Non-mathematician's question about continuum

Discussion:

(too old to reply)

j***@gmail.com

2019-02-04 09:58:31 UTC

I am not a mathematician.
I am not even sure whether I will be able frame this question properly - far from understanding the answers. Please pardon me in advance.

I got this idea from persistence of vision.

To me, various number sets - natural, integers, rationals,... kind of "exist". Can't argue much about them.

Does continuum (and anything that leads to it) really "exist" or our mind cheats into believing us in one? Do "dense enough" linear points "intellectually appear" continuous akin to picture frames "visually appearing" as a movie?

A circle may never be "possible" and hence pi may never be "possible". It may just appear to be possible. Similarly, drawing a hypotenuse on a isosceles right angle triangle may never be "possible". Euclid's straight line may never exist. Such action may just "appear to make sense". May be our intellect is collapsing such constructs into something we tend to take as continuum... probably due to one of:
1. lack of processing bandwidth of mind
2. lack of interesting information
3. survival technique through evolutionary means
4. our minds playing tricks to reduce information
5. ...
(I hope I make sense till this point.)

Is continuum proven or hypothesized?

Once again, please don't flame. I really, really don't know math.

-Bhushit

Bill

2019-02-04 10:32:46 UTC

Permalink

"exist" in what sense. In mathematics, the continuum exists,
period. I would even argue that it exists in a physical sense,
but I would not be interested in debating it--and not every
process is a continuous one. Judging by posts to this this
newsgroup, maybe it exists for some and not for others. ; )
Maybe neuron activity is not continuous, I can't say. But what
are you going to do if it is not. You'll assume it is, for
convenience... When graphing population growth for instance,
they usually use connected curves or lines...

Do "dense enough" linear points "intellectually appear"
continuous akin to picture frames "visually appearing" as a movie?

Post by j***@gmail.com
1. lack of processing bandwidth of mind
2. lack of interesting information
3. survival technique through evolutionary means
4. our minds playing tricks to reduce information
5. ...
(I hope I make sense till this point.)
Is continuum proven or hypothesized?
Once again, please don't flame. I really, really don't know math.
-Bhushit

Jew Lover

2019-02-04 12:44:32 UTC

Permalink

Post by j***@gmail.com
I am not a mathematician.

I am a mathematician - very likely the greatest ever.

Post by j***@gmail.com
I am not even sure whether I will be able frame this question properly - far from understanding the answers. Please pardon me in advance.

No need to apologise. Only the ignorant remain ignorant because they are afraid to ask.

Post by j***@gmail.com
I got this idea from persistence of vision.
To me, various number sets - natural, integers, rationals,... kind of "exist". Can't argue much about them.

Correct. In the sense that they can be realised from the realm of noumena.

There is no such thing as an infinite set. Infinity is a junk concept that has unfortunately found its way in mythmatics (mainstream mathematics).

Chapter 3, 4 and 5 of my free eBook explain all these things and much more.

https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO/view

Post by j***@gmail.com
Does continuum (and anything that leads to it) really "exist" or our mind cheats into believing us in one?

The continuum exists only in terms of magnitudes (NOT numbers because not all magnitudes can be measured). The "real number line" is a myth.

Post by j***@gmail.com
Do "dense enough" linear points "intellectually appear" continuous akin to picture frames "visually appearing" as a movie?

Think about the phrases and expressions which you use. Does "dense enough" make any logical sense to you? How do you measure denseness in terms of the continuum? And what measure constitutes "enough"?

Post by j***@gmail.com
A circle may never be "possible" and hence pi may never be "possible".

False. Both the circle and its symptom pi are well-formed concepts that exist in the realm of noumena and are independent of the human mind or any other mind.

Post by j***@gmail.com
It may just appear to be possible. Similarly, drawing a hypotenuse on a isosceles right angle triangle may never be "possible". Euclid's straight line may never exist.

No. All these things exist and can be systematically derived from NOTHING. My free eBook explains everything. If an alien had to realise these concepts, the alien would realise them in the same logical systematic way (all things being equal in terms of intellect).

No. None of this is true.

Post by j***@gmail.com
Is continuum proven or hypothesized?

Hypothesized in terms of "real numbers" corresponding to unreifiable points on a number line.

However, one can talk about a continuum of magnitudes.

Post by j***@gmail.com
Once again, please don't flame. I really, really don't know math.

This is the only answer that matters to you on this forum. Every other response will be by an idiot who is intellectually inferior to me. Oh, you can safely ignore the comment by Bill - a misguided moron.

Dan Christensen

2019-02-04 18:32:17 UTC

Permalink

Post by Jew Lover

Post by j***@gmail.com
I am not a mathematician.

I am a mathematician - very likely the greatest ever.

John "Troll Boy" Gabriel here is no mathematician. He is a crank and a troll. Even at his advanced age (60+?), he is still struggling with basic, elementary-school arithmetic. As he has repeatedly posted here:

"1/2 not equal to 2/4"
--October 22, 2017

“1/3 does NOT mean 1 divided by 3 and never has meant that”
-- February 8, 2015

"3 =< 4 is nonsense.
--October 28, 2017

"Zero is not a number."
-- Jan. 10, 2017

"0 is not required at all in mathematics, just like negative numbers."
-- Jan. 4, 2017

What a moron!

**************************************

Dear reader,

Are you fed up with trolls like this idiot JG and his endless, repetitive torrents of abuse and misinformation? Join me at a new, lightly moderated math forum at Google Groups. Its open format promises to be a troll-free zone for serious discussions of mathematical topics at all levels.

https://groups.google.com/forum/#!forum/sci-math-moderated

All can visit and read.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

Alan Mackenzie

2019-02-04 15:55:54 UTC

Permalink

You might have thought so. But there are some very energetic trolls on
this newsgroup who attempt to spread confusion and doubt about such
things. They are also known for promulgating mathematical falsehoods.
One of these trolls has already answered you. Beware of them.

Post by j***@gmail.com
Does continuum (and anything that leads to it) really "exist" or our
mind cheats into believing us in one? Do "dense enough" linear points
"intellectually appear" continuous akin to picture frames "visually
appearing" as a movie?

I notice the way you put "exist" into quote marks. :-) From a
mathematical point of view, the continuum simply exists.

Post by j***@gmail.com
A circle may never be "possible" and hence pi may never be "possible".
It may just appear to be possible. Similarly, drawing a hypotenuse on a
isosceles right angle triangle may never be "possible". Euclid's
straight line may never exist. Such action may just "appear to make
sense". May be our intellect is collapsing such constructs into
1. lack of processing bandwidth of mind
2. lack of interesting information
3. survival technique through evolutionary means
4. our minds playing tricks to reduce information
5. ...
(I hope I make sense till this point.)

I hope that, too. ;-) In mathematics, something generally not being
possible means it leads to a contradiction. In that sense, a circle and
that triangle very much are possible and real. So are straight lines.

Post by j***@gmail.com
Is continuum proven or hypothesized?

It is proven, starting from an appropriate set of axioms.

Post by j***@gmail.com
Once again, please don't flame. I really, really don't know math.

I predict you will get flamed, here. Just not by me. Thanks for putting
these points forward.

Post by j***@gmail.com
-Bhushit

--
Alan Mackenzie (Nuremberg, Germany).

FredJeffries

2019-02-04 17:25:27 UTC

Permalink

On Monday, February 4, 2019 at 1:58:38 AM UTC-8, ***@gmail.com wrote:

First of all, I would like to congratulate you and thank you for the most intelligent post to this group in several weeks.

Actually, you CAN (argue about them), as (for instance) Skolem, Godel, Robinson, Tennenbaum, etc have shown.

Post by j***@gmail.com
Does continuum (and anything that leads to it) really "exist" or our mind cheats into believing us in one?

Humans have been arguing over that for thousands of years. I, personally, see no reason to believe a definitive answer to be forthcoming during the rest of my lifetime.

Post by j***@gmail.com
Do "dense enough" linear points "intellectually appear" continuous akin to picture frames "visually appearing" as a movie?
1. lack of processing bandwidth of mind
2. lack of interesting information
3. survival technique through evolutionary means
4. our minds playing tricks to reduce information
5. ...
(I hope I make sense till this point.)

or perhaps metaphorical extrapolation from our experiences with indefinite heaps? Or illusions foisted upon us by our technology?

For the first you might consult Shaughan Lavine's "Understanding the Infinite" (although some mathematical understanding is necessary) and Nunez and Lakoff's "Where Mathematics Comes From"

https://philpapers.org/rec/LAVFM
http://www.cogsci.ucsd.edu/~nunez/web/
https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From

For the latter, Tobias Dantzig and Marshall McLuhan trace the modern emphasis to culture and technology: The repetition of the printing press and the machinery of the industrial revolution evoked the notion of a never-ending successor principle.

Tobias Dantzig, Number: The Language of Science
https://books.google.com/books?id=zVsgAQAAIAAJ

Marshall Mcluhan, The Gutenberg Galaxy
https://books.google.com/books?id=VWspDwAAQBAJ

Post by j***@gmail.com
Is continuum proven

Ah! Proven from WHAT?

Post by j***@gmail.com
or hypothesized?

Perhaps it is merely taken as a working hypothesis to enable us to DO (modern) mathematics.

Or perhaps it has been reverse engineered (by Dedekind, Cantor, Weierstrass, et al) from our culture's "belief" in Laplacian determinism.

Post by j***@gmail.com
Once again, please don't flame. I really, really don't know math.
-Bhushit

FredJeffries

2019-02-06 21:30:22 UTC

Permalink

Post by FredJeffries
First of all, I would like to congratulate you and thank you for the most intelligent post to this group in several weeks.

Actually, you CAN (argue about them), as (for instance) Skolem, Godel, Robinson, Tennenbaum, etc have shown.

Post by j***@gmail.com
Does continuum (and anything that leads to it) really "exist" or our mind cheats into believing us in one?

Humans have been arguing over that for thousands of years. I, personally, see no reason to believe a definitive answer to be forthcoming during the rest of my lifetime.

or perhaps metaphorical extrapolation from our experiences with indefinite heaps? Or illusions foisted upon us by our technology?
For the first you might consult Shaughan Lavine's "Understanding the Infinite" (although some mathematical understanding is necessary) and Nunez and Lakoff's "Where Mathematics Comes From"
https://philpapers.org/rec/LAVFM
http://www.cogsci.ucsd.edu/~nunez/web/
https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From
For the latter, Tobias Dantzig and Marshall McLuhan trace the modern emphasis to culture and technology: The repetition of the printing press and the machinery of the industrial revolution evoked the notion of a never-ending successor principle.
Tobias Dantzig, Number: The Language of Science
https://books.google.com/books?id=zVsgAQAAIAAJ
Marshall Mcluhan, The Gutenberg Galaxy
https://books.google.com/books?id=VWspDwAAQBAJ

Post by j***@gmail.com
Is continuum proven

Ah! Proven from WHAT?

Post by j***@gmail.com
or hypothesized?

Perhaps it is merely taken as a working hypothesis to enable us to DO (modern) mathematics.
Or perhaps it has been reverse engineered (by Dedekind, Cantor, Weierstrass, et al) from our culture's "belief" in Laplacian determinism.

For "real world" implications of the continuum, see Leslie Lamport's discussions of Buridan's Principle: A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time

http://lamport.azurewebsites.net/pubs/pubs.html#buridan
https://en.wikipedia.org/wiki/Buridan%27s_ass#Buridan's_principle

which has applications in electrical engineering

https://en.wikipedia.org/wiki/Buridan%27s_ass#Application_to_digital_logic:_metastability
https://en.wikipedia.org/wiki/Metastability_(electronics)

j4n bur53

2019-02-04 17:40:13 UTC

Permalink

An early account of continuum is attributed to Eudoxos:

Eudoxus, 408 - 355 BC, Ancient Greek
Eudoxus's life was one of extreme struggle and poverty. A
physician and administrator, his main mathematical archivement
was the creation of a theory of geometrical continuum in
which magnitudes and measures could be made as small as
desired by repeated divison of a given length. He gave
birth to the mathematical notion of making deductions
from certain clear postulates, a notion which was refined
by Euclid in "The Elements".

But dividing by 2 would only give dyadic rationals.
So how comes?

Post by j***@gmail.com
I am not a mathematician.
I am not even sure whether I will be able frame this question properly - far from understanding the answers. Please pardon me in advance.
I got this idea from persistence of vision.
To me, various number sets - natural, integers, rationals,... kind of "exist". Can't argue much about them.
Does continuum (and anything that leads to it) really "exist" or our mind cheats into believing us in one? Do "dense enough" linear points "intellectually appear" continuous akin to picture frames "visually appearing" as a movie?
1. lack of processing bandwidth of mind
2. lack of interesting information
3. survival technique through evolutionary means
4. our minds playing tricks to reduce information
5. ...
(I hope I make sense till this point.)
Is continuum proven or hypothesized?
Once again, please don't flame. I really, really don't know math.
-Bhushit

j4n bur53

2019-02-04 17:51:30 UTC

Permalink

Definition V.5 of Euclid seems from Eudoxus,
and allows to compare irrational numbers.

https://www.spektrum.de/wissen/eudoxos-von-knidos-der-schoepfer-der-exhaustionsmethode/1331747

The idea of Definition V.5 leads more or
less directly to Dedekind Cuts.

Post by j4n bur53
Eudoxus, 408 - 355 BC, Ancient Greek
Eudoxus's life was one of extreme struggle and poverty. A
physician and administrator, his main mathematical archivement
was the creation of a theory of geometrical continuum in
which magnitudes and measures could be made as small as
desired by repeated divison of a given length. He gave
birth to the mathematical notion of making deductions
from certain clear postulates, a notion which was refined
by Euclid in "The Elements".
But dividing by 2 would only give dyadic rationals.
So how comes?

Ross A. Finlayson

2019-02-04 22:45:24 UTC

Permalink

Mathematical continuity among the
objects of mathematics, has of
course that modern axiomatics
has a goal of independent,
consistent axioms that define
objects and their relations
then for reasoning about the
object, and particularly only
the objects as formally defined
to exist as follow axioms of the
theory.

Then, some have something like the
integers, following Peano's axioms,
these are "non-logical" as so defined,
where the "logical" or "purely logical"
content is as per the rules of reasoning,
and here of logical and mathematical objects.

In many extended discussions, we talk about
mathematical continuity as the elephant,
as in various guises it looks to users of
the mathematical objects as various things:
as (the completion from more than the) limits
of series that define rational then real numbers
in a particularly way (establishing least upper
bound, Dedekind completeness of the complete
ordered field of rationals), then (the completion
from more than the) density of the rationals,
establishing some saturation property, and the
(completion from more than the) putting the points
in the line is making the points of the line.

So, the modern curriculum, almost universally
builds the complete ordered field or "Dedekind's
definition" (Eudoxus/Cauchy/Dedekind), this
"field continuity".

Then something like Zeno's motion or the background
of motion or sweep, this "line continuity", has a
different formalism, and is usually quite all left
out of the modern curriculum, because it gets into
the definitions of "what are Cartesian functions"
and "what are Cantorian sets", here that functions
aren't necessarily Cartesian, and orderings, vis-a-vis,
sets, ordinary sets from ordinary set theory.

Then that the rationals are almost half as dense
as the reals (to be continuous, i.e., to establish
the fundamental theorems of the integral calculus,
the infinitesimal analysis, real analysis, to
establish a continuous domain), is yet a _third_
definition altogether, of "what is mathematical
continuity".

Then usually in mathematics "continuous" as descriptive
is only applied to functions, that infinitesimal
differences see infinitesimal differences and couched
in the language of "delta-epsilonics", here some
"continuous domain" basically sees that as after a
prototype of a line or ray or unit line segment.

line continuity
field continuity
signal continuity

m***@gmail.com

2019-02-05 03:04:02 UTC

Permalink

Post by Ross A. Finlayson

Mathematical continuity among the
objects of mathematics, has of
course that modern axiomatics
has a goal of independent,
consistent axioms that define
objects and their relations
then for reasoning about the
object, and particularly only
the objects as formally defined
to exist as follow axioms of the
theory.
Then, some have something like the
integers, following Peano's axioms,
these are "non-logical" as so defined,
where the "logical" or "purely logical"
content is as per the rules of reasoning,
and here of logical and mathematical objects.
In many extended discussions, we talk about
mathematical continuity as the elephant,
as in various guises it looks to users of
as (the completion from more than the) limits
of series that define rational then real numbers
in a particularly way (establishing least upper
bound, Dedekind completeness of the complete
ordered field of rationals), then (the completion
from more than the) density of the rationals,
establishing some saturation property, and the
(completion from more than the) putting the points
in the line is making the points of the line.
So, the modern curriculum, almost universally
builds the complete ordered field or "Dedekind's
definition" (Eudoxus/Cauchy/Dedekind), this
"field continuity".
Then something like Zeno's motion or the background
of motion or sweep, this "line continuity", has a
different formalism, and is usually quite all left
out of the modern curriculum, because it gets into
the definitions of "what are Cartesian functions"
and "what are Cantorian sets", here that functions
aren't necessarily Cartesian, and orderings, vis-a-vis,
sets, ordinary sets from ordinary set theory.
Then that the rationals are almost half as dense
as the reals (to be continuous, i.e., to establish
the fundamental theorems of the integral calculus,
the infinitesimal analysis, real analysis, to
establish a continuous domain), is yet a _third_
definition altogether, of "what is mathematical
continuity".
Then usually in mathematics "continuous" as descriptive
is only applied to functions, that infinitesimal
differences see infinitesimal differences and couched
in the language of "delta-epsilonics", here some
"continuous domain" basically sees that as after a
prototype of a line or ray or unit line segment.
line continuity
field continuity
signal continuity

The infinitely small beginning magnitude builds
with itself for complete continuity of quantity.
Einstein needed to represent that math
is the ultimate continuum... his physical Unified field
comes first from it.

God creates gravity.

Mitchell Raemsch

Ross Finlayson

2024-11-12 04:44:14 UTC

Permalink

Post by Ross A. Finlayson

Mathematical continuity among the
objects of mathematics, has of
course that modern axiomatics
has a goal of independent,
consistent axioms that define
objects and their relations
then for reasoning about the
object, and particularly only
the objects as formally defined
to exist as follow axioms of the
theory.
Then, some have something like the
integers, following Peano's axioms,
these are "non-logical" as so defined,
where the "logical" or "purely logical"
content is as per the rules of reasoning,
and here of logical and mathematical objects.
In many extended discussions, we talk about
mathematical continuity as the elephant,
as in various guises it looks to users of
as (the completion from more than the) limits
of series that define rational then real numbers
in a particularly way (establishing least upper
bound, Dedekind completeness of the complete
ordered field of rationals), then (the completion
from more than the) density of the rationals,
establishing some saturation property, and the
(completion from more than the) putting the points
in the line is making the points of the line.
So, the modern curriculum, almost universally
builds the complete ordered field or "Dedekind's
definition" (Eudoxus/Cauchy/Dedekind), this
"field continuity".
Then something like Zeno's motion or the background
of motion or sweep, this "line continuity", has a
different formalism, and is usually quite all left
out of the modern curriculum, because it gets into
the definitions of "what are Cartesian functions"
and "what are Cantorian sets", here that functions
aren't necessarily Cartesian, and orderings, vis-a-vis,
sets, ordinary sets from ordinary set theory.
Then that the rationals are almost half as dense
as the reals (to be continuous, i.e., to establish
the fundamental theorems of the integral calculus,
the infinitesimal analysis, real analysis, to
establish a continuous domain), is yet a _third_
definition altogether, of "what is mathematical
continuity".
Then usually in mathematics "continuous" as descriptive
is only applied to functions, that infinitesimal
differences see infinitesimal differences and couched
in the language of "delta-epsilonics", here some
"continuous domain" basically sees that as after a
prototype of a line or ray or unit line segment.
line continuity
field continuity
signal continuity