Re: Non-mathematician's question about continuum

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

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

>
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
>