The unreasonable effectiveness of beauty in science
Tons of famous scientists emphasize the importance of aesthetics.
For instance, Murray Gell-Mann:
“We have this remarkable experience in this field of fundamental physics that beauty is a very successful criterion for choosing the right theory.”
“The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances. I am far from despising this, but it has nothing to do with science. What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp.”
“Be guided by beauty. I really mean that. Pretty much everything I’ve done has had an aesthetic component, at least to me.
"It is more important to have beauty in one's equations than to have them fit experiment."
Hermann Weyl:
"My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful."
Richard Feynman:
“You can recognize truth by its beauty and simplicity.”
Subrahmanyan Chandrasekhar:
"I would like to suggest that in science, as in art, the sense of beauty is the sense of the appropriate or the fitting. And from this point of view, the pursuit of science is simply the pursuit of elegance, or aesthetic satisfaction."
On the other hand, every time researchers try to formalize this idea and use it as a guiding principle things go horribly wrong.1
Formalizing beauty
The most common ways to formalize what beauty in science mean are: symmetry and naturalness.2
The study of more symmetric or even supersymmetric models has, so far, turned out to be a dead end.
One issue is that there’s an infinite number of ways to make the current best model of nature at fundamental scales more symmetric. Literally.
For example, you can embed the “ugly” gauge symmetry of the standard model SU(3) x SU(2) x U(1) in SU(5) and then argue this symmetry breaks down at higher energies into the puzzle pieces we observe at energy scales we can currently probe.
But you can also embed it into SU(6), SU(7), SU(8) or SU(9999). Or you can embed it in SO(10), SO(14), SO(18), or SO(998). Or you can embed it in E(6), E(7), or E(8).
There are infinitely many options. The story is always the same. All the extra stuff decouples at sufficiently high energies and what we are able to observe at low energies is SU(3) x SU(2) x U(1).
Similarly, there are infinitely many family symmetries you can invoke to unify the three generations of elementary particles.3
Adding the requirement of “simplicity” isn’t of much help either.
The minimal grand-unified model based on SU(5) predicts that protons should decay at a rate above the current experimental threshold. In other words, it’s ruled out by experiments.
Another idea is to augment symmetry by hinting at a vague requirement of “exceptionalness”. To quote, Ed Witten:
Describing nature by a group taken from an infinite family does raise an obvious question – why this group and not another? In addition to the three infinite families, there are five exceptional Lie groups, namely G2, F4,E6,E7, and E8. Since nature is so exceptional, why not describe it using an exceptional Lie group? […] The grand unified theory based on E6 is not clearly superior to the SO(10) model, but it does capture the successes of the SO(10) model “exceptionally.”
This seems like an even weaker argument.
On the other hand, if we take another look at the standard model gauge group it doesn’t seem that bad at all. SU(3) x SU(2) x U(1), while not a “simple group”, has a nice 1-2-3 ring to it and it’s definitely perfectly suitable for a minimal model that is able to describe what we observe in collider experiments.
Symmetry is undoubtedly beautiful but there is no way to rank different symmetries in terms of their aesthetic value.4
Arguments that our models of nature should be “natural” run into similar issues.
First of all, calling anything that we observe "unnatural" is an oxymoron.
Have a look at the Borwein integrals below.
The breakdown of the pattern is undeniably ugly.
But that doesn’t mean it’s wrong.
Secondly, naturalness was the primary reason why many particle physicist were convinced new particles besides the Higgs boson would show up in LHC detectors.
Well, that didn’t happen.
Various measures of naturalness have been invented to argue for and against different models.
Hundreds of papers have been published that propose solutions to the various “naturalness problems” in modern physics (hierarchy problem, flatness problem, baryon asymmetry, strong CP problem).5
There is zero evidence that any of these solutions are actually realized in nature.
So just as symmetry arguments, naturalness has proven to be an unreliable guide for scientific discovery.
In this sense, Sabine Hossenfelder isn’t wrong when she argues that “aesthetic criteria have become a source of cognitive bias leading physics astray”.
This seems pretty weird, right?
On the one hand, so many famous scientist emphasize the importance of beauty in in scientific theories.
On the other hand, all attempts to formalize this idea have led us nowhere.
Here’s how I think the paradox can be resolved.
Effective aesthetics
The beauty that famous scientists talk about is an incompressible concept that can't be turned into a simple algorithm for choosing theories.
Instead, what they describe is related to the idea that big breakthroughs are often the result of someone following hunches and intuitions.
Successful researchers were often simply guided by a general sense of what felt right to them.
Therefore, beauty as a successful guiding principle in science has little to do with easily definable concepts like symmetry or naturalness.
These are just notions people use in hindsight.6
To explain the difference between these different interpretations of beauty as a successful guiding principle let me quote what I wrote about the game Picbreeder.
Picbreeder is a website that allows users to “breed” evolutionary art.
You can select images you like and the system then uses a genetic algorithm to breed new art.
It works analogously to breeding horses except that instead of choosing animals to breed, you choose pictures.
Say all experts agree that breeding the picture of a skull is the top priority right now. […]Experts develop a test to measure the skull-ness of any candidate picture on a scale from 0 to 100.
Only images that show clear progress towards skull-ness are selected. All other images are discarded.
Will this effort succeed?
Look at this sequence of images that actually generated an image of a skull.
None of the intermediate steps would have passed the skull-ness rating test.
In Picbreeder the most interesting images emerge whenever players follow their hunches and intuitions about what looks most interesting at every step.
Whenever you force to generate a certain type of image, you will almost certainly fail.
With this in mind, it’s hardly surprising that attempts to “breed” a beautiful theory by applying criteria like symmetry or naturalness fail.7
To be clear, the next paradigm-shifting theory in physics will be beautiful.
But I very much doubt that whoever discovers it will find it by applying criteria like symmetry or naturalness.
In fact, a theory's beauty is often only recognized long after its initial discovery.
The beautiful, deep symmetry hidden in Mawell’s theory of electrodynamics, for example, was only fully appreciated decades after Maxwell published his equations.
And yet, scientists will rightfully keep emphasizing the importance of beauty as a guiding principle.
We just need to remember that this is code for following unexplainable intuitions and hunches rather than a rigid criterium.
Without irrational preferences scientific progress grinds to a halt.
There is an infinite number of theories that are perfectly compatible with all known data. Hence it’s impossible to systematically explore the full space of theories.
Moreover, applying measures like symmetry or naturalness are more likely than not leading us down dead ends.
Scientific progress requires that someone develops what will seem to everyone else like an unreasonable preference.
They typically can't fully explain their preference or defend it against all initial attacks. The best they can often do is say "it’s beautiful”.
Most of these aesthetic preferences will indeed turn out to be completely unjustified.
But the good thing about science is that, in the long run, the bad stuff doesn’t matter at all.
All that matters is that eventually one aesthetic hunch will be proven correct.
A fantastic read on the topic (even though she reaches quite different conclusions) is Sabine Hossenfelder’s Lost in Math.
There are other ideas to formalize what scientists mean by beauty in science (elegance, simplicity, fruitfulness,…) that are all equally flawed.
The aesthetic value of supersymmetric models is questionable. Conventional unified gauge symmetries allow us to understand all elementary particles (of one generation) as excitations of just one unified quantum field. That’s pretty cool.
Supersymmetry, on the other hand, is not unifying existing particle representations but only known particle representations with yet undiscovered new supersymmetric partner representations. At least to me this makes the theory more ugly.
The standard motivation for supersymmetry is a mishmash of different reasons. A mishmash of reasons should always be treated with skepticism. To quote Peter Thiel:
“If there’s no single reason that can cause you to do something, you should think carefully about whether it’s important or not. Oftentimes we’ll want to do something, and we’ll give multiple reasons for it without thinking hard about them. If you can’t give a single reason that justifies doing something on its own, you should be very wary that you aren’t exercising sufficient intellectual discipline.”
In addition, most of the things achieved by invoking supersymmetry like unification of gauge couplings at high energies can be accomplished for much cheaper.
Some researchers, ahem, went even as far and tried to build models combining grand unified symmetry and naturalness arguments. Beautiful? I think so. Realized in nature? Probably not.
This is no different to when successful entrepreneurs are asked to explain their success. They often claim that they used certain mental frameworks to build their company.
For example, looking at Airbnb we could say that they identified an "underutilized asset" (spare rooms) and connected it with "latent demand" (affordable travel accommodations). But this framework emerged after the fact. The founders initially just thought it was a cool idea to rent air mattresses during conferences.
After all, “the problem with startup advice is all of it is true”.
If you could actually download frameworks for building successful companies into your brain by reading books, we would see a lot more successful startups.
Luck (timing) plays a big role. But so does a sixth sense for promising opportunities that can’t be fully encapsulated using popular startup frameworks.
Attempts to use, for example, naturalness criteria are directly analogues to using tests to measure “skull-ness” in the example above.