathCircleIv2.pdf

I do not know if the conflict can be solved, maybe it can be precisely outlined where is exactly the conflictive zone.

Looks in the following links, maybe, one or two links can be useful :

https://en.wikipedia.org/wiki/Quotient

https://en.wikipedia.org/wiki/Polynomial#Abstract_algebra

https://en.wikipedia.org/wiki/Exponentiation#Generalizations

https://en.wikipedia.org/wiki/Indeterminate_(variable)#Polynomials
https://math.stackexchange.com/questions/495465/what-is-an-indeterminate-in-a-polynomial-ring
https://proofwiki.org/wiki/Definition:Polynomial_Ring/Indeterminate

https://en.wikipedia.org/wiki/Polylogarithmic_function
https://en.wikipedia.org/wiki/Trigonometric_polynomial

https://proofwiki.org/wiki/Definition:Polynomial_Function/Ring
https://proofwiki.org/wiki/Definition:Polynomial_over_Ring
https://proofwiki.org/wiki/Definition:Polynomial
https://proofwiki.org/wiki/Definition:Polynomial_Ring
https://en.wikipedia.org/wiki/Ring_of_polynomial_functions

https://en.wikipedia.org/wiki/Matrix_polynomial
https://en.wikipedia.org/wiki/Polynomial_matrix
https://en.wikipedia.org/wiki/Matrix_ring
https://proofwiki.org/wiki/Ring_of_Square_Matrices_over_Ring_is_Ring

https://commalg.subwiki.org/wiki/Polynomial_ring_over_a_field
https://commalg.subwiki.org/wiki/Polynomial_ring
https://en.wikipedia.org/wiki/Algebra_over_a_field#Associative_algebras_over_rings
https://proofwiki.org/wiki/Definition_of_Polynomial_from_Polynomial_Ring_over_Sequence

https://proofwiki.org/wiki/Polynomial_Ring_of_Sequences_is_Ring
https://proofwiki.org/wiki/Definition:Ring_of_Polynomial_Forms
https://proofwiki.org/wiki/Definition:Ring_(Abstract_Algebra)

https://en.wikipedia.org/wiki/Center_(ring_theory)
https://encyclopediaofmath.org/wiki/Centre_of_a_ring

An Arithmetic for Rooted Trees by Fabrizio Luccio
http://export.arxiv.org/abs/1510.05512

https://en.wikipedia.org/wiki/Binary_operation
https://en.wikipedia.org/wiki/Binary_function
https://en.wikipedia.org/wiki/Partial_function

https://en.wikipedia.org/wiki/Homogeneous_function

https://groupprops.subwiki.org/
https://encyclopediaofmath.org/
https://commalg.subwiki.org/
https://proofwiki.org/
Tim Golden BandTech.com
2020-09-01 11:29:36 UTC
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
[snip rest of ridiculous rant]
X + ...
=
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
This all reads like you are serious, but you clearly have never studied
abstract algebra, and know next to nothing about it - certainly not
enough to criticise even a basic construction which you don't
understand. Or is it some kind of subtle joke? (If so I miss the point.)
Regards,
Mike.
Hi Mike.
a1 X
is not a binary operation. agree?
Assuming a1 is an element of the underlying ring...
a) the polynomial a1 X
[i.e. whose X coefficient is a1, and all other coefficients zero]
b) the product of the polynomial a1
[i.e. whose constant coefficient is a1, and all others zero]
and the polynomial X
This above (b) is it; just as I wrote in the O.P. except I would not call this 'the polynomial a1'
It is true that the polynomial form
0 + a1 X + 0 X X + 0 X X X + ...
is equal to
a1 X
so I see how you come to name it so. These systems of umpteen variables are true across all values and so to select zeros for most of the values is a fine usage. Now we have something simple to discuss.
Further we can use real values for a1, as is the custom in abstract algebra as they often implement the 'polynomial with real coefficients'. It is particularly at this point that full breakage occurs, though the usage of an undefined X in a product, even with itself, is a fairly sore point as well.
X of course is in its 'abstract' form. a1 and X are not in the same set. Yet there is a product being taken. This product ought to get rather some attention since it is another operator... and wasn't it just moments ago in this subject that the ring operators were so carefully laid out?
Where was the discussion of this new non-binary operator and their non-binary sums by the way as we study
a0 + a1 X
this sum can only further the problem, as now we definitely have a real (as chosen above) in sum with a non-real entity.
How such a direct contradiction in a subject that is supposedly pristine can be propagated and absorbed by so many for so long is surely a statement with broader consequences. All that most can do is to deny the breakage. Here at least a lamb has offered itself up. Thanks Mike and I hope you will pardon the rhetoric for there is actual content to discuss here. The strictness of the ring definition; it was well built. The sum and the product are sufficient without the reverse operators. This polynomial stage though; then the quotient and ideal; these things are very dirty. Should mathematicians really be taking up particle/wave duality without explicitly stating it? At least the physicists bother to explain the contradiction before they eat it en mass. The mathematicians cover it up. That is not mathematics at all, and yet the constructions stands freely and mostly unchallenged. The consequences are broad even if non-mathematical. The separation of philosophy from mathematics and from physics might just have a wee bit to do with this. These are false divisions. As the ring provides there is no need for division. We ought to do without it in the name of simplicity.
Post by Mike Terry
[i.e. whose X coefficient is 1, and all others zero]
c) something else?
[e.g. maybe X represents multiplication or something!]
Mike.
zelos...@gmail.com
2020-09-01 11:48:29 UTC
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
[snip rest of ridiculous rant]
X + ...
=
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
This all reads like you are serious, but you clearly have never studied
abstract algebra, and know next to nothing about it - certainly not
enough to criticise even a basic construction which you don't
understand. Or is it some kind of subtle joke? (If so I miss the point.)
Regards,
Mike.
Hi Mike.
a1 X
is not a binary operation. agree?
Assuming a1 is an element of the underlying ring...
a) the polynomial a1 X
[i.e. whose X coefficient is a1, and all other coefficients zero]
b) the product of the polynomial a1
[i.e. whose constant coefficient is a1, and all others zero]
and the polynomial X
This above (b) is it; just as I wrote in the O.P. except I would not call this 'the polynomial a1'
It is true that the polynomial form
0 + a1 X + 0 X X + 0 X X X + ...
is equal to
a1 X
so I see how you come to name it so. These systems of umpteen variables are true across all values and so to select zeros for most of the values is a fine usage. Now we have something simple to discuss.
Further we can use real values for a1, as is the custom in abstract algebra as they often implement the 'polynomial with real coefficients'. It is particularly at this point that full breakage occurs, though the usage of an undefined X in a product, even with itself, is a fairly sore point as well.
X of course is in its 'abstract' form. a1 and X are not in the same set. Yet there is a product being taken. This product ought to get rather some attention since it is another operator... and wasn't it just moments ago in this subject that the ring operators were so carefully laid out?
Where was the discussion of this new non-binary operator and their non-binary sums by the way as we study
a0 + a1 X
this sum can only further the problem, as now we definitely have a real (as chosen above) in sum with a non-real entity.
How such a direct contradiction in a subject that is supposedly pristine can be propagated and absorbed by so many for so long is surely a statement with broader consequences. All that most can do is to deny the breakage. Here at least a lamb has offered itself up. Thanks Mike and I hope you will pardon the rhetoric for there is actual content to discuss here. The strictness of the ring definition; it was well built. The sum and the product are sufficient without the reverse operators. This polynomial stage though; then the quotient and ideal; these things are very dirty. Should mathematicians really be taking up particle/wave duality without explicitly stating it? At least the physicists bother to explain the contradiction before they eat it en mass. The mathematicians cover it up. That is not mathematics at all, and yet the constructions stands freely and mostly unchallenged. The consequences are broad even if non-mathematical. The separation of philosophy from mathematics and from physics might just have a wee bit to do with this. These are false divisions. As the ring provides there is no need for division. We ought to do without it in the name of simplicity.
Post by Mike Terry
[i.e. whose X coefficient is 1, and all others zero]
c) something else?
[e.g. maybe X represents multiplication or something!]
Mike.
again mate, read the formal definition. There is no sum, it is only historic notation.
Mike Terry
2020-09-01 17:42:25 UTC
Post by Tim Golden BandTech.com
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
[snip rest of ridiculous rant]
X + ...
=
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
This all reads like you are serious, but you clearly have never studied
abstract algebra, and know next to nothing about it - certainly not
enough to criticise even a basic construction which you don't
understand. Or is it some kind of subtle joke? (If so I miss the point.)
Regards,
Mike.
Hi Mike.
a1 X
is not a binary operation. agree?
Assuming a1 is an element of the underlying ring...
a) the polynomial a1 X
[i.e. whose X coefficient is a1, and all other coefficients zero]
b) the product of the polynomial a1
[i.e. whose constant coefficient is a1, and all others zero]
and the polynomial X
This above (b) is it; just as I wrote in the O.P. except I would not call this 'the polynomial a1'
OK, so you are saying that you mean the product of a1 and X, so clearly
the product operation involved is a binary operation. [It involves two
operands]

There are now two possibilities: you consider a1 to be a polynomial, in
which case we have regular polynomial multiplication, which is properly
defined, or you consider a1 to be a member of the underlying ring.

It seems your issue is with the second approach. Lets focus on
polynomials over R, which is what you discuss below.

I'll assume you're ok with the definition of a polynomial over R. There
are several equivalent approaches to how these are defined, and I could
expand on this, but they all lead the same way, defining addition and
multiplication /of polynomials/.

But now we have something else: a binary operation taking a real number
and a real polynomial. Of course such an operation needs to be properly
defined to have a meaning, as it's not covered by the definition of
polynomial multiplication. Let's write the op as a binary function sm,
so sm: (R x R[X]) ---> R[x]. [R is the set of real numbers, and R[x]
the set of polynomials over R.] I choose the notation "sm" for "scalar
multiplication".

And here is the definition for sm: if a is in R and p is a polynomial
over R, represented as
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
then
sm(a,p) = (a p_0) + (a p_1) x + (a p_2) x^2 + ... + (a p_n) x^n

There is an important point here. When I wrote p = p_0 + p_1 x + p_2
x^2 + ... + p_n x^n above, I am not writing a long sequence of sums and
products on the right hand side (rhs) of the equals! If I were, you
could correctly claim the definition is circular! I am writing a
polynomial specified in whatever notation polynomials have been
previously defined, except admittedly I've just assumed for convenience
they've been introduced as expressions of this form. (Even though there
are other technical ways of introducing them, authors would typically
make a point that they can be represented in this notation, at least for
typographical convenience.)

So sm is well defined, no problem with this approach either.
Effectively, this is considering R[x] to be a module over its base ring
R, and sm is just like multiplying vectors in a vector space by a scalar
in its base field. [A module is akin to a vector space, except it has a
base /ring/ rather than a base /field/.]

And of course, now that we've defined sm, we can go on and prove basic
things like
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n =
sm(p_0, x^0) + sm(p_1, x^1) + ... + sm(p_n, x^n)

(Here, the + signs on lhs of equals are just part of the notation for a
polynomial p, not an operation sign, while the + signs on rhs ARE the
binary operation acting on R[x].)

And just to add to notation confusion, it is typical tradition to write
scalar multiplication of sm(a, p) simply as a p, as you did initially,
in which case we can also write

sm(p_0, x^0) + sm(p_1, x^1) + ... + sm(p_n, x^n) =
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n

but the meaning of rhs here is quite different from meaning of lhs
above: the rhs here IS a long sequence of (polynomial) sums and
(scalar) products.

Does it seem terrible to you that the notation is ambiguous? The point
is that we realise it is ambiguous, but provably no harm is done, as
we've proved that all the different interpretation lead to the same
result, so no harm is done. This is common practice in mathematics,
balancing simplicity of notation against formal syntactical correctness.
Post by Tim Golden BandTech.com
It is true that the polynomial form
0 + a1 X + 0 X X + 0 X X X + ...
is equal to
a1 X
so I see how you come to name it so. These systems of umpteen variables are true across all values and so to select zeros for most of the values is a fine usage. Now we have something simple to discuss.
Further we can use real values for a1, as is the custom in abstract algebra as they often implement the 'polynomial with real coefficients'. It is particularly at this point that full breakage occurs, though the usage of an undefined X in a product, even with itself, is a fairly sore point as well.
No, there is nothing "undefined" here. You've not properly grasped the
definition of polynomials I think. Perhaps my explanation above will
help, but possibly we need to go back to the basics of how polynomials
are rigorously defined, and the basic operations on them.

I wouldn't mind doing that if you're serious, or alternatively let me
know if you still think something is undefined...
Post by Tim Golden BandTech.com
X of course is in its 'abstract' form. a1 and X are not in the same set. Yet there is a product being taken. This product ought to get rather some attention since it is another operator... and wasn't it just moments ago in this subject that the ring operators were so carefully laid out?
OK, the problem here is either that you missed the definition for the
scalar product, or possibly the author of the text you're using omitted
it for whatever reason. That does not make abstract algebra "wrong" in
any way, and the problem with your OP was that it comes across as a rant.

Well, lets say it was a rant, but for most people if they were trying to
understand a new field of knowledge and didn't follow something, they
would ASK FOR HELP IN EXPLAINING THEIR CONFUSION, rather than rant on
about how the field is a pile of crap etc.. Do you see how the latter
behaviour justifiably invites laughter and (frankly) scorn from readers?

Anyway, I gave the definitions for scalar multiplication above, so
hopefully all is clear now! :)
Post by Tim Golden BandTech.com
Where was the discussion of this new non-binary operator and their non-binary sums by the way as we study
a0 + a1 X
this sum can only further the problem, as now we definitely have a real (as chosen above) in sum with a non-real entity.
In this context, when you consider the sum a0 + a1 X, it is understood
that a0 is the /polynomial/ a0 X^0, and it is simply by tradition that
we often write the shorter a0. Similar to the situation above with sm,
there are basic (provable) results which underly this slight abuse of
notation, rendering it harmless.

Specifically, it is shown that the set of polynomials of the form (a0
x^0) together with polynomial addition and multiplication is
/isomorphic/ to the set of real numbers with real number addition and
multiplication. That is, we have the correspondence

a0 x^0 <----> a0

and we show this correspondence respects the operations of addition and
multiplication appropriate for each side of the correspondence.

Example: (3 x^0) + (5 x^0) = (8 X^0) <----> 8 = 3 + 5, and also
3 x^0 <----> 3 and 5 x^0 <----> 5. (Yes, this is as obvious is it
seems!) So algebraicly R and the set of polynomials of form a x^0
behave exactly the same, and we informally identify them together in day
to day use. (This is like we identify the real number 2 with the
natural number 2, although it can be argued they are conceptually
distinct.)
Post by Tim Golden BandTech.com
How such a direct contradiction in a subject that is supposedly pristine can be propagated and absorbed by so many for so long is surely a statement with broader consequences. All that most can do is to deny the breakage. Here at least a lamb has offered itself up.
I hope I've shown there is no "breakage". At worst there is some minor
abuse of notation going on, which does no harm, and is completely
understood by everybody except you.

It may not be your fault that you missed out on a fuller explanation in
your studies, but your underlying response to this (your attitude in
posting a rant) is down to you...

Regards,
Mike.
Post by Tim Golden BandTech.com
Thanks Mike and I hope you will pardon the rhetoric for there is actual content to discuss here. The strictness of the ring definition; it was well built. The sum and the product are sufficient without the reverse operators. This polynomial stage though; then the quotient and ideal; these things are very dirty. Should mathematicians really be taking up particle/wave duality without explicitly stating it? At least the physicists bother to explain the contradiction before they eat it en mass. The mathematicians cover it up. That is not mathematics at all, and yet the constructions stands freely and mostly unchallenged. The consequences are broad even if non-mathematical. The separation of philosophy from mathematics and from physics might just have a wee bit to do with this. These are false divisions. As the ring provides there is no need for division. We ought to do without it in the name of simplicity.
Post by Mike Terry
[i.e. whose X coefficient is 1, and all others zero]
c) something else?
[e.g. maybe X represents multiplication or something!]
Mike.
Lalo Torres
2020-09-01 20:06:20 UTC
Depending on how you assign(or not assign) what names (of different algebraic structures) to what.
In case you were referring to our little ring has psychological problems, concretely, an identity problem,
and, under some circumstances, the ring pretend to be an algebraic structure, higher in the algebraic hierarchy,
compared to what actually is, specifically, through being named differently of its actual name.
And for binary operation...?
Maybe the usage of the name of an algebraic structure in a compounded name(with several words)

In some way you are requesting a meticulous/thorough examination of the formal definition and several specific examples,
while simultaneously requesting the meaning of each word being used...
Tim Golden BandTech.com
2020-09-02 11:34:59 UTC
Post by Lalo Torres
Depending on how you assign(or not assign) what names (of different algebraic structures) to what.
In case you were referring to our little ring has psychological problems, concretely, an identity problem,
and, under some circumstances, the ring pretend to be an algebraic structure, higher in the algebraic hierarchy,
compared to what actually is, specifically, through being named differently of its actual name.
And for binary operation...?
Maybe the usage of the name of an algebraic structure in a compounded name(with several words)
In some way you are requesting a meticulous/thorough examination of the formal definition and several specific examples,
while simultaneously requesting the meaning of each word being used...
Sorry, but I am not the one who built the curriculum known as abstract algebra. I have my nose rubbed in it by the status quo.
Now I rub their noses in it. You seem to think that I am playing on words as if to make english language into mathematics.
No. That said we are deep down at a very fundamental level. The subject bothers to carefully define operators, and this is a first.
It is good and clean. A compiler could work by these principles, but when that same compiler is fed
a0 + a1 X + a2 X X
is will certainly tell you that there is an error. There are five type match errors present in that statement.
If this statement is to compile not by a mathematician but by a machine then another operator must be presented, and because the subject goes to the trouble for the first few operators it ought to go to the trouble for this other operator or two.

Let's not forget that this form is developing multidimensional systems and it does so from infinite dimension. No you won't find this language in the books. It's my interpretation. Rather, staying within what is in the books is what I am trying to do here. It is clear that with real coefficients
a1 X
is not a valid construction. This problem then extends onto the entire polynomial. Undefined product operator present. Non-binary operation detected. Compilation failed. The entire curriculum is up for grabs here. This is something that the straight A's will not understand.

I think you are correct that we are dealing in a matter of language, but I am not the one who is cheating within a clearly stated language. I am the one pointing my finger at it; rubbing your noses in it. It stinks. It is a broken thing. The breaks are carefully covered up and propagated, just as Zelos attempts here.
Mostowski Collapse
2020-09-02 11:53:59 UTC
"a1 x" is not necessarily multiplicatively, but
additively closed if you include the zero,

with the usual operations of polynomial multiplication and
polynomial addition in variable X and coefficients
from lets say R you get. The result is not a general
validity, since we checked the special case of R[X]:

a1 x + a2 x = (a1 + a2) x

a1 x * a2 x = (a1 * a2) x^2

But if you use a Boolean ring, where x^2 = x, you get,
and its closed:

a1 x * a2 x = (a1 * a2) x^2 = (a1 * a2) x

Thats what Boole used, x^2 = x, or x^2 - x = 0, or:

x*(x-1) = 0

The special case is now Z[X]/(X^2-X). There is nothing
broken. Its only that you have probably learnt logic
and algebra on a cow farm.
Post by Tim Golden BandTech.com
Post by Lalo Torres
Depending on how you assign(or not assign) what names (of different algebraic structures) to what.
In case you were referring to our little ring has psychological problems, concretely, an identity problem,
and, under some circumstances, the ring pretend to be an algebraic structure, higher in the algebraic hierarchy,
compared to what actually is, specifically, through being named differently of its actual name.
And for binary operation...?
Maybe the usage of the name of an algebraic structure in a compounded name(with several words)
In some way you are requesting a meticulous/thorough examination of the formal definition and several specific examples,
while simultaneously requesting the meaning of each word being used...
Sorry, but I am not the one who built the curriculum known as abstract algebra. I have my nose rubbed in it by the status quo.
Now I rub their noses in it. You seem to think that I am playing on words as if to make english language into mathematics.
No. That said we are deep down at a very fundamental level. The subject bothers to carefully define operators, and this is a first.
It is good and clean. A compiler could work by these principles, but when that same compiler is fed
a0 + a1 X + a2 X X
is will certainly tell you that there is an error. There are five type match errors present in that statement.
If this statement is to compile not by a mathematician but by a machine then another operator must be presented, and because the subject goes to the trouble for the first few operators it ought to go to the trouble for this other operator or two.
Let's not forget that this form is developing multidimensional systems and it does so from infinite dimension. No you won't find this language in the books. It's my interpretation. Rather, staying within what is in the books is what I am trying to do here. It is clear that with real coefficients
a1 X
is not a valid construction. This problem then extends onto the entire polynomial. Undefined product operator present. Non-binary operation detected. Compilation failed. The entire curriculum is up for grabs here. This is something that the straight A's will not understand.
I think you are correct that we are dealing in a matter of language, but I am not the one who is cheating within a clearly stated language. I am the one pointing my finger at it; rubbing your noses in it. It stinks. It is a broken thing. The breaks are carefully covered up and propagated, just as Zelos attempts here.
Mostowski Collapse
2020-09-02 11:58:29 UTC
You find x^2 = x here in Boole's work,
just at the beginning:

first principles. page 16
The Mathematical Analysis by George Boole, 1847
http://www.gutenberg.org/ebooks/36884

Have Fun!
Post by Mostowski Collapse
"a1 x" is not necessarily multiplicatively, but
additively closed if you include the zero,
with the usual operations of polynomial multiplication and
polynomial addition in variable X and coefficients
from lets say R you get. The result is not a general
a1 x + a2 x = (a1 + a2) x
a1 x * a2 x = (a1 * a2) x^2
But if you use a Boolean ring, where x^2 = x, you get,
a1 x * a2 x = (a1 * a2) x^2 = (a1 * a2) x
x*(x-1) = 0
The special case is now Z[X]/(X^2-X). There is nothing
broken. Its only that you have probably learnt logic
and algebra on a cow farm.
Post by Tim Golden BandTech.com
Post by Lalo Torres
Depending on how you assign(or not assign) what names (of different algebraic structures) to what.
In case you were referring to our little ring has psychological problems, concretely, an identity problem,
and, under some circumstances, the ring pretend to be an algebraic structure, higher in the algebraic hierarchy,
compared to what actually is, specifically, through being named differently of its actual name.
And for binary operation...?
Maybe the usage of the name of an algebraic structure in a compounded name(with several words)
In some way you are requesting a meticulous/thorough examination of the formal definition and several specific examples,
while simultaneously requesting the meaning of each word being used...
Sorry, but I am not the one who built the curriculum known as abstract algebra. I have my nose rubbed in it by the status quo.
Now I rub their noses in it. You seem to think that I am playing on words as if to make english language into mathematics.
No. That said we are deep down at a very fundamental level. The subject bothers to carefully define operators, and this is a first.
It is good and clean. A compiler could work by these principles, but when that same compiler is fed
a0 + a1 X + a2 X X
is will certainly tell you that there is an error. There are five type match errors present in that statement.
If this statement is to compile not by a mathematician but by a machine then another operator must be presented, and because the subject goes to the trouble for the first few operators it ought to go to the trouble for this other operator or two.
Let's not forget that this form is developing multidimensional systems and it does so from infinite dimension. No you won't find this language in the books. It's my interpretation. Rather, staying within what is in the books is what I am trying to do here. It is clear that with real coefficients
a1 X
is not a valid construction. This problem then extends onto the entire polynomial. Undefined product operator present. Non-binary operation detected. Compilation failed. The entire curriculum is up for grabs here. This is something that the straight A's will not understand.
I think you are correct that we are dealing in a matter of language, but I am not the one who is cheating within a clearly stated language. I am the one pointing my finger at it; rubbing your noses in it. It stinks. It is a broken thing. The breaks are carefully covered up and propagated, just as Zelos attempts here.
Tim Golden BandTech.com
2020-09-02 12:40:11 UTC
Post by Mostowski Collapse
You find x^2 = x here in Boole's work,
first principles. page 16
The Mathematical Analysis by George Boole, 1847
http://www.gutenberg.org/ebooks/36884
Thanks Mostowski.
It does seem to be a good read.
When a1 is a 'real coefficient' Mostowski... is the language of the polynomial voo-doo?
I hope you are not a witch.
Post by Mostowski Collapse
Have Fun!
Post by Mostowski Collapse
"a1 x" is not necessarily multiplicatively, but
additively closed if you include the zero,
with the usual operations of polynomial multiplication and
polynomial addition in variable X and coefficients
from lets say R you get. The result is not a general
a1 x + a2 x = (a1 + a2) x
a1 x * a2 x = (a1 * a2) x^2
But if you use a Boolean ring, where x^2 = x, you get,
a1 x * a2 x = (a1 * a2) x^2 = (a1 * a2) x
x*(x-1) = 0
The special case is now Z[X]/(X^2-X). There is nothing
broken. Its only that you have probably learnt logic
and algebra on a cow farm.
Post by Tim Golden BandTech.com
Post by Lalo Torres
Depending on how you assign(or not assign) what names (of different algebraic structures) to what.
In case you were referring to our little ring has psychological problems, concretely, an identity problem,
and, under some circumstances, the ring pretend to be an algebraic structure, higher in the algebraic hierarchy,
compared to what actually is, specifically, through being named differently of its actual name.
And for binary operation...?
Maybe the usage of the name of an algebraic structure in a compounded name(with several words)
In some way you are requesting a meticulous/thorough examination of the formal definition and several specific examples,
while simultaneously requesting the meaning of each word being used...
Sorry, but I am not the one who built the curriculum known as abstract algebra. I have my nose rubbed in it by the status quo.
Now I rub their noses in it. You seem to think that I am playing on words as if to make english language into mathematics.
No. That said we are deep down at a very fundamental level. The subject bothers to carefully define operators, and this is a first.
It is good and clean. A compiler could work by these principles, but when that same compiler is fed
a0 + a1 X + a2 X X
is will certainly tell you that there is an error. There are five type match errors present in that statement.
If this statement is to compile not by a mathematician but by a machine then another operator must be presented, and because the subject goes to the trouble for the first few operators it ought to go to the trouble for this other operator or two.
Let's not forget that this form is developing multidimensional systems and it does so from infinite dimension. No you won't find this language in the books. It's my interpretation. Rather, staying within what is in the books is what I am trying to do here. It is clear that with real coefficients
a1 X
is not a valid construction. This problem then extends onto the entire polynomial. Undefined product operator present. Non-binary operation detected. Compilation failed. The entire curriculum is up for grabs here. This is something that the straight A's will not understand.
I think you are correct that we are dealing in a matter of language, but I am not the one who is cheating within a clearly stated language. I am the one pointing my finger at it; rubbing your noses in it. It stinks. It is a broken thing. The breaks are carefully covered up and propagated, just as Zelos attempts here.
Tim Golden BandTech.com
2020-09-02 12:16:17 UTC
Post by Mostowski Collapse
"a1 x" is not necessarily multiplicatively, but
Well Mostowski Collapse, a1 x clearly is a product, and that is what I am discussing. Once again somebody skips right over the fundamental construction. There is an operator already in
a1 X .
Is it a binary operator? Does it fit the ring definition? If not is that a notable detail in a subject which goes to this trouble of defining operators? Particularly when a1 is real and those binary operations are defined on reals... do you see what I see? I've just put your nose right on it here. Does it smell?
Post by Mostowski Collapse
additively closed if you include the zero,
with the usual operations of polynomial multiplication and
polynomial addition in variable X and coefficients
from lets say R you get. The result is not a general
a1 x + a2 x = (a1 + a2) x
a1 x * a2 x = (a1 * a2) x^2
But if you use a Boolean ring, where x^2 = x, you get,
a1 x * a2 x = (a1 * a2) x^2 = (a1 * a2) x
x*(x-1) = 0
The special case is now Z[X]/(X^2-X). There is nothing
broken. Its only that you have probably learnt logic
and algebra on a cow farm.
Post by Tim Golden BandTech.com
Post by Lalo Torres
Depending on how you assign(or not assign) what names (of different algebraic structures) to what.
In case you were referring to our little ring has psychological problems, concretely, an identity problem,
and, under some circumstances, the ring pretend to be an algebraic structure, higher in the algebraic hierarchy,
compared to what actually is, specifically, through being named differently of its actual name.
And for binary operation...?
Maybe the usage of the name of an algebraic structure in a compounded name(with several words)
In some way you are requesting a meticulous/thorough examination of the formal definition and several specific examples,
while simultaneously requesting the meaning of each word being used...
Sorry, but I am not the one who built the curriculum known as abstract algebra. I have my nose rubbed in it by the status quo.
Now I rub their noses in it. You seem to think that I am playing on words as if to make english language into mathematics.
No. That said we are deep down at a very fundamental level. The subject bothers to carefully define operators, and this is a first.
It is good and clean. A compiler could work by these principles, but when that same compiler is fed
a0 + a1 X + a2 X X
is will certainly tell you that there is an error. There are five type match errors present in that statement.
If this statement is to compile not by a mathematician but by a machine then another operator must be presented, and because the subject goes to the trouble for the first few operators it ought to go to the trouble for this other operator or two.
Let's not forget that this form is developing multidimensional systems and it does so from infinite dimension. No you won't find this language in the books. It's my interpretation. Rather, staying within what is in the books is what I am trying to do here. It is clear that with real coefficients
a1 X
is not a valid construction. This problem then extends onto the entire polynomial. Undefined product operator present. Non-binary operation detected. Compilation failed. The entire curriculum is up for grabs here. This is something that the straight A's will not understand.
I think you are correct that we are dealing in a matter of language, but I am not the one who is cheating within a clearly stated language. I am the one pointing my finger at it; rubbing your noses in it. It stinks. It is a broken thing. The breaks are carefully covered up and propagated, just as Zelos attempts here.
Mostowski Collapse
2020-09-02 12:23:34 UTC
Of course you can topple something. Even
with a board in front of the head you can
topple something.

Just put a toddler into a room with a tower
of toy boxes, and give him a stick. He will
surely make the tower fell down.

A polynomial a1+a2*X+..+an*X^n from a given
ring R[X] has not more information than
a tuple <a1,a2,..,an>.

Not sure whether this can lead to a revolution.

LoL
Post by Tim Golden BandTech.com
Post by Mostowski Collapse
"a1 x" is not necessarily multiplicatively, but
Well Mostowski Collapse, a1 x clearly is a product, and that is what I am discussing. Once again somebody skips right over the fundamental construction. There is an operator already in
a1 X .
Is it a binary operator? Does it fit the ring definition? If not is that a notable detail in a subject which goes to this trouble of defining operators? Particularly when a1 is real and those binary operations are defined on reals... do you see what I see? I've just put your nose right on it here. Does it smell?
Post by Mostowski Collapse
additively closed if you include the zero,
with the usual operations of polynomial multiplication and
polynomial addition in variable X and coefficients
from lets say R you get. The result is not a general
a1 x + a2 x = (a1 + a2) x
a1 x * a2 x = (a1 * a2) x^2
But if you use a Boolean ring, where x^2 = x, you get,
a1 x * a2 x = (a1 * a2) x^2 = (a1 * a2) x
x*(x-1) = 0
The special case is now Z[X]/(X^2-X). There is nothing
broken. Its only that you have probably learnt logic
and algebra on a cow farm.
Post by Tim Golden BandTech.com
Post by Lalo Torres
Depending on how you assign(or not assign) what names (of different algebraic structures) to what.
In case you were referring to our little ring has psychological problems, concretely, an identity problem,
and, under some circumstances, the ring pretend to be an algebraic structure, higher in the algebraic hierarchy,
compared to what actually is, specifically, through being named differently of its actual name.
And for binary operation...?
Maybe the usage of the name of an algebraic structure in a compounded name(with several words)
In some way you are requesting a meticulous/thorough examination of the formal definition and several specific examples,
while simultaneously requesting the meaning of each word being used...
Sorry, but I am not the one who built the curriculum known as abstract algebra. I have my nose rubbed in it by the status quo.
Now I rub their noses in it. You seem to think that I am playing on words as if to make english language into mathematics.
No. That said we are deep down at a very fundamental level. The subject bothers to carefully define operators, and this is a first.
It is good and clean. A compiler could work by these principles, but when that same compiler is fed
a0 + a1 X + a2 X X
is will certainly tell you that there is an error. There are five type match errors present in that statement.
If this statement is to compile not by a mathematician but by a machine then another operator must be presented, and because the subject goes to the trouble for the first few operators it ought to go to the trouble for this other operator or two.
Let's not forget that this form is developing multidimensional systems and it does so from infinite dimension. No you won't find this language in the books. It's my interpretation. Rather, staying within what is in the books is what I am trying to do here. It is clear that with real coefficients
a1 X
is not a valid construction. This problem then extends onto the entire polynomial. Undefined product operator present. Non-binary operation detected. Compilation failed. The entire curriculum is up for grabs here. This is something that the straight A's will not understand.
I think you are correct that we are dealing in a matter of language, but I am not the one who is cheating within a clearly stated language. I am the one pointing my finger at it; rubbing your noses in it. It stinks. It is a broken thing. The breaks are carefully covered up and propagated, just as Zelos attempts here.
Mostowski Collapse
2020-09-02 12:25:29 UTC
Corr.:

Just put a blind toddler into a room with a
pinata somewhere, and give him a stick. He will
surely make hit the pinata sooner or later.
Post by Mostowski Collapse
Of course you can topple something. Even
with a board in front of the head you can
topple something.
Just put a toddler into a room with a tower
of toy boxes, and give him a stick. He will
surely make the tower fell down.
A polynomial a1+a2*X+..+an*X^n from a given
ring R[X] has not more information than
a tuple <a1,a2,..,an>.
Not sure whether this can lead to a revolution.
LoL
Post by Tim Golden BandTech.com
Post by Mostowski Collapse
"a1 x" is not necessarily multiplicatively, but
Well Mostowski Collapse, a1 x clearly is a product, and that is what I am discussing. Once again somebody skips right over the fundamental construction. There is an operator already in
a1 X .
Is it a binary operator? Does it fit the ring definition? If not is that a notable detail in a subject which goes to this trouble of defining operators? Particularly when a1 is real and those binary operations are defined on reals... do you see what I see? I've just put your nose right on it here. Does it smell?
Post by Mostowski Collapse
additively closed if you include the zero,
with the usual operations of polynomial multiplication and
polynomial addition in variable X and coefficients
from lets say R you get. The result is not a general
a1 x + a2 x = (a1 + a2) x
a1 x * a2 x = (a1 * a2) x^2
But if you use a Boolean ring, where x^2 = x, you get,
a1 x * a2 x = (a1 * a2) x^2 = (a1 * a2) x
x*(x-1) = 0
The special case is now Z[X]/(X^2-X). There is nothing
broken. Its only that you have probably learnt logic
and algebra on a cow farm.
Post by Tim Golden BandTech.com
Post by Lalo Torres
Depending on how you assign(or not assign) what names (of different algebraic structures) to what.
In case you were referring to our little ring has psychological problems, concretely, an identity problem,
and, under some circumstances, the ring pretend to be an algebraic structure, higher in the algebraic hierarchy,
compared to what actually is, specifically, through being named differently of its actual name.
And for binary operation...?
Maybe the usage of the name of an algebraic structure in a compounded name(with several words)
In some way you are requesting a meticulous/thorough examination of the formal definition and several specific examples,
while simultaneously requesting the meaning of each word being used...
Sorry, but I am not the one who built the curriculum known as abstract algebra. I have my nose rubbed in it by the status quo.
Now I rub their noses in it. You seem to think that I am playing on words as if to make english language into mathematics.
No. That said we are deep down at a very fundamental level. The subject bothers to carefully define operators, and this is a first.
It is good and clean. A compiler could work by these principles, but when that same compiler is fed
a0 + a1 X + a2 X X
is will certainly tell you that there is an error. There are five type match errors present in that statement.
If this statement is to compile not by a mathematician but by a machine then another operator must be presented, and because the subject goes to the trouble for the first few operators it ought to go to the trouble for this other operator or two.
Let's not forget that this form is developing multidimensional systems and it does so from infinite dimension. No you won't find this language in the books. It's my interpretation. Rather, staying within what is in the books is what I am trying to do here. It is clear that with real coefficients
a1 X
is not a valid construction. This problem then extends onto the entire polynomial. Undefined product operator present. Non-binary operation detected. Compilation failed. The entire curriculum is up for grabs here. This is something that the straight A's will not understand.
I think you are correct that we are dealing in a matter of language, but I am not the one who is cheating within a clearly stated language. I am the one pointing my finger at it; rubbing your noses in it. It stinks. It is a broken thing. The breaks are carefully covered up and propagated, just as Zelos attempts here.
Tim Golden BandTech.com
2020-09-02 12:46:22 UTC
Post by Mostowski Collapse
Just put a blind toddler into a room with a
pinata somewhere, and give him a stick. He will
surely make hit the pinata sooner or later.
Yes; quite so. Along the way likely some principles will be gained through the stick as well.
As subjects go the toddler stage is down there in the definition of the ring.
Fundamentals; then suddenly up pops a polynomial in abstract X; something familiar yet quite different too.
Is it so wrong to pick apart that polynomial and have a look at what it is made of?
Would it be so wrong to pick apart the real number and see what it is made of?
Mimicry is far more operant amongst mathematicians than is appreciated.
Toddlers too.
Post by Mostowski Collapse
Post by Mostowski Collapse
Of course you can topple something. Even
with a board in front of the head you can
topple something.
Just put a toddler into a room with a tower
of toy boxes, and give him a stick. He will
surely make the tower fell down.
A polynomial a1+a2*X+..+an*X^n from a given
ring R[X] has not more information than
a tuple <a1,a2,..,an>.
Not sure whether this can lead to a revolution.
LoL
Post by Tim Golden BandTech.com
Post by Mostowski Collapse
"a1 x" is not necessarily multiplicatively, but
Well Mostowski Collapse, a1 x clearly is a product, and that is what I am discussing. Once again somebody skips right over the fundamental construction. There is an operator already in
a1 X .
Is it a binary operator? Does it fit the ring definition? If not is that a notable detail in a subject which goes to this trouble of defining operators? Particularly when a1 is real and those binary operations are defined on reals... do you see what I see? I've just put your nose right on it here. Does it smell?
Post by Mostowski Collapse
additively closed if you include the zero,
with the usual operations of polynomial multiplication and
polynomial addition in variable X and coefficients
from lets say R you get. The result is not a general
a1 x + a2 x = (a1 + a2) x
a1 x * a2 x = (a1 * a2) x^2
But if you use a Boolean ring, where x^2 = x, you get,
a1 x * a2 x = (a1 * a2) x^2 = (a1 * a2) x
x*(x-1) = 0
The special case is now Z[X]/(X^2-X). There is nothing
broken. Its only that you have probably learnt logic
and algebra on a cow farm.
Post by Tim Golden BandTech.com
Post by Lalo Torres
Depending on how you assign(or not assign) what names (of different algebraic structures) to what.
In case you were referring to our little ring has psychological problems, concretely, an identity problem,
and, under some circumstances, the ring pretend to be an algebraic structure, higher in the algebraic hierarchy,
compared to what actually is, specifically, through being named differently of its actual name.
And for binary operation...?
Maybe the usage of the name of an algebraic structure in a compounded name(with several words)
In some way you are requesting a meticulous/thorough examination of the formal definition and several specific examples,
while simultaneously requesting the meaning of each word being used...
Sorry, but I am not the one who built the curriculum known as abstract algebra. I have my nose rubbed in it by the status quo.
Now I rub their noses in it. You seem to think that I am playing on words as if to make english language into mathematics.
No. That said we are deep down at a very fundamental level. The subject bothers to carefully define operators, and this is a first.
It is good and clean. A compiler could work by these principles, but when that same compiler is fed
a0 + a1 X + a2 X X
is will certainly tell you that there is an error. There are five type match errors present in that statement.
If this statement is to compile not by a mathematician but by a machine then another operator must be presented, and because the subject goes to the trouble for the first few operators it ought to go to the trouble for this other operator or two.
Let's not forget that this form is developing multidimensional systems and it does so from infinite dimension. No you won't find this language in the books. It's my interpretation. Rather, staying within what is in the books is what I am trying to do here. It is clear that with real coefficients
a1 X
is not a valid construction. This problem then extends onto the entire polynomial. Undefined product operator present. Non-binary operation detected. Compilation failed. The entire curriculum is up for grabs here. This is something that the straight A's will not understand.
I think you are correct that we are dealing in a matter of language, but I am not the one who is cheating within a clearly stated language. I am the one pointing my finger at it; rubbing your noses in it. It stinks. It is a broken thing. The breaks are carefully covered up and propagated, just as Zelos attempts here.
Tim Golden BandTech.com
2020-09-02 11:49:16 UTC
So below here you state:
"At worst there is some minor abuse of notation going on, which does no harm..."
What exactly is the abuse of notation that you are speaking of?
Just a few short lines will do.
Does the abuse you speak of occur when we write
a1 X
where a1 is real and X is not real?
If so then aren't you conceding my point?
Next of course we really ought to consider how minor abuse can lead to major abuse.
For instance if the abuse in
a1 X
is minor then is the abuse in
a0 + a1 X
larger than the abuse in the more minor abusive form?
Post by Mike Terry
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
[snip rest of ridiculous rant]
X + ...
=
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
This all reads like you are serious, but you clearly have never studied
abstract algebra, and know next to nothing about it - certainly not
enough to criticise even a basic construction which you don't
understand. Or is it some kind of subtle joke? (If so I miss the point.)
Regards,
Mike.
Hi Mike.
a1 X
is not a binary operation. agree?
Assuming a1 is an element of the underlying ring...
a) the polynomial a1 X
[i.e. whose X coefficient is a1, and all other coefficients zero]
b) the product of the polynomial a1
[i.e. whose constant coefficient is a1, and all others zero]
and the polynomial X
This above (b) is it; just as I wrote in the O.P. except I would not call this 'the polynomial a1'
OK, so you are saying that you mean the product of a1 and X, so clearly
the product operation involved is a binary operation. [It involves two
operands]
There are now two possibilities: you consider a1 to be a polynomial, in
which case we have regular polynomial multiplication, which is properly
defined, or you consider a1 to be a member of the underlying ring.
It seems your issue is with the second approach. Lets focus on
polynomials over R, which is what you discuss below.
I'll assume you're ok with the definition of a polynomial over R. There
are several equivalent approaches to how these are defined, and I could
expand on this, but they all lead the same way, defining addition and
multiplication /of polynomials/.
But now we have something else: a binary operation taking a real number
and a real polynomial. Of course such an operation needs to be properly
defined to have a meaning, as it's not covered by the definition of
polynomial multiplication. Let's write the op as a binary function sm,
so sm: (R x R[X]) ---> R[x]. [R is the set of real numbers, and R[x]
the set of polynomials over R.] I choose the notation "sm" for "scalar
multiplication".
And here is the definition for sm: if a is in R and p is a polynomial
over R, represented as
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
then
sm(a,p) = (a p_0) + (a p_1) x + (a p_2) x^2 + ... + (a p_n) x^n
There is an important point here. When I wrote p = p_0 + p_1 x + p_2
x^2 + ... + p_n x^n above, I am not writing a long sequence of sums and
products on the right hand side (rhs) of the equals! If I were, you
could correctly claim the definition is circular! I am writing a
polynomial specified in whatever notation polynomials have been
previously defined, except admittedly I've just assumed for convenience
they've been introduced as expressions of this form. (Even though there
are other technical ways of introducing them, authors would typically
make a point that they can be represented in this notation, at least for
typographical convenience.)
So sm is well defined, no problem with this approach either.
Effectively, this is considering R[x] to be a module over its base ring
R, and sm is just like multiplying vectors in a vector space by a scalar
in its base field. [A module is akin to a vector space, except it has a
base /ring/ rather than a base /field/.]
And of course, now that we've defined sm, we can go on and prove basic
things like
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n =
sm(p_0, x^0) + sm(p_1, x^1) + ... + sm(p_n, x^n)
(Here, the + signs on lhs of equals are just part of the notation for a
polynomial p, not an operation sign, while the + signs on rhs ARE the
binary operation acting on R[x].)
And just to add to notation confusion, it is typical tradition to write
scalar multiplication of sm(a, p) simply as a p, as you did initially,
in which case we can also write
sm(p_0, x^0) + sm(p_1, x^1) + ... + sm(p_n, x^n) =
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
but the meaning of rhs here is quite different from meaning of lhs
above: the rhs here IS a long sequence of (polynomial) sums and
(scalar) products.
Does it seem terrible to you that the notation is ambiguous? The point
is that we realise it is ambiguous, but provably no harm is done, as
we've proved that all the different interpretation lead to the same
result, so no harm is done. This is common practice in mathematics,
balancing simplicity of notation against formal syntactical correctness.
It is true that the polynomial form
0 + a1 X + 0 X X + 0 X X X + ...
is equal to
a1 X
so I see how you come to name it so. These systems of umpteen variables are true across all values and so to select zeros for most of the values is a fine usage. Now we have something simple to discuss.
Further we can use real values for a1, as is the custom in abstract algebra as they often implement the 'polynomial with real coefficients'. It is particularly at this point that full breakage occurs, though the usage of an undefined X in a product, even with itself, is a fairly sore point as well.
No, there is nothing "undefined" here. You've not properly grasped the
definition of polynomials I think. Perhaps my explanation above will
help, but possibly we need to go back to the basics of how polynomials
are rigorously defined, and the basic operations on them.
I wouldn't mind doing that if you're serious, or alternatively let me
know if you still think something is undefined...
X of course is in its 'abstract' form. a1 and X are not in the same set. Yet there is a product being taken. This product ought to get rather some attention since it is another operator... and wasn't it just moments ago in this subject that the ring operators were so carefully laid out?
OK, the problem here is either that you missed the definition for the
scalar product, or possibly the author of the text you're using omitted
it for whatever reason. That does not make abstract algebra "wrong" in
any way, and the problem with your OP was that it comes across as a rant.
Well, lets say it was a rant, but for most people if they were trying to
understand a new field of knowledge and didn't follow something, they
would ASK FOR HELP IN EXPLAINING THEIR CONFUSION, rather than rant on
about how the field is a pile of crap etc.. Do you see how the latter
behaviour justifiably invites laughter and (frankly) scorn from readers?
Anyway, I gave the definitions for scalar multiplication above, so
hopefully all is clear now! :)
Where was the discussion of this new non-binary operator and their non-binary sums by the way as we study
a0 + a1 X
this sum can only further the problem, as now we definitely have a real (as chosen above) in sum with a non-real entity.
In this context, when you consider the sum a0 + a1 X, it is understood
that a0 is the /polynomial/ a0 X^0, and it is simply by tradition that
we often write the shorter a0. Similar to the situation above with sm,
there are basic (provable) results which underly this slight abuse of
notation, rendering it harmless.
Specifically, it is shown that the set of polynomials of the form (a0
x^0) together with polynomial addition and multiplication is
/isomorphic/ to the set of real numbers with real number addition and
multiplication. That is, we have the correspondence
a0 x^0 <----> a0
and we show this correspondence respects the operations of addition and
multiplication appropriate for each side of the correspondence.
Example: (3 x^0) + (5 x^0) = (8 X^0) <----> 8 = 3 + 5, and also
3 x^0 <----> 3 and 5 x^0 <----> 5. (Yes, this is as obvious is it
seems!) So algebraicly R and the set of polynomials of form a x^0
behave exactly the same, and we informally identify them together in day
to day use. (This is like we identify the real number 2 with the
natural number 2, although it can be argued they are conceptually
distinct.)
How such a direct contradiction in a subject that is supposedly pristine can be propagated and absorbed by so many for so long is surely a statement with broader consequences. All that most can do is to deny the breakage. Here at least a lamb has offered itself up.
I hope I've shown there is no "breakage". At worst there is some minor
abuse of notation going on, which does no harm, and is completely
understood by everybody except you.
It may not be your fault that you missed out on a fuller explanation in
your studies, but your underlying response to this (your attitude in
posting a rant) is down to you...
Regards,
Mike.
Thanks Mike and I hope you will pardon the rhetoric for there is actual content to discuss here. The strictness of the ring definition; it was well built. The sum and the product are sufficient without the reverse operators. This polynomial stage though; then the quotient and ideal; these things are very dirty. Should mathematicians really be taking up particle/wave duality without explicitly stating it? At least the physicists bother to explain the contradiction before they eat it en mass. The mathematicians cover it up. That is not mathematics at all, and yet the constructions stands freely and mostly unchallenged. The consequences are broad even if non-mathematical. The separation of philosophy from mathematics and from physics might just have a wee bit to do with this. These are false divisions. As the ring provides there is no need for division. We ought to do without it in the name of simplicity.
Post by Mike Terry
[i.e. whose X coefficient is 1, and all others zero]
c) something else?
[e.g. maybe X represents multiplication or something!]
Mike.
Mike Terry
2020-09-02 14:40:55 UTC
Post by Tim Golden BandTech.com
"At worst there is some minor abuse of notation going on, which does no harm..."
What exactly is the abuse of notation that you are speaking of?
Just a few short lines will do.
Certain expressions could be formally interpreted in multiple
(equivalent) ways. In an automated proof system, these would need to be
disambiguated, resulting in a reduction of readability, which is why
mathematicians typically allow such things to go on - no harm is done,
and readability is improved.
Post by Tim Golden BandTech.com
Does the abuse you speak of occur when we write
a1 X
where a1 is real and X is not real?
If you read my answer properly you would know.
Post by Tim Golden BandTech.com
If so then aren't you conceding my point?
No, because no harm is done, and abstract algebra is not "broken" as you
were suggesting, and nothing needs to change. Your rant is just that -
a cranky rant, making ridiculous accusations and claims.
Post by Tim Golden BandTech.com
Next of course we really ought to consider how minor abuse can lead to major abuse.
For instance if the abuse in
a1 X
is minor then is the abuse in
a0 + a1 X
larger than the abuse in the more minor abusive form?
No, about the same, and also totally harmless. So nothing has led to
any "major abuse".

I get the impression you've not properly read what I wrote and
definitely not learned anything from it, so I won't be commenting
further, unless I feel you are trying to genuinely understand something

Mike.
Julio Di Egidio
2020-09-02 15:03:47 UTC
Post by Mike Terry
Post by Tim Golden BandTech.com
"At worst there is some minor abuse of notation going on, which does no harm..."
What exactly is the abuse of notation that you are speaking of?
Just a few short lines will do.
Certain expressions could be formally interpreted in multiple
(equivalent) ways. In an automated proof system, these would need to be
disambiguated, resulting in a reduction of readability, which is why
mathematicians typically allow such things to go on - no harm is done,
and readability is improved.
Alas, as usual, I find your approach totally misguided:

- Mathematicians give careful definitions, then the notation is
neither ambiguous nor problematic, it just depends on the specific
context.

- Indeed, if one is uncomfortable with symbol overloading, then
they better find a different hobby, as maths is full of it: OTOH,
that is certainly not a problem for a machine.

- Moreover, the reality here is that Mr Golden is not complaining
about notation, he apparently is just confused about the definitions
involved. (To dislike the notation, he'd firstly have to agree on
the definitions.)

Bah...

Julio
Tim Golden BandTech.com
2020-09-10 12:17:20 UTC
Post by Julio Di Egidio
Post by Mike Terry
Post by Tim Golden BandTech.com
"At worst there is some minor abuse of notation going on, which does no harm..."
What exactly is the abuse of notation that you are speaking of?
Just a few short lines will do.
Certain expressions could be formally interpreted in multiple
(equivalent) ways. In an automated proof system, these would need to be
disambiguated, resulting in a reduction of readability, which is why
mathematicians typically allow such things to go on - no harm is done,
and readability is improved.
- Mathematicians give careful definitions, then the notation is
neither ambiguous nor problematic, it just depends on the specific
context.
This is the quintessential mathematician statement above here. Within Julio's tract lays no concern of skepticism or scrutiny. These things have been judged by those who came before us. They have been written down. One must preserve the book. That we each are individually burdened with and by the information that we adopt... this seems to have been an overlooked detail within the upbringing of this generation. Why?

It is as if the mathematician can make no error. Can make no mistake. That the current state is the final state. I can assure the onlooker that this is not the case. That infinite requirements on a system that is about to be whittled down to just a few components is not necessary. That the poor student who goes through this topic believing that his mind is weak for its inability to absorb cleanly and clearly the subject at hand... all the while paying a steep price not just mentally but fiscally... follow the money. Oh what's that... you didn't sweat through abstract algebra? Well, that then places you in a camp of humans who are highly unmathematical. We learn as children a language out of the air it would seem. Whatever language; no matter how contorted; the human does in fact work like that. But mathematics is supposed to be something else altogether; yet what it has become is a game of follow the leader with no scrutiny possible on the leader. That role will be carefully passed down to the best mimic.

What Julio of when the notation is ambiguous? What then sir?
Post by Julio Di Egidio
- Indeed, if one is uncomfortable with symbol overloading, then
they better find a different hobby, as maths is full of it: OTOH,
that is certainly not a problem for a machine.
- Moreover, the reality here is that Mr Golden is not complaining
about notation, he apparently is just confused about the definitions
involved. (To dislike the notation, he'd firstly have to agree on
the definitions.)
Bah...
Julio
zelos...@gmail.com
2020-09-10 12:26:06 UTC
Post by Tim Golden BandTech.com
It is as if the mathematician can make no error. Can make no mistake.
They can, have dand does but here there is no mistake.

The mistake is by you, you don't understand formal constructions.
John Mathlseed
2020-09-11 22:37:07 UTC
https://we.tl/t-RVjcpZ3ri9
Lalo T.
2020-09-12 09:02:46 UTC
(A)

Forget also about infinities. Just the cases up to the second, maybe the third power.

(I) A₀*I + A₁*X + A₂*X² + a₃*X³ = A₀*I + A₁*X + A₂*(X*X) + A₃*(X*(X*X))

(II) a₀·I + a₁·X + a₂·X² + a₃·X³ = a₀·I + a₁·X + a₂·(X*X) + a₃·(X*(X*X))

Name (I) as "ring" and name (II) as "pineapple with outsourcing"
If you choose either (I) or (II), in both cases will yield a square matrix as
a result. Here you can change the sets that you are using, but for the sake of
instantiation, we will restrict it, capital letters for square matrices
and lowercase letters to reals numbers, just to emphasize that there are
a mix of different objects, as you requested ( in (II) )

"Ring" has two operations, internal addition '+' and internal multiplication '*'
"Pineapple with outsourcing" has three operations, internal addition '+'
and internal multiplication '*' and external multiplication '·'

Certainly "pineapple with outsourcing" violates a principle of the "ring"

So, up to here, there should be no conflict.

I can say that "pineapple with outsourcing" is not a "ring", but prejudging
a "pineapple with outsourcing" as a "ring" is a bit weird. (here I do not want
to introduce "polynomial rings", it seems that none agreement can be attained
in that area)

(B)

(bI) The topic of Unit Ring https://mathworld.wolfram.com/UnitRing.html

Take a look to the section "Notes_on_the_definition" :
https://en.wikipedia.org/wiki/Ring_%28mathematics%29#Notes_on_the_definition

https://en.wikipedia.org/wiki/Mathematical_folklore
the word "folklore" is listed in https://en.wikipedia.org/wiki/List_of_mathematical_jargon

(bII) coefficients "from" the distributive property :

p = 1 + X + X² + X³ and q = 1 + X + X²

P + q = ...
P x q = ...

(this is just an arbitrary bit, just in case)

(bIII) the topic of "compatibility" ("underground wiring")

(bIV) What are the actual usages of the polynomial inside and outside mathematics ?

(bV) What is the meaning of " X*X " ?

(bVI) Structures With External Binary Operations ("extra structure")
https://www.mathyma.com/mathsNotes/index.php?trg=S1C2_Struct_ExtBinOp

(C) Now, very different is to have objections with 'allowing the entry of
foreign/external elements'. At some point in the thread, another topic, parallel
to te initial topic, begin to being landed.

You may be referring not only to your particular instance (your model example) but, also
an abtraction "behind" your construct. Maybe other way to say it, is looking for
this feature in the abstraction layer.

What is your take on the "four types" ? in :
https://en.wikipedia.org/wiki/External_(mathematics)#Generalizations

It is not arbitrary that you would not accept modules and vector spaces.

Tim, when you say "...and without any need of the Cartesian product.", I
understand it in this "generalized" context.

Mostowsky "kick the wasp nest" towards Logic
Ross brought to the table https://en.wikipedia.org/wiki/Quasi-invariant_measure
I have not really in detail all the posts anyway.

https://en.wikipedia.org/wiki/Quantity_calculus

In some past post/message of another context, you said something about the usage of the word "linear"
https://en.wikipedia.org/wiki/Linear_map

https://math.stackexchange.com/questions/796374/binary-operation-english-terminology

Note the word " Boolean algebra" in :
https://en.wikipedia.org/wiki/Clopen_set

The exact context of why I put the "Code smell" link is because in the past, one (or maybe more) times,
you mention that abstract algebra has something wrong but it still manages to, somehow, keep its functioning.

One question could be 'In which cases the External Binary Operation can be reduced to an Binary Operation ?'

One could search and analyse cases where it is presumably impossible to reduce it to a binary operation.

There are more literature about other study cases?

What is the interpretation from the Physics in this issue ?

Explain the closure property in a polysign context (s1,m1)*(s2,m2) and (s1,m1) @ (s2,m2)
Is it really mandatory to use extra symbols to identify the addition of
magnitudes and the addition of signs besides the symbol '@' (addition of terms)

(D) In which aspects Geometry seems more leveraged towards the "square" instead of the "triangle" ?

In the topic of Area :

https://en.wikipedia.org/wiki/Hyperrectangle
and no word special word for elongated regular hexagons (like "leftangle")
https://en.wikipedia.org/wiki/Parallelogon
https://en.wikipedia.org/wiki/Egyptian_geometry
https://en.wikipedia.org/wiki/Square_metre
https://en.wikipedia.org/wiki/Ancient_Egyptian_units_of_measurement

"Ternary arithmetic, factorization, and the class number one problem" by Aram Bingham
https://arxiv.org/pdf/2002.02059.pdf

In the topic of Angles :

There is a word for 90 degree angles (right angle), is there a word for 60 and 120 degrees angles ?
Where is the terminology for 60/120 degrees angles ?
https://www.inverse.com/article/45157-michael-jackson-anti-gravity-lean-biomechanis

https://en.wikipedia.org/wiki/Perpendicular

In the topic of tillings :

https://en.wikipedia.org/wiki/Square_tiling

https://en.wikipedia.org/wiki/Hexagonal_tiling
https://en.wikipedia.org/wiki/Rhombille_tiling

In the topic of honeycombs :

https://en.wikipedia.org/wiki/Cubic_honeycomb

https://en.wikipedia.org/wiki/Rhombic_dodecahedron
https://en.wikipedia.org/wiki/Rhombic_dodecahedral_honeycomb
https://en.wikipedia.org/wiki/Trigonal_trapezohedron#Asymmetric_variation
https://en.wikipedia.org/wiki/Trigonal_trapezohedral_honeycomb

Pythagorean-like theoprems :

Possible veins in polysigns :

- the user L.walker put the question if Polysign p5 is a field with fractions (instead of reals)

- The user Tommy1729 seems to have some kind of approach toward number theory that include polysigns
https://en.wikipedia.org/wiki/Geometry_of_numbers ??

- Polysigns p5 and the complexity of pentagons

(E) Others topics :

https://en.wikipedia.org/wiki/Tetrachromacy

I am recalling well, Krishnamurti use the phrase "don't institutionalize" (or something similar)
(probably in some documentary about the Schools in India related with him
I do not have the link in hand)

Gears and education :
Tim Golden BandTech.com
2020-09-12 11:47:06 UTC
Post by Lalo T.
(A)
Forget also about infinities. Just the cases up to the second, maybe the third power.
(I) A₀*I + A₁*X + A₂*X² + a₃*X³ = A₀*I + A₁*X + A₂*(X*X) + A₃*(X*(X*X))
(II) a₀·I + a₁·X + a₂·X² + a₃·X³ = a₀·I + a₁·X + a₂·(X*X) + a₃·(X*(X*X))
Name (I) as "ring" and name (II) as "pineapple with outsourcing"
If you choose either (I) or (II), in both cases will yield a square matrix as
a result. Here you can change the sets that you are using, but for the sake of
instantiation, we will restrict it, capital letters for square matrices
and lowercase letters to reals numbers, just to emphasize that there are
a mix of different objects, as you requested ( in (II) )
"Ring" has two operations, internal addition '+' and internal multiplication '*'
"Pineapple with outsourcing" has three operations, internal addition '+'
and internal multiplication '*' and external multiplication '·'
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what? Well, it is a polynomial of old and familiar nature, except it isn't because X is a new thing that lacks definition and set theoretic quality. This freedom taken just after carefully constructing operators within set theory is about as ambiguous as you can go. And it does break down, and their only protection is to distance themselves away from the polynomial and away from finite instances of the polynomial. By induction I believe we can dismiss the insistence upon the infinite polynomial. Of course for those who have gone down this road of admittedly runaway behavior the original sum in the polynomial has to be denied as well.
The sum in the polynomial used to help it be interpreted as a ring. The ability to multiply its terms with one another used to be a quality that helped it look like a ring. Oh, some will want to take these things back and claim them while they claim the sum is not actually present. Of course the black goose rears its head best when it is instantiated completely. Of course it is in the dismantling of the polynomial that the problem is exposed. As a long sum it can be dismantled, at which point those who want to be wishy-washy will back off and say that you cannot do that.
Post by Lalo T.
Certainly "pineapple with outsourcing" violates a principle of the "ring"
No I don't see that an additional operator makes the ring violated. You've claimed that it has clean addition and multiplication.
The point to me is that if you are going to provide another operator as you have done, having carefully developed the first two wouldn't you care to even mention that a third is present? You have; they (the AA people) have not.
Post by Lalo T.
So, up to here, there should be no conflict.
I can say that "pineapple with outsourcing" is not a "ring", but prejudging
a "pineapple with outsourcing" as a "ring" is a bit weird. (here I do not want
to introduce "polynomial rings", it seems that none agreement can be attained
in that area)
(B)
(bI) The topic of Unit Ring https://mathworld.wolfram.com/UnitRing.html
I see this as drivel until an instance of a ring without unity is put before us to work with. They name no such ring and so this is PhD bloat.
Post by Lalo T.
https://en.wikipedia.org/wiki/Ring_%28mathematics%29#Notes_on_the_definition
https://en.wikipedia.org/wiki/Mathematical_folklore
the word "folklore" is listed in https://en.wikipedia.org/wiki/List_of_mathematical_jargon
p = 1 + X + X² + X³ and q = 1 + X + X²
P + q = ...
P x q = ...
(this is just an arbitrary bit, just in case)
(bIII) the topic of "compatibility" ("underground wiring")
(bIV) What are the actual usages of the polynomial inside and outside mathematics ?
(bV) What is the meaning of " X*X " ?
(bVI) Structures With External Binary Operations ("extra structure")
https://www.mathyma.com/mathsNotes/index.php?trg=S1C2_Struct_ExtBinOp
(C) Now, very different is to have objections with 'allowing the entry of
foreign/external elements'. At some point in the thread, another topic, parallel
to te initial topic, begin to being landed.
You may be referring not only to your particular instance (your model example) but, also
an abtraction "behind" your construct. Maybe other way to say it, is looking for
this feature in the abstraction layer.
https://en.wikipedia.org/wiki/External_(mathematics)#Generalizations
I did look at these. No AA X will ever fit in here. They cannot possibly cover the vastness of the AA achievement which spills forth a vapor that permeates everything everywhere. The fog of abstract algebra which none can seem to see through... No, really though the boolean instance is rather peculiar since we can always make mistakes:
( 1.2 )( 0.1 ) = 12.0
1.2 @ 0.1 < 0.1
This is a point of interest in the procedure of falsification. These are what we are seeking out. If we find one in existing mathematics then it ought to be of interest. This also exposes how deep we are here in the stages of mathematics. We are way down deep in operator theory and construction. You can't go much deeper than this, and this ought to be done with tremendous care and if it is not done correctly then you are bound to face problems such as those which I have pointed out. To merely dodge these problems and go ahead with the existing doctrine is a religious choice and places those adherents who expose their own closed nature in a category that is not mathematical at all. They may as well have studied the bible. These are Newtonian mathematicians who will stop only just shy of doing numerical processing on their scripture. At least Jefferson tried to edit the damn thing. Not so for Newton.

I don't have any strong opinion on modules or on vector spaces in terms of this conversation on AA. Whereas you throw the bounds outward and cast a wide net I attempt to push them inward toward a simplistic instance. I have trapped a black swan
1.23 X
which contains an operator that is not consistent with abstract algebra. Now, you say that this is an external operator while you look the other way. If your fix is correct then you have developed another angle of criticism on the subject. This is as it should be, for when a construction falls apart it tends to carry these multiple failings. For some reason you don't care to apply your work onto a simple instance. But you see in this instance X has no set theoretic properties. So how then do the AA people get to call upon 'polynomials with real coefficients' without code smell taking place? This piece of code cannot compile. The best answer is that it is done under threat of a failing grade, not to mention that you are going to pay for it too. Best of all upon falsification of AA ten branches of fix will be strewn forth. It is the perfect position to install the excess of PhDs. There will be great room for new texts and genres of Abstract Algebra such as
Post Abstract Algebra
Pre Abstract Algebra
Nonabstract Algebra
Noninvertible Abstract Algebra
Preabstracted Nonabstract Algebra
....
Post by Lalo T.
It is not arbitrary that you would not accept modules and vector spaces.
Tim, when you say "...and without any need of the Cartesian product.", I
understand it in this "generalized" context.
Mostowsky "kick the wasp nest" towards Logic
Ross brought to the table https://en.wikipedia.org/wiki/Quasi-invariant_measure
I have not really in detail all the posts anyway.
https://en.wikipedia.org/wiki/Quantity_calculus
In some past post/message of another context, you said something about the usage of the word "linear"
https://en.wikipedia.org/wiki/Linear_map
https://math.stackexchange.com/questions/796374/binary-operation-english-terminology
https://en.wikipedia.org/wiki/Clopen_set
The exact context of why I put the "Code smell" link is because in the past, one (or maybe more) times,
you mention that abstract algebra has something wrong but it still manages to, somehow, keep its functioning.
One question could be 'In which cases the External Binary Operation can be reduced to an Binary Operation ?'
One could search and analyse cases where it is presumably impossible to reduce it to a binary operation.
There are more literature about other study cases?
What is the interpretation from the Physics in this issue ?
Is it really mandatory to use extra symbols to identify the addition of
(D) In which aspects Geometry seems more leveraged towards the "square" instead of the "triangle" ?
https://en.wikipedia.org/wiki/Hyperrectangle
and no word special word for elongated regular hexagons (like "leftangle")
https://en.wikipedia.org/wiki/Parallelogon
https://en.wikipedia.org/wiki/Egyptian_geometry
https://en.wikipedia.org/wiki/Square_metre
https://en.wikipedia.org/wiki/Ancient_Egyptian_units_of_measurement
"Ternary arithmetic, factorization, and the class number one problem" by Aram Bingham
https://arxiv.org/pdf/2002.02059.pdf
There is a word for 90 degree angles (right angle), is there a word for 60 and 120 degrees angles ?
Where is the terminology for 60/120 degrees angles ?
https://www.inverse.com/article/45157-michael-jackson-anti-gravity-lean-biomechanis
https://en.wikipedia.org/wiki/Perpendicular
https://en.wikipedia.org/wiki/Square_tiling
https://en.wikipedia.org/wiki/Hexagonal_tiling
https://en.wikipedia.org/wiki/Rhombille_tiling
https://en.wikipedia.org/wiki/Cubic_honeycomb
https://en.wikipedia.org/wiki/Rhombic_dodecahedron
https://en.wikipedia.org/wiki/Rhombic_dodecahedral_honeycomb
https://en.wikipedia.org/wiki/Trigonal_trapezohedron#Asymmetric_variation
https://en.wikipedia.org/wiki/Trigonal_trapezohedral_honeycomb
- the user L.walker put the question if Polysign p5 is a field with fractions (instead of reals)
- The user Tommy1729 seems to have some kind of approach toward number theory that include polysigns
https://en.wikipedia.org/wiki/Geometry_of_numbers ??
- Polysigns p5 and the complexity of pentagons
https://en.wikipedia.org/wiki/Tetrachromacy
I am recalling well, Krishnamurti use the phrase "don't institutionalize" (or something similar)
(probably in some documentary about the Schools in India related with him
I do not have the link in hand)
zelos...@gmail.com
2020-09-14 05:32:27 UTC
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
Post by Tim Golden BandTech.com
Zelos, would you care to name another place where an infinite series is formally required?
If you mean sequence, then real number construction.
Post by Tim Golden BandTech.com
Here however you and your cohorts have committed to admitting that in n the thing does not work out therefor you insist on an infinite series.
No, we work with infinite sequences because that is the fucking formal construction because it covers all cases then.
Post by Tim Golden BandTech.com
Of course these have to be explicitly stated to be in abstract X; something that the curriculum shies away from.
No one shies away from anything, it is unneccisary in early stages to go on about the formal consturction.
Post by Tim Golden BandTech.com
I would not quibble with a real valued X for then the ring definition would hold up particularly well upon instantiating X.
The ring axioms are fullfilled for polynomials, X is just notation, nothing else.
Post by Tim Golden BandTech.com
" polynomials with real coefficients"
Yes, which comes from history and how they were viewed and are viewed outside of formal constructions. So fucking what?
Post by Tim Golden BandTech.com
and clearly the terminology of 'coefficient' does in fact imply multiplication doesn't it?
Again, historical reasons makes it retain the name, just as i is called "imaginary" despite it being just as "real" as real numbers.
Post by Tim Golden BandTech.com
Is it precisely these concerns that cause one to shy away from these simple instances and insist upon an infinite series of coefficients?
Complaining about vocabulary that is admitted to be there due to historical reasons and not a product of the formal construction does not in anyway pose a challange against the formal construction.
Post by Tim Golden BandTech.com
As to what exactly is wrong with a polynomial with finitely many terms: this problem is completely avoided by the AA people here.
You have yet to show ANYTHING is wrong.
Lalo T.
2020-09-14 08:10:27 UTC
Well, One may consider as suspicious "only plans, and not action" as if an architect is
suspicious only doing Architectural diagrams and never actually "building" houses....

https://en.wikipedia.org/wiki/Absorption_law
"...and ordered sets with min and max operations."

https://en.wikipedia.org/wiki/Racks_and_quandles#Racks
"...with binary operations satisfying axioms analogous to the Reidemeister
moves used to manipulate knot diagrams."

https://en.wikipedia.org/wiki/Graph_operations#Binary_operations

https://en.wikipedia.org/wiki/Knot_operation

https://en.wikipedia.org/wiki/String_operations

A Noncommutative Version of the Natural Numbers by Tyler Foster
https://arxiv.org/pdf/1003.2081.pdf

https://mathworld.wolfram.com/AlgebraicLoop.html
https://library.wolfram.com/infocenter/MathSource/6198/

https://en.wikipedia.org/wiki/Parallel_(operator)

https://en.wikipedia.org/wiki/Circuits_over_sets_of_natural_numbers

https://en.wikipedia.org/wiki/Tropical_semiring

On a novel 3D hypercomplex number system by Shlomo Jacobi
https://arxiv.org/pdf/1509.01459.pdf

Santilli Isomathematics for Generalizing Modern Mathematics by Chun-Xuan Jiang
http://www.santilli-foundation.org/docs/10.11648.j.ajmp.s.2015040501.14.pdf

(b) Still, I do not totally agree with you Tim , but, in any case, this issue is worthy of examination :

- Following the Ross's recommendation, I would like to add one book:

Surfaces and Essences , Analogy as the fuel and fire of thinking
by Douglas Hofstadter and Emmanuel Sander
(well suited to this context, intricacies of analogy making and categorization)

- Did W.R.Hamilton choose "good" names to concepts related to quaternions?
https://en.wikipedia.org/wiki/Classical_Hamiltonian_quaternions

- One usage of word polynomial :

https://en.wiktionary.org/wiki/quasipolynomial
https://en.wikipedia.org/wiki/Quasi-polynomial

https://en.wikipedia.org/wiki/Generalization <---

- Set Theory is not the only "foundational" mathematical area

- John Mathlseed wrote " https://we.tl/t-RVjcpZ3ri9 "
Mostowski hashtags ???

- controlled languages https://en.wikipedia.org/wiki/Controlled_natural_language

- One could ask deep enquiries to Wikipedia (I m clueless in this topic)
in case you want to "quarantine" or isolate wikipedia articles
https://en.wikipedia.org/wiki/DBpedia
h***@gmail.com
2020-09-14 10:25:52 UTC
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Tim Golden BandTech.com
2020-09-14 22:35:06 UTC
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.

Worse though the polynomial of abstract algebra is meant explicitly not to perform any operation so as to become a placeholder. This behavior is in direct contradiction with the ring definition. Possibly you can let through polynomials in unspecified coefficients (though this is dubious since no instantiable form will ever be had) and certainly when you try polynomials with real coefficients the black swan can be found readily.

You horand are speaking out your ass here as far as I can tell. Possibly you are one of the great mimics, but you have done nothing for your cause here with such short words. I hope you are taller. Well, possibly you are down there staring the black swan in the face. Nah, can't be.

How a subject could come to be so carefully constructed and yet fail to have any discussion on these terms... how this generation comes to bind to such information without question... clearly academia means to be something rather different than the mathematics of old. Possibly you do not see the abstract X since it is largely avoided. I have no idea. Your first post here is barely a start. Possibly I have started in for you on the polymorphic front. Do you see that X has no type at all?
Lalo T.
2020-09-14 22:55:14 UTC
At last, I m starting to understand those strange programming jokes "Java vs Javascript"

https://en.wikipedia.org/wiki/Java_%28programming_language%29

https://en.wikipedia.org/wiki/JavaScript

https://hazel.org/
Lalo T.
2020-09-15 01:23:51 UTC
https://hazel.org/build/trunk/index.html
https://github.com/hazelgrove/hazel
https://en.wikipedia.org/wiki/C_%28programming_language%29

(I) A₀*I + A₁*X + A₂*X² + a₃*X³ = A₀*I + A₁*X + A₂*(X*X) + A₃*(X*(X*X))
(II) a₀·I + a₁·X + a₂·X² + a₃·X³ = a₀·I + a₁·X + a₂·(X*X) + a₃·(X*(X*X))

a₀·I + a₁·⮽ + a₂·⮽² + a₃·⮽³ = a₀·I + a₁·⮽ + a₂·(⮽⯑⮽) + a₃·(⮽⯑(⮽⯑⮽))

Maybe you can consider that golden_membership(..) is carved in C and
AA_membership(..) is carved in Hazel, but I think both can offer a lot...
(mathematically s
h***@gmail.com
2020-09-15 11:07:36 UTC
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.

Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
Post by Tim Golden BandTech.com
Worse though the polynomial of abstract algebra is meant explicitly not to perform any operation so as to become a placeholder.
That's exactly what header files do, too.
Post by Tim Golden BandTech.com
This behavior is in direct contradiction with the ring definition. Possibly you can let through polynomials in unspecified coefficients (though this is dubious since no instantiable form will ever be had) and certainly when you try polynomials with real coefficients the black swan can be found readily.
You horand are speaking out your ass here
That's as far as I read. Insult me again, and you can go to hell.
Tim Golden BandTech.com
2020-09-15 13:03:20 UTC
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.

I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.

Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?

No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.

PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.

I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.

I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.

That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.

Now don't run away from the black swan horand:
1.23 X
Is this product ring behaved?
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Worse though the polynomial of abstract algebra is meant explicitly not to perform any operation so as to become a placeholder.
That's exactly what header files do, too.
Post by Tim Golden BandTech.com
This behavior is in direct contradiction with the ring definition. Possibly you can let through polynomials in unspecified coefficients (though this is dubious since no instantiable form will ever be had) and certainly when you try polynomials with real coefficients the black swan can be found readily.
You horand are speaking out your ass here
That's as far as I read. Insult me again, and you can go to hell.
h***@gmail.com
2020-09-15 14:53:36 UTC
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
I don't know what you mean by that. It is not an object of the ring of real numbers, but that is not what anybody ever claimed or needed. In fact, polynomials stitch together operations operating on two different rings: The ring of real numbers, and the ring of the polymorphic objects symbolized by X and its powers, including X^0. There are in fact six different operations involved: addition and multiplication in the ring of reals (to gather together the real numbers), addition and multiplication in the ring of the polymorphic objects to explain things such as X^n + X^n and X^n * X^m, scalar multiplication of a real with an object X^n, and the "addition" of mixed expressions such as 1.23 X + 2.34 X^2. If you identify real numbers such as 1.23 with the polynomials 1.23*X^0, then you can collapse these six operations into two, which allows you to work with a single ring, the ring of polynomials. Nothing more, nothing less.
Tim Golden BandTech.com
2020-09-16 10:21:55 UTC
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
I don't know what you mean by that. It is not an object of the ring of real numbers, but that is not what anybody ever claimed or needed. In fact, polynomials stitch together operations operating on two different rings: The ring of real numbers, and the ring of the polymorphic objects symbolized by X and its powers, including X^0. There are in fact six different operations involved: addition and multiplication in the ring of reals (to gather together the real numbers), addition and multiplication in the ring of the polymorphic objects to explain things such as X^n + X^n and X^n * X^m, scalar multiplication of a real with an object X^n, and the "addition" of mixed expressions such as 1.23 X + 2.34 X^2. If you identify real numbers such as 1.23 with the polynomials 1.23*X^0, then you can collapse these six operations into two, which allows you to work with a single ring, the ring of polynomials. Nothing more, nothing less.
Tim Golden BandTech.com
2020-09-16 10:52:23 UTC
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
I don't know what you mean by that. It is not an object of the ring of real numbers, but that is not what anybody ever claimed or needed. In fact, polynomials stitch together operations operating on two different rings: The ring of real numbers, and the ring of the polymorphic objects symbolized by X and its powers, including X^0. There are in fact six different operations involved: addition and multiplication in the ring of reals (to gather together the real numbers), addition and multiplication in the ring of the polymorphic objects to explain things such as X^n + X^n and X^n * X^m, scalar multiplication of a real with an object X^n, and the "addition" of mixed expressions such as 1.23 X + 2.34 X^2. If you identify real numbers such as 1.23 with the polynomials 1.23*X^0, then you can collapse these six operations into two, which allows you to work with a single ring, the ring of polynomials. Nothing more, nothing less.
If you don't know what I mean by that then you've been putting the blinders on here. The polynomial is a sum of such terms, and if the polynomial is ring behaved then so are these terms ring behaved. I don't technically even have to provide this argument, for the specification of zero coefficients would simply wipe the others out and we would still arrive at this conflict. I will ask myself now:
Is 1.23 X a ring behaved product?
No. It is not. It obviously lacks the closure requirement. 1.23 is a real value whereas X is not a real value. Therefore
1.23 X
is not ring behaved. So how does it come that you name eight operators and this is none of them? Or at the very least you mention six which turn into two. Furthermore, what option does this curriculum take when after carefully defining operators it goes ahead and makes usage of
'polynomials with real coefficients'
never discussing the conflict of that usage with its carefully constructed operators? The black swan is a polynomial with a real coefficient. Also though it is merely one term. The product cannot be a ring product. Each term of the polynomial does not inter-operate and so the addition operation of the polynomial cannot meet the ring definition, for the ring strictly specifies that two elements evaluate to one element. That the polynomial is the antithesis of the ring definition... nice discussion they had on that one, eh?

So at this point horand you've looked the black swan in the eye, yet the lights were not on. Now we turn on the lights; what do you see?
Oh maybe you did see but you cannot bring yourself to admit that the closure requirement fails in the case of the black swan. I do see that you put addition in quotes above here in reference to the polynomial terms. I think this puts you into Mike Terry's camp of denying that the terms of the polynomial are actually a sum. Is that where you are going? And would this behavior be because they fail to meet the ring requirements? Why not them simply admit that
1.23 X
is not ring behaved? Isn't it obvious to everyone here that this is the case? There are only two options as far as I can tell. Ahh, but this is the thing: When a construction fails is fails in so many ways that you could go anywhere next. Of course the third option is the usenet dodge, which occurs regularly here. I suppose you could take up zelos's argument that we are dealing in historical details that no longer have any relevance. But then he's been wishy-washy on that attack. It really won't pass the smell test. It's inverted induction for those who insist on an infinity of terms in a system that breaks with just one term.

You see, it is valid to dismantle the polynomial. It is valid to dismantle the real number. It is in fact the pursuit of fundamental mathematics which requires that we question the works that have been done and dismantle those works. If they hold up then all is well. If they do not hold up could it be that the leads into a new way lay there?
h***@gmail.com
2020-09-16 16:06:03 UTC
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But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
I don't know what you mean by that. It is not an object of the ring of real numbers, but that is not what anybody ever claimed or needed. In fact, polynomials stitch together operations operating on two different rings: The ring of real numbers, and the ring of the polymorphic objects symbolized by X and its powers, including X^0. There are in fact six different operations involved: addition and multiplication in the ring of reals (to gather together the real numbers), addition and multiplication in the ring of the polymorphic objects to explain things such as X^n + X^n and X^n * X^m, scalar multiplication of a real with an object X^n, and the "addition" of mixed expressions such as 1.23 X + 2.34 X^2. If you identify real numbers such as 1.23 with the polynomials 1.23*X^0, then you can collapse these six operations into two, which allows you to work with a single ring, the ring of polynomials. Nothing more, nothing less.
Is 1.23 X a ring behaved product?
No. It is not. It obviously lacks the closure requirement. 1.23 is a real value whereas X is not a real value.
1.23 X is *not* an element of the ring of real numbers, and nobody ever claimed that it was. The ring of reals is not the only ring in existence, and it is not the ring in question. 1.23 X *is* an element of the ring of polynomials with real coefficients. So, yes. 1.23 X *is* "ring behaved", and it is derived by a scalar multiple of the monomial X. This is all well-defined and old news.
Tim Golden BandTech.com
2020-09-17 12:04:16 UTC
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But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
I don't know what you mean by that. It is not an object of the ring of real numbers, but that is not what anybody ever claimed or needed. In fact, polynomials stitch together operations operating on two different rings: The ring of real numbers, and the ring of the polymorphic objects symbolized by X and its powers, including X^0. There are in fact six different operations involved: addition and multiplication in the ring of reals (to gather together the real numbers), addition and multiplication in the ring of the polymorphic objects to explain things such as X^n + X^n and X^n * X^m, scalar multiplication of a real with an object X^n, and the "addition" of mixed expressions such as 1.23 X + 2.34 X^2. If you identify real numbers such as 1.23 with the polynomials 1.23*X^0, then you can collapse these six operations into two, which allows you to work with a single ring, the ring of polynomials. Nothing more, nothing less.
Is 1.23 X a ring behaved product?
No. It is not. It obviously lacks the closure requirement. 1.23 is a real value whereas X is not a real value.
1.23 X is *not* an element of the ring of real numbers, and nobody ever claimed that it was. The ring of reals is not the only ring in existence, and it is not the ring in question. 1.23 X *is* an element of the ring of polynomials with real coefficients. So, yes. 1.23 X *is* "ring behaved", and it is derived by a scalar multiple of the monomial X. This is all well-defined and old news.
When X was real valued it was well defined. But with X in this new class of nondistinct types it is obvious that the ring requirement of closure does not apply. You happily call it a "scalar multiple" and this implies multiplication which has just been carefully defined within the ring definition. two elements in the same set can be multiplied to yield an element in that set. What set are you in here? I have dismantled the polynomial. We are down to but one term as you admit we are working with the monomial X. We are looking not at a scalar, for we have already declared that we are working from polynomials with real coefficients. So this is a real value and not just a scalar. It seems so obnoxious to zone in on this here, but yes, this is exactly my point. If the terms of the polynomial fail to meet the ring requirement then so must the entire polynomial fail for it is a sum of such terms. For instance a polynomial with real coefficients
a0 + a1 X
is a real value in sum with a non-real value and therefor this sum cannot be the ring sum and so goes the denial of Mike Terry that this actually is a sum and your own need to put 'sum' in quotes. You obviously have seen the black swan, but to admit that it exists would be a social bad dog kind of experience. What is sad about this is that there is the possibility of an opening here. The notion that these subjects are done and sealed off from further development is not healthy to the future state. We are in fact engaged in a progression, and along the way attempts will be made that can work but they may not work great. I see how AA develops their version of the complex value, but did it really take an infinitude of terms to get there? No. It just requires introducing modulo behavior onto X. Their contorted language starts here though. This is the first contortion. As code sensetive types let's face it: the formal definition of operators sounds pretty good, and it looks pretty clean. I don't like the naming so much. But then here we see that the requirements are offended and yet things still work. At a situation like this there are rather a lot of interpretations that could take place. As to whether this is the actual position of the human race at this time: yes; I believe that this is so. We could put Abstract Algebra up on the mantle along with relativity theory... never to be challenged again... like in the bible... one must preserve the book. All should bow to the greats who came before us. Do not look up on the mantle behind these things to see how they are held there. That is not for you or I to do. Those who maintain the journals will tell us what is good and what is not.

This strategy builds out the next generation of mathematicians as mimics who are fully programmed not to go outside of the bounds of the curriculum that they have 'mastered' (i.e. mimicked). I have come to this subject from another angle and find it to be conflicted. It is not surprising that the same minds which could master either Chinese or English from naught could carry on in a mish-mash of terminology and conflict all the while resolving the matching game and never bothering to look back. It is this stage of stepping back and taking another look... even the PhD student is only encouraged forward to publish and print more gobbledy gook atop the existing mess on into substrata that declare their own bounds of speciality which only those who purvey that particular linguistic ability may contribute to... or be rejected by the lords of that 'industry'. I have only knocked on a few doors and already see what is going on. I don't think that the thing should collapse in its entirety but I do rather think that the perfection of mathematics as the queen of old is gone now. If we aren't picking through a rather large pile of debris I think we may at least see a grand structure whose upkeep will not make it through the next century.

The future state is not going to look like this obnoxious accumulation. Here I find a footing propped with a rotten stick of wood and give it a kick. That is to say that the polynomial with real coefficients and undefined X is not a fair tool when operator theory is undergoing such scrutiny. Some may take the path of revising operator theory, and I suspect that this could yield. There are constructions sitting right under our noses whose smells are so familiar that they are masked out; so obvious yet they sit outside of the known language. Just one of these down here in the fundamental layer could be a game changer. These are the places to look. Those at the top of the pile; well; enjoy the ride.
Tim Golden BandTech.com
2020-09-17 15:16:57 UTC
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But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
I don't know what you mean by that. It is not an object of the ring of real numbers, but that is not what anybody ever claimed or needed. In fact, polynomials stitch together operations operating on two different rings: The ring of real numbers, and the ring of the polymorphic objects symbolized by X and its powers, including X^0. There are in fact six different operations involved: addition and multiplication in the ring of reals (to gather together the real numbers), addition and multiplication in the ring of the polymorphic objects to explain things such as X^n + X^n and X^n * X^m, scalar multiplication of a real with an object X^n, and the "addition" of mixed expressions such as 1.23 X + 2.34 X^2. If you identify real numbers such as 1.23 with the polynomials 1.23*X^0, then you can collapse these six operations into two, which allows you to work with a single ring, the ring of polynomials. Nothing more, nothing less.
Is 1.23 X a ring behaved product?
No. It is not. It obviously lacks the closure requirement. 1.23 is a real value whereas X is not a real value.
1.23 X is *not* an element of the ring of real numbers, and nobody ever claimed that it was. The ring of reals is not the only ring in existence, and it is not the ring in question. 1.23 X *is* an element of the ring of polynomials with real coefficients. So, yes. 1.23 X *is* "ring behaved", and it is derived by a scalar multiple of the monomial X. This is all well-defined and old news.
So Horand, I believe that you have the strongest response here yet. Clearly you have stated that:
1.23 X is ring behaved.
although there are stars on the *is* and quotes on the "ring behaved" which suggest some discomfort on your part with making the statement cleanly. That is fine, but I want to hold you to it, and will assume that those punctuations are your version of rhetoric. I mix mine differently but we all are granted freedom here. We are on an uncensored medium and this is a beautiful thing though many shit upon it. Here we have the nature of the human race most cleanly exposed and I have little doubt that some future orb will gather a full understanding of humans here rather than in some math journal where epithets are not generally published though they may be going on behind the operations of such institutions.

Now to represent my side I will stand by the statement that
1.23 X is not ring behaved.
Should it be the case that one of us is wrong?
If so then shouldn't the falsification be forthcoming?
I happily await your falsification of my details. Those details in short state:
the real value 1.23 multiplied by the value X does not fit the ring definition ( i.e. is not ring behaved) because X and 1.23 are not elements of the same set. 1.23 is a real value whereas X is not a real value. The closure requirement of the ring definition is not met.

Now, clearly my statement conflicts with your statement, and so logically I should be able to debug your statements and find an error. As such this error is likely a quotable series of words. I do see you are being very careful yet I should attempt this on your statements just as you should attempt this on my statements. This would help to complete the argument. One of us has made a mistake. You being on the status quo side of things are admittedly likely to be correct given that this topic is one of 'pure mathematics'. However this probability means little in the logic of falsification. On my side and in its slender probability I need only provide one black swan to falsify the statement that all swans are white. Thus the slenderest instance suffices for the side of least probability. Of course I have chosen the black swan
1.23 X
and you have allowed the analysis of it far more strongly than anyone else here. Most simply dodge it. Peter for a moment had the thing by the neck but let it go. You have made friends with the thing and declare it to be white:
"So, yes. 1.23 X *is* "ring behaved", and it is derived by a scalar multiple of the monomial X. This is all well-defined and old news."
Now in the expression
1.23 X
we do see one multiplication taking place right? This is where you say 'scalar multiple'. So we are looking clearly at a product of two elements:
1.23 (real valued), X (no value; little discussion other than as a placeholder)
We may as well write
Multiply( 1.23, X )
and to use your own code analogy we could type this Product() function:
avoid1 Multiply( real a, avoid2 X)
where avoid1 and avoid2 are poorly defined types. I may stand to be corrected here as to your own type specifications but no matter what they are the function Multiply() here goes strongly against the closure requirement of the ring definition. The ring definition clearly calls for
type1 Multiply( type1 v1, type1 v2 ) [ Closure requirement of binary operators ]
and forms exactly such a strong template. This sort of behavior is automatically attempted in modern C++ compilers and if there is no function of this type (or the operator * if we chose it) then (forgoing type conversions) we would see a compiler error. By the ring definition alone we ought to only (ever) see products of this form. That a product not matching the ring definition could be introduced into the subject of abstract algebra directly after developing this strict operator theory (for the first time in mathematics) could seem devastating to some minds who felt awoke... but apparently not so for all here but me.

I feel quite certain that these avoid* types are not polynomial types for if we had:
Polynomial Multiply( real a, Polynomial X )
we would find that the real value gets multiplied with distribution through the entire series of terms. To correct this action we could try:
Polynomial Multiply( real a, int index, Polynomial X )
but this then is three arguments and none of them are of the same type. Still, this is closer isn't it to the operation that you are describing? How did I get closer to what you are describing while getting farther away from the ring definition? I must be really lost here, eh?

Poor Zelos is still back trying to compile an infinite length polynomial in real time. Come along Zelos and pet the black swan for a change. Horand is doing marvelously isn't he?
h***@gmail.com
2020-09-17 19:14:28 UTC
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
I don't know what you mean by that. It is not an object of the ring of real numbers, but that is not what anybody ever claimed or needed. In fact, polynomials stitch together operations operating on two different rings: The ring of real numbers, and the ring of the polymorphic objects symbolized by X and its powers, including X^0. There are in fact six different operations involved: addition and multiplication in the ring of reals (to gather together the real numbers), addition and multiplication in the ring of the polymorphic objects to explain things such as X^n + X^n and X^n * X^m, scalar multiplication of a real with an object X^n, and the "addition" of mixed expressions such as 1.23 X + 2.34 X^2. If you identify real numbers such as 1.23 with the polynomials 1.23*X^0, then you can collapse these six operations into two, which allows you to work with a single ring, the ring of polynomials. Nothing more, nothing less.
Is 1.23 X a ring behaved product?
No. It is not. It obviously lacks the closure requirement. 1.23 is a real value whereas X is not a real value.
1.23 X is *not* an element of the ring of real numbers, and nobody ever claimed that it was. The ring of reals is not the only ring in existence, and it is not the ring in question. 1.23 X *is* an element of the ring of polynomials with real coefficients. So, yes. 1.23 X *is* "ring behaved", and it is derived by a scalar multiple of the monomial X. This is all well-defined and old news.
1.23 X is ring behaved.
although there are stars on the *is* and quotes on the "ring behaved" which suggest some discomfort on your part with making the statement cleanly.
Not at all. The stars are there for emphasis, and the quotes were to show terminology that *you* (emphasis again!) introduced. I don't know that there is anything else aside from you who knows what "ring-behaved" means.

That is fine, but I want to hold you to it, and will assume that those punctuations are your version of rhetoric. I mix mine differently but we all are granted freedom here. We are on an uncensored medium and this is a beautiful thing though many shit upon it. Here we have the nature of the human race most cleanly exposed and I have little doubt that some future orb will gather a full understanding of humans here rather than in some math journal where epithets are not generally published though they may be going on behind the operations of such institutions.
Post by Tim Golden BandTech.com
Now to represent my side I will stand by the statement that
1.23 X is not ring behaved.
Should it be the case that one of us is wrong?
If so then shouldn't the falsification be forthcoming?
the real value 1.23 multiplied by the value X does not fit the ring definition ( i.e. is not ring behaved) because X and 1.23 are not elements of the same set.
This is not true. The ring contains X and because of the ring properties also X^2, X^3,..., as well as X^0
Post by Tim Golden BandTech.com
1.23 is a real value whereas X is not a real value. The closure requirement of the ring definition is not met.
This is a fundamental misunderstanding on your part. The ring of polynomials is *not* (emphasis, get it?) the field of real numbers. *If you want to*, you can think of the polynomial ring as the ring generated from the real numbers plus one object (X), which is anything other than a real number. Then you have to properly *define* addition and multiplication on this ring, of course, but *that is not hard*. For any real numbers a and b a+b and a*b are defined as the results of the corresponding addition and multiplication in the reals; X+a = a+X for every real number a, X*a = a*X for every real number a, X*X = X^2, etc.
Post by Tim Golden BandTech.com
Now, clearly my statement conflicts with your statement, and so logically I should be able to debug your statements and find an error. Well yes, either that, or your statement is simply wrong.
We may as well write
Multiply( 1.23, X )
and to use your own code analogy we could type this Product() function:
avoid1 Multiply( real a, avoid2 X)

This is not a good way to think about it, and your terminology (avoid1, avoid2) shows your prejudices. If you want to write
Multiply( 1.23, X )
it is better to set it up as
polynomial: extends real;
and define addition and multiplication as addition and multiplication of polynomials. I don't have the time to write the entire program for you, but an important component of the implementation would be
isReal (polynomial x)
which returns true ix x is a real number and false otherwise.

Probably the easiest way to do this would be to represent polynomials as strings, because you can then search for occurrences of '+' or '-' to split the polynomial (recursively) into a monomial and a remainder polynomial, and work out how the monomials of one string need to interact with the monomials of another string during the addition and multiplication.

At this level of abstraction you simply define multiply (X^n, X^m) as X^(n+m)
whenever n and m are nonnegative integers, and identify X^0 with the real number 1 (or its representation as a string).

Then your black swan is white, and I hope you now finally understand the concept of milk.
Lalo T.
2020-09-17 19:51:41 UTC
- What are the issues with "simulating" one type of computing inside another
type of computing ? (also in the case where the "simulated" computing is a
"superset" of the computing in use...)

"Of course if you mention the word 'polynomial' to a high school student they
will feel very comfortable with their knowledge of them"

Well, suppose that is not an infinite polynomial. Even with a finite usual
polynomial function, "pixel systems" are finite, and also are finite the
memories that save the values to plot the function (with or without plotting),
but enough to trick the eyes in believing that is continuous. What happen if the
polynomial function is evaluated just in rationals or naturals (with the coeffs
being real, natural or rationals)

- I suppose it would be fine (if one is a generalist) dont read material
produced by specialists, otherwise I would be contributing to foster the
specialist culture in one way or another.

Is this point, just an oversimplification ?
what is the practical degree of sub-specialization ?
What is your take on different types of funding?
(state, private, self-funding, crowdfunding, random funding)
https://en.wikipedia.org/wiki/Kleroterion

- In your perspective on the usage of the word "polynomial". Would it be fine
restrict the coefficients only to naturals ?

- take for example the gaussian integers ("integers" in complex numbers)

What would be your "level 0" or "reference height" to allow me append the
word "polynomial" to the name of an object that use gaussian integers ?

Do the "prime" elements of some instance of an algebrac structure has some
importance from your perspective ?
https://en.wikipedia.org/wiki/Table_of_Gaussian_integer_factorizations

- what is the role of the Distributive Property here ?

- Would be fine to linguistically emphasise the difference between "polynomial"
and "polynomial function" ? Is it proper the usage of the word "polynomial" in
"polynomial function" ?
in https://en.wikipedia.org/wiki/Polynomial flip though the section etimology
and definition.

What is your take on "polynomial equations" ?
https://en.wikipedia.org/wiki/Algebraic_equation
Lalo T.
2020-09-17 22:43:18 UTC
A house and the blueprint of a house
An instance of the blueprint and the blueprint

(or alternatively use a generalization vs specific)

One also may check the chronological appearance of the concepts of
"polynomial", "polynomial equation" and "polynomial function"

I guess that you may disagree with :

"Polynomials appear in many areas of mathematics and science. For example, they
are used to form polynomial equations "

in https://en.wikipedia.org/wiki/Polynomial
zelos...@gmail.com
2020-09-18 07:36:34 UTC
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
I don't know what you mean by that. It is not an object of the ring of real numbers, but that is not what anybody ever claimed or needed. In fact, polynomials stitch together operations operating on two different rings: The ring of real numbers, and the ring of the polymorphic objects symbolized by X and its powers, including X^0. There are in fact six different operations involved: addition and multiplication in the ring of reals (to gather together the real numbers), addition and multiplication in the ring of the polymorphic objects to explain things such as X^n + X^n and X^n * X^m, scalar multiplication of a real with an object X^n, and the "addition" of mixed expressions such as 1.23 X + 2.34 X^2. If you identify real numbers such as 1.23 with the polynomials 1.23*X^0, then you can collapse these six operations into two, which allows you to work with a single ring, the ring of polynomials. Nothing more, nothing less.
Is 1.23 X a ring behaved product?
No. It is not. It obviously lacks the closure requirement. 1.23 is a real value whereas X is not a real value.
1.23 X is *not* an element of the ring of real numbers, and nobody ever claimed that it was. The ring of reals is not the only ring in existence, and it is not the ring in question. 1.23 X *is* an element of the ring of polynomials with real coefficients. So, yes. 1.23 X *is* "ring behaved", and it is derived by a scalar multiple of the monomial X. This is all well-defined and old news.
1.23 X is ring behaved.
although there are stars on the *is* and quotes on the "ring behaved" which suggest some discomfort on your part with making the statement cleanly. That is fine, but I want to hold you to it, and will assume that those punctuations are your version of rhetoric. I mix mine differently but we all are granted freedom here. We are on an uncensored medium and this is a beautiful thing though many shit upon it. Here we have the nature of the human race most cleanly exposed and I have little doubt that some future orb will gather a full understanding of humans here rather than in some math journal where epithets are not generally published though they may be going on behind the operations of such institutions.
Now to represent my side I will stand by the statement that
1.23 X is not ring behaved.
Should it be the case that one of us is wrong?
If so then shouldn't the falsification be forthcoming?
the real value 1.23 multiplied by the value X does not fit the ring definition ( i.e. is not ring behaved) because X and 1.23 are not elements of the same set. 1.23 is a real value whereas X is not a real value. The closure requirement of the ring definition is not met.
Now, clearly my statement conflicts with your statement, and so logically I should be able to debug your statements and find an error. As such this error is likely a quotable series of words. I do see you are being very careful yet I should attempt this on your statements just as you should attempt this on my statements. This would help to complete the argument. One of us has made a mistake. You being on the status quo side of things are admittedly likely to be correct given that this topic is one of 'pure mathematics'. However this probability means little in the logic of falsification. On my side and in its slender probability I need only provide one black swan to falsify the statement that all swans are white. Thus the slenderest instance suffices for the side of least probability. Of course I have chosen the black swan
1.23 X
"So, yes. 1.23 X *is* "ring behaved", and it is derived by a scalar multiple of the monomial X. This is all well-defined and old news."
Now in the expression
1.23 X
1.23 (real valued), X (no value; little discussion other than as a placeholder)
We may as well write
Multiply( 1.23, X )
avoid1 Multiply( real a, avoid2 X)
where avoid1 and avoid2 are poorly defined types. I may stand to be corrected here as to your own type specifications but no matter what they are the function Multiply() here goes strongly against the closure requirement of the ring definition. The ring definition clearly calls for
type1 Multiply( type1 v1, type1 v2 ) [ Closure requirement of binary operators ]
and forms exactly such a strong template. This sort of behavior is automatically attempted in modern C++ compilers and if there is no function of this type (or the operator * if we chose it) then (forgoing type conversions) we would see a compiler error. By the ring definition alone we ought to only (ever) see products of this form. That a product not matching the ring definition could be introduced into the subject of abstract algebra directly after developing this strict operator theory (for the first time in mathematics) could seem devastating to some minds who felt awoke... but apparently not so for all here but me.
Polynomial Multiply( real a, Polynomial X )
Polynomial Multiply( real a, int index, Polynomial X )
but this then is three arguments and none of them are of the same type. Still, this is closer isn't it to the operation that you are describing? How did I get closer to what you are describing while getting farther away from the ring definition? I must be really lost here, eh?
Poor Zelos is still back trying to compile an infinite length polynomial in real time. Come along Zelos and pet the black swan for a change. Horand is doing marvelously isn't he?
Poor Zelos is still back trying to compile an infinite length polynomial in real time. Come along Zelos and pet the black swan for a change. Horand is doing marvelously isn't he?
Not at all you imbecile, no one is doing anything in time here. We are talking about mathematics. The issue is you focus on notation, rather than meaning.
Lalo T.
2020-09-18 08:50:13 UTC
"...would be a social bad dog kind of experience"
To this effects, it seems there is an area of knowledge dedicated to these kind of endeavors, kind of dealing with the "invisible" social pressure
Tim Golden BandTech.com
2020-09-18 11:51:57 UTC
Post by ***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
I don't know what you mean by that. It is not an object of the ring of real numbers, but that is not what anybody ever claimed or needed. In fact, polynomials stitch together operations operating on two different rings: The ring of real numbers, and the ring of the polymorphic objects symbolized by X and its powers, including X^0. There are in fact six different operations involved: addition and multiplication in the ring of reals (to gather together the real numbers), addition and multiplication in the ring of the polymorphic objects to explain things such as X^n + X^n and X^n * X^m, scalar multiplication of a real with an object X^n, and the "addition" of mixed expressions such as 1.23 X + 2.34 X^2. If you identify real numbers such as 1.23 with the polynomials 1.23*X^0, then you can collapse these six operations into two, which allows you to work with a single ring, the ring of polynomials. Nothing more, nothing less.
Is 1.23 X a ring behaved product?
No. It is not. It obviously lacks the closure requirement. 1.23 is a real value whereas X is not a real value.
1.23 X is *not* an element of the ring of real numbers, and nobody ever claimed that it was. The ring of reals is not the only ring in existence, and it is not the ring in question. 1.23 X *is* an element of the ring of polynomials with real coefficients. So, yes. 1.23 X *is* "ring behaved", and it is derived by a scalar multiple of the monomial X. This is all well-defined and old news.
1.23 X is ring behaved.
although there are stars on the *is* and quotes on the "ring behaved" which suggest some discomfort on your part with making the statement cleanly. That is fine, but I want to hold you to it, and will assume that those punctuations are your version of rhetoric. I mix mine differently but we all are granted freedom here. We are on an uncensored medium and this is a beautiful thing though many shit upon it. Here we have the nature of the human race most cleanly exposed and I have little doubt that some future orb will gather a full understanding of humans here rather than in some math journal where epithets are not generally published though they may be going on behind the operations of such institutions.
Now to represent my side I will stand by the statement that
1.23 X is not ring behaved.
Should it be the case that one of us is wrong?
If so then shouldn't the falsification be forthcoming?
the real value 1.23 multiplied by the value X does not fit the ring definition ( i.e. is not ring behaved) because X and 1.23 are not elements of the same set. 1.23 is a real value whereas X is not a real value. The closure requirement of the ring definition is not met.
Now, clearly my statement conflicts with your statement, and so logically I should be able to debug your statements and find an error. As such this error is likely a quotable series of words. I do see you are being very careful yet I should attempt this on your statements just as you should attempt this on my statements. This would help to complete the argument. One of us has made a mistake. You being on the status quo side of things are admittedly likely to be correct given that this topic is one of 'pure mathematics'. However this probability means little in the logic of falsification. On my side and in its slender probability I need only provide one black swan to falsify the statement that all swans are white. Thus the slenderest instance suffices for the side of least probability. Of course I have chosen the black swan
1.23 X
"So, yes. 1.23 X *is* "ring behaved", and it is derived by a scalar multiple of the monomial X. This is all well-defined and old news."
Now in the expression
1.23 X
1.23 (real valued), X (no value; little discussion other than as a placeholder)
We may as well write
Multiply( 1.23, X )
avoid1 Multiply( real a, avoid2 X)
where avoid1 and avoid2 are poorly defined types. I may stand to be corrected here as to your own type specifications but no matter what they are the function Multiply() here goes strongly against the closure requirement of the ring definition. The ring definition clearly calls for
type1 Multiply( type1 v1, type1 v2 ) [ Closure requirement of binary operators ]
and forms exactly such a strong template. This sort of behavior is automatically attempted in modern C++ compilers and if there is no function of this type (or the operator * if we chose it) then (forgoing type conversions) we would see a compiler error. By the ring definition alone we ought to only (ever) see products of this form. That a product not matching the ring definition could be introduced into the subject of abstract algebra directly after developing this strict operator theory (for the first time in mathematics) could seem devastating to some minds who felt awoke... but apparently not so for all here but me.
Polynomial Multiply( real a, Polynomial X )
Polynomial Multiply( real a, int index, Polynomial X )
but this then is three arguments and none of them are of the same type. Still, this is closer isn't it to the operation that you are describing? How did I get closer to what you are describing while getting farther away from the ring definition? I must be really lost here, eh?
Poor Zelos is still back trying to compile an infinite length polynomial in real time. Come along Zelos and pet the black swan for a change. Horand is doing marvelously isn't he?
Poor Zelos is still back trying to compile an infinite length polynomial in real time. Come along Zelos and pet the black swan for a change. Horand is doing marvelously isn't he?
Not at all you imbecile, no one is doing anything in time here. We are talking about mathematics. The issue is you focus on notation, rather than meaning.
I would hope that when we are discussing operator theory that we have cleanly distinguished operators from values or elements. Because the operators have been carefully declared within set theoretic terms we ought to be able to apply these distinctions onto
1.23 X
and we should admit that this expression carries one multiplication operation on two elements: 1.23 and X.

There actually is a similar criticism on real numbers, where the signs of values
- 0.01, + 0.12, 1.23
are obscured by the usage of the '+' sign as the sum operator. And then of course the last instance above exposes the implied positive value when the sign is missing. Whether our own conventions and notation matter: yes, they certainly do. Especially as we have descended into formal operator theory there ought to be no looseness whether by imbeciles or by high priests of the subject. If an imbecile should provide the priest with a black swan ought the priest to take it into consideration? A new form of number system that has no need of the conflicted abstractions within AA is right under your noses here but it will take an imbecile to find it. Your noses are too high in the air.

I don't accept Horand's claim that the set
{ R, X } (where R is the set of real numbers and X is an 'object')
is legitimate for it suggests then that the black swan ought to resolve to a single element. This is why instantiation is relevant. So long as one remains in a polyvariate form such as
a0 + a1 X
then clearly the expression cannot evaluate simply due to the unknown values. This is true regardless of the ring definition. I might as well have just written
a0 a1
where again these are real values under discussion. Since we have descended so low as to enter operator theory formally should we declare an offense on the usage of variables for denying the operator its primal duty? I have not gone here, yet this is a possible opening. Because we have descended so low as to formalize operators could it be that formalizing variables becomes necessary? Such as multiple variable types? Here at least there would be some hope of cleaning up this topic as for instance the variable a(n) clearly is distinct from the X which has no meaning in the case of abstract algebra other than as a rotational placeholder that refuses to evaluate. It is definitely provable that the polynomial usage within abstract algebra flies in the face of the ring definition and is antithetical to it. At the very least I have exposed this much here. For Zelos there is no room to grow. The priests would not like us tampering with their curriculum why? Because the whole thing is up for grabs once you do so. I can readily declare this area open and indeed due for development; particularly in light of PhD bloat.

Nicely enough the study of sign of the real number does lead into concrete instantiation of general dimensional algebraically behaved types. Sure they can take variable assignments such as z1, z2, z3
z1 ( z2 @ z3 ) = z1 z2 @ z1 z3
but these variables do in fact allow for instantiation of a concrete type which preexisted their usage. Possibly this is a criterion for the usage of variables. So long as a(n) are real valued they imply that actual substitution of actual concrete values is possible, but so long as a(n) remain undeclared then they have to be back a level from the status of a variable that allows concrete instances. These sorts of ideas are probably going to have to enter this curriculum in order for it to resolve, yet the troubles that we see in two sides arriving at
1.23 X is ring behaved
1.23 X is not ring behaved
don't really bode well for the existing curriculum.
A lack of instantiation exists throughout mathematics and is leveraged as a means of accommodating PhD bloat.
That is a broad form whereas here there is an attempt at specificity. It is a sad state of affairs when my own proof that
1.23 X is not ring behaved
goes unfalsified by the math studs here but I do credit Horand with facing the black swan directly. Shortly Zelos will be backing off on his infinite requirements no different than on the 'historical' interpretation. Ah, but then he'll be back to it again. I'd rather take up the elemental form with you all since it does seem to fill out the coverage a bit more. I am certain that the elements of the polynomial form are ubiquitous. I am happy to dismantle the polynomial into those elements, yet to claim that they are ring behaved will never fly. They are the antithesis of ring behaved. They refuse to operate.

As far as I know, sets contain elements and operators work on those elements. These are fundamental concepts that are being stretched beyond the formality that the subject claims. No care is taken to study the polynomial form. Worse yet they attempt to make it untouchable by insisting formally on an infinite series form. That is a sure coverup. Meanwhile the antique form that I work in still is kicking around; even at MIT. Yeah, they've got the bloat problem real bad there.
h***@gmail.com
2020-09-18 13:39:54 UTC
Post by Tim Golden BandTech.com
I would hope that when we are discussing operator theory that we have cleanly distinguished operators from values or elements. Because the operators have been carefully declared within set theoretic terms we ought to be able to apply these distinctions onto
1.23 X
and we should admit that this expression carries one multiplication operation on two elements: 1.23 and X.
It does. Every real number is in the ring generated by R and the singleton {X}. 1.23 * X is a product of two ring elements; it is in the ring.
Post by Tim Golden BandTech.com
There actually is a similar criticism on real numbers, where the signs of values
- 0.01, + 0.12, 1.23
are obscured by the usage of the '+' sign as the sum operator. And then of course the last instance above exposes the implied positive value when the sign is missing. Whether our own conventions and notation matter: yes, they certainly do. Especially as we have descended into formal operator theory there ought to be no looseness whether by imbeciles or by high priests of the subject. If an imbecile should provide the priest with a black swan ought the priest to take it into consideration? A new form of number system that has no need of the conflicted abstractions within AA is right under your noses here but it will take an imbecile to find it. Your noses are too high in the air.
I don't accept Horand's claim that the set
{ R, X } (where R is the set of real numbers and X is an 'object')
is legitimate for it suggests then that the black swan ought to resolve to a single element. This is why instantiation is relevant. So long as one remains in a polyvariate form such as
a0 + a1 X
then clearly the expression cannot evaluate simply due to the unknown values. This is true regardless of the ring definition.
And here's the rub. You are confusing the two concepts of resolution and evaluation. Your black swan 1.23X is *not* a real number, but a polynomial. To that degree it is resolved, even though you may not even have instantiated yet the abstract operations of addition and multiplication in the ring. Even if you think of X as a (variable) real number, you do not get a single value for the polynomial (or monomial in this case).

Resolution comes into play most forcefully when you are setting up the abstract rules for adding or multiplying two polynomials 1.23X + 2.34X^2 and 3.45 + 4.56X^3.

Evaluation depends on what you envision X to stand for. X could be a real variable, a complex variable, a matrix variable (with or without any number of additional constraints), etc., and even then you can only evaluate the polynomial when you substitute *one* particular value for X.
Post by Tim Golden BandTech.com
I might as well have just written
a0 a1
where again these are real values under discussion.
That is just a little too loose, but you *could* write the ordered pair (a0,a1) and *define* it to mean a0 + a1X. (In fact, someone upthread suggested that to you previously.) You also have to be careful about polynomials such as a0 + a2X^2. Simply writing a0 a2 won't cut it, but the ordered triple (a0, 0, a2) describes the situation perfectly.
Post by Tim Golden BandTech.com
Since we have descended so low as to enter operator theory formally should we declare an offense on the usage of variables for denying the operator its primal duty? I have not gone here, yet this is a possible opening. Because we have descended so low as to formalize operators could it be that formalizing variables becomes necessary? Such as multiple variable types? Here at least there would be some hope of cleaning up this topic as for instance the variable a(n) clearly is distinct from the X which has no meaning in the case of abstract algebra other than as a rotational placeholder that refuses to evaluate. It is definitely provable that the polynomial usage within abstract algebra flies in the face of the ring definition and is antithetical to it. At the very least I have exposed this much here. For Zelos there is no room to grow. The priests would not like us tampering with their curriculum why? Because the whole thing is up for grabs once you do so. I can readily declare this area open and indeed due for development; particularly in light of PhD bloat.
Nicely enough the study of sign of the real number does lead into concrete instantiation of general dimensional algebraically behaved types. Sure they can take variable assignments such as z1, z2, z3
but these variables do in fact allow for instantiation of a concrete type which preexisted their usage. Possibly this is a criterion for the usage of variables. So long as a(n) are real valued they imply that actual substitution of actual concrete values is possible, but so long as a(n) remain undeclared then they have to be back a level from the status of a variable that allows concrete instances. These sorts of ideas are probably going to have to enter this curriculum in order for it to resolve, yet the troubles that we see in two sides arriving at
1.23 X is ring behaved
1.23 X is not ring behaved
don't really bode well for the existing curriculum.
A lack of instantiation exists throughout mathematics and is leveraged as a means of accommodating PhD bloat.
That is a broad form whereas here there is an attempt at specificity. It is a sad state of affairs when my own proof that
1.23 X is not ring behaved
goes unfalsified by the math studs here but I do credit Horand with facing the black swan directly. Shortly Zelos will be backing off on his infinite requirements no different than on the 'historical' interpretation. Ah, but then he'll be back to it again. I'd rather take up the elemental form with you all since it does seem to fill out the coverage a bit more. I am certain that the elements of the polynomial form are ubiquitous. I am happy to dismantle the polynomial into those elements, yet to claim that they are ring behaved will never fly. They are the antithesis of ring behaved. They refuse to operate.
As far as I know, sets contain elements and operators work on those elements. These are fundamental concepts that are being stretched beyond the formality that the subject claims. No care is taken to study the polynomial form. Worse yet they attempt to make it untouchable by insisting formally on an infinite series form. That is a sure coverup. Meanwhile the antique form that I work in still is kicking around; even at MIT. Yeah, they've got the bloat problem real bad there.
Lalo T.
2020-09-18 23:27:50 UTC
(1)
Post by Lalo T.
Which style would you follow in polysigns ? with a,b,c,d magnitudes
(-)a # (+)b # (*)c # (#)d
where '#' addition operation, '-' '+' '*' '#' as signs
(II) - a + b * c # d
where the arithmetical information migrates from the "operand space" towards the "operation space"
(algebraic-structure-like object)
When one look for how to get (I) from (II) [ or (II) from (I) ], one goes up in
the abstraction stairs, make some changes, and the return to the base level.

(2) If we are good at mimicking socially, then we are geniuses mimicking
our family.

(3) related to language :

https://en.wikipedia.org/wiki/Template:Arithmetic_operations <---
https://en.wikipedia.org/wiki/Algebraic_operation#Arithmetic_vs_algebraic_operations
https://en.wikipedia.org/wiki/Algebraic_expression#Algebraic_and_other_mathematical_expressions
https://en.wikipedia.org/wiki/Algebraic_expression
https://en.wikipedia.org/wiki/Polynomial_ring#Polynomial_expression
https://en.wikipedia.org/wiki/Elementary_algebra

(4) miscellaneous in Types :

https://en.wikipedia.org/wiki/Setoid <---
https://en.wikipedia.org/wiki/Quotient_type
https://en.wikipedia.org/wiki/Algebraic_data_type
https://en.wikipedia.org/wiki/Product_type
http://www.cse.chalmers.se/research/group/logic/TypesSS05/Extra/palmgren.pdf

(5) it may be considered go off on a tangent, but, just in case :

Understanding Polysign Numbers the Standard Way by Hagen von Eitzen
http://www.von-eitzen.de/math/PolysignNumbers.pdf
(mainly, section 2)

https://mathworld.wolfram.com/EquivalenceClass.html
https://en.wikipedia.org/wiki/Equivalence_class
https://groupprops.subwiki.org/wiki/Conjugacy_class
https://math.stackexchange.com/questions/594458/definition-of-quotient-set

https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)
https://en.wikipedia.org/wiki/Quotient_module
https://en.wikipedia.org/wiki/Quotient_ring
https://mathworld.wolfram.com/QuotientGroup.html

https://mathworld.wolfram.com/Subgroup.html
https://mathworld.wolfram.com/Coset.html
https://groupprops.subwiki.org/wiki/Proper_subgroup
https://groupprops.subwiki.org/wiki/Normal_subgroup
https://groupprops.subwiki.org/wiki/Proper_normal_subgroup

https://en.wikipedia.org/wiki/Subset#Definitions
https://en.wikipedia.org/wiki/Subobject
https://en.wikipedia.org/wiki/Class_(set_theory)

https://en.wikipedia.org/wiki/Module_homomorphism
https://en.wikipedia.org/wiki/Ring_homomorphism
https://en.wikipedia.org/wiki/Algebra_homomorphism
https://en.wikipedia.org/wiki/Associative_algebra#Definition
https://math.stackexchange.com/questions/2132812/r-algebra-homomorphisms
https://math.stackexchange.com/questions/970368/what-is-an-r-algebra
https://en.wikibooks.org/wiki/Category:Book:Abstract_Algebra
https://en.wikipedia.org/wiki/Kernel_(algebra)
https://encyclopediaofmath.org/wiki/Ideal
https://en.wikipedia.org/wiki/Bilinear_map
https://en.wikipedia.org/wiki/Injective_function
https://mathworld.wolfram.com/Isomorphism.html
https://en.wikipedia.org/wiki/Image_(mathematics)
https://en.wikipedia.org/wiki/Iota

http://math.chapman.edu/~jipsen/structures/doku.php/equivalence_relations
https://simple.wikipedia.org/wiki/Equivalence_relation
http://math.chapman.edu/~jipsen/structures/doku.php/reflexive_relations
http://math.chapman.edu/~jipsen/structures/doku.php/symmetric_relations
http://math.chapman.edu/~jipsen/structures/doku.php/transitive_relations

https://en.wikipedia.org/wiki/Root_of_unity
https://mathworld.wolfram.com/CyclotomicPolynomial.html
https://mathworld.wolfram.com/MonicPolynomial.html
https://mathworld.wolfram.com/deMoivreNumber.html
https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem
https://mathworld.wolfram.com/RelativelyPrime.html
https://mathworld.wolfram.com/ChineseRemainderTheorem.html
https://mathworld.wolfram.com/DirectSum.html
https://en.wikipedia.org/wiki/Direct_sum
Tim Golden BandTech.com
2020-09-19 17:24:52 UTC
Sat 19 Sep 2020 12:50:58 PM EDT
Post by Lalo T.
(1)
Which style would you follow in polysigns ? with a,b,c,d magnitudes
(-)a # (+)b # (*)c # (#)d
where '#' addition operation, '-' '+' '*' '#' as signs
(II) - a + b * c # d
where the arithmetical information migrates from the "operand space" towards the "operation space"
(algebraic-structure-like object)
When one look for how to get (I) from (II) [ or (II) from (I) ], one goes up in
the abstraction stairs, make some changes, and the return to the base level.
(2) If we are good at mimicking socially, then we are geniuses mimicking
our family.
I don't see why you post obnoxious quantities of links. We all can jump around on the internet and gather an obnoxious number of links. This does nothing to further any position. I guess though it is proof of the lack of censorship here that you have the freedom to do so. Still, I likewise have the freedom to complain.

Yes, we are mimicry geniuses. Our native tounges vary widely and we gain them out of thin air. We do so by mimicry. It matters very little how coherent they are. Thus there is cause to doubt the best mimics as the best analysts, for they have grown attached to a structure once they have been trained onto it. That mathematics could be the same; yes, this is a plausable theory. Also though that our linguistic abilities may not actually be complete; that these abilities are a matter of genetics; a FOXP2 gene has been implicated. Of course we are only just learning these details.
https://en.wikipedia.org/wiki/FOXP2
If however we are working our best within a genetic linguistic limitation then even our greatest mathematicians (and I do not include myself amongst them) could be operating within a limitation that is preventing all of us from getting to the truth of reality.

Certainly the accumulation of information in this age is daunting and if you don't feel that, well, I am sorry but you are going to be chasing your tail until you are dead. Even Einstein did so. Probably I will too. Why do we do it? I would think we could all admit that we'd like to make a contribution to the pile... but man look at the size of the thing now. I'm sorry, but at this point our service is as badly needed in dismantling the pile as much as it is adding on to it. Academia is a rather different business than what we as individuals ought to be attempting; especially as free individuals who are not compartmentalized into it as dependents on its financial and social structure. So I appreciate the rebels here. King Bassam; Amy666; Tommy; and Lalo... whoever is willing to attempt a break well break a leg! And if your leg doesn't break then you are onto something. Can't really recommend chasing down 0.999... though. Not primes either. Not even pi.

In (1) above here you have failed to specify that you are in P4. Within polysign the signs are reused accross domains. This term 'domain' is getting abused in mathematics ( e.g. https://en.wikipedia.org/wiki/Domain_(ring_theory) ) and I mean it in its broadest sense really as in the phrase
a king's domain
as in the space that he has to work. This is the nature of the polysign family where noone really is king but every instance has a specific domain
P1
P2
P3
P4
...
Pn
which generally I write horizontally
P1 P2 P3 P4 ... Pn
and actually may be taken as a sustaining product though generally we are concerned with one on this family at a time and particularly any value such as
- 1.1 + 2.1
or
+ a * b
should already have the attribute of one of these domains. Certainly it is true that the meaning of
+ 2.1
in P3 is quite different than its meaning in P2 or P4 for that matter. This may seem distasteful at first, but when you've generalized sign there are notational issues that our old conventions do not accommodate. It is rather odd that the real value happens to fit so well into a generalized sign system, and in truth the system of notation that I use allows for this, but if for instance the symbols minus and plus were reversed the consistency would be lost as the number of strokes that it takes to draw those signs is mnemonic. The only conflict with conventional notation is the usage of '+' to imply addition, and I suppose if you want to take the '-' as inverse as well, though I don't generally state that. I am more in line actually with the operator theory of abstract algebra that the sum and product are sufficient operators. Clearly upon introducing a third sign '*' one can logically see that the old meanings cannot hold.

Still though as you are picking at notation and I am claiming an ability to stay with old conventions we still see the logical error of confusing operators and values within sign mechanics. Formalizing sign so that
s x (where s is sign and x is magnitude)
exposes two elemental types then have we formed a product here? It is much like the polynomial product with one exception: these values do truly take instantiation! They are already modulo behaved! They are elemental. They fit well with set theory. They are already the real numbers at P2. Out of the gate and with a hammer and a chisel we have general dimensional rotational geometries demanded by these number systems. They are primitive and they stand freely. They expose that the real number is not fundamental. They expose a construction of the complex number as P3 from the same rules that build the real number P2. This alone is cause for celebration... but wait, there's more. There is so much here and it all modifies aspects of human understanding. That emergent spacetime could come from pure arithmetic is the most beautiful consquence of all. That these details are denied and that such tripe as abstract algebra is upheld exposes the frailty of the human race.

Yes, mimicry is operant here. Heavily. As much as mimicry provides the propagation of information there is as well an impedance factor to details which go against said mimicked information. The lack of tolerance by mathematical types to information that goes against their training is a plenty good instance of this impedance, but just as we can name our linguistic abilities as a positive instance learned from childhood we can readily instance the foraging for foods and a lack of humans who eat poisonous varieties as an evolutionary source of such impedance. The two together are like getting forked in a game of chess. Another F word for you. Have you witnessed the position of Zelos making his move then taking it back only to make it again? That sort of infinity is not even a strange loop. His own impedance has him locked up. It's not a pretty thing, but it is my duty here to cause it. Without this behavior for others to gaze upon I have nothing. It is thanks to those who engage here and attempt a defense of the existing system that I can have a voice in a new scale. Rather it is through this process that I can be heard. It is close to the Socratic method. Clearly participation is required of both sides. Those who fail to participate really ought to be taken to bear given the depth of the focus. In that abstract algebra is a move on a chess board it is a weak move. In hindsight of polysign it is an overly complicated strategy. In that math carries any game theoretic process should we discuss cheating and game theory? That I am in a capitalist nation that treats the world as its domain no doubt factors in here. That academia might suffer under capitalism is an obvious state. Enough. We'll take this one up another time.
Post by Lalo T.
https://en.wikipedia.org/wiki/Template:Arithmetic_operations <---
https://en.wikipedia.org/wiki/Algebraic_operation#Arithmetic_vs_algebraic_operations
https://en.wikipedia.org/wiki/Algebraic_expression#Algebraic_and_other_mathematical_expressions
https://en.wikipedia.org/wiki/Algebraic_expression
https://en.wikipedia.org/wiki/Polynomial_ring#Polynomial_expression
https://en.wikipedia.org/wiki/Elementary_algebra
https://en.wikipedia.org/wiki/Setoid <---
https://en.wikipedia.org/wiki/Quotient_type
https://en.wikipedia.org/wiki/Algebraic_data_type
https://en.wikipedia.org/wiki/Product_type
http://www.cse.chalmers.se/research/group/logic/TypesSS05/Extra/palmgren.pdf
Understanding Polysign Numbers the Standard Way by Hagen von Eitzen
http://www.von-eitzen.de/math/PolysignNumbers.pdf
(mainly, section 2)
https://mathworld.wolfram.com/EquivalenceClass.html
https://en.wikipedia.org/wiki/Equivalence_class
https://groupprops.subwiki.org/wiki/Conjugacy_class
https://math.stackexchange.com/questions/594458/definition-of-quotient-set
https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)
https://en.wikipedia.org/wiki/Quotient_module
https://en.wikipedia.org/wiki/Quotient_ring
https://mathworld.wolfram.com/QuotientGroup.html
https://mathworld.wolfram.com/Subgroup.html
https://mathworld.wolfram.com/Coset.html
https://groupprops.subwiki.org/wiki/Proper_subgroup
https://groupprops.subwiki.org/wiki/Normal_subgroup
https://groupprops.subwiki.org/wiki/Proper_normal_subgroup
https://en.wikipedia.org/wiki/Subset#Definitions
https://en.wikipedia.org/wiki/Subobject
https://en.wikipedia.org/wiki/Class_(set_theory)
https://en.wikipedia.org/wiki/Module_homomorphism
https://en.wikipedia.org/wiki/Ring_homomorphism
https://en.wikipedia.org/wiki/Algebra_homomorphism
https://en.wikipedia.org/wiki/Associative_algebra#Definition
https://math.stackexchange.com/questions/2132812/r-algebra-homomorphisms
https://math.stackexchange.com/questions/970368/what-is-an-r-algebra
https://en.wikibooks.org/wiki/Category:Book:Abstract_Algebra
https://en.wikipedia.org/wiki/Kernel_(algebra)
https://encyclopediaofmath.org/wiki/Ideal
https://en.wikipedia.org/wiki/Bilinear_map
https://en.wikipedia.org/wiki/Injective_function
https://mathworld.wolfram.com/Isomorphism.html
https://en.wikipedia.org/wiki/Image_(mathematics)
https://en.wikipedia.org/wiki/Iota
http://math.chapman.edu/~jipsen/structures/doku.php/equivalence_relations
https://simple.wikipedia.org/wiki/Equivalence_relation
http://math.chapman.edu/~jipsen/structures/doku.php/reflexive_relations
http://math.chapman.edu/~jipsen/structures/doku.php/symmetric_relations
http://math.chapman.edu/~jipsen/structures/doku.php/transitive_relations
https://en.wikipedia.org/wiki/Root_of_unity
https://mathworld.wolfram.com/CyclotomicPolynomial.html
https://mathworld.wolfram.com/MonicPolynomial.html
https://mathworld.wolfram.com/deMoivreNumber.html
https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem
https://mathworld.wolfram.com/RelativelyPrime.html
https://mathworld.wolfram.com/ChineseRemainderTheorem.html
https://mathworld.wolfram.com/DirectSum.html
https://en.wikipedia.org/wiki/Direct_sum
Lalo T.
2020-09-20 08:13:18 UTC
- https://en.wikipedia.org/wiki/Indeterminate_(variable)

- Sometime I just put links (or a family of them), sometimes I give context of a certain link.
I dont stuff with links because 'the reader will be not able to google them',
in a very few cases will be trouble, or cases where concepts with the same name
in nearby contexts (like in https://en.wikipedia.org/wiki/Homogeneity_(disambiguation)#In_mathematics )
Everyone here knows how to google, but even that take one, maybe two seconds.
The risk of being called a "stuffer" is acceptable.

- in https://en.wikipedia.org/wiki/Setoid :
" Setoids are studied especially in proof theory and in type-theoretic foundations
of mathematics. Often in mathematics, when one defines an equivalence relation
on a set, one immediately forms the quotient set (turning equivalence into
equality). In contrast, setoids may be used when a difference between identity
and equivalence must be maintained, often with an interpretation of intensional
equality (the equality on the original set) and extensional equality (the
equivalence relation, or the equality on the quotient set). "

- Who first defined polynomials as sequences?
https://hsm.stackexchange.com/questions/9710/who-first-defined-polynomials-as-sequences

- When were polynomial equations first factored?
https://hsm.stackexchange.com/questions/8068/when-were-polynomial-equations-first-factored

- How did Ruffini discover his method of polynomial division?
https://hsm.stackexchange.com/questions/8121/how-did-ruffini-discover-his-method-of-polynomial-division

- in the page 165 (perspective of a chief editor of a journal)
http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf

- In the link https://en.wikipedia.org/wiki/Template:Arithmetic_operations
is for the difference between formats :

(participant) (participant) = ...

(surgeon) patient = .... (André Warusfel used this analogy)
Tim Golden BandTech.com
2020-09-20 12:32:39 UTC
Post by Lalo T.
- https://en.wikipedia.org/wiki/Indeterminate_(variable)
- Sometime I just put links (or a family of them), sometimes I give context of a certain link.
I dont stuff with links because 'the reader will be not able to google them',
in a very few cases will be trouble, or cases where concepts with the same name
in nearby contexts (like in https://en.wikipedia.org/wiki/Homogeneity_(disambiguation)#In_mathematics )
Everyone here knows how to google, but even that take one, maybe two seconds.
The risk of being called a "stuffer" is acceptable.
" Setoids are studied especially in proof theory and in type-theoretic foundations
of mathematics. Often in mathematics, when one defines an equivalence relation
on a set, one immediately forms the quotient set (turning equivalence into
equality). In contrast, setoids may be used when a difference between identity
and equivalence must be maintained, often with an interpretation of intensional
equality (the equality on the original set) and extensional equality (the
equivalence relation, or the equality on the quotient set). "
- Who first defined polynomials as sequences?
https://hsm.stackexchange.com/questions/9710/who-first-defined-polynomials-as-sequences
- When were polynomial equations first factored?
https://hsm.stackexchange.com/questions/8068/when-were-polynomial-equations-first-factored
- How did Ruffini discover his method of polynomial division?
https://hsm.stackexchange.com/questions/8121/how-did-ruffini-discover-his-method-of-polynomial-division
For all of these uses of the term 'polynomial' you now have to wonder which polynomial they are speaking of. As far as I know the polynomial of abstract algebra still to this day goes confused. The only terminology that I see is of an 'indeterminate' polynomial as far as the X specification goes. Just how indeterminate? There are two levels: X is a variable in the domain of the coefficient, or X is an abstract type whose quality goes ignored even by the subject that created it shortly after it bothered to formalize operators. There is no usage of 'abstract X' though it appears as if there ought to be. It would help greatly a random reader of links such as yours above to distinguish just what polynomial is implied by the word polynomial. Oh what's that? X squared actually means you multiply X by X? Oh no, that is just a placeholder here. Do not even think of doing that when I say polynomial here. That was the old way. The meek student bows to the master and wonders about this conflict of information. How can this word come to mean two completely different things? I would suggest that the proper naming would have been something like
dimensionomial
so that it carries the familiarity of polynomial but distinguishes itself cleanly from the polynomial of old. The choice to maintain this confusion is strategic I believe. It is a sign of the pushing of a poorly built concept. The avoidance of the problems that I highlight here as the black swan
1.23 X
whose operator is definitely not binary and yet whose ring properties are solidly claimed by abstract algebra is the grand dodge. This then leads into an insistence on an infinity of coefficients why? This is just another dodge. This then leads to the denial of the polynomial sum actually being a binary operator sum, at which point we do indeed have the falsification. At this point Every polynomial term fails the formalization. Thanks to Mike Terry we have that. Poor Zelos, still moving back and forth between the 'historical' notation and his infinite sum can't find any difference at all there, but he'll keep moving back and forth as a form of dodge to protect his own mental well-being. One must preserve the book. Peter petered out really early, though I did have high hopes because he was staring right at the thing. Horand is now devolving in a dodge of the ring definition which he so strongly upholds on the polynomial form. It is a long way around, but I'm pretty sure he's going to reverse himself anytime now. Certainly I do not anticipate that any here including you will come over to my side. This is standard feature of such debates. I just keep putting the black swan under your noses here. I suppose being black it is harder to see, but I'm doing my best to put some light onto it. Well, this does have to include the background in order for their to be any contrast so good on you to fill out some of that background.

Have you found any distinction in terminology that allows you to discover whether the X in a polynomial is the abstract form or the indeterminate form? It is as if the polynomial became god and no body can question its form. This does bring me around to the complications of the ideal and quotient. Why couldn't they have simple declared
X X = - 1
in order to expose the complex numbers? We see the infinite series collapse here. We see the modulo form wrap. Ah, but we see an abuse of the confusion that they have leveraged on the usage of the term 'polynomial'. This confusion must be wrapped in more confusion so that those frustrated by the level of obscurity will either walk away or just shut up. I suggest that this is the actual result of this topic. Students who drop their math 'career' at this level are likely the finest students.

Perhaps it is just too silly that X turns out to be sqrt(-1) even after all of this careful construction. So in order to veil it and the failings of the binary operators in the polynomial construction they've taken a long and painful route. I once climbed a mountain with such a path. So difficult that upon going up two thirds of the way you hit a really tricky spot but realize you have to go forward through it because going back is no solution. This again is the state of the art here.

Thanks for the less obnoxious series of links. I did at least check a few out.
Post by Lalo T.
- in the page 165 (perspective of a chief editor of a journal)
http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
- In the link https://en.wikipedia.org/wiki/Template:Arithmetic_operations
(participant) (participant) = ...
(surgeon) patient = .... (André Warusfel used this analogy)
Tim Golden BandTech.com
2020-09-19 15:52:03 UTC
Post by h***@gmail.com
Post by Tim Golden BandTech.com
I would hope that when we are discussing operator theory that we have cleanly distinguished operators from values or elements. Because the operators have been carefully declared within set theoretic terms we ought to be able to apply these distinctions onto
1.23 X
and we should admit that this expression carries one multiplication operation on two elements: 1.23 and X.
It does. Every real number is in the ring generated by R and the singleton {X}. 1.23 * X is a product of two ring elements; it is in the ring.
Post by Tim Golden BandTech.com
There actually is a similar criticism on real numbers, where the signs of values
- 0.01, + 0.12, 1.23
are obscured by the usage of the '+' sign as the sum operator. And then of course the last instance above exposes the implied positive value when the sign is missing. Whether our own conventions and notation matter: yes, they certainly do. Especially as we have descended into formal operator theory there ought to be no looseness whether by imbeciles or by high priests of the subject. If an imbecile should provide the priest with a black swan ought the priest to take it into consideration? A new form of number system that has no need of the conflicted abstractions within AA is right under your noses here but it will take an imbecile to find it. Your noses are too high in the air.
I don't accept Horand's claim that the set
{ R, X } (where R is the set of real numbers and X is an 'object')
is legitimate for it suggests then that the black swan ought to resolve to a single element. This is why instantiation is relevant. So long as one remains in a polyvariate form such as
a0 + a1 X
then clearly the expression cannot evaluate simply due to the unknown values. This is true regardless of the ring definition.
And here's the rub. You are confusing the two concepts of resolution and evaluation. Your black swan 1.23X is *not* a real number, but a polynomial. To that degree it is resolved, even though you may not even have instantiated yet the abstract operations of addition and multiplication in the ring. Even if you think of X as a (variable) real number, you do not get a single value for the polynomial (or monomial in this case).
Resolution comes into play most forcefully when you are setting up the abstract rules for adding or multiplying two polynomials 1.23X + 2.34X^2 and 3.45 + 4.56X^3.
Evaluation depends on what you envision X to stand for. X could be a real variable, a complex variable, a matrix variable (with or without any number of additional constraints), etc., and even then you can only evaluate the polynomial when you substitute *one* particular value for X.
Post by Tim Golden BandTech.com
I might as well have just written
a0 a1
where again these are real values under discussion.
That is just a little too loose, but you *could* write the ordered pair (a0,a1) and *define* it to mean a0 + a1X. (In fact, someone upthread suggested that to you previously.) You also have to be careful about polynomials such as a0 + a2X^2. Simply writing a0 a2 won't cut it, but the ordered triple (a0, 0, a2) describes the situation perfectly.
Post by Tim Golden BandTech.com
Since we have descended so low as to enter operator theory formally should we declare an offense on the usage of variables for denying the operator its primal duty? I have not gone here, yet this is a possible opening. Because we have descended so low as to formalize operators could it be that formalizing variables becomes necessary? Such as multiple variable types? Here at least there would be some hope of cleaning up this topic as for instance the variable a(n) clearly is distinct from the X which has no meaning in the case of abstract algebra other than as a rotational placeholder that refuses to evaluate. It is definitely provable that the polynomial usage within abstract algebra flies in the face of the ring definition and is antithetical to it. At the very least I have exposed this much here. For Zelos there is no room to grow. The priests would not like us tampering with their curriculum why? Because the whole thing is up for grabs once you do so. I can readily declare this area open and indeed due for development; particularly in light of PhD bloat.
Nicely enough the study of sign of the real number does lead into concrete instantiation of general dimensional algebraically behaved types. Sure they can take variable assignments such as z1, z2, z3
but these variables do in fact allow for instantiation of a concrete type which preexisted their usage. Possibly this is a criterion for the usage of variables. So long as a(n) are real valued they imply that actual substitution of actual concrete values is possible, but so long as a(n) remain undeclared then they have to be back a level from the status of a variable that allows concrete instances. These sorts of ideas are probably going to have to enter this curriculum in order for it to resolve, yet the troubles that we see in two sides arriving at
1.23 X is ring behaved
1.23 X is not ring behaved
don't really bode well for the existing curriculum.
A lack of instantiation exists throughout mathematics and is leveraged as a means of accommodating PhD bloat.
That is a broad form whereas here there is an attempt at specificity. It is a sad state of affairs when my own proof that
1.23 X is not ring behaved
goes unfalsified by the math studs here but I do credit Horand with facing the black swan directly. Shortly Zelos will be backing off on his infinite requirements no different than on the 'historical' interpretation. Ah, but then he'll be back to it again. I'd rather take up the elemental form with you all since it does seem to fill out the coverage a bit more. I am certain that the elements of the polynomial form are ubiquitous. I am happy to dismantle the polynomial into those elements, yet to claim that they are ring behaved will never fly. They are the antithesis of ring behaved. They refuse to operate.
As far as I know, sets contain elements and operators work on those elements. These are fundamental concepts that are being stretched beyond the formality that the subject claims. No care is taken to study the polynomial form. Worse yet they attempt to make it untouchable by insisting formally on an infinite series form. That is a sure coverup. Meanwhile the antique form that I work in still is kicking around; even at MIT. Yeah, they've got the bloat problem real bad there.
Again I credit you with upholding the status quo position strongly. But your lack of attention to the binary operator and the closure requirement is a strong indicator that your interpretation cannot hold up. This might send you down Zelos's path of insisting that the problem is with a lack of infinite length polynomial; an inverted induction that was only built in order to defray the weakness that we are onto here. He blames me for the formalities but I can assure him they are not of my making. As if his 'formal' notation does any favors to clear things up. Those lies Horand you have dodged, and I wonder if you have any comments regarding these stalwarts of the curriculum of abstract algebra? Especially the claim that the addition operator within the polynomial is not actually addition such as Mike Terry's interpretation... I'm afraid you are about to take this path.

I find it peculiar that you would care to uphold that the real values unioned with X form the set of the binary operations that allow the terminology 'ring behaved' to be applied to the polynomial with real coefficients. But I also see that the only way that you are allowing this formal set
{ R, X } (where R are the real numbers and X is the placeholder of abstract algebra)
is to claim that the polynomial form
a0 + a1 X + a2 X X + ...
is the form on which these elements play out to form that ring. Yet the preservation of this very form is antithetical to the ring definition. By definition these X's cannot allow for binary operations. They act as preservatives. They halt such action. Yes, on the larger scale of polynomials we do see that addition and multiplication play out, but this behavior does not prevent us from dismantling the polynomial. It is here that I am working. It is here that your claim of the elemental form breaks. It is by the closure requirement of these binary operators; a topic you are avoiding; that is at the heart of the ring definition; that you and the curriculum are offending.

For the sake of an alternate route to get to the dismantled polynomial let's start from scratch with the assumption that we have elements of a ring
a0, a1, a2, a3, ...
so that we are in the abstracted form here of untyped yet declaratively behaved 'elements' a(n) in the ring R1. I do believe that some will find puzzling conflicts within this language yet these are well used and abused by the modern mathematician. Indeed it is through these terms that such piles of accumulation can build making room for ever more publication space that PhD bloats require. Here we are popping just one of these elder bloats that seem to have fused into a powerful skin. Rhetoric aside (for a little while) if we build a sum of these elements
a2 + a3 + a4
then by induction we can admit that a sum of three elements yields a single element in the set as well as a sum of two elements, which is all that is defined. Really to even expand such a simple point out to such a long paragraph with so much denigrating graphic language seems a fraud, yet isn't this also the form of the polynomial? Whether it has infinite terms or not, we may as well simply regard just a few terms for study, and since those terms are elements of the ring there should be no problem admitting them and their contents as worthy of study. Yet all status quo representatives discourage this view. This is exactly where the rubber meets the bloat. The burnout of the bloat occurs on a black swan
1.23 X
and at this point we have no need of even an infinitude of other terms. By the method of dissection above we don't even need to declare them as zero coefficients, though this zero assignment probably is the simplest route. Now we're on a fancier form of falsification. That's a lot of efs in a little space. If you are a student of this subject you might not be mumbling many f words under your breath, but a teacher or worse yet an author of one of those money making texts... let the f words flow freely f*cktards.

The insistence on the polynomial form is not so much a requirement as it is a construction that can be freely built. There is nothing in it really that is formal. It's most general form and likely most compact general form is
Sum over n of [ a sub n X ^ n ] (please visualize a sigma sum and proper notation; indexing how you like)
yet (again) by the ring definition the terms are just as usable as is the entire expression. The problem really is that the language we are using of variables and worse yet untyped variables has brought us to some high land where the very notion of elemental values has completely disappeared. But when we utter
"polynomials with real coefficients"
does this then turn X real? No! not under the abstract algebra system. It is out of these sorts that things like the complex numbers will be recovered. It's really quite a long way around... especially if you are of Zelos' ilk. He's having trouble with an 'out of RAM' error at his first instantiation so that he can never get to the type match conflicts that he coded. Very clever position to take.

The formal set of elements of the real numbers and the element X is not going to fly. Dismantling the polynomial is where the conflict occurs.
1.23 X
is not an elemental form. You cannot simply grant it elemental form because it is composed of two elements. Now you have bowed to claiming the set of reals and one X as the elemental set. Now apply the binary operator requirement to these elements. We do not get a single element. I have no idea how you can come to claim the polynomial form as elemental when it clearly is composed of numerous operators. We cannot compose elemental forms from operators. Thus we are landing in a fairly simple fundamental issue here. In fact this issue does creep into other places, such as the construction of the complex numbers out of consideration of sqrt(-1). We are conditioned to consider an operator on a constant value as a constant value, yet there is an operator present and so the elemental status is not really present.

I think at this point we can shift the conversation down to this set theoretic question of whether the polynomial is an elemental form. Particularly where you have selected a set of elements the answer is no. Unfortunately the scrutiny that we are onto here does require that we take care with notions of variables and set containment that have been pushed farther than simplistic set theory allows. The abstract algebraist is working off of a set like
( grapes, apples, oranges, cherries, bananas, ..., electricity )
if Horand's interpretation is correct. That this topic is taken as a part of 'pure' mathematics is a fraud. It is trickery and it is inauthentic. Horand your own line of argumentation is practically self-falsifying. As I dismantle the polynomial you will continue to insist that I have not dismantled the polynomial. This forces us onto this elemental awareness I believe. The polynomial can appear to be ring behaved in its general form when the coefficients are still undeclared, for there is no possibility of doing any elemental mechanics... well this then does become a broader criticism on much more of mathematics doesn't it? Any mathematics which makes use of elemental set theory within its basis cannot be instantiated until the set on which it works is instantiated. Possibly the very usage of language such as 'the ring R' is invalid since the claim of specificity has gone an order beyond the variable which was a familiar and usable type under stricter set theory. Polynomials in the ring R have gone into la-la land with an untyped X and an unspecified coefficient. Yes, they harken back to a simpler form that we learned in high school building parabolas and so forth, but that this complicated and farcical construction is the yield of higher math is not a path or method to be proud of. How could the subject of abstract algebra occur without an abstract type? Because the abstract type breaks. Because it is no type at all. Because the very existence of set theory is a matter of categorizing values and invalidating values which do not belong in a well built set. How does it come that the real numbers were so carefully constructed with regard to set theory and then comes this? No. My polysign numbers are far superior to this tripe. The falsification of abstract algebra requires just one black swan. It seems that at this point we have a black and white swan. It will do, but I believe that it is fully black. This is mathematics we are doing right?

Your real duty Horand is to falsify my claim of a black swan. That we can each interpret the same expression as conflicted and as unconflicted is going to be a sore point to the subject isn't it? How come you dodge the ring requirements so readily? They are the crux. That the polynomial is composed of so many operations internal to it does deny it elemental status doesn't it? What then are elements to you? What are the consequences of this construction that lead you to defend it so strongly? Can't those results be achieved without this run-around? Is it really so hard to just grant X a modulo quality? I have done this for sign and it works perfectly well. Ultimately what it exposes is that the real numbers themselves are not fundamental; that their modulo-two type is just one step in the midst of a family:
P1, P2, P3, P4, ...
These constructions do not rely upon any 'abstract' methods. These values are readily instantiated. That I find that P3 are the complex numbers, and that P3 build out from the same exact rules that build P2; just up one in the progression; that there is even a P1 beneath these whose paradox exactly matches that of time... yet the religious qualities of the modern mathematician who has attached to the four hundred year programming of the real number as fundamental has to come up here into this obnoxious curriculum to find an answer. Really it adds up to just what much of my rhetoric points out. It is a fun read too, I hope. That said we can only go on with any readership here for so long. It seems that we have stalled out rather than covered any new ground in the last few go-arounds. I think the last substantial piece was your admission of the set of real numbers and X to formalize the polynomials with real coefficients, but it just does not work. It's like you've come part way yet you are landing yourself in the contradiction and denying it. You pay no attention to the role of binary operators which are exactly implied by the usage of 'ring'.

Can we admit that we are down deep in the subject of mathematics here? That a fault in this region ought to be of importance? That we are in so deep that matters of notation and convention are definitely important? That an opening in this region could carry consequences throughout the entirety of not just the subject at hand but in philosophy and physics as well? Those divisions are false. They allow the mathematician to play escapist, but there really is no escapement. We are prisoners of spacetime; here; now; engaged in a progression that is supposedly one of the high points of the human race, and yet in the religious form that is being practiced lays its downfall. Though my position appears to be one of destruction it is ultimately a declaration of openness. We all ought to formally declare these problems as open every day as much as they seem closed and done. One of these methods leads into a progression. The other to stagnation. I have a stepping stone that predicts arithmetic correspondence to spacetime. It suggests that electromagnetic behavior will be demoted and that a simplified form will exist when spacetime takes on more structure. This flies in the face of an isotropic spacetime; the first cosmological principal. Yes, I can go so far as to declare that wrong and indeed they make a farce by insist on an averaging to cover their asses. The means are indirect, yet what leads me through these interpretations is obvious. It's been under everyone's noses for the last four hundred years. So simple as to be constructed with a hammer and a chisel while you all insist on the swiss army knife.
h***@gmail.com
2020-09-20 13:13:29 UTC
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
I would hope that when we are discussing operator theory that we have cleanly distinguished operators from values or elements. Because the operators have been carefully declared within set theoretic terms we ought to be able to apply these distinctions onto
1.23 X
and we should admit that this expression carries one multiplication operation on two elements: 1.23 and X.
It does. Every real number is in the ring generated by R and the singleton {X}. 1.23 * X is a product of two ring elements; it is in the ring.
Post by Tim Golden BandTech.com
There actually is a similar criticism on real numbers, where the signs of values
- 0.01, + 0.12, 1.23
are obscured by the usage of the '+' sign as the sum operator. And then of course the last instance above exposes the implied positive value when the sign is missing. Whether our own conventions and notation matter: yes, they certainly do. Especially as we have descended into formal operator theory there ought to be no looseness whether by imbeciles or by high priests of the subject. If an imbecile should provide the priest with a black swan ought the priest to take it into consideration? A new form of number system that has no need of the conflicted abstractions within AA is right under your noses here but it will take an imbecile to find it. Your noses are too high in the air.
I don't accept Horand's claim that the set
{ R, X } (where R is the set of real numbers and X is an 'object')
is legitimate for it suggests then that the black swan ought to resolve to a single element. This is why instantiation is relevant. So long as one remains in a polyvariate form such as
a0 + a1 X
then clearly the expression cannot evaluate simply due to the unknown values. This is true regardless of the ring definition.
And here's the rub. You are confusing the two concepts of resolution and evaluation. Your black swan 1.23X is *not* a real number, but a polynomial. To that degree it is resolved, even though you may not even have instantiated yet the abstract operations of addition and multiplication in the ring. Even if you think of X as a (variable) real number, you do not get a single value for the polynomial (or monomial in this case).
Resolution comes into play most forcefully when you are setting up the abstract rules for adding or multiplying two polynomials 1.23X + 2.34X^2 and 3.45 + 4.56X^3.
Evaluation depends on what you envision X to stand for. X could be a real variable, a complex variable, a matrix variable (with or without any number of additional constraints), etc., and even then you can only evaluate the polynomial when you substitute *one* particular value for X.
Post by Tim Golden BandTech.com
I might as well have just written
a0 a1
where again these are real values under discussion.
That is just a little too loose, but you *could* write the ordered pair (a0,a1) and *define* it to mean a0 + a1X. (In fact, someone upthread suggested that to you previously.) You also have to be careful about polynomials such as a0 + a2X^2. Simply writing a0 a2 won't cut it, but the ordered triple (a0, 0, a2) describes the situation perfectly.
Post by Tim Golden BandTech.com
Since we have descended so low as to enter operator theory formally should we declare an offense on the usage of variables for denying the operator its primal duty? I have not gone here, yet this is a possible opening. Because we have descended so low as to formalize operators could it be that formalizing variables becomes necessary? Such as multiple variable types? Here at least there would be some hope of cleaning up this topic as for instance the variable a(n) clearly is distinct from the X which has no meaning in the case of abstract algebra other than as a rotational placeholder that refuses to evaluate. It is definitely provable that the polynomial usage within abstract algebra flies in the face of the ring definition and is antithetical to it. At the very least I have exposed this much here. For Zelos there is no room to grow. The priests would not like us tampering with their curriculum why? Because the whole thing is up for grabs once you do so. I can readily declare this area open and indeed due for development; particularly in light of PhD bloat.
Nicely enough the study of sign of the real number does lead into concrete instantiation of general dimensional algebraically behaved types. Sure they can take variable assignments such as z1, z2, z3
but these variables do in fact allow for instantiation of a concrete type which preexisted their usage. Possibly this is a criterion for the usage of variables. So long as a(n) are real valued they imply that actual substitution of actual concrete values is possible, but so long as a(n) remain undeclared then they have to be back a level from the status of a variable that allows concrete instances. These sorts of ideas are probably going to have to enter this curriculum in order for it to resolve, yet the troubles that we see in two sides arriving at
1.23 X is ring behaved
1.23 X is not ring behaved
don't really bode well for the existing curriculum.
A lack of instantiation exists throughout mathematics and is leveraged as a means of accommodating PhD bloat.
That is a broad form whereas here there is an attempt at specificity. It is a sad state of affairs when my own proof that
1.23 X is not ring behaved
goes unfalsified by the math studs here but I do credit Horand with facing the black swan directly. Shortly Zelos will be backing off on his infinite requirements no different than on the 'historical' interpretation. Ah, but then he'll be back to it again. I'd rather take up the elemental form with you all since it does seem to fill out the coverage a bit more. I am certain that the elements of the polynomial form are ubiquitous. I am happy to dismantle the polynomial into those elements, yet to claim that they are ring behaved will never fly. They are the antithesis of ring behaved. They refuse to operate.
As far as I know, sets contain elements and operators work on those elements. These are fundamental concepts that are being stretched beyond the formality that the subject claims. No care is taken to study the polynomial form. Worse yet they attempt to make it untouchable by insisting formally on an infinite series form. That is a sure coverup. Meanwhile the antique form that I work in still is kicking around; even at MIT. Yeah, they've got the bloat problem real bad there.
Again I credit you with upholding the status quo position strongly. But your lack of attention to the binary operator and the closure requirement is a strong indicator that your interpretation cannot hold up.
The point, of course, as I wrote before, is that all results about polynomials where the X are real-valued variables extend to situations where the X are complex-valued variables, or real-valued matrices, or even other situations as long as *you* (the user) can define a sensible way to multiply an *instance* of such an object by a scalar (that does not even have to be real), and how to multiply the same instance of X by itself enough times to evaluate X^n whenever needed. Since the results about *formally* adding two polynomials and multiplying two polynomials will hold however you think of X (subject to the closure and distributive properties), you do *not* need to formulate a separate theory of complex-valued or matrix-valued polynomials. The abstract version of the theory is A. perfectly sufficient and B. much more economical.
Post by Tim Golden BandTech.com
Your real duty Horand is to falsify my claim of a black swan.
I have no duty to you at all, sir.

You are free to disregard anything I write, as I am free to disregard your "black swan". (I have dealt with your swan to my own satisfaction. You are again free to disregard what I wrote.)
Tim Golden BandTech.com
2020-09-20 14:40:56 UTC
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
I would hope that when we are discussing operator theory that we have cleanly distinguished operators from values or elements. Because the operators have been carefully declared within set theoretic terms we ought to be able to apply these distinctions onto
1.23 X
and we should admit that this expression carries one multiplication operation on two elements: 1.23 and X.
It does. Every real number is in the ring generated by R and the singleton {X}. 1.23 * X is a product of two ring elements; it is in the ring.
Post by Tim Golden BandTech.com
There actually is a similar criticism on real numbers, where the signs of values
- 0.01, + 0.12, 1.23
are obscured by the usage of the '+' sign as the sum operator. And then of course the last instance above exposes the implied positive value when the sign is missing. Whether our own conventions and notation matter: yes, they certainly do. Especially as we have descended into formal operator theory there ought to be no looseness whether by imbeciles or by high priests of the subject. If an imbecile should provide the priest with a black swan ought the priest to take it into consideration? A new form of number system that has no need of the conflicted abstractions within AA is right under your noses here but it will take an imbecile to find it. Your noses are too high in the air.
I don't accept Horand's claim that the set
{ R, X } (where R is the set of real numbers and X is an 'object')
is legitimate for it suggests then that the black swan ought to resolve to a single element. This is why instantiation is relevant. So long as one remains in a polyvariate form such as
a0 + a1 X
then clearly the expression cannot evaluate simply due to the unknown values. This is true regardless of the ring definition.
And here's the rub. You are confusing the two concepts of resolution and evaluation. Your black swan 1.23X is *not* a real number, but a polynomial. To that degree it is resolved, even though you may not even have instantiated yet the abstract operations of addition and multiplication in the ring. Even if you think of X as a (variable) real number, you do not get a single value for the polynomial (or monomial in this case).
Resolution comes into play most forcefully when you are setting up the abstract rules for adding or multiplying two polynomials 1.23X + 2.34X^2 and 3.45 + 4.56X^3.
Evaluation depends on what you envision X to stand for. X could be a real variable, a complex variable, a matrix variable (with or without any number of additional constraints), etc., and even then you can only evaluate the polynomial when you substitute *one* particular value for X.
Post by Tim Golden BandTech.com
I might as well have just written
a0 a1
where again these are real values under discussion.
That is just a little too loose, but you *could* write the ordered pair (a0,a1) and *define* it to mean a0 + a1X. (In fact, someone upthread suggested that to you previously.) You also have to be careful about polynomials such as a0 + a2X^2. Simply writing a0 a2 won't cut it, but the ordered triple (a0, 0, a2) describes the situation perfectly.
Post by Tim Golden BandTech.com
Since we have descended so low as to enter operator theory formally should we declare an offense on the usage of variables for denying the operator its primal duty? I have not gone here, yet this is a possible opening. Because we have descended so low as to formalize operators could it be that formalizing variables becomes necessary? Such as multiple variable types? Here at least there would be some hope of cleaning up this topic as for instance the variable a(n) clearly is distinct from the X which has no meaning in the case of abstract algebra other than as a rotational placeholder that refuses to evaluate. It is definitely provable that the polynomial usage within abstract algebra flies in the face of the ring definition and is antithetical to it. At the very least I have exposed this much here. For Zelos there is no room to grow. The priests would not like us tampering with their curriculum why? Because the whole thing is up for grabs once you do so. I can readily declare this area open and indeed due for development; particularly in light of PhD bloat.
Nicely enough the study of sign of the real number does lead into concrete instantiation of general dimensional algebraically behaved types. Sure they can take variable assignments such as z1, z2, z3
but these variables do in fact allow for instantiation of a concrete type which preexisted their usage. Possibly this is a criterion for the usage of variables. So long as a(n) are real valued they imply that actual substitution of actual concrete values is possible, but so long as a(n) remain undeclared then they have to be back a level from the status of a variable that allows concrete instances. These sorts of ideas are probably going to have to enter this curriculum in order for it to resolve, yet the troubles that we see in two sides arriving at
1.23 X is ring behaved
1.23 X is not ring behaved
don't really bode well for the existing curriculum.
A lack of instantiation exists throughout mathematics and is leveraged as a means of accommodating PhD bloat.
That is a broad form whereas here there is an attempt at specificity. It is a sad state of affairs when my own proof that
1.23 X is not ring behaved
goes unfalsified by the math studs here but I do credit Horand with facing the black swan directly. Shortly Zelos will be backing off on his infinite requirements no different than on the 'historical' interpretation. Ah, but then he'll be back to it again. I'd rather take up the elemental form with you all since it does seem to fill out the coverage a bit more. I am certain that the elements of the polynomial form are ubiquitous. I am happy to dismantle the polynomial into those elements, yet to claim that they are ring behaved will never fly. They are the antithesis of ring behaved. They refuse to operate.
As far as I know, sets contain elements and operators work on those elements. These are fundamental concepts that are being stretched beyond the formality that the subject claims. No care is taken to study the polynomial form. Worse yet they attempt to make it untouchable by insisting formally on an infinite series form. That is a sure coverup. Meanwhile the antique form that I work in still is kicking around; even at MIT. Yeah, they've got the bloat problem real bad there.
Again I credit you with upholding the status quo position strongly. But your lack of attention to the binary operator and the closure requirement is a strong indicator that your interpretation cannot hold up.
The point, of course, as I wrote before, is that all results about polynomials where the X are real-valued variables extend to situations where the X are complex-valued variables, or real-valued matrices, or even other situations as long as *you* (the user) can define a sensible way to multiply an *instance* of such an object by a scalar (that does not even have to be real), and how to multiply the same instance of X by itself enough times to evaluate X^n whenever needed. Since the results about *formally* adding two polynomials and multiplying two polynomials will hold however you think of X (subject to the closure and distributive properties), you do *not* need to formulate a separate theory of complex-valued or matrix-valued polynomials. The abstract version of the theory is A. perfectly sufficient and B. much more economical.
Post by Tim Golden BandTech.com
Your real duty Horand is to falsify my claim of a black swan.
I have no duty to you at all, sir.
You are free to disregard anything I write, as I am free to disregard your "black swan". (I have dealt with your swan to my own satisfaction. You are again free to disregard what I wrote.)
Once again Horand you have dodged the binary operator within your own interpretation. The polynomial as a god for you is established, and yet your very abuse of it here is exposed. The multiplication of X by itself is again offensive to the binary operator requirements since
( X )( X ) = X X
is no operation at all. This is definitely not a binary operation. I find that your own inability to go into what an elemental form is; for instance that elements and operators within a formal system are distinct things. It makes perfect sense that you end in this dodgy position. Still I have to credit you with the best defense yet. Also though by the way as we cross the various defenses that have taken place we see unique forms and claims. Why should this be so? And how many of them falsify the black swan? None as far as I can tell. Methods of falsification do go poorly understood it seems within this arena. Why you might wonder? Could PhD bloat have anything to do with this? It certainly would account for the situation. Human nature as well has to be included as a culprit. After all we witness other false belief systems propagated for thousands of years which are far more easily falsified, and these often involving wars and tortures to this day by my own nation. To kill a commie for you mommy then confuse socialists with communists and dictators while your two party system is so much closer to their one party system than most of the socialist countries in existence... you can see how little the math matters. I am not one to take the terms 'ideology' or 'fundamentalism' as negative connotations. It is a matter of what the ideology or fundamentals being practiced are that matters. Here your own methods are careful and yet I do not believe that you are being honest. You are holding your ground at the position of the complete polynomial whereas I have already gone beneath this position to dismantle it and investigate its parts. You stay at your position, and I do see your position and that product and sum out at that level appear hunky dory. Yet as a sum of terms which are ring behaved my position of dismantling the structure that you refuse to dismantle is valid. This explains how Mike Terry can come to claim that the polynomial sum is not actually a sum. This explains how Zelos can come to insist that the polynomial form always has an infinite number of terms (though this inverted induction is really more dodgy it appears more convincing than denying the sum as a sum).

That you do not care to either falsify or support these prior arguments in this thread I can take as a signal. This signal fits perfectly with your own method of defense, and yet because there really is no redundancy to your arguments with theirs I could as well take your position as refuting theirs. You are happy to work on the swan
1.23 X
which you see as white and I see as black. This is more than can be said of either of them. Peter did attempt it, but I can't even remember how he landed up in the cage that once held the black swan. Fear I suppose. The very crux of my falsification hinges on the binary operator requirement of the ring definition. Your own avoidance of this is indicative. I am happy that you are resigned at this position of halting here. I would also like to try and be done with this thread. Obviously if I am correct this issue will deserve quite a lot more attention and serious discussion. It is rather grandiose of me to claim to be restructuring the academic curriculum. I have no actual rights in this situation. I merely put forth a falsification here for others to gaze upon. This block of information will hopefully be available in the future for some orb or other to parse.

I do believe in the freedom to construct and yet it may also be true that the options are limited. Still I have evidence that there do remain constructions yet to be discovered and I predict that as they unfold they will have nothing at all to do with abstract algebra. This subject does not form a strong basis. It is flimsy. It cheats its way through. It hides problematic details. It is PhD bloat. The very lack of detail in speaking of an indeterminate polynomial as if it covers abstract algebra's usage is extremely poor form. Still, thank you for coming forward and being such a firm backboard. That's much better than a waterboard.
h***@gmail.com
2020-09-20 15:48:37 UTC
[...]
Post by Tim Golden BandTech.com
Post by h***@gmail.com
You are free to disregard anything I write, as I am free to disregard your "black swan". (I have dealt with your swan to my own satisfaction. You are again free to disregard what I wrote.)
Once again Horand you have dodged the binary operator within your own interpretation. The polynomial as a god for you is established, and yet your very abuse of it here is exposed. The multiplication of X by itself is again offensive to the binary operator requirements since
( X )( X ) = X X
is no operation at all.
It is a formal operation on strings, if you will. But there is more to it, because the addition or multiplication of the *abstract* polynomials (1.23X + 2.34X^2) + (3.45 + 4.56X^3) will give the same expression whether you think of X as a placeholder for a real number or some other object. (Remember the topic of this thread is *abstract algebra*, not the real numbers.) The rules for finding the sum or product of the two given polynomials does *not* depend on the nature of X, so you should not pretend that it must.

1.23X is well formed (i.e., a *white* object in your parlance), whether X is a real-valued, complex-valued or matrix-valued variable. And X is a variable, *not* a constant. Once you have instantiated the type of X, you can define the binary expression that allows you to multiply a real number and any *value* you substitute for X to obtain a *value* for the polynomial that is associated with the particular *value* you substitute for X. Nothing arcane about that. The big difference, of course, is that this scalar multiplication is *different* for real-valued X and for matrix-valued X, but that should not come as a big surprise. Likewise the product of X*X (or the evaluation of same for specific instantiations of X) is defined differently when you think of X as real-valued and matrix valued objects.

I believe you are making a mountain out of, well, not even a molehill, but a speck of dust or even less.
Tim Golden BandTech.com
2020-09-20 22:24:28 UTC
Post by h***@gmail.com
[...]
Post by Tim Golden BandTech.com
Post by h***@gmail.com
You are free to disregard anything I write, as I am free to disregard your "black swan". (I have dealt with your swan to my own satisfaction. You are again free to disregard what I wrote.)
Once again Horand you have dodged the binary operator within your own interpretation. The polynomial as a god for you is established, and yet your very abuse of it here is exposed. The multiplication of X by itself is again offensive to the binary operator requirements since
( X )( X ) = X X
is no operation at all.
It is a formal operation on strings, if you will. But there is more to it, because the addition or multiplication of the *abstract* polynomials (1.23X + 2.34X^2) + (3.45 + 4.56X^3) will give the same expression whether you think of X as a placeholder for a real number or some other object. (Remember the topic of this thread is *abstract algebra*, not the real numbers.) The rules for finding the sum or product of the two given polynomials does *not* depend on the nature of X, so you should not pretend that it must.
1.23X is well formed (i.e., a *white* object in your parlance), whether X is a real-valued, complex-valued or matrix-valued variable. And X is a variable, *not* a constant. Once you have instantiated the type of X, you can define the binary expression that allows you to multiply a real number and any *value* you substitute for X to obtain a *value* for the polynomial that is associated with the particular *value* you substitute for X. Nothing arcane about that. The big difference, of course, is that this scalar multiplication is *different* for real-valued X and for matrix-valued X, but that should not come as a big surprise. Likewise the product of X*X (or the evaluation of same for specific instantiations of X) is defined differently when you think of X as real-valued and matrix valued objects.
I believe you are making a mountain out of, well, not even a molehill, but a speck of dust or even less.
Yes, certainly
1.23 X
is just a speck, but of course the molehill is built of specks like this, and the mountains too; at least the AA mountains are.

And again you fail to apply the binary operator criterion. The only operations that will satisfy the binary operator have the property of closure that was only just laid out formally in the subject:
"An operator defined on a set S which takes two elements from S as inputs and returns a single element of S."
- https://mathworld.wolfram.com/BinaryOperator.html

The polynomial with real coefficients
1.23 X
is not ring behaved. It is a black swan.

Again, at this point of our discussion the proper route is to get to this detail of distinguishing elements from operations. Clearly the structure of the polynomial contains both in large quantity potentially. Dismantling the polynomial is the only way to apply the ring definition carefully. It is easily done under the guise that it is ring behaved, but the conclusion is that it is not ring behaved. This contradiction really is the black of the black swan. That you stonewall at the point of declaring it white and are refusing to perform the necessary analysis is not such a strong stance. And here we are working at such a fundamental level as merely to instantiate the object that is under discussion. Well, again, I stress that this is likely the cause of downfall of rather a lot of mathematical work. As mathematicians started getting into the real values as a's rather than as real values the step was known as the usage of a variable. Fine, but now these a's in multiplicity are getting applied to something that has gone far beyond the status of variable, and in that we are discussing fundamental issues this lack of attention to the magical X of the AA polynomial is the sore point. It is the opposite of what the ring definition requires. It has no set membership and it refuses to operate. Now this late in the game you want to instantiate X. This is not part of the claim. I am back working on the uninstantiated form of X that is claimed to be ring behaved. The "polynomial with real coefficients" is in use at that stage and is claimed to be ring behaved.You see it and I see it at the large scale but as you say when you get down to the speck it reverses itself. We have landed in a logical fallacy. In hindsight the entire polynomial fails term by term.
h***@gmail.com
2020-09-21 10:09:29 UTC
Post by Lalo T.
Post by h***@gmail.com
[...]
Post by Tim Golden BandTech.com
Post by h***@gmail.com
You are free to disregard anything I write, as I am free to disregard your "black swan". (I have dealt with your swan to my own satisfaction. You are again free to disregard what I wrote.)
Once again Horand you have dodged the binary operator within your own interpretation. The polynomial as a god for you is established, and yet your very abuse of it here is exposed. The multiplication of X by itself is again offensive to the binary operator requirements since
( X )( X ) = X X
is no operation at all.
It is a formal operation on strings, if you will. But there is more to it, because the addition or multiplication of the *abstract* polynomials (1.23X + 2.34X^2) + (3.45 + 4.56X^3) will give the same expression whether you think of X as a placeholder for a real number or some other object. (Remember the topic of this thread is *abstract algebra*, not the real numbers.) The rules for finding the sum or product of the two given polynomials does *not* depend on the nature of X, so you should not pretend that it must.
1.23X is well formed (i.e., a *white* object in your parlance), whether X is a real-valued, complex-valued or matrix-valued variable. And X is a variable, *not* a constant. Once you have instantiated the type of X, you can define the binary expression that allows you to multiply a real number and any *value* you substitute for X to obtain a *value* for the polynomial that is associated with the particular *value* you substitute for X. Nothing arcane about that. The big difference, of course, is that this scalar multiplication is *different* for real-valued X and for matrix-valued X, but that should not come as a big surprise. Likewise the product of X*X (or the evaluation of same for specific instantiations of X) is defined differently when you think of X as real-valued and matrix valued objects.
I believe you are making a mountain out of, well, not even a molehill, but a speck of dust or even less.
Yes, certainly
1.23 X
is just a speck, but of course the molehill is built of specks like this, and the mountains too; at least the AA mountains are.
"An operator defined on a set S which takes two elements from S as inputs and returns a single element of S."
- https://mathworld.wolfram.com/BinaryOperator.html
The polynomial with real coefficients
1.23 X
is not ring behaved. It is a black swan.
1.23X is not a black swan. Again I appeal to your knowledge of computer science.

The function
multiply (real a, real b)
is not a real number, no matter how much you might try to convince yourself or others that it should be. However,
c = multiply (1.23, 2.34)
*returns* a real number c, given real-valued inputs a and b. Likewise the function
Multiply (real a, polymorphic B)
is not a real number or a polymorphic object. However, implemented properly,
c = Multiply (1.23, 2.34)
returns a real number, and
C = multiply (1.23, [[2.34, 3.45] [4.56, 5.67]])
returns a 2x2 matrix. And if you define a further function
swan(polymorphic B)
and implement it to return
C = Multiply (1.23, B)
guess what happens? Multiply (1.23, B) is your swan 1.23X, but it is decidedly white.
Tim Golden BandTech.com
2020-09-21 12:35:18 UTC
Post by h***@gmail.com
Post by Lalo T.
Post by h***@gmail.com
[...]
Post by Tim Golden BandTech.com
Post by h***@gmail.com
You are free to disregard anything I write, as I am free to disregard your "black swan". (I have dealt with your swan to my own satisfaction. You are again free to disregard what I wrote.)
Once again Horand you have dodged the binary operator within your own interpretation. The polynomial as a god for you is established, and yet your very abuse of it here is exposed. The multiplication of X by itself is again offensive to the binary operator requirements since
( X )( X ) = X X
is no operation at all.
It is a formal operation on strings, if you will. But there is more to it, because the addition or multiplication of the *abstract* polynomials (1.23X + 2.34X^2) + (3.45 + 4.56X^3) will give the same expression whether you think of X as a placeholder for a real number or some other object. (Remember the topic of this thread is *abstract algebra*, not the real numbers.) The rules for finding the sum or product of the two given polynomials does *not* depend on the nature of X, so you should not pretend that it must.
1.23X is well formed (i.e., a *white* object in your parlance), whether X is a real-valued, complex-valued or matrix-valued variable. And X is a variable, *not* a constant. Once you have instantiated the type of X, you can define the binary expression that allows you to multiply a real number and any *value* you substitute for X to obtain a *value* for the polynomial that is associated with the particular *value* you substitute for X. Nothing arcane about that. The big difference, of course, is that this scalar multiplication is *different* for real-valued X and for matrix-valued X, but that should not come as a big surprise. Likewise the product of X*X (or the evaluation of same for specific instantiations of X) is defined differently when you think of X as real-valued and matrix valued objects.
I believe you are making a mountain out of, well, not even a molehill, but a speck of dust or even less.
Yes, certainly
1.23 X
is just a speck, but of course the molehill is built of specks like this, and the mountains too; at least the AA mountains are.
"An operator defined on a set S which takes two elements from S as inputs and returns a single element of S."
- https://mathworld.wolfram.com/BinaryOperator.html
The polynomial with real coefficients
1.23 X
is not ring behaved. It is a black swan.
1.23X is not a black swan. Again I appeal to your knowledge of computer science.
The function
multiply (real a, real b)
is not a real number, no matter how much you might try to convince yourself or others that it should be. However,
c = multiply (1.23, 2.34)
*returns* a real number c, given real-valued inputs a and b. Likewise the function
Multiply (real a, polymorphic B)
is not a real number or a polymorphic object. However, implemented properly,
c = Multiply (1.23, 2.34)
returns a real number, and
C = multiply (1.23, [[2.34, 3.45] [4.56, 5.67]])
returns a 2x2 matrix. And if you define a further function
swan(polymorphic B)
and implement it to return
C = Multiply (1.23, B)
guess what happens? Multiply (1.23, B) is your swan 1.23X, but it is decidedly white.
I guess we are covering some ground here so I will try to stay on with you authentically here. Again though, the ring definition uses binary operators which are strictly of the same type. It is as if you would recommend relaxing the ring definition. We don't need the higher types that you are going into to falsify the subject. Already stopping at the polynomial with real coefficients the claimed ability to recover the complex numbers is a well used instance of the abstract algebra system. To require that next level of the curriculum where you are going is not helpful. I could accuse you of a grand dodge here. In the name of simplicity you can't get much simpler than
1.23 X .
The fact is that you refuse to apply the ring definition even onto your more complicated forms. If it fails here then that is enough. I guess possibly a restatement of your position is that until X is instantiated we cannot say anything of it? But then how can it be claimed to be ring behaved? Really this only exposes the dodge that you are going into. This sort of hazing is commonplace here. Who is hazing who though? It's sort of like through a thick haze one mathematician says to the other 'oh, yes, I see it now' and that then is declared a good enough for the journals. Any and all mathematical minds see modulo behavior. It's implementation is productive. I can guarantee that. And I can guarantee that without any use at all of this tripe known as abstract algebra. The proper course is simply to grant it. There is no need of this long conflicted run-around. In hindsight it was already granted in the real value as sign.

As you want to go farther, there are two options on the multiply that you perform. One is an attempt to subset that real value into the matrix form and the other is to distribute it through the matrix universally. A very simple awareness of dimension allows some informational analysis without getting too carried away, but it is up to you to construct whichever multiply you are going to do. As to which is the ring behaved version: you will have to cast the real value into a Matrix, at which point the type specifications have changed and this is only going to raise the confusion level. This is the same as multiplying a real value by a complex value. Are the reals a subset of the complex numbers? Even if they are the operation still does not meet the strict ring requirements. I believe some of these variants were covered by Lalo who is probably up to over several hundred links here so I'll not be able to refer you to the correct link without doing some work.

OK, so I've stretched a little bit into the ground that you are trying to cover here even though I don't recommend it. Likewise I don't recommend going into the ideal or the quotient. The language is so obfuscated that the simple constructional freedom of choosing
X X = - 1
requires stunts and flips to collapse the polynomial. No, we would be done already if this were allowed. I'm not going to try to apply the ring definition onto this assignment of a one dimensional real value to an undefined position holder X squared. The first transgression is the one that I am focused on. Beyond the first transgression lay many more transgressions. This is the nature of a corrupt regime. It props itself with diversions as for instance the requirement of an infinite number of terms in the polynomial, or Russiagate for that matter. I do suggest to you that there must have been quite some marketing effort to get this subject nailed down and those nails pierced some flesh in the process.

Anyways this instance of the recovery of the complex numbers from the polynomial with real coefficients exposes the lack of need to ratchet up the complexity here as you would like to do. All the while, and this is about the fourth or fifth go around now, you continue to ignore the binary operator
https://en.wikipedia.org/wiki/Binary_operation
and all that we need to do is witness that the polynomial has the opposite behavior of this concept while the AA curriculum upholds it as a shining example. This is the start of the conflict.

Using the ring definition as a guide we can rule out the scalar multiplication of a real value as the operation implied in you own text above, yet you avoid the ring definition altogether. Again what I see is a diversion that is exactly consistent with the methods of abstract algebra. That is fine; you play on their team. But within game theory when cheating enters the mix, well, obviously the rules of the game no longer matter. This I can now accurately accuse you of. I keep calling forth the fundamental rule of the ring definition and you keep avoiding and offending it. Your own prior responses here form a record.

I am not in need of a remedial lesson on abstract algebra. That said I stopped quite short of the full curriculum too. I want to stress again that where this subjects starts out is at an extremely fundamental level of formalizing operators. This is of great interest to me. Yet not long after the ring is defined the care with which it was developed is thrown out. This is exactly the position that you are upholding and it amazes me that such a strong mind would care to do so in the face of accountability. As you so readily throw up the ring this and the ring that the ring the other then what about the ring definition sir? What happened to it in your own analysis above here?

As to what is elemental within a matrix... as to what is elemental in a complex value... all of these ultimately come back to real values as the actual elemental type. Higher structures have gone beyond elemental forms and they have to reflect that. We can no longer simply assign them a real value. That has to take a formal type interpretation. In some regards this is an arbitrary procedure. We can arbitrarily assign that real value to a first position if the thing is even linear but again here this choice is ultimately arbitrary as is the product operation itself. Yes there typically will be one obvious standard but this standard is not the only option. Really the possibility for structural mixups here is where you are headed I feel quite sure. By the time you throw in multidimensional polynomials onto matrices you'll have something right? You want to go there Horand to get yourself even further away from the ring definition don't you? Is that hazy enough for you? Certainly the loss of elemental type will arrive eventually right?
h***@gmail.com
2020-09-21 13:10:11 UTC
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Lalo T.
Post by h***@gmail.com
[...]
Post by Tim Golden BandTech.com
Post by h***@gmail.com
You are free to disregard anything I write, as I am free to disregard your "black swan". (I have dealt with your swan to my own satisfaction. You are again free to disregard what I wrote.)
Once again Horand you have dodged the binary operator within your own interpretation. The polynomial as a god for you is established, and yet your very abuse of it here is exposed. The multiplication of X by itself is again offensive to the binary operator requirements since
( X )( X ) = X X
is no operation at all.
It is a formal operation on strings, if you will. But there is more to it, because the addition or multiplication of the *abstract* polynomials (1.23X + 2.34X^2) + (3.45 + 4.56X^3) will give the same expression whether you think of X as a placeholder for a real number or some other object. (Remember the topic of this thread is *abstract algebra*, not the real numbers.) The rules for finding the sum or product of the two given polynomials does *not* depend on the nature of X, so you should not pretend that it must.
1.23X is well formed (i.e., a *white* object in your parlance), whether X is a real-valued, complex-valued or matrix-valued variable. And X is a variable, *not* a constant. Once you have instantiated the type of X, you can define the binary expression that allows you to multiply a real number and any *value* you substitute for X to obtain a *value* for the polynomial that is associated with the particular *value* you substitute for X. Nothing arcane about that. The big difference, of course, is that this scalar multiplication is *different* for real-valued X and for matrix-valued X, but that should not come as a big surprise. Likewise the product of X*X (or the evaluation of same for specific instantiations of X) is defined differently when you think of X as real-valued and matrix valued objects.
I believe you are making a mountain out of, well, not even a molehill, but a speck of dust or even less.
Yes, certainly
1.23 X
is just a speck, but of course the molehill is built of specks like this, and the mountains too; at least the AA mountains are.
"An operator defined on a set S which takes two elements from S as inputs and returns a single element of S."
- https://mathworld.wolfram.com/BinaryOperator.html
The polynomial with real coefficients
1.23 X
is not ring behaved. It is a black swan.
1.23X is not a black swan. Again I appeal to your knowledge of computer science.
The function
multiply (real a, real b)
is not a real number, no matter how much you might try to convince yourself or others that it should be. However,
c = multiply (1.23, 2.34)
*returns* a real number c, given real-valued inputs a and b. Likewise the function
Multiply (real a, polymorphic B)
is not a real number or a polymorphic object. However, implemented properly,
c = Multiply (1.23, 2.34)
returns a real number, and
C = multiply (1.23, [[2.34, 3.45] [4.56, 5.67]])
returns a 2x2 matrix. And if you define a further function
swan(polymorphic B)
and implement it to return
C = Multiply (1.23, B)
guess what happens? Multiply (1.23, B) is your swan 1.23X, but it is decidedly white.
I guess we are covering some ground here so I will try to stay on with you authentically here.
This is not a very encouraging statement. It entails the subtle threat that you might start yanking my chain if at some point you feel you are no longer interested.. I'd rather you stopped responding if that ever turns out to be the case.
Post by Tim Golden BandTech.com
Again though, the ring definition uses binary operators which are strictly of the same type.
Again. There is *no* requirement that 1.23 and X are of the same type. For starters, X is decidedly *not* a real number. It is just a placeholder. But, as I explained before, X does not have to be a placeholder for a real number. 1.23X is a perfectly good polynomial in the ring of 2x2 matrices.
Post by Tim Golden BandTech.com
It is as if you would recommend relaxing the ring definition.
No relaxation of the ring definition, but most certainly a relaxation of what the underlying set is allowed to be. You still need to be able to add and multiply instances of X, both with themselves and with real numbers (or any other scalar field you want to choose).
Post by Tim Golden BandTech.com
We don't need the higher types that you are going into to falsify the subject. Already stopping at the polynomial with real coefficients the claimed ability to recover the complex numbers is a well used instance of the abstract algebra system. To require that next level of the curriculum where you are going is not helpful. I could accuse you of a grand dodge here. In the name of simplicity you can't get much simpler than
1.23 X .
The fact is that you refuse to apply the ring definition even onto your more complicated forms. If it fails here then that is enough.
Well, what I am trying to tell you, the ring definition does not fail. If you think of X as representing a real-valued variable, then cool. You still cannot do much else with 1.23X other than write it down. (Try, you can formally add or multiply with another polynomial, but that happens on an equally formal level.) *After* you instantiate X with a particular value, and *only* then, can you treat 1.23X as an expression and evaluate it. If X is real-valued, then the valuation amounts to a multiplication of real numbers. But it's only the *instantiation* that makes it so.
Post by Tim Golden BandTech.com
I guess possibly a restatement of your position is that until X is instantiated we cannot say anything of it? But then how can it be claimed to be ring behaved?
I don't think anybody claims that it has to be "ring behaved" a priori. However, the evaluation has to follow addition, multiplication and distributive laws, which forces the instances of X to live in a ring, but you still have considerable flexibility in deciding what this ring looks like.
Post by Tim Golden BandTech.com
Really this only exposes the dodge that you are going into. This sort of hazing is commonplace here. Who is hazing who though?
To me this question has a very clear answer. I said earlier that I object to you trying to yank my chain, and it is becoming more and more obvious that that is exactly what you are doing. (That is why I deleted the rest.)
zelos...@gmail.com
2020-09-22 05:13:49 UTC
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Lalo T.
Post by h***@gmail.com
[...]
Post by Tim Golden BandTech.com
Post by h***@gmail.com
You are free to disregard anything I write, as I am free to disregard your "black swan". (I have dealt with your swan to my own satisfaction. You are again free to disregard what I wrote.)
Once again Horand you have dodged the binary operator within your own interpretation. The polynomial as a god for you is established, and yet your very abuse of it here is exposed. The multiplication of X by itself is again offensive to the binary operator requirements since
( X )( X ) = X X
is no operation at all.
It is a formal operation on strings, if you will. But there is more to it, because the addition or multiplication of the *abstract* polynomials (1.23X + 2.34X^2) + (3.45 + 4.56X^3) will give the same expression whether you think of X as a placeholder for a real number or some other object. (Remember the topic of this thread is *abstract algebra*, not the real numbers.) The rules for finding the sum or product of the two given polynomials does *not* depend on the nature of X, so you should not pretend that it must.
1.23X is well formed (i.e., a *white* object in your parlance), whether X is a real-valued, complex-valued or matrix-valued variable. And X is a variable, *not* a constant. Once you have instantiated the type of X, you can define the binary expression that allows you to multiply a real number and any *value* you substitute for X to obtain a *value* for the polynomial that is associated with the particular *value* you substitute for X. Nothing arcane about that. The big difference, of course, is that this scalar multiplication is *different* for real-valued X and for matrix-valued X, but that should not come as a big surprise. Likewise the product of X*X (or the evaluation of same for specific instantiations of X) is defined differently when you think of X as real-valued and matrix valued objects.
I believe you are making a mountain out of, well, not even a molehill, but a speck of dust or even less.
Yes, certainly
1.23 X
is just a speck, but of course the molehill is built of specks like this, and the mountains too; at least the AA mountains are.
"An operator defined on a set S which takes two elements from S as inputs and returns a single element of S."
- https://mathworld.wolfram.com/BinaryOperator.html
The polynomial with real coefficients
1.23 X
is not ring behaved. It is a black swan.
1.23X is not a black swan. Again I appeal to your knowledge of computer science.
The function
multiply (real a, real b)
is not a real number, no matter how much you might try to convince yourself or others that it should be. However,
c = multiply (1.23, 2.34)
*returns* a real number c, given real-valued inputs a and b. Likewise the function
Multiply (real a, polymorphic B)
is not a real number or a polymorphic object. However, implemented properly,
c = Multiply (1.23, 2.34)
returns a real number, and
C = multiply (1.23, [[2.34, 3.45] [4.56, 5.67]])
returns a 2x2 matrix. And if you define a further function
swan(polymorphic B)
and implement it to return
C = Multiply (1.23, B)
guess what happens? Multiply (1.23, B) is your swan 1.23X, but it is decidedly white.
I guess we are covering some ground here so I will try to stay on with you authentically here. Again though, the ring definition uses binary operators which are strictly of the same type. It is as if you would recommend relaxing the ring definition. We don't need the higher types that you are going into to falsify the subject. Already stopping at the polynomial with real coefficients the claimed ability to recover the complex numbers is a well used instance of the abstract algebra system. To require that next level of the curriculum where you are going is not helpful. I could accuse you of a grand dodge here. In the name of simplicity you can't get much simpler than
1.23 X .
The fact is that you refuse to apply the ring definition even onto your more complicated forms. If it fails here then that is enough. I guess possibly a restatement of your position is that until X is instantiated we cannot say anything of it? But then how can it be claimed to be ring behaved? Really this only exposes the dodge that you are going into. This sort of hazing is commonplace here. Who is hazing who though? It's sort of like through a thick haze one mathematician says to the other 'oh, yes, I see it now' and that then is declared a good enough for the journals. Any and all mathematical minds see modulo behavior. It's implementation is productive. I can guarantee that. And I can guarantee that without any use at all of this tripe known as abstract algebra. The proper course is simply to grant it. There is no need of this long conflicted run-around. In hindsight it was already granted in the real value as sign.
As you want to go farther, there are two options on the multiply that you perform. One is an attempt to subset that real value into the matrix form and the other is to distribute it through the matrix universally. A very simple awareness of dimension allows some informational analysis without getting too carried away, but it is up to you to construct whichever multiply you are going to do. As to which is the ring behaved version: you will have to cast the real value into a Matrix, at which point the type specifications have changed and this is only going to raise the confusion level. This is the same as multiplying a real value by a complex value. Are the reals a subset of the complex numbers? Even if they are the operation still does not meet the strict ring requirements. I believe some of these variants were covered by Lalo who is probably up to over several hundred links here so I'll not be able to refer you to the correct link without doing some work.
OK, so I've stretched a little bit into the ground that you are trying to cover here even though I don't recommend it. Likewise I don't recommend going into the ideal or the quotient. The language is so obfuscated that the simple constructional freedom of choosing
X X = - 1
requires stunts and flips to collapse the polynomial. No, we would be done already if this were allowed. I'm not going to try to apply the ring definition onto this assignment of a one dimensional real value to an undefined position holder X squared. The first transgression is the one that I am focused on. Beyond the first transgression lay many more transgressions. This is the nature of a corrupt regime. It props itself with diversions as for instance the requirement of an infinite number of terms in the polynomial, or Russiagate for that matter. I do suggest to you that there must have been quite some marketing effort to get this subject nailed down and those nails pierced some flesh in the process.
Anyways this instance of the recovery of the complex numbers from the polynomial with real coefficients exposes the lack of need to ratchet up the complexity here as you would like to do. All the while, and this is about the fourth or fifth go around now, you continue to ignore the binary operator
https://en.wikipedia.org/wiki/Binary_operation
and all that we need to do is witness that the polynomial has the opposite behavior of this concept while the AA curriculum upholds it as a shining example. This is the start of the conflict.
Using the ring definition as a guide we can rule out the scalar multiplication of a real value as the operation implied in you own text above, yet you avoid the ring definition altogether. Again what I see is a diversion that is exactly consistent with the methods of abstract algebra. That is fine; you play on their team. But within game theory when cheating enters the mix, well, obviously the rules of the game no longer matter. This I can now accurately accuse you of. I keep calling forth the fundamental rule of the ring definition and you keep avoiding and offending it. Your own prior responses here form a record.
I am not in need of a remedial lesson on abstract algebra. That said I stopped quite short of the full curriculum too. I want to stress again that where this subjects starts out is at an extremely fundamental level of formalizing operators. This is of great interest to me. Yet not long after the ring is defined the care with which it was developed is thrown out. This is exactly the position that you are upholding and it amazes me that such a strong mind would care to do so in the face of accountability. As you so readily throw up the ring this and the ring that the ring the other then what about the ring definition sir? What happened to it in your own analysis above here?
As to what is elemental within a matrix... as to what is elemental in a complex value... all of these ultimately come back to real values as the actual elemental type. Higher structures have gone beyond elemental forms and they have to reflect that. We can no longer simply assign them a real value. That has to take a formal type interpretation. In some regards this is an arbitrary procedure. We can arbitrarily assign that real value to a first position if the thing is even linear but again here this choice is ultimately arbitrary as is the product operation itself. Yes there typically will be one obvious standard but this standard is not the only option. Really the possibility for structural mixups here is where you are headed I feel quite sure. By the time you throw in multidimensional polynomials onto matrices you'll have something right? You want to go there Horand to get yourself even further away from the ring definition don't you? Is that hazy enough for you? Certainly the loss of elemental type will arrive eventually right?
No one is refusing anything but you to understand that notation does not equal anything.
1.23X has no multiplication in it.
Lalo T.
2020-09-22 07:05:20 UTC
https://mathworld.wolfram.com/Congruence.html
https://en.wikipedia.org/wiki/Modulo_operation

b ≡ c (mod a) vs e mod f = g
equivalence relation binary operation
zelos...@gmail.com
2020-09-21 05:48:11 UTC
Post by Tim Golden BandTech.com
and we should admit that this expression carries one multiplication operation on two elements: 1.23 and X.
Except it doesn't, there is no multiplication going on there. 1.23X is just notation for (0,1.23,...) in the formal construction you've been shown before.
Lalo T.
2020-09-21 07:47:29 UTC
How to work with polynomials in difficult classes?
https://matheducators.stackexchange.com/questions/10471/how-to-work-with-polynomials-in-difficult-classes

Different motivations like sportsmanship, curiosity,
art, pragmatism or critique to foundations. What is your take on this ?

When did usage of the word polynomial become standard?
https://hsm.stackexchange.com/questions/6763/when-did-usage-of-the-word-polynomial-become-standard
Tim Golden BandTech.com
2020-09-21 13:17:30 UTC
Post by Lalo T.
How to work with polynomials in difficult classes?
https://matheducators.stackexchange.com/questions/10471/how-to-work-with-polynomials-in-difficult-classes
Different motivations like sportsmanship, curiosity,
art, pragmatism or critique to foundations. What is your take on this ?
When did usage of the word polynomial become standard?
https://hsm.stackexchange.com/questions/6763/when-did-usage-of-the-word-polynomial-become-standard
I don't see how they could leave Descartes out. La Geometrie Book II and III are loaded. Roots are throughout Book III. Parabolas in Book II.
Though the usage of the word 'polynome' is not there...

"But it is not my purpose to write a large book. I am trying rather to include much in a few words, as will perhaps be inferred from what I have done, if it is considered that, while reducing to a single construction all the problems of one class, I have at the same time given a method of transforming then into an infinity of others, and thus of solving each in an infinite number of ways; that, furthermore, having constructed all plane problems by the cutting of a circle by a straight line, and all solid problems by the cutting of a circle by a parabola; and, finally, all that are but one degree more complex by cutting a circle by a curve but one degree higher than the parabola, it is only necessary to follow the same general method to construct all problems, more and more complex, ad infinitum; for in the case of a mathematical progression, whenever the first two or three terms are given, it is easy to find the rest.
I hope that posterity will judge my kindly, not only as to the things which I have explained, but also as to those which I have intentionally omitted so as to leave to others the pleasure of discovery.

FIN.

BY THE GRACE AND PRIVELEGE of the very Christian King, it is permitted to the author of the book entitled Discourse on Method, etc., together with Dioptrics, Meteorology, and Geometry, etc., to have printed wherever he wishes, within or without the Kingdom of France, and during the period of ten consecutive years, beginning on the day when the printing is completed, without any publisher (except the one whom he selects) printing it or causing it to be printed, under any pretext or disguise, or selling or delivering any other impression except that which has been allowed, under penalty of a fine of a thousand livres, the confiscation of all the copies, etc. This is more fully set forth in the letters given at Paris, on the fourth day of May, 1637, signed by the King and his counsel, Ceberet, and sealed with the great seal of yellow wax on a simple ribbon.
The author has given permission to Jan Maire, bookseller at Leyden, to print the said book and enjoy the said privilege for the time and under the conditions agreed upon between them.
The printing is completed the eighth day of June, 1637."

But you see Lalo, the distinction of this polynomial versus the one in use within abstract algebra is a distinction that goes ignored to this day it seems.It is an entirely different puzzle as to when X became abstract and abstruse. I can assure you that Descartes was not working with an abusive polynomial. Certainly if he did he would make careful note of it; unlike today's mathemagicians.
Lalo T.
2020-09-21 22:51:46 UTC
(1) In the excellent book "A History of Algebra" (by Bartel Leendert van der Waerden) :

" Gauss notes that Euler obtained this pair of equations by using complex
numbers. Gauss avoids complex numbers: he derives (..) and (...) directly
from the assumption that the polynomial X has a linear factor "x ± r" or a

" In his first proof Gauss does not introduse complex numbers. He proves
the fundamental theorem in the following form:
Every polynomial X with real coefficients can be factored into linear and
(gauss)

" In a later treatise "Meditationes algebraicae" (Oxford 1770) Waring derives
another method for expressing symmetrie polynomials. "
(Edward Waring)

" Galois supposes the polynomial F x to be irreducible modulo p "
(galois)

"..the complex numbers as residue classes of polynomials in |R[x] modulo x² + 1"
(cauchy)

Mr. Waerden does not refer if the word polynomial is or not in the original
articles, but, independently of the usage of the word "polynomial", notably, he wrote :

" Modern algebra begins with Evariste Galois. With Galois, the character of
algebra changed radically. Before Galois, the efforts of algebrists were mainly
directed towards the solution of algebraic equations... "

(2) in https://hsm.stackexchange.com/questions/6763/when-did-usage-of-the-word-polynomial-become-standard
the user Conifold mention :

A History of Mathematics by Florian Cajori (page 139)
https://archive.org/details/ahistorymathema02cajogoog/mode/2up <--

Viète (1579)

"logistica speciosa"
https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te#The_logic_of_species

Another article :
The Literal Calculus of Viete and Descartes by I. G. Bashmakova and G. S. Smirnova
https://historiamatecuaciones.files.wordpress.com/2012/07/the-literal-calculus-of-viete-and-descartes.pdf

In the wikipedia article of Viete https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te :
" A few weeks before his death, he wrote a final thesis on issues of
cryptography " (check source...)
He maybe has something with letters in mind...

(some reader with knowledge in cryptography to give his opinion on this...?)

"Verbal arithmetic can be useful as a motivation and source of exercises in the teaching of algebra."
https://en.wikipedia.org/wiki/Verbal_arithmetic
(this article has nothing to do with Viete)

(3)

in https://en.wikipedia.org/wiki/Algebraic_expression#Algebraic_and_other_mathematical_expressions
are listed several names : Arithmetic expressions, Polynomial expressions,
Algebraic expressions, Closed-form expressions, Analytic expressions and
Mathematical expressions

the usage of the word "polynomial" in the following articles :

https://en.wikipedia.org/wiki/Integer-valued_polynomial
https://en.wikipedia.org/wiki/Ring_of_polynomial_functions

Maybe looking for the etimology of the word "polynomial"
The words of mathematics : an etymological dictionary of mathematical terms used in English
by Steven Schwartzman
https://archive.org/details/wordsofmathemati0000schw

"A system of polynomial equations (sometimes simply a polynomial system)..."
in https://en.wikipedia.org/wiki/System_of_polynomial_equations
( https://en.wikipedia.org/wiki/System_of_equations )

"..relates the circular functions and hyperbolic functions without explicitly using complex numbers."
https://en.wikipedia.org/wiki/Gudermannian_function (Christoph Gudermann)

https://polynomialshistory.weebly.com/history.html

(4) other named topics of this thread

https://en.wikipedia.org/wiki/Indeterminate
https://en.wikipedia.org/wiki/Free_variables_and_bound_variables
(without explicit alluding to other thread)
https://en.wikipedia.org/wiki/Closure_(computer_programming)
https://hazel.org/
https://en.wikipedia.org/wiki/Interface_(Java)
bassam karzeddin
2020-09-22 11:05:48 UTC
On Tuesday, September 22, 2020 at 1:51:56 AM UTC+3, Lalo T. wrote:

[Snip the entire Garbage ....]

If one is truly interested to know the simplest elementary facts about the modern and completely broken algebra before it started, then (she/he) must understand why do the whole mathematical official communities *GLOBALLY* can't tolerate the ever simplest questions about the three oldest historical unsolved problems of the ancient Greeks in our alleged free mind world of today's world of acadimea and not only in mathematics but also in all theoretical sciences such as pure physics, logic and philosophy as well ...

See how simply such a very innocent question must be closed immeadeatly from being visible by innocent school students by those described above (just because they did understand immeadeatly the size of taragedy about those mathematical scandels of the Greeks problems that had never been well-understood as they are too simple to well-understand truly even by middle school students, interested people in true knowledge and amateures as well.

And what was that horrible and forbidden question to ask in such an opened minded mathematical comunity as Stalk Exchange in the history section ... ?

https://hsm.stackexchange.com/questions/11917/who-was-the-first-person-in-the-history-that-constructed-exactly-the-cube-root-o

Of course, that is most likely invisible to general readers where they wanted them so delibratly not to notice anything very important that should have been well-undestood by the Greeks themselves before any others

However, here is the **CLOSED** question again and again:

Q: Who was the first person in the history that constructed exactly the cube root of two and not necessarily by the Greek tools but by any means?

Where are the simplist answers were provided in many other ** Deleted** answers and not only in their sites but here on sci. math, Qoura, etc,

since this site regulations of sci. math is immoderate by human arbitrary reactions like those of official and moderated sites for mathematics despite the fact that the vast majorities of general common acadimic mainstreams (here on sci. math) in mathematics, physics, logic and philosohy are almost the same identical types of those official and well-moderated sites

Here is the text provided with that forbidden qoestion in **MATHEMATICS**

where the person who asks usually such simple questions was banned as a result of their worries from people realizing **CORRECTLY** those oldest problems that breaks immeadeatly the entire modern algebra with its all dependent sciences instead of many too lengthy ways that people are increasingly and recently finding **SUDDENLY** many solid unjustifiable
and undeniable contradictions nowadays as we see here by the OP and in many other lengthy threads as well

But at the same time, we can also observe almost the same gange of most likely a hirred gropup of acadimic trolls and imbiciles who keep arguing aimelessly against well-proven facts that one is too embarrased to keep repeating it endlessly for all types of modern acadimic believers

Note also the many hiding charectures (under many fake names) behind those who try tirelessly to keep all people as clueless and so ignorant about the entire elementary issues ... in mathematics and alike

The origin of this problem is the impossibility of doubling the cube problem by an unmarked straight edge and a compass with a known number of steps

But it seems very evident that many historical mathematicians had claimed to do solve it but by other tools and means where that was generally accepted by mathematicians since the cube root of two is considered as a real (algebraic) number nowadays and globally I think

But it seems to me that in every claimed method there was something fallacious that passed away overheads of everyone's attention where then that number was considered as a real existing number at the end

But in comparison with the very similar story of discovering the square root of two, where that was simply based upon a very great theorem of Pythagoras, which as a great revolution of discovering true irrational numbers that are strictly "constructible"

But was there any similar theorem or so in mathematics that discovered the cube root of two as an exact distance even without construction or that was only unproven conclusion due to the very high density of many uncountable cube numbers around 2, were then one can't properly distinguish between them

In other words, the square root operation is generally valid for any distant number, but the cube root operation seems valid only to cube numbers where its cube root is simply a constructible number, which doesn't add anything meaningful to human truer knowledge as a result

Now, is the cube root operation is generally valid for prime numbers?

Since I found in many elementary methods and lastly even from the truthness of Fermat's last theorem that is impossible existence, that is to say: \$(x^3 = p)\$, where \$p\$ is a prime number, implies, (x: is no existing distance, and hence no number)

So, can anyone kindly show any historical proof for this cube root two number?
**************************

***Notice: You are never allowed to see mt many older and very relevent questions and very short answers (provided strictly by myslf) in their well-gurded site with all its corrupted branches and sections despite the fact that few topics are still remaining and being visible for general readers.

At all cases, they can't hide the well-proven facts by the spiders threads any more .... since people started suddenly attacking them in every wrong established concept with too many hot issues that would finally impose itself more strongly with the days passing until they would finally arrive at the most simple elementary fact that the real (existing) number is only described as a "constructible number" in mathematics where tools of constructions are never related to any serious problem

Good luck to know and well-understand many things in mathematics that you certainly never like to hear about nor wanting to beileave by all the means

Bassam Karzeddin
Tim Golden BandTech.com
2020-09-22 13:44:11 UTC
Post by bassam karzeddin
[Snip the entire Garbage ....]
If one is truly interested to know the simplest elementary facts about the modern and completely broken algebra before it started, then (she/he) must understand why do the whole mathematical official communities *GLOBALLY* can't tolerate the ever simplest questions about the three oldest historical unsolved problems of the ancient Greeks in our alleged free mind world of today's world of acadimea and not only in mathematics but also in all theoretical sciences such as pure physics, logic and philosophy as well ...
See how simply such a very innocent question must be closed immeadeatly from being visible by innocent school students by those described above (just because they did understand immeadeatly the size of taragedy about those mathematical scandels of the Greeks problems that had never been well-understood as they are too simple to well-understand truly even by middle school students, interested people in true knowledge and amateures as well.
And what was that horrible and forbidden question to ask in such an opened minded mathematical comunity as Stalk Exchange in the history section ... ?
https://hsm.stackexchange.com/questions/11917/who-was-the-first-person-in-the-history-that-constructed-exactly-the-cube-root-o
Of course, that is most likely invisible to general readers where they wanted them so delibratly not to notice anything very important that should have been well-undestood by the Greeks themselves before any others
Q: Who was the first person in the history that constructed exactly the cube root of two and not necessarily by the Greek tools but by any means?
Where are the simplist answers were provided in many other ** Deleted** answers and not only in their sites but here on sci. math, Qoura, etc,
since this site regulations of sci. math is immoderate by human arbitrary reactions like those of official and moderated sites for mathematics despite the fact that the vast majorities of general common acadimic mainstreams (here on sci. math) in mathematics, physics, logic and philosohy are almost the same identical types of those official and well-moderated sites
Here is the text provided with that forbidden qoestion in **MATHEMATICS**
where the person who asks usually such simple questions was banned as a result of their worries from people realizing **CORRECTLY** those oldest problems that breaks immeadeatly the entire modern algebra with its all dependent sciences instead of many too lengthy ways that people are increasingly and recently finding **SUDDENLY** many solid unjustifiable
and undeniable contradictions nowadays as we see here by the OP and in many other lengthy threads as well
But at the same time, we can also observe almost the same gange of most likely a hirred gropup of acadimic trolls and imbiciles who keep arguing aimelessly against well-proven facts that one is too embarrased to keep repeating it endlessly for all types of modern acadimic believers
Note also the many hiding charectures (under many fake names) behind those who try tirelessly to keep all people as clueless and so ignorant about the entire elementary issues ... in mathematics and alike
The origin of this problem is the impossibility of doubling the cube problem by an unmarked straight edge and a compass with a known number of steps
But it seems very evident that many historical mathematicians had claimed to do solve it but by other tools and means where that was generally accepted by mathematicians since the cube root of two is considered as a real (algebraic) number nowadays and globally I think
But it seems to me that in every claimed method there was something fallacious that passed away overheads of everyone's attention where then that number was considered as a real existing number at the end
But in comparison with the very similar story of discovering the square root of two, where that was simply based upon a very great theorem of Pythagoras, which as a great revolution of discovering true irrational numbers that are strictly "constructible"
But was there any similar theorem or so in mathematics that discovered the cube root of two as an exact distance even without construction or that was only unproven conclusion due to the very high density of many uncountable cube numbers around 2, were then one can't properly distinguish between them
In other words, the square root operation is generally valid for any distant number, but the cube root operation seems valid only to cube numbers where its cube root is simply a constructible number, which doesn't add anything meaningful to human truer knowledge as a result
Now, is the cube root operation is generally valid for prime numbers?
Since I found in many elementary methods and lastly even from the truthness of Fermat's last theorem that is impossible existence, that is to say: \$(x^3 = p)\$, where \$p\$ is a prime number, implies, (x: is no existing distance, and hence no number)
So, can anyone kindly show any historical proof for this cube root two number?
**************************
***Notice: You are never allowed to see mt many older and very relevent questions and very short answers (provided strictly by myslf) in their well-gurded site with all its corrupted branches and sections despite the fact that few topics are still remaining and being visible for general readers.
At all cases, they can't hide the well-proven facts by the spiders threads any more .... since people started suddenly attacking them in every wrong established concept with too many hot issues that would finally impose itself more strongly with the days passing until they would finally arrive at the most simple elementary fact that the real (existing) number is only described as a "constructible number" in mathematics where tools of constructions are never related to any serious problem
Good luck to know and well-understand many things in mathematics that you certainly never like to hear about nor wanting to believe by all the means
Bassam Karzeddin
Well King Bassam, I would hope you could at least come down into operator theory here at least a little bit. By the rules of abstract algebra just two operators are needed. They are the sum and the product. To what degree then a square root or a cube root is fundamental becomes troubling. Though we can write the thing to what degree is the notation a value or is it a value and an operator? You see, the wool is pulled over our eyes if we insist that we are looking at a raw value when we see that division sign with a notch in it and a simple value under it. Especially as the square root of four was so readily evaluated. Sage has an answer for you:
sage: 2 ** ( 1.0/3 )
1.25992104989487
What exactly is wrong with this answer?

I am about as stuck as you are but we'll be stuck until we get our way right? Well covering some new ground ought to appeal to you here. So I've come into your topic here much as it seems a misfit to this thread. Now how about you reciprocate? Though I can't agree with Horand or his methods at least he interacts. You seem to have isolated yourself into a very narrow range. Come have a look at the black swan.
zelos...@gmail.com
2020-09-17 05:34:08 UTC
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
I don't know what you mean by that. It is not an object of the ring of real numbers, but that is not what anybody ever claimed or needed. In fact, polynomials stitch together operations operating on two different rings: The ring of real numbers, and the ring of the polymorphic objects symbolized by X and its powers, including X^0. There are in fact six different operations involved: addition and multiplication in the ring of reals (to gather together the real numbers), addition and multiplication in the ring of the polymorphic objects to explain things such as X^n + X^n and X^n * X^m, scalar multiplication of a real with an object X^n, and the "addition" of mixed expressions such as 1.23 X + 2.34 X^2. If you identify real numbers such as 1.23 with the polynomials 1.23*X^0, then you can collapse these six operations into two, which allows you to work with a single ring, the ring of polynomials. Nothing more, nothing less.
Is 1.23 X a ring behaved product?
No. It is not. It obviously lacks the closure requirement. 1.23 is a real value whereas X is not a real value. Therefore
1.23 X
is not ring behaved. So how does it come that you name eight operators and this is none of them? Or at the very least you mention six which turn into two. Furthermore, what option does this curriculum take when after carefully defining operators it goes ahead and makes usage of
'polynomials with real coefficients'
never discussing the conflict of that usage with its carefully constructed operators? The black swan is a polynomial with a real coefficient. Also though it is merely one term. The product cannot be a ring product. Each term of the polynomial does not inter-operate and so the addition operation of the polynomial cannot meet the ring definition, for the ring strictly specifies that two elements evaluate to one element. That the polynomial is the antithesis of the ring definition... nice discussion they had on that one, eh?
So at this point horand you've looked the black swan in the eye, yet the lights were not on. Now we turn on the lights; what do you see?
Oh maybe you did see but you cannot bring yourself to admit that the closure requirement fails in the case of the black swan. I do see that you put addition in quotes above here in reference to the polynomial terms. I think this puts you into Mike Terry's camp of denying that the terms of the polynomial are actually a sum. Is that where you are going? And would this behavior be because they fail to meet the ring requirements? Why not them simply admit that
1.23 X
is not ring behaved? Isn't it obvious to everyone here that this is the case? There are only two options as far as I can tell. Ahh, but this is the thing: When a construction fails is fails in so many ways that you could go anywhere next. Of course the third option is the usenet dodge, which occurs regularly here. I suppose you could take up zelos's argument that we are dealing in historical details that no longer have any relevance. But then he's been wishy-washy on that attack. It really won't pass the smell test. It's inverted induction for those who insist on an infinity of terms in a system that breaks with just one term.
You see, it is valid to dismantle the polynomial. It is valid to dismantle the real number. It is in fact the pursuit of fundamental mathematics which requires that we question the works that have been done and dismantle those works. If they hold up then all is well. If they do not hold up could it be that the leads into a new way lay there?
can you stop complaining about notation already?
Lalo T.
2020-09-17 07:41:19 UTC
Other reasons of the difficulty of polysigns could be the rarity that
"cancellating terms" is not mandatory , or because seems like a mix between
homogeneous and the usual coordinates. But it could be other reasons.

Scimath to the classroom...(to the university classroom?)

...analogy with pixels

https://en.wikipedia.org/wiki/Pixel_geometry
https://en.wikipedia.org/wiki/Sub-pixel_resolution
https://en.wikipedia.org/wiki/Dexel
https://inkjetinsight.com/knowledge-base/apparent-resolution-even-mean/
https://en.wikipedia.org/wiki/Pixel_connectivity
https://en.wikipedia.org/wiki/Rasterisation
https://www.warmplace.ru/soft/pixilang/
https://bmao.tech/PixelPlusPlus/
https://en.wikipedia.org/wiki/Quasi-isometry#Examples

You will not be using time sculping a program to catch the eyeshifting of frisky
reptilians, similarly, when I buy a camera and I take a photo of a natural
landscape, I dont expect, that the photo will show a chair that is
not in the landscape (too much improvement to the image)

I could proclaim algebraic numbers and you, floating point (or some other).
I m reading the new messages, reading some old messages of other users,
wandering in the topics of this thread, observing how different users approach
distinct aspects, wondering if having expertise in programming would make
a reader to have one view or another view on the topic.

https://en.wikipedia.org/wiki/Frink_(programming_language)
http://witheve.com/deepdives/literate.html

...briefly passing by some definition

" Let R be a commutative ring and let I be an ideal of R whose index is 2.
Elements of the coset 0+I may be called even, while elements of the coset 1+I
may be called odd. As an example, let R = Z(2) be the localization of Z at the
prime ideal (2). Then an element of R is even or odd if and only if its
numerator is so in Z. "
in https://en.wikipedia.org/wiki/Parity_(mathematics)#Higher_dimensions_and_more_general_classes_of_numbers

...trying to figuring out how one can accept or dont accept the action of
projecting this or that property of numbers into other structures

...attempting to find some analogy in computing topics

https://en.wikipedia.org/wiki/Unconventional_computing
https://mathworld.wolfram.com/Braid.html

...knitting?

A Compiler for 3D Machine Knitting
by Jim McCann, Lea Albaugh, Vidya Narayanan, April Grow, Wojciech Matusik,
Jen Mankoff and Jessica Hodgins
https://la.disneyresearch.com/publication/machine-knitting-compiler/

"Languages for 3D Industrial Knitting" by Lea Albaugh
https://morphingmatter.cs.cmu.edu/machine-knitting-soft-actuation/

...and then, returning to the "defined vs undefined" issue.

a₀·I + a₁·X + a₂·(X*X) + a₃·(X*(X*X))

a₀⚬‽ � a₁⚬⮽ � a₂⚬(⮽⯑⮽) � a₃⚬(⮽⯑(⮽⯑⮽))

You could consider carve a compound of two compilers, a conventional compiler
and a "pen-and-paper" compiler, where the two compilers are allowed to read
simultaneously the same text, and also, examining each other.

How do you choose one compiler over the other?

Can a painter go outside of the scope of painting techniques ?
Tim Golden BandTech.com
2020-09-17 10:51:40 UTC
Post by Lalo T.
Other reasons of the difficulty of polysigns could be the rarity that
"cancellating terms" is not mandatory , or because seems like a mix between
homogeneous and the usual coordinates. But it could be other reasons.
Scimath to the classroom...(to the university classroom?)
...analogy with pixels
https://en.wikipedia.org/wiki/Pixel_geometry
https://en.wikipedia.org/wiki/Sub-pixel_resolution
https://en.wikipedia.org/wiki/Dexel
https://inkjetinsight.com/knowledge-base/apparent-resolution-even-mean/
https://en.wikipedia.org/wiki/Pixel_connectivity
https://en.wikipedia.org/wiki/Rasterisation
https://www.warmplace.ru/soft/pixilang/
https://bmao.tech/PixelPlusPlus/
https://en.wikipedia.org/wiki/Quasi-isometry#Examples
You will not be using time sculping a program to catch the eyeshifting of frisky
reptilians, similarly, when I buy a camera and I take a photo of a natural
landscape, I dont expect, that the photo will show a chair that is
not in the landscape (too much improvement to the image)
I could proclaim algebraic numbers and you, floating point (or some other).
I m reading the new messages, reading some old messages of other users,
wandering in the topics of this thread, observing how different users approach
distinct aspects, wondering if having expertise in programming would make
a reader to have one view or another view on the topic.
https://en.wikipedia.org/wiki/Frink_(programming_language)
http://witheve.com/deepdives/literate.html
...briefly passing by some definition
" Let R be a commutative ring and let I be an ideal of R whose index is 2.
Elements of the coset 0+I may be called even, while elements of the coset 1+I
may be called odd. As an example, let R = Z(2) be the localization of Z at the
prime ideal (2). Then an element of R is even or odd if and only if its
numerator is so in Z. "
in https://en.wikipedia.org/wiki/Parity_(mathematics)#Higher_dimensions_and_more_general_classes_of_numbers
...trying to figuring out how one can accept or dont accept the action of
projecting this or that property of numbers into other structures
...attempting to find some analogy in computing topics
https://en.wikipedia.org/wiki/Unconventional_computing
https://mathworld.wolfram.com/Braid.html
...knitting?
A Compiler for 3D Machine Knitting
by Jim McCann, Lea Albaugh, Vidya Narayanan, April Grow, Wojciech Matusik,
Jen Mankoff and Jessica Hodgins
https://la.disneyresearch.com/publication/machine-knitting-compiler/
"Languages for 3D Industrial Knitting" by Lea Albaugh
https://morphingmatter.cs.cmu.edu/machine-knitting-soft-actuation/
...and then, returning to the "defined vs undefined" issue.
a₀·I + a₁·X + a₂·(X*X) + a₃·(X*(X*X))
a₀⚬‽ � a₁⚬⮽ � a₂⚬(⮽⯑⮽) � a₃⚬(⮽⯑(⮽⯑⮽))
You could consider carve a compound of two compilers, a conventional compiler
and a "pen-and-paper" compiler, where the two compilers are allowed to read
simultaneously the same text, and also, examining each other.
How do you choose one compiler over the other?
Can a painter go outside of the scope of painting techniques ?
https://1.bp.blogspot.com/-003bVyndUwI/TV27Oxs8SoI/AAAAAAAAAJs/Bco2EV6E3uo/w1200-h630-p-k-no-nu/velazquez-las-meninas.jpg
Given you interest in compilers I would think you could see that insistence on an infinite length polynomial is bogus. Why and how this has become a necessary part of the strictest interpretation of AA is unknown apparently. There is no fail point on finite polynomials under product and sum.
The fail point is inside the terms.
Peter
2020-09-15 15:04:52 UTC
Post by Tim Golden BandTech.com
The point horand is that the multiplication you speak of is not ring behaved.
Have you ever defined "ring behaved"?
Lalo T.
2020-09-15 22:24:00 UTC
In the list https://github.com/ajinkyakulkarni/awesome-compilers

One finds :

http://www.is.ocha.ac.jp/~asai/Black/

Programming Should Eat Itself

With arithmetical polynomials, no problem.
In the initial post, is it a requirement not jump into the metalevel ?
Lalo T.
2020-09-16 04:51:50 UTC
(A) "...then I am dumbfounded that any would insist that polysign is not remarkable "

If the Algebra Representation and Metrics trivialize Polysigns, then, do not
look them through that lenses. Any mathematical knowledge can be "trivialized"
or "alternatively", be seen as a boring knowledge. But in the hands of who?

Possible difficulties:

(I) phonetic termination of the extra signs
minuS, pluS, ...

(II) the election of symbol '-' that is accepted collectively
as some kind of indication of additive inverse (notation collision)
If I m recalling well, it could be an similar problem in some trivalent logics
depending how you implement your symbols ("symbol retrocompatibility")
https://aymara.org/ternary/ternary.pdf

(III) the arithmetical increment in p2 can be attained either through addition
or product. But "poly-reciprocals" dont seem to have "physical reference"...

(B)
- Can general binary operators be sensibly reduced
to "internal binary operation" + unary operator ?
- Can a sub-area of Graph Theory without any "allusion to numbers" exist ?

(C) addition and multiplication in P3 numbers

p = -1 @ +2 @ *3
q = -4 @ +5

pⓐq := (-1 @ +2 @ *3) @ (-4 @ +5) = (-5 @ +7 @ *3)

p✖q := (-1 @ +2 @ *3)(-4 @ +5)=(+4 @ *8 @ -12 @ *5 @ -10 @ +15)= -22 @ +19 @ *13

(D) How do you enquiry into this topics ?

If we consider the "secured zone" as the areas of mathematics that have a direct
counterpart ("physical reference") in the "arithmetic of pixels and electrons"
(not just a computational implementation or "digitalized maths")

What are the reasons behind this concept of "secured area"?

Could I think that doing maths beyond this "secured area" is like
using Descriptive Geometry instead of three-dimensional geometry ? (explain...)
https://en.wikipedia.org/wiki/Descriptive_geometry

https://www.strayalpha.com/otm/

https://en.wikipedia.org/wiki/Real_computation

In G.E.B. Douglas Hofstadter, poetically, wrote :
"...this theory of types, which we might also call the "theory of the abolition of Strange Loops" "

"...but this operation has not been defined..."
So far, the differences in opinions seem to come from different compilers...
Tim Golden BandTech.com
2020-09-16 12:37:12 UTC
Post by Lalo T.
(A) "...then I am dumbfounded that any would insist that polysign is not remarkable "
If the Algebra Representation and Metrics trivialize Polysigns, then, do not
look them through that lenses. Any mathematical knowledge can be "trivialized"
or "alternatively", be seen as a boring knowledge. But in the hands of who?
(I) phonetic termination of the extra signs
minuS, pluS, ...
(II) the election of symbol '-' that is accepted collectively
as some kind of indication of additive inverse (notation collision)
If I m recalling well, it could be an similar problem in some trivalent logics
depending how you implement your symbols ("symbol retrocompatibility")
https://aymara.org/ternary/ternary.pdf
(III) the arithmetical increment in p2 can be attained either through addition
or product. But "poly-reciprocals" dont seem to have "physical reference"...
(B)
- Can general binary operators be sensibly reduced
to "internal binary operation" + unary operator ?
- Can a sub-area of Graph Theory without any "allusion to numbers" exist ?
(C) addition and multiplication in P3 numbers
(D) How do you enquiry into this topics ?
If we consider the "secured zone" as the areas of mathematics that have a direct
counterpart ("physical reference") in the "arithmetic of pixels and electrons"
(not just a computational implementation or "digitalized maths")
What are the reasons behind this concept of "secured area"?
Could I think that doing maths beyond this "secured area" is like
using Descriptive Geometry instead of three-dimensional geometry ? (explain...)
https://en.wikipedia.org/wiki/Descriptive_geometry
https://www.strayalpha.com/otm/
https://en.wikipedia.org/wiki/Real_computation
"...this theory of types, which we might also call the "theory of the abolition of Strange Loops" "
"...but this operation has not been defined..."
So far, the differences in opinions seem to come from different compilers...
I enjoy your cryptic writing, but really as you speak of some 'secured area' all that we have is the sum. The product and its geometry are already outside of physical correspondence. It is algebraic. It is rotational. But any geometric interpretation is mysterious. Rather, if you did find some correspondence then it would be worth mentioning. P2 obviously is already well established. P3 indirectly is well established throughout physics and engineering, though there the mysteriousness does creep in and often enough as P2 resultant is all that they are after. That P1 ought to have been established by now as the representative of time; well this is on the table as far as propagation goes. That P4 is at a breakpoint
P1 P2 P3 | P4 P5 ...
allows for multiple emergent spacetime claims. Which one is correct? In the past I've focused mostly on
P1 P2 P3
but I'm becoming more adapted to
P1 P2 P3 | P4 .

Did I ever declare a prize for the expression of electromagnetism within polysign? That's got to be worth \$100. Not sure how to constrain it though. I guess it has to involve emergent spacetime as well since these are the indicators. Really the value of this work is about 9 orders of magnitude more valuable so you might just want to keep the check. Certainly there could be multiple winners. Anyway, as the prize suggests it is a prediction of polysign that unified spacetime which carries some of the electromagnetic properties within the spacetime basis ought then to simplify the laws of electromagnetism. Of course in the grand scheme other properties are likely to slip in.

To try and stoke this fire we could throw mass as a one-signed phenomenon, charge as a two-signed phenomenon, and then next we land in a three-signed phenomenon and then possibly a four signed as well. That quarks take triples with 2/3 type of interpretation does look promising. This is where I am headed. Math for math's sake leads to abstract algebra. PhD bloat. It's not just the math department that suffers. Imagine when you delete the three. It'll be quite a smorgasbord! Can you imagine allowing your physicist prof to go off on his personal philosophy rant in class? Then go on to criticize this that or the other mathematician that he finds fault with? Shall we admit that it all is wrapped up in human politics as well? This is how difficult it is for us as elements of spacetime to find the basis. Most here merely want to escape it I think. There really is no escape. We are all prisoners of spacetime.
zelos...@gmail.com
2020-09-16 05:37:30 UTC
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Post by h***@gmail.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
He seems to know about computer science, yet has trouble understanding polymorphic functions. Sad, really.
Well horand for each new type there has to be some code otherwise you get a compiler error. This is actually not a bad slant on the subject, yet you see as you mention it you only build support for my own argument.
Actually, I do not. Abstract algebra only gives the header information, if you will. The actual implementation is left for the user.
Your black swan is the polynomial 1.23 X, right? This is a scalar multiplication, *NOT* a multiplication of two reals, and it is therefore implemented differently for different types of X. The same is true for the computation of the powers of X, and for the addition of different monomials.
The point horand is that the multiplication you speak of is not ring behaved. You have just broken abstract algebra. You see? The closure requirement of the product is offended by the black swan 1.23 X . The polynomial is claimed to be ring behaved but its terms are not ring behaved; this is the falsification. As a sum of terms the terms have to be ring behaved, or else the polynomial cannot possibly be ring behaved. Therefor according to abstract algebra the ring behaved polynomial is not a ring behaved polynomial.
I believe the correct posture here includes an analysis of the terminology of 'elemental'. As far as I can tell the elements of a polynomial are a set containing the coefficients (which we've specified to be real values) and X. Just one X which is guaranteed not to operate. It is a preserver of some sort which lacks any type. It is the antithesis of the ring definition. To avoid this analysis is a major failing point of abstract algebra. This elemental set denies the existence of operators as defined within the ring definiiton. Of course when things are broken there are rather a lot of ways to state how they are broken. This analysis I would call the set analysis as an attempt to clearly state the set of elements which build the polynomial.
Here is inverted induction: A belief has formed that the way out of this conundrum is to insist on an infinite series of terms, but this really does nothing but raise the confusion level. Upon raising the confusion level then the further confusing ideal and quotient will be placed before the people paying the professor's way. I think if the course were optional there would be rather a lot of drops in this region of it, and that would be a mark of the best students; that is to say that the best students will fail this course. Academia has this sort of effect, and there must be more room made for more publications for more PhDs to fill out the void so that we can all swim in this accumulation. Sure it smells a bit, and you certainly shouldn't drink it, but swimming in it is our great pursuit right? We'll get stronger for it right?
No horand, the compiler won't run on header files alone. By the laws of abstract algebra don't you dare instantiate a type for X. That will not work. Of course, by the time you manage to implement this topic in code you will find no need of an infinite length polynomial. You will find that they are simply rotating values around. In a ring? No, that word has already been taken by these mathematicians to mean something very much not ring-like. Then there is the term 'group' which has no standard group property whatsoever. I am so fortunate that the term 'polysign' has only been taken so far by a manufacturer of plastic signs. Still their nature as geometric algebra (or was it algebraic geometry) has long since been stolen. They (polysign) certainly lack all of the abstraction that one finds in modern mathematics.
PhD bloat covers quite a lot I think. Let's say you are a coder and you face a wave of new employees who all have to make some code in order for them to succeed. Would you think you might suffer some code bloat at that job? By the time the terabyte project is achieved the petabyte project will be right around the corner. I just downloaded Sage which is a free Mathematica and its already at 7.1 GB. In sight of accumulation we must understand that a philosophy (which ought never to have been separated out from mathematics) of simplicity that can still be a guiding force. It is possible that software people will be the great falsifiers of the accumulated mathematics simply because we have spent so many hours debugging. Generally it is the invalid assumption which is the failing point. Of course our time with the compiler is also meaningful... something which the mathematician lacks.
I know that my form here is rather crude. I do apologize to you for the swear word. Still why we should practice deference without any critical energy onto this subject or any other for that matter... do you see the danger of math as religion? It is already formed. We are humans doing these pursuits. No human mathematician can actually cut themselves a ticket out of this position. We are social animals. Academia selects the best mimics. Possibly I am over-reacting, but without it I don't see anything changing.
I do approve of your usage of code analogy, yet type safety is the celebrated form whose success is proven. Abstract algebra flies in the face of this at the polynomial construction. At the ring definition it did study and insist on type safety. Claims that the polynomial is ring behaved are perversions of the truth. As to whether you fit in there... we'll see.
That your initial attack on my position was the polymorphic function: the mistake that you have made in using this fodder only builds support for my position. Though it is loose and round-about this same falsification format should be applied onto my position. In other words you should falsify my words quite directly if I am incorrect. A mistake in my statements should show itself right? I see you are unafraid to address the black swan directly. That is very good. You are just the second here to do so as I recall. Most cannot even get that far. Mathematicians have grown an aversion to instantiation. We as coders know instantiation intimately. There would be nothing going on without it. Well, this applies to modern mathematics as well. Where you see a lack of instantiation... where there is no code to smell... these are forms of harmless code bloat by PhDs whose tutors granted them a place to jump through a fake hoop. This no doubt is the best way to accommodate them.
1.23 X
Is this product ring behaved?
Post by h***@gmail.com
Post by Tim Golden BandTech.com
Worse though the polynomial of abstract algebra is meant explicitly not to perform any operation so as to become a placeholder.
That's exactly what header files do, too.
Post by Tim Golden BandTech.com
This behavior is in direct contradiction with the ring definition. Possibly you can let through polynomials in unspecified coefficients (though this is dubious since no instantiable form will ever be had) and certainly when you try polynomials with real coefficients the black swan can be found readily.
You horand are speaking out your ass here
That's as far as I read. Insult me again, and you can go to hell.
Can you stop complaining on notation and actually use the formal definition?
Tim Golden BandTech.com
2020-09-14 13:24:49 UTC
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
Post by Tim Golden BandTech.com
Zelos, would you care to name another place where an infinite series is formally required?
If you mean sequence, then real number construction.
Post by Tim Golden BandTech.com
Here however you and your cohorts have committed to admitting that in n the thing does not work out therefor you insist on an infinite series.
No, we work with infinite sequences because that is the fucking formal construction because it covers all cases then.
Post by Tim Golden BandTech.com
Of course these have to be explicitly stated to be in abstract X; something that the curriculum shies away from.
No one shies away from anything, it is unneccisary in early stages to go on about the formal consturction.
Post by Tim Golden BandTech.com
I would not quibble with a real valued X for then the ring definition would hold up particularly well upon instantiating X.
The ring axioms are fullfilled for polynomials, X is just notation, nothing else.
Post by Tim Golden BandTech.com
" polynomials with real coefficients"
Yes, which comes from history and how they were viewed and are viewed outside of formal constructions. So fucking what?
Post by Tim Golden BandTech.com
and clearly the terminology of 'coefficient' does in fact imply multiplication doesn't it?
Again, historical reasons makes it retain the name, just as i is called "imaginary" despite it being just as "real" as real numbers.
Post by Tim Golden BandTech.com
Is it precisely these concerns that cause one to shy away from these simple instances and insist upon an infinite series of coefficients?
Complaining about vocabulary that is admitted to be there due to historical reasons and not a product of the formal construction does not in anyway pose a challange against the formal construction.
Post by Tim Golden BandTech.com
As to what exactly is wrong with a polynomial with finitely many terms: this problem is completely avoided by the AA people here.
You have yet to show ANYTHING is wrong.
No, Zelos. You are lying. The curriculum of AA is a carefully planted set of lies which on the far side appear to be quite something. To suggest that something holds:
"because it is the formal construction"
is rather like pythagoras making a declaration that certain numbers cannot exist. That this is the grounds of refutation that you have used in order to falsify my own logic above here seems to be going ignored here by you now. This is excellent.Was it you earlier who called the polynomial form
a0 + a1 X + a2 XX + ...
historical notation? Ah, but when that bit was challenged you denied it... then went back to that interpretation and here it is again. Historical reasons make it retain the name? The name polynomial? Or the name coefficient? Are you now claiming that neither of these concepts are an integral part of the subject? Clearly in the simplest terms
(a0 + a1 X)( b0 + b1 X) = a0b0 +(a1b0 + a1b0) X + a1b1 X
these are well behaved in low n such as n=1 here. What exactly does insisting on an infinite series do? What exactly is wrong with these expressions that causes you to hinge a rejection upon an infinite requirement? This behavior of yours (and of the topic at hand) I call inverted induction. These specimens above do little to destroy abstract algebra. It is when we call upon the polynomial with real coefficients and we study it in the simplest of terms... Then we see the black swan. The claim that an undefined X gets multiplied by a real value is present in the utterance of the
Polynomial With Real Coefficients
and so to instantiate one of these such as
1.23 X
and ponder the operation that is taking place which offends the ring requirements is the topic which goes denied by the master of disaster Zelos. He cannot address it. He will call it a historical blip. He will swear that the proper form is an infinite series of zero terms attached on to this nonzero term. His friend Mike Terry will claim that all those zero terms in sum with this term are not actually sums! This detail really was already icing on the cake. In effect by requiring the infinite series of sums they think that you cannot break back down to such a simple construction as above. As to how one could possibly arrive at an infinite construction in the first place... That would be induction from these lower places that I inhabit. But that infinite form then would be built upon the grounds that the form works in n. To claim that the form which doesn't work in n=1 works in its infinite case is inverted induction. It is a fraud and it is as much on you Zelos as it is on humanity. You have met the black swan and you have denied its presence. That is not how fasification works. Should I present to you a black swan you should falsify the claim by finding errors in my logic or construction. As to who is in the dark here... who is bringing light onto the subject here... who is hiding in the shadows here...

Now, again, because the polynomial is a sum of terms by its own ring behavior we can arrive at a single term. This is as it must be. These are the elemental pieces of the polynomial form. To claim that an infinite length sum is elemental while its pieces lay there before you to dismantle is blasphemy. To claim a piece of the sum such as
1.23 X
is elemental while it clearly does not operate and clearly offends the ring definition is the cause of all of the drivel for infinite series, sum-denial, historical put downs, and whatever comes next. I do hope you are working on another one Zelos, because you've already used up your present material, denied your present material, and reengaged in you present material. If you'd like I can climb through this thread and find your own inversions on the 'historical notation' slant. No; the standard polynomial notation is equivalent to the series notation. It is not unique and if anything it is more explicit. You would rather play blind while staring the black swan in the face. Inverted induction will not serve you in this battle. Pulling off the blindfold might help though.

To help belay my situation I call on:
http://abstract.ups.edu/aata/section-poly-rings.html
This 'historical' notation apparently is still in use at MIT:

and I suggest that this be our 'formal language' which you deny Zelos. Would you call this a falsification? Of your claim? Really I hate to seem bitchy, yet you've brought it out in me. Why don't you try feeding the black swan? Maybe just a little bit of white cracker? Maybe it will turn white again. Maybe it bites the hand that feeds it. One thing I feel comfortable with: Thomas W. Judson has more integrity than you do; he completely dodges the ideal, the quotient, the nature of an undefined X in the polynomial; could this really be MIT's version of abstract algebra? Yes, it is. Judson is nonconfrontational and so abides by the laws of PhD bloat; he just floats his own boat by stopping short. Just how indeterminate his X is really isn't discussed at all. Of course your own X goes very well dodged too. What of this? Could we entice Judson into an open conversation here?
zelos...@gmail.com
2020-09-15 05:52:16 UTC
Post by Tim Golden BandTech.com
No, Zelos. You are lying
I have told no lies here.
Post by Tim Golden BandTech.com
The curriculum of AA is a carefully planted set of lies which on the far side appear to be quite something.
There are no lies here in abstract algebra.
Post by Tim Golden BandTech.com
is rather like pythagoras making a declaration that certain numbers cannot exist
The formal construction is THE way it is done in mathematics. What youd raw from it is what is true, not what you draw from informal notions.
Post by Tim Golden BandTech.com
Was it you earlier who called the polynomial form a0 + a1 X + a2 XX + ... historical notation?
Yes, because the notation comes from history and is kept because of it and that it is more intuitive.
Post by Tim Golden BandTech.com
Ah, but when that bit was challenged you denied it
No, I have never denied it.
Post by Tim Golden BandTech.com
What exactly does insisting on an infinite series do?
SEQUENCE, not series. The definition uses a SEQUENCE.
Post by Tim Golden BandTech.com
The claim that an undefined X gets multiplied by a real value is present in the utterance of the Polynomial With Real Coefficients
That's not what the fucking formal definition says AT FUCKING ALL!

That is NOTATION, based on HISTORY and the intuitive/INFORMAL way to think of it.

the abstract algebra formal definition has no "undefined" or the likes, it is all sequences in |R^|N
Post by Tim Golden BandTech.com
He will call it a historical blip
Because the notation comes from history, it is not the formal one.
Post by Tim Golden BandTech.com
Now, again, because the polynomial is a sum of terms by its own ring behavior we can arrive at a single term
Polynomials are not formally defiend as sums of anything!
Yes, it is perfectly fine to use old notation as long as we understand it is all fucking notation and not the formal construction!
Post by Tim Golden BandTech.com
and I suggest that this be our 'formal language' which you deny Zelos.
Notation is not formal language, we are not going to base shit on notation.
Post by Tim Golden BandTech.com
Just how indeterminate his X is really isn't discussed at all
Shut up about the X, in the formal construction of polynomials, that I provided before, show me where there was any "indeterminate X"

You can't because there is none. That is why your complain is void.
Lalo T.
2020-09-15 07:27:33 UTC
(a) Recycling some topics :

https://en.wikipedia.org/wiki/Polynomial#Polynomial_functions
https://en.wikipedia.org/wiki/Matrix_polynomial

https://en.wikipedia.org/wiki/Algebraic_equation
https://mathworld.wolfram.com/MatrixEquation.html

https://phoneia.com/en/education/exercises-of-arithmetic-polynomials/

https://en.wikipedia.org/wiki/Type_theory
https://en.wikipedia.org/wiki/Intuitionistic_type_theory
https://en.wikipedia.org/wiki/Polymorphism_(computer_science)

(b)

In the line of your link "The Preservation of Scripture" ...

Dr. Mony Vital: A Lifestyle of True Freedom (1/2)

The man who was eating metal
https://en.wikipedia.org/wiki/Michel_Lotito

...these links are obviously decontextualized, but I want to make a point...

(C)

Live Functional Programming with Typed Holes
https://arxiv.org/pdf/1805.00155.pdf

I must admit that I had to ask for some help in these computation topics
( not my area )

Ok Tim, make an analogy using C, Wolfram Language and Hazel
and give your two cents on this issue.

https://en.wikipedia.org/wiki/Wolfram_Language
https://en.wikipedia.org/wiki/C_%28programming_language%29
https://hazel.org/

https://www.shapeways.com/product/RBVF4Z6JY/modern-art-d3-3-sided-die
https://mathgrrl.com/hacktastic/2018/01/three-sided-cylinder-coins/

Does high (or low) flexibility compensate ?
bassam karzeddin
2020-09-14 17:35:01 UTC
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
Post by Tim Golden BandTech.com
Zelos, would you care to name another place where an infinite series is formally required?
If you mean sequence, then real number construction.
Post by Tim Golden BandTech.com
Here however you and your cohorts have committed to admitting that in n the thing does not work out therefor you insist on an infinite series.
No, we work with infinite sequences because that is the fucking formal construction because it covers all cases then.
Post by Tim Golden BandTech.com
Of course these have to be explicitly stated to be in abstract X; something that the curriculum shies away from.
No one shies away from anything, it is unneccisary in early stages to go on about the formal consturction.
Post by Tim Golden BandTech.com
I would not quibble with a real valued X for then the ring definition would hold up particularly well upon instantiating X.
The ring axioms are fullfilled for polynomials, X is just notation, nothing else.
Post by Tim Golden BandTech.com
" polynomials with real coefficients"
Yes, which comes from history and how they were viewed and are viewed outside of formal constructions. So fucking what?
Post by Tim Golden BandTech.com
and clearly the terminology of 'coefficient' does in fact imply multiplication doesn't it?
Again, historical reasons makes it retain the name, just as i is called "imaginary" despite it being just as "real" as real numbers.
Post by Tim Golden BandTech.com
Is it precisely these concerns that cause one to shy away from these simple instances and insist upon an infinite series of coefficients?
Complaining about vocabulary that is admitted to be there due to historical reasons and not a product of the formal construction does not in anyway pose a challange against the formal construction.
Post by Tim Golden BandTech.com
As to what exactly is wrong with a polynomial with finitely many terms: this problem is completely avoided by the AA people here.
You have yet to show ANYTHING is wrong.
Would you please stop your complete utter sense "Zero"

People are certainly complaining about your mental incurable illness

Don't jump in any discussion randomly where you have no idea what *SH*T* you are babeling MORON

bkk
zelos...@gmail.com
2020-09-15 05:42:42 UTC
Post by bassam karzeddin
Post by ***@gmail.com
Post by Tim Golden BandTech.com
But the polynomial with real coefficients has no need of distinction amongst these operators. This is done merely by the utterance of 'real coefficient' which implies multiplication of an old and familiar type, yet multiply by what?
Can you fucking stop this? This is no argument that targets the formal construction!
Post by Tim Golden BandTech.com
Zelos, would you care to name another place where an infinite series is formally required?
If you mean sequence, then real number construction.
Post by Tim Golden BandTech.com
Here however you and your cohorts have committed to admitting that in n the thing does not work out therefor you insist on an infinite series.
No, we work with infinite sequences because that is the fucking formal construction because it covers all cases then.
Post by Tim Golden BandTech.com
Of course these have to be explicitly stated to be in abstract X; something that the curriculum shies away from.
No one shies away from anything, it is unneccisary in early stages to go on about the formal consturction.
Post by Tim Golden BandTech.com
I would not quibble with a real valued X for then the ring definition would hold up particularly well upon instantiating X.
The ring axioms are fullfilled for polynomials, X is just notation, nothing else.
Post by Tim Golden BandTech.com
" polynomials with real coefficients"
Yes, which comes from history and how they were viewed and are viewed outside of formal constructions. So fucking what?
Post by Tim Golden BandTech.com
and clearly the terminology of 'coefficient' does in fact imply multiplication doesn't it?
Again, historical reasons makes it retain the name, just as i is called "imaginary" despite it being just as "real" as real numbers.
Post by Tim Golden BandTech.com
Is it precisely these concerns that cause one to shy away from these simple instances and insist upon an infinite series of coefficients?
Complaining about vocabulary that is admitted to be there due to historical reasons and not a product of the formal construction does not in anyway pose a challange against the formal construction.
Post by Tim Golden BandTech.com
As to what exactly is wrong with a polynomial with finitely many terms: this problem is completely avoided by the AA people here.
You have yet to show ANYTHING is wrong.
Would you please stop your complete utter sense "Zero"
People are certainly complaining about your mental incurable illness
Don't jump in any discussion randomly where you have no idea what *SH*T* you are babeling MORON
bkk
No, peopel complain about that with you.

I won't stop correcting you or anyone on here.
Tim Golden BandTech.com
2020-09-12 15:11:10 UTC
Post by ***@gmail.com
Post by Tim Golden BandTech.com
It is as if the mathematician can make no error. Can make no mistake.
They can, have dand does but here there is no mistake.
The mistake is by you, you don't understand formal constructions.
Zelos, would you care to name another place where an infinite series is formally required?
I most math, something that works in n can be brought out to an infinite series by induction to n+1.
Here however you and your cohorts have committed to admitting that in n the thing does not work out therefor you insist on an infinite series.
This forms yet another attack on the subject and this is as it should be. I will call this version of the failing of abstract algebra
inverted induction
and certainly my side does require refinement. I would hope that you might engage here in the defense and in so doing possibly the argument will form.

An ordinary polynomial such as
1.0 + 2.0 X
is a fine instance of a polynomial in n where n is one. Of course so is the black swan
1.23 X
which is a simpler expression to study. Of course these have to be explicitly stated to be in abstract X; something that the curriculum shies away from. I would not quibble with a real valued X for then the ring definition would hold up particularly well upon instantiating X. The terminology used within the subject of Abstract Algebra (AA) is:
" polynomials with real coefficients"
and clearly the terminology of 'coefficient' does in fact imply multiplication doesn't it? Is it precisely these concerns that cause one to shy away from these simple instances and insist upon an infinite series of coefficients? Yet by induction we can prove that the real coefficient in n is not satisfactory and so we can prove that it is not satisfactory unconditionally; whether or not the polynomial has infinitely many terms.

This is my first attempt here at formalizing this line of reasoning. I do reserve the right to improve it, but I would thing that some means of falsification ought to present itself here by which my correction ought to be made. I may as well have attacked
1.23 X ^ 100
as an invalid term, but this is not really the relevant point of the argument here. The point is that by your insisting on infinitely many terms you actually have done nothing to remedy the basic problem that caused that movement. Induction works rather differently.

As to what exactly is wrong with a polynomial with finitely many terms: this problem is completely avoided by the AA people here.
Lalo T.
2020-09-13 06:57:21 UTC
well, that explains a lot of things.

https://en.wikipedia.org/wiki/Polynomial#Polynomial_functions
https://en.wikipedia.org/wiki/Ring_of_polynomial_functions

Suppose that in order for a mathematical object to be worthy of using the
word "polynomial" in its name, has to be put to two test.

Golden_membership( mathematicalobject , "some name" )

AA_membership( mathematicalobject , "some name" )

Here, the membership is referred as in :
https://en.wikipedia.org/wiki/Fuzzy_set
This is, a membership with "degrees", concretely, a number between 0 and 1.

Alternatively, the functions also accepts mathematical statements.

Let's go to the action :

In the link https://en.wikipedia.org/wiki/Matrix_polynomial

take the second phrase and use it as input.

statement = " a₀·I + a₁·X + a₂·(X*X) + a₃·(X*(X*X)) "

(a) Golden_membership( statement , "polynomial" ) = 0,4

(b) AA_membership( statement, "polynomial") = 0.8

We can observe they have different values.

If we inspect the functioning, both functions give similar weight to :

(1) : separation of monomials through "internal addition"

but, the Golden_membership(..) give 0 points to :

(2) : indeterminate 'X' and coefficients being objects of different nature

In a context of "Polynomialism", if one build a new mathematical object that
has some resemblance with the usual polynomials(with real numbers), I would
definitely want to call this object "polynomial whatever". It is more cool.

But, is this taking the analogy too far?

the Golden_membership(..) says that giving points to (2) is inflated/exaggerated
AA_membership(..) says that (2) is perfectly well

In this fashion, we will have to examine all the properties of usual polynomials
and if a new mathematical object is discovered, examinate it under the light
of the properties of the usual polynomials, relative to the two functions
(in case we have the intention of call it "polynomial.....")

What is the origin of polynomials and notation for them?
https://hsm.stackexchange.com/questions/2504/what-is-the-origin-of-polynomials-and-notation-for-them/2509#2509

Why is (2) really important ?
To what degree an object is "polynomiesque" or a "polynomial-like" object?
to what degree an object is worthy of a name with the word "polynomial"?
Is the "external operation issue" a totally foreign topic in this context ?
To what degree is the "external binary operation" a vacuous operation ?

https://en.wikipedia.org/wiki/Function_composition

This issue belongs to mathematics, and language.

Suppose we alter (2) , and create (3)

(3) : indeterminate 'X' and coefficients being objects of the same type

statement = " A₀*I + A₁*X + A₂*(X*X) + A₃*(X*(X*X)) "

Where coefficients and X are a specified object, not necessarily real numbers

(c) Golden_membership( statement , "polynomial" ) = ?

(d) AA_membership( statement, "polynomial") = ??
Tim Golden BandTech.com
2020-09-13 11:25:42 UTC
Post by Lalo T.
well, that explains a lot of things.
https://en.wikipedia.org/wiki/Polynomial#Polynomial_functions
https://en.wikipedia.org/wiki/Ring_of_polynomial_functions
Suppose that in order for a mathematical object to be worthy of using the
word "polynomial" in its name, has to be put to two test.
Golden_membership( mathematicalobject , "some name" )
AA_membership( mathematicalobject , "some name" )
https://en.wikipedia.org/wiki/Fuzzy_set
This is, a membership with "degrees", concretely, a number between 0 and 1.
Alternatively, the functions also accepts mathematical statements.
In the link https://en.wikipedia.org/wiki/Matrix_polynomial
take the second phrase and use it as input.
statement = " a₀·I + a₁·X + a₂·(X*X) + a₃·(X*(X*X)) "
(a) Golden_membership( statement , "polynomial" ) = 0,4
(b) AA_membership( statement, "polynomial") = 0.8
We can observe they have different values.
(1) : separation of monomials through "internal addition"
(2) : indeterminate 'X' and coefficients being objects of different nature
In a context of "Polynomialism", if one build a new mathematical object that
has some resemblance with the usual polynomials(with real numbers), I would
definitely want to call this object "polynomial whatever". It is more cool.
But, is this taking the analogy too far?
the Golden_membership(..) says that giving points to (2) is inflated/exaggerated
AA_membership(..) says that (2) is perfectly well
In this fashion, we will have to examine all the properties of usual polynomials
and if a new mathematical object is discovered, examinate it under the light
of the properties of the usual polynomials, relative to the two functions
(in case we have the intention of call it "polynomial.....")
What is the origin of polynomials and notation for them?
https://hsm.stackexchange.com/questions/2504/what-is-the-origin-of-polynomials-and-notation-for-them/2509#2509
Why is (2) really important ?
To what degree an object is "polynomiesque" or a "polynomial-like" object?
to what degree an object is worthy of a name with the word "polynomial"?
Is the "external operation issue" a totally foreign topic in this context ?
To what degree is the "external binary operation" a vacuous operation ?
https://en.wikipedia.org/wiki/Function_composition
This issue belongs to mathematics, and language.
Suppose we alter (2) , and create (3)
(3) : indeterminate 'X' and coefficients being objects of the same type
statement = " A₀*I + A₁*X + A₂*(X*X) + A₃*(X*(X*X)) "
Where coefficients and X are a specified object, not necessarily real numbers
(c) Golden_membership( statement , "polynomial" ) = ?
(d) AA_membership( statement, "polynomial") = ??
What they are getting out of the polynomial is rotational behavior. It's very troubling trying to read language like:
"Given an ordinary, scalar-valued polynomial"
which is at the beginning of your link to matrix polynomials. There are at least two types of thing in an unspecified polynomial and so any description which lacks specificity in two types is ambiguous in sight of the freedoms that have been taken in abstract algebra (aa).
This is of course for polynomials in one variable X. Taking these into general dimension x,y,z,... is only going to obscure things further.
I do think that going broad as you do does help paint what a conundrum the field of mathematics has become. My approach is to take on the simplest forms. If these fail then so does the rest. If these simple forms carry a broader generalization that has been overlooked then all the better. This is the case with the real number. It's sign already performs a modulo two rotational behavior and we readily build the modulo three version and so forth in n without any need to insist on all of this gobbledy gook.

aa has contaminated the original usage of the polynomial as for instance in recovering a computable sine function from a series. Beautiful stuff. When they took X as abstract and never mentioned it carefully they took a rather large step, and yet you see it goes under the rug quickly. Now any time that you read text which discusses polynomials you have to wonder which one the text is talking about, and quite possibly the text is leveraging both forms at once since the ambiguity is so well laid away. Such usage of course will be ambiguous and invalid. I see no current language distinction between these usages. As far as I can tell they and the mathematicians using them have no care. Of course if you mention the word 'polynomial' to a high school student they will feel very comfortable with their knowledge of them. Likewise a student in the curriculum of engineering such as myself will be quite comfortable with polynomials. But what this word meant to those who have not taken the abusive abstract algebra curriculum is starkly different. We interpret them in the one dimensional sense whereas aa has gone infinite dimensional according to the masters here.

Due to the ordinary usage by high school students I would then the term 'real polynomial' would be implied anytime the word is used without further specificity, however, even this usage is ambiguous since for instance:
"A real polynomial is a polynomial with real coefficients." - https://en.wikipedia.org/wiki/Polynomial,
https://mathworld.wolfram.com/RealPolynomial.html

say nothing about the X portion. I would think we are able to use real valued X, but because these definitions have explicitly left it out there can be no conclusion drawn in light of an overarching principle of consistency within mathematics. This consistency of course must cover all branches. The branch which is abstract algebra has done the offense of making the language obscure especially to those who are not steeped in its version of truth. Beyond this the namings of 'group' and 'ring' are misnomers. Come on... group? for an operator that provides consistency with addition? Yielding one element out of two elements is a group? No it is not. A group is a distinct gathering of distinct objects which do not operate on each other thus preserving their group nature, whereas if these objects were not distinct such as watery fluids upon their combination they would mix into a single melange. Thus the non-group quality of the mathematical term 'group' is fully established. Ring certainly ought to be reserved for something better than two of these flops.

Beyond the naming of these operations as generics... while their actual meanings are known as addition and multiplication... this seeming purity which has been achieved is immediately extinguished by the usage of the polynomial and the lack of discussion of its transgressions. These issues go ingored so that the student can then swim well weighted down through the ideal and the quotient. Push them through quickly here and those that don't drown will never look back.
Lalo T.
2020-09-14 00:35:49 UTC
( ┌ ┐ ( ┌ ┐ ┌ ┐ ) )
... + 3 · ( | 5 7 | * ( | 5 7 | * | 5 7 | ) ) + ....
( | 6 8 | ( | 6 8 | | 6 8 | ) )
( └ ┘ ( └ ┘ └ ┘ ) )

... + 3·(X*(X*X)) + ....

...no instances, due to difficulties with the format...
Tim Golden BandTech.com
2020-09-07 12:21:09 UTC
Post by Mike Terry
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
[snip rest of ridiculous rant]
X + ...
=
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
This all reads like you are serious, but you clearly have never studied
abstract algebra, and know next to nothing about it - certainly not
enough to criticise even a basic construction which you don't
understand. Or is it some kind of subtle joke? (If so I miss the point.)
Regards,
Mike.
Hi Mike.
a1 X
is not a binary operation. agree?
Assuming a1 is an element of the underlying ring...
a) the polynomial a1 X
[i.e. whose X coefficient is a1, and all other coefficients zero]
b) the product of the polynomial a1
[i.e. whose constant coefficient is a1, and all others zero]
and the polynomial X
This above (b) is it; just as I wrote in the O.P. except I would not call this 'the polynomial a1'
OK, so you are saying that you mean the product of a1 and X, so clearly
the product operation involved is a binary operation. [It involves two
operands]
There are now two possibilities: you consider a1 to be a polynomial, in
which case we have regular polynomial multiplication, which is properly
defined, or you consider a1 to be a member of the underlying ring.
It seems your issue is with the second approach. Lets focus on
polynomials over R, which is what you discuss below.
I'll assume you're ok with the definition of a polynomial over R. There
are several equivalent approaches to how these are defined, and I could
expand on this, but they all lead the same way, defining addition and
multiplication /of polynomials/.
But now we have something else: a binary operation taking a real number
and a real polynomial. Of course such an operation needs to be properly
defined to have a meaning, as it's not covered by the definition of
polynomial multiplication. Let's write the op as a binary function sm,
so sm: (R x R[X]) ---> R[x]. [R is the set of real numbers, and R[x]
the set of polynomials over R.] I choose the notation "sm" for "scalar
multiplication".
And here is the definition for sm: if a is in R and p is a polynomial
over R, represented as
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
then
sm(a,p) = (a p_0) + (a p_1) x + (a p_2) x^2 + ... + (a p_n) x^n
There is an important point here. When I wrote p = p_0 + p_1 x + p_2
x^2 + ... + p_n x^n above, I am not writing a long sequence of sums and
products on the right hand side (rhs) of the equals! If I were, you
could correctly claim the definition is circular!
I've got to quote you here:

" When I wrote
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
above, I am not writing a long sequence of sums and products " - Mike Terry

Mike this is extremely weak. If you did not mean to have a sum of products then why would you write a sum of products?
Why would the operators work out as a sum of products? Why would you make use of a 'polynomial with real coefficients'? You have falsified the subject here by denying that the binary operators work within the polynomial, and yet you disown this detail for your own convenience. This sort of wheedling does expose not just minor abuse. Of course this current abuse that I speak of is you attempting some sort of fixup on a failed construction. Obviously for you to go this far suggests that in secret you have gazed upon the black swan. That you would go this far to save the thing it falsifies is a fine indication. I certainly do not claim any circular definition. The very name 'polynomial' means many terms. It existed prior to abstract algebra and possibly because of this it gets less scrutiny by the beloved suppliers of book fees. The ring sum is so strongly implied that for you to deny that the '+' signs in the construction are addition (the ring defined operator) is a sure loser. I just love how others pop in here to make such corrections. By disowning the polynomial you feel that you have corrected the subject... perfect status quo position. Thanks Mike.

Circular definitions are actually far less offensive than constructions which falsify their own axioms. All of your argumentation does not confront the black swan
1.23 X
This is why I've avoided it till now. When someone provides a falsification of a subject the falsification has to be wrong in order for you to have refuted it. This means that you have to work in terms of this expression, which you have not done here. All that you have done, and all that others have done, is provide their own interpretation on the standing subject. It would seem as if (taking your statement above here seriously) that what you are attempting is a claim that the polynomial may not be dismantled. This is a false claim, and your own attempt to deny that the sums are actual ring sums is quite a position for you to land in... all the while admitting 'minor abuse' of the propped system. Again, you have supported my argument by dodging it and further you have landed yourself in a mess of goose shit. You've slipped in it and it is all over your clothing and your face here.

I'd love a link to a text which denies that the polynomial expression in use in abstract algebra is not actually a sum of terms. Thanks Mike. You've really demonstrated what a hack abstract algebra actually is. The quotient and ideal use similar language as you are using here. You are so brilliant that maybe the goose shit will burn off.

May the black swan live on.
Post by Mike Terry
I am writing a
polynomial specified in whatever notation polynomials have been
previously defined, except admittedly I've just assumed for convenience
they've been introduced as expressions of this form. (Even though there
are other technical ways of introducing them, authors would typically
make a point that they can be represented in this notation, at least for
typographical convenience.)
So sm is well defined, no problem with this approach either.
Effectively, this is considering R[x] to be a module over its base ring
R, and sm is just like multiplying vectors in a vector space by a scalar
in its base field. [A module is akin to a vector space, except it has a
base /ring/ rather than a base /field/.]
And of course, now that we've defined sm, we can go on and prove basic
things like
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n =
sm(p_0, x^0) + sm(p_1, x^1) + ... + sm(p_n, x^n)
(Here, the + signs on lhs of equals are just part of the notation for a
polynomial p, not an operation sign, while the + signs on rhs ARE the
binary operation acting on R[x].)
And just to add to notation confusion, it is typical tradition to write
scalar multiplication of sm(a, p) simply as a p, as you did initially,
in which case we can also write
sm(p_0, x^0) + sm(p_1, x^1) + ... + sm(p_n, x^n) =
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
but the meaning of rhs here is quite different from meaning of lhs
above: the rhs here IS a long sequence of (polynomial) sums and
(scalar) products.
Does it seem terrible to you that the notation is ambiguous? The point
is that we realise it is ambiguous, but provably no harm is done, as
we've proved that all the different interpretation lead to the same
result, so no harm is done. This is common practice in mathematics,
balancing simplicity of notation against formal syntactical correctness.
It is true that the polynomial form
0 + a1 X + 0 X X + 0 X X X + ...
is equal to
a1 X
so I see how you come to name it so. These systems of umpteen variables are true across all values and so to select zeros for most of the values is a fine usage. Now we have something simple to discuss.
Further we can use real values for a1, as is the custom in abstract algebra as they often implement the 'polynomial with real coefficients'. It is particularly at this point that full breakage occurs, though the usage of an undefined X in a product, even with itself, is a fairly sore point as well.
No, there is nothing "undefined" here. You've not properly grasped the
definition of polynomials I think. Perhaps my explanation above will
help, but possibly we need to go back to the basics of how polynomials
are rigorously defined, and the basic operations on them.
I wouldn't mind doing that if you're serious, or alternatively let me
know if you still think something is undefined...
X of course is in its 'abstract' form. a1 and X are not in the same set. Yet there is a product being taken. This product ought to get rather some attention since it is another operator... and wasn't it just moments ago in this subject that the ring operators were so carefully laid out?
OK, the problem here is either that you missed the definition for the
scalar product, or possibly the author of the text you're using omitted
it for whatever reason. That does not make abstract algebra "wrong" in
any way, and the problem with your OP was that it comes across as a rant.
Well, lets say it was a rant, but for most people if they were trying to
understand a new field of knowledge and didn't follow something, they
would ASK FOR HELP IN EXPLAINING THEIR CONFUSION, rather than rant on
about how the field is a pile of crap etc.. Do you see how the latter
behaviour justifiably invites laughter and (frankly) scorn from readers?
Anyway, I gave the definitions for scalar multiplication above, so
hopefully all is clear now! :)
Where was the discussion of this new non-binary operator and their non-binary sums by the way as we study
a0 + a1 X
this sum can only further the problem, as now we definitely have a real (as chosen above) in sum with a non-real entity.
In this context, when you consider the sum a0 + a1 X, it is understood
that a0 is the /polynomial/ a0 X^0, and it is simply by tradition that
we often write the shorter a0. Similar to the situation above with sm,
there are basic (provable) results which underly this slight abuse of
notation, rendering it harmless.
Specifically, it is shown that the set of polynomials of the form (a0
x^0) together with polynomial addition and multiplication is
/isomorphic/ to the set of real numbers with real number addition and
multiplication. That is, we have the correspondence
a0 x^0 <----> a0
and we show this correspondence respects the operations of addition and
multiplication appropriate for each side of the correspondence.
Example: (3 x^0) + (5 x^0) = (8 X^0) <----> 8 = 3 + 5, and also
3 x^0 <----> 3 and 5 x^0 <----> 5. (Yes, this is as obvious is it
seems!) So algebraicly R and the set of polynomials of form a x^0
behave exactly the same, and we informally identify them together in day
to day use. (This is like we identify the real number 2 with the
natural number 2, although it can be argued they are conceptually
distinct.)
How such a direct contradiction in a subject that is supposedly pristine can be propagated and absorbed by so many for so long is surely a statement with broader consequences. All that most can do is to deny the breakage. Here at least a lamb has offered itself up.
I hope I've shown there is no "breakage". At worst there is some minor
abuse of notation going on, which does no harm, and is completely
understood by everybody except you.
It may not be your fault that you missed out on a fuller explanation in
your studies, but your underlying response to this (your attitude in
posting a rant) is down to you...
Regards,
Mike.
Thanks Mike and I hope you will pardon the rhetoric for there is actual content to discuss here. The strictness of the ring definition; it was well built. The sum and the product are sufficient without the reverse operators. This polynomial stage though; then the quotient and ideal; these things are very dirty. Should mathematicians really be taking up particle/wave duality without explicitly stating it? At least the physicists bother to explain the contradiction before they eat it en mass. The mathematicians cover it up. That is not mathematics at all, and yet the constructions stands freely and mostly unchallenged. The consequences are broad even if non-mathematical. The separation of philosophy from mathematics and from physics might just have a wee bit to do with this. These are false divisions. As the ring provides there is no need for division. We ought to do without it in the name of simplicity.
Post by Mike Terry
[i.e. whose X coefficient is 1, and all others zero]
c) something else?
[e.g. maybe X represents multiplication or something!]
Mike.
Mike Terry
2020-09-07 16:24:24 UTC
<snip>
Post by Mike Terry
Post by Mike Terry
Post by Mike Terry
Post by Tim Golden BandTech.com
Hi Mike.
a1 X
is not a binary operation. agree?
Assuming a1 is an element of the underlying ring...
a) the polynomial a1 X
[i.e. whose X coefficient is a1, and all other coefficients zero]
b) the product of the polynomial a1
[i.e. whose constant coefficient is a1, and all others zero]
and the polynomial X
This above (b) is it; just as I wrote in the O.P. except I would not call this 'the polynomial a1'
OK, so you are saying that you mean the product of a1 and X, so clearly
the product operation involved is a binary operation. [It involves two
operands]
There are now two possibilities: you consider a1 to be a polynomial, in
which case we have regular polynomial multiplication, which is properly
defined, or you consider a1 to be a member of the underlying ring.
It seems your issue is with the second approach. Lets focus on
polynomials over R, which is what you discuss below.
I'll assume you're ok with the definition of a polynomial over R. There
are several equivalent approaches to how these are defined, and I could
expand on this, but they all lead the same way, defining addition and
multiplication /of polynomials/.
But now we have something else: a binary operation taking a real number
and a real polynomial. Of course such an operation needs to be properly
defined to have a meaning, as it's not covered by the definition of
polynomial multiplication. Let's write the op as a binary function sm,
so sm: (R x R[X]) ---> R[x]. [R is the set of real numbers, and R[x]
the set of polynomials over R.] I choose the notation "sm" for "scalar
multiplication".
And here is the definition for sm: if a is in R and p is a polynomial
over R, represented as
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
then
sm(a,p) = (a p_0) + (a p_1) x + (a p_2) x^2 + ... + (a p_n) x^n
There is an important point here. When I wrote p = p_0 + p_1 x + p_2
x^2 + ... + p_n x^n above, I am not writing a long sequence of sums and
products on the right hand side (rhs) of the equals! If I were, you
could correctly claim the definition is circular!
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
above, I am not writing a long sequence of sums and products " - Mike Terry
Mike this is extremely weak. If you did not mean to have a sum of products then why would you write a sum of products?
I didn't write a sum of products, I wrote a polynomial specification, as
I explained.

Look, I think the definition of a polynomial as a "formal expression (a0
x^0 + a1 x^1 + ... + an x^n)" is confusing you. So let's use a clearer
definition without any x, and I'll restate what I said using the new
notation:

Def: a real polynomial is an infinite sequence of real numbers, of
which all but finitely many are zero.

So here are two examples of polynomials:

p = (3,2,1)
q = (0,1)

[ where I write ")" after finitely many terms, it is taken as an
indication that all remaining terms are zero from that point on. (3,2,1)
means the infinite sequence (3,2,1,0,0,0,0,0,0,...)]

Where I previously /specified/ a polynomial as 17x^0 + 4x^1 +23x^2, in
this new notation it would be specified as (17,4, 23). Of course, we
define addition and multiplication of polynomials in the expected way,
but there is NO MENTIONING OF ANY x anywhere. So e.g. with the above p, q,

p + q = (3,3,1)
pq = (0,3,2,1)

(I will give proper defs if you don't get how the defs would go)

With these definitions, the set of all real polynomials under the
defined ops of addition and multiplication form a ring: the "polynomial
ring over R" which I'll write as R[]. The additive identity is seen to
be (0) and the multiplicative identity is (1).

Now let me restate the argument which you think is extremely weak (but
didn't understand) using this clearer notation: [the meaning of what I
write is unchanged, just clearer notation]

<rewrite>
And here is the definition for sm: if a is in R and p is a polynomial
over R, represented as

p = (p_0, p_1, p_2, ... pn)

then

sm(a,p) = (a p_0, a p_1, a p_2, ...a p_n)

There is an important point here. When I wrote

p = (a p_0, a p_1, a p_2, ...a p_n)

above, I am not writing a long sequence of sums
and products on the right hand side (rhs) of the equals!
If I were, you could correctly claim the definition is circular!
</rewrite>

So now when you read the rewrite it should be totally obvious: OF COURSE
(a p_0, a p_1, a p_2, ...a p_n) isn't a sequence of sums of products -
it's a FINITE SEQUENCE OF REAL NUMBERS, aka a polynomial. Remember that
a and p_0 are both just real numbers, and so (a p_0) is a real number
too viz. the product of a and p_0, and same with other components,
Post by Mike Terry
Why would the operators work out as a sum of products? Why would you make use of a 'polynomial with real coefficients'? You have falsified the subject here by denying that the binary operators work within the polynomial, and yet you disown this detail for your own convenience.
Don't be ridiculous. I explained in my OP the distinction between a
polynomial specification and an expression which IS the sum of products.
Let me make it doubly clear, using the clearer notation above.

If we want to write an expression which represents a sum of products,
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n, then we first have to be clear
about what x is in this expression. In my notation, what is meant by x
in that expression would be the polynomial (0,1). I.e. we DEFINE:

Def: x = (0,1) [to be doubly clear, THIS x IS A POLYNOMIAL]

Note that with rules for polynomial multiplication we have:

x^0 = (1) [the identity of the polynomial ring]
X^1 = x = (0,1)
x^2 = x x = (0,1)(0,1) = (0,0,1)
x^3 = ... = (0,0,0,1)
and so on.

So let's concentrate on one term (p_2 x^2). This is the scalar product
of p_2 and x^2. Making this clear by using the sm() function defined
earlier:
p_2 x^2 = sm(p_2, x^2 )
= sm(p_2, (0,0,1) )
= (p_2 0, p_2 0, p_2 1)
= (0, 0, p_2) as we might expect

So the "sum of products" expression p_0 + p_1 x + p_2 x^2 + ... + p_n
x^n unambiguously evaluates like this:

p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
= (p_0) +
(0, p_1) +
(0, 0, p_2) + .... +
(0, 0, 0, ... p_n)
= (p_0, p_1, p_2, ... p_n)

So I am not denying anything - you just don't understand the subject
matter properly. :)

Summary: (p_0, p_1, p_2, ... p_n) is NOT a sum of products, it's a
sequence of real numbers (aka polynomial). With x defined as
polynomial (0,1), the expression p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
IS a sum of products, which evaluates routinely to the polynomial (p_0,
p_1, p_2, ... p_n).

I've skipped the rest of your post which goes ranty, so I couldn't be
bothered to read. If there were any serious (non-ranty) questions,
you'll have to repeat them again and I'll try to answer.

Mike.
Tim Golden BandTech.com
2020-09-09 11:33:13 UTC
Post by Mike Terry
<snip>
Post by Mike Terry
Post by Mike Terry
Post by Mike Terry
Post by Tim Golden BandTech.com
Hi Mike.
a1 X
is not a binary operation. agree?
Assuming a1 is an element of the underlying ring...
a) the polynomial a1 X
[i.e. whose X coefficient is a1, and all other coefficients zero]
b) the product of the polynomial a1
[i.e. whose constant coefficient is a1, and all others zero]
and the polynomial X
This above (b) is it; just as I wrote in the O.P. except I would not call this 'the polynomial a1'
OK, so you are saying that you mean the product of a1 and X, so clearly
the product operation involved is a binary operation. [It involves two
operands]
There are now two possibilities: you consider a1 to be a polynomial, in
which case we have regular polynomial multiplication, which is properly
defined, or you consider a1 to be a member of the underlying ring.
It seems your issue is with the second approach. Lets focus on
polynomials over R, which is what you discuss below.
I'll assume you're ok with the definition of a polynomial over R. There
are several equivalent approaches to how these are defined, and I could
expand on this, but they all lead the same way, defining addition and
multiplication /of polynomials/.
But now we have something else: a binary operation taking a real number
and a real polynomial. Of course such an operation needs to be properly
defined to have a meaning, as it's not covered by the definition of
polynomial multiplication. Let's write the op as a binary function sm,
so sm: (R x R[X]) ---> R[x]. [R is the set of real numbers, and R[x]
the set of polynomials over R.] I choose the notation "sm" for "scalar
multiplication".
And here is the definition for sm: if a is in R and p is a polynomial
over R, represented as
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
then
sm(a,p) = (a p_0) + (a p_1) x + (a p_2) x^2 + ... + (a p_n) x^n
There is an important point here. When I wrote p = p_0 + p_1 x + p_2
x^2 + ... + p_n x^n above, I am not writing a long sequence of sums and
products on the right hand side (rhs) of the equals! If I were, you
could correctly claim the definition is circular!
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
above, I am not writing a long sequence of sums and products " - Mike Terry
Mike this is extremely weak. If you did not mean to have a sum of products then why would you write a sum of products?
I didn't write a sum of products, I wrote a polynomial specification, as
I explained.
Look, I think the definition of a polynomial as a "formal expression (a0
x^0 + a1 x^1 + ... + an x^n)" is confusing you. So let's use a clearer
definition without any x, and I'll restate what I said using the new
Def: a real polynomial is an infinite sequence of real numbers, of
which all but finitely many are zero.
p = (3,2,1)
q = (0,1)
[ where I write ")" after finitely many terms, it is taken as an
indication that all remaining terms are zero from that point on. (3,2,1)
means the infinite sequence (3,2,1,0,0,0,0,0,0,...)]
Where I previously /specified/ a polynomial as 17x^0 + 4x^1 +23x^2, in
this new notation it would be specified as (17,4, 23). Of course, we
define addition and multiplication of polynomials in the expected way,
but there is NO MENTIONING OF ANY x anywhere. So e.g. with the above p, q,
p + q = (3,3,1)
pq = (0,3,2,1)
(I will give proper defs if you don't get how the defs would go)
With these definitions, the set of all real polynomials under the
defined ops of addition and multiplication form a ring: the "polynomial
ring over R" which I'll write as R[]. The additive identity is seen to
be (0) and the multiplicative identity is (1).
Now let me restate the argument which you think is extremely weak (but
didn't understand) using this clearer notation: [the meaning of what I
write is unchanged, just clearer notation]
<rewrite>
And here is the definition for sm: if a is in R and p is a polynomial
over R, represented as
p = (p_0, p_1, p_2, ... pn)
then
sm(a,p) = (a p_0, a p_1, a p_2, ...a p_n)
p = (a p_0, a p_1, a p_2, ...a p_n)
above, I am not writing a long sequence of sums
and products on the right hand side (rhs) of the equals!
If I were, you could correctly claim the definition is circular!
</rewrite>
So now when you read the rewrite it should be totally obvious: OF COURSE
(a p_0, a p_1, a p_2, ...a p_n) isn't a sequence of sums of products -
it's a FINITE SEQUENCE OF REAL NUMBERS, aka a polynomial. Remember that
a and p_0 are both just real numbers, and so (a p_0) is a real number
too viz. the product of a and p_0, and same with other components,
Post by Mike Terry
Why would the operators work out as a sum of products? Why would you make use of a 'polynomial with real coefficients'? You have falsified the subject here by denying that the binary operators work within the polynomial, and yet you disown this detail for your own convenience.
Don't be ridiculous. I explained in my OP the distinction between a
polynomial specification and an expression which IS the sum of products.
Let me make it doubly clear, using the clearer notation above.
If we want to write an expression which represents a sum of products,
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n, then we first have to be clear
about what x is in this expression. In my notation, what is meant by x
Def: x = (0,1) [to be doubly clear, THIS x IS A POLYNOMIAL]
x^0 = (1) [the identity of the polynomial ring]
X^1 = x = (0,1)
x^2 = x x = (0,1)(0,1) = (0,0,1)
x^3 = ... = (0,0,0,1)
and so on.
So let's concentrate on one term (p_2 x^2). This is the scalar product
of p_2 and x^2. Making this clear by using the sm() function defined
p_2 x^2 = sm(p_2, x^2 )
= sm(p_2, (0,0,1) )
= (p_2 0, p_2 0, p_2 1)
= (0, 0, p_2) as we might expect
So the "sum of products" expression p_0 + p_1 x + p_2 x^2 + ... + p_n
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
= (p_0) +
(0, p_1) +
(0, 0, p_2) + .... +
(0, 0, 0, ... p_n)
= (p_0, p_1, p_2, ... p_n)
So I am not denying anything - you just don't understand the subject
matter properly. :)
Summary: (p_0, p_1, p_2, ... p_n) is NOT a sum of products, it's a
sequence of real numbers (aka polynomial). With x defined as
polynomial (0,1), the expression p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
IS a sum of products, which evaluates routinely to the polynomial (p_0,
p_1, p_2, ... p_n).
I've skipped the rest of your post which goes ranty, so I couldn't be
bothered to read. If there were any serious (non-ranty) questions,
you'll have to repeat them again and I'll try to answer.
Mike.
I think that upon insisting on the infinite ordered series outright and no usage of X that you've really gone beyond the
"polynomial with real coefficients"
You are now dealing in an infinite series of real values; they are no longer coefficients. They do have a product behavior remeniscent of the polynomial, but they are no longer polynomials. You yourself have separated them out as distinct. Should you return to the polynomial stage they fail. This is why you have bothered in the first place to insist upon this isolation. Far from a 'minor abuse' this is critical to the notation. Now, looking at the product is clearly where the productive results occur, and to get them back down the subject has to invoke a modulo effect onto the construction; otherwise the meaning of the products can only send the results upward in dimension... other than a few lowly terms that won't budge.

I can only feed the cynicism of future students here. To develop a system such as
R[X] / (1 + X X)
as the complex numbers is not such a great achievement. Oh, what's this? Those X's there... did you manage to delete those from the curriculum yet Mike? These then would be the minor abuses that you spoke of not long ago? Really, you still have failed to admit an instance of that minor abuse. Could this possibly be one?
zelos...@gmail.com
2020-09-09 11:53:44 UTC
I think that upon insisting on the infinite ordered series outright and no usage of X that you've really gone beyond the "polynomial with real coefficients"
No he hasn't because that is the FORMAL DEFINITION OF POLYNOMIALS!

and its not a series it is a SEQUENCE!
You are now dealing in an infinite series of real values; they are no longer coefficients.
It is how you define polynomials formally for fuck sake!
They do have a product behavior remeniscent of the polynomial, but they are no longer polynomials.
Except THEY ARE! They are THE definition of polynomials and how they are FORMALLY constructed!
Really, you still have failed to admit an instance of that minor abuse. Could this possibly be one?
Thats fucking notation you imbecile! Stop complaining about notation and understand the actual construction!
Tim Golden BandTech.com
2020-09-09 12:33:18 UTC
Post by ***@gmail.com
I think that upon insisting on the infinite ordered series outright and no usage of X that you've really gone beyond the "polynomial with real coefficients"
No he hasn't because that is the FORMAL DEFINITION OF POLYNOMIALS!
and its not a series it is a SEQUENCE!
You are now dealing in an infinite series of real values; they are no longer coefficients.
It is how you define polynomials formally for fuck sake!
They do have a product behavior remeniscent of the polynomial, but they are no longer polynomials.
Except THEY ARE! They are THE definition of polynomials and how they are FORMALLY constructed!
Really, you still have failed to admit an instance of that minor abuse. Could this possibly be one?
Thats fucking notation you imbecile! Stop complaining about notation and understand the actual construction!
I can only imagine the struggle taking place in your head Zelos. For you to stoop to pure rhetoric is a mark of an internal conflict. Possibly this is me getting into your head here, but I assure that it is not me personally, but an informational attack on a construction which you have adopted emotionally as a pristine and good thing. I come along and see that this supposedly pristine and healthy patient who came in for a goose bite is going out in a wheelchair.

Now out to the broader topic: the term 'polynomial' certainly did precede the term 'abstract algebra'. That the term has been reused with abuse within this subject is a cause for concern. That the product mimics the polynomial would be a fine observation, but to call an infinite series of values with a sum and product defined a polynomial is a reuse of an old familiar term and indeed upon going into that form as when I write
1.23 X (the black swan that bit the patient on the leg)
then we see that it fails immediately to actually behave as a polynomial of old. Thus criticism of the subject at hand and the passing on of this subject to unsuspecting well endowed (if even by banks to whom they will be indebted the next half of their lives if all goes well) school children
who are familiar with polynomials which build ellipses and other interesting curves; whose definition are not at all the same; yet the word the same; is valid criticism. Attempts to invalidate the criticism only build my case. Simplicity I would argue is a much finer trait than what you practice. Truth and its pursuit and its intermediate expression by humans is what we are engaged in hopefully here. That some prefer pure mimicry is not my problem. They and their arguments stand here a test of time that possibly future generations will take on. As to what the replacement for academia actually is... I think it includes the practice of mathematical construction in its free form, for this is exactly what is lacking for the first sixteen years or so of the educational system. To suddenly then grant your students this new ability after all that time of having one right answer and having fused all of those creative neurons down to said accumulated patterns and their mimicry cannot possibly be the correct choice. I suspect that early exposure to these higher maths would also be wise, but you see this early exposure would declare openly and outright that they all are merely attempts and that freedom to construct better is yours. It may not be easy, and not every student may have success, and if you choose to be a great mimic first then that is your choice, but ultimately that behavior must end. We each must take on the burden of judgement. There is no other way forward.
zelos...@gmail.com
2020-09-10 05:18:33 UTC
Post by Tim Golden BandTech.com
I can only imagine the struggle taking place in your head Zelos.
There is no struggle because I can understand this :)
Post by Tim Golden BandTech.com
but an informational attack on a construction which you have adopted emotionally as a pristine and good thing
You have come with nothing but complaints based on notation, not based on formal construction.
Post by Tim Golden BandTech.com
Now out to the broader topic: the term 'polynomial' certainly did precede the term 'abstract algebra'.
Yes, hence I say, historical notation
Post by Tim Golden BandTech.com
That the term has been reused with abuse within this subject is a cause for concern.
There is no abuse going on in the formal construction, it is just formalized from our intuitive historical notion.
Post by Tim Golden BandTech.com
That the product mimics the polynomial would be a fine observation, but to call an infinite series of values with a sum and product defined a polynomial is a reuse of an old familiar term and indeed upon going into that form as when I write
Again, not a series, it is a SEQUENCE and it is THE FORMAL CONSTRUCTION of polynomials. Do you understand what a formal construction is?
Post by Tim Golden BandTech.com
1.23 X (the black swan that bit the patient on the leg)
which is notation for (0,1.23,0,0,0,...)
Post by Tim Golden BandTech.com
then we see that it fails immediately to actually behave as a polynomial of old.
Not at all, it is a polynomial and behaves just like one.

It can be multiplied by a polynomial and we gain a polynomial, it can be added to polynomials and we get a polynomial. It is perfectly fine a polynomial.
Post by Tim Golden BandTech.com
Attempts to invalidate the criticism only build my case.
Except you've never made any criticism beyond notation. What in 1.23X, aka (0,1.23,0,0,0,...), does not make it behave like a polynomial? Nothing!
Mike Terry
2020-09-09 15:05:32 UTC
Post by Tim Golden BandTech.com
Post by Mike Terry
<snip>
Post by Mike Terry
Post by Mike Terry
Post by Mike Terry
Post by Tim Golden BandTech.com
Hi Mike.
a1 X
is not a binary operation. agree?
Assuming a1 is an element of the underlying ring...
a) the polynomial a1 X
[i.e. whose X coefficient is a1, and all other coefficients zero]
b) the product of the polynomial a1
[i.e. whose constant coefficient is a1, and all others zero]
and the polynomial X
This above (b) is it; just as I wrote in the O.P. except I would not call this 'the polynomial a1'
OK, so you are saying that you mean the product of a1 and X, so clearly
the product operation involved is a binary operation. [It involves two
operands]
There are now two possibilities: you consider a1 to be a polynomial, in
which case we have regular polynomial multiplication, which is properly
defined, or you consider a1 to be a member of the underlying ring.
It seems your issue is with the second approach. Lets focus on
polynomials over R, which is what you discuss below.
I'll assume you're ok with the definition of a polynomial over R. There
are several equivalent approaches to how these are defined, and I could
expand on this, but they all lead the same way, defining addition and
multiplication /of polynomials/.
But now we have something else: a binary operation taking a real number
and a real polynomial. Of course such an operation needs to be properly
defined to have a meaning, as it's not covered by the definition of
polynomial multiplication. Let's write the op as a binary function sm,
so sm: (R x R[X]) ---> R[x]. [R is the set of real numbers, and R[x]
the set of polynomials over R.] I choose the notation "sm" for "scalar
multiplication".
And here is the definition for sm: if a is in R and p is a polynomial
over R, represented as
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
then
sm(a,p) = (a p_0) + (a p_1) x + (a p_2) x^2 + ... + (a p_n) x^n
There is an important point here. When I wrote p = p_0 + p_1 x + p_2
x^2 + ... + p_n x^n above, I am not writing a long sequence of sums and
products on the right hand side (rhs) of the equals! If I were, you
could correctly claim the definition is circular!
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
above, I am not writing a long sequence of sums and products " - Mike Terry
Mike this is extremely weak. If you did not mean to have a sum of products then why would you write a sum of products?
I didn't write a sum of products, I wrote a polynomial specification, as
I explained.
Look, I think the definition of a polynomial as a "formal expression (a0
x^0 + a1 x^1 + ... + an x^n)" is confusing you. So let's use a clearer
definition without any x, and I'll restate what I said using the new
Def: a real polynomial is an infinite sequence of real numbers, of
which all but finitely many are zero.
p = (3,2,1)
q = (0,1)
[ where I write ")" after finitely many terms, it is taken as an
indication that all remaining terms are zero from that point on. (3,2,1)
means the infinite sequence (3,2,1,0,0,0,0,0,0,...)]
Where I previously /specified/ a polynomial as 17x^0 + 4x^1 +23x^2, in
this new notation it would be specified as (17,4, 23). Of course, we
define addition and multiplication of polynomials in the expected way,
but there is NO MENTIONING OF ANY x anywhere. So e.g. with the above p, q,
p + q = (3,3,1)
pq = (0,3,2,1)
(I will give proper defs if you don't get how the defs would go)
With these definitions, the set of all real polynomials under the
defined ops of addition and multiplication form a ring: the "polynomial
ring over R" which I'll write as R[]. The additive identity is seen to
be (0) and the multiplicative identity is (1).
Now let me restate the argument which you think is extremely weak (but
didn't understand) using this clearer notation: [the meaning of what I
write is unchanged, just clearer notation]
<rewrite>
And here is the definition for sm: if a is in R and p is a polynomial
over R, represented as
p = (p_0, p_1, p_2, ... pn)
then
sm(a,p) = (a p_0, a p_1, a p_2, ...a p_n)
p = (a p_0, a p_1, a p_2, ...a p_n)
above, I am not writing a long sequence of sums
and products on the right hand side (rhs) of the equals!
If I were, you could correctly claim the definition is circular!
</rewrite>
So now when you read the rewrite it should be totally obvious: OF COURSE
(a p_0, a p_1, a p_2, ...a p_n) isn't a sequence of sums of products -
it's a FINITE SEQUENCE OF REAL NUMBERS, aka a polynomial. Remember that
a and p_0 are both just real numbers, and so (a p_0) is a real number
too viz. the product of a and p_0, and same with other components,
Post by Mike Terry
Why would the operators work out as a sum of products? Why would you make use of a 'polynomial with real coefficients'? You have falsified the subject here by denying that the binary operators work within the polynomial, and yet you disown this detail for your own convenience.
Don't be ridiculous. I explained in my OP the distinction between a
polynomial specification and an expression which IS the sum of products.
Let me make it doubly clear, using the clearer notation above.
If we want to write an expression which represents a sum of products,
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n, then we first have to be clear
about what x is in this expression. In my notation, what is meant by x
Def: x = (0,1) [to be doubly clear, THIS x IS A POLYNOMIAL]
x^0 = (1) [the identity of the polynomial ring]
X^1 = x = (0,1)
x^2 = x x = (0,1)(0,1) = (0,0,1)
x^3 = ... = (0,0,0,1)
and so on.
So let's concentrate on one term (p_2 x^2). This is the scalar product
of p_2 and x^2. Making this clear by using the sm() function defined
p_2 x^2 = sm(p_2, x^2 )
= sm(p_2, (0,0,1) )
= (p_2 0, p_2 0, p_2 1)
= (0, 0, p_2) as we might expect
So the "sum of products" expression p_0 + p_1 x + p_2 x^2 + ... + p_n
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
= (p_0) +
(0, p_1) +
(0, 0, p_2) + .... +
(0, 0, 0, ... p_n)
= (p_0, p_1, p_2, ... p_n)
So I am not denying anything - you just don't understand the subject
matter properly. :)
Summary: (p_0, p_1, p_2, ... p_n) is NOT a sum of products, it's a
sequence of real numbers (aka polynomial). With x defined as
polynomial (0,1), the expression p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
IS a sum of products, which evaluates routinely to the polynomial (p_0,
p_1, p_2, ... p_n).
I've skipped the rest of your post which goes ranty, so I couldn't be
bothered to read. If there were any serious (non-ranty) questions,
you'll have to repeat them again and I'll try to answer.
Mike.
I think that upon insisting on the infinite ordered series outright and no usage of X that you've really gone beyond the
"polynomial with real coefficients"
You are now dealing in an infinite series of real values; they are no longer coefficients.
No, what I have defined is exectly "polynomials with real coefficients".
If we have a polynomial (p0, p1, ... pn), then p0 p1, ...pn ARE its
coefficients. That's what we mean by the polynomial's coefficents.
Post by Tim Golden BandTech.com
They do have a product behavior remeniscent of the polynomial, but they are no longer polynomials.
The important thing about polynomials is exactly their behaviour, not
the notation employed for them. So YES they are polynomials. And my
approach amounts to exactly the same thing as the "formal power series"
approach, provided that is correctly interpreted.

You don't think they are polynomials and want to argue? You admit you
are a student who doesn't currently know anything about abstract
algebra, so learn! :)
Post by Tim Golden BandTech.com
You yourself have separated them out as distinct. Should you return to the polynomial stage they fail. This is why you have bothered in the first place to insist upon this isolation.
What is "the polynomial stage"? I have not left anything, and where is
the supposed failure?
Post by Tim Golden BandTech.com
Far from a 'minor abuse' this is critical to the notation.
Obviously, the notation itself is "critical to the notation". Also
there are no minor abuses in what I wrote. Where do you think there's a
minor abuse?
Post by Tim Golden BandTech.com
Now, looking at the product is clearly where the productive results occur, and to get them back down the subject has to invoke a modulo effect onto the construction;
Gibberish. What modulo effect??? Perhaps you actually mean something
with this, but you've not explained any context for it to make sense.
Post by Tim Golden BandTech.com
otherwise the meaning of the products can only send the results upward in dimension...
Of course. It is a well know fact about polynomials that the degree of
a product of two polynomials is the sum of the degrees of the individual
products. So of course degrees of product go up in dimension, unless
one of the polynomials is of degree 0 (or is zero). This is not a flaw
- it's exactly WHAT WE EXPECT. :)
Post by Tim Golden BandTech.com
other than a few lowly terms that won't budge.
I can only feed the cynicism of future students here. To develop a system such as
R[X] / (1 + X X)
as the complex numbers is not such a great achievement. Oh, what's this? Those X's there... did you manage to delete those from the curriculum yet Mike?
In my clearer notation, that would be

R[] / < (1,0,1) >

no X. Note you need <> because you want to say the ideal generated by

You don't say why you think future students would be cynical about this.
You are just projecting here, based on your own misunderstandings.
Post by Tim Golden BandTech.com
These then would be the minor abuses that you spoke of not long ago?
No. The "minor abuses of notation" do not occur with my notation (which
is totally equivalent to the "formal finite power series" with which
you've become confused).
Post by Tim Golden BandTech.com
Really, you still have failed to admit an instance of that minor abuse. Could this possibly be one?
Um, No. :)

Seems to me that you are actively struggling not to learn here!

My recomendation to you is:
1) Forget the "formal power series" approach, at least for now.
2) Go back to the beginning and study polynomials again, but
exclusively using the clearer notation I showed. (You have
not gone so far yet that this would be an unreasonable effort.)
3) When you have come to understand that this approach is
consistent, unambiguous, and genuinely does reflect the usual
intuitions we have for polynomials (and that there are no
"abuses of notation" and that abstract algebra
is not "broken"), you could (if you wish) go back and examine
the "formal power series" approach again. But in doing this
you should always keep in mind that anything expressed in
that formalism can/should be taken as corresponding to the
clearer notation you will have learned. Then you will see
that in fact nothing has changed, and where you were
confused before all will be clear!

Regards,
Mike.
Lalo T.
2020-09-09 21:02:42 UTC
Ok, Tim, give me one last chance. If I can not produce a better approximation/approaching towards
this conflict in the present message, I will admit that "Abstract Algebra is a giant pile of nonsensical crap...to the end of times".

For a moment, forget almost all the posts of the thread (including polynomial rings), except two, the post about matrices, and
one of the last posts of Mostowski Collapse.

(a) One one hand, keep in sight the entire matricial post, but mainly, this fragment of the post with matrices :

(a1) λ₀·I + λ₁·X + λ₂·X² + ... + λₙ₋₁·Xⁿ⁻¹ + λₙ·Xⁿ where λ are 'Scalars', X matrices

(a2) A₀*I + A₁*X + A₂*X² + ... + Aₙ₋₁*Xⁿ⁻¹ + Aₙ*Xⁿ where A are 'Scalar Matrices', and X matrices

(b) On the other hand, in the Mostowski Collapse' post :

take :

" There are two ways, todo what you want todo, I mean
to have these pseudo rings, with non-closed operations. "

concretely

"pseudo rings" in https://en.wikipedia.org/wiki/Pseudo-ring

take the third definition :

"An abelian group (A,+) equipped with a subgroup B and a multiplication B × A → A making B a ring and A a B-module."

and keep it in sight also, the ring definition.

Does this post approach the conflictive zone,somehow, at least a bit ?
Mostowski Collapse
2020-09-09 21:22:09 UTC
I wasn't refering to this pseudo-ring.
I meant ring axioms without closure.
Something like here:

S. Kochen; E. P. Specker (1967). "The problem
of hidden variables in quantum mechanics".
Journal of Mathematics and Mechanics.
https://www.jstor.org/stable/24902153

They use a logic, where p∧q is only defined
when p⊙q, i.e. p and q commeasurable.
The only defined is realized by relativizing

axiom, similar like Dan relativizes via
U(a). But the guard is p⊙q. Some astonishing
things come out of that, and they have

a physical reading. You still find people
writing about these things, like this
here from 01.06.2020:

Partial Boolean algebras and the logical
exclusivity principle
Samson Abramsky - Rui Soares Barbosa
https://wdi.centralesupelec.fr/users/valiron/qplmfps/papers/qs08t2.pdf

Have Fun!
Post by Lalo T.
" There are two ways, todo what you want todo, I mean
to have these pseudo rings, with non-closed operations. "
"pseudo rings" in https://en.wikipedia.org/wiki/Pseudo-ring
Lalo T.
2020-09-10 03:19:02 UTC
You are right, I err again.

Tim, in a "flip a coin" bet, habitually, I would expect win or lose.
But sometimes may happen that the coin unexpectedly lands in its third side, or
alternatively, the coin lands in a rambling bird. An unexpected event...
Tim Golden BandTech.com
2020-09-10 11:52:40 UTC
Post by Lalo T.
Ok, Tim, give me one last chance. If I can not produce a better approximation/approaching towards
this conflict in the present message, I will admit that "Abstract Algebra is a giant pile of nonsensical crap...to the end of times".
For a moment, forget almost all the posts of the thread (including polynomial rings), except two, the post about matrices, and
one of the last posts of Mostowski Collapse.
(a1) λ₀·I + λ₁·X + λ₂·X² + ... + λₙ₋₁·Xⁿ⁻¹ + λₙ·Xⁿ where λ are 'Scalars', X matrices
(a2) A₀*I + A₁*X + A₂*X² + ... + Aₙ₋₁*Xⁿ⁻¹ + Aₙ*Xⁿ where A are 'Scalar Matrices', and X matrices
Let's suppose you had a construction such as that above, and I pointed out an error when n=5. You then go on to insist the n must be infinite. Would that be a fair bit of math you did? Yet this is exactly what the upholders of AA do. All the while swearing up and down that there is no problem... I do not understand how you have not caught a whiff of it. Really, the problem is a human problem. We must be terrible mathematicians. The existence of the polysign numbers and the fact that so many greats able to cover so much ground overlooked them as well... by the way they work in n. The biblical nature of academia is fully established within the branch known as Abstract Algebra.
Post by Lalo T.
" There are two ways, todo what you want todo, I mean
to have these pseudo rings, with non-closed operations. "
concretely
"pseudo rings" in https://en.wikipedia.org/wiki/Pseudo-ring
"An abelian group (A,+) equipped with a subgroup B and a multiplication B × A → A making B a ring and A a B-module."
and keep it in sight also, the ring definition.
Does this post approach the conflictive zone,somehow, at least a bit ?
zelos...@gmail.com
2020-09-08 05:29:46 UTC
Post by Mike Terry
Post by Mike Terry
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
[snip rest of ridiculous rant]
X + ...
=
Post by Mike Terry
Post by Tim Golden BandTech.com
Post by Mike Terry
This all reads like you are serious, but you clearly have never studied
abstract algebra, and know next to nothing about it - certainly not
enough to criticise even a basic construction which you don't
understand. Or is it some kind of subtle joke? (If so I miss the point.)
Regards,
Mike.
Hi Mike.
a1 X
is not a binary operation. agree?
Assuming a1 is an element of the underlying ring...
a) the polynomial a1 X
[i.e. whose X coefficient is a1, and all other coefficients zero]
b) the product of the polynomial a1
[i.e. whose constant coefficient is a1, and all others zero]
and the polynomial X
This above (b) is it; just as I wrote in the O.P. except I would not call this 'the polynomial a1'
OK, so you are saying that you mean the product of a1 and X, so clearly
the product operation involved is a binary operation. [It involves two
operands]
There are now two possibilities: you consider a1 to be a polynomial, in
which case we have regular polynomial multiplication, which is properly
defined, or you consider a1 to be a member of the underlying ring.
It seems your issue is with the second approach. Lets focus on
polynomials over R, which is what you discuss below.
I'll assume you're ok with the definition of a polynomial over R. There
are several equivalent approaches to how these are defined, and I could
expand on this, but they all lead the same way, defining addition and
multiplication /of polynomials/.
But now we have something else: a binary operation taking a real number
and a real polynomial. Of course such an operation needs to be properly
defined to have a meaning, as it's not covered by the definition of
polynomial multiplication. Let's write the op as a binary function sm,
so sm: (R x R[X]) ---> R[x]. [R is the set of real numbers, and R[x]
the set of polynomials over R.] I choose the notation "sm" for "scalar
multiplication".
And here is the definition for sm: if a is in R and p is a polynomial
over R, represented as
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
then
sm(a,p) = (a p_0) + (a p_1) x + (a p_2) x^2 + ... + (a p_n) x^n
There is an important point here. When I wrote p = p_0 + p_1 x + p_2
x^2 + ... + p_n x^n above, I am not writing a long sequence of sums and
products on the right hand side (rhs) of the equals! If I were, you
could correctly claim the definition is circular!
p = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
above, I am not writing a long sequence of sums and products " - Mike Terry
Mike this is extremely weak. If you did not mean to have a sum of products then why would you write a sum of products?
Why would the operators work out as a sum of products? Why would you make use of a 'polynomial with real coefficients'? You have falsified the subject here by denying that the binary operators work within the polynomial, and yet you disown this detail for your own convenience. This sort of wheedling does expose not just minor abuse. Of course this current abuse that I speak of is you attempting some sort of fixup on a failed construction. Obviously for you to go this far suggests that in secret you have gazed upon the black swan. That you would go this far to save the thing it falsifies is a fine indication. I certainly do not claim any circular definition. The very name 'polynomial' means many terms. It existed prior to abstract algebra and possibly because of this it gets less scrutiny by the beloved suppliers of book fees. The ring sum is so strongly implied that for you to deny that the '+' signs in the construction are addition (the ring defined operator) is a sure loser. I just love how others pop in here to make such corrections. By disowning the polynomial you feel that you have corrected the subject... perfect status quo position. Thanks Mike.
Circular definitions are actually far less offensive than constructions which falsify their own axioms. All of your argumentation does not confront the black swan
1.23 X
This is why I've avoided it till now. When someone provides a falsification of a subject the falsification has to be wrong in order for you to have refuted it. This means that you have to work in terms of this expression, which you have not done here. All that you have done, and all that others have done, is provide their own interpretation on the standing subject. It would seem as if (taking your statement above here seriously) that what you are attempting is a claim that the polynomial may not be dismantled. This is a false claim, and your own attempt to deny that the sums are actual ring sums is quite a position for you to land in... all the while admitting 'minor abuse' of the propped system. Again, you have supported my argument by dodging it and further you have landed yourself in a mess of goose shit. You've slipped in it and it is all over your clothing and your face here.
I'd love a link to a text which denies that the polynomial expression in use in abstract algebra is not actually a sum of terms. Thanks Mike. You've really demonstrated what a hack abstract algebra actually is. The quotient and ideal use similar language as you are using here. You are so brilliant that maybe the goose shit will burn off.
May the black swan live on.
Post by Mike Terry
I am writing a
polynomial specified in whatever notation polynomials have been
previously defined, except admittedly I've just assumed for convenience
they've been introduced as expressions of this form. (Even though there
are other technical ways of introducing them, authors would typically
make a point that they can be represented in this notation, at least for
typographical convenience.)
So sm is well defined, no problem with this approach either.
Effectively, this is considering R[x] to be a module over its base ring
R, and sm is just like multiplying vectors in a vector space by a scalar
in its base field. [A module is akin to a vector space, except it has a
base /ring/ rather than a base /field/.]
And of course, now that we've defined sm, we can go on and prove basic
things like
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n =
sm(p_0, x^0) + sm(p_1, x^1) + ... + sm(p_n, x^n)
(Here, the + signs on lhs of equals are just part of the notation for a
polynomial p, not an operation sign, while the + signs on rhs ARE the
binary operation acting on R[x].)
And just to add to notation confusion, it is typical tradition to write
scalar multiplication of sm(a, p) simply as a p, as you did initially,
in which case we can also write
sm(p_0, x^0) + sm(p_1, x^1) + ... + sm(p_n, x^n) =
p_0 + p_1 x + p_2 x^2 + ... + p_n x^n
but the meaning of rhs here is quite different from meaning of lhs
above: the rhs here IS a long sequence of (polynomial) sums and
(scalar) products.
Does it seem terrible to you that the notation is ambiguous? The point
is that we realise it is ambiguous, but provably no harm is done, as
we've proved that all the different interpretation lead to the same
result, so no harm is done. This is common practice in mathematics,
balancing simplicity of notation against formal syntactical correctness.
It is true that the polynomial form
0 + a1 X + 0 X X + 0 X X X + ...
is equal to
a1 X
so I see how you come to name it so. These systems of umpteen variables are true across all values and so to select zeros for most of the values is a fine usage. Now we have something simple to discuss.
Further we can use real values for a1, as is the custom in abstract algebra as they often implement the 'polynomial with real coefficients'. It is particularly at this point that full breakage occurs, though the usage of an undefined X in a product, even with itself, is a fairly sore point as well.
No, there is nothing "undefined" here. You've not properly grasped the
definition of polynomials I think. Perhaps my explanation above will
help, but possibly we need to go back to the basics of how polynomials
are rigorously defined, and the basic operations on them.
I wouldn't mind doing that if you're serious, or alternatively let me
know if you still think something is undefined...
X of course is in its 'abstract' form. a1 and X are not in the same set. Yet there is a product being taken. This product ought to get rather some attention since it is another operator... and wasn't it just moments ago in this subject that the ring operators were so carefully laid out?
OK, the problem here is either that you missed the definition for the
scalar product, or possibly the author of the text you're using omitted
it for whatever reason. That does not make abstract algebra "wrong" in
any way, and the problem with your OP was that it comes across as a rant.
Well, lets say it was a rant, but for most people if they were trying to
understand a new field of knowledge and didn't follow something, they
would ASK FOR HELP IN EXPLAINING THEIR CONFUSION, rather than rant on
about how the field is a pile of crap etc.. Do you see how the latter
behaviour justifiably invites laughter and (frankly) scorn from readers?
Anyway, I gave the definitions for scalar multiplication above, so
hopefully all is clear now! :)
Where was the discussion of this new non-binary operator and their non-binary sums by the way as we study
a0 + a1 X
this sum can only further the problem, as now we definitely have a real (as chosen above) in sum with a non-real entity.
In this context, when you consider the sum a0 + a1 X, it is understood
that a0 is the /polynomial/ a0 X^0, and it is simply by tradition that
we often write the shorter a0. Similar to the situation above with sm,
there are basic (provable) results which underly this slight abuse of
notation, rendering it harmless.
Specifically, it is shown that the set of polynomials of the form (a0
x^0) together with polynomial addition and multiplication is
/isomorphic/ to the set of real numbers with real number addition and
multiplication. That is, we have the correspondence
a0 x^0 <----> a0
and we show this correspondence respects the operations of addition and
multiplication appropriate for each side of the correspondence.
Example: (3 x^0) + (5 x^0) = (8 X^0) <----> 8 = 3 + 5, and also
3 x^0 <----> 3 and 5 x^0 <----> 5. (Yes, this is as obvious is it
seems!) So algebraicly R and the set of polynomials of form a x^0
behave exactly the same, and we informally identify them together in day
to day use. (This is like we identify the real number 2 with the
natural number 2, although it can be argued they are conceptually
distinct.)
How such a direct contradiction in a subject that is supposedly pristine can be propagated and absorbed by so many for so long is surely a statement with broader consequences. All that most can do is to deny the breakage. Here at least a lamb has offered itself up.
I hope I've shown there is no "breakage". At worst there is some minor
abuse of notation going on, which does no harm, and is completely
understood by everybody except you.
It may not be your fault that you missed out on a fuller explanation in
your studies, but your underlying response to this (your attitude in
posting a rant) is down to you...
Regards,
Mike.
Thanks Mike and I hope you will pardon the rhetoric for there is actual content to discuss here. The strictness of the ring definition; it was well built. The sum and the product are sufficient without the reverse operators. This polynomial stage though; then the quotient and ideal; these things are very dirty. Should mathematicians really be taking up particle/wave duality without explicitly stating it? At least the physicists bother to explain the contradiction before they eat it en mass. The mathematicians cover it up. That is not mathematics at all, and yet the constructions stands freely and mostly unchallenged. The consequences are broad even if non-mathematical. The separation of philosophy from mathematics and from physics might just have a wee bit to do with this. These are false divisions. As the ring provides there is no need for division. We ought to do without it in the name of simplicity.
Post by Mike Terry
[i.e. whose X coefficient is 1, and all others zero]
c) something else?
[e.g. maybe X represents multiplication or something!]
Mike.
my god you are a dumb shit. 1.23X is just (0,1.23,0,0,0,0,...) in the construction, there is nothing bad behaving with it, multiply it with any polynomial and you get another polynomial as I have demonstrated!
Lalo T.
2020-09-08 07:23:01 UTC
In your initial post, between the 'P.O.S.I.W.I.D. part' and the 'Statement-of-Mission part', I do
not see the Falsification. Here, P.O.S.I.W.I.D + Statements-of-Mission = strange loop.
The thing is, this strange loop is not arbitrary, and what is missing is to outline/describe a bit more
precisely its source...

Having said that, It can be perceived as "endow peculiarity" to 'External Binary Operations' or alternatively,
"bringing to the table" the topic of Closure and put it under DETAILED EXAMINATION.
https://en.wikipedia.org/wiki/External_(mathematics)#Generalizations

Where is the sector in evaluation ? Well, at least in :
https://en.wikipedia.org/wiki/Algebraic_structure#Two_sets_with_operations
Could be potencially this, at least, a point of bifurcation? I do not know

One of the topics you named made remember http://www.ams.org/notices/201005/rtx100500608p.pdf

I think that your Model Example' and Polynomial Rings' are not the same structure, that is the reason that
there will be no success in trying to overlap or squash one against the other.

Do you consider the minus symbol '-' an unary operator in :
https://proofwiki.org/wiki/Definition:Inverse_(Abstract_Algebra)/Inverse
??

I guess you will not agree with describe a polysign term/rudiment
as (s,m) instead of (s)m and/or s(m) ...? (where s: sign and m:magnitude)

Would you separate X into Xₘ and Xₛ in polynomials and algebra?

(1) P(X) = X² + 3x -5

(2) Q(Xₘ;Xₛ) = -(Xₘ)² @ Xₛ(5Xₘ) @ -2
where 'Xₘ' a unknown magnitude, 'Xₛ' an unknown sign and '@' is addition

(3) X = A <--> X - A = 0

(4) Xₛ(Xₘ) = Aₛ(Aₘ) <--> Xₛ(Xₘ) @ [Aₛ(Aₘ)]' = 0
... 'Aₘ' as an unknown magnitude, 'Aₛ' as an unknown sign and " [..]' " as additive inverse

Which style would you follow in polysigns ? with a,b,c,d magnitudes

(I) (-)a @ (+)b @ (*)c @ (#)d
where '@' addition operation, '-' '+' '*' '#' as signs

(-)a # (+)b # (*)c # (#)d
where '#' addition operation, '-' '+' '*' '#' as signs

(II) - a + b * c # d
where the arithmetical information migrates from the "operand space" towards the "operation space"
(algebraic-structure-like object)
zelos...@gmail.com
2020-09-01 05:40:14 UTC
Post by Tim Golden BandTech.com
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
Because its about rings, not real numbers.
Post by Tim Golden BandTech.com
After the careful construction of the ring from two groups, both operators obeying the closure principle
Not all rings have both operations as groups.
Post by Tim Golden BandTech.com
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
how is that not a polynomial?
Lalo Torres
2020-09-01 07:02:14 UTC
" Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together "

One can think of the ring K[X] as arising from K by adding one new element X that is external to K, commutes with all elements of K, and has no other specific properties. (This may be used for defining polynomial rings.) from https://en.wikipedia.org/wiki/Polynomial_ring#Definition_(univariate_case)

" Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect "

Expand this point. This may have several meaning, specify...

" The sad part is that the weakness of the construction cannot be challenged "

If you are somehow speaking against a construct, you could come up with some alternative, even with a basic and primitive construct.
Otherwise, you leaves us into choose "mid-air with no-ground under our feet" or "our everyday customs". Since the mind is lazy almost
always prefer the last option. Different thing is to provide an alternative.

I guess that a majority of pro-academia depend on their academic status, but this is not a generalization to all.
Not all pro-academia people depend on their status to pay the bread, or a reputation to be defended.
It is just that some people are in a situation, lined up with academia, without much to lose if the wind change the direction...
Lalo Torres
2020-09-01 08:43:46 UTC
On the other hand, I could accept 'X as apples with An as apples',
I could reject 'X as pineapples with An as apples'
And indeed accept 'X as fruits with An as apples'

In which case you could give more examples regarding what you think can be mixed, or not mixed...
Tim Golden BandTech.com
2020-09-03 19:34:52 UTC
Post by Lalo Torres
On the other hand, I could accept 'X as apples with An as apples',
I could reject 'X as pineapples with An as apples'
And indeed accept 'X as fruits with An as apples'
In which case you could give more examples regarding what you think can be mixed, or not mixed...
It is such a simplistic instance that I put out here, and yet none of these people who I can accept have mastered abstract algebra can come down to such a simple level. In this case it is not for me really to say what I think can and cannot be mixed. The requirements have already been formalized and I am working within those formalities on an extremely simple instance. The polynomial construction fails the ring requirements under these conditions; particularly when we specify real coefficients for the still abstract polynomial in X where X has no type constraint whatsoever. I make one simple black swan as
1.23 X
and none here will address it. This expression above offends the ring requirements. It is this simple. Instantiation breaks the long back of abstract algebra.
Mostowski Collapse
2020-09-03 19:45:41 UTC
The instantiation already breaks if you
plug in decimal numbers with two digits.

1.23 * 9.87 = 12.1401

The result has 4 digits and not anymore
two digits. We can say this little example

already Fucked Up Beyond All Recognition
abstract algebra and renders it totally useless.
Post by Tim Golden BandTech.com
Post by Lalo Torres
On the other hand, I could accept 'X as apples with An as apples',
I could reject 'X as pineapples with An as apples'
And indeed accept 'X as fruits with An as apples'
In which case you could give more examples regarding what you think can be mixed, or not mixed...
It is such a simplistic instance that I put out here, and yet none of these people who I can accept have mastered abstract algebra can come down to such a simple level. In this case it is not for me really to say what I think can and cannot be mixed. The requirements have already been formalized and I am working within those formalities on an extremely simple instance. The polynomial construction fails the ring requirements under these conditions; particularly when we specify real coefficients for the still abstract polynomial in X where X has no type constraint whatsoever. I make one simple black swan as
1.23 X
and none here will address it. This expression above offends the ring requirements. It is this simple. Instantiation breaks the long back of abstract algebra.
Mostowski Collapse
2020-09-03 19:49:36 UTC
Or as a friend of mine used to say,
when the ice machine didn't work

since he didn't fill water,
abstract algebra is borked.

https://www.urbandictionary.com/define.php?term=borked

LMAO!
Post by Mostowski Collapse
The instantiation already breaks if you
plug in decimal numbers with two digits.
1.23 * 9.87 = 12.1401
The result has 4 digits and not anymore
two digits. We can say this little example
already Fucked Up Beyond All Recognition
abstract algebra and renders it totally useless.
Post by Tim Golden BandTech.com
Post by Lalo Torres
On the other hand, I could accept 'X as apples with An as apples',
I could reject 'X as pineapples with An as apples'
And indeed accept 'X as fruits with An as apples'
In which case you could give more examples regarding what you think can be mixed, or not mixed...
It is such a simplistic instance that I put out here, and yet none of these people who I can accept have mastered abstract algebra can come down to such a simple level. In this case it is not for me really to say what I think can and cannot be mixed. The requirements have already been formalized and I am working within those formalities on an extremely simple instance. The polynomial construction fails the ring requirements under these conditions; particularly when we specify real coefficients for the still abstract polynomial in X where X has no type constraint whatsoever. I make one simple black swan as
1.23 X
and none here will address it. This expression above offends the ring requirements. It is this simple. Instantiation breaks the long back of abstract algebra.
Tim Golden BandTech.com
2020-09-03 20:01:59 UTC
Post by Mostowski Collapse
The instantiation already breaks if you
plug in decimal numbers with two digits.
1.23 * 9.87 = 12.1401
The result has 4 digits and not anymore
two digits. We can say this little example
already Fucked Up Beyond All Recognition
Well at least I've now learned what FUBAR means.
I can't really agree with your statement here though.
I guess u r far less serious than I had anticipated.
You are on a diversion from a diversion here.
I've stuck your nose right under the black swan's ass,
it left you a nice load on the tip of your nose,
and still you won't wipe it.
Post by Mostowski Collapse
abstract algebra and renders it totally useless.
Post by Lalo Torres
On the other hand, I could accept 'X as apples with An as apples',
I could reject 'X as pineapples with An as apples'
And indeed accept 'X as fruits with An as apples'
In which case you could give more examples regarding what you think can be mixed, or not mixed...
It is such a simplistic instance that I put out here, and yet none of these people who I can accept have mastered abstract algebra can come down to such a simple level. In this case it is not for me really to say what I think can and cannot be mixed. The requirements have already been formalized and I am working within those formalities on an extremely simple instance. The polynomial construction fails the ring requirements under these conditions; particularly when we specify real coefficients for the still abstract polynomial in X where X has no type constraint whatsoever. I make one simple black swan as
1.23 X
and none here will address it. This expression above offends the ring requirements. It is This must be true only mthis simple. Instantiation breaks the long back of abstract algebra.
Mostowski Collapse
2020-09-03 20:21:55 UTC
We can say abstract algebra is plastered
with black swans, possible all dead.
Its very difficult to not step on them
and navigate through this swamp call

abstract algebra. This is because the
slient majority of undergraduates doesn't
oppose, they swallow all the fake science
presented by teachers, youtube videos

and text book. Lets make mathematics great
again and drain this swamp.
Post by Tim Golden BandTech.com
Post by Mostowski Collapse
The instantiation already breaks if you
plug in decimal numbers with two digits.
1.23 * 9.87 = 12.1401
The result has 4 digits and not anymore
two digits. We can say this little example
already Fucked Up Beyond All Recognition
Well at least I've now learned what FUBAR means.
I can't really agree with your statement here though.
I guess u r far less serious than I had anticipated.
You are on a diversion from a diversion here.
I've stuck your nose right under the black swan's ass,
it left you a nice load on the tip of your nose,
and still you won't wipe it.
Lalo T.
2020-09-03 21:49:00 UTC
I know Tim is completely able to operate the construction and obtain numbers from it.
Also he will not have problems with operating "more composite" objects like:

https://en.wikipedia.org/wiki/Quaternion#Matrix_representations
https://en.wikipedia.org/wiki/Supermatrix
https://en.wikipedia.org/wiki/Split-octonion#Zorn's_vector-matrix_algebra
https://en.wikipedia.org/wiki/Polynomial_matrix
https://en.wikipedia.org/wiki/Zhegalkin_polynomial
https://en.wikipedia.org/wiki/Matrix_polynomial

...or objecs that changing the usual two operations of a structure, like in tropical geometry,
Note that arithmetic in tropical geometry the "addition operation" does not have inverse, hence, a semiring, but it serves to this purpose.
https://circles.math.ucla.edu/circles/lib/data/Handout-2121-1848.pdf (tropical polynomial)
I put this example just for the sake, that you can replace the usual operations of certain algebraic structures usually perceived as the usual addition and product in of numbers,
in this case, this object has two operations, where the "addition operation" is the min operation(or max operation) and the "product operation" is the usual addition.

As Julio said, Tim does not have any problem with notation, and I agree with that.

It is just the case that Tim does not accept the definition (the very contruction).

But let's proceed.

An external binary operation is a binary function from K × S to S. This differs from a binary operation on a set in the sense in that K need not be S; its elements come from outside...
from :
https://en.wikipedia.org/wiki/External_(mathematics)#Generalizations
https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations
https://en.wikipedia.org/wiki/Scalar_multiplication#Interpretation}

Now, suppose we take, for example, matrices of mxm (squares matrices)
https://en.wikipedia.org/wiki/Square_matrix

...being its element real numbers. We endow this matrices with the operation "Entrywise sum"

So far we have (matrices of mxm, entrywise sum '+') and this is an abelian group:

https://proofwiki.org/wiki/Definition:Group
https://proofwiki.org/wiki/Definition:Abelian_Group

We proceed to endow our object with another operation, concretely, matrix multiplication :

https://en.wikipedia.org/wiki/Matrix_multiplication#Definition
https://en.wikipedia.org/wiki/File:Matrix_multiplication_diagram_2.svg

So far we have (matrices of mxm, entrywise sum '+', matrix multiplication '*') and this is a "Unit Ring with zero divisors".

https://mathworld.wolfram.com/UnitRing.html

For the present purpose we are insterested in the "Unit Ring" part.

Instantiating your statement to square matrices :

a1 X <---here a1 is a real number and X a square matrix where its elements are real numbers.

Here we use an specific object (matrices of mxm, entrywise sum '+', scalar multiplication '·') !!!
We say here that our matrices of mxm is provided with a structure of a Vector Space.

a1·X <--it is defined
where symbol ' · ' is the scalar multiplication between a real and a square matrix (not the Dot product)

Now, It happens that in :

a0 I + a1 X + a2 X X + a3 X X X + ...
a0·(I) + a1·(X) + a2·(X*X) + a3·(X*X*X) ...

Tim request, the application of the ring definition :

https://proofwiki.org/wiki/Definition:Ring_(Abstract_Algebra)

Our object is :

(a) (matrices of mxm, entrywise sum '+', matrix multiplication '*')

but in the other definition we have :

" The polynomial ring in X over K is equipped with an addition, a multiplication AND SCALAR MULTIPLICATION that make it a commutative algebra "

this is :

(b) (matrices of mxm, entrywise sum '+', matrix multiplication '*',scalar multiplication '·')

Which indeed, is a structure with three operations, not just two. Note that (b) is algebraically bigger than (a)
("less primitive than")

Certainly Tim rejects (b) as a ring, due to the third operation (scalar multiplication ' · ' )
and also alluding to the ring closure property, concretely, the operation ' · ',
which start from the set of :

"the cartesian product of 'matrices of mxm' with ' with 'real numbers' "

ending in the set of :

'matrices of mxm'

Aₘₓₘ X r ---> Aₘₓₘ (here ' X ' symbol is the cartesian product)

Tim could use the words "perceptual blindness", "ignorance" or "doublethink" on us, since we SEEM to reject his statements.

Note the words "scalar-valued polynomial" in :
https://en.wikipedia.org/wiki/Matrix_polynomial

As a curiosity we can get the same effect in :

(1) λ₀·I + λ₁·X + λ₂·X² + ... + λₙ₋₁·Xⁿ⁻¹ + λₙ·Xⁿ where λ are Scalars, X matrices

(2) A₀*I + A₁*X + A₂*X² + ... + Aₙ₋₁*Xⁿ⁻¹ + Aₙ*Xⁿ where A are Scalar Matrices, and X matrices

https://en.wikipedia.org/wiki/Center_(ring_theory) (in square matrices)

In this context, the scalar multiplication is well-defined, and also an atractive operation to have in a structure...
One likes to "scale" things. Which bring to the table the question of the origin of the scalar multiplication (used as a external operation, also a bigwig/kingpin )
In the case where is it possible to use the "second operation" of a ring to get the same results, equivalent to asking why dont use the "internal" product of a ring if
we get the same effects, why do I have to outsource an operation that my team can do for themself ?

Note in "Buildings, polytopes and tropical convexity" by Annette Werner
https://www.uni-frankfurt.de/50581267/tropical_geometry11.pdf
and look the word "tropical scalar multiplication" in page 2
(semi-module structure)
Mostowski Collapse
2020-09-03 22:44:35 UTC
Doesn't work for multivariante polynomials.
Matrix multiplication is not commutative.

You don't have in general:

/* is not generally valid */

X*Y = Y*X

https://en.wikipedia.org/wiki/Matrix_multiplication#Non-commutativity
Post by Lalo T.
a1 X <---here a1 is a real number and X a square matrix where its elements are real numbers.
Lalo T.
2020-09-03 23:29:27 UTC
Yes, certainly I inject the definition "The polynomial ring, K[X], in X over a field (or, more generally, a COMMUTATIVE ring) K can be defined..."

...and after, I used a non-commutative ring, which is mistake.

Certainly, Tim's initial post on this thread use rings, not commutative rings.
Lalo T.
2020-09-04 02:19:12 UTC
In the post, what I am contending is that the conflict between Tim and everyone else, in this thread, is only apparent.
I think it could be cleared through careful and detailed examination, with specific examples.

In addition, that the Scalar Multiplication operation can be seen, in some instances as a mix of :

https://en.wikipedia.org/wiki/Alien_hand_syndrome
https://www.investopedia.com/terms/o/outsourcing.asp

"Design Decision" or not, the point here is that this last topic has some tentacles towards
Theoretical Physics (topic which I am ignorant of) in terms of the physical significance of maths,
while attempting to isolate the scalar multiplication "landscape".

In any case, my intention with the last topic is not as much, divert the attention of the thread.
In one of the his statements Tim says that something is stinking. I do not agree with him in this regard,
that "a0 X" is ill-formed formula, and I think it is a well-formed formula.

Although as a consecuence, this thread give me the chance to appreciate from another angle the Vector War debates of 150 years ago.
Lalo T.
2020-09-04 03:33:23 UTC
Long story short, I think the smelling thing (stink or aroma) is not the definitions nor all constructions on top of these definitions,
rather, the motivations behind the definitions, which in part, are beyond the scope of Mathematics...
Tim Golden BandTech.com
2020-09-04 12:40:27 UTC
Post by Lalo T.
In the post, what I am contending is that the conflict between Tim and everyone else, in this thread, is only apparent.
I think it could be cleared through careful and detailed examination, with specific examples.
https://en.wikipedia.org/wiki/Alien_hand_syndrome
https://www.investopedia.com/terms/o/outsourcing.asp
"Design Decision" or not, the point here is that this last topic has some tentacles towards
Theoretical Physics (topic which I am ignorant of) in terms of the physical significance of maths,
while attempting to isolate the scalar multiplication "landscape".
In any case, my intention with the last topic is not as much, divert the attention of the thread.
In one of the his statements Tim says that something is stinking. I do not agree with him in this regard,
that "a0 X" is ill-formed formula, and I think it is a well-formed formula.
OK, so Lalo can come to the black swan and state that it is a well-formed formula.
That's nice. At least you have had a visit to the thing and attempted to look at it.
Next part of the puzzle is to actually address whether "a0 X" is a binary operation...
Is it ring behaved?

Thus far you have not invoked any of the language that makes up the ground work of abstract algebra.
And Lalo you must know by now that the X is not real and that a0 is real.
So you are staring the swan in the face. Is it a black swan?
Look again please. The full polynomial is simply a long sum of such individual parts. The conflict worsens here, but that is not the problem. Just one black swan will do in the name of falsification... a detail that none here deny. They simply avoid it altogether. Perhaps usenet is providing a new form: proof by vacuum. The usenet dodge is something very familiar to me. Really I write more for the onlookers. We have to get some kind of entertainment into the thread. To me the best is a mix of strong rhetoric with direct content. But still I think focus is more important here than the branching meandering that is your habit. Whatever: you are free here in an uncensored distributed medium that is something future bots will look over to at least gain an understanding of the human race and its individual inhabitants.

Thank you so much for even going this far. It exposes the others as WIMPS as in the weakly interacting type. None has come as far as you. Even though Mostowski has it on his nose he still cannot smell it. I wouldn't mind hearing about the vector war which no doubt requires the usage of ordered values. This is another area where polysign shine. Also though the lack of need for the cartesian product, whose use in the ring definition is problematic as well. I know that is a bit cryptic so let's go ahead and have a diversion from your own diversions here. In ordinary physical terms we do not generally multiply values on a real line and land with a value on a real line. We instead land such things in units; say meters; whose product then yield square meters. Thus the space of the result is not at all the space of the sources. Within the formal definition of the binary operator we often will see
f : S X S -> S
which is exactly inverted from the ordinary physical sense. You see the abuse of dimensionality by mathematicians is profound and so the interdimensional interpretation has quite a lot of room. Abstract algebra goes on to develop two dimensional numbers from infinite dimensional numbers. Brilliant eh?

I do think it is valid to cast doubt on the accumulation of information that is modern mathematics. And yes, notation does matter. It is as if we have granted humans compiler level integrity by naming them 'mathematician' but it is by no means true. To me this break in the fundamentals is exciting and because we are down in so low and deep any modification at this level is bound to have extensive side effects. Kaboom!
Post by Lalo T.
Although as a consecuence, this thread give me the chance to appreciate from another angle the Vector War debates of 150 years ago.
Peter
2020-09-04 14:09:06 UTC
Post by Tim Golden BandTech.com
Post by Lalo T.
In the post, what I am contending is that the conflict between Tim and everyone else, in this thread, is only apparent.
I think it could be cleared through careful and detailed examination, with specific examples.
https://en.wikipedia.org/wiki/Alien_hand_syndrome
https://www.investopedia.com/terms/o/outsourcing.asp
"Design Decision" or not, the point here is that this last topic has some tentacles towards
Theoretical Physics (topic which I am ignorant of) in terms of the physical significance of maths,
while attempting to isolate the scalar multiplication "landscape".
In any case, my intention with the last topic is not as much, divert the attention of the thread.
In one of the his statements Tim says that something is stinking. I do not agree with him in this regard,
that "a0 X" is ill-formed formula, and I think it is a well-formed formula.
OK, so Lalo can come to the black swan and state that it is a well-formed formula.
That's nice. At least you have had a visit to the thing and attempted to look at it.
Next part of the puzzle is to actually address whether "a0 X" is a binary operation...
No it isn't. In a ring the binary operations are addition and
multiplication. "a0 X" is the result of multiplying the ring element a0
and the ring element X (that X is premultiplied n=by an invisible 1) but
it isn't a binary operation. There is obviously a difference between
binary operations and what results when two elements are combined by the
operation.
Post by Tim Golden BandTech.com
Is it ring behaved?
Thus far you have not invoked any of the language that makes up the ground work of abstract algebra.
And Lalo you must know by now that the X is not real and that a0 is real.
So you are staring the swan in the face. Is it a black swan?
Look again please. The full polynomial is simply a long sum of such individual parts. The conflict worsens here, but that is not the problem. Just one black swan will do in the name of falsification... a detail that none here deny. They simply avoid it altogether. Perhaps usenet is providing a new form: proof by vacuum. The usenet dodge is something very familiar to me. Really I write more for the onlookers. We have to get some kind of entertainment into the thread. To me the best is a mix of strong rhetoric with direct content. But still I think focus is more important here than the branching meandering that is your habit. Whatever: you are free here in an uncensored distributed medium that is something future bots will look over to at least gain an understanding of the human race and its individual inhabitants.
Thank you so much for even going this far. It exposes the others as WIMPS as in the weakly interacting type. None has come as far as you. Even though Mostowski has it on his nose he still cannot smell it. I wouldn't mind hearing about the vector war which no doubt requires the usage of ordered values. This is another area where polysign shine. Also though the lack of need for the cartesian product, whose use in the ring definition is problematic as well. I know that is a bit cryptic so let's go ahead and have a diversion from your own diversions here. In ordinary physical terms we do not generally multiply values on a real line and land with a value on a real line. We instead land such things in units; say meters; whose product then yield square meters. Thus the space of the result is not at all the space of the sources. Within the formal definition of the binary operator we often will see
f : S X S -> S
which is exactly inverted from the ordinary physical sense. You see the abuse of dimensionality by mathematicians is profound and so the interdimensional interpretation has quite a lot of room. Abstract algebra goes on to develop two dimensional numbers from infinite dimensional numbers. Brilliant eh?
I do think it is valid to cast doubt on the accumulation of information that is modern mathematics. And yes, notation does matter. It is as if we have granted humans compiler level integrity by naming them 'mathematician' but it is by no means true. To me this break in the fundamentals is exciting and because we are down in so low and deep any modification at this level is bound to have extensive side effects. Kaboom!
Post by Lalo T.
Although as a consecuence, this thread give me the chance to appreciate from another angle the Vector War debates of 150 years ago.
Tim Golden BandTech.com
2020-09-01 11:49:43 UTC
Post by Lalo Torres
" Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together "
One can think of the ring K[X] as arising from K by adding one new element X that is external to K, commutes with all elements of K, and has no other specific properties. (This may be used for defining polynomial rings.) from https://en.wikipedia.org/wiki/Polynomial_ring#Definition_(univariate_case)
Nice find. It is not pretty, but worst of all the notion of binary operators is still in conflict with this construction. I would suggest that this univariate definition is inherently conflicted. It actually breaks the ring definition now formally. Before it was a cover up; now under this definition it is an out and out lie. Still it is an excellent find and does deserve a follow up within this thread. So long as they maintain K as some abstract thing you see it doesn't quite fully fall apart. Upon instantiating the reals with this X in the univariate mode then we break the operators since
a1 X
does not resolve. Thus the operators cannot be binary operators and so the ring definition is explicitly broken. We are in the same position with a slight variation in interpretation, but the failing is the same.
Post by Lalo Torres
" Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect "
Expand this point. This may have several meaning, specify...
" The sad part is that the weakness of the construction cannot be challenged "
If you are somehow speaking against a construct, you could come up with some alternative, even with a basic and primitive construct.
Otherwise, you leaves us into choose "mid-air with no-ground under our feet" or "our everyday customs". Since the mind is lazy almost
always prefer the last option. Different thing is to provide an alternative.
I guess that a majority of pro-academia depend on their academic status, but this is not a generalization to all.
Not all pro-academia people depend on their status to pay the bread, or a reputation to be defended.
It is just that some people are in a situation, lined up with academia, without much to lose if the wind change the direction...
I think I'd rather just remain within the falsification of abstract algebra at the polynomial stage rather than state its replacement.
The falsification is free-standing and needs no such patch.
On academia; well; it makes a good punching bag. Of course some of my language is rhetoric. But also there is content. My belief is that the two stand together to make strong points. Some of the rhetoric stands true without the math content. I don't need to falsify abstract algebra to criticize the academic system, but it helps. I don't need to criticize the academic system to falsify abstract algebra, but the connection is fairly clear. Anyway we are free here to speak our minds even if some go sour. Those who post pure rhetoric without any content are truly suspect. Fortunately Mike has allowed a discussion with content to take place, though his opening was pure rhetoric. We'll see how it goes.
Julio Di Egidio
2020-09-01 12:33:33 UTC
Post by Tim Golden BandTech.com
a1 X
does not resolve. Thus the operators cannot be binary operators and so
the ring definition is explicitly broken. We are in the same position
with a slight variation in interpretation, but the failing is the same.
The "ring definition" is simply irrelevant there. The coefficients come
from a ring, but X, the "indeterminate" in the formation of a polynomial,
*as well as the operations* involved in that structure formation, are
simply *formal terms* standing for the place of the coefficients in a
sequence: there is no evaluation involved.

As for evaluation, we actually first go from the polynomial to a
polynomial function where critical is that the domain of the function
is itself a ring and contains the ring of coefficients, then we do
evaluation of the function, where there is now no problem with taking
the operations involved as operations: on the function domain ring...

HTH,

Julio
Tim Golden BandTech.com
2020-09-02 12:11:11 UTC
Post by Tim Golden BandTech.com
a1 X
does not resolve. Thus the operators cannot be binary operators and so
the ring definition is explicitly broken. We are in the same position
with a slight variation in interpretation, but the failing is the same.
The "ring definition" is simply irrelevant there. The coefficients come
from a ring, but X, the "indeterminate" in the formation of a polynomial,
*as well as the operations* involved in that structure formation, are
simply *formal terms* standing for the place of the coefficients in a
sequence: there is no evaluation involved.
As for evaluation, we actually first go from the polynomial to a
polynomial function where critical is that the domain of the function
is itself a ring and contains the ring of coefficients, then we do
evaluation of the function, where there is now no problem with taking
the operations involved as operations: on the function domain ring...
HTH,
Julio
Hi Julio. I appreciate that you will uphold the standard interpretation fairly well. So let's get on with it. I particularly am discussing polynomials with real coefficients, so a(n) are real valued. This is commonly done within the curriculum. So we start with the acclaimed ring behaved
a0 + a1 X + a2 X X + ...
Now we choose zero values for a(n) except we'll keep a1 variable. Thus the ring behaved polynomial condenses to
a1 X
which is not ring behaved since the product is not a binary operation. This is the conflict in its entirety, though the extensions go to considering
a0 + a1 X
since this addition operation is not binary and so is undefined. Clearly we have a real being added to a nonreal. Is this addition a binary operator or isn't it? If it is not a binary operator then should this operator be more explicitly defined within abstract algebra since the subject has bothered to define operators so carefully?

This procedure then extends onto the a2 term and so forth. In effect the entire polynomial is constructed of undefined operators.

Instantiation is often actually a problem in higher maths. Sure we can construct all sorts of fancy ideas and notations on paper. Humans excel at it. Quite often I myself will sit down to design something and it gets complicated. Quite often the final solution is considerably simpler. Instantiation of the high ideas that 'mathematicians' work on often descend down to fairly trivial instances with no fascinating instances actually constructed. I do put abstract algebra here, except that it is more badly broken. Quite oddly it does some work. Well, this is true of humans since we came down out of the trees. We've gotten by and we continue to develop in a progression. Often this progression is prompted by past mistakes. Academia has such a massive and breathless accumulation now... could it be time to topple some of it?
Julio Di Egidio
2020-09-02 13:41:11 UTC
Post by Tim Golden BandTech.com
Post by Tim Golden BandTech.com
a1 X
does not resolve. Thus the operators cannot be binary operators and so
the ring definition is explicitly broken. We are in the same position
with a slight variation in interpretation, but the failing is the same.
The "ring definition" is simply irrelevant there. The coefficients come
from a ring, but X, the "indeterminate" in the formation of a polynomial,
*as well as the operations* involved in that structure formation, are
simply *formal terms* standing for the place of the coefficients in a
sequence: there is no evaluation involved.
As for evaluation, we actually first go from the polynomial to a
polynomial function where critical is that the domain of the function
is itself a ring and contains the ring of coefficients, then we do
evaluation of the function, where there is now no problem with taking
the operations involved as operations: on the function domain ring...
Hi Julio. I appreciate that you will uphold the standard interpretation
fairly well. So let's get on with it. I particularly am discussing
polynomials with real coefficients, so a(n) are real valued. This is
commonly done within the curriculum. So we start with the acclaimed ring
behaved
a0 + a1 X + a2 X X + ...
You keep equivocating on what "ring-behaved" refers to. It's
P[x] = a0+a1*X+... (the whole expression) that is the member of a
ring, a ring of polynomials. And it's the a_i themselves that
are members of a ring, of scalars, e.g. the real numbers. But X
is NOT a member of a ring, and the operations *in the expression*
of P[X] are NOT ring operations, they are just *formal symbols*
that stand for the places of the coefficients in *a sequence*.

And what is a sequence if not a vector? As Wikipedia puts it:
<< The set of polynomials in X_1, ..., X_n, denoted K[X_1, ...,
X_n], is thus *a vector space* (or a free module, if K is a ring)
that has the monomials as a basis. >>

So, as for the ring operations *of polynomials*: << Addition and
scalar multiplication of polynomials are those of a vector space
or free module equipped by a specific basis (here the basis of the
monomials). [...] The multiplication is [...] The verification
of the axioms of an *associative algebra* is straightforward. >>

<https://en.wikipedia.org/wiki/Polynomial_ring#Definition_(multivariate_case)>
Post by Tim Golden BandTech.com
This procedure then extends onto the a2 term and so forth. In effect
the entire polynomial is constructed of undefined operators.
Rather, "+" means different things in different contexts: it's you
who are not taking the definition for serious. And then you might
at best complain that that notation is ambiguous, but, arguably,
it's not even so: that notation is rather conducive to polynomial
functions.

And I am not even at all an expert in that field: (not only I do
apologise in advance for any imprecision in my above statements,)
there may very well be more stringent reasons that justify the
use of exactly that notation. Anyway, even should you dislike
the notation, it is a fact that the maths is just fine.

Indeed, I won't insist: HTH and good luck.

Julio
Peter
2020-09-04 14:03:29 UTC
[...] I particularly am discussing polynomials with real
coefficients, so a(n) are real valued. This is commonly done within
the curriculum. So we> a(n) are real valued. This is commonly done
within the curriculum. So we start with the acclaimed ring behaved
a0 + a1 X + a2 X X + ...
Now we choose zero values for a(n) except we'll
keep a1 variable. Thus the ring behaved polynomial condenses to
a1 X
That is one polynomial in the ring R[X]. Since R[X] *is* a ring it will
also contain (a1)^2 XX.

Your phrase "ring behaved polynomial" is odd. It suggests to me that
you think if R[X] is a ring then any element of R[X] is a ring, which is
potty. The real numbers constitute a ring, 4.5 doesn't.
which is not ring behaved since the product is not a binary
operation. This is the conflict in its entirety, though the
extensions go to considering
a0 + a1 X
since this addition operation is not binary and so is undefined.
Tim Golden BandTech.com
2020-09-04 14:43:25 UTC
[...] I particularly am discussing polynomials with real
coefficients, so a(n) are real valued. This is commonly done within
the curriculum. So we> a(n) are real valued. This is commonly done
within the curriculum. So we start with the acclaimed ring behaved
a0 + a1 X + a2 X X + ...
Now we choose zero values for a(n) except we'll
keep a1 variable. Thus the ring behaved polynomial condenses to
a1 X
That is one polynomial in the ring R[X]. Since R[X] *is* a ring it will
also contain (a1)^2 XX.
Your phrase "ring behaved polynomial" is odd. It suggests to me that
you think if R[X] is a ring then any element of R[X] is a ring, which is
potty. The real numbers constitute a ring, 4.5 doesn't.
Come now; you must see that instantiation is always an option. We should replace your word 'any' above with 'every' and then I might agree with your statement partially. It is as it you would deny that
(4.5)(0.1) = 0.45
is not an instance of a ring operation. These are all real values. The reals are acclaimed ring status. So it as if your philosophy states that one must never instantiate a concrete instance in abstract algebra. Now that would be some axiom and it would solidify the status of the subject.

I will falsify your words more carefully in the near future. I do appreciate that you are one of two now who has at least approached the black swan.
Good on you Peter. Yet within your refutation you are somewhat declaring that you can approach the black swan no further. Still, I believe I have enough to falsify you, and that is exactly as it should be on my own language if I am wrong. You are attempting this so I do see this as a potentially productive debate. Thank you for joining in. Best of all with careful language our statements here stand freely and if they are flawed the flaw can be amplified. I will try not to dodge as others do here. Please do provide amplification on my own errors as aggressively as possible. Of course simplicity is helpful, but then too variations matter I think. I've settled onto a concrete instance of
1.23 X
to study and I can easily back off of this but I do suspect this is the simplest of forms to study. There are two operands and one operator right?
which is not ring behaved since the product is not a binary
operation. This is the conflict in its entirety, though the
extensions go to considering
a0 + a1 X
since this addition operation is not binary and so is undefined.
Peter
2020-09-04 15:11:03 UTC
Post by Tim Golden BandTech.com
[...] I particularly am discussing polynomials with real
coefficients, so a(n) are real valued. This is commonly done within
the curriculum. So we> a(n) are real valued. This is commonly done
within the curriculum. So we start with the acclaimed ring behaved
a0 + a1 X + a2 X X + ...
Now we choose zero values for a(n) except we'll
keep a1 variable. Thus the ring behaved polynomial condenses to
a1 X
That is one polynomial in the ring R[X]. Since R[X] *is* a ring it will
also contain (a1)^2 XX.
Your phrase "ring behaved polynomial" is odd. It suggests to me that
you think if R[X] is a ring then any element of R[X] is a ring, which is
potty. The real numbers constitute a ring, 4.5 doesn't.
Come now; you must see that instantiation is always an option. We should replace your word 'any' above with 'every' and then I might agree with your statement partially. It is as it you would deny that
(4.5)(0.1) = 0.45
is not an instance of a ring operation. These are all real values. The reals are acclaimed ring status. So it as if your philosophy states that one must never instantiate a concrete instance in abstract algebra. Now that would be some axiom and it would solidify the status of the subject.
I will falsify your words more carefully in the near future. I do appreciate that you are one of two now who has at least approached the black swan.
Good on you Peter. Yet within your refutation you are somewhat declaring that you can approach the black swan no further. Still, I believe I have enough to falsify you, and that is exactly as it should be on my own language if I am wrong. You are attempting this so I do see this as a potentially productive debate. Thank you for joining in. Best of all with careful language our statements here stand freely and if they are flawed the flaw can be amplified. I will try not to dodge as others do here. Please do provide amplification on my own errors as aggressively as possible. Of course simplicity is helpful, but then too variations matter I think. I've settled onto a concrete instance of
1.23 X
to study and I can easily back off of this but I do suspect this is the simplest of forms to study. There are two operands and one operator right?
which is not ring behaved since the product is not a binary
operation. This is the conflict in its entirety, though the
extensions go to considering
a0 + a1 X
since this addition operation is not binary and so is undefined.
a0 +(that's the ring addition) a1 X = a0 + a1 X

is an example of two ring element a0 and a1 X adding to make one ring
element.

If you read a book on algebra you'll see a proof that the set of
polynomials with one indeterminate (here X) and coefficients from a ring
is itself a ring under addition and multiplication of polynomials.
Tim Golden BandTech.com
2020-09-05 15:45:38 UTC
Post by Peter
[...] I particularly am discussing polynomials with real
coefficients, so a(n) are real valued. This is commonly done within
the curriculum. So we> a(n) are real valued. This is commonly done
within the curriculum. So we start with the acclaimed ring behaved
a0 + a1 X + a2 X X + ...
Now we choose zero values for a(n) except we'll
keep a1 variable. Thus the ring behaved polynomial condenses to
a1 X
That is one polynomial in the ring R[X]. Since R[X] *is* a ring it will
also contain (a1)^2 XX.
Your phrase "ring behaved polynomial" is odd. It suggests to me that
you think if R[X] is a ring then any element of R[X] is a ring, which is
potty. The real numbers constitute a ring, 4.5 doesn't.
Come now; you must see that instantiation is always an option. We should replace your word 'any' above with 'every' and then I might agree with your statement partially. It is as it you would deny that
(4.5)(0.1) = 0.45
is not an instance of a ring operation. These are all real values. The reals are acclaimed ring status. So it as if your philosophy states that one must never instantiate a concrete instance in abstract algebra. Now that would be some axiom and it would solidify the status of the subject.
I will falsify your words more carefully in the near future. I do appreciate that you are one of two now who has at least approached the black swan.
Good on you Peter. Yet within your refutation you are somewhat declaring that you can approach the black swan no further. Still, I believe I have enough to falsify you, and that is exactly as it should be on my own language if I am wrong. You are attempting this so I do see this as a potentially productive debate. Thank you for joining in. Best of all with careful language our statements here stand freely and if they are flawed the flaw can be amplified. I will try not to dodge as others do here. Please do provide amplification on my own errors as aggressively as possible. Of course simplicity is helpful, but then too variations matter I think. I've settled onto a concrete instance of
1.23 X
to study and I can easily back off of this but I do suspect this is the simplest of forms to study. There are two operands and one operator right?
which is not ring behaved since the product is not a binary
operation. This is the conflict in its entirety, though the
extensions go to considering
a0 + a1 X
since this addition operation is not binary and so is undefined.
a0 +(that's the ring addition) a1 X = a0 + a1 X
is an example of two ring element a0 and a1 X adding to make one ring
element.
Then how come you now have three elements with two operators?
This goes in contradiction to the ring definition. It states that two elements will resolve to one element.
I do appreciate what you are saying and I do see that you are obeying the standard interpretation.
Can we please specify real coefficients. Will this be alright with you?
In other words in the expression above a0 and a1 are both real valued.
The notion of an element as a singular thing seems such a fundamental concept and yet possibly this ray of argumentation that you are on here will lead us down into set theory fundamentals. I do suspect that if elements are composed of multiple elements then they are not elementary. In other words your argument above is false. And so in covering this ground you in effect are falsifying abstract algebra under this interpretation.

Of course it starts to feel a bit silly right? We are being such math weenies here. Yet we have whittled down now to a fairly simple and by the way alternative position on elements in set theory. I do still believe that one of the real elements can be
1.23
and that the ring behaviors can still be applied. I guess though that rather than confuse our responses I'll wait for your response here and maybe we can pick up this trail of concrete instantiation as well here off of this subthread. Here is that statement of yours transcribed here:
Your phrase "ring behaved polynomial" is odd.
It suggests to me that you think if R[X] is a ring
then any element of R[X] is a ring, which is potty.
The real numbers constitute a ring, 4.5 doesn't.

I have to admit that I just did a google search on '"ring behaved" polynomial' and get only 54 results so I accept your criticism though I don't see any real abuse of language here. Logically I am finding that the supposed ring behaved polynomial is not ring behaved. I suppose some would simply say 'is a ring' or 'is not a ring' and you are on the side of 'is a ring' and I am on the side of 'is not a ring' But isn't your dialect above really explaining why I use this terminology? The fact that for real values
( 0.01) ( 1.23 ) = 0.00123
is an expression that is consistent with the ring definition and so is ring behaved though clearly these are elements of the set and not the set of reals itself, which is the ring. Oddly enough we are right back to this sticking point of elemental quality within a set. Had I stated that
( 0.01) ( 1.23 ) = #0.00123
where the # indicates a value that is outside of the set of real numbers, then we would say that this expression violates the ring requirements so long as we still kept the declaration of real valued antecedents. I would say then that this is not a ring behaved expression. I do not care to engage is more highfaluting language which no doubt exists for such concepts but lacks instantiable types. We do know the real value well and we do know that the curriculum of abstract algebra does make use of the polynomial with real coefficients. I do point out here the conflict of this construction and particularly claims that these are ring behaved.

Peter you are the strongest respondent to this thread. I appreciate that you are taking this on and that you are taking me seriously. In effect you expose better than I the weakness of the other responses here. The informational concepts ought to be free-standing regardless of these human interactions that involve personal slights and geese crapping on people's noses and so forth. Please feel free to throw in a few yourself since you are on usenet here. Poor Mostowski still can't smell it and it's been days. I don't think he sees it either. Just let it dry and scrape it off Mostowski. Any day now.
Post by Peter
If you read a book on algebra you'll see a proof that the set of
polynomials with one indeterminate (here X) and coefficients from a ring
is itself a ring under addition and multiplication of polynomials.
Peter
2020-09-06 16:40:55 UTC
Post by Tim Golden BandTech.com
Post by Peter
[...] I particularly am discussing polynomials with real
coefficients, so a(n) are real valued. This is commonly done within
the curriculum. So we> a(n) are real valued. This is commonly done
within the curriculum. So we start with the acclaimed ring behaved
a0 + a1 X + a2 X X + ...
Now we choose zero values for a(n) except we'll
keep a1 variable. Thus the ring behaved polynomial condenses to
a1 X
That is one polynomial in the ring R[X]. Since R[X] *is* a ring it will
also contain (a1)^2 XX.
Your phrase "ring behaved polynomial" is odd. It suggests to me that
you think if R[X] is a ring then any element of R[X] is a ring, which is
potty. The real numbers constitute a ring, 4.5 doesn't.
Come now; you must see that instantiation is always an option. We should replace your word 'any' above with 'every' and then I might agree with your statement partially. It is as it you would deny that
(4.5)(0.1) = 0.45
is not an instance of a ring operation. These are all real values. The reals are acclaimed ring status. So it as if your philosophy states that one must never instantiate a concrete instance in abstract algebra. Now that would be some axiom and it would solidify the status of the subject.
I will falsify your words more carefully in the near future. I do appreciate that you are one of two now who has at least approached the black swan.
Good on you Peter. Yet within your refutation you are somewhat declaring that you can approach the black swan no further. Still, I believe I have enough to falsify you, and that is exactly as it should be on my own language if I am wrong. You are attempting this so I do see this as a potentially productive debate. Thank you for joining in. Best of all with careful language our statements here stand freely and if they are flawed the flaw can be amplified. I will try not to dodge as others do here. Please do provide amplification on my own errors as aggressively as possible. Of course simplicity is helpful, but then too variations matter I think. I've settled onto a concrete instance of
1.23 X
to study and I can easily back off of this but I do suspect this is the simplest of forms to study. There are two operands and one operator right?
which is not ring behaved since the product is not a binary
operation. This is the conflict in its entirety, though the
extensions go to considering
a0 + a1 X
since this addition operation is not binary and so is undefined.
a0 +(that's the ring addition) a1 X = a0 + a1 X
is an example of two ring element a0 and a1 X adding to make one ring
element.
Then how come you now have three elements with two operators?
This goes in contradiction to the ring definition. It states that two elements will resolve to one element.
I do appreciate what you are saying and I do see that you are obeying the standard interpretation.
Can we please specify real coefficients. Will this be alright with you?
Yes. I would it be alright with you for you to stop using meaningless
phrases such as

the ring definition [...] states that two elements will
resolve to one element.
Post by Tim Golden BandTech.com
In other words in the expression above a0 and a1 are both real valued.
The notion of an element as a singular thing seems such a fundamental concept and yet possibly this ray of argumentation that you are on here will lead us down into set theory fundamentals. I do suspect that if elements are composed of multiple elements then they are not elementary. In other words your argument above is false. And so in covering this ground you in effect are falsifying abstract algebra under this interpretation.
Of course it starts to feel a bit silly right? We are being such math weenies here. Yet we have whittled down now to a fairly simple and by the way alternative position on elements in set theory. I do still believe that one of the real elements can be
1.23
Your phrase "ring behaved polynomial" is odd.
It suggests to me that you think if R[X] is a ring
then any element of R[X] is a ring, which is potty.
The real numbers constitute a ring, 4.5 doesn't.
I have to admit that I just did a google search on '"ring behaved" polynomial' and get only 54 results so I accept your criticism though I don't see any real abuse of language here. Logically I am finding that the supposed ring behaved polynomial is not ring behaved. I suppose some would simply say 'is a ring' or 'is not a ring' and you are on the side of 'is a ring' and I am on the side of 'is not a ring' But isn't your dialect above really explaining why I use this terminology? The fact that for real values
( 0.01) ( 1.23 ) = 0.00123
is an expression that is consistent with the ring definition and so is ring behaved
There you go again. How about sticking to the language which one finds
in algebra texts? Have you ever read one?
Post by Tim Golden BandTech.com
though clearly these are elements of the set and not the set of reals itself, which is the ring. Oddly enough we are right back to this sticking point of elemental quality within a set. Had I stated that
( 0.01) ( 1.23 ) = #0.00123
where the # indicates a value that is outside of the set of real numbers, then we would say that this expression violates the ring requirements so long as we still kept the declaration of real valued antecedents. I would say then that this is not a ring behaved expression. I do not care to engage is more highfaluting language which no doubt exists for such concepts but lacks instantiable types. We do know the real value well and we do know that the curriculum of abstract algebra does make use of the polynomial with real coefficients. I do point out here the conflict of this construction and particularly claims that these are ring behaved.
Peter you are the strongest respondent to this thread. I appreciate that you are taking this on and that you are taking me seriously.
What makes you think that?
Post by Tim Golden BandTech.com
In effect you expose better than I the weakness of the other responses here. The informational concepts ought to be free-standing regardless of these human interactions that involve personal slights and geese crapping on people's noses and so forth. Please feel free to throw in a few yourself since you are on usenet here. Poor Mostowski still can't smell it and it's been days. I don't think he sees it either. Just let it dry and scrape it off Mostowski. Any day now.
Post by Peter
If you read a book on algebra you'll see a proof that the set of
polynomials with one indeterminate (here X) and coefficients from a ring
is itself a ring under addition and multiplication of polynomials.
Tim Golden BandTech.com
2020-09-06 20:25:43 UTC
Post by Tim Golden BandTech.com
Post by Peter
[...] I particularly am discussing polynomials with real
coefficients, so a(n) are real valued. This is commonly done within
the curriculum. So we> a(n) are real valued. This is commonly done
within the curriculum. So we start with the acclaimed ring behaved
a0 + a1 X + a2 X X + ...
Now we choose zero values for a(n) except we'll
keep a1 variable. Thus the ring behaved polynomial condenses to
a1 X
That is one polynomial in the ring R[X]. Since R[X] *is* a ring it will
also contain (a1)^2 XX.
Your phrase "ring behaved polynomial" is odd. It suggests to me that
you think if R[X] is a ring then any element of R[X] is a ring, which is
potty. The real numbers constitute a ring, 4.5 doesn't.
Come now; you must see that instantiation is always an option. We should replace your word 'any' above with 'every' and then I might agree with your statement partially. It is as it you would deny that
(4.5)(0.1) = 0.45
is not an instance of a ring operation. These are all real values. The reals are acclaimed ring status. So it as if your philosophy states that one must never instantiate a concrete instance in abstract algebra. Now that would be some axiom and it would solidify the status of the subject.
I will falsify your words more carefully in the near future. I do appreciate that you are one of two now who has at least approached the black swan.
Good on you Peter. Yet within your refutation you are somewhat declaring that you can approach the black swan no further. Still, I believe I have enough to falsify you, and that is exactly as it should be on my own language if I am wrong. You are attempting this so I do see this as a potentially productive debate. Thank you for joining in. Best of all with careful language our statements here stand freely and if they are flawed the flaw can be amplified. I will try not to dodge as others do here. Please do provide amplification on my own errors as aggressively as possible. Of course simplicity is helpful, but then too variations matter I think. I've settled onto a concrete instance of
1.23 X
to study and I can easily back off of this but I do suspect this is the simplest of forms to study. There are two operands and one operator right?
which is not ring behaved since the product is not a binary
operation. This is the conflict in its entirety, though the
extensions go to considering
a0 + a1 X
since this addition operation is not binary and so is undefined.
a0 +(that's the ring addition) a1 X = a0 + a1 X
is an example of two ring element a0 and a1 X adding to make one ring
element.
Then how come you now have three elements with two operators?
This goes in contradiction to the ring definition. It states that two elements will resolve to one element.
I do appreciate what you are saying and I do see that you are obeying the standard interpretation.
Can we please specify real coefficients. Will this be alright with you?
Yes. I would it be alright with you for you to stop using meaningless
phrases such as
the ring definition [...] states that two elements will
resolve to one element.
Post by Tim Golden BandTech.com
In other words in the expression above a0 and a1 are both real valued.
The notion of an element as a singular thing seems such a fundamental concept and yet possibly this ray of argumentation that you are on here will lead us down into set theory fundamentals. I do suspect that if elements are composed of multiple elements then they are not elementary. In other words your argument above is false. And so in covering this ground you in effect are falsifying abstract algebra under this interpretation.
Of course it starts to feel a bit silly right? We are being such math weenies here. Yet we have whittled down now to a fairly simple and by the way alternative position on elements in set theory. I do still believe that one of the real elements can be
1.23
Your phrase "ring behaved polynomial" is odd.
It suggests to me that you think if R[X] is a ring
then any element of R[X] is a ring, which is potty.
The real numbers constitute a ring, 4.5 doesn't.
I have to admit that I just did a google search on '"ring behaved" polynomial' and get only 54 results so I accept your criticism though I don't see any real abuse of language here. Logically I am finding that the supposed ring behaved polynomial is not ring behaved. I suppose some would simply say 'is a ring' or 'is not a ring' and you are on the side of 'is a ring' and I am on the side of 'is not a ring' But isn't your dialect above really explaining why I use this terminology? The fact that for real values
( 0.01) ( 1.23 ) = 0.00123
is an expression that is consistent with the ring definition and so is ring behaved
There you go again. How about sticking to the language which one finds
in algebra texts? Have you ever read one?
We are not in algebra per se. We are in the curriculum of abstract algebra. I have no idea how you could come to disagree with this language particularly when you find no actual fault with the language. I have linked to numerous online references here; not as numerous as Lalo though. It is almost as if you want to deny that the real numbers do fit the ring definition. We are simply multiplying two real values above. Why are you in denial of this simplistic instance? You need to understand Peter that abstract algebra is radically different from ordinary algebra that we learn in high school. Your request that I stay within the language of some abstract algebra text will inherently deny falsification won't it? They are in the business of propagating this topic. Possibly you are misunderstanding the nature of the polynomial within AA. It is not at all the polynomial that we learn in high school algebra where x takes real values. That polynomial in real x is fine. The AA version is not so fine. So again, when I write:
( 0.01) ( 1.23 ) = 0.00123
where all these values are real numbers we have a concrete instance of multiplication that satisfies the ring definition. These are elements in the reals that I've written above. Again I will source for you the definition of a binary operator, which is required by ring definitions:
https://en.wikipedia.org/wiki/Binary_operation
and particularly
https://en.wikipedia.org/wiki/Closure_(mathematics)
which isn't really strongly stressed on wikipedia, but is in many renditions incuded right in the ring requirements. Possibly of importance for you to understand what is going on:
"The closure axiom is already implied by the condition that +/• be a binary operation. Some authors therefore omit this axiom. "
- https://en.wikipedia.org/wiki/Ring_(mathematics)#ref_c
By definition the product
1.23 X
simply cannot satisfy the closure requirement. It is a one line falsification. Yes, I can expound and take a few alternate forms, but really this crux is about all that there is to my argument. This black swan is so simple. I show you the black swan and you run away and hide in a cage. I will not lock the door. But it is up to you to step out and face the black swan.

We have to allow for the long form sum of the polynomial
a + b + c + d + e + f + ...
and witness that while there are umpteen operators and terms present in this expression it is implied that they all satisfy the closure requirement; otherwise they would break the ring requirements. In order to dismantle the polynomial; something that the AA texts fail to do; we simply need to study a few of these terms. One of these terms really is a fine instance of the black swan that falsifies the abstract algebra polynomial with real coefficients. Once again up on your head with its ass ready to dump on you:
1.23 X

s x
where s is sign and x is magnitude. What results? Quite a bit, and it actually overlaps with abstract algebra. But you see because I dismantled the real value and found its substructure; its elemental nature; the real number itself is no longer fundamental for me. Indeed the complex numbers come right along as three-signed numbers with no additional rules required. The real numbers are no more fundamental than are the complex numbers. They exist side by side as members in a family of number systems
P1, P2, P3, P4, ...
These systems inherently demand their dimensional geometry without orthogonality and without any need of the Cartesian product. They stand freely and primitively without any need for additional language. Through polysign numbers I was put onto abstract algebra. The bizarre stage of the polynomial development and its awkward quotient and ideal is not at all something to be proud of. Some texts even go into defense mode prior to bringing these details up. It is very clear that through all of the contorted language the polynomial is an infinite dimensional construction and by forcing modulo behavior on it reduced dimension is gotten. In effect they celebrate building a two dimensional complex value out of an infinite dimensional series. Great. All the while sitting on top of the pristine ring definition... which they completely obliterated when they built the polynomial. They are all telling you lies. The entire field is a bunch of PhDs looking for financial compensation. The room to publish must expand exponentially right along with the human population. This is the academic system that has brought you abstract algebra and which will continue to uphold it as if it has great integrity. To what degree is the mathematics community authentically brainwashed and to what degree are the highest performing mimics inherently susceptible to brainwashing? This level of philosophy I actually do arrive at through my work. We are humans doing mathematics. We cannot write ourselves a free ticket out of humanity. The very separation of the subject matter out into physics, philosophy, and mathematics; the queen; may merely be a matter of making room for more PhDs. That procedure has gone on and on now. These divides are false. These subjects are one. After all of your escapism into mathematics when you step outside the door and witness the world in amazement and ponder the weakness of the human condition then you will have arrived at a truth that is far beneath the subject at hand. We are the greatest mimics on the planet, and as a result we struggle very hard to find the truth.
Post by Tim Golden BandTech.com
though clearly these are elements of the set and not the set of reals itself, which is the ring. Oddly enough we are right back to this sticking point of elemental quality within a set. Had I stated that
( 0.01) ( 1.23 ) = #0.00123
where the # indicates a value that is outside of the set of real numbers, then we would say that this expression violates the ring requirements so long as we still kept the declaration of real valued antecedents. I would say then that this is not a ring behaved expression. I do not care to engage is more highfaluting language which no doubt exists for such concepts but lacks instantiable types. We do know the real value well and we do know that the curriculum of abstract algebra does make use of the polynomial with real coefficients. I do point out here the conflict of this construction and particularly claims that these are ring behaved.
Peter you are the strongest respondent to this thread. I appreciate that you are taking this on and that you are taking me seriously.
What makes you think that?
Post by Tim Golden BandTech.com
In effect you expose better than I the weakness of the other responses here. The informational concepts ought to be free-standing regardless of these human interactions that involve personal slights and geese crapping on people's noses and so forth. Please feel free to throw in a few yourself since you are on usenet here. Poor Mostowski still can't smell it and it's been days. I don't think he sees it either. Just let it dry and scrape it off Mostowski. Any day now.
Post by Peter
If you read a book on algebra you'll see a proof that the set of
polynomials with one indeterminate (here X) and coefficients from a ring
is itself a ring under addition and multiplication of polynomials.
Lalo T.
2020-09-06 21:01:29 UTC
In the context of your initial thesis, it is not possible ( that you, with compiler integrity quality standard,
pointing to facts, and we also pointing to facts) the existence of opposition. If it is possible, then the playground is not accurate.

What I mean by "Language Issues" is, for example, in :
https://en.wikipedia.org/wiki/Ring_(mathematics)#Definition
in the phrase "A ring is a set R" the symbol 'R' is not for real numbers (just an example)

I take for granted that you already distinguish operations of the rings from how its elements are
constituted. Either way, that is not the core issue.

I can not use the three-word forbidden concept starting with "Exte... ".
Otherwise I would get caught in your "clopenness" bait and I would disrespect
your R-BO-C (Ring-BinaryOperation-Closure )

I'm requesting your examination of three points :

(1) The polynomial ring example (non-evaluated, not mixing)

(2) The matricial example (evaluated, mixing and not mixing)

(3) The context where you originally pick your model example (???)

Let's focus more on (3)

" How is it that I am the only one here who does go to the trouble of this sort of language? "

I guess that you pick your model example from somewhere. This does not appear out of the blue.
Tim, you mentioned that through Polysigns you saw this issue. Bring to the table the EXACT
POINT that inspire your questioning, an the discussion will move towards there...
Also, you must point out, why do I can, or can not, mix reals in your set, and details related.

An arbitrary question would be : Is the case that while examining the relationship between
RxC and P4, you were requesting your R-BO-C, without mentioning the forbidden concept ?

https://en.wikipedia.org/wiki/Module_(mathematics)
note " A module over a ring is a generalization of the notion of VECTOR SPACE over a field "
https://en.wikipedia.org/wiki/Vector_space#History
https://en.wikipedia.org/wiki/Linear_map

"As for a resolution as you are working towards above; that is good, but as to the initial falsification"
Presumably those two points are different. Maybe the other topic if just off-topic. Maybe yes, maybe no...
"Can we dispense with scalar multiplication when it is possible ?" is presumably off-topic.

Well, frisky reptilians or not, I have not stated that the refutation of your falsification have been performed.
Current options :

(a) thesis is true

(b) thesis is false

(c) non-falsifiable in a boring way

(d) non-falsifiable in an interesting way

I would not further define (c) and (d), or I 'll get into trouble.

"In the philosophy of science, falsifiability or refutability is the capacity for a statement, theory or
hypothesis to be contradicted by evidence. For example, the statement "All swans are white" is falsifiable
because one can observe that black swans exist." ( from : https://en.wikipedia.org/wiki/Falsifiability )

It would be not accurate on my side to say "I don't agree with you Tim".
I would like to say that "I do not agree with you and you are not accurate".
But first, to disagree (or agree) I have check the place where the complaint is originated.
Maybe the place has not issues and all is fine, ok, but the thing is the place is under examination.
So far, suspicion on (d). The only certainty that I have, is that something is going on...

These "two" oddities (although of different nature) pretty near of each other seem to indicate
the presence of a wandering Strange Loop in the surrounding area. Before pressing more on the
last point, state your new conditions, or I will give up and I ll be absorved by the "proof by vacuum".
...and wing-beaten
Peter
2020-09-01 11:24:10 UTC
Post by Tim Golden BandTech.com
After the careful construction of the ring from two groups,
You seem not to know what a ring is.
Tim Golden BandTech.com
2020-09-01 11:51:52 UTC
Post by Peter
Post by Tim Golden BandTech.com
After the careful construction of the ring from two groups,
You seem not to know what a ring is.
U R A lamb so long as you post no content.
Awaiting Falsification.
zelos...@gmail.com
2020-09-01 11:27:32 UTC
Post by Tim Golden BandTech.com
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
It distinctly disobeys the closure requirement
You might want to learn the more formal definition because the whole shebang about X is an artifact of history more than anything formal in abstract algebra. let (G,0, + ) be a commutative monoid, adn R a ring, then we define the polynomials to be R[G] to be the set {x e R^G: n e G & x(n)=0 for almost all.}, define addition as (x+y)(n)=x(n)+y(n) and multiplication as (x*y)(n)=sum_{i=0}^n x(i)*y(m), where i+m=n.

So it works perfectly fine, it is closed and satisfies the ring axioms.

the whole thing about X and all is just an artifact of history, and in our usual case, G is the additive monoid of natural numbers.
Tim Golden BandTech.com
2020-09-01 12:11:23 UTC
Post by ***@gmail.com
Post by Tim Golden BandTech.com
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
It distinctly disobeys the closure requirement
You might want to learn the more formal definition because the whole shebang about X is an artifact of history more than anything formal in abstract algebra. let (G,0, + ) be a commutative monoid, adn R a ring, then we define the polynomials to be R[G] to be the set {x e R^G: n e G & x(n)=0 for almost all.}, define addition as (x+y)(n)=x(n)+y(n) and multiplication as (x*y)(n)=sum_{i=0}^n x(i)*y(m), where i+m=n.
So it works perfectly fine, it is closed and satisfies the ring axioms.
the whole thing about X and all is just an artifact of history, and in our usual case, G is the additive monoid of natural numbers.
Sorry, no, the ring definition does not include an additional argument (n). The ring operators are quite simple and well established. If effect here you are also actually admitting that my falsification does hold, since you have not falsified it. You have failed to address closure and the notion of a binary operator. It looks as if you have established a trinary operator, but you've waived your hands and raised the complexity in order to avoid the conflict. How about we look within your definition at
a1 X
and see what sort of operation this is. Shall we say that n = 1 here? What sort of operator is this? Then let's have a look at
a0 + a1 X
You see the problem remains. Are you saying that abstract algebra is historically conflicted? That the works that do not use your notation are bad?
In effect then I suppose we might be in agreement. I'm not really so sure, but I would like to understand the distinction that you make better. It sounds a bit like when complex numbers stopped using sqrt(-1) to develop i and just went ahead and defined a two dimensional product and sum. It so happens that my own fix to these things develops the complex numbers with the same rules that develop the real number and so all of these constructions are needless as the system works in general dimension. The modulo behaved systems already have their kernel within the sign of the real value. It just needs generalization. But that is polysign and this is a falsification of abstract algebra. You've pivoted, but in your pivot you are admitting the weakness that most deny. I wish you could expound and bring it back down to the simplest of instances that I've put under your nose here. I really doubt that more fancy language is going to explain away the problem. I do think that this is an open problem. All problems are ultimately. They have to be. And this failing is a fine instance of why this is the case.
zelos...@gmail.com
2020-09-02 05:29:13 UTC
Post by Tim Golden BandTech.com
Sorry, no, the ring definition does not include an additional argument (n)
I didn't say that rings do, I said for that specific one :)
Post by Tim Golden BandTech.com
The ring operators are quite simple and well established
It all depends on the ring and "simplicity" again depends on teh ring, we use a different definition for polynomial rings. There is no "the operations" for rings. Only that there is AN operation we call addition and AN operation that we cann multiplication.
Post by Tim Golden BandTech.com
f effect here you are also actually admitting that my falsification does hold, since you have not falsified it
I point out that your complain is based on a misunderstanding because in formal definition, the whole X is nothing but notation.
Post by Tim Golden BandTech.com
You have failed to address closure and the notion of a binary operator.
I actually did adress it, I gave you the two binary operators for polynomial, defined them and you can see they are closed.
Post by Tim Golden BandTech.com
It looks as if you have established a trinary operator, but you've waived your hands and raised the complexity in order to avoid the conflict
I haven't made a ternary one, you just don't understand how it is formalyl done.
Post by Tim Golden BandTech.com
and see what sort of operation this is.
Thats no operation, it is notation. A polynomial is a function of |R^|N such that for almost all n e |N, p(n)=0.
Post by Tim Golden BandTech.com
You see the problem remains.
As I pointed out, thats not the formal definition, I gave you the fucking formal definition. Why did you not read it?
Post by Tim Golden BandTech.com
Are you saying that abstract algebra is historically conflicted? That the works that do not use your notation are bad?
What I am saying is that the formal definition of polynomials do not deal with X, unknowns or anything of the like. the usage of X is an artifact of history where it started.
Post by Tim Golden BandTech.com
It sounds a bit like when complex numbers stopped using sqrt(-1) to develop i and just went ahead and defined a two dimensional product and sum.
Complex numbers didn't have a proper formal construction initially, but they eventually did get it.
Post by Tim Golden BandTech.com
You've pivoted, but in your pivot you are admitting the weakness that most deny.
There is no weakness here, there is only you not understanding the differens between notation based on history vs formal abstract algebra definition/construction.
Post by Tim Golden BandTech.com
I really doubt that more fancy language is going to explain away the problem.
I use precise correct language, nothing "fancy". However you calling it fancy does indicate you do not know what it is saying.
Post by Tim Golden BandTech.com
I do think that this is an open problem. All problems are ultimately. They have to be. And this failing is a fine instance of why this is the case.
You might think it but there isn't.

"a_i X, what is the operation?" is not a problem question because it is just NOTATION, there is no operation there. Nothing being done. It is just a different way of writing (0,a_i,0,0,0,...)
which is the formal construction.
Tim Golden BandTech.com
2020-09-02 11:13:28 UTC
Post by ***@gmail.com
Post by Tim Golden BandTech.com
Sorry, no, the ring definition does not include an additional argument (n)
I didn't say that rings do, I said for that specific one :)
Post by Tim Golden BandTech.com
The ring operators are quite simple and well established
It all depends on the ring and "simplicity" again depends on teh ring, we use a different definition for polynomial rings. There is no "the operations" for rings. Only that there is AN operation we call addition and AN operation that we cann multiplication.
Sorry, but 'the operations' are very carefully defined within the ring definition. They are definitely binary operators, and when we have algebraically behaved systems they typically fit it quite well. These operators are explicitly defined to take two elements from a set and yield an element in that same set.

"In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. " - https://en.wikipedia.org/wiki/Ring_(mathematics)

I have no idea why you would wheedle here. Indeed without your ordered notation you are essentially admitting that my claim is accurate. Furthermore the ordered notation has not replaced the traditional notation. Possibly it is in transition, but by no means am I discussing antiquated notation. That you push for this interpretation is indicative of the accuracy of my falsification, which continues to go unaddressed within your own interpretation. Therefor I have to place your own language, no matter how authentically you have written it, into the category of a dodge rather than a direct attack onto my falsification. In taking this step you have essentially built support for my argument though it is indirect.

As we discuss things at this fundamental level of detail to claim that something is 'just notation' is not at all a fair defense. If our notation is conflicted then an attack on notation is of importance. This is after all mathematics that we are discussing right? I do agree generally with your interpretation of the meaning of X. In my own language I am happy to see it as a dimensional placeholder. To then implement it into the ring operators as
a0 + a1 X + a2 X X + a3 X X X + ...
ought to be as offensive to you as it is to me. In effect you are admitting this though it is as if you whisper it here, all the while saying 'NO' with plenty of lung.
Post by ***@gmail.com
Post by Tim Golden BandTech.com
f effect here you are also actually admitting that my falsification does hold, since you have not falsified it
I point out that your complain is based on a misunderstanding because in formal definition, the whole X is nothing but notation.
Post by Tim Golden BandTech.com
You have failed to address closure and the notion of a binary operator.
I actually did adress it, I gave you the two binary operators for polynomial, defined them and you can see they are closed.
Post by Tim Golden BandTech.com
It looks as if you have established a trinary operator, but you've waived your hands and raised the complexity in order to avoid the conflict
I haven't made a ternary one, you just don't understand how it is formalyl done.
Post by Tim Golden BandTech.com
and see what sort of operation this is.
Thats no operation, it is notation. A polynomial is a function of |R^|N such that for almost all n e |N, p(n)=0.
Post by Tim Golden BandTech.com
You see the problem remains.
As I pointed out, thats not the formal definition, I gave you the fucking formal definition. Why did you not read it?
Post by Tim Golden BandTech.com
Are you saying that abstract algebra is historically conflicted? That the works that do not use your notation are bad?
What I am saying is that the formal definition of polynomials do not deal with X, unknowns or anything of the like. the usage of X is an artifact of history where it started.
Post by Tim Golden BandTech.com
It sounds a bit like when complex numbers stopped using sqrt(-1) to develop i and just went ahead and defined a two dimensional product and sum.
Complex numbers didn't have a proper formal construction initially, but they eventually did get it.
Post by Tim Golden BandTech.com
You've pivoted, but in your pivot you are admitting the weakness that most deny.
There is no weakness here, there is only you not understanding the differens between notation based on history vs formal abstract algebra definition/construction.
Post by Tim Golden BandTech.com
I really doubt that more fancy language is going to explain away the problem.
I use precise correct language, nothing "fancy". However you calling it fancy does indicate you do not know what it is saying.
Post by Tim Golden BandTech.com
I do think that this is an open problem. All problems are ultimately. They have to be. And this failing is a fine instance of why this is the case.
You might think it but there isn't.
"a_i X, what is the operation?" is not a problem question because it is just NOTATION, there is no operation there. Nothing being done. It is just a different way of writing (0,a_i,0,0,0,...)
which is the formal construction.
zelos...@gmail.com
2020-09-02 13:40:38 UTC
Post by Tim Golden BandTech.com
Sorry, but 'the operations' are very carefully defined within the ring definition
No, they aren't. They are just stated to have the distirbutive relation and that addition is commutative.
Post by Tim Golden BandTech.com
They are definitely binary operators
Yes, but what operation it is does not matter as long as its binary and sates said qualities.
Post by Tim Golden BandTech.com
These operators are explicitly defined to take two elements from a set and yield an element in that same set.
Yes, but one ring has two operands and another has two different ones.
Post by Tim Golden BandTech.com
Furthermore the ordered notation has not replaced the traditional notation.
It is a sequence or infinite cartesian product which are the same essentially. And yes, it hasn't replaced it due to history, so fucking what?
Post by Tim Golden BandTech.com
Possibly it is in transition, but by no means am I discussing antiquated notation.
It won't and you ARE! your complaint about a_1 X is complaint about notation!
Post by Tim Golden BandTech.com
That you push for this interpretation is indicative of the accuracy of my falsification
No, it is demonstrating you are complaining about notation, not formal construction which makes your point moot.
Post by Tim Golden BandTech.com
Therefor I have to place your own language, no matter how authentically you have written it, into the category of a dodge rather than a direct attack onto my falsification. In taking this step you have essentially built support for my argument though it is indirect.
No, it is you being ignorant.

Your complaint about a_i X is INVALID because it is about NOTATION and not formal definition.
Post by Tim Golden BandTech.com
As we discuss things at this fundamental level of detail to claim that something is 'just notation' is not at all a fair defense.
When your complaint is about ntoation, saying it is just notation IS a method to show your complaint is invalid.
Post by Tim Golden BandTech.com
If our notation is conflicted then an attack on notation is of importance.
Notation is important but this is a historical artifact and it doesn't matter ultimately if you have some intelligence.
Post by Tim Golden BandTech.com
This is after all mathematics that we are discussing right? I do agree generally with your interpretation of the meaning of X. In my own language I am happy to see it as a dimensional placeholder. To then implement it into the ring operators as
a0 + a1 X + a2 X X + a3 X X X + ...
Post by Tim Golden BandTech.com
ought to be as offensive to you as it is to me. In effect you are admitting this though it is as if you whisper it here, all the while saying 'NO' with plenty of lung.
It is just notation, there is nothing offensive about it.
Ross A. Finlayson
2020-09-02 15:59:30 UTC
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.

I'm sure the modular arithmetic and abstract algebra is familiar to everybody.

I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.

Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.

In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.

"To formalize this argument

we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.

This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.

Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.

The entire polynomials fails under their own formalities. "

Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)

I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.

Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)

So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.

Just more, though - not different.

Thanks, it's OK.
Mostowski Collapse
2020-09-02 16:10:54 UTC
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.

Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.

I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
Mostowski Collapse
2020-09-02 16:21:51 UTC
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it

must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.

Take disjunction and negation defined as:

x v y := x + y - x*y

~x := 1 - x

Then take this tautology:

x v ~x = x + (1 - x) - x*(1 - x)

= 1 - x + x^2

= 1 - x + x

= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
Lalo T.
2020-09-02 18:19:58 UTC
Ok, manage the email thing. I 'am curious to how far the reasoning can be extended.

With a bit of luck this will not be a lumpy rug
http://rugs.droogkast.com/lumpy-rug/

You seem more centered in the Polynomial Ring concept and Ring of Polynomial Functions concept (in regarding to the Ring concept)

https://en.wikipedia.org/wiki/Scalar_multiplication#Interpretation
https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations
https://en.wikipedia.org/wiki/External_(mathematics)

https://commalg.subwiki.org/wiki/Polynomial_ring
https://en.wikipedia.org/wiki/Polynomial#Polynomial_functions
https://en.wikipedia.org/wiki/Ring_of_polynomial_functions

https://encyclopediaofmath.org/wiki/Unital_ring

I will try to do another approach (in the case of matrices)

1. a₀·1 + a₁·x + a₂·x² + ... + aₙ₋₁·xⁿ⁻¹ + aₙ·xⁿ = 0

2. A₀·1 + A₁·x + A₂·x² + ... + Aₙ₋₁·xⁿ⁻¹ + Aₙ·xⁿ = O

3. a₀·I + a₁·X + a₂·X² + ... + aₙ₋₁·Xⁿ⁻¹ + aₙ·Xⁿ = O

4. A₀*I + A₁*X + A₂*X² + ... + Aₙ₋₁*Xⁿ⁻¹ + Aₙ*Xⁿ = O

https://en.wikipedia.org/wiki/Scalar_multiplication#Scalar_multiplication_of_matrices
https://en.wikipedia.org/wiki/Diagonal_matrix#Scalar_matrix
https://en.wikipedia.org/wiki/Center_(ring_theory)

Do you believe, looking points (a) and (b), that the scalar multiplication of matrices is non-primitive as operation, and the scalar multiplication of matrices is, indeed, equivalent to use the usual matrix multiplication with "scalar matrices" ? (as if the scalar product were a non-essential operation (since you consider it as an alien operation) )

(a) λ₀·I + λ₁·X + λ₂·X² + ... + λₙ₋₁·Xⁿ⁻¹ + λₙ·Xⁿ = O where '·' is scalar product

equivalent to :

(b) A₀*I + A₁*X + A₂*X² + ... + Aₙ₋₁*Xⁿ⁻¹ + Aₙ*Xⁿ = O where '*' is matrix product

???

Note that instead of use the scalar product ' λ·I = A '
to define our "Scalar Matrices" A (the coefficients in (b) ), we directly define the elements of the Scalar Matrices A in (b)

A = [aₕₖ]ₘₓₘ :
a matrix A of dimensions mxm (m times m), subscript h indicates the row and subscript k indicates the column

where an element aₕₖ = λ if h = k , and aₕₖ = 0 if h =/= k

Would it be fine if we just kick out the scalar product in the case of matrices, and only use matrix multiplication ?

(...And in this respect, the same question for any structure that use scalar multiplication? )
Lalo T.
2020-09-02 21:30:44 UTC
https://hsm.stackexchange.com/questions/11235/who-started-calling-the-matrix-multiplication-multiplication
The vector algebra war: a historical perspective https://arxiv.org/abs/1509.00501v2

Tim, if you requesting a revision of the concept of scalar, with respect to his interaction/relationship with other algebraic structures...

https://en.wikipedia.org/wiki/Scalar_(mathematics)#Etymology
https://en.wikipedia.org/wiki/Scalar_multiplication
https://en.wikipedia.org/wiki/Scaling_(geometry)

...we will end up re-examining and inspecting stuff from mid-19th century or before.

I suppose you think in the context of an abstract X, and the interaction a2 with X² in 'a2 X X'
You indeed will not consider too problematic if we say ' a2*X*X ', but you will consider non well-formulated the statement ' a2·(X*X) '
If somehow, is always the case, or in a number of cases (since the '·' operation seems to have an independent status) that for some
unknown reason, the '·' operation can be reduced to merely the '*' operation. Would the conflict disappear under your quality standard of compiler integrity ?

cheers
Mostowski Collapse
2020-09-03 10:29:20 UTC
Boole also introduced, it can be justified:

∀P^A == A[P/0]*A[P/1]

Its a bold step since Boolean algebras can have more than
2 elements. Not sure whether Quine was aware about this.
Post by Mostowski Collapse
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it
must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.
x v y := x + y - x*y
~x := 1 - x
x v ~x = x + (1 - x) - x*(1 - x)
= 1 - x + x^2
= 1 - x + x
= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
Mostowski Collapse
2020-09-03 10:31:35 UTC
It also shows that the fundamental theorem
does not hold in Boolean algebras in general.

For a binary Boolean algebra this here has
two roots x=0 and x=1:

x*(1-x) = 0

But in a Boolean algebra with more than two
values, it has much more roots.
Post by Mostowski Collapse
∀P^A == A[P/0]*A[P/1]
Its a bold step since Boolean algebras can have more than
2 elements. Not sure whether Quine was aware about this.
Post by Mostowski Collapse
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it
must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.
x v y := x + y - x*y
~x := 1 - x
x v ~x = x + (1 - x) - x*(1 - x)
= 1 - x + x^2
= 1 - x + x
= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
Ross A. Finlayson
2020-09-04 15:56:34 UTC
Post by Mostowski Collapse
It also shows that the fundamental theorem
does not hold in Boolean algebras in general.
For a binary Boolean algebra this here has
x*(1-x) = 0
But in a Boolean algebra with more than two
values, it has much more roots.
Post by Mostowski Collapse
∀P^A == A[P/0]*A[P/1]
Its a bold step since Boolean algebras can have more than
2 elements. Not sure whether Quine was aware about this.
Post by Mostowski Collapse
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it
must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.
x v y := x + y - x*y
~x := 1 - x
x v ~x = x + (1 - x) - x*(1 - x)
= 1 - x + x^2
= 1 - x + x
= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
"It has much more roots."
Lalo T.
2020-09-04 23:13:13 UTC
"...polynomial evaluated at .."

Nice to see that the issue has some leverage. Ok Tim, let me add to the pile...

I wonder if this debate can be effectively solved in the realm of Abstract Algebra when Vector Spaces are already present.
My bet is that this conflict can not be solved (or dissolved) in this context (solutions mainly using abstract algebra).
Maybe resorting to other foundational areas. I'm open to be corrected, in case this issue can effectively be solved inside
Abstract Algebra, in which case, I will retract my words.

I bet that our apparent black swan, it is being victim of an ultra-deformulation mechanism, originated by the bullets of an exotic handgun
being used by a frisky reptilian, with the unwanted effect of a misbehavior on the side of the photons lying in the vicinity of our swan.
Of course, this situation being on top of the misbehaviour of our already bad-tempered swan. I think that will be enough some strikes with the mace,
to our frisky friend, not too rough (exercise some tough love) to indicate that we will not tolerate pranks at those levels. Not too hard also,
because animals have rights, and, undercover agents of pro-animal organizations are attentive to this kind of situations, ready to jump...

https://www.flickr.com/photos/elsie/3181428441/
https://www.dailymail.co.uk/news/article-2154839/Black-swan-stands-gatecrashing-group-600-white-ones-ancient-swannery.html

Events with swans have been reported: https://www.youtube.com/watch?v=2OJA2Tw5bzg

Since my intention now is to state that it is not the case that :

(1) "the falsification of abstract algebra" is true

(2) "this conflict is solvable in the usual abstract algebra" is true

(3) "a0 I + a1 X + a2 X X + .. is ill-formed " is true

( where it is not my intention : https://en.wikipedia.org/wiki/Minority_Report_(film) )

...then my own position here would be stepping out of Post-Scalar-Multiplication Abstract Algebra, for the sake of preserving
the structural coherence of the building, or alternatively, look for some Scalar-Free building, in order to make the refutability, or even the formulation of your statement impossible,
although, at an ridiculously expensive price, of course, since I state that the conflict is pointing to scalars and vector spaces.

If I am quoting well, Roger B. refer to "indexed vs ordered".

For the sake of diversity on this dispute over the ring, here is missing the words of wisdom of a true veteran in rings, namely, "The Lord of the Rings", to share his grasp of the issue,
and of course, give his/her two cents on it.

Consider write, in the future, an epic novel about swans, rings and frisky reptilians, and allocate part of the profits to pro-animal organizations (in case you wish).

If we were is a maze, in a competition where only one can get out, and weapons are provided, before proceed to the combat, better check first if the walls can be broken, with the possibililty
of finding a tunnel.

While restricting to the three concepts that you use in your initial post, to some extent I give you the benefit of doubt, but not, at the "Abstract Algebra Falsification" part...
Tim Golden BandTech.com
2020-09-05 14:59:40 UTC
Post by Lalo T.
"...polynomial evaluated at .."
Nice to see that the issue has some leverage. Ok Tim, let me add to the pile...
I wonder if this debate can be effectively solved in the realm of Abstract Algebra when Vector Spaces are already present.
My bet is that this conflict can not be solved (or dissolved) in this context (solutions mainly using abstract algebra).
Maybe resorting to other foundational areas. I'm open to be corrected, in case this issue can effectively be solved inside
Abstract Algebra, in which case, I will retract my words.
I bet that our apparent black swan, it is being victim of an ultra-deformulation mechanism, originated by the bullets of an exotic handgun
being used by a frisky reptilian, with the unwanted effect of a misbehavior on the side of the photons lying in the vicinity of our swan.
Of course, this situation being on top of the misbehaviour of our already bad-tempered swan. I think that will be enough some strikes with the mace,
to our frisky friend, not too rough (exercise some tough love) to indicate that we will not tolerate pranks at those levels. Not too hard also,
because animals have rights, and, undercover agents of pro-animal organizations are attentive to this kind of situations, ready to jump...
https://www.wikihow.com/images/9/9a/Color-Step-7-32.jpg
http://pixeljoint.com/files/icons/full/bicho_lanza.gif
https://www.flickr.com/photos/elsie/3181428441/
https://www.dailymail.co.uk/news/article-2154839/Black-swan-stands-gatecrashing-group-600-white-ones-ancient-swannery.html
Events with swans have been reported: https://www.youtube.com/watch?v=2OJA2Tw5bzg
(1) "the falsification of abstract algebra" is true
(2) "this conflict is solvable in the usual abstract algebra" is true
(3) "a0 I + a1 X + a2 X X + .. is ill-formed " is true
( where it is not my intention : https://en.wikipedia.org/wiki/Minority_Report_(film) )
...then my own position here would be stepping out of Post-Scalar-Multiplication Abstract Algebra, for the sake of preserving
the structural coherence of the building, or alternatively, look for some Scalar-Free building, in order to make the refutability, or even the formulation of your statement impossible,
although, at an ridiculously expensive price, of course, since I state that the conflict is pointing to scalars and vector spaces.
If I am quoting well, Roger B. refer to "indexed vs ordered".
For the sake of diversity on this dispute over the ring, here is missing the words of wisdom of a true veteran in rings, namely, "The Lord of the Rings", to share his grasp of the issue,
and of course, give his/her two cents on it.
Consider write, in the future, an epic novel about swans, rings and frisky reptilians, and allocate part of the profits to pro-animal organizations (in case you wish).
If we were is a maze, in a competition where only one can get out, and weapons are provided, before proceed to the combat, better check first if the walls can be broken, with the possibililty
of finding a tunnel.
While restricting to the three concepts that you use in your initial post, to some extent I give you the benefit of doubt, but not, at the "Abstract Algebra Falsification" part...
Nice piece Lalo. I see some here are like
If I could just find one for Mostowski...
I can't stand all the reptilean theories going around, but it is well proven that we mammals do still have reptillian brain components. Supposedly the fight or flee instinct is embedded there. The amygdala reaction...

It does seem to beg the question of what if Tim is accurate here... what are the consequences? This math was an attempt at a pristine formulation. Clearly it is not pristine. That much is already admitted by others here though they can only whisper it under their breath. Minor abuse; historical notation; these are not my own terms. My terms are considerably stronger. Falsification as a practice seems quite foreign to these users of sci.math. Can it be done so quickly and so shortly as uttering " the polynomial with real coefficients 1.23 X cannot be ring behaved"

Obviously coverage of this claim will have to consider what the elements of the operator are. The ring requires elements in the same set and a singular result which belongs in that same set. If this operator does not fit this requirement then it obviously has offended the ring definition. How is it that I am the only one here who does go to the trouble of this sort of language? The rest are lost up in the high lands of the infinite polynomial. Is it really wise what they are doing? When their infinite construction is composed of a sum that is obeying (supposedly) the ring definition and whose products (supposedly) obey the ring definition? No. We go to the elemental stage and study these things. A conflict is found and the dodge begins all over again. As for a resolution as you are working towards above; that is good, but as to the initial falsification: this part is so important that it has to be established on its own terms and in the simplest language possible. Of course then things do fall apart and we can expand out the attack, but the initial attack has to be down in there in the fundamental and kept quite simple I think. As far as I can tell we can firstly falsify the multiplicative operator with
1.23 X
and then we can go on to falsify the additive operator with
0.01 + 1.23 X
and from there we pretty well have collapsed the usage of the polynomial with real coefficients as a farce which has been pushed onto numerous disbelieving students who for the sake of an A must mimic on and on and on no matter how bad the basis is; at their own cost to boot. Even just the text book fee alone makes it a bad deal. But then pay the guy who shoves this crap down their throats... well, you might say; that sounds a bit harsh. But if I am correct than all of this rhetoric holds up just fine and the guy who came before that guy who shoved the white goose shit down his throat and so on. Ahhh... back to a time when possibly this subject was treated as an open problem... should we remain there at treating these sorts of systems as an open problem then I would say that the maturity of the subject at hand could be established. Instead it is treated as a closed book. One must preserve the book...
Tim Golden BandTech.com
2020-09-03 13:22:33 UTC
Post by Mostowski Collapse
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it
must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.
x v y := x + y - x*y
~x := 1 - x
Well, I do perk up a little bit when you mention negation here, but since you define it using a minus its original form is arguably still present.
Still for boolean systems it squares with logic, yet that zero would be the inverse of one does not take much extensive meaning beyond that.
Doesn't this detail really expose the uniqueness of the boolean system? All that notation for such a simple form... It really doesn't seem right.
Still, I have weighed in on it, and now I'd like you to weigh in on this thread Mostowski Collapse.

Perhaps just one more variation should be played out on the polynomial of abstract algebra. The polynomial
a(0) + a(1) X a(2) X X + a(3) X X X + ...
is claimed to be ring behaved for all a(n). Normally mathematicians stay in this infinite form for quite a bit longer before they do any work. The sum and the product of multiple polynomials expose that they still fit this polynomial form (though the product and its indexing is fishy). At this point they feel that they have established the ring behavior of the expression. At some later point they will introduce the polynomial ring with real coefficients. Thus the form that I speak of is in use in the curriculum.

I'm stretching this out a bit since everyone here wants to dodge the simplest detail. A falsification is a falsification, no matter how much icing you throw on top of the cake. Still I can go to a simpler form and that is the variation that I will try here. We've selected real coefficients for the polynomial and still believe it to be ring behaved. Now we select the real coefficients. This would be the first concrete instance of a polynomial; a stage which the ordinary curriculum skips right over; no different than many 'high' mathematics courses do of their own constructions, if indeed any instance can ever be found. So let's really cement that polynomial to
0 + 1.23 X + 0 X X + 0 X X X + 0...
which is a fairly simple instance and now the supposed ring expression becomes
1.23 X
where 1.23 is a real value and X is not a real value. Therefore this first concrete instance of the polynomial form is not ring behaved, as it conflicts with the ring operator definitions. It fails to meet the closure requirement under product. This is a formal falsification. The failure of others here to meet this falsification head on is unacceptable. The math is not general if it cannot withstand instantiation.

I am amazed at the amount of dodge taking place on this thread. Can anyone take this on directly? Apparently not, since if they could have they would have. When somebody provides a falsification you have to return a refutation. In other words I should have made a mistake in my work here. No one has stepped up and pointed out an error in my work. Instead they just go on back to the curriculum as if I posted a message on needing some help understanding the subject.

Falsification methods... I need only provide one black swan to falsify the statement that all swans are white. Somehow you all fail to observe this effect, and this then again brings us into a broader statement on philosophy and the modern human. Could it be that at your computers sipping your coffee you are staring at the black swan in disbelief? Who here can utter a statement about the black swan? None so far except myself. Why this is so; a very fascinating system that we are caught up in. This is sci.math and we are on an uncensored medium. All here are self-censoring. If they witness the black swan even they will have gone too far. Put your hands up before your eyes and ask yourself if you are doing math. Or are you merely a good mimic?
Post by Mostowski Collapse
x v ~x = x + (1 - x) - x*(1 - x)
= 1 - x + x^2
= 1 - x + x
= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
Julio Di Egidio
2020-09-03 14:48:00 UTC
Post by Tim Golden BandTech.com
I am amazed at the amount of dodge taking place on this thread.
Can anyone take this on directly? Apparently not, since if they
could have they would have. When somebody provides a falsification
you have to return a refutation. In other words I should have made
a mistake in my work here. No one has stepped up and pointed out an
error in my work. Instead they just go on back to the curriculum as
if I posted a message on needing some help understanding the subject.
years now, so let me be more blatant: you are the typical self-deluded
crank in permanent denial, and that's all there is to it. -- That said,
whether you need help or not I'll rather let you decide for yourself.

*Plonk*

Julio
Mostowski Collapse
2020-09-03 14:51:19 UTC
This here is not a polynomial:

a(0) + a(1) X a(2) X X + a(3) X X X + ...

Polynomials from R[X] are finite. If p(X) is a polynomial,
then the function deg(p(X)) is defined and
it has a value from the natural numbers.

https://en.wikipedia.org/wiki/Degree_of_a_polynomial

What you indicated was possibly the ring of
formal power series R[[X]]. For more information
see here:

https://en.wikipedia.org/wiki/Formal_power_series

That monomials in a fixed degree are not closed
for multiplication was already noted. The meaning
of the word mono-mial and poly-nomial are

easy to understand. A mono-mial has a single
non-zero coefficient. A poly-nomial can have
multiple non-zero coefficients.

Mono-mials are closed over multiplication:

aX^n * bX^m = (a*b)X^(n+m)

The degree behaves as follows:

deg(aX^n) = n

deg(aX^m) = m

deg(aX^n * bX^m) = deg((a*b)X^(n+m)) = n+m

Requiring the monomials over a fixed degree k
are closed, would require that:

deg(aX^k * bX^k) = deg((a*b)X^(2k)) = 2k = k

Which is only possible for k=0. With the monomials
of degree k=0, respectively the polynomials of
degree k=0 you can recover the domain R of R[X].

So R[X] embeds R through polynomials of
degree k=0. Thats what is usually closed over
multiplication when the variables are free,

when we do not have some constraint like X^2-X=0,
as in a Boolean polynomial.
Post by Tim Golden BandTech.com
Post by Mostowski Collapse
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it
must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.
x v y := x + y - x*y
~x := 1 - x
Well, I do perk up a little bit when you mention negation here, but since you define it using a minus its original form is arguably still present.
Still for boolean systems it squares with logic, yet that zero would be the inverse of one does not take much extensive meaning beyond that.
Doesn't this detail really expose the uniqueness of the boolean system? All that notation for such a simple form... It really doesn't seem right.
Still, I have weighed in on it, and now I'd like you to weigh in on this thread Mostowski Collapse.
Perhaps just one more variation should be played out on the polynomial of abstract algebra. The polynomial
a(0) + a(1) X a(2) X X + a(3) X X X + ...
is claimed to be ring behaved for all a(n). Normally mathematicians stay in this infinite form for quite a bit longer before they do any work. The sum and the product of multiple polynomials expose that they still fit this polynomial form (though the product and its indexing is fishy). At this point they feel that they have established the ring behavior of the expression. At some later point they will introduce the polynomial ring with real coefficients. Thus the form that I speak of is in use in the curriculum.
I'm stretching this out a bit since everyone here wants to dodge the simplest detail. A falsification is a falsification, no matter how much icing you throw on top of the cake. Still I can go to a simpler form and that is the variation that I will try here. We've selected real coefficients for the polynomial and still believe it to be ring behaved. Now we select the real coefficients. This would be the first concrete instance of a polynomial; a stage which the ordinary curriculum skips right over; no different than many 'high' mathematics courses do of their own constructions, if indeed any instance can ever be found. So let's really cement that polynomial to
0 + 1.23 X + 0 X X + 0 X X X + 0...
which is a fairly simple instance and now the supposed ring expression becomes
1.23 X
where 1.23 is a real value and X is not a real value. Therefore this first concrete instance of the polynomial form is not ring behaved, as it conflicts with the ring operator definitions. It fails to meet the closure requirement under product. This is a formal falsification. The failure of others here to meet this falsification head on is unacceptable. The math is not general if it cannot withstand instantiation.
I am amazed at the amount of dodge taking place on this thread. Can anyone take this on directly? Apparently not, since if they could have they would have. When somebody provides a falsification you have to return a refutation. In other words I should have made a mistake in my work here. No one has stepped up and pointed out an error in my work. Instead they just go on back to the curriculum as if I posted a message on needing some help understanding the subject.
Falsification methods... I need only provide one black swan to falsify the statement that all swans are white. Somehow you all fail to observe this effect, and this then again brings us into a broader statement on philosophy and the modern human. Could it be that at your computers sipping your coffee you are staring at the black swan in disbelief? Who here can utter a statement about the black swan? None so far except myself. Why this is so; a very fascinating system that we are caught up in. This is sci.math and we are on an uncensored medium. All here are self-censoring. If they witness the black swan even they will have gone too far. Put your hands up before your eyes and ask yourself if you are doing math. Or are you merely a good mimic?
Post by Mostowski Collapse
x v ~x = x + (1 - x) - x*(1 - x)
= 1 - x + x^2
= 1 - x + x
= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
Mostowski Collapse
2020-09-03 15:03:02 UTC
If you have more constraints, more polynomials
or monomials get closed concerning multiplication.
This is easy to see, if you have a constraint:

p(X) = a0 + a1 X + ... + an X^n = 0 (*)

Then you can already reduce every polynomial
with degree larger or equal than n. This is
easy to see, simply note that a constraint (*)

as above also implies, provided the coefficients
can divide:

-a0 - a1 X - ... - an-1 X^(n-1)
X^n = -------------------------------
an

a0 a1 an-1
= - --- - --- X - ... - ---- X^(n-1) (**)
an an an

So when ever you have a polynomial of degree
larger or equal than n you can reduce it by one
degree and so on, until it has a degree

below n. This polynomial rewriting possibly
doesn't work for formal power series. Not
sure. But you see a difference between

polynomials and formal power series.
Things get also more complicated in multi-
variante polynomials such as R[X][Y],

because the coefficients from R[X] do
not divide anymore.
Post by Tim Golden BandTech.com
a(0) + a(1) X a(2) X X + a(3) X X X + ...
Polynomials from R[X] are finite. If p(X) is a polynomial,
then the function deg(p(X)) is defined and
it has a value from the natural numbers.
https://en.wikipedia.org/wiki/Degree_of_a_polynomial
What you indicated was possibly the ring of
formal power series R[[X]]. For more information
https://en.wikipedia.org/wiki/Formal_power_series
That monomials in a fixed degree are not closed
for multiplication was already noted. The meaning
of the word mono-mial and poly-nomial are
easy to understand. A mono-mial has a single
non-zero coefficient. A poly-nomial can have
multiple non-zero coefficients.
aX^n * bX^m = (a*b)X^(n+m)
deg(aX^n) = n
deg(aX^m) = m
deg(aX^n * bX^m) = deg((a*b)X^(n+m)) = n+m
Requiring the monomials over a fixed degree k
deg(aX^k * bX^k) = deg((a*b)X^(2k)) = 2k = k
Which is only possible for k=0. With the monomials
of degree k=0, respectively the polynomials of
degree k=0 you can recover the domain R of R[X].
So R[X] embeds R through polynomials of
degree k=0. Thats what is usually closed over
multiplication when the variables are free,
when we do not have some constraint like X^2-X=0,
as in a Boolean polynomial.
Post by Tim Golden BandTech.com
Post by Mostowski Collapse
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it
must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.
x v y := x + y - x*y
~x := 1 - x
Well, I do perk up a little bit when you mention negation here, but since you define it using a minus its original form is arguably still present.
Still for boolean systems it squares with logic, yet that zero would be the inverse of one does not take much extensive meaning beyond that.
Doesn't this detail really expose the uniqueness of the boolean system? All that notation for such a simple form... It really doesn't seem right.
Still, I have weighed in on it, and now I'd like you to weigh in on this thread Mostowski Collapse.
Perhaps just one more variation should be played out on the polynomial of abstract algebra. The polynomial
a(0) + a(1) X a(2) X X + a(3) X X X + ...
is claimed to be ring behaved for all a(n). Normally mathematicians stay in this infinite form for quite a bit longer before they do any work. The sum and the product of multiple polynomials expose that they still fit this polynomial form (though the product and its indexing is fishy). At this point they feel that they have established the ring behavior of the expression. At some later point they will introduce the polynomial ring with real coefficients. Thus the form that I speak of is in use in the curriculum.
I'm stretching this out a bit since everyone here wants to dodge the simplest detail. A falsification is a falsification, no matter how much icing you throw on top of the cake. Still I can go to a simpler form and that is the variation that I will try here. We've selected real coefficients for the polynomial and still believe it to be ring behaved. Now we select the real coefficients. This would be the first concrete instance of a polynomial; a stage which the ordinary curriculum skips right over; no different than many 'high' mathematics courses do of their own constructions, if indeed any instance can ever be found. So let's really cement that polynomial to
0 + 1.23 X + 0 X X + 0 X X X + 0...
which is a fairly simple instance and now the supposed ring expression becomes
1.23 X
where 1.23 is a real value and X is not a real value. Therefore this first concrete instance of the polynomial form is not ring behaved, as it conflicts with the ring operator definitions. It fails to meet the closure requirement under product. This is a formal falsification. The failure of others here to meet this falsification head on is unacceptable. The math is not general if it cannot withstand instantiation.
I am amazed at the amount of dodge taking place on this thread. Can anyone take this on directly? Apparently not, since if they could have they would have. When somebody provides a falsification you have to return a refutation. In other words I should have made a mistake in my work here. No one has stepped up and pointed out an error in my work. Instead they just go on back to the curriculum as if I posted a message on needing some help understanding the subject.
Falsification methods... I need only provide one black swan to falsify the statement that all swans are white. Somehow you all fail to observe this effect, and this then again brings us into a broader statement on philosophy and the modern human. Could it be that at your computers sipping your coffee you are staring at the black swan in disbelief? Who here can utter a statement about the black swan? None so far except myself. Why this is so; a very fascinating system that we are caught up in. This is sci.math and we are on an uncensored medium. All here are self-censoring. If they witness the black swan even they will have gone too far. Put your hands up before your eyes and ask yourself if you are doing math. Or are you merely a good mimic?
Post by Mostowski Collapse
x v ~x = x + (1 - x) - x*(1 - x)
= 1 - x + x^2
= 1 - x + x
= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
Mostowski Collapse
2020-09-03 15:08:32 UTC
In case you are interesed in multivariate
polynomials, you might like this book:

Using Algebraic Geometry
von David A. Cox , John Little , Donal O'Shea (Autor)
https://www.amazon.de/dp/0387207066
Post by Mostowski Collapse
If you have more constraints, more polynomials
or monomials get closed concerning multiplication.
p(X) = a0 + a1 X + ... + an X^n = 0 (*)
Then you can already reduce every polynomial
with degree larger or equal than n. This is
easy to see, simply note that a constraint (*)
as above also implies, provided the coefficients
-a0 - a1 X - ... - an-1 X^(n-1)
X^n = -------------------------------
an
a0 a1 an-1
= - --- - --- X - ... - ---- X^(n-1) (**)
an an an
So when ever you have a polynomial of degree
larger or equal than n you can reduce it by one
degree and so on, until it has a degree
below n. This polynomial rewriting possibly
doesn't work for formal power series. Not
sure. But you see a difference between
polynomials and formal power series.
Things get also more complicated in multi-
variante polynomials such as R[X][Y],
because the coefficients from R[X] do
not divide anymore.
Post by Tim Golden BandTech.com
a(0) + a(1) X a(2) X X + a(3) X X X + ...
Polynomials from R[X] are finite. If p(X) is a polynomial,
then the function deg(p(X)) is defined and
it has a value from the natural numbers.
https://en.wikipedia.org/wiki/Degree_of_a_polynomial
What you indicated was possibly the ring of
formal power series R[[X]]. For more information
https://en.wikipedia.org/wiki/Formal_power_series
That monomials in a fixed degree are not closed
for multiplication was already noted. The meaning
of the word mono-mial and poly-nomial are
easy to understand. A mono-mial has a single
non-zero coefficient. A poly-nomial can have
multiple non-zero coefficients.
aX^n * bX^m = (a*b)X^(n+m)
deg(aX^n) = n
deg(aX^m) = m
deg(aX^n * bX^m) = deg((a*b)X^(n+m)) = n+m
Requiring the monomials over a fixed degree k
deg(aX^k * bX^k) = deg((a*b)X^(2k)) = 2k = k
Which is only possible for k=0. With the monomials
of degree k=0, respectively the polynomials of
degree k=0 you can recover the domain R of R[X].
So R[X] embeds R through polynomials of
degree k=0. Thats what is usually closed over
multiplication when the variables are free,
when we do not have some constraint like X^2-X=0,
as in a Boolean polynomial.
Post by Tim Golden BandTech.com
Post by Mostowski Collapse
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it
must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.
x v y := x + y - x*y
~x := 1 - x
Well, I do perk up a little bit when you mention negation here, but since you define it using a minus its original form is arguably still present.
Still for boolean systems it squares with logic, yet that zero would be the inverse of one does not take much extensive meaning beyond that.
Doesn't this detail really expose the uniqueness of the boolean system? All that notation for such a simple form... It really doesn't seem right.
Still, I have weighed in on it, and now I'd like you to weigh in on this thread Mostowski Collapse.
Perhaps just one more variation should be played out on the polynomial of abstract algebra. The polynomial
a(0) + a(1) X a(2) X X + a(3) X X X + ...
is claimed to be ring behaved for all a(n). Normally mathematicians stay in this infinite form for quite a bit longer before they do any work. The sum and the product of multiple polynomials expose that they still fit this polynomial form (though the product and its indexing is fishy). At this point they feel that they have established the ring behavior of the expression. At some later point they will introduce the polynomial ring with real coefficients. Thus the form that I speak of is in use in the curriculum.
I'm stretching this out a bit since everyone here wants to dodge the simplest detail. A falsification is a falsification, no matter how much icing you throw on top of the cake. Still I can go to a simpler form and that is the variation that I will try here. We've selected real coefficients for the polynomial and still believe it to be ring behaved. Now we select the real coefficients. This would be the first concrete instance of a polynomial; a stage which the ordinary curriculum skips right over; no different than many 'high' mathematics courses do of their own constructions, if indeed any instance can ever be found. So let's really cement that polynomial to
0 + 1.23 X + 0 X X + 0 X X X + 0...
which is a fairly simple instance and now the supposed ring expression becomes
1.23 X
where 1.23 is a real value and X is not a real value. Therefore this first concrete instance of the polynomial form is not ring behaved, as it conflicts with the ring operator definitions. It fails to meet the closure requirement under product. This is a formal falsification. The failure of others here to meet this falsification head on is unacceptable. The math is not general if it cannot withstand instantiation.
I am amazed at the amount of dodge taking place on this thread. Can anyone take this on directly? Apparently not, since if they could have they would have. When somebody provides a falsification you have to return a refutation. In other words I should have made a mistake in my work here. No one has stepped up and pointed out an error in my work. Instead they just go on back to the curriculum as if I posted a message on needing some help understanding the subject.
Falsification methods... I need only provide one black swan to falsify the statement that all swans are white. Somehow you all fail to observe this effect, and this then again brings us into a broader statement on philosophy and the modern human. Could it be that at your computers sipping your coffee you are staring at the black swan in disbelief? Who here can utter a statement about the black swan? None so far except myself. Why this is so; a very fascinating system that we are caught up in. This is sci.math and we are on an uncensored medium. All here are self-censoring. If they witness the black swan even they will have gone too far. Put your hands up before your eyes and ask yourself if you are doing math. Or are you merely a good mimic?
Post by Mostowski Collapse
x v ~x = x + (1 - x) - x*(1 - x)
= 1 - x + x^2
= 1 - x + x
= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
Mostowski Collapse
2020-09-03 15:21:47 UTC
Here is an example of a formal power series
that you cannot reduce for example versus
X^2-X=0. Its simply 1/(1-X) formally:

1
----- = 1 + X + X^2 + X^3 + ...
1 - X

If you reduce with X^2-X=0 respectively
X^2=X, you get:

= 1 + X + X + X + ...

= 1 + (1 + 1 + 1 + ...) X

So you get a coefficient that tends to
infinity, but such coefficients are not
found in R.
Post by Mostowski Collapse
In case you are interesed in multivariate
Using Algebraic Geometry
von David A. Cox , John Little , Donal O'Shea (Autor)
https://www.amazon.de/dp/0387207066
Post by Mostowski Collapse
If you have more constraints, more polynomials
or monomials get closed concerning multiplication.
p(X) = a0 + a1 X + ... + an X^n = 0 (*)
Then you can already reduce every polynomial
with degree larger or equal than n. This is
easy to see, simply note that a constraint (*)
as above also implies, provided the coefficients
-a0 - a1 X - ... - an-1 X^(n-1)
X^n = -------------------------------
an
a0 a1 an-1
= - --- - --- X - ... - ---- X^(n-1) (**)
an an an
So when ever you have a polynomial of degree
larger or equal than n you can reduce it by one
degree and so on, until it has a degree
below n. This polynomial rewriting possibly
doesn't work for formal power series. Not
sure. But you see a difference between
polynomials and formal power series.
Things get also more complicated in multi-
variante polynomials such as R[X][Y],
because the coefficients from R[X] do
not divide anymore.
Post by Tim Golden BandTech.com
a(0) + a(1) X a(2) X X + a(3) X X X + ...
Polynomials from R[X] are finite. If p(X) is a polynomial,
then the function deg(p(X)) is defined and
it has a value from the natural numbers.
https://en.wikipedia.org/wiki/Degree_of_a_polynomial
What you indicated was possibly the ring of
formal power series R[[X]]. For more information
https://en.wikipedia.org/wiki/Formal_power_series
That monomials in a fixed degree are not closed
for multiplication was already noted. The meaning
of the word mono-mial and poly-nomial are
easy to understand. A mono-mial has a single
non-zero coefficient. A poly-nomial can have
multiple non-zero coefficients.
aX^n * bX^m = (a*b)X^(n+m)
deg(aX^n) = n
deg(aX^m) = m
deg(aX^n * bX^m) = deg((a*b)X^(n+m)) = n+m
Requiring the monomials over a fixed degree k
deg(aX^k * bX^k) = deg((a*b)X^(2k)) = 2k = k
Which is only possible for k=0. With the monomials
of degree k=0, respectively the polynomials of
degree k=0 you can recover the domain R of R[X].
So R[X] embeds R through polynomials of
degree k=0. Thats what is usually closed over
multiplication when the variables are free,
when we do not have some constraint like X^2-X=0,
as in a Boolean polynomial.
Post by Tim Golden BandTech.com
Post by Mostowski Collapse
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it
must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.
x v y := x + y - x*y
~x := 1 - x
Well, I do perk up a little bit when you mention negation here, but since you define it using a minus its original form is arguably still present.
Still for boolean systems it squares with logic, yet that zero would be the inverse of one does not take much extensive meaning beyond that.
Doesn't this detail really expose the uniqueness of the boolean system? All that notation for such a simple form... It really doesn't seem right.
Still, I have weighed in on it, and now I'd like you to weigh in on this thread Mostowski Collapse.
Perhaps just one more variation should be played out on the polynomial of abstract algebra. The polynomial
a(0) + a(1) X a(2) X X + a(3) X X X + ...
is claimed to be ring behaved for all a(n). Normally mathematicians stay in this infinite form for quite a bit longer before they do any work. The sum and the product of multiple polynomials expose that they still fit this polynomial form (though the product and its indexing is fishy). At this point they feel that they have established the ring behavior of the expression. At some later point they will introduce the polynomial ring with real coefficients. Thus the form that I speak of is in use in the curriculum.
I'm stretching this out a bit since everyone here wants to dodge the simplest detail. A falsification is a falsification, no matter how much icing you throw on top of the cake. Still I can go to a simpler form and that is the variation that I will try here. We've selected real coefficients for the polynomial and still believe it to be ring behaved. Now we select the real coefficients. This would be the first concrete instance of a polynomial; a stage which the ordinary curriculum skips right over; no different than many 'high' mathematics courses do of their own constructions, if indeed any instance can ever be found. So let's really cement that polynomial to
0 + 1.23 X + 0 X X + 0 X X X + 0...
which is a fairly simple instance and now the supposed ring expression becomes
1.23 X
where 1.23 is a real value and X is not a real value. Therefore this first concrete instance of the polynomial form is not ring behaved, as it conflicts with the ring operator definitions. It fails to meet the closure requirement under product. This is a formal falsification. The failure of others here to meet this falsification head on is unacceptable. The math is not general if it cannot withstand instantiation.
I am amazed at the amount of dodge taking place on this thread. Can anyone take this on directly? Apparently not, since if they could have they would have. When somebody provides a falsification you have to return a refutation. In other words I should have made a mistake in my work here. No one has stepped up and pointed out an error in my work. Instead they just go on back to the curriculum as if I posted a message on needing some help understanding the subject.
Falsification methods... I need only provide one black swan to falsify the statement that all swans are white. Somehow you all fail to observe this effect, and this then again brings us into a broader statement on philosophy and the modern human. Could it be that at your computers sipping your coffee you are staring at the black swan in disbelief? Who here can utter a statement about the black swan? None so far except myself. Why this is so; a very fascinating system that we are caught up in. This is sci.math and we are on an uncensored medium. All here are self-censoring. If they witness the black swan even they will have gone too far. Put your hands up before your eyes and ask yourself if you are doing math. Or are you merely a good mimic?
Post by Mostowski Collapse
x v ~x = x + (1 - x) - x*(1 - x)
= 1 - x + x^2
= 1 - x + x
= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
Mostowski Collapse
2020-09-03 15:25:58 UTC
You could also view it as the observation that
for polynomials p(x) the substitution p(1)

always exists, whereas for formal power series f(x)
the substitution f(1) need not exist.

LoL
Post by Mostowski Collapse
Here is an example of a formal power series
that you cannot reduce for example versus
1
----- = 1 + X + X^2 + X^3 + ...
1 - X
If you reduce with X^2-X=0 respectively
= 1 + X + X + X + ...
= 1 + (1 + 1 + 1 + ...) X
So you get a coefficient that tends to
infinity, but such coefficients are not
found in R.
Post by Mostowski Collapse
In case you are interesed in multivariate
Using Algebraic Geometry
von David A. Cox , John Little , Donal O'Shea (Autor)
https://www.amazon.de/dp/0387207066
Post by Mostowski Collapse
If you have more constraints, more polynomials
or monomials get closed concerning multiplication.
p(X) = a0 + a1 X + ... + an X^n = 0 (*)
Then you can already reduce every polynomial
with degree larger or equal than n. This is
easy to see, simply note that a constraint (*)
as above also implies, provided the coefficients
-a0 - a1 X - ... - an-1 X^(n-1)
X^n = -------------------------------
an
a0 a1 an-1
= - --- - --- X - ... - ---- X^(n-1) (**)
an an an
So when ever you have a polynomial of degree
larger or equal than n you can reduce it by one
degree and so on, until it has a degree
below n. This polynomial rewriting possibly
doesn't work for formal power series. Not
sure. But you see a difference between
polynomials and formal power series.
Things get also more complicated in multi-
variante polynomials such as R[X][Y],
because the coefficients from R[X] do
not divide anymore.
Post by Tim Golden BandTech.com
a(0) + a(1) X a(2) X X + a(3) X X X + ...
Polynomials from R[X] are finite. If p(X) is a polynomial,
then the function deg(p(X)) is defined and
it has a value from the natural numbers.
https://en.wikipedia.org/wiki/Degree_of_a_polynomial
What you indicated was possibly the ring of
formal power series R[[X]]. For more information
https://en.wikipedia.org/wiki/Formal_power_series
That monomials in a fixed degree are not closed
for multiplication was already noted. The meaning
of the word mono-mial and poly-nomial are
easy to understand. A mono-mial has a single
non-zero coefficient. A poly-nomial can have
multiple non-zero coefficients.
aX^n * bX^m = (a*b)X^(n+m)
deg(aX^n) = n
deg(aX^m) = m
deg(aX^n * bX^m) = deg((a*b)X^(n+m)) = n+m
Requiring the monomials over a fixed degree k
deg(aX^k * bX^k) = deg((a*b)X^(2k)) = 2k = k
Which is only possible for k=0. With the monomials
of degree k=0, respectively the polynomials of
degree k=0 you can recover the domain R of R[X].
So R[X] embeds R through polynomials of
degree k=0. Thats what is usually closed over
multiplication when the variables are free,
when we do not have some constraint like X^2-X=0,
as in a Boolean polynomial.
Post by Tim Golden BandTech.com
Post by Mostowski Collapse
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it
must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.
x v y := x + y - x*y
~x := 1 - x
Well, I do perk up a little bit when you mention negation here, but since you define it using a minus its original form is arguably still present.
Still for boolean systems it squares with logic, yet that zero would be the inverse of one does not take much extensive meaning beyond that.
Doesn't this detail really expose the uniqueness of the boolean system? All that notation for such a simple form... It really doesn't seem right.
Still, I have weighed in on it, and now I'd like you to weigh in on this thread Mostowski Collapse.
Perhaps just one more variation should be played out on the polynomial of abstract algebra. The polynomial
a(0) + a(1) X a(2) X X + a(3) X X X + ...
is claimed to be ring behaved for all a(n). Normally mathematicians stay in this infinite form for quite a bit longer before they do any work. The sum and the product of multiple polynomials expose that they still fit this polynomial form (though the product and its indexing is fishy). At this point they feel that they have established the ring behavior of the expression. At some later point they will introduce the polynomial ring with real coefficients. Thus the form that I speak of is in use in the curriculum.
I'm stretching this out a bit since everyone here wants to dodge the simplest detail. A falsification is a falsification, no matter how much icing you throw on top of the cake. Still I can go to a simpler form and that is the variation that I will try here. We've selected real coefficients for the polynomial and still believe it to be ring behaved. Now we select the real coefficients. This would be the first concrete instance of a polynomial; a stage which the ordinary curriculum skips right over; no different than many 'high' mathematics courses do of their own constructions, if indeed any instance can ever be found. So let's really cement that polynomial to
0 + 1.23 X + 0 X X + 0 X X X + 0...
which is a fairly simple instance and now the supposed ring expression becomes
1.23 X
where 1.23 is a real value and X is not a real value. Therefore this first concrete instance of the polynomial form is not ring behaved, as it conflicts with the ring operator definitions. It fails to meet the closure requirement under product. This is a formal falsification. The failure of others here to meet this falsification head on is unacceptable. The math is not general if it cannot withstand instantiation.
I am amazed at the amount of dodge taking place on this thread. Can anyone take this on directly? Apparently not, since if they could have they would have. When somebody provides a falsification you have to return a refutation. In other words I should have made a mistake in my work here. No one has stepped up and pointed out an error in my work. Instead they just go on back to the curriculum as if I posted a message on needing some help understanding the subject.
Falsification methods... I need only provide one black swan to falsify the statement that all swans are white. Somehow you all fail to observe this effect, and this then again brings us into a broader statement on philosophy and the modern human. Could it be that at your computers sipping your coffee you are staring at the black swan in disbelief? Who here can utter a statement about the black swan? None so far except myself. Why this is so; a very fascinating system that we are caught up in. This is sci.math and we are on an uncensored medium. All here are self-censoring. If they witness the black swan even they will have gone too far. Put your hands up before your eyes and ask yourself if you are doing math. Or are you merely a good mimic?
Post by Mostowski Collapse
x v ~x = x + (1 - x) - x*(1 - x)
= 1 - x + x^2
= 1 - x + x
= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
Tim Golden BandTech.com
2020-09-03 19:57:41 UTC
If you could dodge the black swan any harder Mostowski I'd have to call you out.
A simple question for you:
Is
1.23 X
an instance of a polynomial with real coefficients? The X here of course is the abstract form of abstract algebra.
The value 1.23 is a real number. Is this valid?

If I could simplify down any farther I would, but I'm afraid I've reached the bottom here. Next though of course I'd ask you if
0.12 + 1.23 X
is a valid polynomial expression within abstract algebra? Of course again these values are real except for the X which has no actual set definition at all. Why should all here deny such simple forms? Are they too simple? Are they beneath you? Does the terminology 'fundamental' have anything to do with simplicity? Possibly you are a bot and merely work out variations on the higher side of things. If so I really think you should try and compile this low level disproof.

None here seem capable of even discussing the act of falsification. Mimicry rules this day.

Two black swans for Mostowski. It seems you see no swans at all.
Post by Mostowski Collapse
You could also view it as the observation that
for polynomials p(x) the substitution p(1)
always exists, whereas for formal power series f(x)
the substitution f(1) need not exist.
LoL
Post by Mostowski Collapse
Here is an example of a formal power series
that you cannot reduce for example versus
1
----- = 1 + X + X^2 + X^3 + ...
1 - X
If you reduce with X^2-X=0 respectively
= 1 + X + X + X + ...
= 1 + (1 + 1 + 1 + ...) X
So you get a coefficient that tends to
infinity, but such coefficients are not
found in R.
Post by Mostowski Collapse
In case you are interesed in multivariate
Using Algebraic Geometry
von David A. Cox , John Little , Donal O'Shea (Autor)
https://www.amazon.de/dp/0387207066
Post by Mostowski Collapse
If you have more constraints, more polynomials
or monomials get closed concerning multiplication.
p(X) = a0 + a1 X + ... + an X^n = 0 (*)
Then you can already reduce every polynomial
with degree larger or equal than n. This is
easy to see, simply note that a constraint (*)
as above also implies, provided the coefficients
-a0 - a1 X - ... - an-1 X^(n-1)
X^n = -------------------------------
an
a0 a1 an-1
= - --- - --- X - ... - ---- X^(n-1) (**)
an an an
So when ever you have a polynomial of degree
larger or equal than n you can reduce it by one
degree and so on, until it has a degree
below n. This polynomial rewriting possibly
doesn't work for formal power series. Not
sure. But you see a difference between
polynomials and formal power series.
Things get also more complicated in multi-
variante polynomials such as R[X][Y],
because the coefficients from R[X] do
not divide anymore.
a(0) + a(1) X a(2) X X + a(3) X X X + ...
Polynomials from R[X] are finite. If p(X) is a polynomial,
then the function deg(p(X)) is defined and
it has a value from the natural numbers.
https://en.wikipedia.org/wiki/Degree_of_a_polynomial
What you indicated was possibly the ring of
formal power series R[[X]]. For more information
https://en.wikipedia.org/wiki/Formal_power_series
That monomials in a fixed degree are not closed
for multiplication was already noted. The meaning
of the word mono-mial and poly-nomial are
easy to understand. A mono-mial has a single
non-zero coefficient. A poly-nomial can have
multiple non-zero coefficients.
aX^n * bX^m = (a*b)X^(n+m)
deg(aX^n) = n
deg(aX^m) = m
deg(aX^n * bX^m) = deg((a*b)X^(n+m)) = n+m
Requiring the monomials over a fixed degree k
deg(aX^k * bX^k) = deg((a*b)X^(2k)) = 2k = k
Which is only possible for k=0. With the monomials
of degree k=0, respectively the polynomials of
degree k=0 you can recover the domain R of R[X].
So R[X] embeds R through polynomials of
degree k=0. Thats what is usually closed over
multiplication when the variables are free,
when we do not have some constraint like X^2-X=0,
as in a Boolean polynomial.
Post by Tim Golden BandTech.com
Post by Mostowski Collapse
A Boolean polynomials with no roots, is a tautology.
If it has no roots, i.e. never becomes zero, it
must be always 1, hence it is a tautology. That it
also reduces algebraically to 1, is quite a feat.
x v y := x + y - x*y
~x := 1 - x
Well, I do perk up a little bit when you mention negation here, but since you define it using a minus its original form is arguably still present.
Still for boolean systems it squares with logic, yet that zero would be the inverse of one does not take much extensive meaning beyond that.
Doesn't this detail really expose the uniqueness of the boolean system? All that notation for such a simple form... It really doesn't seem right.
Still, I have weighed in on it, and now I'd like you to weigh in on this thread Mostowski Collapse.
Perhaps just one more variation should be played out on the polynomial of abstract algebra. The polynomial
a(0) + a(1) X a(2) X X + a(3) X X X + ...
is claimed to be ring behaved for all a(n). Normally mathematicians stay in this infinite form for quite a bit longer before they do any work. The sum and the product of multiple polynomials expose that they still fit this polynomial form (though the product and its indexing is fishy). At this point they feel that they have established the ring behavior of the expression. At some later point they will introduce the polynomial ring with real coefficients. Thus the form that I speak of is in use in the curriculum.
I'm stretching this out a bit since everyone here wants to dodge the simplest detail. A falsification is a falsification, no matter how much icing you throw on top of the cake. Still I can go to a simpler form and that is the variation that I will try here. We've selected real coefficients for the polynomial and still believe it to be ring behaved. Now we select the real coefficients. This would be the first concrete instance of a polynomial; a stage which the ordinary curriculum skips right over; no different than many 'high' mathematics courses do of their own constructions, if indeed any instance can ever be found. So let's really cement that polynomial to
0 + 1.23 X + 0 X X + 0 X X X + 0...
which is a fairly simple instance and now the supposed ring expression becomes
1.23 X
where 1.23 is a real value and X is not a real value. Therefore this first concrete instance of the polynomial form is not ring behaved, as it conflicts with the ring operator definitions. It fails to meet the closure requirement under product. This is a formal falsification. The failure of others here to meet this falsification head on is unacceptable. The math is not general if it cannot withstand instantiation.
I am amazed at the amount of dodge taking place on this thread. Can anyone take this on directly? Apparently not, since if they could have they would have. When somebody provides a falsification you have to return a refutation. In other words I should have made a mistake in my work here. No one has stepped up and pointed out an error in my work. Instead they just go on back to the curriculum as if I posted a message on needing some help understanding the subject.
Falsification methods... I need only provide one black swan to falsify the statement that all swans are white. Somehow you all fail to observe this effect, and this then again brings us into a broader statement on philosophy and the modern human. Could it be that at your computers sipping your coffee you are staring at the black swan in disbelief? Who here can utter a statement about the black swan? None so far except myself. Why this is so; a very fascinating system that we are caught up in. This is sci.math and we are on an uncensored medium. All here are self-censoring. If they witness the black swan even they will have gone too far. Put your hands up before your eyes and ask yourself if you are doing math. Or are you merely a good mimic?
Post by Mostowski Collapse
x v ~x = x + (1 - x) - x*(1 - x)
= 1 - x + x^2
= 1 - x + x
= 1
Post by Mostowski Collapse
In Boolean polynomials, interestingly the fundamental
theorem of algebra, still holds somehow.
Because a Boolean polynomial in one variable with no roots,
i.e. that never becomes zero, reduces to 1.
I guess thats why Boole was so excited.
Post by Ross A. Finlayson
Post by Tim Golden BandTech.com
Mon 31 Aug 2020 08:47:53 AM EDT
Abstract Algebra Broken
So this issue I can only repeat myself on so many times. Still I'll try another variation.
Possibly eventually you will be able to at least regurgitate my own position to yourself since my position is completely unchanged.
https://en.wikipedia.org/wiki/Closure_(mathematics)
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Binary_operation
These links all fit together. The ring construction is at the base of the curriculum of abstract algebra. It is treating the operators with extreme care. They are so carefully defined that they don't even require the usage of the terms 'multiply' and 'add' though that is exactly what these operators do in the real numbers.
After the careful construction of the ring from two groups, both operators obeying the closure principle, the polynomial is introduced with an 'X' that is not real, and then those polynomials are simply assigned real coefficients, but this operation has not been defined, for the multiplication of a real by a non-real value X is undefined within their own operator theory that was just so carefully constructed. The term
a1 X
is not a binary operation. It distinctly disobeys the closure requirement. Thus abstract algebra has only managed to rebuild a conundrum that even DesCartes trailed off on in his book Rules For The Direction Of The Mind. The sad part is that the weakness of the construction cannot be challenged. Your own impedance is a fine instance of a subconscious power that doctrine holds over us. We are en mass hypnotized. This obvious blunder within Abstract Algebra is a fine instance. It has nothing to do with polysign. Well, at least this argument holds with or without polysign. I just happen to have seen it through polysign, which brought me to Abstract Algebra. Still it is true that the ultimate behavior of the polynomial that they are after is the modulo effect, which is painfully constructed though the 'quotient' and the 'ideal' which to this day do not make sense to me. I suspect that the cause of the contaminated language is the flaws which they dodge. I have bumped into an online AA book that alludes to it but it has been some time.
To formalize this argument we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one. This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle. Now likewise each term in the polynomial can be attacked and after all they are a sum which in the ring definition requires that they all be in one set together. The entire polynomials fails under their own formalities.
Your own denial of this is fine, for this is also true of anyone else who has bumped into this falsification. Clearly the extent of the problem if I am correct does develop a philosophical statement on the modern human and its culture. The burden of credibility; particularly in a subject such as mathematics; the main burden of the modern student is of mimicry; something which humans excel at. Mimic the complexities of this subject and you can achieve an A grade and a future within academia. Deny the credibility of this subject and you will be shunned out of the system. Even within our own heads this bad dog relation is playing out as we are social animals and so the biblical nature of academia is established not only from without but from within. The developments of academia are more a method of fitting more and more PhDs than it is a pursuit of the truth. Such a branch as abstract algebra with its imperfections goes unchallenged in this arena for good reason. As one PhD steps on another PhDs subject as false the whole system will collapse its extraordinary accumulation. Should it collapse down to truth? I doubt it. I think we are more like monkeys in the tree still trying to figure out fundamental principles. We are only part way and we've landed ourselves in this morass.
We need to maintain these subjects as alive rather than as dead and pickled to perfection. Our past leaders should not be placed up on a mantle as unapproachable. The burden of their work is on us.
It's great, if you found something mathematics owes, i.e. that it gives.
I'm sure the modular arithmetic and abstract algebra is familiar to everybody.
I'm very interested in your opinion about these number systems,
that you know about for example machine arithmetic and polysign
numbers, of course that abstract algebra besides cancellation or
opposition (division, the reciprocal, or inverse) then is linear then
that in the non-linear, as what results in the vertical or pole,
that it's OK and algebra does carry over, boxing up the terms
and circling the others, as what arrive in the results in the
evaluation, of what are the terms, of the abstract algebra.
Otherwise how we'd read this as your humble acknowledgment
that you found something that needs fixing - that what you have
in mind you also know that for all the intents and purposes of
abstract algebra, it will be OK in those terms.
In the operator space there then using for example polysign numbers
for points with algebra's pointing to their neighbors, or about moments,
when you find abstract algebra all broken it's usually in terms of
simple matters of scale - that the variables weren't in terms.
"To formalize this argument
we should really take the acclaimed ring behaved
a0 + a1 X + a2 XX + ...
then assign an = 0 when n not equal to one.
This then leaves us the ring behaved
a1 X
which is not ring behaved because it offends the closure principle.
Now likewise
each term in the polynomial can be attacked and
after all they are a sum which
in the ring definition requires
that they all be in one set together.
The entire polynomials fails under their own formalities. "
Now, I actually agree here though for defining "the entire polynomials fails", as,
as together that of course it's read in the un-usual way "the, entire polynomial, 's",
that "fails" only means "doesn't hold up". (Like you say, in the terms.)
I.e. the polynomials entirely hold up the fundamental theorem
of algebra about the existence of roots.
Then initial terms and such moments of characteristics,
that is for example all assigned to numerical methods
then what of course adding numerical methods to non-linear
or non-parametric modeling, often multiplies instead of
cancels, the error term. (The other term after the approximation
the first term of which in the numerical method is correct.)
So, you'd have to write that out some more,
to make it so logically all the conditions arrive
at what you say.
Just more, though - not different.
Thanks, it's OK.
zelos...@gmail.com
2020-09-04 05:25:10 UTC
Post by Tim Golden BandTech.com
an instance of a polynomial with real coefficients? The X here of course is the abstract form of abstract algebra.
it is a notation of it, yes.
Post by Tim Golden BandTech.com
The value 1.23 is a real number. Is this valid?
Yes
Post by Tim Golden BandTech.com
is a valid polynomial expression within abstract algebra?
Yes
Post by Tim Golden BandTech.com
Of course again these values are real except for the X which has no actual set definition at all.
As I pointed out already, the X is a historical NOTATION that we use, it means n=1 for our polynomial sequence as I defined earlier. just as X^2 means n=2, etc
Post by Tim Golden BandTech.com
Why should all here deny such simple forms? Are they too simple? Are they beneath you? Does the terminology 'fundamental' have anything to do with simplicity? Possibly you are a bot and merely work out variations on the higher side of things. If so I really think you should try and compile this low level disproof
Complainin about notation, that I already showed you is just notation, is not a disproof, it is however proof of your ignorance.
Lalo T.
2020-09-04 05:54:26 UTC
https://proofwiki.org/wiki/Definition:Scalar
https://proofwiki.org/wiki/Definition:Scalar_Ring

https://mathworld.wolfram.com/ScalarMultiplication.html
https://en.wikipedia.org/wiki/Scalar_multiplication
https://encyclopediaofmath.org/wiki/Scalar

https://en.wikipedia.org/wiki/Classical_Hamiltonian_quaternions
Tim Golden BandTech.com
2020-09-04 11:46:11 UTC
Post by ***@gmail.com
Post by Tim Golden BandTech.com
an instance of a polynomial with real coefficients? The X here of course is the abstract form of abstract algebra.
it is a notation of it, yes.
Post by Tim Golden BandTech.com
The value 1.23 is a real number. Is this valid?
Yes
Post by Tim Golden BandTech.com
is a valid polynomial expression within abstract algebra?
Yes
Post by Tim Golden BandTech.com
Of course again these values are real except for the X which has no actual set definition at all.
As I pointed out already, the X is a historical NOTATION that we use, it means n=1 for our polynomial sequence as I defined earlier. just as X^2 means n=2, etc
Post by Tim Golden BandTech.com
Why should all here deny such simple forms? Are they too simple? Are they beneath you? Does the terminology 'fundamental' have anything to do with simplicity? Possibly you are a bot and merely work out variations on the higher side of things. If so I really think you should try and compile this low level disproof
Complainin about notation, that I already showed you is just notation, is not a disproof, it is however proof of your ignorance.
Zelos I am not the one who only just wrote out formalized operators. This is supposedly key to this topic. More fundamental really than the polynomial. The polynomial is claimed to be ring behaved, but my assertion here proves it not to be ring behaved.

It is as if you deny my ability to dismantle the polynomial in the way that I do and yet the way that I do is just as it is constructed. It is a long series of sums. This is how you can come to claim its ring behavior and yet when we study the individual terms we land in this conclusion that these expressions are not ring behaved. Their internal structure is broken.

And now again you pivot back to claiming that this is 'historical' notation but didn't we already cover this above? This is not antiquated notation. It is still in use. It is as if you can see the conflict; the black swan. Only one is needed sir. That there are many and that there are still white swans is not of concern to the process of falsification. One black swan will do:
1.23 X
is not ring behaved. It cannot be. It goes in direct contradiction to the closure requirement of the binary operator. Thus just as soon as the abstracted algebraics goes to use their new construction with 'real coefficients' they broke away from their own rules. Your own inability Zelos to come to speak about the black swan above and admit that it is not ring behaved is proven by the elongated and diversionary terms on which you speak in this thread. This must be about the fifth go around now. I do appreciate you staying on but it could be helpful couldn't it if you could come to facing the music of the quacker that is biting you in the but? Poor Mostowski has it on the nose. You are caught running away. You must face the black swan to halt the attack.
Mostowski Collapse
2020-09-04 12:44:33 UTC
We can only say that the axioms do not
so easily satisfy a submodel property.

If you have N ⊆ M, and M satisfies a
closure property

∀x,y(x e M /\ y e M => x*y e M)

then its not automatic that N also
satisfies a closure property

∀x,y(x e N /\ y e N => x*y e N)

So only some of the N ⊆ M are also
closed. And among the closed only

some satisfy the original axioms.
In Grouptheory such a Group has a special
name, its called Subgroup:

https://en.wikipedia.org/wiki/Subgroup

You can also call N ⊆ M of a group M,
which is not a subgroup a black swan.
But this is a little overdramatic.
Post by Tim Golden BandTech.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
an instance of a polynomial with real coefficients? The X here of course is the abstract form of abstract algebra.
it is a notation of it, yes.
Post by Tim Golden BandTech.com
The value 1.23 is a real number. Is this valid?
Yes
Post by Tim Golden BandTech.com
is a valid polynomial expression within abstract algebra?
Yes
Post by Tim Golden BandTech.com
Of course again these values are real except for the X which has no actual set definition at all.
As I pointed out already, the X is a historical NOTATION that we use, it means n=1 for our polynomial sequence as I defined earlier. just as X^2 means n=2, etc
Post by Tim Golden BandTech.com
Why should all here deny such simple forms? Are they too simple? Are they beneath you? Does the terminology 'fundamental' have anything to do with simplicity? Possibly you are a bot and merely work out variations on the higher side of things. If so I really think you should try and compile this low level disproof
Complainin about notation, that I already showed you is just notation, is not a disproof, it is however proof of your ignorance.
Zelos I am not the one who only just wrote out formalized operators. This is supposedly key to this topic. More fundamental really than the polynomial. The polynomial is claimed to be ring behaved, but my assertion here proves it not to be ring behaved.
It is as if you deny my ability to dismantle the polynomial in the way that I do and yet the way that I do is just as it is constructed. It is a long series of sums. This is how you can come to claim its ring behavior and yet when we study the individual terms we land in this conclusion that these expressions are not ring behaved. Their internal structure is broken.
1.23 X
is not ring behaved. It cannot be. It goes in direct contradiction to the closure requirement of the binary operator. Thus just as soon as the abstracted algebraics goes to use their new construction with 'real coefficients' they broke away from their own rules. Your own inability Zelos to come to speak about the black swan above and admit that it is not ring behaved is proven by the elongated and diversionary terms on which you speak in this thread. This must be about the fifth go around now. I do appreciate you staying on but it could be helpful couldn't it if you could come to facing the music of the quacker that is biting you in the but? Poor Mostowski has it on the nose. You are caught running away. You must face the black swan to halt the attack.
Mostowski Collapse
2020-09-04 12:53:05 UTC
A subring of a ring (R, +, ∗, 0, 1) is a subset S
of R that is both a subgroup of (R, +, 0) and a
submonoid of (R, ∗, 1).

We can now say:

White Swan: S is subgroup and submonoid

Yellow Swan: S is subgroup but not submonoid

Blue Swan: S is not subgroup but submonoid

Black Swan: S is neither subgroup nor submonoid

Example R = polynomials, S = {a1 X e R}. Thats a
yellow swan, and not a black swan.
Post by Mostowski Collapse
We can only say that the axioms do not
so easily satisfy a submodel property.
If you have N ⊆ M, and M satisfies a
closure property
∀x,y(x e M /\ y e M => x*y e M)
then its not automatic that N also
satisfies a closure property
∀x,y(x e N /\ y e N => x*y e N)
So only some of the N ⊆ M are also
closed. And among the closed only
some satisfy the original axioms.
In Grouptheory such a Group has a special
https://en.wikipedia.org/wiki/Subgroup
You can also call N ⊆ M of a group M,
which is not a subgroup a black swan.
But this is a little overdramatic.
Post by Tim Golden BandTech.com
Post by ***@gmail.com
Post by Tim Golden BandTech.com
an instance of a polynomial with real coefficients? The X here of course is the abstract form of abstract algebra.
it is a notation of it, yes.
Post by Tim Golden BandTech.com
The value 1.23 is a real number. Is this valid?
Yes
Post by Tim Golden BandTech.com
is a valid polynomial expression within abstract algebra?
Yes
Post by Tim Golden BandTech.com
Of course again these values are real except for the X which has no actual set definition at all.
As I pointed out already, the X is a historical NOTATION that we use, it means n=1 for our polynomial sequence as I defined earlier. just as X^2 means n=2, etc
Post by Tim Golden BandTech.com
Why should all here deny such simple forms? Are they too simple? Are they beneath you? Does the terminology 'fundamental' have anything to do with simplicity? Possibly you are a bot and merely work out variations on the higher side of things. If so I really think you should try and compile this low level disproof
Complainin about notation, that I already showed you is just notation, is not a disproof, it is however proof of your ignorance.
Zelos I am not the one who only just wrote out formalized operators. This is supposedly key to this topic. More fundamental really than the polynomial. The polynomial is claimed to be ring behaved, but my assertion here proves it not to be ring behaved.
It is as if you deny my ability to dismantle the polynomial in the way that I do and yet the way that I do is just as it is constructed. It is a long series of sums. This is how you can come to claim its ring behavior and yet when we study the individual terms we land in this conclusion that these expressions are not ring behaved. Their internal structure is broken.
1.23 X
is not ring behaved. It cannot be. It goes in direct contradiction to the closure requirement of the binary operator. Thus just as soon as the abstracted algebraics goes to use their new construction with 'real coefficients' they broke away from their own rules. Your own inability Zelos to come to speak about the black swan above and admit that it is not ring behaved is proven by the elongated and diversionary terms on which you speak in this thread. This must be about the fifth go around now. I do appreciate you staying on but it could be helpful couldn't it if you could come to facing the music of the quacker t