Lost in Math, Beautiful Theories, and a Racehorse Analogy

Estimated read time (minus contemplative pauses): 15 min.

hokusai_fibonacci_golden-ratioThe latest episode of the Rationally Speaking podcast features an interesting conversation with theoretical physicist Sabine Hossenfelder about her new book, Lost in Math: How Beauty Leads Physics Astray. I look forward to reading it. In the meanwhile, I’d like to ponder an example brought up in the discussion, involving evaluating racehorses based on a criterion of beauty. First, a quick look at what the book’s about.

Here’s the Amazon description:

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.

While Hossenfelder also brings up concerns about funding and groupthink, the podcast discussion’s emphasis is on beauty, which she summarizes as having three general aspects: Simplicity, Naturalness, and Elegance. (For an explanation of these, see her blogpost “What Do Physicists Mean When They Say the Laws of Nature Are Beautiful?“) I find Hossenfelder’s arguments persuasive; that is, given what I know of them so far and as an outsider who’s been suspicious for some time of a field that’s been making the sorts of incredible claims for which theoretical physicists have been getting attention in recent decades.

Still, it’s worth noting the following comment in a Science Magazine review of the book by Djuna Croon (a theoretical particle physics and cosmology postdoc): “Different concepts are conflated throughout the book (for example, technical naturalness, which has a statistical meaning, and mathematical elegance) and are somewhat mockingly referred to collectively as ‘beauty.'” (Read Hossenfelder’s response to the review here, where you’ll also find a followup comment from Croon.)

From what I’ve so far seen of the book and Hossenfelder’s blog, it doesn’t seem to me that her use of the word “beauty” is mocking in the sense of attributing a meaning to it that would make other physicists who employ the word appear silly. On the contrary, she seems to take the word seriously (see again her blog post about the three aspects of beauty). She very well may, however, at times take a mocking tone when criticizing the fact (according to her) that physicists in general mistakenly, and perhaps at times smugly, rely on that word, however defined, as a basis for theory selection or elevation.

How much, and in what way, any given physicist relies on the word, I don’t know. When asked by podcast host Julia Galef if what she means by “beauty” is a “beauty-is-in-the-eye-of-the-beholder thing” or, rather, if there’s “a consensus among physicists about what constitutes a beautiful theory,” Hossenfelder responds: “Interestingly enough, there’s mostly consensus about it. So, let me put ahead that I will not try to tell anyone what ‘beauty’ means. I’m just trying to summarize what physicists seem to mean when they speak of beauty when they say a theory is beautiful.”

I would imagine that such a consensus would not rule out the possibility for a physicist to connect that sort of beauty to a less restrictive, more general sort of beauty—specialists do this often enough with their technical words, hopping between specialized and ordinary usage as it suits them, blurring distinguishing lines in the process; this must be even easier to do with a non-technical word, Hossenfelder’s three aspects notwithstanding (indeed, she points out in the podcast discussion that there’s no reason these three aspects can’t apply in other domains, such as music, and her blogpost features a minimalist photograph).

That in mind, I just today, by coincidence, came across the following passages in Lawrence Krauss’s 2017 book, The Greatest Story Ever Told—So Far: Why Are We Here?, two chapter titles of which contain the word “beauty”; maybe these words align with the perspective Hossenfelder is critiquing:

There is remarkable poetry in nature, as there often is in human dramas. And in my favorite epic poems from ancient Greece, written even as Plato was writing about his cave, there emerges a common theme: the discovery of a beautiful treasure previously hidden from view, unearthed by a small and fortunate band of unlikely travelers, who, after its discovery, are changed forever.

Oh, to be so lucky. That possibility drove me to study physics, because the romance of possibly discovering some new and beautiful hidden corner of nature for the first time had an irresistible allure. This story is all about those moments when the poetry of nature merges with the poetry of human existence.

Much poetry exists in almost every aspect of the episodes I am about to describe, but to see it clearly requires the proper perspective. Today, in the second decade of the twenty-first century, we might easily agree about which of the great theories of the twentieth century are most beautiful. But to appreciate the real drama of the progress of science, one has to understand that, at the time they are proposed, beautiful theories often aren’t as seductive as they are years later—like a fine wine, or a distant love.

So it was that the ideas of Yang and Mills, and Schwinger and the rest, based on the mathematical poetry of gauge symmetry, failed at the time to inspire or compete with the idea that quantum field theory, with quantum electrodynamics as its most beautiful poster child, wasn’t a productive approach to describe the other forces in nature—the weak and strong nuclear forces. For forces such as these, operating on short ranges appropriate to the scale of atomic nuclei, many felt that new rules must apply, and that the old techniques were misplaced.

So too the subsequent attempts by Nambu and Anderson to apply ideas from the physics of materials—called many-body physics, or condensed matter physics—to the subatomic realm were dismissed by many particle physicists, who deeply distrusted whether this emerging field could provide any new insights for “fundamental” physics. The skepticism in the community was expressed by the delightful theorist Victor Weisskopf, who was reported to have said at a seminar at Cornell, “Particle physicists are so desperate these days that they have to borrow from the new things coming up in many-body physics… Perhaps something will come of it.” (Krauss 201–203, Kindle Edition) …

And at the heart of all of the forces governing the dynamical behavior of everything we can observe is a beautiful mathematical framework called gauge symmetry. (Krauss 247) …

It is then possible, at least, to argue that when the weak interaction Higgs field condenses in empty space, this could stimulate the condensation of another Higgs-like field with properties that would allow it to store just the right energy to explain the observed inflation of the universe today. The mathematics required to make this happen is pretty contrived—the model is ugly. But who knows? Maybe it is ugly because we haven’t found the correct framework in which to embed it. (Krauss 298)

Now on to the promised racehorse example. In Chapter 2 of Lost in Math, Hossenfelder introduces and critiques an analogy attributed to Steven Weinberg:

Weinberg, who was awarded a Nobel Prize for unifying the electromagnetic and weak interaction, likes to make an analogy with horse breeding: “[The horse breeder] looks at a horse and says ‘That’s a beautiful horse.’ While he or she may be expressing a purely aesthetic emotion, I think there’s more to it than that. The horse breeder has seen lots of horses, and from experience with horses knows that that’s the kind of horse that wins races.”

But just as experience with horses doesn’t help when building a race car, experience with last century’s theories might not be of much help conceiving better ones. And without justification from experience beauty remains as subjective as ever. This apparent clash with the scientific method is acknowledged by today’s physicists, but using aesthetic criteria has nevertheless become widely accepted practice. And the more removed from experimental test a subject area is, the more relevant the aesthetic appeal of its theories. (Hossenfelder 26, Kindle Edition)

The quote is from a NOVA interview with Weinberg called “Viewpoints on String Theory,” in which he goes on to say:

String theories in particular have gotten much more rigid as time has passed, which is good. You don’t want a theory that accounts for any conceivable set of data; you want a theory that predicts that the data must be just so, because then you will have explained why the world is the way it is. That’s a kind of beauty that you also see in works of art, perhaps in a sonata of Chopin, for example. You have the sense that a note has been struck wrong even if you’ve never heard the piece before. The kind of beauty that we search for in physics really does work as a guide, and it is a large part of what attracts people to string theory. And I’m betting that they’re right.

In Chapter 5 of Lost in Math, Hossenfelder asks Weinberg about the horse analogy, in particular that it “seems to suggest that paying attention to beauty in theory construction is based on experience,” to which Weinberg responds, “Yes, I think it is,” and then goes into a lengthy elaboration starting with the Hellenistic Greeks and talking, according to Hossenfelder, “like a book, almost print-ready” (Hossenfelder 96).

I suggest you read the book for the full thrust of his comments, but I think this is a passage worth highlighting, and coheres with the thoughts on rigidity expressed in the above NOVA quote:

“Our sense of beauty has changed. And as I described it in my book, the beauty that we seek now, not in art, not in home decoration—or in horse breeding—but the beauty we seek in physical theories is a beauty of rigidity. We would like theories that to the greatest extent possible could not be varied without leading to impossibilities, like mathematical inconsistencies. (Hossenfelder 98)

I won’t say any more about that here. Rather, I’m interested in how one might respond to Weinberg’s horse analogy. Hossenfelder’s response, as noted above, is that racehorse experience doesn’t help you with racecars. That sounds right to me. But my first thought was that there are good reasons to suspect experience-borne aesthetic response to racehorses insufficient for dealing even with racehorses.

Well, unless maybe everyone’s relying on their aesthetic intuition. But then everyone is engaging in a kind of sophisticated guesswork. The point is that there seem to be better ways. For example, measuring the size of a horse’s left ventricle. That’s the story told, anyway, in Chapter 3 of Seth Stephens-Davidowitz‘s 2017 book Everybody Lies: Big Data, New Data, and What the Internet Can Tell Us About Who We Really Are. The long and short of it is that, in 2013, an average-sized, reddish-brown horse of good, not great, pedigree and with a scratch on his ankle that could have been seen as evidence of an injury was nevertheless picked out by Jeff Seder and his team as “the best horse of the year and, quite possibly, the decade” (Stephens-Davidowitz 64, Kindle Edition). In 2015, the reddish-brown horse, racing under the name American Pharaoh, “became the first horse in more than three decades to win the Triple Crown” (Stephens-Davidowitz 64).

Seder, who heads a firm called EQB, doesn’t take the usual approach of analyzing gaits, examining visually, or evaluating what’s apparently considered most significant: pedigree. Stephens-Davidowitz notes that pedigree fails to pick out nearly as many champions as one might initially expect, but that this really shouldn’t be surprising (scouts for professional human athletes, for example, don’t work this way at all). As for the other things horse agents look for, it seems to come down to personal style and intuition; at any rate, Stephens-Davidowitz found little agreement among the agents he spoke with.

Seder, it turns out, uses data, and spent decades perfecting his craft. To no avail, he collected data on nostril size, EKGs, muscle-volume, and excrement. And then he got the idea to measure horses’ internal organs using a portable ultrasound he made himself:

The results were remarkable. He found that the size of the heart, and particularly the size of the left ventricle, was a massive predictor of a horse’s success, the single most important variable. Another organ that mattered was the spleen: horses with small spleens earned virtually nothing.

Seder had a couple more hits. He digitized thousands of videos of horses galloping and found that certain gaits did correlate with racetrack success. He also discovered that some two-year-old horses wheeze after running one-eighth of a mile. Such horses sometimes sell for as much as a million dollars, but Seder’s data told him that the wheezers virtually never pan out. He thus assigns an assistant to sit near the finish line and weed out the wheezers.

Of about a thousand horses at the Ocala auction, roughly ten will pass all of Seder’s tests. He ignores pedigree entirely, except as it will influence the price a horse will sell for. (Stephens-Davidowitz 69)

And so it seems, beauty alone—even for expert sensitivities—is not the best indicator of which horse to put in the race. But I don’t know horse racing. At all. Has Seder since amassed the best record around? The track record listed on his website looks good to me: www.eqb.com/trackrecord.html. And this 2017 Inverse article seems promising: “Racehorse Big Data Unlocks the Formula for Human Superathletes” (subhead: “Jeff Seder is changing what elite race horses look like, inside and out”).

But wait. Need his track record be the best around to vindicate his methods as the best around? It was, to at least some degree, luck that he was the one called that day to evaluate the reddish-brown horse. (And had that one not been there, indeed had it never been born, Seder he would have recommended no horse from the group in question, as so few pass his tests; indeed, the reddish-brown horse was the only one he recommended that day of the 152 on auction.)

I suppose once everyone has the perfect racehorse detection technology, it’s just a matter of lucking out and finding the right horse to test. Maybe that will come down to some sort of intuition, if not one based on beauty. Or perhaps it will redefine what counts as beautiful in that domain. Indeed, it wasn’t hard to find examples of American Pharaoh being called beautiful—e.g., here and here. But I don’t know if they mean “beautiful” in the sense that a racehorse evaluator would mean it, or more in the way a layperson would call a horse beautiful; especially one that’s a winner.

Speaking of which, that second link is to a list ranking racehorses based on user votes. Today, American Pharaoh comes in there at 15. He initially wasn’t picked for his beauty, but for his data profile. And those who are in the top five aren’t in such elite spots because they were beautiful, but due to their performance. Suppose we were able to go back and time and put American Pharaoh and the top five horses on that list in front of horse-picking agents. If they rated the top five horses as more beautiful than American Pharaoh, and if those horses outperformed American Pharaoh, would this simply mean that those agents got lucky (e.g., that Secretariat was beautiful and happened to be fast)? It can’t just be that, can it? And which horse, I wonder, would Seder have chosen if he had to pick only one?

It’s tempting to say that we’ll just have to wait and see if data can bring us the hands-down best horse ever—that is the most beautiful; or, more precisely, that breaks all records, thereby proving itself the most beautiful. But that’s wrong: it’s not data that makes the horse singularly great, rather it’s data that can help find that horse, should the stars align so that such a horse is born. At any rate, that’s the short term. Let’s skip over what might happen once data and technology make genetic engineering more precise than the current methods (e.g., selective breeding).

In the meanwhile, the job of the data-based horse agent comes down to correctly running the right numbers then, finally, running the horses. This is somewhat consistent with Weinberg’s metaphor in the sense that, over the centuries, holding theories up to the real world is has trained the contemporary physicist’s aesthetic sensitivities; but the metaphor loses steam when it’s meant to justify theories in which, so to speak, there is no physical race in which to insert the horses—when they aren’t testable or meant to be predictive. The sense then becomes that intuitions have evolved to stand in for the race.

It seems this concern underlies Hossenfelder’s push against choosing theories for their beauty rather than for their ability to make predictions and, well, for their failure to deal with racecars. And the above Krauss passage seems consistent with this worry: he may endorse beauty, but points out that even in the relatively recent past, prevailing aesthetic intuitions missed the mark. The idea this leads to, perhaps, is that intuitions may adjust to the needs for consistency across theories as they gain traction within the field; no predictive power required (at least not as a necessity for theory-making). Maybe that’s fine. Maybe it’s the best we can hope for. Or maybe it’s even a mark of progress (once our theories are strong enough, if correct, it seems they would entail things to which we otherwise have no access).

Who knows. But it certainly does strike me that, if you’re going to have a network of interdependently verifying theories, you should be extra-sensitive to the fact that all it takes is a few corrective tweaks at an early stage of the network’s development to later find yourself with an intricately delicate, if self-sustaining and apparently sturdy, fiction. Maybe this thought suggests a benefit to according with a beauty standard after all: such theories are easier to track. Maybe we’ll get lucky and the right theories will turn out to be beautiful; if not, perhaps they’ll elude us whatever standards we adopt. There’s no ultrasound for measuring the innards of an infinity of big bangs.

I should probably read Hossenfelder’s book, at the very least, before speculating further.

So I’ll summarize:

1. To support physicists’ reliance on beauty when choosing theories, Weinberg gives the analogy of the best racehorses being picked out by the developed aesthetic sensitivities of expert breeders.

2. Hossenfelder pushes against the analogy on the grounds that intuitions about racehorses don’t help with racecars. I find this persuasive, especially if racecars are around the corner.

3. But I also think Weinberg’s analogy might fail due to racehorse intuitions not even being the best way of choosing racehorses. Collecting and evaluating data about surprising features of a horse might turn out to much better correlate with winning races.

4. So, Weinberg might want to pick a more appropriately illustrative analogy, if indeed there is one out there. If there isn’t one, then we’re left with the question of whether that fact suggests (a) progress, (b) stagnation, or (c) simply that analogies are hard.

If (c) wins, this provides a kind of trivial vindication of the analogy, but leads one to wonder what the point of the analogy is. If, on the other hand, Weinberg remains dedicated to the horse analogy, or some better chosen one, then one’s response to the analogy, provided one understands it, must count in some significant way as a response to the thing analogized (whether or not one understands it).

That in mind, next up is another of Weinberg’s analogy attempts: the above-cited Chopin example, in which we ponder the question of whether the feelings invoked by the skillful play of tension and release in music—of the sort that can be jolted by an errant note and is indirectly exploited even in dodecaphonic and aleatoric works—signal something true about the world external to human consciousness, or merely result from an accident of human biology; and, furthermore, whether our answers to that question detract from or support Weinberg’s claims, given, to mention just one complication, that the Chopin example seems to rely on the (potentially, though surely unintentionally, offensive?) idea that some earlier version of the (musically sophisticated or at least musically inclined?) human specimen, say, a Hellenistic Greek would, despite sharing essentially our biology, fail to recognize an out-of-place note, not because of anything physically inherent in the music (but wasn’t that largely the point of the music example?) or biology, but rather due to objectively less-well-developed musical aesthetic sensitivities than those of the more recent human (oh wait, this was the point of the example… right?).

We’ll save that thorny path for another day.


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