Love and Math

by Frenkel, Edward

  Like Penrose, I believe that the Platonic world of mathematics is separate from both the physical world and the mental world. For example, consider Fermat’s Last Theorem. Penrose asks rhetorically in his book whether “we take the view that Fermat’s assertion was always true, long before Fermat actually made it, or [that] its validity [is] a purely cultural matter, dependent upon whatever might be the subjective standards of the community of human mathematicians?”10 Relying on the time-honored tradition of argument by reductio ad absurdum, Penrose shows that embracing the subjective interpretation quickly leads us to assertions that are “patently absurd,” underscoring the independence of mathematical knowledge of any human activity.

  Kurt Gödel, whose work – especially, the celebrated incompleteness theorems – revolutionized mathematical logic, was an unabashed proponent of this view. He wrote that mathematical concepts “form an objective reality of their own, which we cannot create or change, but only perceive and describe.”11 In other words, “mathematics describes a non-sensual reality, which exists independently both of the acts and of the dispositions of the human mind and is only perceived, and probably perceived very incompletely, by the human mind.”12

  The Platonic world of mathematics also exists independently of physical reality. For example, as we discussed in Chapter 16, the apparatus of gauge theories was originally developed by mathematicians without any reference to physics. In fact, it turns out that only three of those models describe known forces of nature (electromagnetic, weak, and strong). They correspond to three specific Lie groups (the circle group, SU(2), and SU(3), respectively), even though there is a gauge theory for any Lie group. The gauge theories associated to the Lie groups other than those three are perfectly sound mathematically, but there are no known connections between them and the real world. Furthermore, we have talked about the supersymmetric extensions of these gauge theories, which we can analyze mathematically even though supersymmetry has not been found in nature, and quite possibly is not present there at all. Similar models also make sense mathematically in a space-time that has dimension different from four. There are plenty of other examples of rich mathematical theories that are not directly linked to any kind of physical reality.

  In his book Shadows of the Mind, Roger Penrose talks about the triangle: the physical world, the mental world, and the Platonic world of math.13 They are separate but deeply intertwined with each other. We still don’t fully understand how they are linked together, but one thing is clear: each of them affects our lives in profound ways. However, while we all appreciate the significance of the physical and mental worlds, many of us remain blissfully ignorant of the world of mathematics. I believe that when we awaken to this hidden reality and use its untapped powers, this will lead to a shift in our society on the order of the Industrial Revolution.

  In my view, it is the objectivity of mathematical knowledge that is the source of its limitless possibilities. This quality distinguishes mathematics from any other type of human endeavor. I believe that understanding what is behind this quality will shed light on the deepest mysteries of physical reality, consciousness, and interrelations between them. In other words, the closer we are to the Platonic world of math, the more power we will have to understand the world around us and our place in it.

  Luckily, nothing can stop us from delving deeper into this Platonic reality and integrating it into our lives. What’s truly remarkable is mathematics’ inherent democracy: while some parts of the physical and mental worlds may be perceived or interpreted differently by different people or may not even be accessible to some of us, mathematical concepts and equations are perceived in the same way and belong to all of us equally. No one can have a monopoly on mathematical knowledge; no one can claim a mathematical formula or idea as his or her invention; no one can patent a formula! Albert Einstein, for example, would not be able to patent his formula E = mc2. That’s because, if correct, a mathematical formula expresses an eternal truth about the universe. Hence no one can claim ownership of it; it is ours to share.14 Rich or poor, black or white, young or old – no one can take these formulas away from us. Nothing in this world is so profound and elegant, and yet so available to all.

  Following Mishima, the centerpiece of the austere decor in Rites of Love and Math was a large calligraphy hanging on the wall. In Mishima’s film the calligraphy read shisei: sincerity. His film was about sincerity and honor. Ours was about the truth, so naturally we thought our calligraphy should say truth. And we decided to do it not in Japanese, but in Russian.

  The word “truth” has two translations into Russian. The more familiar pravda refers to factual truth, such as a news item (hence the name of the official newspaper of the Communist Party of the USSR). The other one, istina, means deeper, philosophical truth. For example, the statement that the group of symmetries of a round table is a circle is pravda, but the statement of the Langlands Program (in the cases in which it has been proved) is istina. Clearly, the truth for which the Mathematician sacrifices himself in the movie is istina.

  In our film, we wanted to reflect on the moral aspect of mathematical knowledge: a formula with so much power may well have a flip side and could potentially be used for evil. Think of a group of theoretical physicists at the beginning of the twentieth century trying to understand the structure of the atom. What they thought was a pure and noble scientific pursuit inadvertently led them to the discovery of atomic energy. It brought us a lot of good, but also destruction and death. Likewise, a mathematical formula discovered as part of our quest for knowledge could prove to be harmful. Although scientists should be free to pursue their ideas, I also believe that it is our responsibility to do everything in our power to ensure that the formulas we discover are not used for evil. That’s why in our film the Mathematician is prepared to die to protect the formula from falling into the wrong hands. Tattooing is his way to hide the formula and at the same time ensure that it survives.

  Because I have never had a tattoo, I had to learn about the process. These days tattoos are made with a machine, but historically (in Japan) tattoos were engraved with a bamboo stick – a longer, more painful process. I’ve been told it’s still possible to find tattoo parlors in Japan that use this ancient technique. This is how we presented it in the film.

  Which formula should play the role of “formula of love” was a big question. It had to be sufficiently complicated (it’s a formula of love, after all) but also aesthetically pleasing. We wanted to convey that a mathematical formula could be beautiful in content as well as form. And I wanted it to be my formula.

  Doing “casting” for the formula of love, I stumbled on this:

  It appears as formula (5.7) in a hundred-page paper Instantons Beyond Topological Theory I, which I wrote in 2006 with two good friends, Andrey Losev and Nikita Nekrasov.15

  This equation can seem forbidding enough that if we had made a film in which I wrote this formula on a blackboard and tried to explain its meaning, most people would have probably walked out of the theater. But seeing it in the form of a tattoo elicited a totally different reaction. It really got under everyone’s skin: everyone wanted to know what it meant.

  So what does it mean? Our paper was the first installment in a series we wrote about a new approach to quantum field theories with “instantons” – these are configurations of fields with remarkable properties. Althought quantum field theories have been successful in accurately describing the interaction between elementary particles, there are many important phenomena that are still poorly understood. For example, according to the Standard Model, protons and neutrons consist of three quarks each, which cannot be separated. In physics, this phenomenon is known as confinement. Its proper theoretical explanation is still lacking, and many physicists believe that instantons hold the key to solving this mystery. However, in the conventional approach to quantum field theories, instantons are elusive.

  We proposed a novel approach to quantum field theories that we hoped would help us un
derstand better instantons’ powerful effects. The above formula expresses a surprising identity between two ways to compute a correlation function in one of our theories.16 Little did we know at the time we discovered it that it would soon be slated to play the role of formula of love.

  Oriane Giraud, our special effects artist, liked the formula, but said it was too involved for a tattoo. I simplified the notation, and here’s how it appears in our film:

  The tattoo scene in the film was meant to represent the passion involved in doing mathematical research. While he is making the tattoo, the Mathematician completely shuts himself off from the world. To him, the formula really becomes a question of life and death.

  Shooting this scene took us many hours. It was psychologically and physically draining both for me and for Kayshonne Insixieng May, the actress playing Mariko. We finished this scene close to midnight on our last day of shooting. It was an emotional moment for our crew of about thirty, after everything we had been through together.

  The film’s premiere was in April 2010, sponsored by Fondation Sciences Mathématiques de Paris, at Max Linder Panorama theater, one of best in Paris. It was a success. The first articles about the film started to appear. Le Monde called Rites of Love and Math “a stunning short film” that “offers an unusual romantic vision of mathematicians.”17 And the New Scientist wrote:18

  It is beautiful to look at.... If Frenkel’s goal was to bring more people to maths, he can congratulate himself on a job well done. The formula of love, which is actually a simplified version of an equation he published in a 2006 paper on quantum field theory entitled “Instantons beyond topological theory I,” will probably soon have been seen – if not understood – by a far larger audience than it would otherwise ever have reached.

  In the words of the popular French magazine Tangente Sup,19 the film “will intrigue those who think of mathematics as the absolute opposite of art and poetry.” In an insert accompanying the article, Hervé Lehning wrote:

  In the mathematical research of Edward Frenkel, symmetry and duality are of great importance. They are related to the Langlands Program which aims to establish a bridge between the theory of numbers and representations of certain groups. This very abstract subject actually has applications, for example in cryptography.... If the idea of duality is so important to Edward Frenkel, one could ask whether he sees a duality between love and mathematics, as the title of his film would suggest. His answer to this question is clear. For him, mathematical research is like a love story.

  Since then, the film has been shown at film festivals in France, Spain, and California; in Paris, Kyoto, Madrid, Santa Barbara, Bilbao, Venice... The screenings and the ensuing publicity gave me the opportunity to see some of the differences between the “two cultures.” At first, this came as culture shock. My mathematics can be fully understood only by a small number of people; sometimes, no more than a dozen in the whole world at first. Furthermore, because each mathematical formula represents an objective truth, there is in essence only one way to interpret that truth. My mathematical work is therefore perceived in the same way by everyone who reads it. In contrast, our film was intended for a wide audience: thousands were being exposed to it. And, of course, they all interpreted it in their own ways.

  What I learned from this is that the viewer is always part of an artistic project; at the end of the day, it’s all in the eye of the beholder. A creator has no power over viewers’ perceptions. But of course, this is something we can benefit from because when we share our views we all get enriched.

  In our film, we attempted to create a synthesis of the two cultures by speaking about mathematics with an artist’s sensibility. At the beginning of the film, Mariko is writing a love poem to the Mathematician.20 When, at the end of the film, he tattoos the formula, this is his way to reciprocate: for him, the formula is an expression of his love. It can carry the same passion and emotional charge as a poem, so this was our way to show the parallel between mathematics and poetry. For the Mathematician, it’s his gift of love, the product of his creation, passion, imagination. It’s as if he is writing a love letter to her – remember the young Galois writing his equations on the eve of his death.

  But who is she? In the framework of the mythical world we envisioned, she is the incarnation of Mathematical Truth (hence her name Mariko, “truth” in Japanese, and that’s why the word istina is calligraphed on the painting hanging on the wall). The Mathematician’s love for her is meant to represent his love for Mathematics and Truth, for which he sacrifices himself. But she has to survive and carry his formula, as she would their child. Mathematical Truth is eternal.

  Can mathematics be a language of love? Some viewers were uneasy about the idea of a “formula of love.” For example, someone said to me after watching the film: “Logic and feelings don’t always get along. That’s why we say that love is blind. So how could a formula of love possibly work?” Indeed, our feelings and emotions often appear to us as irrational (though cognitive scientists will tell you that some aspects of this apparent irrationality can actually be described using mathematics). Therefore, I don’t believe that there is a formula that describes or explains love. When I talk about a connection between love and math, I don’t mean to say that love can be reduced to math. Rather, my point is that there is a lot more to math than most of us realize. Among other things, mathematics gives us a rationale and an additional capacity to love each other and the world around us. A mathematical formula does not explain love, but it can carry a charge of love.

  As poet Norma Farber wrote,21

  Make me no easy love...

  Move me from case to case.

  Mathematics moves us “from case to case,” and herein lies its deep and largely untapped spiritual function.

  Albert Einstein wrote:22 “Every one who is seriously involved in the pursuit of science becomes convinced that some spirit is manifest in the laws of the Universe – a spirit vastly superior to that of man, and one in the face of which we with our modest powers must feel humble.” And Isaac Newton expressed his feelings this way:23 “to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”

  My dream is that one day we will all awaken to this hidden reality. We may then perhaps be able to set aside our differences and focus on the profound truths that unite us. Then, we will all be like children playing on the seashore, marveling at the dazzling beauty and harmony we discover, share, and cherish together.

  Epilogue

  My plane is landing at Logan Airport in Boston. It is January 2012. I am coming to the Annual Joint Meeting of the American Mathematical Society (AMS) and Mathematical Association of America, invited to deliver the 2012 AMS Colloquium Lectures. These lectures have been given annually since 1896. Looking at the list of past speakers and the subjects of their lectures is like revisiting the history of mathematics of the past century: John von Neumann, Shiing-Shen Chern, Michael Atiyah, Raoul Bott, Robert Langlands, Edward Witten, and many other great mathematicians. I feel honored and humbled to be part of this tradition.

  Coming back to Boston brings up memories. My first landing at Logan was in September 1989, when I came to Harvard – to paraphrase the famous movie title, From Russia with Math. I was twenty-one then, not knowing yet what to expect, what was to come. Three months later, growing up fast in those turbulent times, I was back at Logan to see off my mentor Boris Feigin who was returning to Moscow, wondering when I would see him again. In fact, our mathematical collaboration and friendship continued and flourished.

  My stay at Harvard turned out to be much longer than I expected: getting my Ph.D. the following year, being elected to the Harvard Society of Fellows, and at the end of my term there, being appointed Associate Professor at Harvard. Then, five years after my arrival in Boston, I anxiously waited at Logan for my parents and my sister’s fami
ly to arrive, to join me and settle in America. They have been living in the Boston area since then, but I left in 1997, after the University of California at Berkeley made me an offer I couldn’t refuse.

  I still visit Boston regularly to see my family. In fact, my parents’ place is just a few blocks from the Hynes Convention Center, where the Joint Mathematics Meeting convenes, so they will have a chance to see me in action for the first time. What a beautiful gift – to be able to share this experience with my family. “Welcome home!”

  The Joint Meeting has more than 7,000 registered participants – most likely, the biggest math gathering ever. Many of them came to my lectures, held in a giant ballroom. My parents, my sister, and my niece are in the front row. The lectures are about my recent joint work with Robert Langlands and Ngô Bao Châu. It is the result of our three-year collaboration, our attempt to develop further the ideas of the Langlands Program.1

  “What if we were to make a film about the Langlands Program?” I ask the audience. “Then, as any screenwriter would tell you, we would have to grapple with questions like these: What’s at stake? Who are the characters? What’s the story line? What are the conflicts? How are they resolved?”

  People in the audience are smiling. I talk about André Weil and his Rosetta stone. We go on a journey through different continents of the world of mathematics, examining mysterious connections between them.

  Every click of the remote brings up the next slide of my presentation beamed to four giant screens. Each describes a small step in our never-ending quest for knowledge. We are pondering eternal questions of truth and beauty. And the more we learn about mathematics – this magic hidden universe – the more we realize how little we know, how much more lies ahead. Our journey continues.

  Acknowledgments