Alan J. Cain, Form & Number, 2025-02-23, 1267pp. https://archive.org/details/cain_formandnumber_ebook_large

This book is a definitive comprehensive introduction to the topic of mathematical beauty. The existing literature on this fascinating topic is notoriously unsystematic: in mathematics, big names drop fleeting anecdotal remarks about beauty in random places and quotable tidbits are recycled with frustrating superficiality, while in philosophy what theorizing there is about mathematical aesthetics is often driven by concerns that are not relatable to working mathematicians. Cain’s guide is an invaluable help in seeing the order in this sprawling, scattershot literature. With encyclopedic coverage, meticulous references, and lovingly crafted presentation, Cain’s book means that a systematic starting point for all future work on this topic is now available for the first time. The book can certainly be recommended as pleasure reading for anyone with an interest in beautiful mathematics and what people have said about it. But what I wish most of all is that mathematicians and philosophers who set out to opine on this topic in the future make sure to read this book first, as this is bound to elevate the academic discourse on this subject to a more critically aware and informed level — which would be a most welcome development.

The subject is treated in three parts. Historia introduces many key characters and plot points organically through a pleasant chronological narrative — although Cain perhaps risks casting his net too broadly here: any tenuous connection to beauty warrants inclusion of a topic, so that the Historia almost ends up being closer to a full-fledged survey history of mathematics than a focussed historical perspective on aesthetics. Still, giving Historia a place of pride is a welcome corrective to the 20th-century myopia suffered by all too many modern mathematicians and analytic philosophers alike.

Historia is followed by Theoria — the heart of the matter as I see it. Mathematical beauty is easier to experience than to define, but great minds have tried. Cain’s clear and comprehensive survey of attempts to pin down philosophically the elusive essence of mathematical beauty is an excellent guide to a scattered literature. Cain brings together in one convenient and careful survey virtually all significant writings on mathematical aesthetics, which has never been done before. Cain himself essentially remains a neutral reporter — even treating his own previous writings on this topic with journalistic objectivity — which enhances the usefulness of this survey as the definitive literature review on the topic. Of course it is to be hoped that such a survey is only a prolegomenon, and that many a reader will now try their own hand at Theoria, well informed by the efforts of their predecessors.

Finally, Physikos, or the role of mathematical beauty in modern science, is no doubt an important part of the story which is furthermore rich in connections to profound questions such as “unreasonable effectiveness.” Nevertheless, as someone wishing to see the concept of mathematical beauty thrive, I worry that the recent infatuation of certain physicists with this notion has done more harm than good. “Beauty” is the enemy in Sabine Hossenfelder’s book Lost in Math: How Beauty Leads Physics Astray. But to Hossenfelder, and perhaps the physicists she is criticizing, “beauty” is just what one calls the part of science that is not empirical: theoretical physics that has no applications and goes beyond what can be empirically tested is virtually by definition designated as having so-called “beauty” as its goal. Thus all manner of dubious things get classified under beauty by default rather than merit, like unwanted leftovers being dumped in a compost. This lazy categorization can only bring the noble notion of mathematical beauty into disrepute, due to no fault of its own.

Throughout the book, the quality of exposition and care for detail is notable. The book has many beautiful original figures and is an evident labor of love. The careful references and conscientious source-checking are a valuable service to the reader. And although the book is a fair and balanced survey rather than an argumentative contribution to the debate, Cain is not afraid to be critical where appropriate, such as debunking claims that Neolithic carved stones instantiate all five regular polyhedra, that Penrose-style aperiodic tilings can be found in Islamic geometrical art, or that Hardy was proud of the uselessness of pure mathematics. Also welcome and refreshing is Cain’s open-mindedness in including lesser-known authors on par with the famous bigwigs (such as Hardy or Einstein) who are often cited more for their status than for the merit of their actual ideas on beauty.

Now, if I may engage in some Theoria, let’s see how inviting it is to draw on the store of ideas in this book to shape one’s own characterization of beauty by way of synthesis. In my opinion, philosophical thought on mathematical beauty in the 20th century has been doomed to fail, because analytic philosophers have desperately tried to “save objectivity” by reducing mathematical beauty to something — anything — that can be formally captured in some way or other (such as symmetry, logical simplicity, empirical success, etc.). In my opinion, however, the answer should not be sought in objective formal properties but in human cognition: beautiful mathematics is mathematics that “fits the mind” well. Analytic philosophers have not wanted to touch such theories with a ten-foot pole, since they would rather not have a theory of beauty at all than admit that mathematics involves an element not reducible to immaculate objective rationality. But the record of testimony of mathematicians past and present suggests that they are on the wrong track.

Beauty obviously has to do with simplicity and aha-moments: “aesthetic experience is the enjoyment of self-evidence” (Whiteside); “in a fraction of a second one is convinced” (Connes). (All quotations in this review are from Cain’s book.) But of course trivial things are not beautiful. There must be a balance of simplicity and complexity: variety against boredom, uniformity against fatigue, as Crousaz put it. Thus a beautiful proof is not merely simple but unexpectedly simple, or simple in relation to what it achieves: it “leaves the reader wondering how it is possible that so much can be done with so little” (Mordell); “Complexity and perhaps toil are foreseen, and yet there comes a revelation that the situation is much simpler and easier than anticipated” (Penrose, paraphrased by Cain).

Thus beauty is mental compression. When what should be complex and burdensome becomes easy by utilizing cognitive modules that enable a high “compression factor,” then that is beautiful. The ugliest mathematics is that which the mind cannot compress at all: brute-force calculations, case-by-case checking, and so on. An ugly proof can only be checked step by step, since we do not possess any cognitive module that can compress the chain of thought into an “aha.” By contrast, when “the mind can, without effort, take in the whole without neglecting the details” (Poincaré), then a proof is beautiful.

Thus for instance the famous computer-verified proof of the four-color theorem by extensive case-checking is surely ugly, as has often been observed. “God would never permit the best proof of such a beautiful theorem to be so ugly,” as one initial reaction went. Actually on this point I disagree somewhat with Cain’s own take. Cain’s own theory of ugliness associates it with the literary notion of deus ex machina: the abrupt intervention of events that are not natural within the narrative. Thus he argues that “in narrative, the parallel [of the computer proof of the four-color theorem] would be a leap ahead in a story from being confronted with a problem to having solved it, with no inkling of the solution,” and this is what makes it ugly. But I am not convinced that this is the right narrative parallel. The cinematic equivalent of having a computer check thousands of cases is perhaps seeing Rocky do three pushups to “Eye of the Tiger” in a training montage and the viewer being left to infer that he did a thousand more without this being shown. This is not a deus ex machina, because it does not involve any extraneous elements that are not organic within the story. So lack of cognitive compression, rather than narrative deus, is a better way to capture what makes this proof ugly, in my view.

The emphasis on cognitive compression fits beautifully with the psychological analysis of mathematical discovery by Poincaré and Hadamard (as brought out for instance in the interesting but little-known PhD thesis by Ewa Bigaj, which Cain discusses in some detail). They both emphasized the role of the unconscious: after working on a problem and failing to solve it, a mathematician may be struck, perhaps at an unexpected later moment, by a sudden flash of insight where the idea of a solution is perceived. Evidently, the unconscious part of the mind continued to work on the problem even after conscious attention has ceased and then reported back when it found an answer. This unconscious process is a “preliminary sifting” of “the vast space of possible combinations of ideas,” and both Poincaré and Hadamard emphasized that the selection principle for what gets sent to the conscious mind is an “aesthetic sense.” This is perhaps a cognitive necessity: the “bandwidth” of communication between the unconscious and the conscious does not allow complex proofs to be transmitted whole, and hence it is precisely the ideas that are compressible into an “aha” that are cognitively viable in this communication interface. Thus the idea of beauty as cognitive compression not only captures common judgements of beauty and ugliness well but is also well motivated as a natural solution to a cognitive engineering problem.

Having highlighted some themes that speak to me particularly, I reiterate that Cain’s book, where every perspective under the sun is thoughtfully discussed in an even-handed way, is an ideal starting point for readers to make up their own mind on what mathematical beauty is, and thereby be able to say, with Plutarch: “Such is the nature of virtue, truth, and the beauty of mathematics, and with what of these do your trappings of wealth compare?”