sociology and pure math

[sobriquet]

Does he have a point?

I think he's right that AI is about to take math to a higher level
beyond what we can comprehend from our current point of view that is
most likely misguided and full of historical misconceptions.

The question should be, what's taking these silly AI researchers so long?

We don't even have AI yet that is able to come up with concepts from
scratch for toy problems like the game of go.

[FromTheRafters]

Post by sobriquet
Does he have a point?
http://youtu.be/YhN4X56E7iM
I think he's right that AI is about to take math to a higher level beyond
what we can comprehend from our current point of view that is most likely
misguided and full of historical misconceptions.
The question should be, what's taking these silly AI researchers so long?

It's complicated.

Post by sobriquet
We don't even have AI yet that is able to come up with concepts from scratch
for toy problems like the game of go.

Yes, but that failure, if it is one, is not because of the inability to
complete a supertask in the real world. Let the essential workings of
the task *be* the representation.

I have seen some of his other presentations and he is, IMO, a good
teacher. He manages to teach contemporary mathematics while mostly
keeping his opinions of infinity as a side issue.

I think he is wrong about infinite representations. IMO we construct
our notion of number as needed for our mathematics. We had to extend
the naturals to include zero, and had to further extend them to
integers to accomodate addition's inverse operation. We then had to
extend them to rationals to accomodate multiplication's inverse
operation. Then to accomodate exponentiation's inverse, we invented the
reals. There is nothing wrong with having real number representations
resemble supertasks.

[sobriquet]

Post by FromTheRafters

Post by sobriquet
Does he have a point?
http://youtu.be/YhN4X56E7iM
I think he's right that AI is about to take math to a higher level
beyond what we can comprehend from our current point of view that is
most likely misguided and full of historical misconceptions.
The question should be, what's taking these silly AI researchers so long?

It's complicated.

Post by sobriquet
We don't even have AI yet that is able to come up with concepts from
scratch for toy problems like the game of go.

Yes, but that failure, if it is one, is not because of the inability to
complete a supertask in the real world. Let the essential workings of
the task *be* the representation.
I have seen some of his other presentations and he is, IMO, a good
teacher. He manages to teach contemporary mathematics while mostly
keeping his opinions of infinity as a side issue.
I think he is wrong about infinite representations. IMO we construct our
notion of number as needed for our mathematics. We had to extend the
naturals to include zero, and had to further extend them to integers to
accomodate addition's inverse operation. We then had to extend them to
rationals to accomodate multiplication's inverse operation. Then to
accomodate exponentiation's inverse, we invented the reals. There is
nothing wrong with having real number representations resemble supertasks.

I think one possible objection is that if we assume irrational numbers
like square root of 2 to exist, that presumes that we can in principle
compute it to arbitrary levels of detail. But from physics we know that
reality seems to be discrete rather than continuous for the most part.
So ultimately it seems that the universe is composed of an extremely
large but finite number of discrete elements, so however optimal we
represent things, we'll run out of elements if we try to spell things
out explicitly to unbounded levels of detail. A computation might
involve computing things to a level of detail beyond what we would be
able to compute given the limitations of the universe.

Just like if you want to store digits of the square root of 2 on a usb
stick. At some point you run out of space and you would need a bigger
usb stick to store more digits. With an infinite number of digits, you
would need a usb stick with infinite storage space and they don't exist
and it's unlikely they can exist. So it seems reasonable to claim that
the square root of 2 can't exist as an explicit number with all of its
digits at our disposal. It just exists as a concept (the diagonal length
of a square with unit side length) just like the concept of pi as the
ratio between circumference to the diameter of a circle, rather than the
concept of pi as an infinite string of digits in some number system.

[FromTheRafters]

Post by FromTheRafters

Post by sobriquet
Does he have a point?
http://youtu.be/YhN4X56E7iM
I think he's right that AI is about to take math to a higher level beyond
what we can comprehend from our current point of view that is most likely
misguided and full of historical misconceptions.
The question should be, what's taking these silly AI researchers so long?

It's complicated.

Post by sobriquet
We don't even have AI yet that is able to come up with concepts from
scratch for toy problems like the game of go.

Yes, but that failure, if it is one, is not because of the inability to
complete a supertask in the real world. Let the essential workings of the
task *be* the representation.
I have seen some of his other presentations and he is, IMO, a good teacher.
He manages to teach contemporary mathematics while mostly keeping his
opinions of infinity as a side issue.
I think he is wrong about infinite representations. IMO we construct our
notion of number as needed for our mathematics. We had to extend the
naturals to include zero, and had to further extend them to integers to
accomodate addition's inverse operation. We then had to extend them to
rationals to accomodate multiplication's inverse operation. Then to
accomodate exponentiation's inverse, we invented the reals. There is
nothing wrong with having real number representations resemble supertasks.

I think one possible objection is that if we assume irrational numbers like
square root of 2 to exist, that presumes that we can in principle compute it
to arbitrary levels of detail. But from physics we know that reality seems to
be discrete rather than continuous for the most part.
So ultimately it seems that the universe is composed of an extremely large
but finite number of discrete elements, so however optimal we represent
things, we'll run out of elements if we try to spell things out explicitly to
unbounded levels of detail. A computation might involve computing things to a
level of detail beyond what we would be able to compute given the limitations
of the universe.
Just like if you want to store digits of the square root of 2 on a usb stick.
At some point you run out of space and you would need a bigger usb stick to
store more digits. With an infinite number of digits, you would need a usb
stick with infinite storage space and they don't exist
and it's unlikely they can exist. So it seems reasonable to claim that
the square root of 2 can't exist as an explicit number with all of its digits
at our disposal. It just exists as a concept (the diagonal length of a square
with unit side length) just like the concept of pi as the ratio between
circumference to the diameter of a circle, rather than the concept of pi as
an infinite string of digits in some number system.

Indeed! I feel that a number's value is in its formula and it matters
not if other representations fall short of philosophical goals.

Funny, and good point, that our mathematical model of reality achieves
better results if it is done with discrete quanta rather than
continuous values. God may not throw dice, but it sure looks like he
plays with blocks.

[Ross Finlayson]

Post by FromTheRafters

Post by sobriquet

Post by FromTheRafters

Post by sobriquet
Does he have a point?
http://youtu.be/YhN4X56E7iM
I think he's right that AI is about to take math to a higher level
beyond what we can comprehend from our current point of view that is
most likely misguided and full of historical misconceptions.
The question should be, what's taking these silly AI researchers so long?

It's complicated.

Post by sobriquet
We don't even have AI yet that is able to come up with concepts from
scratch for toy problems like the game of go.

Yes, but that failure, if it is one, is not because of the inability
to complete a supertask in the real world. Let the essential workings
of the task *be* the representation.
I have seen some of his other presentations and he is, IMO, a good
teacher. He manages to teach contemporary mathematics while mostly
keeping his opinions of infinity as a side issue.
I think he is wrong about infinite representations. IMO we construct
our notion of number as needed for our mathematics. We had to extend
the naturals to include zero, and had to further extend them to
integers to accomodate addition's inverse operation. We then had to
extend them to rationals to accomodate multiplication's inverse
operation. Then to accomodate exponentiation's inverse, we invented
the reals. There is nothing wrong with having real number
representations resemble supertasks.

I think one possible objection is that if we assume irrational numbers
like square root of 2 to exist, that presumes that we can in principle
compute it to arbitrary levels of detail. But from physics we know
that reality seems to be discrete rather than continuous for the most
part.
So ultimately it seems that the universe is composed of an extremely
large but finite number of discrete elements, so however optimal we
represent things, we'll run out of elements if we try to spell things
out explicitly to unbounded levels of detail. A computation might
involve computing things to a level of detail beyond what we would be
able to compute given the limitations of the universe.
Just like if you want to store digits of the square root of 2 on a usb
stick. At some point you run out of space and you would need a bigger
usb stick to store more digits. With an infinite number of digits, you
would need a usb stick with infinite storage space and they don't exist
and it's unlikely they can exist. So it seems reasonable to claim that
the square root of 2 can't exist as an explicit number with all of its
digits at our disposal. It just exists as a concept (the diagonal
length of a square with unit side length) just like the concept of pi
as the ratio between circumference to the diameter of a circle, rather
than the concept of pi as an infinite string of digits in some number
system.

Indeed! I feel that a number's value is in its formula and it matters
not if other representations fall short of philosophical goals.
Funny, and good point, that our mathematical model of reality achieves
better results if it is done with discrete quanta rather than continuous
values. God may not throw dice, but it sure looks like he plays with
blocks.

Seems you don't know "hidden variables" of the wave equation was
renamed "supplementary variables", about that after of course the
tremendous success after the ultraviolet catastrophe after blackbody
radiation the spectral lines and electron physics defining particle
physics and old quantum mechanics, then that Copenhagen new quantum
mechanics is exactly because pilot wave is superclassical continuum
mechanics, that a new sort of infrared catastrophe is making for
continuum mechanics again and a real wave equation for things like
beta decay, muon physics, neutrino physics, and the conceit of the
particle in particle/wave duality as wave/particle dichotomy in
the theory of wave resonances, harmonics, and potential theory,
about a theory of sum-of-histories sum-of-potentials.

There's a pretty remarkable back story about the theory of
the objects of a mathematical universe, it's called "mathematical
Platonism" and historically most philosophers of mathematics
have adopted it at some point.

Otherwise it helps to make for a very generous reading of the
likes of Archimedes Plutonium, a retro-finitist of bounded
inference who nevertheless expresses relations in what would
otherwise be infinite expressions the completions of the closure
of forms, about a potential and practical and effective and actual
infinity, and what is about insurmountable yet insufficient bounds.

Or, you know, not, ....


Seems the point is that "mathematics the theory higher mathematics
already is a higher level".

[Phil Carmody]

...

Post by FromTheRafters

Post by sobriquet
I think one possible objection is that if we assume irrational
numbers like square root of 2 to exist, that presumes that we can in
principle compute it to arbitrary levels of detail. But from physics
we know that reality seems to be discrete rather than continuous for
the most part.

Indeed! I feel that a number's value is in its formula and it matters
not if other representations fall short of philosophical goals.
Funny, and good point, that our mathematical model of reality achieves
better results if it is done with discrete quanta rather than
continuous values. God may not throw dice, but it sure looks like he
plays with blocks.

Yeah, and that 1x1 block, when sawn in 2 diagonally, or even just
suitably marked, gives you a physical sqrt(2).

Phil

[Ross Finlayson]

Post by Phil Carmody
...

Post by FromTheRafters

Post by sobriquet
I think one possible objection is that if we assume irrational
numbers like square root of 2 to exist, that presumes that we can in
principle compute it to arbitrary levels of detail. But from physics
we know that reality seems to be discrete rather than continuous for
the most part.

Indeed! I feel that a number's value is in its formula and it matters
not if other representations fall short of philosophical goals.
Funny, and good point, that our mathematical model of reality achieves
better results if it is done with discrete quanta rather than
continuous values. God may not throw dice, but it sure looks like he
plays with blocks.

Yeah, and that 1x1 block, when sawn in 2 diagonally, or even just
suitably marked, gives you a physical sqrt(2).
Phil

That may help illustrate one of the ideas about the Planck length,
that if the Planck length were some smallest distance, then
if there was a metric and norm, and right angles and straight
lines, then a right isosceles triangle with equal sides Planck
length as "natural units", would have an irrational length root2.

So, when one has a theory with some granular least finite distance,
it's as though there weren't either straight lines or right angles,
and then it sort of results not being a continuous manifold,
with a metric and usually a norm.

So, Planck length is sort of a conceit, a concession to the
concept of particles fundamentally, where particle theory
is sort of a conceit, a concession to the concept of
smallest things in an otherwise continuous milieu.

In this way one can see that it's sort of exactly like
positing some least real number in mathematics' field,
the complete ordered field, some least distance in physics'
field, the continuous manifold of space in space-time.

Then, the idea goes, that superstrings are just as smaller
than atoms as atoms are smaller than us, making a brief
reprieve from otherwise the inconsistency of the smooth
continuous manifold, being grainy, or as reticulated.

Anyways pure and applied mathematics have that abstraction
is pure mathematics and application is reflection on
physical models, so it's mostly always both, deliberately.

Such retro-finitism is a no-go in terms of the actual theory
of the actual objects of mathematics, while ultra-finitism
makes for a sort of sub-field of mathematics if done
conscientiously, it doesn't get a free pass back to
continuity, though.


About combinatorial explosion there's combinatorial enumeration.
Then we get counting arguments and the usual standard law of
large numbers, then that there are other laws of large numbers,
arrives at from examining the space of all combinations,
then that deductive inference arrives at how things get
there after the infinitely-divisible, the infinitely-divided.

[Chris M. Thomasson]

Post by sobriquet

Post by FromTheRafters

Post by sobriquet
Does he have a point?
http://youtu.be/YhN4X56E7iM
I think he's right that AI is about to take math to a higher level
beyond what we can comprehend from our current point of view that is
most likely misguided and full of historical misconceptions.
The question should be, what's taking these silly AI researchers so long?

It's complicated.

Post by sobriquet
We don't even have AI yet that is able to come up with concepts from
scratch for toy problems like the game of go.

Yes, but that failure, if it is one, is not because of the inability
to complete a supertask in the real world. Let the essential workings
of the task *be* the representation.
I have seen some of his other presentations and he is, IMO, a good
teacher. He manages to teach contemporary mathematics while mostly
keeping his opinions of infinity as a side issue.
I think he is wrong about infinite representations. IMO we construct
our notion of number as needed for our mathematics. We had to extend
the naturals to include zero, and had to further extend them to
integers to accomodate addition's inverse operation. We then had to
extend them to rationals to accomodate multiplication's inverse
operation. Then to accomodate exponentiation's inverse, we invented
the reals. There is nothing wrong with having real number
representations resemble supertasks.

I think one possible objection is that if we assume irrational numbers
like square root of 2 to exist, that presumes that we can in principle
compute it to arbitrary levels of detail.

We can. It's all about the numbers... ;^)

In principle we can say we need 43520054 digits of precision, and be
able to do it? However, we are finite beings, so shit happens.


But from physics we know that

Post by sobriquet
reality seems to be discrete rather than continuous for the most part.
So ultimately it seems that the universe is composed of an extremely
large but finite number of discrete elements, so however optimal we
represent things, we'll run out of elements if we try to spell things
out explicitly to unbounded levels of detail. A computation might
involve computing things to a level of detail beyond what we would be
able to compute given the limitations of the universe.
Just like if you want to store digits of the square root of 2 on a usb
stick. At some point you run out of space and you would need a bigger
usb stick to store more digits. With an infinite number of digits, you
would need a usb stick with infinite storage space and they don't exist
and it's unlikely they can exist. So it seems reasonable to claim that
the square root of 2 can't exist as an explicit number with all of its
digits at our disposal. It just exists as a concept (the diagonal length
of a square with unit side length) just like the concept of pi as the
ratio between circumference to the diameter of a circle, rather than the
concept of pi as an infinite string of digits in some number system.

[Jim Burns]

Post by sobriquet
Does he have a point?

http://youtu.be/YhN4X56E7iM
|
| I think currently what we've done is
| we've managed to insulate ourselves from
| mathematical reality.
| And for me, that means
| being in touch with computation.
| I don't want mathematics to be
| an exercise in philosophy.

I consider
the verification of the validity of a proof to be
a computation.
I think that makes all or nearly.all of
what Wildberger considers problematic
what Wildberger sees going from strength to strength.

Am I outside the mainstream of thought on that?

My philosophy of mathematics,
boiled down to its essentials:

A claim is one of true or false.

In a finite sequence of claims,
if each claim is not.first.false,
then each claim is not.false.

A claim in a sequence might be externally verified
to be not.false (and thus, not.first.false)
by definitions, by axioms, by theorems
external to that finite sequence of claims.

A claim in a sequence might be internally verified
to be not.first.false.
Famously, Q in ⟨P P⇒Q Q⟩ is an example of this.
Is Q true? What does Q even mean?
Such questions don't matter.
Q is not.first false in ⟨P P⇒Q Q⟩

If Q is in a finite only.not.first.false sequence
then Q is true.
And I still don't know what Q means.

If
we see in front of us
a finite sequences of claims which is
only.not.first.false about each one of infinitely.many
then
each claim in that sequence is
true of each one of infinitely.many,
even the claims that are only internally verified
(the interesting claims, learned from that sequence).

We have not laid hands on each one of infinitely.many,
so I'm guessing Wildberger is troubled by my philosophy,
but we have "laid hands on" _each claim_

My philosophy is that logic about
differentiable manifolds or inaccessible cardinals
is primarily logic about finite sequences of claims.
And _the claims_ are accessible to finite beings.


There are many claims in physics about which
it would be perfectly reasonable to ask
how could we _possibly_ know that?

I think that my philosophy (hands.on claims)
answers that question.

A favorite example of mine is the size of
the cosmos, outside our observable universe
Starting from the assumption that
there is nothing unusual about our speck of dust,
we measure the (observable) curvature of the universe
and extrapolate outside the observable.

That method requires us to reason about
that which we cannot, even in principle, observe.
How can we possibly know that?

My answer is:
by assembling finite only.not.first.false
sequences of claims, which we can observe.

Post by sobriquet
I think he's right that
AI is about to take math to a higher level
beyond what we can comprehend
from our current point of view
that is most likely misguided and
full of historical misconceptions.

I don't know. We'll have to see whether
the offspring of our minds slip free of our history.

Each generation of human children,
offspring of our minds and bodies,
struggle to slip free of their parents' history.
Each generation succeeds some and fails some,
I think.

If we don't build into our AI an ability to slip free
of history at least as well as human children,
then I think the AI mathpocalypse won't come,
but I also think that, if we don't do that,
this whole AI thing going on will be found
to be not worth the effort put into it,
and will eventually be thought a passing fad.