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On Mathematical Beauty in Physics (unimelb.edu.au)
80 points by Anon84 59 days ago | hide | past | web | favorite | 54 comments

> as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen

There is a fair bit of mysticism that gets bandied about along these lines--and I can certainly see why a mathematical platonist might marvel at the spooky correspondence between beautiful mathematics and to-be-discovered physical theories.

But if you can let go of the platonism, then I think a rather straightforward explanation emerges:

A mathematician is one who explores and names the various ways that a human mind can stretch and fold. I guess it's kind of like yoga. The ways of folding that best reflect whatever it is that makes the human mind, well, human--those ones we call beautiful. Nature is the only thing not of human construction that we fold our minds to fit (you might call this process 'perception')--but we are still only able to fold in human-like ways.

So the mathematician catalogs the ways we can see, and the physicist catalogs the things we do see. It doesn't strike me as especially spooky that the physicist finds things that correspond nicely to the mathematician's list.

Except that sometimes the resonances are extremely peculiar. It's not terribly surprising that differential geometry was the right language for describing physics on spacetime. It's a bit more surprising (to me anyways) that the same probability distribution describes both the energy levels of complicated nuclei and the distribution of zeros on the critical line of the Riemann zeta function.

The spooky thing is that the physicist that "finds things that correspond nicely to the mathematician's list" is so freaking good at predicting nature and yielding it to his will.

People always looked at nature through the lens of their minds and came up with creative explanations for reality (like the gods/4 elements/mysticism etc) the difference is that they were much worse at any actual real-world predictions/control compared to modern day physicists.

So we much be on to something here.

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.

> the Penrose singularity theorems in General relativity were impossible until Einstein's theory was formulated correctly with rigorous mathematics

I wonder if you would please justify that statement, specifically and only with respect to one of the things called the [Hawking-]Penrose singularity theorem? In particular, what was sufficiently unrigorous or alternatively missing from classical General Relativity that blocked a reasonable choice -- especially the early 1960s ones (example and commentary below) -- of such a theorem?

[Penrose 1965] Gravitational Collapse and Space-Time Singularities https://doi.org/10.1103/PhysRevLett.14.57 (2.5 pages)

(You can also find a copy in the usual place)

(I submit the reason it took until 1964 for a Penrose singularity theorem is explained by the first two lines of [Penrose 1965]: the surprising discovery that QSO 3C 273's highly extragalactic redshift z ~ 0.16 having been published in 1963 and QSO 3C 147's z ~ 0.55 following in 1964 strongly suggested SMBH activity. In other words the underlined statement in the fourth paragraph did not follow any sort of mathematical development (ADM, for instance) but rather was motivated by one of the most provocative observations of nature I can think of off the top of my head, up there with the (also 1964) discovery of the CMB.)

I think OP's argument is that string theory is probably in roughly the same state as the Bohr model of the hydrogen atom: It's got some of the essential tensions right, but it's probably wrong in important details and it's missing a framework in which it all just makes sense.

In a similar vein, I wouldn't expect that the singularity theorems necessarily hold for some of Einstein's early attempts at GR.

Couldn't agree more.

I expected this to be an a new article or report of Sabine Hossenfelder's work, but apparently the author S is someone else?

Hossenfelder's counterpoint, disussed on HN in the past:


"A contrarian argues that modern physicists' obsession with beauty has given us wonderful math but bad science

Whether pondering black holes or predicting discoveries at CERN, physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones. This is why, Sabine Hossenfelder argues, we have not seen a major breakthrough in the foundations of physics for more than four decades. The belief in beauty has become so dogmatic that it now conflicts with scientific objectivity: observation has been unable to confirm mindboggling theories, like supersymmetry or grand unification, invented by physicists based on aesthetic criteria. Worse, these "too good to not be true" theories are actually untestable and they have left the field in a cul-de-sac. To escape, physicists must rethink their methods. Only by embracing reality as it is can science discover the truth."

Hossenfelder's blog: http://backreaction.blogspot.com/

Yep, this is Stella (someone); click on the author's tiny blue S and be rewarded with similar articles.


I think it’s mostly a case of the drunk looking for his keys under the lamp post because that’s where the light is.

We have no idea how much there is to know about reality. We do know how to describe some parts with mathematics, but who knows how much there is out there which is not possible to describe with mathematics.

We may just be discovering infinitesimally small part ls of reality which math happens to describe well.

Math is a description of our reality. If we discovered something that math couldn't explain we'd expand math to explain it.

We know about lots of things that math doesn't explain. AFAICT there are reasonable information-theoretic grounds for suspecting that many if not most things we might want to ask about won't yield the the kind of explanations we find satisfying. Irreducible complexity is a thing, at least apparently, and among documented phenomena there is a spectrum of amenability to formal modelling. Statements expositing an 'unreasonable effectiveness of mathematics' or TFA's aesthetic rendition don't seem to amount to more than 'some phenomena admit simple explanations'. Much more often the best effective mathematical explanation is a lukewarm admixture of awkward cludges, or is simply non-existent.

How does that square with something like godel's incompleteness therom?

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.

That's not what Godel showed.

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 even if his ideas are wrong gives you something to think about.

Keeping in mind that Tegmark's idea is 2,500 years old and the basis of Western civilization. Pythagoras was the biggest philosophical influence on Plato. And, according to Aristotle (150 years later), Pythagoras believed that "all is number".

Thales (the "first" Greek philosopher) believed that the underlying principle of the world was water, a kind of flowing force. Anaximenes, his student, believed the underlying principle was air, a sort of bouncing around set of particles.

Pythagoras believed that underneath everything, the most basic underlying principle was math. And not just that: he believed that the natural harmonies in mathematics led to the natural harmonies in the kosmos (a term he is credited with introducing, along with the term "philosopher").

Pythagoras had some insider knowledge, while Thales et al just speculated.

I'm curious what you mean, insider knowledge!

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.

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...

Everybody who writes about Semir Zeki (PI on that paper) and his work is polite about him. I can only assume that this is because he is a) a likeable guy and b) the professional aestheticians and artists who could make negative comments think it wiser not to get involved in any way lest they be accused of punching down.

Euler's identity is pretty obviously something elegant, and it's widely recognized. But Zeki seems to overlook the importance of acculturation in the appreciation of these things. The idea that a mind-blowing Ramanujan formula is ugly seems particularly problematic. Sure, it is surprising, complex, and apparently arbitrary, and most test subjects won't have any insight into how it works. But why should those people (let's say they are "ignorant" when it comes to Ramanujan's results) be expected to have any opinion on the formula? Any judgement they come up with will be superficial.

This problem runs through most of Zeki's work. His experiments take test subjects who are, as it were, preloaded with certain aesthetic experiences and norms, and attempt to derive universal cognitive principles of beauty from the result. What he doesn't do is try to discover just what those aesthetic prejudices are. He doesn't try to extract from the test subjects the key features of their aesthetic indoctrination. In the math situation, it's totally conceivable that a group of test subjects who were educated to find every Ramanujan formula transcendently fascinating would rate those formulas as beautiful.

The problem is more conspicuous when his research looks at art. Since he hasn't studied art himself, he really doesn't seem to appreciate the basic reality: that some people have learned to 'get' certain styles or genres of art, while others haven't had the good fortune to learn to appreciate those same styles. People walk around with extremely complex acquired aesthetic processing capabilities in their heads. Everyone's is different, and while Zeki may be able to find some universal regularities, (like Chomsky's hypothesized universal innate grammar) that wouldn't be a neural explanation of aesthetics any more than the success of Chomsky's programme would tell us about the nature of literature.

It's a frivolous exercise from the point of view of anyone seriously interested in art (or math).

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.

There is more to a beautiful equation than the shape of the symbols used to describe it. The beauty of Euler's identity is the relationship between 5 fundamental constants (0, 1, e, i, pi). It's simple, elegant and far reaching.

The relationship is the same regardless of the notation.

Mostly serious comment: I'm not sure why the form e^{i\pi}=-1 isn't better. Only it doesn't have 0, but is it worse, less beautiful? (It doesn't seem to have the same "relationship between 5 fundamental constants", although it adds the negative number realm, to the imaginary and transcendental–neat.) Would E-mc^2=0 be similarly be better than E=mc^2, because it has an additional "fundamental constant"?

People fetishize the formula but the beautiful idea is that multiplication/exponentiation can be expanded in such a way that it describes oscillatory relations. This is how eg. eigenvalues get to play a role in models of harmonic resonance. Or how AC impedance naturally generalizes DC resistance.

Try to imagine complex interest rates. Now try to make them matrix-valued. It works. It all works.

I don't want to start an argument about tau versus pi, but I like a tau form (or modification) of Eulor's Identy: [e^(ikτ) = 1] for all integer values of [k]. This gets across rather well that this formula expresses a complete turn around a unit circle. You can't get something quite equivalent using pi.

I even more prefer the full form of Euler's Formula: [e^(ix) = cos(x) + isin(x)]. The real beauty of Euler's Formula I think is that it shows an equivalence between an algebraic function and a trigonometric function.

(Note that I'm only a mildly learned laymen when it comes to mathematics. Any experts in math should feel very free to tell me why I'm wrong.)

As others have said, it is beautiful because of the relationship between e and pi. This is mind blowing. What do these 2 constants have in common? Not a thing. I feel the same way about the pi / 4 = 3/4 x 5/4 x .... infinite series. Why a relationship between pi and prime numbers?

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

Why would a sense of beauty be related to natural truth? Well, say that everyone is born with a randomly assigned set of aesthetic preferences. The people whose aesthetic preferences just so happen to align with a theory that's about to be discovered will be more likely to guess at it in advance, and they will have lots of career success once experiments confirm those guesses. This selects after the fact for people whose aesthetic preferences are luckily aligned with future discoveries. Anyone who prefers the aesthetics of bad theories over good ones will either waste their time or switch to a different field, and nobody will care what they thought was beautiful at the time.

In the end, when it comes time to vote on which theories are beautiful, the theorists who thought that the best theories were beautiful will say "I always thought it was beautiful," and everyone will listen to them because their randomly assigned bias towards the good theories would have led them to respected positions. The ones who always disliked the good theories will not be around, or will have a less respected voice.

That's my theory for how the word "beauty" gets preferentially assigned to good (as in true) theories. One implication of this theory would be that even though claims of beauty tend to align with known-true theories, they aren't useful for telling in advance which theories are going to be true: the people whose preferences have randomly attached them to the bad theories today won't be silenced until the experiments come out.

It also suggests that ugly theories are more likely to be overlooked. Since the profession selects for people who gravitate to a certain type of theory (what they call beautiful), there might be ugly-but-true theories waiting to be discovered.

Your 'theory' is just the survivorship bias.

Just a note on the author's comment under Goya's painting. She advances that "unlike painting", in physics beauty can only be appreciated by those who understand their meaning. I disagree there's a difference in these disciplines, case in point, I knew Goya's painting long before becoming interested in grecoroman mythology, and I plainly didn't like it. Today I can appreciate the brutal tone of the painting knowing the story behind, and that has turned my view of the opus.

Another shocking example is El Expolio by El Greco, shocking technique but doubtful setting... until the author's intention is explained (or understood without further word), at that point it just becomes beautiful.

And in the same way that it's not necessary to know the story behind in art (you could like a painting for technique, looks alone), a layperson could like a mathematical formula for purely aesthetic reasons.

So all in all, I don't think mathematics and physics stand out in this regard.

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 Heegner numbers are a good gateway into a brutally ugly and weird corner of analysis, with lots of almost-numbers and almost-facts: https://en.wikipedia.org/wiki/Heegner_number

In higher (n-)category theory, there is not a single canonical way to describe higher categories, leading to lots of careful use of "strict" and "weak" qualifiers, as well as many different non-equivalent flavors of higher category: https://ncatlab.org/nlab/show/semi-strict+infinity-category

Speaking of categories, the number of finite categories, or other finite abstract algebraic objects, is not a beautiful combinatoric theorem, but instead a nasty study of computer-aided searches: https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttw...

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.

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.

Good point, proofs that require brute force enumeration of all possibilities are not nice, but the physicists who use the results to describe some symmetries still think the results are beautiful.

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.

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.

“The artist, the musical composer, the architect, the scientist all feel a fundamental need to discover and create something new that is whole and total, harmonious and beautiful.”

“Having seen that the perception of harmony and totality need not be a purely private kind of judgement, one can now understand in a new light the fact that the really great scientists have, without exception, all seen in the structural process of nature a vast harmony of order of indescribable beauty… Indeed, every great scientific theory was in reality founded on such perception of some very general and fundamental feature of the harmony of nature’s order. Such perceptions, when expressed systematically and formally, are called “laws of nature.”

-- David Bohm, On Creativity

Quoting a teacher about math-related assignments.

"The answer has to be beautiful, being correct is obvious".

I wonder if anyone would find Einstein's field equation aesthetically pleasing.

To me, General Relativity has some pleasing qualities: it's mathematically complete, and it's very flexible.

As to the Einstein Field Equations (EFEs) themselves, the aesthetically pleasing quality is the terse notation brought about by the Einstein summation convention, abstraction into the Einstein tensor G_{\mu\nu}, the stress-energy-momentum tensor T_{\mu\nu} and other objects, and so forth (it's even shorter with geometrized units c = G = 1, ignoring the cosmological constant (say if one relies on thin shells and junctions[1]), \pi = 1, and/or 0 on the RHS).

This terseness can hide several pages of partial derivatives that look like http://4.bp.blogspot.com/-0e2Zl5QrRiA/UZfL29xVTZI/AAAAAAAAAF... (from NCSA's (offline) numerical relativity mathmine1.html originally) -- there will be sixteen of those for the Ricci tensor, plus some more pages for the other parts of the left-hand-side, although by introducing lots of symmetries and simplifications we can cut down by quite a bit.

One can get a feel for how these are generated with the example at https://www.maplesoft.com/support/help/Maple/view.aspx?path=... (eqn 12 for the metric at eqn 6)

Hartle's (anathe)Mathematica examples https://web.physics.ucsb.edu/~gravitybook/math/curvature.pdf are nice but don't take you to an interesting Ricci tensor.

Instead, perhaps see the answer https://mathematica.stackexchange.com/a/8908

Frankly, the mechanics of working with known solutions of the EFEs can be a pain. Perturbing against those can be even more of a pain. Junction conditions[1] between different exact solutions can be a super pain. Modern symbolic computing systems help a lot: https://en.wikipedia.org/wiki/Tensor_software (sigh, that needs updating, e.g. GRTensorII->GRTensorIII https://github.com/grtensor/grtensor/wiki )

- --

[1] https://arxiv.org/abs/gr-qc/9510052v3 since absorbed into GRTensorIII

Very much so.

Yes for sure.

Just in time I started reading ``Drawing Physics'' by Don S Lemons

It's an interesting opinion. And just like the concept of beauty that it mentions, it is subjective.

Math is a human language, written and interpreted by humans. Without human interpretation, math means nothing, just like any other language. And just like any physical model.

We cannot separate ourselves from nature, hence we cannot "truly" state anything about nature in an absolute sense, only relative to us.

I wish physics (and math for that matter), were a little bit more like computer science, in which usually there's an "interpreting machine" (a lot of times a Turing machine), which in a way acts as the explicit observer of the system, without which the system just doesn't make sense (literally, as sense can only be given to something by an interpreting-observer).

Aren't measurements of some natural system objective? We just convert/summarize these measurements to representations that facilitate, to humans, a better understanding of Nature.

Who performs the measurements? What instruments are used to perform those measurements?

If there's someone or something performing the measurements, then they depend on that someone or something to interpret them.

You are right that measurements and models facilitate a better human understanding of nature/ourselves. However, it is important to keep in mind that these things are maps, not the terrain.

A measurement can be the interaction between two sufficiently decoupled subsystems.

But I don't really understand why there needs to be some "interpreter." Nature could be some computer/system that is minimizing Action over all possible trajectories. Humans attempt to communicate an approximation to this truth by writing down nonlinear differential equations.

> But I don't really understand why there needs to be some "interpreter."

More than there needing to be, it's just all we have.

It's impossible for us to know what the universe without us is like. All of our knowledge has been gathered and interpreted by us.

Maybe there is a reality that doesn't depend on us, but how could we ever know that?

We cannot remove ourselves from our experiences, so how could we ever know what those experiences/reality would be like without us?

How could you ever know what truth is, if there is no you to know it (or approximate it)?

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