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 was 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. Hey everyone, we're doubling up on comments
today - last week I was offline in Yellowstone communing with Bison and supervolcanos. Huge shoutout to David who let me and Bahar
look through his scope to see one of the famous Yellowstone wolves that was busy restabilizing
the ecosystem. And like digging for worms or something. Amazing place. So today we're covering our episode on the
future circular collider and on how we know the composition of stars. In fact let's start with the stars. Cognitive Failure asked a great PhD level
question on stellar absorption lines. Basically, why do we see specific wavelengths
missing from starlight due to electrons absorbing those wavelengths in atoms? Shouldn't those same electrons then drop back
down in energy level, emitting the same wavelengths they absorbed? So the answer is yes they absolutely do - and
in some cases you see extra light at those special wavelengths - what we call emission
lines, in some cases less light - absorption lines. It depends on what the background light looks
like. If the atoms in question are between us and
a source of light that's bright at all wavelengths, then we see absorption - that's because although
those atoms to reemit the absorbed energy, they do so in random directions AND potentially
through a cascade of energy level drops that may not be the same as the initial energy
level jump. That's the case when looking through the Sun's
atmosphere at it's bright interior. On the other hand, if you have a cloud of
gas hanging out in space like a nebula - it will be illuminated by stars but you're not
looking directly AT those stars through the cloud. In that case you'll see all of those photons
produced when absorbed light is reemitted - emission lines. You won't see absorption unless you look directly
through the cloud at one of those stars. OK, on to the future circular collider - well
actually Awesome Octagon dropped some knowledge on a different next-generation particle collider
that's worth sharing. The Belle II experiment that just started
taking data on Japan's superKEKB electron-positron collider. This is nowhere near as big as even the large
hadron collider, but it does achieve higher luminosities - more collisions per second. The point is that there are some insanely
rare interactions and extremely subtle deviations from the predictions of the standard model. To detect these you need more collisions,
not higher energies. That said, I want to add that we don't know
in which direction the new knowledge is hiding, so we need to explore multiple paths. Regarding the value of pursuing crazy ambitious
science projects - DG put it better than I can: If Humans only did what looked to be
the most immediately important task we'd still be living in trees, if not extinct. Deep Time goes in both directions. I love the idea of future deep time. I hope it actually happens. For humans, I mean. John Constantine offers a different type of
wisdom regarding the motivation for building the FCC: Their demon overlords have demanded
a larger summoning circle. Yes, and this time we'll summon a God. Particle. In fact countless of them. Bwa ha ha.
