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I understand entirely if "love" isn't something that math evokes in you. It didn't for me, either, until I was in my early twenties.
As a high school student, I felt relieved when I completed the required math credits and could skip calculus my senior year. Had anyone told me I'd eventually major in mathematics — even pursue a doctorate in it! — I'd have laughed in their face.
I remember the moment things changed. It was 2005, and I was living in Los Angeles. I'd grown my hair out and was playing keyboards in a rock band. My roommate — a boisterous Londoner and the band's drummer — had read Bill Bryson's A Short History of Nearly Everything* and encouraged me to read it. I didn't.
I mean, I was going to. I went to Barnes & Noble and headed straight for the physics section. I saw the Bill Bryson book on the shelf, but something else caught my eye: The Elegant Universe* by Brian Greene. I probably read a hundred pages of Greene's book in the store. I forgot all about Bryson and walked out of Barnes & Noble that day with a new hunger to understand something called "string theory."
As it turns out, physics is a pretty effective gateway drug to the world of mathematics. Within two years of reading The Elegant Universe, I'd left the world of music and entered university ready to study physics. But thanks to a superb calculus teacher, I caught the math bug instead. I graduated four years later with a degree in mathematics. I never even took a physics course.
At the same time that I became interested in string theory, the U.S. government was funding a group of mathematicians and theoretical physicists to connect ideas from string theory to a deep problem in mathematics called the Langlands Program. And one of the principal researchers — a mathematician named Edward Frenkel — had, like me, begun his mathematical life with an interest in physics.
As I read Frenkel's 2013 book Love & Math*, I couldn't help but smile as I thought about how, at the same time that a 20-year-old kid in Los Angeles was just starting his mathematical life enamored by string theory, there was a mathematician at Princeton exploring related ideas and experiencing the culmination of a life of study brought on by the same impulse I'd felt: a desire to understand the universe.
I've never met Edward Frenkel and probably never will (although I may have once been in the same room with him — we both attended the 2012 Joint Mathematics Meetings). But I can't help but feel a connection to his story. Reading Love & Math kicked up a whirlwind of emotions and rekindled an interest in an old flame.
Too Precious To Keep
Part autobiography, part math explainer, and part romance novel, Love & Math* is, at its core, an invitation for everyone to experience what mathematicians have largely kept to themselves.
Frenkel's goal is to "unlock the power and beauty of mathematics, and enable you to enter this magical world… even if you are the sort of person who has never used the words 'math' and 'love' in the same sentence." He aims to level the playing field:
It's too precious to be given away to the "initiated few." It belongs to all of us.
Frenkel alludes to elitism, but his desire to democratize mathematics may be rooted in something far more insidious. See, Frenkel grew up in Soviet Russia during the 1970s and 1980s. And although his mathematical talents were obvious, state-sponsored anti-Semitism (Frenkel's father is Jewish) kept him from attending the best Russian universities.
In one incredible story, Frenkel recounts how during his entrance exam to the Mekh-Mat, the "flagship mathematics program of the USSR," one of his interviewers asked him to define a circle. Frenkel answered correctly, "A circle is the set of points on the plane equidistant from a given point." The man told him he was wrong, that a circle is "the set of all points on the plane equidistant from a given point." Frenkel notes that this "sounded like an excessive parsing of words." But, really, it was just racism.
A Human Pursuit
The autobiographical aspects of Love & Math* are superbly well-written. They tell the story of a young man from Russia who, against all odds, persevered and fulfilled his dream of becoming a professional mathematician. In doing so, the reader glimpses far more than Frenkel's life story. You get to experience the drama — the humanity — that's inherent in doing mathematics.
In one chapter, Frenkel dramatizes a discussion with another Russian mathematician, Vladimir Drinfeld, that shaped his mathematical career. I mean "dramatizes" quite literally: Frenkel presents the discussion as a screenplay. It's one of the best portrayals I've encountered of the social component of mathematical research. It reminded me of the talks I'd have with my advisor in her office while working on my thesis.
Frenkel even describes the emotional experience of mathematical research, beginning with the frustration often encountered when first tackling a challenging problem:
It didn't come easily to me. It never does. I tried many different methods. As each of them failed, I felt increasingly frustrated and anxious.
This was Frenkel's first "real" crack at solving a research problem, and, as so many people do, he experienced imposter syndrome:
I questioned whether I could be a mathematician.
It's significant to me that Frenkel chose to include the anguish and doubt he experienced while working on a math problem. So often, people envision the big players in math as consummate geniuses. But, I'd bet money that even the likes of Gauss and Euler had their faith shaken from time to time. They were mere mortals, after all.
Frenkel spent "endless hours" with his problem, toiling over nights and weekends. His struggle led to insomnia, giving him a taste of the "side effects" that unhealthy obsession can ignite.
After months of sleepless nights and self-doubt, the answer came "suddenly, as if in a stroke of black magic." Frenkel describes the feeling of finally solving the problem as a "high." I think most people can relate to this. It's not any different than the dopamine rush you get from completing a challenging video game. But then Frenkel describes what, in my mind, is the true allure of mathematics:
For the first time in my life, I had in my possession something that no one else in the world had. I was able to say something new about the universe.
Humans are explorers by nature. Curiosity has taken us to the ocean floor. It has landed us on the moon. However, few locations remain to explore on our planet. The sciences offer a means to investigate the physical universe in new ways. Still, not everyone can afford space telescopes, particle colliders, and electron microscopes.
In contrast, mathematics provides the opportunity to explore the universe with a single tool available free of charge to every living person: your mind.
Exploring The Cosmos Without Leaving Home
Interwoven among the autobiographical sections of Love & Math*, Frenkel tells another story: the story of the math behind the Langlands Program. Named after Canadian mathematician Robert Langlands, the Langlands Program connects disparate ideas in mathematics with a kind of "Rosetta Stone."
As Frenkel's father put it, the details of the Langlands Program are "quite heavy." Frenkel admits, "even among specialists, very few people know the nuts and bolts of all the elements." That doesn't stop Frenkel from giving some descriptions of what's going on. And although the math in Love & Math can sometimes leave your head swimming, it serves a purpose:
My point is not for you to learn [it] all. Rather, I want to indicate the logical connections between these objects and show the creative process of scientists studying them: what drives them, how they learn from each other, how the knowledge they acquire is used to advance our understanding of the key questions.
Frenkel expertly guides the reader through concepts typically reserved for upper-level undergraduate — and even graduate-level — mathematics courses. And he does so in a way that meets readers where they are, without assuming much background.
This ability to distill technical concepts into everyday language is apparently a skill Frenkel learned from one of his mentors, Israel Gelfand. In the book's preface, Frenkel shares how Gelfand described this technique:
People think they don't understand math, but it's all about how you explain it to them. 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.
This approach works, and Love & Math* is the proof. In Chapter 2, The Essence of Symmetry, Frenkel describes the concept of a symmetry group by examining a round table. Most people know what symmetry is, but mathematicians use the word "group" differently than we do in everyday English. Everyone has seen a round table and can easily imagine interacting with it. So when Frenkel dives into things like the group of rotational symmetries, you have a concrete image to work with.
Many times, Frenkel presents concepts as a narrative. Rather than slap you across the face with definitions and theorems, Frenkel shares how he learned the concepts himself. He explains rotational symmetries, for example, in a dialog between himself and his teenage mentor, Evgeny Evgenievich Petrov, a professor at a local college and friend of Frenkel's parents.
Frenkel readily shares his own confusion during these conversations, which helps put the reader at ease. One can imagine expressing their confusion to Frenkel and him responding with a smile: "Hey, these things aren't always easy to understand, and it's okay to struggle. But it's worth the effort, and thinking of it this way helped me finally grasp it."
All this is not to say that you won't encounter any serious math. As Frenkel points out, "It is perfectly fine to skip those parts that look confusing or tedious at the first reading (this is what I often do myself). Coming back to those parts later, equipped with newly gained knowledge, you might find the materials easier to follow. But that is usually not necessary in order to be able to follow what comes next."
Even I, with the advantage of formal training in mathematics, had to mull over several parts of the text. And although this might seem scary, I think it adds richness to the book. You don't need to understand everything at first, or even at all, to get the most value from Love & Math*: an appreciation of what mathematicians do and how they do it. But if you want to go deeper, it's there for you.
The book's climax deals with a curious property of mathematics: ideas developed without any ties to the physical world sometimes, and often mysteriously, materialize as descriptions of some natural process. For the Langlands Program, this connection comes from quantum mechanics and, in particular, supersymmetry.
From 2004 to 2013, Frenkel led a research group investigating connections between supersymmetry and the Langlands Program. Although supersymmetry has yet to be tested experimentally, it's a model for understanding the universe. Somewhat miraculously, elements of supersymmetry correspond nicely to elements in the Langlands Program. It's as if all of these abstractions that, so far, exist only in the collective minds of mathematicians yearn to break free of the bonds of solipsism and claim their material existence.
Whether or not supersymmetry accurately describes the universe, the connection to the Langlands Program is real. And it embodies Frenkel's point that pursuing mathematics can lead to a profound understanding of the world around us. Mathematicians are, in some ways, like astronauts who can journey across the cosmos without ever leaving Earth.
A Charge of Love
There's a quote I've seen attributed to the German mathematician David Hilbert who, upon learning that a student had dropped his class to study poetry, apparently retorted, "You know, for a mathematician, he did not have enough imagination. But he has become a poet, and now he is fine."
I get what Hilbert is saying. Too often, the role of creativity in mathematics is understated, if even mentioned at all. I've met people who think that my interest in math means that I'm an excellent accountant. But, if the story is, in fact, true, then I can't help but wonder if Hilbert lacked the imagination required to reach certain students. Perhaps he needed a mentor, like Israel Gelfand, who saw communicating mathematics as an exercise in empathy.
Empathy. That's a word I keep returning to in my mind when I try to pinpoint what I enjoyed most about Love & Math*. Frenkel writes with genuine empathy — a kindness that seems to be born from an appreciation for the love of his parents, mentors, and all of the people that helped make his dream of becoming a mathematician possible, even when an anti-Semitic government put up roadblocks every step of the way.
Love & Math's final chapter, titled Searching For the Formula of Love, is both a memoir of Frenkel's excursions into the arts (he co-wrote a screenplay, The Two Body Problem, and a short film, Rites of Love and Math) and a manifesto for Frenkel's vision of a future where widespread appreciation for mathematics is normal. I admire Frenkel's willingness to pursue artistic interests. I wouldn't be surprised if his experience with film and theater helped develop the impressive storytelling he displays in Love & Math.
In Rites of Love and Math, a mathematician discovers "the formula for love." But there's a dark side, and, not unlike the discovery of atomic energy, the mathematician finds that his formula can be used for evil. "He realizes that he has to hide the formula," Frenkel writes, "to protect it from falling into the wrong hands. And he decides to tattoo it on the body of the woman he loves."
The film is an allegory for the "deeply personal experience" of creating new mathematics:
It requires love and dedication, a struggle with the unknown and oneself, which elicits strong emotions. And the formulas you discover really do get under your skin, just like the tattooing in the film.
Rites of Math and Love appeals to a general audience. By "emphasizing the human and spiritual elements of mathematics," Frenkel hoped to "inspire viewer's curiosity." In many ways, Love & Math does the same thing, except it doesn't shy away from digging into technical details. It's not just a love letter to mathematics; it's an invitation for readers to get their hands dirty and, possibly, discover something to love about math for themselves.
And so, Love & Math* serves two purposes: first, it's an account of the human, emotional experience of doing mathematics, and second, it's an introduction to the kinds of problems that unite people across disciplines and probe the secrets of the universe.
A mathematical formula does not explain love, but it can carry a charge of love.
"Among other things," Frenkel writes, "mathematics gives us a rationale and an additional capacity to love each other and the world around us."
Thanks to Jeremy Alm for reading the draft of this post and providing valuable feedback.
- I was once shunned by a fellow graduate student at Texas A&M because my mathematical interests weren't pure enough for him. I was primarily interested in graph theory and combinatorics, which was apparently too "elementary" for this person's liking. For some people, knowing mathematics isn't enough. You must care about the correct mathematics. ↩︎
- I never wrote a dissertation. I left graduate school before completing my Ph.D. I did, however, write an extensive thesis as an undergraduate based on original research in graph theory. ↩︎
- It's true. They're dead. ↩︎
- Frenkel was fortunate to have a mentor at such a young age, and he met him thanks to the dedication and support of his parents. I, too, benefited enormously from the support of my parents, especially my mom, who sought out and found a fantastic mentor for me as a budding pianist. I'm now a parent myself. Reading about Frenkel's experience reminded me that I have a duty to not only be the best father I can be but also to seek out mentors who can guide my children in their interests better than I can. ↩︎
- I don't know if this story is true. The only reference I could find was on the r/MathQuotes reddit, and even there the author admits to having no source. ↩︎
- In 2012, while returning from the Joint Mathematics Meetings (the same one where Frenkel himself gave a series of lectures about his research in quantum physics and the Langlands Program), one of the plane passengers sitting next to me, after learning that I was a mathematics student, remarked: "Oh, you must love QuickBooks, then." I do not. ↩︎