
On Mathematical Beauty in Physics - Anon84
https://blogs.unimelb.edu.au/sciencecommunication/2017/09/24/on-mathematical-beauty-in-physics/
======
__MatrixMan__
> 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.

~~~
auntienomen
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.

------
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.

~~~
raattgift
> 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](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.)

~~~
auntienomen
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.

------
gowld
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:

[https://www.amazon.com/Lost-Math-Beauty-Physics-
Astray/dp/04...](https://www.amazon.com/Lost-Math-Beauty-Physics-
Astray/dp/0465094252)

"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/](http://backreaction.blogspot.com/)

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

[https://blogs.unimelb.edu.au/sciencecommunication/author/sst...](https://blogs.unimelb.edu.au/sciencecommunication/author/sstella1/)

------
empath75
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.

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

~~~
ses1984
How does that square with something like godel's incompleteness therom?

~~~
613style
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.

~~~
ses1984
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.

~~~
lidHanteyk
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.

------
certmd
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.

~~~
dr_dshiv
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").

~~~
joes223
Pythagoras had some insider knowledge, while Thales et al just speculated.

~~~
dr_dshiv
I'm curious what you mean, insider knowledge!

------
univalent
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...](https://www.nature.com/news/equations-are-art-inside-a-
mathematician-s-brain-1.14825)

~~~
pnx
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.

~~~
matt_j
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.

~~~
yesenadam
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"?

~~~
thanatropism
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.

------
whatshisface
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.

~~~
tlb
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.

------
nachexnachex
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.

------
chr1
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.

~~~
klyrs
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.

~~~
Mugwort
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.

------
dr_dshiv
“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

------
GuB-42
Quoting a teacher about math-related assignments.

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

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

~~~
raattgift
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...](http://4.bp.blogspot.com/-0e2Zl5QrRiA/UZfL29xVTZI/AAAAAAAAAF4/Puo69vLzHn0/s1600/eqn_1.jpg)
(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=...](https://www.maplesoft.com/support/help/Maple/view.aspx?path=Physics/Einstein)
(eqn 12 for the metric at eqn 6)

Hartle's (anathe)Mathematica examples
[https://web.physics.ucsb.edu/~gravitybook/math/curvature.pdf](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](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](https://en.wikipedia.org/wiki/Tensor_software)
(sigh, that needs updating, e.g. GRTensorII->GRTensorIII
[https://github.com/grtensor/grtensor/wiki](https://github.com/grtensor/grtensor/wiki)
)

\- --

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

------
jestinjoy1
Just in time I started reading ``Drawing Physics'' by Don S Lemons

------
ta1234567890
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).

~~~
mrfox321
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.

~~~
ta1234567890
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.

~~~
mrfox321
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.

~~~
ta1234567890
> 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)?

