愛德華·弗倫克爾：現實是悖論——數學、物理、真理與愛

Lex Fridman Podcast #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love Machine transcription by Qwen3-ASR 0.6B (MLX, local, Apache-2.0). May contain recognition errors. The following is a conversation with Edward Frankel, one of the greatest living mathematicians, doing research on the interface of mathematics and quantum physics, with an emphasis on the Langlands program, which he describes as a grand unified theory of mathematics. He also is the author of Love and Math, The Heart of Hidden Reality. And now a quick few second mention of each sponsor. Check them out in the description. It's the best way to support this podcast. We got House of Academias for healthy snacks, Shopify for e commerce and ExpressVPN for security and privacy on the internet. Choose wisely, my friends. Also, if you want to work with our amazing team, we're always hiring. Go to lexfreedman dot com slash hiring. And now onto the full ad reads. As always, no ads in the middle. 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I've used them for long, long, long, long time before they were ever a sponsor. So we flirted for a while, we dated for a while, and now we're fully committed together and married and not looking for any extramarital affairs in case you're wondering. I'm happy, I'm happy on Linux, I'm happy on Windows, on Android, on every single iPhone, everything I got, every device got ExpressVPN on it, and I'm happy. And if you want to be happy like me, go to expressvpn dot com slash lexpod for an extra three months free. This is the Lex Freedman podcast. To support it, please check out our sponsors in the description. And now, dear friends, here's Edward Frankel. You open your book, Love and Math, with a question: How does one become a mathematician? There are many ways that this can happen. Let me tell you how it happened to me. So how did it happen to you? So first of all, I grew up in the Soviet Union, in a small town near Moscow, called Kolomna, and I was a smart kid, you know, in school, but mathematics was probably my least favorite subject. Not because I couldn't do it, I was you know a straight A student and I could do all the problems easily, but I thought it was incredibly boring. And since the only math I knew was what was presented at school, I thought that was it, and I was like, "What kind of boring subject is this?" So what I really liked was physics, and especially quantum physics. So I was buying, I was I would go to a bookstore and buy popular books about elementary particles and atoms and things like that and read them, you know, devour them. And so I thought my dream was to become a theoretical physicist and to delve into this finer structure of the universe, you know. So then something happened when I was 15 years old. It turns out that a friend of my parents was a mathematician who was a professor at the local college. It was a small college preparing educators and teachers. It's a provincial town. Imagine it's like 117 kilometers from Moscow, which would be some like 70 miles, I guess. You do the math. I like how you remember the number exactly. Yeah, it's not funny how we remember numbers. Yeah, so his name was Yevgeny Yevgenyevich Petrov. Yeah, and if this doesn't remind you of the great works of Russian literature, then you haven't read them. Like war and peace, you know, like with the patronymic names, but this was all real, this was all happening. So my mom one day by chance met Yevgeny Yevgenyevich and told him about me. That I was a bright kid and interested in physics, and he said, "Oh, I want to meet him. I'm going to convert him into math." And my mom was like, "Math, he doesn't like mathematics." So they said, "Okay, let's let's see what I can do." So I went to see him. So I'm about 15 and a bit a bit arrogant, I would say, you know, like average teenager. So he says to me, "So I hear that you are interested in in physics and elementary particles." I was like, "Yeah, sure." And said, "For example, do you know about quarks?" And I said, "Yes, of course I know about quarks. Quarks are the you know constituents of particles like protons and neutrons, and it was one of the greatest discoveries in theoretical physics in the 60s that those particles were not elementary but in fact had the smaller parts. And he said, 'Oh, so then you probably know representation theory of the group SU three.' That's like as you would. So in fact, I wanted to know what was what were the underpinnings of those theories. I knew the story, I knew the narrative, I knew kind of the basic story of what these particles look like, but how did physicists come up with these ideas? How were they able to theorize them? And so I remembered, you know, like it was yesterday. So he pulls out a book and it's kind of like, like a Bible, you know, like like a substantial book and he opens them, it in the somewhere in the middle and there I see the diagrams that I saw in popular books, but in popular books there was no explanation. And now I see all these weird symbols and equations. It's clear that it is explained in there. Oh my god. He said, "You think what they teach you at school is mathematics? It's like no, this is real mathematics." So I was instantly converted. That you understand the underpinnings of physical reality, you have to understand what SU three is. You have to learn what are groups, what is group SU three, what are representations of SU three. There was a coherent and beautiful. I could appreciate the beauty even though I could not understand heads and tails of it. But you were drawn to the the methodology, the the the machinery of how such understanding could be attained. Well, in that respect, I think what I was really craving was a deeper understanding. And up to that point, the deepest that I could see was for those diagrams, but for that story that you know a proton consists of three quarks and a neutron consists of three quarks and they're called up and down and so on. But I didn't know that there was actually underneath, beneath the surface, there was this mathematical theory. If you can just linger on it, what drew you to quantum mechanics? Is there some romantic notion of understanding the universe? What is interesting to you? Is it the puzzle of it or is it like the philosophical thing? Now I am looking back. Yeah, so whatever I say about Edward at 15. Yeah, he's colored by my you know all my experiences that happen in in the meantime. I should say current views and so on. For the people who may not know you, I think your book and your presentations kind of revealed that that 15 year old is still in there somewhere. Well, I take it as a compliment. Some of the joy he's probably still here now. Yes, in some way. Yeah, I think it was a joy of discovery and the joy of going deeper into the kind of the to the root to the to the deepest structures of universe to the secrets, the secrets. And we may not discover all of them. We may not be able to understand, but we're going to try and go as far and as deep as we can. I think that's what. Was the motivating factor in this? Yeah, there's this mystery. There's this dark room, and there's a few of these mathematical physicists. They're able to shine a flashlight briefly into there. We'll talk about it, but it also kind of makes me sad that there are so few of your kind that have the the flashlight to look into the room. It's interesting. I don't think there are so few to be honest because I think I find a lot of people are actually interested. If you to talk, if you talk to people, you know like some people you wouldn't expect to to be interested in this from all walks of life from people of all kinds of professions. I tell them I'm a mathematician and I'm a mathematician, okay. So that's a separate story. A lot of people I think have been traumatized by their experience in their math classes. We can talk about it later. But then they ask me what kind of research I do and. I mentioned that I I work on the interface of math and quantum physics, and their eyes light up and say, "Oh, quantum physics or like Einstein's relativity. I'm really curious about it. I watch this podcast or I watch that podcast, you know, and I've learned this. It's like, what do you think about that? " So I actually find that actually physicists are doing great job educating the public, so to speak, in terms of. Um, popular books and videos and so on. Mathematicians are behind that we're catching we're starting to catch up a little bit. Have been starting last ten years, but we are still behind. But I think people are people are curious. Science is a is still very much you know something that people want to learn because that's our kind of the best way we know to establish some sort of objective reality, whatever that might be. Yeah, to figure out this whole puzzle, to figure out the secrets of the universe whole things that we can agree on, kind of you know like even though for me at this point I always you know make an argument that our physical theories always change, they get updated. So you had Newton's theory of gravity. Then Einstein's theory, you know, superseded it, but in mathematics, it seems that theories don't change. Pythagoras theorem has been the same for the last 2500 years. X squared plus y squared equals z squared. We don't expect that next year suddenly it will be z cubed, you know. So and so that to me is actually even more hints, even more at how how much we are connected to each other because Pythagoras theorem, if you think about it or any other mathematical theorem. Means the same thing to anyone in the world today, regardless of their cultural, you know, bringing religion, you know, ideas, ideal ideology, gender, whatever nationality, race, whatever, right? And it has meant the same to everyone everywhere and most likely will mean the same. So that's to me kind of an antidote to the kind of divisiveness that we sometimes observe these days, where it seems that we can't agree on anything. To the political complexity of two plus two equals five in Georgia, was 1984. I was in the Soviet Union in 1984, and so in many ways, I see that it was present, the novel was present, but we still have not found a dictator who would actually say two plus two equals five and would demand their citizens to repeat that. The night is still young. Has not happened yet. It does feel like math and physics are both sneaking up to a deep truth from slightly different angles, and you stand at the crossroads or at the intersection of the two. It's interesting to ask, what do you think is the difference between physics and mathematics in the way physics and mathematics look at the world. There is actually an essential difference, which is that physicists are interested in describing this universe. Okay, mathematicians are interested in describing all possible mathematical universes of which. You know, and some of our work of I still consider myself more of a mathematician than a physicist. My first love for physics, notwithstanding. Mathematicians are in a way we have more diversity. If you if you you might say, so we we are accepting, for instance. Our universe has three spatial dimensions in one time dimension, right? So what I mean is that allegedly, allegedly observed that we can observe today, right? So of course there are theories where there are some hidden dimensions as well. Well, let's just say observed observed dimensions. So this tabletop has two dimensions because you can have two axis, two coordinate axes, x and y, but then there is also a third one to describe the space of this room. And then there is a time time dimension. So realistic theories of physics have to be about spaces of three three dimensions or spacetimes of four dimensions. But mathematically, we are just as interested in theories in 10 space time dimensions or 11 or 25 or whatever or infinite. Dimensional spaces, you know, so that's the difference. On the other hand, I have to give it to the physicists. We don't have the same satisfaction that they have of having their theories confirmed by an experiment. We don't yet to play with big machines like LHC in Geneva, Large Hadron Collider that recently discovered you know the Higgs boson and some other things. For us, it's all like a mental exercise in some sense. We do we prove things by using rules of logic, and that's our way of confirming experimental confirmation, if you will. But I think we kind of I kind of envy a little bit my friends physicists that they they get to they get to experience this sort of this big toys, you know, and play with them. But it does seem that sometimes as you've spoken about abstract mathematical concepts map to reality, and it seems to happen quite a bit. That's right. So the mathematics is underpins physics, obviously. It's a language, the book of nature, as Galileo famously said, is written in the language of mathematics. And and you know the letters in it are the circles, triangles, and squares, and those who don't know the language I'm paraphrasing. Are left to wander in a dark labyrinth. That's a famous quote from Galileo, which is very true and has become even more true more recently in theoretical physics in the most in the most um sort of. Far out parts of the theoretical physics that have to do with elementary particles and and as well as the the structure of the cosmos at the large scale. What do you think of Max Tegmark who wrote the book Mathematical Universe? So do you think just lingering on that point? You think at the end of the day. The future generations will all be mathematicians, meaning meaning the ones that deeply understand the way the universe works at the core is just. Mathematics. At the core of, you know. I would say mathematics is one half of the core. So the book is called Love and Math. Okay, so these are the two pillars. Yeah. In my view. Yes. In other words, you can't cover everything by math. So mathematics gives you tools, it gives you way of a kind of a clear. Vision, but mathematics by itself is not enough for one to have a harmonious and. And balanced life, you know, so I, I am suspicious of any theory that. Declares that everything is mathematics. So math can generate things that are beautiful, but it can't explain why. It's beautiful. Math, you could say, is a way to discern patterns to find regularities in the universe and both physical and mental universe. The mathematics explores the mind as much as it explores. The physical world around us, and it helps us to find those patterns, which kind of which makes our perception more sophisticated, our ability to perceive things such as beauty. You know, and it sharpens our ability to see beauty to understand beauty. So our world becomes more complex. From thinking that our. That our that Earth is flat, we go to realizing that it is round, that it is shaped as a sphere, so that we can actually travel around the Earth, you know, so there isn't a place where we hit the end, so to speak. And then proceeding in the same vein, then Einstein's general relativity theory tells us that. Our space time is not flat either. This is much harder to to imagine that bend a bend three dimensional or four dimensional three dimensional space or four dimensional space time because this idea that the space around us is flat is so deeply entrenched. And yet we know from the from this theory. And from the experiments that have confirmed it, that a ray of light bends around a star as if being attracted by the by the force of gravity. But in fact, the force of gravity is the bending. It's just that it's not only the bending of the space, it's also the bending of space time. There is a curvature. Not only between spatial dimensions, the way parallels and meridians come together in a small scale, they look like perpendicular lines, but if you zoom out, you see that the space. Are they curving the space? They are sort of the the tracks along which the space gets curved. That's that's that would be the curvature of spatial dimensions. But in fact, now throwing time and one time, imagine a sphere which lives which has one of the meridians corresponding to time. And parallels corresponding to space. I can't imagine it, but I can write mathematical formula expressing that curvature. And that's in fact that curvature is responsible for the force of gravity attraction between the sort of simplest instantiation of it attraction between two planets. Or between two human beings. Yeah, the time bending time, is not very nice that what that theory did to time, because. It feels like the marching of time forward is fundamental to our human experience. The arrow of time marching forward nicely, seems to be the only way we can understand the universe, and the fact that you can start. Up to now, up to now, there are people who claim that they can that they have they possess other ways of experiencing it. So truly can visualize messing with time. Well, messing with time, but not necessarily messing with time because one point of view is that you know. I think who was it? I think William Blake. Who wrote that eternity loves time production? So one point of view is that it is eternity which is fundamental where time stands still, which our mind conceptualizes. As the time, so but in fact you know it's not something mystical. If you think about when you about it, when you really absorbed in something. Time does the stand still, and then you look at the clock and it's like oh my god two hours have passed, and it felt like a couple of seconds. When you are absorbed, when you are in love, when you are passionate about something, when you are creating something. You are we lose ourselves and we lose in the sense of time and space for that matter. You see, so there is only that which is happening that creative process. So I think that this this is familiar to all of us. And we may be actually the closest to the truth at that moment. The truth. And so yes, so then there is a point of view that this is where we are the we are who we are. At our sort of fundamental at our fundamental level, and after that the mind comes in. And tries to conceptualize it. It's like oh because I was writing something, um, I was writing a book, I was painting this painting, or maybe I was watching this painting and got totally absorbed in it. Or I fell in love with this person. That's what happened. But in the moment when it's happening, you are not thinking about it. You are just there. Yeah, we construct narratives around the set of memories that that seem to have happened in sequence, or at least that's the way we tell ourselves that, and we also have a bunch of weird human things like consciousness and the experience of free will that we chose. A set of actions as the time unrolled forward. Right. And we are intelligent conscious agents making taking those actions, but what if all of that is just. In illusion. In illusion and a nice narrative we tell ourselves. That's a really difficult thing to internalize. And imagine imagine that to make it really catch twenty two. That our imagine that our minds are set up in such a way. Yeah. That they can't approach the world or experience otherwise, so in other words to understand to see that from a more. Um, kind of all encompassing point of view, we have to step out of the mind. Well, I wonder what's the more honest way to look at things. But I think we like to be to play with time. I think we like to play with these experiences with all the drama of it with all the memories with all the tribulations. I think we love it. We love it. Otherwise, we wouldn't be doing it. I think that's or Earth loves it. The evolutionary process somehow loves it. Whatever whatever this thing that's being created here on Earth, it seems to. Like to create, like to allow its children to play with certain. Truths that they hold subjective truths that are useful for the competition or whatever this dance that we call life broadly defined not just humans. And and you know I am glad you mentioned that because, what I find fascinating is that the greatest scientists are on record saying that when they were making their. Discoveries they felt like children, so Isaac Newton said to myself, "I only appear as a child playing on the seashore and every once in a while finding a prettier pebble or a prettier shell." Whilst I think some he said something like the infinite ocean of knowledge lay. Lay was lying before me. Alexander Grothendieck, who probably it was the greatest mathematician of the second half of the twentieth century, the French mathematician Alexander Grothendieck. Wrote that discovery is a privilege of a child, a child who is not afraid to be wrong once again to look like an idiot, you know to to to try this and that, and I am paraphrasing and go through trial and error. That is for them, in other words, for them. That innocence. Of a child who is not afraid, who has not yet been told that it cannot be done. Okay, that was essential to scientific pursuit to scientific discovery. And now and now also compared to Pablo Picasso, a great artist, right? So who said every every child is an artist? The question is to how to preserve that as we grow up. Do you struggle with that? You are one of the most respected mathematicians in the world. You are Berkeley. You are like this this this statue. You are supposed to be very like you know. I have an tower. Yeah, sometimes I joke. I say I take I take an elevator to the top of the ivory tower every day. Yeah. And you are supposed to speak like royalty. Do you struggle to let strip all of that away to rediscover the child when you are thinking about problems when you are teaching when you are thinking about the world. Absolutely, I mean that's part of being human because when we grow up, I mean all of them all of these great scientists. I think they were so great in part because they were able they maintained that connection, okay, in that fascination that vulnerability. Um, that spontaneity you know and. Um, kind of looking at the world through the eyes of a child, but it's difficult because you know you go through education system and for many of us. Uh, it's not especially helpful for maintaining that connection that we kind of like. We are being told certain things that we accept, take for granted, and so on and little by little and also we get hit every time we act different, okay every time we act. That doesn't in a way that doesn't fit sort of the pattern. We get punished by the teachers, get punished by. Parents and so on. And don't get respect when you act childlike in in your thinking when you are fearless and. Looking like an idiot. That's right. Because there is a hierarchy. Nobody wants to look like an idiot. You know once you start growing up or you think you are growing up. Yeah. At the beginning, you don't even think you don't think in these terms. You just play, you are just playing, and you are open to possibilities to these infinite possibilities that this world presents to us. So how do we? I am not saying that. Education system should not be also kind of taming that a little bit. Obviously, the goal is balance. That acquiring knowledge, so that. We can be more mature and more discerning, more discriminating in terms of our approach to the world in terms of our connections to the world and people and so on. But how do we do that while also preserving that innocence, of a child. And my guess is that there is no formula for this. It is a life is an answer. Every life, every human being is one particular answer to how do we find balance. That's one imperfect approximation, approximate solution. To the problem, but we can look we can look up to the great ones. Who have credentials in the sense that they have shown and they have proved that they have done something that other humans appreciate our civilization appreciates, say Isaac Newton or Alexander Grothendieck or Pablo Picasso. So they have established their right. To speak about these matters, and we cannot dismiss them as mere madmen. They say, "Okay, well if if the same thing was said by somebody who never achieved anything in in that in their in their field of endeavor, you would be it would be easy for us to dismiss it. But when it comes from someone like Isaac Newton. " We take notice, so I think there is something important that they they teach us, and especially today in this age of AI. Of course, there is a big elephant in the room always, which is called AI. Yeah, right. And so I know that you are an expert on the subject. And we are we are living now in these very interesting times of new AI systems coming online pretty much every couple of weeks. So I kind of to me that whole debate about. What is it? What is artificial intelligence? Where is it going? What should we do about it? Needs an influx of this type of. Considerations that we've just been talking about, that for instance the idea that. Inspiration creativity doesn't come from accumulation of knowledge because obviously a child a child has not yet accumulated knowledge and yet the great ones are on record saying that a child has a capacity. To to create, and an adult credits the inner child. The inner child. Yeah. For this capacity to create, as an adult, you see, that's kind of weird if we take the point of view. That everything is computation, everything is accumulation of knowledge that just bigger and bigger datasets, finer and finer neural networks, and then we will be able to replicate human consciousness. If we take that point of view, then what I just said kind of doesn't fit. Because obviously a child has not been fed any training data as far as we know. Yet they are perfectly capable of of you know of distinction between cats and dogs for instance and stuff like that, but much more than that they are also capable of that. You know wide-eyed in a sort of perspective, so does can it really be captured that perspective that sense of awe? Can it really be captured by computation alone? I actually I don't know the answer, so I am not sort of trying to. To present a particular point of view, I am trying to question any theory that starts out by saying. Life is this or consciousness is this, because when you look more closely, you recognize that there are some other things at play. Which do not quite fit the narrative. And it's hard to know where they come from. It's it's also possible that the evolutionary process has created. Is the very it is computation and the and the child is actually not a blank slate but the result of one of the most incredible. Several billion year old computations that that had explored all kinds of aspect. Of of life on Earth, of of war and love and terror and ambition and violence and invention all of that from the bacteria to today. So like that young child is not is not a blank slate. They coming they they they actually hold within them the knowledge of several billions of years. Right, the question is whether as a child you carry that in the form of the kind of computational algorithms that we are aware today. You see what what strikes me as unlikely. Is that how should I put it? How interesting that you know we you you are a computer scientist and there are other people coming I have studied computer science so I know a little bit. And so it's tempting to say, oh, the whole world is computer science or is based can be explained by computer science. Yes, why because it makes me feel good because I have mastered it I have learned it my ego is very happy. Yeah, and people come to me and and and they look up to me and they revere me kind of like priests in old in old days when the religion was paramount when when you would be would tend to explain things in theological religious terms. Today science has progressed. There are fewer people who kind of buy into religious official religion, you know. So. We have this urge, I suppose, to to explain and to know and to dissect and to analyze and to conceptualize, which is a wonderful quality that we have and we should definitely. Pursue that, but I find it a little bit unlikely. That the universe is just exactly what I have learned, and not something that I don't know. You see. Well, there is a lot of interesting aspects of the current large language models. That one perspective of it, I think speaks to the love and math that you talk to, which is. They're trained on human data from the internet, so at its best a large language model like GPT four. Captures the magic of the human condition on its full display, its full complexity. And it's mimicking it's trying to compress all the weirdness of humans. Of all the debates and discussions, the perspectives, all the different ways that people approach solving different problems, all of that compressed. So we live we're each individual ants. We only have like we have a family we interact with a few little ants, and here comes AI that's able to summarize like a Tldr report. Of humanity, and that's the beauty of it. So I embrace it. What I wonder, I am very impressed by it. I wonder if it can be very impressive, meaning way more impressive in being able to. Fake or simulate or emulate a human. Fake. I am glad you you mentioned that because that just seems to be the mantra. Fake it till you make it. Yeah, isn't it? Isn't that what we all do though? No, well yes we do that, but we also do other things. We can be truly in love. We can be truly inspired when it is not fake. I do believe, call me romantic, okay, but I do believe. And this is a very good. I am glad you are putting it in the storms because I've had conversations like that. That yeah, fake it till you make it, but that's like that's what humans do. Yes, we do that, but not all the time. So and that is debatable because also I speak from my own experience and that's where the first person perspective comes in the subjective view. I cannot prove to you, for instance, or anyone else, that there are certain moments in my life where I am genuine. I am pure, so to speak, when it's not faking it. But I do I do have a tremendous certainty of it, and that's a subjective certainty. Now I am as a scientist I am also trained to give more credibility to objective arguments that in other things that can be reproduced things that I can demonstrate that I can show. But as I get older, there we go. As I get more mature, hopefully you know. I am starting to question why I am not giving as much credibility to my subjective understanding of the world—the kind of the first person perspective when actually modern science. Has already sold on that, you know quantum mechanics. Has shown unambiguously that the observer is always involved in the observation. Likewise, Godel's incompleteness theorems to me. Show that how essential is the observer of a mathematical theory. For one thing, that's the one who chooses the axioms. And we can talk about this in more detail. Likewise, Einstein's relativity where. Time is relative to the observer, for instance. That's brilliant. You are just describing in all of these different scales the observer with the observer means. So we, science of 19th century had the. From modern perspective, and I don't want to offend anybody, had the delusion that somehow you could analyze the world being completely detached from it. We now know after the landmark achievements of the first half of the 20th century that this is nonsense. That is simply not true, and this has been experimentally proved time and time again. So to me, I am thinking maybe it's a hint that I should take my first person perspective seriously as well and not just rely on kind of objective phenomena things that can be proved in a in a in a traditional. Sort of objective way by setting up an experiment that can be repeated many times. Maybe I fall in love in the part, the deepest love of my life perhaps, perhaps hasn't happened yet. Perhaps I will fall in love and but it's unique it's a unique event you can't reproduce it necessarily you see. So so in that sense, you see how these things are closely connected. I think that if you if we are declaring from the outset that all there is to life is. You know computation in the form of neural networks or something like this, however sophisticated they might be. I think we are from the outset denying to ourselves the possibility that yes there is a side of me which is not faking it. Yes there is a side of me which cannot be captured by logic and reason. And you know what another great scientist said, Blaise Pascal. He said the heart has its reasons, of which the reason knows nothing. And then he also said the last step of reason is to grasp that there are infinitely many things beyond reason. How interesting! This was not a theologian, this was not a priest, this was not a spiritual guru. It was a hardcore scientist who actually developed I think one of the very first calculators. How interesting that this guy also was able to impart on us that wisdom. Now you can always say that's not the case. But why should we from the outset exclude this possibility that there is something to what. He was saying that is my question. I am not taking sides. What I am trying to do is to shake a little bit the debate because most mathematicians that I know and computer scientists even more so they are kind of already sold on this. We are just you know reminds me of this famous Lord Kelvin's quote from the end of 19th century. There is some debate whether he actually said that, but never let a good story you know stand in the way of truth. He said physics is basically finished. All that remains is more more precise measurement. So I find a lot of my colleagues are happy to say. Everything's finished. We already got we got