Why Does Mathematics Work?

tuco · **#21** by **tuco** » May 05, 2019 10:36 am

I can.

Spearthrower · **#22** by **Spearthrower** » May 05, 2019 10:48 am

Thommo wrote:
tuco wrote:Now, can we imagine a universe where pi wasn't like the pi we know? Of course, we can.

Can we?

I expect we can imagine the notion of universes existing where pi isn't the value we know, but I doubt anyone can actually imagine pi not being the value we know, or what that would result in for a universe.

Again though, strange as it is, mathematics might be able to let us try it.

Thommo · **#23** by **Thommo** » May 05, 2019 11:13 am

I'm not sure I'm better informed by those answers.

The point I was making is that the curvature of a surface might lead to local observations of values such that the ratios of the length of a locus of fixed radius around a point to that radius does not come out to pi (indeed it might not be constant over a given space at all), but the general mathematical properties of surfaces allow us to embed an N-surface into an N+m-space*, and IIRC we can recover our cartesian properties in that superspace.

If there is a mathematical structure that cannot be so embedded it's not clear what it is or what approach "imagining" is supposed to denote. Can we specify some of the mathematical properties this universe has?

*An example of that would be that the Klein bottle can be embedded in R4.

ETA: A bit of googling shows this is the result I was trying to recall:
https://en.wikipedia.org/wiki/Whitney_embedding_theorem

In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:

The strong Whitney embedding theorem states that any smooth real m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2m-space (R2m), if m > 0. This is the best linear bound on the smallest-dimensional Euclidean space that all m-dimensional manifolds embed in, as the real projective spaces of dimension m cannot be embedded into real (2m − 1)-space if m is a power of two (as can be seen from a characteristic class argument, also due to Whitney).

Spearthrower · **#24** by **Spearthrower** » May 23, 2019 5:09 am

Weinberg - Is Mathematics Invented or Discovered?

newolder · **#25** by **newolder** » May 23, 2019 7:24 am

Tangentially related article - A Different Kind of Theory of Everything by Natalie Wolchover in the New Yorker discusses recent thoughts about the nature of the questions that require answering (by physicists) now and in the days to come (Nima et al's, "What does the volume of the amplituhedron tell us about scattering amplitudes?" &c) ...

Spearthrower · **#26** by **Spearthrower** » May 23, 2019 9:36 am

find the question to which the universe is the answer

I like it! :thumbup:

Calilasseia · **#27** by **Calilasseia** » May 23, 2019 11:02 am

My view, is that the reason mathematics enjoys its applicability, is this.

The moment one has any set of reliably repeatable interactions and well-defined entities taking part in those interactions, a mathematical model can be constructed thereof practically by definition. Because at bottom, mathematics is the study of reliably repeatable interactions and well-defined entities taking part therein.

TopCat · **#28** by **TopCat** » May 23, 2019 12:23 pm

Calilasseia wrote:Because at bottom, mathematics is the study of reliably repeatable interactions and well-defined entities taking part therein.

Isn't that true only of a subset of mathematics?

That is, there is lots of maths that doesn't describe observable things. Or is your definition of repeated interactions broader than that?

Perhaps I should say "doesn't yet describe observable things". Didn't Heisenberg independently invent matrix mechanics in his formulation of quantum theory?

Thommo · **#29** by **Thommo** » May 23, 2019 12:41 pm

TopCat wrote:
Calilasseia wrote:Because at bottom, mathematics is the study of reliably repeatable interactions and well-defined entities taking part therein.

Isn't that true only of a subset of mathematics?

You are correct. There are all sorts of obvious examples of mathematics that resemble that definition in few meaningful ways. Set theory, the study of proper classes, geometric objects like the Klein bottle or a well known result like the Banach–Tarski paradox to name but a few.

None of those govern interactions at all, or indeed real world objects, or anything you might term repeatable in any sensible way.

To be fair I don't think there's any recognised good definition of mathematics at all. In general it's not the sort of thing mathematicians worry about.

Macdoc · **#30** by **Macdoc** » May 23, 2019 9:20 pm

I like this

It is a logical system that builds an internally consistent set of rules that are amazingly useful in describing the world around us. ... You could argue that mathematics is a construct of the human mind.

that latter bit might e considered a tautology

tuco · **#31** by **tuco** » May 24, 2019 3:05 am

The question was not why does mathematics work but why the success of mathematics in its role in physics appears so baffling?

Spearthrower · **#32** by **Spearthrower** » May 24, 2019 3:25 am

No, it was quite clearly 'Why Does Mathematics Work?", and the other formulation isn't really a question.

Cito di Pense · **#33** by **Cito di Pense** » May 24, 2019 4:02 am

Spearthrower wrote:
Cito di Pense wrote:Why ask why?

I think it ends up being the most intriguing of questions.

One can read Wigner's essay as an elaborate joke at the expense of philosophers, but it's just an essay and not a proof of anything. In what follows, I have not disclosed everything I noted when I re-read Wigner's essay and watched the video.

In connection with this, I recently ran across some of Leslie Lamport's essays on how to write proofs. Mathematics is a language and proofs can be written clearly or obtusely (in the latter case, so that only someone who knows how to add the intervening steps of the proof can follow it). So the proof either works or it doesn't, depending on who's reading it.

We still go back to the question and ask, why can we write a proof that anyone can understand? and Susskind answers this by saying that we co-evolved with a world we can reason about. Then we can apply an anthropic principle again and say we would not be here if the world was a lot more disorganized than it is.

Rumraket · **#34** by **Rumraket** » May 24, 2019 8:38 am

I've always felt this question is close to nonsensical. Why does mathematics work? Because it can't not. In a world where there are quantitative relationships of any kind that don't constantly change infinitely fast, mathematics will obtain and can be used. It can't be any other way.

What would it take for mathematics not to work? Try to imagine a world in which there are no quantitative relationships of any kind. No distances, no numbers of entities, entities without any imaginable mathematically analyzeable properties (such that they can't, for example, be enumerated).
It should become obvious that such a world cannot even be imagined. Any world with any contents whatsoever of which there is even one, or more entities in existence, will have aspects of it that can be described by mathematics.

The questions of mathematics's success is frankly uninteresting. There's nothing here to be explained.

Macdoc · **#35** by **Macdoc** » May 24, 2019 9:43 am

(such that they can't, for example, be enumerated).

Yet Pi cannot be fully enumerated yet is very useful and what it is approximating can be easily seen.

Mathematics is a useful, transmittable way/language of preserving a description of the universe that can be acted on by others...or a CAM machine. IMNSHO

Rumraket · **#36** by **Rumraket** » May 24, 2019 12:05 pm

Macdoc wrote:
(such that they can't, for example, be enumerated).

Yet Pi cannot be fully enumerated yet is very useful and what it is approximating can be easily seen.

Mathematics is a useful, transmittable way/language of preserving a description of the universe that can be acted on by others...or a CAM machine. IMNSHO

I don't know what it means to "enumerate pi". Pi is itself a quantitative relationship between the radius and circumference of a circle. If circles is a logically coherent concept, then Pi obtains and so does all of geometry. It can't not. The fact that the base 10 representation of Pi has infinitely many decimals is besides the point.

My point is, try to imagine a world where there are "things" in it of any kind, which you could not count. A world where there are trees that can't be counted. You see a tree, but it would not make sense to say "there is one tree". You can't imagine such a world. If there are things in it, then there is either one, or more, or some fraction of them.

Or what about a world where there are no quantitative relationships between things in it. A world where it would not be true to say that there are more, or less, or equal amounts/numbers/magnitudes of A and B, yet both A and B exists. How could this be the case? It can't, the concept is unimaginable.

Mathematics "works" because the alternative is necessarily logically incoherent.

Spearthrower · **#37** by **Spearthrower** » May 24, 2019 1:08 pm

I think there are different levels to the conversation; weak anthropocentricism is one, enumerating discrete entities is one slice of mathematics, but there are other aspects to it which still contain some fairly complicated questions. The relationship between observed quantities and their interactions are pretty much banal in terms of 'why mathematics' - but positing extra dimensions which aren't observable but either appear necessary or just merely useful, and describing interactions within those notional realms even while remaining unobservable, and then predictions arising from that apparently offering real world utility in explaining factors of our space where we can observe interactions but can't explain them solely by observable quantities... all of which is where it becomes harder to explain why mathematics is so effective. As both videos note, in some instances mathematicians arrive - in the abstract - at extremely complex notions that physicists later find essential in dealing with observable quantities.

Macdoc · **#38** by **Macdoc** » May 24, 2019 1:26 pm

Rum

Deconstructing reality in useable quanta is a good tool but like logic, math is a descriptive construct.

A kid can draw a circle without using math.
Math can approximate what was drawn

"logically incoherent" puts a bit too much "must have" into play.
The kid can still create, use and manipulate the circle with neither logic nor math and perhaps not even with a word for what it is.

surreptitious57 · **#39** by **surreptitious57** » May 25, 2019 5:07 pm

The question should not be why does mathematics work but why does it work as well as it does
Why does a completely abstract concept describe physical reality to such a degree of precision

Macdoc · **#40** by **Macdoc** » May 25, 2019 6:36 pm

You need to parse that a bit. Quantasizing certain elements of reality allows consistency over time and distance say to build a boat to similar dimensions...feet and fingers were handy. :roll:

tho some natural variation was involved.

I'm more intrigued by what math might predict about reality that we haven't or cannot observe. Einstein had an incredible thought experiment avbout light and moving objects ....has the math to present a hypothesis, gets confirmation from a transit of venus and realization hits the science community somewhere along the dissemination of a new theory of physics he described could result in a device of mass destruction .....and was.

Unintended consequences driven by the equations. :cheers:

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