Certainly mathematics is a social enterprise. And what constitutes a proof is a kind of consensus among those who practice mathematics. However, when I discovered last night a fact about attracting fixed points in polynomials that minor discovery immediately assumed mathematical existence, regardless of whether it is publicized. And it is possible someone else had arrived at this trivial conclusion, so it might have had mathematical existence already. But, in the larger social scheme there is a kind of mathematical existence based upon an agreement that a revelation is important.

https://math.stackexchange.com/questions/1052299/what-is-a-simple-example-of-an-unprovable-statement

Some mathematicians love this sort of thing. Not me.

Can you create a kind of "staircase" that converges uniformly to the hypotenuse and preserves arc length, so no "paradox?" :chin:

But the point made in the book, (and I think it is correct), is that each formula (other than those applying probabilities), will only lead to one inevitable outcome from any specific start point. That is Determinism. — Gary Enfield

Of course. But who knows what lies beyond mathematical horizons? Certainly in my modest experience mathematical research strongly favors determinism, except for conjectures that currently lie beyond proof. I've reached stages arising from the study of parabolic fixed points in the complex plane where I suspect a conclusion, but cannot reach it logically. This doesn't mean the problem can't be solved, but work stops at a point beyond which two possibilities seem to exist, although computer experiments strongly favor one.

This isn't exactly what you are getting at, and is not the quandary in QM, but it illustrates subtleties in the use of the word "inevitable."

1. Define concepts rigorously.
2. Do no conceptual harm.
:chin:

Interesting take on the situation. (For the uninitiated, functionals map functions to numbers)

Under what circumstances might the past be altered? — jgill

None, since the past is determinate. — aletheist

Wouldn't be too sure of that. Speculative notions like the multiple universe theory might have a bearing. If it were possible to somehow influence the past, changes might simply flow into an alternate universe, avoiding the Grandfather Paradox.

What we currently know of space and time may seem quaint and naive by future generations. :cool:

We can make truthful 'is' and 'is not' statements concerning the past, because the past has already occurred and is therefore determinate — Metaphysician Undercover

Under what circumstances might the past be altered? The Grandfather Paradox? :chin:

Well, the formal answer is that the limit towards which the sequence is converging is not an element of the sequence — SophistiCat

Not quite, but close. The sequence actually converges uniformly to the hypotenuse, but arc length is not necessarily preserved under uniform convergence.

Note the outer corner points seem to generate a line as n increases, but is the eventual line entirely composed of a countable set of points? How can this be? :chin:

That doesn't seem so paradoxical to me. Is there another, perhaps more mathematical, paradox buried in this optical illusion? Cabbage farmer

It's called a paradox in the literature. Note the outer corner points seem to generate a line as n increases, but is the eventual line entirely composed of a countable set of points?

You can obtain the result of the other "paradox" by drawing a symmetrical sawtooth graph on [0,1] that collapses as n increases, and whose length increases without bound. I leave this as an exercise for those interested. Is the ultimate graph differentiable anywhere?

I'm beginning to sound like a math prof. Sorry! :cool:

The first order reaction is mainly instinctual, emotional, and prone to conditioning. The second order reaction is where meta-beliefs take place. — Shawn

Not being a philosopher I interpret this as instinctive (fight or flee), and thoughtful (typical low-pressured problem solving). Am I missing a subtlety? Is this philosophy? :roll:

Sorry to interrupt your discussion, but even though I am a retired math prof I continue to learn about mathematics results by following some of these threads. My thanks to the participants.

Fishfry in particular has opened my eyes to modern set theory, but others have as well. And for game theory, I knew that Nash had used attractive fixed points but I now learn he employed a result I was unaware of, a set theory extension of Brouwer's Fixed Point theorem by Kakutani:

https://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem

Brouwer's Theorem provides an existence result, but doesn't give an algorithm for reaching this point. I am quite familiar with Banach's Fixed Point Theorem (having generalized his result for infinite compositions of functions rather than iteration of a single function - there are dozens of generalizations!) which does describe a simple algorithm.

( https://www.coloradomesa.edu/math-stat/catcf/papers/banach-extension-theorem.pdf }

OK, I'm done. :cool:

The language of the conscious mind, which developed from picture, to symbol, to letter is not one that the sub conscious understands. It understands stories and visualisation as shown by hypnosis and the like. — Antidote

Can you give a reference to this claim, something beyond an Eastern religious doctrine? In my opinion, as humans we reach our potential not by avoiding an aspect of mind, but by living in a kind of balance between the various aspects.

If traditional maths is essentially Determinism, as it produces single inevitable outcomes, (other than when probabilities are deployed), then how should we interpret the multiple outcomes? — Gary Enfield

Mathematics and physics are interwoven, but basically they are separate disciplines. Normally in physics when existing math seems inadequate physicists search for or create new mathematics that might apply, I.e., has predictive qualities. Feynman, following Dirac's work, developed a "path integral" which is not entirely kosher in math departments' faculty lounges, but has predictive value. It can be interpreted as a kind of functional integral. :cool:

Does thinking therefore add anything to understanding, or does an absense of thought allow insight to arise? — Antidote

Mathematical research problems are frequently resolved by diligent thought for a period of time, then relaxing the mind and going about one's daily routine, allowing the subconscious to produce results. However, the subconscious is not infallible and what bubbles up can be disappointing! :cool:

I know very little about game theory other than Nash's work involved attracting fixed points (which I've dabbled with), but I'll be interested in what your friends have to say. Maybe there is a simple explanation. Fishfry? FDrake? Others?

In any sufficiently complex game, given enough iterations, it can be demonstrated that both players become hyper-rational, and thus a winning strategy cannot be entertained. — Shawn

Would you provide a source for this assertion, please. Thanks.

B) The wiggly curve that uniformly converges to the line segment [0,1] is more entertaining, for its length becomes infinite. — jgill

This I don't buy. The combined length of the line segments that represent the lengths respectively of the steps and risers that comprise the stairs is just two. There is no (other) "wiggly" line. There are only steps, however large or small, and together they cross, as steps, a distance of one horizontally and one vertically. — tim wood

Wrong paradox, Tim. The wiggly curve converges uniformly to the line segment [0,1] while its length tends to infinity. Sorry I don't have the image.


Back to the Diagonal:

Imagine a straight line that rests on the outer stair corners, from the bottom most step to the topmost step. As the steps shrink that line more closely resembles the hypotenuse of the large triangle of length square root of two. Now, what can you say about all those outer corner points, which lie on this evolving line, as the number of steps becomes infinite?

There is no "more and more," although there can be be a lot of steps — tim wood

As individual steps shrink in size, the inside corner point - the part of the step furthest from the imaginary limiting line if that line is visualized as above the steps - grows closer to that line. So, yes, there is "more and more". However, the total or accumulated error remains large.

The wiggly curve that uniformly converges to the line segment [0,1] is more entertaining, for its length becomes infinite.

The word "paradox" has two meanings: 1) something that is true but self-contradictory, and 2) something that is true and seems self-contradictory, but in fact isn't — Daz

Yes. These two instances rely upon point of perspective, as Mr. Wood explains. Tarski-Banach seems to depend upon the controversial Axiom of Choice, which doesn't come into play here.

What happens to all the corner points in the stairs as the number of steps increases without bound? :chin:

I've wondered if Tegmark is really serious. :chin:

I tend to look at this through the lens of approximation theory which at times breaks down a process defined over an interval, to minute steps. Riemann integration theory requires such analysis, in which the area under a continuous curve is approximated by thin rectangles, then the error in approximation shrinks to zero as the number of rectangles increases without bound.

One of the celebrated theorems in this subject is Stone-Weierstrass, which says continuous functions on an interval can be approximated by polynomials.

If you were to cite any degree of magnification, I could find an n such that the approximating figures would appear to be the curve in question. :nerd:

I know, silly games.

"Seems to more and more approximate . . . "

The key to dissolving the apparent paradox is to calculate the error in approximation for each tiny right triangle, then add them up. If each step has $\Delta x=\Delta y=\tfrac{1}{n}$ then the increment of error is
$\Delta \varepsilon =\left( 2-\sqrt{2} \right)\Delta x\text{ }\Rightarrow \text{ }Total\text{ }error=n\Delta \varepsilon =2-\sqrt{2}$

Each member of the sequence of functions defined on [0,1],

${{f}_{n}}(x)=\tfrac{1}{10\sqrt{n}}Sin(n\pi x),\text{ }x\in [0,1]$,

is a smooth, wiggly curve oscillating about the line segment [0,1], getting closer and closer to that line as n increases in value. The length of the nth curve is

${{L}_{n}}\gt\tfrac{\sqrt{n}}{5}\text{ }\to \text{ }\infty$,

OK, thanks. Another concept I was unfamiliar with! :smile:

Is this what you are talking about? :

https://www.encyclopediaofmath.org/index.php/Density_of_a_set

Take a line segment of length one, then the circumference of the circle is also one: 1=pi*D means the diameter is irrational. :smile:

Or not??

It also figures into Quantum theory by way of Feynman diagrams.

Sorry for the intrusion. Just checking a link and now can't get rid of it! See my new thread. :sad:

Here is a possibility for VP that Biden once mentioned:

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

Indeed, and if time is not composed of instants, then it must be continuous. — aletheist

Would that imply time is also "smooth?" Or space/time?

Strictly speaking, math is just like theoretical physics: it's a branch of philosophy. — Gregory

Oh no, no, no . . . anything but that! :scream:

I think mathematicians should consider philosophy thru considering substance. — Gregory

There's always the danger of abuse. :gasp:

Once again I attempt to get the conversation about time moving in a slightly different direction:

Wiki: "With an incomplete theory of quantum gravity, it is impossible to be certain what spacetime would look like at small scales. However, there is no reason that spacetime needs to be fundamentally smooth. It is possible that instead, in a quantum theory of gravity, spacetime would consist of many small, ever-changing regions in which space and time are not definite, but fluctuate in a foam-like manner.[3]

Wheeler suggested that the Heisenberg uncertainty principle might imply that over sufficiently small distances and sufficiently brief intervals of time, the "very geometry of spacetime fluctuates".[4] These fluctuations could be large enough to cause significant departures from the smooth spacetime seen at macroscopic scales, giving spacetime a "foamy" character."

And I know mathematicians can hate philosophers :) — Gregory

Not really. We are quite tolerant, and chuckle as we do when a cute puppy barks. :smile:

Indeed, mathematics is 75% invented, 25% discovered — Gregory

The controversy among mathematicians whether math is discovered or created has finally been laid to rest. Thank you, Gregory! :cool:

I am 83, and from my perspective both Biden and Sanders are too old for the presidency. The Democratic choice for vice president is far more important. Biden may win the top job, but the VP will gradually take over behind the scene. So who might that be? :chin:

Since numbers by themselves sit in their own world within the human imagination, you can always make up your own rules as you go with postulations, right? Go ahead. That's pure mathematics research right there. — flame2

If you say so. :roll:

The forum exists. I have my doubts about philosophy. :chin:

Here's a thought. These arguments about the nature of time go on interminably without clarification, repeated over and over. Why not try a different approach:

Momentum, from physics, is defined as mass times velocity: momentum=mass x ds/dt.
Why not try to define "momentum" and "mass" and "distance" for the evolution of ongoing events, then define an increment of time by: dt=mass x ds/mom? :chin:

Don't ask me how. Someway define the process of multiplication and division in the equation.

Or return to formal logic and proceed.

consistent with my contention that continuous motion is the fundamental physical reality, while discrete positions in space and instants in time are artificial creations of thought to facilitate describing such motion — aletheist

Here is Peter Lynds' paper in which he postulates that there are no instants of time. This was published in Foundations of Physics and generated heated arguments, with a strong contingent of physicists declaring it nonsense. But who knows? Lynds is or was a college dropout.

https://arxiv.org/ftp/physics/papers/0310/0310055.pdf

We can talk about infinity in percentages though. — Zelebg

Novel notion, Z! How would you describe 50% of infinity? :chin:

Having spent an entire math career not concerned with these things, I now find there is literally no escape: https://en.wikipedia.org/wiki/Reverse_mathematics

I wasn't even aware of this before opening the brochure I got in the mail today from the Princeton Press.

:worry: