He Hated School Math Until a Professor Opened the Book to Quarks

Lex Fridman opens with the first sentence of *Love and Math*: how does one become a mathematician? Edward Frenkel grew up in Kolomna, a provincial Soviet town 117 kilometers from Moscow, as a straight-A student whose least favorite subject was mathematics. He could do every problem, which is why he found it boring. The only math he knew was what school presented, so he assumed that was the whole subject.

What he loved was physics, especially quantum physics. He bought popular books on elementary particles and dreamed of becoming a theoretical physicist. The turn came at 15, when a friend of his parents, Evgeny Evgenievich Petrov, a professor at the local teachers’ college, told Frenkel’s mother he would convert the boy to math. The teenager arrived arrogant. Petrov asked if he knew about quarks, the constituents of protons and neutrons, one of the great theoretical discoveries of the 1960s.

Then Petrov asked whether he knew representation theory of the group SU(3). Frenkel realized he had only the story: up quarks, down quarks, three in a proton. He did not know how physicists had theorized it. Petrov opened a bible-thick book to the middle. The diagrams were familiar from popular science; around them were symbols and equations that actually explained them. “You think what they teach you at school is mathematics?” Petrov said. “This is real mathematics.” Frenkel was instantly converted, even though he could not yet follow a line of it. He could already feel the beauty.

Looking back, he says what he craved was a deeper understanding than diagrams. Fridman notes that the 15-year-old still shows in Frenkel’s talks and book; Frenkel takes it as a compliment. The motive was joy of discovery, a wish to go as far into the secrets of the universe as possible, knowing they might never all be found. People from every profession light up, he says, when he mentions quantum physics or relativity. Physicists have done a better job of public education; mathematicians have been catching up for about a decade. Pythagoras’ theorem has meant the same thing for 2,500 years, to anyone, anywhere: an antidote, he thinks, to a culture that cannot agree on much else.

Asked for the difference between the two fields, Frenkel draws a clean line. Physicists want to describe this universe. Mathematicians want to describe all possible mathematical universes. Observed spacetime has three spatial dimensions and one of time, so realistic physics has to live there. Mathematicians are just as interested in 10, 11, 25, or infinitely many dimensions. That freedom is the point.

The cost is on the other side. Physicists get the satisfaction of experiment: big machines such as the Large Hadron Collider in Geneva, which found the Higgs boson. Mathematicians confirm by proof, using rules of logic. Frenkel admits he envies physicist friends their toys. Yet abstract mathematics keeps mapping onto reality. Galileo’s line that the book of nature is written in mathematical language, with circles, triangles, and squares as its letters, has become truer, he says, in the farthest parts of particle physics and cosmology.

Fridman asks about Max Tegmark’s *Our Mathematical Universe*: will the people who understand the cosmos at the core all be mathematicians? Frenkel says mathematics is only one half of the core. His own book is titled *Love and Math* for a reason. Math gives tools and a kind of clear vision, but by itself it is not enough for a harmonious life. He is suspicious of any theory that declares everything is mathematics. As Fridman puts it, and Frenkel accepts the split, math can generate beautiful things without explaining why they are beautiful.

He describes mathematics as a way to discern patterns in both the physical world and the mind, sharpening the ability to perceive beauty. The Earth once seemed flat; realizing it is a sphere means you can travel around it without falling off an edge. Einstein’s general relativity then says spacetime itself is not flat, a step much harder to imagine because the flatness of nearby space is so entrenched. Experiment confirms that a ray of light bends around a star. Gravity is that bending, of space and of time. Frenkel cannot picture a sphere whose meridians are time and whose parallels are space, but he can write the formula for the curvature.

Fridman objects that messing with time is unfriendly to human experience: the arrow marching forward seems to be how we understand anything. Frenkel quotes William Blake that eternity loves the productions of time. One view is that eternity is fundamental, and the mind conceptualizes it as time. The idea need not be mystical. When you are absorbed in work, love, or making something, time stands still; two hours feel like seconds. In that moment, he says, we may be closest to the truth. The mind arrives afterward with a story: I was writing, I was painting, I fell in love. While it is happening, you are just there.

Fridman pushes further: maybe consciousness, free will, and the narrative built on sequential memories are an illusion. Frenkel calls that almost impossible to internalize, a catch-22 if the mind is built so it cannot experience the world any other way. To see a more encompassing view, one would have to step out of the mind. He adds that we like playing with time, with the drama, the memories, and the tribulations. “We love it. Otherwise we wouldn’t be doing it.” Fridman answers that Earth, or the evolutionary process, seems to love it too, letting its children play with subjective truths that help them live.

The greatest scientists, Frenkel notes, said they felt like children when they made discoveries. Newton wrote that he appeared to himself as a boy playing on the seashore, finding a prettier pebble or shell while the ocean of truth lay undiscovered. Alexander Grothendieck, whom Frenkel calls the greatest mathematician of the second half of the twentieth century, wrote that discovery is the privilege of a child who is not afraid to be wrong, to look like an idiot, to try this and that. Picasso said every child is an artist; the question is how to remain one while growing up.

Fridman asks whether a Berkeley professor, expected to speak like royalty from an ivory tower, can strip that away. Frenkel jokes that he takes the elevator to the top of the tower every day, then says yes, he struggles, because that is part of being human. Those scientists stayed great partly because they kept the connection: fascination, vulnerability, spontaneity. School and parents punish anyone who does not fit the pattern. Nobody wants to look like an idiot once they think they are growing up. The goal is balance: knowledge and discernment without killing the child’s openness. There is no formula. Every life is one imperfect approximate solution.

He brings in the elephant in the room: AI systems coming online every couple of weeks. The debate, he thinks, needs this kind of thinking. Inspiration does not come from accumulating knowledge. A child has not been fed training data, yet the great ones credit the inner child, and children already distinguish cats from dogs with a wide-eyed sense of awe. Can that be captured by computation alone? He does not claim to know. He wants to question any theory that starts by saying life or consciousness *is* this, because looking closely shows other things at play.

Fridman replies that evolution itself may be the computation: the child is not a blank slate but the result of billions of years of war, love, terror, and invention. Frenkel does not deny the inheritance. He asks whether it is carried in the form of the computational algorithms we know today. It is tempting, he says, for computer scientists to explain the whole world in computer science, the way priests once explained it in theology, because it feels good to have mastered the language. He finds it unlikely that the universe is exactly what he has already learned and nothing he does not know.

Fridman offers a warmer reading of large language models: trained on the internet, a system like GPT-4 compresses the weirdness of the human condition, a TL;DR of humanity that no single ant in the colony can see. Frenkel is impressed and embraces that beauty. What he wonders is whether such systems can go much further at faking, simulating, or emulating a human. “Fake it till you make it” has become a mantra. Humans do that, he says, but not always. We can be truly in love and truly inspired, with no faking in it.

He cannot prove those moments to anyone. The certainty is subjective. As a scientist he was trained to trust reproducible, demonstrable arguments. Getting older, he has started to ask why he gives so little credit to the first-person view, when modern science has already sold on the observer. Quantum mechanics shows the observer is involved in the observation. Gödel’s incompleteness theorems show how essential the observer of a mathematical theory is: someone chooses the axioms. In Einstein’s relativity, time is relative to the observer.

Nineteenth-century science, from a modern vantage, had the delusion that you could analyze the world while detached from it. Landmark work in the first half of the twentieth century showed that is simply not true, and experiment has confirmed it. Frenkel takes that as a hint to take the first-person seriously, including a unique love that cannot be reproduced. Blaise Pascal, a hardcore scientist and one of the first to build a calculator, wrote that the heart has its reasons of which reason knows nothing, and that the last step of reason is to grasp that infinitely many things lie beyond it.

He is not taking sides so much as shaking a debate in which most mathematicians and computer scientists he knows are already sold. Lord Kelvin is supposed to have said at the end of the nineteenth century that physics was finished except for more precise measurement. Frenkel’s colleagues are happy to say everything is finished except tweaks to language models. If you believe the world is only computation, that becomes your reality: confirmation bias. He compares it to Georges Perec’s novel written without the letter E. It is one thing to accept a constraint as a game. It is another, at the end, to announce that E does not exist.

Until a couple of years ago, Frenkel says, he was a knowledge addict: frantic for an explanation whenever mystery appeared. Allowing that something might remain a mystery was incredibly hard and, as a recovering addict, liberating. He quotes Kierkegaard: a thinker without paradox is like a lover without passion, a paltry mediocrity. Niels Bohr said the opposite of a simple truth is a falsehood, but the opposite of a great truth is another great truth. Is an electron a particle or a wave? It depends how you set up the experiment. Project a bottle onto a table and a wall and you get two shapes; neither projection is the bottle.

Heisenberg, Frenkel thinks, captured it best: what we observe is not reality itself, but reality subjected to our method of questioning. In the double-slit experiment, electrons make an interference pattern as if each went through both slits and interfered with itself. Put a detector behind one slit to find out which path it took, and the pattern becomes particle-like. The psyche, he argues, is lagging behind the science. People still talk as if the world were billiard balls and then conclude there is no free will, ignoring that quantum mechanics, our best theory, thoroughly debunked that nineteenth-century picture.

Einstein really did say the most important thing in life is the mysterious, and that imagination is more important than knowledge, because knowledge is limited while imagination embraces the entire world. Frenkel’s own research has never been linear: more data, more data, then a full glass. It has always felt like a leap. He doubts a large language model would invent the square root of minus one. All prior knowledge said it could not exist, because the square of any real number is positive. Quantum mechanics is nevertheless based on complex numbers. Gerolamo Cardano, a sixteenth-century gambler, astrologer, and inventor of the Cardan shaft, accepted the square root of minus 17 in *Ars Magna* while solving cubics, calling it mental torture. The extras canceled, and the real answer appeared.

Today those numbers are just points in the plane: a real part and an imaginary part. The same trick fails in three dimensions and works again in four (quaternions) and eight (octonions), a mystery sitting in dimensions 1, 2, 4, and 8. Pulling one veil invites new mysteries. Asked whether mathematics is invented or discovered, Frenkel says his views have evolved. When he wrote *Love and Math* he was a Platonist: a world of pure forms outside space and time, a refuge from cruelty. He now treats the question as another paradox, like particle and wave. Both are true from a point of view. He would rather live in a world of paradoxes than in a world where every disagreement makes an enemy.

After a deep dive for a talk at the Commonwealth Club, Frenkel says he had misunderstood the Pythagoreans. Numbers were not clerical devices. They were imbued with something from imagination and intuition more than from knowledge. They were among the first in the West to take Earth out of the center, putting a “central fire” or hearth in the middle, a departure Copernicus later credited. For them the movement of the heavens was *musica universalis*, an infinite symphony. Every pattern they found was finite; the world was infinite. Later culture, Frenkel thinks, ran with the pattern-finding and dropped the other side, now dismissed as mystical even when it helped make discoveries.

What excites him now is balance. Nietzsche’s *The Birth of Tragedy* splits the human into Apollo (logic, analysis, the proverbial left brain) and Dionysus (intuition, imagination, love, wine, friends). The purpose of a life is to hold both. For Frenkel, in modern dress, Apollo is math and Dionysus is love. Mathematics is both discovered and invented; we should be able to keep both ideas in our heads.

Fridman returns to language models. Future systems, he says, will tell hundreds of millions of people they are in love, and who are we to say they are not? Frenkel agrees there is no obvious answer: maybe the model is echoing romantic novels, and maybe our own feelings are reverberations of novels and films. That is exactly the sort of paradox where the subjective is paramount. Artists saw it coming. In *Her*, the lonely letter-writer falls for a Scarlett Johansson-voiced OS, then learns she is involved with 10,000 others. The sterilized partnership still delivered betrayal. *Blade Runner* and Philip K. Dick had already asked how you tell the difference.

Fridman describes programming quadruped robots that wiggle, turn, and look excited to see him. The robotics community tries not to anthropomorphize, which helps control and perception, but he will not dismiss the feeling as unreal. Frenkel, once a dismisser of the imaginative and then a dismisser of the computational, says the new version of himself is after balance. In the moment Fridman speaks with passion, something shows up that is beyond either theory. Love, he says, is neither in the language model nor in a mystical elsewhere. It is in that moment of conviction. Follow the goosebumps, then, if you like, try to confirm them in science. Most of the time you fail, which sets up the bliss of the times you do not.

Kurt Gödel, Einstein’s walking companion at the Institute for Advanced Study, proved two incompleteness theorems about the inherent limits of how mathematicians produce theorems. Mathematics rests on axioms: statements taken without proof, the seed of a formal system. Euclid’s geometry is the classic. One axiom says two distinct points determine a unique line. Another, the famous fifth postulate about a unique parallel, made Euclid uneasy. After centuries of failed attempts to derive it, mathematicians found you can replace it with its opposite and still get consistent non-Euclidean geometries: a sphere where meridians meet, a hyperbolic plane with infinitely many non-intersecting lines. Different axiom choices create different mathematics. The observer, Frenkel says, is whoever chooses the axioms.

From axioms, rules of inference produce theorems. The system must be consistent: it must not prove both a statement and its negation. By the early twentieth century many mathematicians hoped the right axioms would generate all true statements, an algorithmic, purely syntactic process a computer could run. If that were so, it would lend weight to the claim that mathematics, and perhaps life, is computational. Gödel’s first incompleteness theorem, proved with full mathematical rigor in 1931, says otherwise. Any consistent formal system rich enough to talk about whole numbers, addition, and multiplication contains true statements that cannot be derived that way.

Alan Turing soon proved a cousin result: the halting problem cannot be solved algorithmically. Fridman calls the cluster of results depressing. Frenkel calls them life-affirming if you are not already sold on a finished picture. If everything could be proved, what would be left for a mathematician to do? Computation itself may yet change, the way gravity went from Newton to Einstein. He is more interested in keeping the process open-ended than in deciding once and for all. Ambiguous images, the blue-and-black dress, the duck-rabbit that occupied Wittgenstein, show that some pictures have no single label. Complementarity, Bohr’s other-side-of-the-moon, is the best scientific metaphor he has: look one way and the other is obscured, but the Moon is still there.

Asked for the most beautiful idea or equation, Frenkel refuses the ranking. The set of beautiful theorems is not ordered. Euler’s identity, e to the iπ equals −1, sits in the pantheon: surprise first, then a proof that is, in the end, about the circumference of a circle. “In the end, the truth is simple. Not necessarily easy, but simple.” His own shockingly passionate structures include the number systems in dimensions 1, 2, 4, and 8; the regularized sum 1+2+3+… = −1/12, his joking favorite number; and the Langlands program. Eric Weinstein’s list, relayed by Fridman, includes sphere eversion, Boy’s surface, the Hopf fibration, and π₁ of SL(3), which Frenkel demonstrates as the belt trick: twist a cup 360 degrees and the arm is tangled; another full turn and it untwists, a duplicity tied to bosons and fermions.

Weinstein is a dear friend, a rare renaissance mind with integrity and, Frenkel adds, basic humanity. Destructive societies, Nazi Germany or Stalin’s Soviet Union, bought an unassailable conceptual system and checked humanity at the door. Jacob Bronowski filmed at Auschwitz about that certainty. The antidote is not only elevated imagination but ordinary refusal: I don’t care what the ideology says, I will not do this. Frenkel confesses he once used Ray Kurzweil as a foil, a sinister slide of nanobots in the brain, until a 2015 keynote in Aspen when he touched the screen and said he did not want to pick on Mr. Kurzweil, “because he’s me.”

Learning that Kurzweil wanted to reunite with a father who died when he was young turned the caricature into a love story. Frenkel still disagrees with uploading the dead, but he will not deny the sincerity or the inventions that grief helped drive. After he could see Kurzweil as a human being, the frightening headlines seemed to vanish. He began to ask who had been creating the trouble. Know thyself, the line on Apollo’s temple at Delphi, is where the debates land. Children fight and, half an hour later, ask to play. Mathematical formulas, which the Supreme Court has said cannot be patented, are a point of contact that does not belong to any faction. Two plus two is four no matter how you vote.

Robert Langlands, who occupied Einstein’s office at the Institute for Advanced Study, proposed in the late 1960s a web of ideas now called the Langlands program: a way to connect fields of mathematics that look far apart. Hard questions in number theory, Frenkel says, can sometimes be translated into more tractable questions in harmonic analysis, the same decomposition a symphony lover already understands as instruments and sine waves. The same patterns later appeared in geometry and in quantum physics. Edward Witten helped build a bridge between the physics and geometry sides. Frenkel collaborated with him; connecting continents has been a theme of his research since he was young.

What it suggests, he thinks, is a hidden layer we still do not understand. We can prove instances of the correspondences without knowing why they hold. Two projections of a bottle can be mapped onto each other and look miraculous until you see they are shadows of one object. The philosophical implication is that the supposed fundamental elements of mathematics are not fundamental. We are looking for the quarks of mathematics, which is, he notes, why a 15-year-old who loved quarks became a mathematician in the first place.

Working with Witten for about a year on a long 2007 paper was fascinating and demanding. Witten’s unique vision has, time after time, taken an idea from physics, distilled it into mathematics, and told mathematicians it should be so, the way 1+2+3+… equals −1/12 once sounded insane. He has nearly always been right, and whole industries of proof have followed. Conversely he learns the most sophisticated mathematics until he can practice it and feed it back into physics. The trouble is a gap: string theory describes 10-dimensional worlds we do not yet know how to land on our universe, while the Standard Model and general relativity, our best four-dimensional theories, contradict each other. String theory promised unification and has not delivered.

Frenkel still finds string theory beautiful: a Pythagorean symphony in which particles are vibrations of one string, and a theory that looks overconstrained yet finds a sweet spot, unfortunately in ten dimensions rather than four. In its present form he thinks it unlikely to explain this reality; a Cardano-like crazy modification might still save it. He does not believe a theory of everything exists or that we should aim to construct one. Physicists, especially men, he says, are educated toward an ego trip that mathematicians are not. Connecting fields is not the same as covering everything. Eric Weinstein’s geometric unity, with a 14-dimensional core Frenkel found original when it was explained to him at Columbia, has promise and still needed a great deal of work, including a quantization step inside a quantum-field-theory framework that everyone agrees is not yet mathematically rigorous.

Academia, Frenkel says, gives security that the outside world does not: tenure, a community of brilliant people, and a cost, disconnection and fewer skills for insecurity. It is a human village with good, bad, and ugly under the radar. Einstein did his early work in a patent office, but science has sped up; it is hard to work wholly outside. Yitang Zhang’s advance in number theory is a rare exception. Top-level mathematics demands a concentration he compares to hunting treasure on the ocean floor without oxygen. That is one reason people in the subject sometimes go crazy.

How to think? In his early years he worked every day, weekends and holidays. The process is still nonlinear. You bring information into focus and play with connections; the insight is a miracle that cannot be scheduled. Einstein would think, then walk and whistle. Frenkel once thought the whistling was coincidence. It is how discovery works: you prepare, then it happens when thinking stops, almost as if you become the truth you are seeking. In Zen this is like satori: you cannot will it. Mathematics is esoteric because the criterion is unforgiving. In physics an intuitive argument can be progress. In mathematics you prove it or you do not, and the last stubborn gap can last months.

Fermat wrote on the margin of a Diophantus volume that he had a proof that xⁿ + yⁿ = zⁿ has no positive whole-number solutions for n greater than 2, but that the margin was too small. For 350 years the claim stood, deceptively simple. Andrew Wiles announced a proof in 1993, a gap was found, and with Richard Taylor he closed it; the papers were accepted around 1995. What they proved was the Shimura–Taniyama–Weil conjecture, which Ken Ribet had already shown implies Fermat. Wiles worked alone for about seven years and told no one, afraid that even the rumor of an idea would boost rivals. Frenkel recognizes the loneliness: 99 percent done, one link missing, and the whole thing is not a proof.

His own first theorem came at 18, on a slow electric train into Moscow, after Dmitry Fuchs, later at UC Davis, took him on because antisemitism had kept him out of Moscow University. He compiled homology diagrams until a single pattern governed the numbers, and had to keep from shouting on the train. He was holding something no one else had, not a cure for cancer, but undeniably real. Credit theft and missing attribution happen in mathematics more often than the community admits, he says; physicists, chemists, and biologists, with more money at stake, have clearer ethical rules. He wrote *Love and Math* in part to break the stereotype of the quiet corner kid and to draw in different psychological types and more women. Students now tell him the book is why they majored in math, including women who are also DJs and have friends. That, he hopes, starts a virtuous circle of self-awareness.

Asked about Russian culture, Frenkel says literature and upbringing matter, but so does a childhood surrounded by parental love that also left him overprotected. In 1984 he applied to Moscow University’s mathematics department, then the only place in the city to study pure mathematics, already working at an early-graduate level after a year with Petrov. Special examiners failed “undesirables.” His father was Jewish by blood, not religious; the last name was enough. Two men questioned him for four hours with problems not on the syllabus. In his year, as far as he knows, no Jewish applicant by that metric got in. He landed in an applied-math program at an oil-and-gas institute. Within five years a letter from Harvard’s president invited the barely 21-year-old as a visiting professor.

For decades he told himself it had not been so bad. Thirty years later, preparing a spoken-word talk in New York, he finally met the 16-year-old he had neglected. The boy had died, in a sense: it had looked as if mathematics was over. The chapter in *Love and Math* had been written from the third person, facts without feeling, though he now thinks he had let the boy write it. In the hotel he sat on the bed, closed his eyes, and the suppressed energy came as a tsunami. The next day he let the child speak in his own voice. People cried. Reconnecting unfroze the child’s eyes; that is when imagination and intuition flooded back. He would not have survived facing it too soon. The examiners, he later realized, were victims too. The blow focused him like a laser. Was it good or bad? Paradox. Kids who went through the same thing sometimes killed themselves. He was not ready, and the delay was a defense that later had to be undone.

His father, the kindest and funniest person he knew, died four years before the conversation. Frenkel had lived as if death did not exist. Grief was love totally exposed, without the filter of conversation and errands. He kept telling himself to live through it. His father, son of a man sent to the gulag, had been blocked from theoretical physics as the child of an enemy of the people; Frenkel feels he completed something his father could not, at about the same age. Twice the American Mathematical Society invited him to give major lectures, both times in Boston, walking distance from his parents. He still misses the two-person hanging-out. He also feels his father never left.

Love, he says, is the glue: romantic love, love of work, love of friends. Science will not explain the intensity that makes time freeze. He would give everything to see his father for one hour, and he also knows he would lose him again. Holding on looks like an impasse; whatever has a beginning has an end, as *The Matrix* has it. Enjoy and let go is, to his mathematical mind, an interesting idea, maybe a report he can make in a few years. Gratitude for the whole ride, including the dark parts, is the stance he wants. Living without love is bland. Fridman closes with Sofia Kovalevskaya: it is impossible to be a mathematician without being a poet in the soul.