A Theory That Fits Anything Is Not Even Wrong

Peter Woit is a theoretical physicist at Columbia University, an outspoken critic of string theory, and the author of the blog and book titled *Not Even Wrong*. Lex Fridman opens by asking a question the guest lives inside every day: what, exactly, is the difference between physics and mathematics? The conventional answer is tidy. Mathematics makes rigorous claims about abstract objects and proves them; physics runs experiments and tests models.

Woit does not deny the textbook split. He just thinks it skips the interesting part. What people actually do under each label already overlaps, and that overlap, he says, is where the deepest work sits. Push the history far enough back, to someone like Newton, and the job titles collapse. Mathematicians claim Newton as a mathematician; physicists claim him as a physicist. Newton himself, Woit suspects, would have said he was a philosopher.

The distinction is a modern bureaucratic fact, not an ancient law of nature. After the twentieth century both fields grew into huge professional systems, but the borderland never went away. It thickened. Woit's colleagues in a mathematics department, he notes, are doing all sorts of different things, and most of those things still brush against physics. He is personally interested in one slice of that overlap, the slice that touches the most fundamental ideas in both subjects.

The overlap Woit cares about is not the old slogan that mathematics supplies tools for physics. It is a claim about the skeleton of the laws themselves. The most successful fundamental theories, he says, are built from a specific mathematical architecture: geometry, connections, curvature, spinors, the Dirac equation. Those objects are not rented calculators. They are the theory. They give physicists a unifying way of thinking, and they are also what mathematicians have been stitching together for a century and a half.

Fridman brings up Edward Frenkel's *Love and Math*, a book Woit recommends even while admitting that its group theory is hard to make accessible. Frenkel treats the mathematics he loves as a grand unified theory of mathematics, pulling geometry, number theory, and representation theory into unexpected contact. Woit's fascination is that the objects in that unification and the objects in the physicists' unification are more or less the same. The deepest ideas on both sides look intimately connected.

He is wary of selling this as everyday geometry. The ideas are alien to the three-dimensional world we grow up in, closer to a long apprenticeship than to triangles and cubes. That arcane quality is what first hooked him as a young student: it felt like a semi-mystical experience you could earn by study, except that it actually worked. Asked to sneak up on the picture for a five-year-old, he will not pretend it lives on a surface you can hold. Geometry is a large part of it, he says, and a striking recent turn in number theory is that the ideas fit naturally into a four-dimensional setting, the same count as space-time.

Fridman had expected to be told that unification demands many more dimensions. Woit's view goes the other way, against a large part of the unification literature. Everything we actually know, he claims, is about four dimensions. The structures seen in fundamental physics and in number theory can be understood there. Physicists, for decades, have tried to formulate theories in higher dimensions and then been stuck with a problem of their own making: how do you get rid of the extras, given that we only ever see four?

Fridman objects that mathematics is allowed to invent abstract objects on the way to a proof. Woit agrees, then draws a temperamental line. Mathematicians love generality. Given an idea, the first reflex is to ask in what greatest generality it can be stated, so for most geometry it does not matter whether you are in one, two, three, four, or seventeen dimensions. Physicists have to care about the specific example. Using extra dimensions to explain structure is tempting; the attempt typically founders because you cannot explain why we only see four.

There is a consolation prize. Fermat's Last Theorem is much easier for n equals three than in the general case, and physicists get a similar break: they live in a universe with n equals four and do not have to solve the general case. Even when Woit teaches mathematics students, he teaches unlike most mathematicians. He hunts for the crucial example that shows why a new technique is worth using. He is less interested in a precise theorem about exactly when it works and when it does not. Boil an idea down to the simplest situation in which it happens, he says, and you almost always get insight.

That taste for examples does not mean he can draw the objects in three-space. Even simple cases in his subject refuse to live in a room you can gesture around; capturing what matters usually means four or more dimensions. He uses pad and paper, and a blackboard in the office, but the drawings are abstract diagrams of how things fit, not pictures of a world you could walk through. Fridman finds that sad, a mystery the mind will never experience directly. Woit relocates the sadness. Most of these questions, he says, can be given a true answer. The scarce resource is time: you would have to sit down and work at it for hours, days, and weeks before your mind wraps around what is going on. Life is limited, so you decide what to spend it on.

Pressed for a beautiful idea in a few hours, Woit almost starts on groups, representations, and quantum mechanics, then stops himself. He wrote a whole textbook on that, and they do not have time. Fridman points instead to a paper of Woit's aimed at a grand unified theory of mathematics and physics. The key claim, Woit says, is the one they have already been circling: successful unification in physics over the last fifty years and the mathematics in Frenkel's book are the same kind of mathematics. You should not try to unify only one side. New ideas in fundamental physics should go deeper into mathematics, and mathematics should take inspiration from those physical successes.

Inside mathematics the name for this gathering is the Langlands program. In the 1960s Robert Langlands saw that what was starting to work in number theory could be recast in terms of symmetry, groups, and representations, in a way close to geometry. In the 1980s and 1990s, geometric Langlands asked what the same story said if you forgot the number theory and looked only at geometry, and it produced new geometric insight. A few months before this conversation, Peter Scholze and Laurent Fargues posted a major paper that took a local number-theory problem, something about behavior near a prime, and turned it into exactly the kind of pure geometry geometric Langlands people had been doing. By generalizing the mathematics they reformulated it, and it worked.

Fridman, who works near theoretical computer science, says he still cannot hear the music, and that makes him sad. Woit is not consoling. Even most mathematicians do not actually understand this. The group who really grasp all of it, including the Scholze-Fargues paper, is very small. Talk to mathematicians and physicists and many will admit the same embarrassment: the material is in their own field, they have a PhD, and it is still pretty clearly far beyond them.

A smaller idea, one Woit says he is embarrassed not to have understood until he started teaching math classes, is how algebra and geometry lock together. Whenever someone hands you an algebra, objects you can add and multiply, you should ask whether it is the space of functions on some geometry. The integers look like the least geometric algebra imaginable, yet they are functions on the space of primes: given an integer, evaluate it at a prime by taking it modulo that prime. Ten is zero at two and one at three, and you keep going. Conversely, given a space, stop staring at its points and study the functions on it. That two-way move is the basis of algebraic geometry. Much of modern number theory, and a good deal of geometry and topology, is built on it. Woit calls it beautiful because it packages a huge amount of power into something very simple.

Fridman asks him to formalize beauty. Woit's working definition is operational. An idea is beautiful when it packages a huge amount of power and information into something very simple. You can almost measure it: what non-trivial things does the idea tell you, versus how simply you can express it. You can also watch an idea get uglier. It does not quite do what you want, so you throw something else in, and you keep doing that until you get the outcome you wanted. That is the tell.

Given two theories, one simpler and one more complicated, which is more likely to capture the fabric of reality? History, Woit says, favors the simpler one, often simpler in a surprising way. The best theories, properly understood, are much simpler than you would have expected. His own belief is that there is something about the universe that is simple, even though our minds are limited and might only ever see the part of the world we have any hope of seeing.

Sabine Hossenfelder's book agrees that the final answer will be simple and argues that a local taste for beauty and simplicity has led physicists astray. Woit read it with real engagement, disagreed with a lot, then found they agreed at the end. He does not think beauty is the culprit. Physicists often say "math" when they mean a complicated calculation, not the interesting ideas. The real failure mode is falling in love. You have an idea, you think it is great, and like any love affair you become sure the object of your affections is more beautiful than others would think. People did not find a simple, powerful idea that then proved useless. They fooled themselves that a weaker idea was simpler and more beautiful than it was.

Physics still has an external check: experiment. If a simple idea fails and a more complicated version works, that can be the world. Hossenfelder's emphasis, which he shares, is that people developing better theories should concentrate on self-consistency, not on whether an idea feels beautiful. Dark matter and dark energy are inconsistencies with the world we see. For quantum gravity, where the effects are tiny, the traditional route has been a search for a fully consistent model of quantum mechanics plus general relativity. The danger, Woit says, is that every program succeeds on consistency. He thinks that is roughly what happened to string theory: it found a consistent theory of quantum gravity and then found 10^500 solutions, an essentially infinite set you cannot tell apart by experiment. Consistency is necessary. It is not enough.

Asked what string theory is, and why his blog calls it not even wrong, Woit starts with a warning: the term is now hard to pin down. The original meaning is clearer. From 1984, people felt they had a unified quantum theory of the Standard Model and gravity, obtained by replacing particles moving in space-time with one-dimensional loops, strings, that have infinitely many degrees of freedom. You could make sense of that quantum theory, but only if space-time was ten-dimensional. Then you had to get rid of six dimensions.

The hope that excited people in 1984 and 1985 was that consistency would force those six dimensions to be a Calabi-Yau manifold, and that there would be very few examples. Pick one of a limited set of ways to make six dimensions small, and you would get an effective four-dimensional theory that looked like the real world. The long story, and the point of Woit's book, is that this was a failure. There ended up being too many ways to do it. You could never pin the idea down well enough to extract a falsifiable prediction. It looked good at first; the more people looked, the less power it carried.

Because the core idea did not really work, people kept trying new things and wandered through a rich space of mathematics and physics. Beautiful mathematics came out of it, including two-dimensional conformal field theory, quantum field theories in one space and one time dimension with enormous symmetry, some of it the same mathematics that appears in Langlands. Woit's complaint is an inverse relation: the more interesting and fruitful the new idea, the less it looked like the real world. The technical hole he has written about is the perturbative definition. Perturbative string theory really only works in ten dimensions. To dump six of them you have to assume something beyond the formulation you understand, and the simplest such assumption, the Calabi-Yau postulate, did not work out.

Compactification means declaring that six dimensions are so small we cannot see them. The mathematical trouble is that "all possible six-dimensional spaces" is close to an unanswerable question; without a very strong principle that picks out specific ones, you can get anything you want. The whole aim of the extra-dimension theory, Woit notes, is to make its effects unobservable, which is an inherently dubious project: you add a lot of stuff and then explain why we do not see it. Experiments can look. Extra dimensions would change the inverse-square law of gravity at short distances; people have measured, and they have not seen a failure. You can always say the extras are still smaller. Fridman leaves a door open for new tools of observation. Woit does not slam it, but nothing of that kind has worked yet.

Fridman asks about Ed Witten, often called one of the most brilliant mathematicians and physicists alive. Woit met him first as a postdoc and calls him a truly remarkable figure, quite a bit smarter than just about anyone else and more hard-working, a frightening combination. Witten has revolutionized areas of mathematics and the relation between mathematics and physics. Most of that greatest work, Woit argues, has little or nothing to do with string theory. Witten does not have a Nobel Prize, while many people nowhere near as smart or hard-working do. Woit's reading is misfortune of timing: the field had gotten much tougher, and no one, however smart, was finding a new idea that worked physically enough to win a Nobel. Witten did get a Fields Medal for mathematical work, something previously almost inconceivable for an outsider to mathematics.

The Standard Model came together around 1973 and is insanely successful. Many of the people who built it got Nobels. Since then, attempts to do better, to find a more unified theory, have not worked. The theory is too good. What Witten has been trying to do with string theory is something no one has managed since 1973. Asked what is broken about the Standard Model, Woit names the usual headline first: gravity is not in it, and people want a unified quantum field theory that includes gravitational interactions. He then points at a different blemish. Forget the Higgs field, and the rest of the theory is tightly constrained, with very few free parameters and a few integers that tell you what the theory is. Introduce the Higgs to make it a theory of the real world, and 20 or 30 parameters appear, telling you the masses and what will happen.

One response is that the world is just like that, or that we live in a multiverse and those numbers differ in every universe. Another, which Woit finds more reasonable, is that the clutter is a clue: there is a deeper idea about where the Higgs comes from that would explain the numbers, and that is what people should look for. Playing devil's advocate, he will grant string theory this much: it was a fruitful source of ideas in mathematics and in parts of physics. Witten and others say an idea that inspiring must have something right about it. That, Woit thinks, is the strongest argument they have. It is not the same as a neighboring path to a theory of everything. Nobody has ever produced the hoped-for non-perturbative theory, and arguing with "string theorists" has become oddly frustrating because the term is less and less well defined. Some people who use it have never worked on a theory that contained any strings.

What happened sociologically, Woit says, is that a fairly well-defined proposal did not work, people branched in all directions, and "string theorist" became a tribal label for those trained in that lineage. Fridman pushes back: language evolves, and the phrase might now mean the broader effort to unify the laws at high dimensions. Neural networks, he notes, went in and out of fashion, and "deep learning" is often a sociological tag for whatever currently works. Woit grants the pattern, then refuses the wash. The initial understanding was specific: ten-dimensional superstrings compactified in six dimensions. The really bad thing, to his mind, is that it is hard to get people to admit, at least publicly, that that was a failure.

De facto, people stopped doing that compactification program and started doing more interesting things. Publicly they keep talking about string theory, pointing back to the 1984 idea as the starting point of the tribe. Graduate students still take courses on it. The public still hears it as the guiding inspiration. Having your origin story be something that never worked out, Woit says, deforms the whole hope of making progress. That, more than any one calculation, is the worst thing that has happened in the field: a lack of transparency about what failed, so there is no firm ground under the next step.

Why is he sure the original idea failed? It was seductive. At the time it looked as if there were only a few Calabi-Yaus that would work, a specific model on specific six-dimensional spaces that would give you everything. Witten and others got excited for an understandable reason. As people learned more, they realized there were more and more things you could do with those six dimensions. The failure mode is emptiness. You can fit quantum mechanics, gravity, anything you want. The idea ends up telling you nothing because it is consistent with just about anything. String theorists now argue that constraints might still emerge, or that we live in a multiverse and everything is true anyway. Those are ways of holding on. For a theory of everything, Woit still wants contact with the real world: an unexpected experimental result, or a way to calculate the unexplained numbers. That is the gold standard.

Is a theory of everything a physics problem, a math problem, or an engineering problem? Woit's guess is a combination of physics and mathematics, not a well-defined calculation waiting for more computer power. We may be missing physical ideas, mathematical tools, or both. If the theory includes gravity, it should describe black holes, and a huge focus of quantum-gravity work has been the paradoxes that appear when black holes meet quantum mechanics. What the slogan seeks, in a reductionist sense, is a description of all the right objects and how they interact at the most fundamental level. Getting from that to tomorrow's weather is another kind of problem, maybe engineering, maybe new science, maybe machine learning.

Fridman dislikes the hubris in the name. A theory of the basic laws would not explain why food tastes delicious, or life, or psychology, or artificial intelligence. Each layer wants its own theory of everything. Woit agrees completely. The terminology is bad and hubristic. Understanding how anything emerges from the underlying theory is hopeless if you only have that theory. Different levels need completely different ideas. He cites Philip Anderson's slogan that more is different: even if you understand perfectly how each piece interacts with the others, what the whole collection will do is a different problem.

Stephen Wolfram's physics project, simple rules on hypergraphs from which space and time emerge, gets "pretty much zero sympathy." Woit spent a little time looking and does not see it going anywhere. If the bottom of the world is cellular automata, then the beautiful, powerful geometric ideas of his whole career are a mirage randomly emerging from something simple-minded, and he has been shown no serious evidence. From one mathematical point of view, quantum mechanics is already as simple as it gets: the objects you write down in a quantum theory are the same objects at the deepest levels of mathematics. Cellular automata do not fit those structures. Fridman argues that the standard should not yet be whether the program solves a physics problem, because we lack good mathematics for how complex behavior comes from simple rules. Woit agrees that is a truly great problem, and that it might one day connect to what he finds fruitful. He will not spend more time on it. Timing, and a limited life, decide.

Eric Weinstein's Geometric Unity and Garrett Lisi's E8 theory, in Woit's view, founder at the same place string theory does. You unify by putting everything in a bigger structure, then you cannot get back down to the Standard Model. The groups we actually see are small: U(1), SU(2), SU(3). Almost immediately after 1973, people stuffed them into SU(5) or SO(10) and faced the same question: why does the world not show the larger symmetry? Why do more physicists not look carefully at Weinstein or Lisi? Woit has heard Weinstein explain it in person and has looked at the paper. To really understand exactly what is going on would take time he does not have. The ugly side effect of being a string-theory critic is the inbox: every day someone writes that since he dislikes string theory, he must agree with their theory of everything. Most of those letters are from people who do not know what they are doing. A continuum runs up to serious attempts. At some point he still has to stop.

Only in the last couple of years, Woit says, has he really come to appreciate Roger Penrose's twistor theory and to see how to do something with it. One idea is a different account of what space-time points are, and it only really works in four dimensions. Instead of taking points as fundamental, you work with twistors. The physical degrees of freedom live on light rays. Ask what you actually know about the world when you open your eyes: you see a sphere, the sphere of light rays coming into your eyes. Penrose's claim is that a point in space-time *is* that sphere. You formulate physics on the space of those spheres.

Penrose and others built a beautiful formalism for moving between the usual language and twistor space. Certain things worked very well. By the 1980s and 1990s they were somewhat stuck, a little like string theorists: the ideas could be developed in interesting directions, but they were not producing new insight into physics. What has Woit excited is a way he thinks actually does work in that direction. Mathematically you have to think in complex numbers, not just reals, which is natural for mathematicians and harder for everyone else. In the simplest case twistor space is four complex dimensions, and a point in space-time is a two-complex-dimensional plane inside it. As you move in space-time, those planes move.

Spinors arrived independently in physics and in mathematics, and they are why this rewrite pays off. Physicists doing quantum mechanics found a doubling: put an electron in a strong magnetic field and you get two energy levels, not one, with the split growing as the field grows. Particles are not described by one complex number but by a pair. Mathematicians, asking what can be consistently rotated in three dimensions, found vectors and everything built from them, and then found something else you cannot build from vectors: objects described by two complex numbers, the spinors. You can make a vector from two spinors; you cannot make a spinor from vectors. That is a lower layer of geometry, spinor geometry, still mostly skipped in graduate courses because it is hard to do correctly if your whole setup is vectors.

Rotate a spinor 360 degrees and it does not come back to itself; it becomes minus what it was. Fridman pictures a vector walking a Möbius strip and ending upside down. Woit says that picture does not capture it. The simplest description is two complex numbers, four real numbers, and it matters that they are complex, so visualization is blocked at the root. In twistor theory the puzzle becomes almost tautological. The space of spinors at a point *is* the point: the complex two-plane you already used to define a location in space-time. Without twistors, describing electrons takes a complicated ritual. With them, what you want is already how you described the point. That is one reason Woit thinks the program is promising. It does not just look pretty. In places it makes the physics simpler.

Does he actually hope this leads to a theory of everything? He does. The piece he does not think anyone had looked at from this angle is how quantum theory treats time. A standard move is to multiply the time coordinate by the square root of minus one and make it purely imaginary. In ordinary time the formulas are singular: even simple questions need delicate limits, taken from the right direction, or they fail. After the substitution the formulas are simpler and much better behaved. What Woit had not appreciated until recently is how dramatically that changes the structure of the theory, including how symmetries act, which is not at all what he first expected.

People usually treat the move as a mathematical trick to make formulas work out. His proposal is to take the trick as really fundamental. Twistor theory, he says, lets you talk about ordinary time and time times the square root of minus one at once; both fit. You end up with structures that look a lot like the Standard Model. That is the one stretch of the conversation where he sounds least like a critic and most like someone with work on the desk that he believes can succeed.

Fridman asks him, on the record, whether twenty years from now a Nobel will go to something directly connected to a theory of everything, and whether string theory might be it. String theory, definitely not. He would not take that bet, and he suspects that even if you asked string theorists now, some would say no. Twenty years earlier, when Michio Kaku was making public bets, some would have taken it; in 1984 a bunch would have said sure. Whether some other theory of everything succeeds he still finds relatively unlikely, mainly because it has become so hard to do higher-energy experiments. Collider technology is easier to forecast than people's ideas. Twenty years on, he claims, you will have the LHC with an order of magnitude more data at the same energy, and no higher-energy accelerator running successfully. Machine learning will help with technical problems. It will not completely change how physicists look at the underlying theory. Quantum computers may crack Philip Anderson's problem, strong-coupling behavior you can write down and cannot calculate. They will not tell you what the right underlying theory is.

Does he find the simulation hypothesis useful? Not really; it does not lead anywhere he can use. Fridman cares about the engineering of worlds you would stay inside anyway. Woit is willing to call those fascinating questions for the species and then repeats the constraint that has structured the whole interview: life is too short, and he is glad other people are thinking about it. Aliens get the same shrug. He has no idea how many intelligent civilizations are out there. Unless the government is covering something up, there is no evidence yet, and no particular obstruction either. He started as a kid with astronomy and science fiction; over the years the actual science he was learning felt weirder and more fascinating than the fiction, so he stopped reading much of it.

The books he will recommend are not the popular volumes that treat string theory as a triumph. Heisenberg writing about the birth of quantum mechanics; *The Second Creation* by Robert P. Crease and Charles C. Mann, a history up to the Standard Model in 1973. After that date, he says, there has not been much that actually worked, so books about the glorious years since 1973 are mostly glorying in things that do not really work. Getting children excited is fine if the exposition is honest. Getting them excited about ideas that are wrong, he argues, discredits the scientific enterprise. He has a very specific complaint about people coming on podcasts and telling the world that string theory is a huge success. It is not true, and when the public realizes that, the loss of scientific credibility is a real social problem.

What high-school students think about string theory does not matter much to him. What the smartest undergraduates and graduate students at Columbia, Harvard, or Princeton think matters a great deal: which textbooks they read, which story they hear about where the field is going, whether they are misled into believing they must learn that story to work. Diversity of ideas, Fridman adds, is powerful, and parking for decades in one pocket of thought is unhealthy even if the idea later turns out to be a piece of something bigger. Woit agrees, and points to Hossenfelder and to Lee Smolin on how research is organized. Humility helps you know when you are wrong. A serious lack of humility is also required if you believe you will make progress. You need both modes, and you need to switch.

Asked for the meaning of life, Woit notices that people put that question to physicists, who write books and teach courses about it, and never to mathematicians. If the public realized how little difference there is between what mathematicians do and what a lot of theoretical physicists do, they might see that looking for life-wisdom in that corner is a bit misguided. He has led a pleasant, privileged life without many of the challenges people mean when they ask about meaning, so he is the wrong person to ask. He is 64. There is probably less ahead than behind, which makes him want to finish some things he might otherwise run out of time for. Day to day he enjoys a pleasant life, including the mathematics, and often thinks things are going well. Fridman closes with Richard Feynman: the first principle is that you must not fool yourself, and you are the easiest person to fool.