As Plato suggested, there is something erotic about the drive that propels us toward knowledge; we are haunted by an almost sensuous yearning to understand. That drive is alluded to in the title of Edward Frenkel’s book, Love and Math: The Heart of Hidden Reality. The emphasis—not “love of math” but “love and math”—may have also helped inspire some of the attention his book has received since its 2013 publication, with translations prepared in at least 16 languages. Turning to it now, after the heat of first encounters and early reviews has passed, may lead to a more sober assessment of its strengths and unsettling failures, while illuminating aspects of an enduring romance.
“Mathematics and science in general,” Frenkel writes:
are often presented as cold and sterile. In truth, the process of creating new mathematics is a passionate pursuit, a deeply personal experience, just like creating art and music.”1
We can glimpse the passionate pursuit of these subjects; it is evident in the book’s account of Frenkel’s life. We see too, though less clearly, an analogy to art and music: the main example is a film Frenkel wrote and starred in, playing a mathematician who discovers the essence of love; he applies that knowledge in a nude love scene and expresses it in a formula he tattoos on his beloved’s belly, just before he commits suicide.
But that is getting ahead of ourselves. Even the bizarre requires some context.
Frenkel is a distinguished mathematician who came to maturity in the last days of the Soviet Union; he was invited to become a visiting professor at Harvard at the age of twenty-one and now teaches at the University of California, Berkeley. His two themes—mathematical passion and artistry—come into play in two interwoven narratives, one of how he became a mathematician, the other of how his own mathematical work developed. It is research which, he writes, “can be fully understood only by a small number of people; sometimes, no more than a dozen in the whole world at first.”2
And his ambition? “Dear reader,” Frenkel writes:
with this book I want to do for you what my teachers and mentors did for me: unlock the power and beauty of mathematics, and enable you to enter this magical world the way I did.3
But how can he bring the reader into such a magical world if it is so arcane and difficult? The problem, he suggests, is that most people’s exposure to mathematics does not go beyond elementary algebra and geometry. That tedious routine gives the wrong impression. It is like taking an art class “in which you were only taught how to paint a fence.”4 So students get turned off. Frenkel too was bored until a family friend told him what he is now telling us.
Edward Frenkel grew up in the small town of Kolomna, about seventy miles from Moscow. The lives of his parents—both engineers—were shaped by totalitarian thuggery. Frenkel’s grandfather had been arrested in 1948 on the charge that he planned to blow up an automobile plant; he was sentenced to hard labor in a coal mine. Frenkel’s father dreamt of becoming a theoretical physicist but found his path blocked as the son of an “enemy of the people.” Then, when his son, Edward Frenkel, dreamt of becoming a theoretical mathematician, he also found his path blocked, as the son of a Jew.
At the age of sixteen, he applied to Moscow State University. He triumphed in the written examinations, but for the simpler oral examinations, he was singled out for special treatment: a grilling that lasted for hours and included mathematical questions far beyond the typical understanding of entering students. The exam room emptied, the time grew late, and he was left alone with his interrogators. Frenkel, knowing that similar techniques had been used against other Jewish students, felt certain of failure. He withdrew his application. Only then did one examiner privately acknowledge to him that his performance was “really impressive” and advised him to try the Moscow Institute of Oil and Gas, which had an applied mathematics program. “They take students like you there.”5
And so they did. The Oil and Gas Institute was a refuge for Jewish mathematicians. The school was colloquially known as the Kerosinka, the word for a kerosene-burning space heater. Frenkel found extraordinary mentors and pursued mathematics with unswerving dedication, even scaling fences to attend lectures at Moscow State University.
The mathematicians who recognized Frenkel’s gifts included Dmitry Fuchs, who often tutored talented young Jews denied entry to the university. Fuchs was active in an informal evening school colloquially known as the Jewish People’s University, an enterprise that was not without risks; one of its organizers, after being brought in for interrogation by the KGB, happened to be run over by a truck. But Frenkel thrived, ultimately finding his way into Israel Gelfand’s seminar. Gelfand, a student of the great Andrei Kolmogorov, considered himself the Mozart of mathematics (and probably thought he was being modest). Frenkel was “star-struck.” “I could swear,” he writes, “that I saw a halo around Gelfand’s head.”6 Though Frenkel doesn’t mention it, Gelfand himself must have had angelic protection; an eminence at Moscow State University, he was also a Jew.
Frenkel gives a fascinating account of the late 1980s, describing a remarkable group of Jewish mathematicians without institutional homes, living in the shadows, holding mundane jobs while privately pursuing their passion. “Mathematics,” Frenkel writes, “became an outpost of freedom in the face of an oppressive regime.”7 And it conferred a sense of liberation, along with a thrill so intense that in recalling his first major insight, Frenkel even now jumbles metaphors:
Suddenly, as if in a stroke of black magic, it all became clear to me. The jigsaw puzzle was complete, and the final image was revealed to me, full of elegance and beauty, in a moment that I will always remember and cherish. It was an incredible feeling of high that made all those sleepless nights worthwhile … like the first kiss, it was very special.8
The suddenness of revelation and the sense of unexpected harmony are familiar from other historic accounts of mathematical discovery. It was, Frenkel writes, like suddenly coming on a vision of a majestic mountain after a long journey. “You catch your breath, take in its majestic beauty, and all you can say is ‘Wow!’”9
If that sounds like vernacular American, it may be because in the 1990s, after Frenkel had made his way to the United States, he learned English by watching David Letterman every night.
These autobiographical accounts are immensely charming, free of puffery and self-importance. Among Frenkel’s gifts is also the ability to treat mathematics as a social activity, not a solitary one. He is a collaborator, generous to colleagues, relishing conversation and, no doubt, his celebrity. This book is not only meant to promote mathematics.
But that is, of course, one of its ambitions. Frenkel cites a comment of Gelfand’s: “People think they don’t understand math, but it’s all about how you explain it to them.”10 Gelfand provided a peculiarly Russian example:
If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.11
And, yes, sometimes recourse to experience is helpful in grasping abstractions. Frenkel also decides to use experience as a resource, and we are asked to follow in his footsteps. He explains about this book:
I wrote it for readers without any background in mathematics. If you think that math is hard, that you won’t get it, if you are terrified by math, but at the same time curious whether there is something there worth knowing–then this book is for you.12
Frenkel sets out to seduce the reader as he was once seduced.
At first, the courtship shows great promise. Frenkel follows Gelfand’s example and begins with ordinary experience. As a young student being offered a glimpse of what real mathematics is about, he was asked to define the concept of symmetry and apply it to rotations of circular and square tables. There are table rotations, he sees, that seem to reproduce the table’s appearance, and table rotations that do not. Rotate a square table 90 degrees and it will look pretty much the same, but not if it were turned 22 degrees. Transformations that preserve shape in this way are called “symmetries” and they have very particular properties. Any one of them can be produced by a sequence of the others; there are transformations that leave things unchanged and transformations that undo those already made. These properties define a mathematical object known as a “group.” This is one of the first abstract concepts Frenkel learns and it becomes important in his later work.
Why would there be any interest in the idea? It is useful, in part, because it creates a realm whose elements, while being manipulated and transformed, stay within defined bounds. Nothing intrudes, nothing breaks out. A group, in one sense, is a closed collection of interacting elements. Think, for example, of the hours on a clock, Frenkel suggests; add or subtract them (as intervals of time) and the result remains a member of this group of numbers (e.g., 3+11=2).
There are also more profound examples. Frenkel soon turns to sub-atomic particles, another of his interests. In the 1950s, many elementary particles were discovered, but they seemed to have no relationship to each other. Both Murray Gell-Mann and Yuval Ne’eman guessed they could be classified into families of eight or ten particles, following what Gell-Mann fancifully called “The Eight Fold Way” (alluding to a doctrine of Buddhism). This seemed to make sense of particle interactions, but where did the idea come from? Gell-Mann thought he recognized the structure of a particular group (known as SU(3)). Based on the nature of that group, he also posited the existence of another particle: the quark.
Frenkel writes:
These particles were predicted not on the basis of empirical data, but on the basis of mathematical symmetry patterns. This was a purely theoretical prediction, made within the framework of a a sophisticated mathematical theory of representations of the group SU(3). It took physicists years to master this theory… but it is now the bread and butter of elementary particle physics. Not only did it provide a classification of hadrons, it also led to the discovery of quarks, which forever changed our understanding of physical reality.13
These examples also serve as a model for Frenkel’s own induction into mathematical thinking, and show how he would like to lead the reader toward understanding. Broadly speaking: begin with an observation derived from experience (the symmetries of turning tables). Look for patterns and principles (the idea of groups). Discover similar patterns and principles in other realms (the hours on a clock). Connect two seemingly unrelated worlds (tables and clocks). Repeat. Examine vastly different kinds of groups, categorize them, and make still more connections. With each elaboration, abstractions increase, but so do structural connections. What we learn about one system can then be applied in a very different one. It is a bit like Lewis Carroll’s riddle “Why is a raven like a writing desk?” only here it is “How are turning tables like sub-atomic particles?”
These examples also give a sense of Frenkel’s interests, which include algebraic geometry and topology—the ways in which abstract “spaces” and their transformations can be categorized—accompanied by forays into quantum physics.
Up until now, “readers without any background in mathematics”—Frenkel’s ideal audience—might very well be seduced, even though Frenkel never steps back enough to explain the procedure clearly. He is too eager, off and running, or rather, off and racing, thinking that somehow, having made a simple example clear, everything else will quickly fall into place. He discusses “braid groups” (which resemble groups of threads connecting pegs) and Betti numbers (which are related to groups and abstract spaces). Neither is made clear enough for readers to share the pleasure Frenkel took in his first significant discovery:
I found that for each divisor of the natural number n (the number of threads in the braids we are considering), there is a Betti number of the group Bʹn that is equal to the celebrated “Euler function” of that divisor.14
The levels of abstraction have mounted precipitously, and no doubt for non-specialists, bewilderingly. Frenkel does the same thing again when describing his doctoral work, beginning calmly with an overlong exploration of derivatives and then frantically pulling us through a series of abstractions climaxing with a “miraculous” finding: “I was able to construct representations of the Kac-Moody algebra of G parametrized by the opers corresponding to the Langlands dual group LG.”15
A clear explanation apparently defies even Frenkel’s best efforts. And soon enough he is juggling Hitchin moduli spaces, sheaves, Galois groups, Lie algebras, A-branes, B-branes and fibrations, surely dropping readers along the way. Again and again, elementary concepts are outlined with specificity and care only to be followed by whirlwinds of abstractions. Perhaps that is how Frenkel’s own mind works, but what of his ideal readers with no mathematical experience? Pretty early on, recalling the erotic promise of the title, they may soon be fast-forwarding, looking for the good parts.
At one point, Frenkel cites Goethe’s gibe “Mathematicians are like Frenchmen. Whatever you say to them they translate into their own language, and forthwith it is something entirely different.”16 And in part that is what is happening here. Unless you are fairly fluent in the way mathematical language works, the argument will soon be lost. The presentation does nothing to ameliorate the terror Frenkel said he aimed to overcome. And the text might as well be like a painted fence in an art class. Even working through these discussions slowly with the mathematically detailed footnotes, I found too many explanations opaque and lacking illustrative examples. This does not necessarily bother Frenkel. He doesn’t expect complete understanding.
It is perfectly OK if something is unclear. That’s how I feel 90 percent of the time when I do mathematics, so welcome to my world! The feeling of confusion (even frustration, sometimes) is an essential part of being a mathematician.17
Yes, sure, but if the purpose is to inspire love of mathematics and to convey its pleasures, something else is desired.
But there is something that Frenkel does accomplish in these expositions. Certain landmarks stand out. The procedure at work in the discussion of symmetry and groups keeps recurring on an ever larger scale. And by discovering appropriate abstract structures that govern the behavior of very different worlds, objects that may have appeared distinct turn out to be similar, and both, perhaps, may turn out to be manifestations of something else. By studying one thing, you get insight into another and sometimes find unusual relationships connecting vastly different fields of study: number theory, the analysis of surfaces, and certain kinds of functions.
This theme is also related to Frenkel’s main preoccupation with what is known as the Langlands program, a series of ideas developed by Robert Langlands in 1967 to link different areas of mathematics. This enterprise—on a vast scale—is wonderfully typical of the mathematical procedures I have been describing: discovering large scale analogies and mappings between very different worlds, revealing their shared structure.
Langlands’ suggestions have stimulated much research ever since, inspiring an almost utopian dream of a Grand Unified Theory: a revelation of the “heart of hidden reality,” as the book’s subtitle puts it, in which principles underlying all of mathematics will be revealed. (This recalls the flavor of mathematical ambitions at the beginning of the twentieth century until they were dashed by Kurt Gödel).
Something else preoccupies Frenkel. “How can it be,” Albert Einstein once asked, “that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?” How does abstract mathematics end up being so applicable in the real world, the way a particular group (SU(3)) was found to be relevant to the behavior of sub-atomic particles? Frenkel ends up pursuing that idea in his research as well, trying to connect mathematical objects he has studied with sub-atomic physics.
I think it was a mistake to frame this as a proselytizing book for lay readers, let alone to suggest as Frenkel does at the book’s beginning, that studying this kind of mathematics might be a “source of power, wealth, and progress.”18 I disagree too that studying high school algebra and geometry is like painting a fence or that it reveals nothing about “real” mathematics. But we do get from Frenkel an overall impression of how powerful mathematical thinking can be.
But the most peculiar aspect of Love & Math is that having immersed the reader in such layers of abstraction, Frenkel ends the book with a chapter about a short film he made with the director Reine Graves: Rites of Love and Math.19 It is here that Eros comes into full play.
The film was inspired by Yukio Mishima’s The Rite of Love and Death (1966), which ends with a Japanese officer in the 1930s, played by Mishima himself, committing ritual suicide, a scene that is queasily close to how the author brought his own life to a bloody finale.20
Frenkel says of Mishima, “His vision of the intimate link between love and death does not appeal to me.”21 But if it doesn’t, it is difficult to understand Frenkel’s fascination. Mishima’s film is like an aestheticized pornographic snuff film: at its climax, the officer, having found the correct spot above his groin, inserts his sword and, in sweaty pain, grips his intestines as they fall out with splashes of blood while his young lover watches, preparing to follow his lead.
Frenkel’s film is almost slavishly imitative; it too takes place on a Japanese Noh stage, is shot with no dialogue, and is accompanied by Richard Wagner’s musical evocation of love and death from “Tristan und Isolde.” The narratives’ back-stories though are different. Mishima’s is about the honor of an officer whose participation in a treasonous plot has not been discovered, but who is expected to execute his condemned fellow conspirators, thus providing some high-concept moral gloss for the self-inflicted blood-letting. Frenkel’s narrative back story seems drawn from a 1950s sci-fi film, if indeed it makes any sense at all: “A mathematician creates a formula of love … but then discovers the flip side of the formula: it can be used for evil as well as good.”22 He doesn’t want it to fall in the wrong hands but still wishes to preserve it, so he secretly tattoos it on his Japanese lover. He then commits suicide so he doesn’t fall into the hands of the enemy.
Frenkel himself plays the Mishima-esque mathematician. In one scene, accompanied by Wagnerian climaxes, he and his lover are naked, writhing in pleasure. Later, when tattoo time comes, the Mathematician drills as his lover moans in pain and passion.
Frenkel suggests that the film “attempted to create a synthesis of the two cultures by speaking about mathematics with an artist’s sensibility.”23 He writes:
We envisioned it as an allegory, showing that a mathematical formula can be beautiful, like a poem, a painting, or a piece of music. The idea was to appeal not to the cerebral but rather to the intuitive and visceral. Let the viewers first feel rather than understand it. We thought emphasizing the human and spiritual elements of mathematics would help inspire viewer’s [sic] curiosity. Mathematics and science in general are often presented as cold and sterile. In truth, the process of creating new mathematics is a passionate pursuit, a deeply personal experience, just like creating art and music. It requires love and dedication.24
And the result of this attempt to show mathematics’ beauty, its spiritual elements, and its relationship to the arts? In almost every respect, it is cartoonish, confused, self-important, and pretentious. In other words, it is everything that I have said Frenkel’s mathematical presentations at their best are not. The fact that he does not recognize this is startling. Has he been seduced by the lures of Western celebrity? Having achieved success in one realm, does he so wildly yearn for conquests in another? At any rate when such an accomplished mathematician declares that “this film is an attempt to introduce mathematics, to communicate about it, in a totally new way,” many have seemed prepared to take him at his word; the film attracted the sponsorship of the Fondation Sciences Mathématiques de Paris. It also inspired reviews in international scientific publications.
The film thrusts mathematics into a Romantic dreamscape, as if, perhaps, Frenkel were playing the lead in Hans-Jürgen Syberberg’s version of “Parsifal.”25 But the grotesqueries may not be unrelated to the mathematics that preoccupies Frenkel. It is intent, like the Langlands program, on connecting disparate realms, finding “mysterious analogies and metaphors.”26 Frenkel wants the film to suggest a “synthesis between the two cultures,” to show that “mathematics is a passionate pursuit, a deeply personal experience, just like creating art and music.”27
So what, exactly, is the mapping being made between art and mathematics? Here it is a crude allegory that Frenkel describes. The woman represents mathematical truth. The formula for love is tattooed as an expression of love. The unseen evil enemies who want to misuse the formula brings up “the moral aspect of mathematical knowledge.”28 Or, if I might put it more formally, Bm (the beauty of mathematics) = Ba (the beauty of art) and Dk (desire for knowledge) = Ds (desire for sex).
Unfortunately, the kind of beauty mathematics can display has little to do with the images of desire, pain, death, and defacement that characterize the film. The metaphysical Eros that pulls the mathematician toward knowledge is not the Eros that the naked figures on the screen display before the mathematician’s suicide (which is, thankfully, not quite as picturesque as Mishima’s).
Sometimes, a theoretical mind like Frenkel’s—so used to building abstractions out of analogies and making connections between disparate realms—mistakenly applies similar techniques to the world of experience. Analogies become identities and metaphors are made literal. We get crude equivalences instead of subtle parallels. The result can be formulaic, in the aesthetic rather than the mathematical sense. This leads to exaggeration and distortion. It is why so many brilliant theoretical thinkers have seemed stilted or witless when they pose as visionaries in other arenas. Perhaps something like that is behind this film with its klutzy abstractions and crude symbols.
It turns out, of course, that there are many ways art is not like mathematics. It does not simply play with metaphors, models and analogies. It makes connections that mathematics cannot (and vice versa). And erotic yearning does not always resemble yearning for knowledge. In Frenkel’s book and film the author misjudges the inspirational qualities of the first and the artistic qualities of the second. As a result, while his prodigious pursuits of mathematical satisfactions yield some pleasure, they promise no lasting contentment.
