The Truth About Beauty in Physics (SCRIPT) (Patreon)

The great physicist Herman Weyl once said, “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.” But is this actually good advice for doing physics?

Many physicists believe that, in fact, you don’t need to choose between beauty and truth - there’s this idea that mathematical beauty is a powerful guiding force towards truth - the more beautiful the equations of a proposed law of physics, the more likely it is to truly represent reality. But it’s also been argued that modern theoretical physics, and in particular string theory, has been overly transfixed by the allure of beauty for decades, and that perhaps this is the reason for the lack of major breakthroughs in the last half century. Indeed, Hermann Weyl himself was led astray by his adherence to beauty, as we’ll see.

Hermann Weyl was hardly the first to be guided - and misguided by this abstract notion of beauty in trying to guess the mathematical behavior of the world. A prime example is our effort to understand the motion of the planets. In the first effort, by Claudius Ptolemy, the planets orbited the Earth in complicated systems of circles embedded within circles - what we call epicycles - needed to explain retrograde motion. It was pretty messy. Ugly even. Nicholaus Copernicus found more beauty of simplicity by placing the Earth along with all planets in simple circular orbits around the Sun. However Copernicus was unable to actually improve on the precision of Ptolemy’s predictions - or at least to do so, he had to add epicycles of his own, which countered the elegance of his original model.

It turns out both Ptolemy and Copernicus were lured by the same bias towards beauty - here, the mathematical perfection of the circle. But the planets move in ellipses, not circles - as deduced a century after Copernicus by Johanne Kepler. Kepler’s three laws of planetary motion are arguably much less elegant than Copernicus’ simple model - a touch uglier. So did Copernicus ride the beauty train one stop too far? More like he rode it in the wrong direction. There IS an extremely simple, elegant law underlying planetary motion - it's Newton’s Law of Universal Gravitation. Not only can Kepler’s complicated laws be derived from Newton, but Newtonian gravity makes predictions far beyond the motion of the planets.

But even Newtonian gravity proved to be a special case of a much deeper law. Einstein’s equations of general relativity give even better precision than Newton’s law - perhaps perfect precision - in their description of gravity. And general relativity is also more explanatory than Newton - it tells us that gravity results from the warping of space and time. Yet the equations of general relativity are notoriously complex. Most who’ve studied the field equations deeply find it supremely elegant - but it’s not straightforward to define where that sense of elegance comes from.

Ptolemy, Copernicus, Kepler, Newton, Einstein - each followed their own sense of mathematical beauty in this long quest to understand gravity. But the process seems ... unsteady - treacherous even. So why is this pursuit sometimes so powerful, but sometimes so fraught? To understand that, we need to think about what mathematical beauty even means. Just as with beauty in any field, it’s fundamentally a subjective sense, and so difficult to define. But let’s think about some of the ways in which math can be considered beautiful.

For Ptolemy and Copernicus it was a sense that certain mathematical forms are intrinsically more beautiful or perfect than others - in their case circular orbits - and so surely nature must preferentially choose these for its laws. Even Kepler fell for that one - he tried to relate his “messy” elliptical orbits to the more perfect Platonic solids.

It’s the geometric symmetry of the circle or the platonic solid that seems beautiful. But in physics, when we talk about symmetry we mean that a physical law is unchanged by some transformation - whether shifts in time, space, angle, or something more abstract like the phase of the wavefunction. The universe does obey deep, underlying symmetries that are reflected on all scales of complexity. We make powerful use of these symmetries to derive our laws of physics, and so perhaps it’s not surprising these equations possess some of the beauty of the underlying symmetries.

A law of physics might also be considered beautiful if it reduces to a compact expression - a small number of physical properties linked in a mathematically simple way. If an ugly set of expressions reduces to one or few simple laws that explain a wide range of phenomena, that feels extremely compelling.

This is connected to the principle of parsimony, also known as Occam’s Razor. It can be stated as follows "other things being equal, simpler explanations are generally better than more complex ones". You can describe any complex phenomenon with a mathematical model if you’re willing to give your model enough moving parts - but that doesn’t mean it holds more truth or explanatory power. Add enough epicycles, and Ptolemy’s circles within circles within circles could describe the motion of anything, from a planet to a particle of air - but it wouldn’t explain that motion.

At its heart, Occam’s Razor comes from the idea that the extreme complexity we observe in our world - in which there are many, seemingly disconnected phenomena - emerges from the action of few simple underlying causes. And that’s an observational fact. Newton’s law of gravitation is a great example, and we can be sure that Newton himself felt a sense of aesthetic pleasure when he realized that the motions of both the moon and an apple could be explained by one piece of high-school algebra.

So is base reality represented by the simplest possible interaction or relationship, expressible with the simplest conceivable mathematical statement? And if so, does pursuing mathematical parsimony guide us inexorably towards the most elementary driving forces? The step from Kepler to Newton suggests so, but then why does complexity seem to increase when you go one level deeper from Newton to Einstein?

This is probably a good time for a quote from Einstein himself: “The supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” In other words, no matter how pretty and parsimonious our theory, if it doesn’t match reality as measured by experimental data, then we’ve gone too far.

One of Einstein’s contemporaries, and one of the very few that might be considered his intellectual peer, seems to have disagreed. Paul Dirac, one of the principal founders of quantum mechanics, said “It is more important to have beauty in one's equation than to have them fit experiment.” By which he meant that an experiment might give incorrect results, and so incorrectly discredit a true hypothesis. But if a physical law is mathematically ugly, then you can be sure it’s wrong, experiment or no.

Dirac’s most famous result came from exactly this pursuit of mathematical beauty. He sought to develop a quantum mechanical wave equation that agreed with Einstein’s special relativity. We’ve talked about how he did this before, but the result was that Dirac’s algebra looked like a dog’s breakfast of cross-terms that could only be described as ugly. But then a simple modification to the founding assumptions caused this mess to collapse into a supremely elegant form. The resulting Dirac equation is the entirely correct relativistic quantum description of the behavior of the electron.  Dirac knew he was on the right track based on an abstract sense that the underlying laws of the universe SHOULD be elegant. 

Oh, by the way, the modified assumption that led to Dirac’s miraculous algebraic convergence? He allowed the electron to have states with negative energy levels - not technically possible, but we now understand these as corresponding to the antimatter counterpart of the electron. Dirac’s pursuit of pure mathematical elegance led him to predict the existence of antimatter, which was discovered only a couple of years after the 1929 publication of his equation. 

Nobel laureate physicist Frank Wilcek talks about a form of mathematical beauty that he calls the exuberance of a theory - its productivity. The equations of physical law are beautiful if you get out more than you put in. If it predicts or describes more aspects of the world than were used to derive the law in the first place. The Dirac equation is a good example. Same with Newton’s gravity, derived from observations of apples and the moon, but it predicts the motions of galaxies.

And Maxwell’s equations, which parsimoniously unite electricity and magnetism but also predict the existence of electromagnetic waves - of light. And then there’s Einstein’s field equations of general relativity. They were derived following the simplest thought experiments - he imagined falling off a roof, or a photon bouncing between mirrors - but the resulting theory predicts black holes, gravitational waves, and even the big bang.

The beauty of predictive power was a big part of what drove Hermann Weyl. A couple of years after Einstein presented his general theory of relativity - Weyl found a simple, elegant way to unify Einsteinian gravity with the other force of nature known at the time - electromagnetism. Just add an additional degree of freedom - actually a symmetry - at each point in space and electromagnetism appears, almost miraculously. We’ve discussed the details before. But Weyl’s theory made some predictions that simply did not reflect the real universe - ultimately it wa’s just plain wrong. Weyl had a hard time accepting that such a beautiful theory could be wrong, and he modified his idea to fix the issues - added some metaphorical epicycles - but that robbed the initial idea of its elegance, just as they had for Copernicus.

So now we come to string theory. The first compellingly beautiful aspect of string theory is that gravity, in the form of the Einstein field equations - automatically emerged from it. We described how in a previous video. But there are other things too - for example, disparate versions of string theory seem to miraculously converge into one master theory. This seemed too mathematically neat to be a coincidence. String theorists find their math beautiful, even if it is far from being simple. And yet it hasn’t managed to produce a testable prediction - besides the whole gravity thing of course. So did string theory fall for the same sort of misguided obsession with beauty as did Weyl?

Well, I don’t know. But let’s get back to Weyl anyway. Actually, Weyl’s “incorrect” attempt to integrate gravity and electromagnetism wasn’t such a failure. His idea of introducing a new symmetry to space was translated to adding a new symmetry to the wavefunction in quantum mechanics. The result was the same - the electromagnetic field popped out like magic. And Weyl’s idea evolved into what we now call gauge symmetry, and it’s the basis for the slightly ugly but fantastically successful standard model of particle physics. 

So perhaps mathematical beauty and convergence DOES provide a reliable indicator that we’re moving in the right direction, but it’s less of a sure compass, and more a hint - getting warmer, getting colder - and much cleverness still has to be applied in interpreting that hint. 

And so perhaps the mathematical wonders of string theory DO reflect something true about reality, but we’re struggling with how to interpret it all. Ultimately, our sense of beauty can’t be cleanly defined - not in art and not in physics. It results from the hidden workings of our brains - many factors contribute subconsciously to this qualitative sense of ... yes, that feels right, or that stirs me. And that imprecise subjectivity may be a reason to pay attention to our sense of beauty in physics, rather than reject it. It symbolizes the synthesis of our unconscious intelligence - our intuition. In science we’re skeptical of pure intuition, and rightly so. Alone it can lead us astray. But it also points us true, if we take care to apply scientific rigor in between leaps of intuition. Used judiciously, the intuition of beauty may ultimately point the way to the deepest truths, leading to the most beautifully fundamental explanations of space time.