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On Mathematical Beauty in Physics

blogs.unimelb.edu.au

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Re: On Mathematical Beauty in Physics

#31

Post by Mugwort »

I'll say something regarding string theory which has been accused of leading physicists astray because of its "mathematical beauty". First of all, string theory in its present form is NOT mathematically beautiful. The mathematical theories used in string theory are in fact beautiful but the way they are stitched together is an UGLY Frankenstein mess. It's essentially the same ugly math that particle physics was built upon in the 60s and 70s. Lie algebras, conformal fields, algebraic topology etc. are all very beautiful maths but they rest upon this ugly mess of correlation functions, vertex operators, BRST invariant (gulp) Lagrangians and other brick-a-brack nobody should be proud of but gets the job done.

The problem with string theory is 21st century mathematics is still in its infancy. I have reasonable confidence that string theory is essentially correct only that the present maths and our understanding of QM (which needs improvement) is definitely not up to the task. Many topics like the Penrose singularity theorems in General relativity were impossible until Einstein's theory was formulated correctly with rigorous mathematics. IMHO this is the case with string theory. It simply won't work with the tools we have.

Another thing about String theory is that it very well could be a complete description of physics on anti-deSitter space. That would be progress but it still wouldn't be a unified field theory. We would need to find a more general theory which allows for deSitter space.

So there's my opinion. If anyone doubts mathematical beauty leads us to better things, plain old vanilla Classical mechanics is looking stronger than ever before and is still yielding interesting physics. Part of the reason for this is that the mathematical foundation of CM were still not understood properly until the mid 20th century. Give the other theories time to catch up. Mathematical beauty is the best guide we have. If strings turn out to be wrong then that's fine. Mathematical beauty is still the best guide we have. If we listen closely enough to what the equations are really saying then we'll find something better.

Re: On Mathematical Beauty in Physics

#32

Post by klyrs »

chr1 wrote: post #26

Does anyone know example of math that is not beautiful? As far as i can tell everything that explains something more complex in terms of simpler rules is called beautiful, but that is all the math! So it's not very surprising that math in physics is beautiful too.

The classification of finite simple groups. From an outsider's perspective, it's ream upon ream of brutal case analysis. Some will certainly find beauty in the thousands of pages; the historicity nature of the ~150y effort is undeniable and awe-inspiring.

Classification theorems, in general, are frequently beautiful in their structure. But they often involve delving deep into rabbit holes, in order to prove the non-existence of the impossible.

Re: On Mathematical Beauty in Physics

#33

Post by Mugwort »

chr1 wrote: post #26

Does anyone know example of math that is not beautiful? As far as i can tell everything that explains something more complex in terms of simpler rules is called beautiful, but that is all the math! So it's not very surprising that math in physics is beautiful too.

The representation theory of Kac-Moody algebras. There are some issues with integration theories that need clearing up, should we really be ditching the Riemann integral for the Lebesgue integral so quickly without a closer look. Recently someone wrote a good book on linear algebra that puts of "ugly" determinants for as long as possible. Much of the problem is making what we already know cleaner and better. Some theories like set theory don't have a unique set of axioms and have lots of loose ends. General topology is a closed subject but there are still potentially better ways to formulate convergence.

Re: On Mathematical Beauty in Physics

#34

Post by Mugwort »

Mugwort wrote: post #33
chr1 wrote: post #26

Does anyone know example of math that is not beautiful? As far as i can tell everything that explains something more complex in terms of simpler rules is called beautiful, but that is all the math! So it's not very surprising that math in physics is beautiful too.

The representation theory of Kac-Moody algebras. There are some issues with integration theories that need clearing up, should we really be ditching the Riemann integral for the Lebesgue integral so quickly without a closer look. Recently someone wrote a good book on linear algebra that puts of "ugly" determinants for as long as possible. Much of the problem is making what we already know cleaner and better. Some the…

My point is the stuff we already know so well still could use some polishing and everybody should be learning newer and better approaches to classical subjects.

Re: On Mathematical Beauty in Physics

#36

Post by Mugwort »

klyrs wrote: post #32
chr1 wrote: post #26

Does anyone know example of math that is not beautiful? As far as i can tell everything that explains something more complex in terms of simpler rules is called beautiful, but that is all the math! So it's not very surprising that math in physics is beautiful too.

The classification of finite simple groups. From an outsider's perspective, it's ream upon ream of brutal case analysis. Some will certainly find beauty in the thousands of pages; the historicity nature of the ~150y effort is undeniable and awe-inspiring. Classification theorems, in general, are frequently beautiful in their structure. But they often involve delving deep into rabbit holes, in order to prove the non-e…

Agreed. Few people ever read and follow those proofs. Another related issue is proofs by computer e.g. the four coloring problem. No human will ever understand every step the computer is doing in these proofs but they can understand the program doing the proof and why it must be correct. If we insist on proofs like Euclid's proof that there are infinite primes or sqrt(2) is irrational then these computer proofs are clearly different. If we must use a computer is the proof somehow "ugly"? I don't think so but maybe I'm wrong.

Re: On Mathematical Beauty in Physics

#37

Post by blackholesRhot »

Mugwort wrote: post #31

I'll say something regarding string theory which has been accused of leading physicists astray because of its "mathematical beauty". First of all, string theory in its present form is NOT mathematically beautiful. The mathematical theories used in string theory are in fact beautiful but the way they are stitched together is an UGLY Frankenstein mess. It's essentially the same ugly math that particle physics was built…

Couldn't agree more.

Re: On Mathematical Beauty in Physics

#38

Post by jxy »

pnx wrote: post #25
univalent wrote: post #3

Reminds me of this paper in Nature from some time back. Especially, the Euler's identity graph from the paper is pretty amazing. https://www.nature.com/news/equations-are-art-inside-a-mathe...

I find it odd that people think that equation is beautiful. Rewriting it in polish notation/lisp you end up with: (= (+ (exp (* i pi)) 1) 0) Which is neither beautiful nor very enlightening.

Given notation as a tool of thought, it is really about which notation you are most familiar with. No doubt some people would find lisp notation enlightening, and some people would find the following beautiful.

        0 = 1 + * ○ 0j1

Re: On Mathematical Beauty in Physics

#39

Post by dorchadas »

certmd wrote: post #9

I found the book "Our Mathematical Universe" by Max Tegmark, a physicist, a pretty good read. His idea is basically that the universe isn't explained by math, it IS math. To get to what he means by that he provides a good review of theoretical physics/cosmology/quantum mechanics. No idea how other physicists think of his work but for someone not in the field who has some interest I thought it was a good overview and…

This reminds me of the project being worked on by Cohl Furey at Cambridge to basically explain physics out of the four normed divison algebras. I'm really interested to see where that leads, and kinda want to head into that field myself.

Re: On Mathematical Beauty in Physics

#40

Post by lidHanteyk »

ses1984 wrote: post #27
613style wrote: post #22

Earlier quoted context omitted.

Don't confuse the terrain with the map. The formal systems which underly mathematics describing the universe are incomplete (containing true but unprovable statements), but that is not a statement about the universe.

I'm not sure I see the point. Or rather, I see The tautology. We discover something math can't explain. By definition, you can't expand math to explain it. You can try, and if you succeed you were wrong all along, in that your discovery could have been explained by math you just couldn't prove it at the time.

That's not exactly what Gödelian incompleteness means; it means that there are gaps within each formal system. Arithmetic is still there, beyond reality, beyond each formal system's ability to describe in full.
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