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There is a famous story about Einstein that he used to, you know, go, think, think and then go for a walk and like he would whistle, and sometimes so. I remember the first time I heard this story I thought how interesting, so the coincidence, that he came to him when he was whistling. But in fact it's not. This is how it works, in some sense, that you have to prepare for it, but then the moment it happens, when you stop thinking, actually it's okay, the moment of Discovery is the moment when thinking stops and you know, you kind of, you kind of almost become that truth that you're seeking. Show more

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The following is a conversation with Edward Frankel, one of the greatest living mathematicians, doing research on the interface of mathematics and quant 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. This is the Lex Friedman podcast. To support it, please check out our sponsors in the description. And now, dear friends, here's Edward Franco. 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 columna, and I was a smart kid, you know, in school. But mathematics was probably my least favorite subject. Show more

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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 quantphysics. So I was buying, 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. Show more

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It's a provincial Town. Imagine it's like a 117 kilometers from Moscow, which would be something 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 evgeny evgenievich 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 patronym nickname. Yeah, but this was all real, this was all happening. So my mom One Day by chance met is ganovic and told him about me. But that was this bright kid and interested in physics, and he said, oh, I want to meet him, I'm going to convert him into math. Show more

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And my mom's like, nah, my ass, he doesn't like mathematics. So they said, okay, let's, let's see what they 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, Elementary particles. I said, yeah, sure, for example, do you know what quirks? And I said, yes, of course I know what quarks. Show more

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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 su3, this is like as you worked. So in fact I wanted to know what was, what were the underpinnings of those theories? I knew the story, I knew the narrative, a new kind of this basic story of what this particles looked 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 it's like a Bible, you know, like a, like a substantial book, and he opens 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. Show more

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You have to understand what su3 is. You have to learn what are groups, what this group su3, what are representations of sc3. 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 all such understanding could be attained. Show more

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Well, in retrospect, 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 the neutron consists of trick works 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 link around it. What Drew you to quant mechanics? Is there some romantic notion of understanding the universe? Well, 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. I think it is a conflict, some of the joy. He's probably still here now. Show more

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Yes, yeah, 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, so the deepest structures of the universe, the secrets, the, 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, we'll talk about it. But it also kind of makes me sad that there's 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 do 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 the 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 I mentioned that I I work on the interface of math and quantphysics- and their eyes light up and say, oh, quantphysics, or like Einstein's relativity, I'm really curious about it. Show more

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I watched 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 that actually, physicists are doing great job at educating the public, so to speak, and in terms of popular books and videos and so on. Mathematicians are behind. That we're starting to catch up a little bit, have been starting last 10 years, but when we're still behind. But I think people are, people are curious. Science is a is still a 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 that the Universe holds, 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 theorists always change, they get updated. Show more

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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 home, how how much we are connected to each other. Because dagger's theory, 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 and George Orwell's 1984. I was in the Soviet Union in 1984 and so in many ways I see that it was President, the novel was present, but we still have not found the dictator who would actually say two plus two equals five and would demand their citizens to repeat that. The Knight is still young, has not happened yet, but 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. Show more

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Okay, mathematicians are interested describing all possible mathematical universes of which you know and some of our work. 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 special spatial dimensions and one time Dimension, right? So what I mean is that allegedly, allegedly observed, but that way I can observe today, right? So of course, there are theories where there are some hidden Dimensions as well. Well, let's just say, absorb to observe dimensions. So this tabletop has two Dimensions, because you can have two axes, two coordinate axes, now, X and Y, but then there is also a third one to describe the space of this room. Show more

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And then there's a Time Time Dimension. So realistic theories of physics have to be about spaces of of three, three dimensions or space time, so four dimensions. But mathematically we are just as interested in theories in 10 space time Dimensions, or 11 or 25, or whatever, or 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. Show more

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We don't get to play with big machines like LHC in Geneva, A 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 visits that they, they get to, they get to experience this sort of these 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. Show more

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That's right. So the mathematics is the underpins physics. Obviously it's a language. The, the book of nature, famously said it's written in the language of mathematics and the and the. You know the. 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 the dark Labyrinth. That's a famous quote from Galileo, which is very true and has become even more true more recently in the in theoretical physics, in the more in the most actor 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 tag Mark or 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 it just mathematics at the core of? Show more

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You know, I would say mathematics is one half of the core. So the book is called love and math. Yeah, 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 up, 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, Matthew could say, is a way to discern patterns, to find regularities in the universe, and both physical and mental Universe. Show more

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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, to see beauty, to understand Beauty. So a world becomes more complex. From thinking that our, that, our, that Earth is flat, we go to realizing that it is round, that is shape 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 the band, a band 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 this, from this Theory and from the experiments that have confirmed it, that a array 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 depending on the space, it's also the bending of space-time. There is a curvature not only between special spatial dimensions, the way parallels and meridians come together. Show more

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In a small scale they look like perpendicular lines, but if you zoom out you see that the spacer at a curving the space, they are sort of the, the tracks along which the space gets curved. That's, that's the. That would be the curvature of spatial Dimensions. But in fact, now throw in time and one time, imagine a sphere which lives, which has one of the meridians correspond to time and the perilous crisp into space. I can't imagine it, but you can, 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, all between two human beings at the time, bending time. It's 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 there of time. Marching forward nicely seems to be the only way we can understand the universe and the fact that you can start now. Up to now, there are people who claim that they can, that they have, they possess other ways of of experiencing it, so truly can visualize messing with time. Show more

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Well, messing with time, but not necessarily messing with time. Because one point of view is that you know, I think, who, 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 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're in love, when you are passionate about something, when you're creating something you're, we lose ourselves and we lose 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. Show more

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So, yes, so then there is a point of view that this is where we are, 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, 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're not thinking about it, you're 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. Show more

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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 taste, taking those actions. But what if all of that is just an illusion? An illusion and a nice narrative would tell ourselves. Sure, that's a really difficult thing. And imagine, imagine that to make it really Catch-22, that I'll imagine that our minds and 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 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. Show more

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I think this 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, yeah, truths that they hold this, subjective, truths that are useful for the competition, or whatever this dance that we call Life broadly Define, not just humans. And and you know, I'm 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 appeared 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 says something like the infinite ocean of knowledge lady was lying before me who, probably it was the greatest mathematician of the second half of the 20th century. The French mathematician Alexander grotonic wrote that discover is a privilege of a child, the child who is not afraid to be wrong once again, to build, to look like an idiot, you know to, to try this and that and 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're one of the most respected mathematicians in the world. Your Berkeley, you're like this. This is the stature you're supposed to be, very like, you know. Yeah, sometimes I joke, I say I, I think I take an elevator to the top of the Eiffel Tower every day, yeah, and you're supposed to speak like royalty. Show more

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Do you struggle to let those strip all of that away, to ReDiscover the child when you're thinking about problems, when you're teaching, when you're 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 to maintain that connection, okay, in that Fascination, that vulnerability, that spontaneity, you know, and 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 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 that's 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 your thinking, when you are fearless and looking like an idiot, that's right because there's a hierarchy. Nobody wants to look like an idiot. You know, once you start growing up- or you think you're growing up, yeah, in the beginning you don't even think of you, don't think in these terms, you just play. Show more

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You're just playing and you are open to possibilities, to these infinite possibilities that this world presents to us. So how do we- I'm 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 we 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 alive, is an answer. Every life, every human being is one particular answer to how do we find balance. That's once imperfect approximation, approximate solution. But we could look, we can look up to the great ones, yeah, 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 grotonic or Pablo Picasso. So they have established their rights to speak about this matters and we could not dismiss them as mere Madman. Show more

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They say, okay, well, if the same thing was said by somebody who never achieved anything in in that, in their, in their field of endeavor, you will 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 it's something important that they teach us, and especially today, in this age of AI. Of course, there's a big elephant in the room always- which is called AI. Yeah, right, and so I know that you are an expert in this subject and we are going- we're living now in this 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? Show more

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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, 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, at an adult, credits the inner child, the inner child, yeah, for this capacity to create as an adult. Show more

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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 data sets, 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're perfectly capable of of, you know, or distinguishing between cats and dogs, for instance, and stuff like that, but much more than that. They're also capable of that you know, wide-eyed, in the sort of perspective. Show more

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So does. Can it really be captured, that perspective, that sense of, or can it really be captured by computation alone? I actually I don't know the answer, so I'm not sort of trying to to present a particular point of view interesting 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. Show more

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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 aspects 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 call me they're. They're actually. They 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. Show more

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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 come. 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, and people come to me and and they look up to me and they Revere me kind of like priests in the old, in old days, when the religion was Paramount, wonder when you would be would tend to explain things in theological, religious terms. Show more

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Today, science has progressed, there are fewer people who kind of buy into religion, 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's 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 the human data from the internet. So at its best, a large language model like gpt4 captures the magic of The Human Condition on its full display. It's full complexity, it's just 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, but I wonder. I'm very impressed by it, I wonder if they can be very impressive- meaning way more impressive- in being able to fake or simulate or emulate a human right. I'm glad you mentioned that, because that's just, it seems to be the Mantra. Show more

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It's just fake, 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 a romantic, okay, but I do believe, and this is a very good. I'm glad you're putting it in these terms, 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 a certainty of it, and that's a subjective certainty. Now I am as a scientist. I'm also trained to give more credibility to objective arguments that another, 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'm 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 I actually modern science has already sold on that. Show more

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You know, quantmechanics has shown unambiguously that the Observer is always involved in the observation. Likewise, yodel's incompleteness ethereums to me a show that. How essential is the Observer of the 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're just describing all of these different scales, The Observer, what they Observer, science. So we signs of 19th century had the. From Modern perspective- and I don't want to offend anybody- I had the delusion that somehow you could analyze the world being completely detached from it. We now know, after the 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. Show more

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So to me, I'm 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 a party, you know, the deepest love, my, of my life Perhaps Perhaps hasn't happened yet. Perhaps I will fall in love and this, 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 outside, denying to ourselves the possibility that, yes, there is the 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: bless 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. Show more

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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'm not taking sides. What I'm trying to do is to shake a little bit the debate, because most mathematicians that I know, and computer scientists even more so, they're 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's some debate whether he actually said that, but never let a good story. You know, he said: physics is basically finished. Yeah, All That Remains is more more precise measurement. So I find a lot of my colleagues are happy to say, yeah, everything's finished, we already got, we got it, we got it maybe little tweaks in in the, in our large language models, you know. So now here's my question. I'm kind of Playing devil's advocate a little bit, because I don't see the other side represented that much and I'm saying, okay, could it be also that if you believe in that, that becomes your reality, that you can kind of put yourself in a box where everything is competition and then you start seeing things as as being such? It's confirmation bias, if you will. You know. This also reminds me, you know, I think a good analogy is: it's a friend of mine, Philip caution, told me that in France there is this literary movement which is called ulipo o u, l IPO, and it's a bunch of writers and mathematicians who create works of literature where, in which they basically impose certain constraints. A good example of this is a novel which is called the void or disappearance, by a writer named George pereck, which is a 300 page novel in French with no no, which never uses the letter e, which is the most used, widely used letter of the French language. Show more

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So in other words, he said these parameters for himself. I'm going to write a book where I don't use this letter, which is a great, you know, it's a great experiment and I upload it, but that's what. It's one thing to do that and to kind of show his gamesmanship, if you will, and and his proclivity and his ability as a writer. But it's another thing if, at the end of writing this book, when you finish the book, he would say letter e actually doesn't exist, and try to convince us that in fact, French, French language does not have that letter, simply because he was able to go so far without using it. You see, so self-imposed limitation. That's how I see it and I wonder why we should do that. Do we really need, do we really feel the urge to say: the world is like that, the world can be explained this way or that way. And I'm saying it. You know, it's a personal question for me because I am addicted to knowledge myself. I, you know. Hi, my name is Edward and I'm an English addict. Okay, I'm being serious, I'm not being facetious. Up until very recently, maybe a couple years ago, I simply did not feel comfortable if I could not say the given answer- explanation. It's like, oh, there has to be some explanation and I try to frantically search for it, just for somebody like me. I know heard, you know a left Brainiac and you know that's kind of typical, typical for a scientists, for mathematician. It is incredibly hard just to allow the possibility that it's a mystery and not to feel the urge to get the answer. It is incredibly hard but it's possible and it is liberating, it's recovering, is recovering addict to knowledge. Let me say what you gain from it. For instance, I understand the value of paradoxes, I I appreciate paradoxes more, and you know to. To use another philosopher, Soren kirkegar, the Danish philosopher, said, I think, or without, Paradox is like a lover without passion, a paltry mediocrity. What's a good line? All right. So, and you know Niels Bohr. Niels Bohr said in similar Vein, The Great Danish also something about that, something about Danes. I think it all started with scramble it. You know. He said the opposite of a simple truth is a falsity. But the opposite of a great truth is another great truth. In other words, things are not black and white. You know they are not, and I would even venture to say the most interesting, the interesting things in life are like that, the ones which are ambiguous. It's an electron, a particle or a wave, it depends how you set up an experiment, it will reveal itself as this or that, depending on how you set up an experiment. This bottle, if you project it down onto the table, you will see more or less a square. It will project it onto wall, you will see a different shape. Show more

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A naive question would be: is it this or that? Because we understand that it's neither. But both projections reveal something, the real, different sides of it. A paradox is like that. It's only paradoxical if we, if we are confined in a particular Vision, if we are wedded to a particular point of view. It's a Harbinger, if you will, of a possibility of seeing things in a more in, in a as they are, as a more sophisticated than we thought before. You see, this is such a difficult idea for science to Grapple with. That, you know. I don't know how. There's so many ways to describe this, but you could say, maybe, that the subjective experience of the world from an observer is actually fundamental. But we know that our best physical theories tell us that unambiguously. Show more

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In quantmechanics actually, you know, Heisenberg, I think, captured it the best when he said what we observe is not reality itself, but reality subjected to our method of questioning. When I talk about electrons, for instance, so that there is a very specific way in which you, in which this is realized. There is a so-called double slit experiment, right? So for those who don't know, it's you, you have a, you have a screen and and you have an emitter from which you send, you kind of shoot, electrons, and in between you put another screen which has two vertical slits parallel to each other. If we were shooting, you know, tennis balls, each ball would go through one slit or another and then hit the screen behind this or that slit, so you would have, let's say, they colored, they painted, so they'll be sort of bumps or or or spots of paint behind this or that. Show more

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But that's not what happens. When we shoot electrons, we see an interference pattern, as if we were actually sending a wave, so that each electron, it seems like the, each electron goes through both slits at once and then and then has the audacity to interfere with it, with itself, where at some points, you know, two crests would amplify and at some points, aggress in the draft would cancel each other. Show more

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Yet. So that suggests okay, so no, electron is a wave, not so fast. Because if you put a detector behind one of the slits and you say: I'm going to, I'm going to capture you, I am going to find out which lid you went through, the pattern will change and it will look like the particles. So that's a very concrete realization of the idea that depending on how we set up an experiment, we will see different results. And the problem, the problem is that our psyche, I feel, kind of lagging, is lagging behind, in part because maybe our scientists are not doing such a great job. So I take responsibility for this, that why haven't I explained this properly? You know I I tried, you know in a bunch of talks and so on. Show more

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So now I'm talking about this again, our psychic kind of lagging behind. We're still, even though our science has progressed so much, from the certainty and the determinism and and all of that of the 19th century, our psyche is somehow still attached to those ideas, the ideas of causality, of this naive determinism that that the war, the world has a bunch of billiard balls hitching each other, driven by some blind forces. You know, that's not at all like this and we've known this for over for well, for about 100 years at least. You know, and you call this self-imposed limitation. It is a self-imposed limitation when we, when we pretend that, that, for instance, that this naive ideas of 19th century physics are still valid, and and then start applying them to our lives and then also derive conclusions from it. And, for instance, people say there is no free will, why? Show more

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Or because the world is just a bunch of billiard balls, where is the free will? But, excuse me, didn't you get the memo that this has been debunked thoroughly by the so-called quant mechanics, which is our best scientific theory. This is not some, some kind of or some kind of you know, concoction of of a, of a Madman. This is our scientific theory, which has been confirmed by experiment. So we should pay attention to that. So, but of course, it's not just self-imposed limitation. Unfortunately, in this case there is a big issue of Education. So a lot of people are not aware of it through no fault of their own, because they were never properly taught that, because our system is broken. Education system is broken, especially in math, and then our. Show more

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So where do we get them? Do you get information? You get information from our scientists, who actually write popular books and so on, which is a great, you know great, thing that they do. But a lot of scientists, somehow, when it comes to explaining the laws of physics, they are doing a fantastic job talking about this phenomenon, for instance, double slit experiments and things like that, but then, you know, interviewed by science managers in about three wheel and so on, they revert back to 19, 19th century physics as if those developments actually never happened. So to me, this is single most important sort of issue in our Popular Science. The idea that somehow there is this world out there but it's complete, has nothing to do with me. So I can, I can reveal in the intricacies of this particles and their interactions, but but completely ignore what implications this has for my own relationship to the physical reality, to my own life. You know, because it's kind of scary, I guess you know. But also, what are the tools with which we can talk about the Observer, the subjective view in reality, what are the tools of which we could talk about rigorously, talk about Free Will and Consciousness? What are the tools of mathematics that allow that? Show more

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I don't think we have those tools because we haven't been taught properly. So actually, tools are there, for instance, I think. Well, here we have to. I have to say my conviction is that everybody knows, in the heart of hearts, everybody knows that there is, that there is something in our football, there is something mysterious. And in fact, you know, somehow, immediately, I feel that you know the impulse to quote somebody on this because, as if, as if my own opinion doesn't occur, there's a long dead expert that has. When Einstein said that he doesn't like, how, see, look at me, I am supposed to do like this smart, intelligent person, I am afraid to say it and own it myself. I have to find a confirmation, I have to find an authority who agrees with me, and in fact it's not so difficult to find, because Albert Einstein literally said: the most important thing in life is the mysterious. Okay, he actually said that there are some quotes which are attributed to him, which he never said. Show more

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But this he did, I investigated. Okay, so, but more importantly, you know, how do you feel about it? I think that everybody knows. But in other words, he also said, Einstein, imagination is more important than knowledge. Okay, and explain, for knowledge is always limited, whereas the imagination Embraces the entire world, giving birth to Evolution. It is, strictly speaking, a real factor in scientific research, he says. And he says: I am enough of an artist to follow my intuition and Imagination. That's Albert Einstein again. So, and I feel the same way, to be honest, if I think about my own mathematical research, it's never linear, it's never like: give me more data, give me more data, give me more data. Boom, the glass is full, and then I come up with a discovery. Show more

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No, it's always. It always is always felt as a jump, as a leap, and I I have actually been studying various examples in a history of mathematics of some fundamental discoveries like Discovery, complex numbers like square root of negative one. I wonder if a large language model could actually ever come up with the idea that square root, square root of negative one, is something that is essential or meaningful, because if all the information that you get, the all the knowledge that had been accumulated up to that point, tells you that you cannot have a square root of a negative number, why? Because if you had such a square root, we know that if, then we would have to. If you square it, you get a negative number, but we know that if you square any real number, positive or negative, you will always get a positive number. Show more

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So, Checkmate, you know it's over. Square root of negative one doesn't exist. Yet we know that these numbers make sense they're called complex numbers and in fact quantmechanics is based on complex numbers. They are in essential and indispensable for quantmechanics. Could one discover that? So to me that sounds like I discontinue it in the process of discovery. It's a jump, It's a departure. It is like a child who is experimenting. It's like a child who says: I'm not afraid to be an idiot. Everybody says the adults are saying square root of negative number doesn't exist. But guess what? I'm going to accept it and I'm going to play with it and I'm going to see what happens. This is literally how they were discovered. There was an Italian mathematician, astronomer, astrologer he was he. He made money apparently by compiling astrological sort of readings for for the elite you know of his ears. This is 16th century, everyone does example all around. Interesting guy, I'm sure we would have an interesting conversation with him. Gerolamo cardano, he's all. He also invented the what's called cardan shaft, so which is an essential component of of a car, we say in Russian. So. So he wrote a book which is called our ass Magna, which is a great art of algebra, and he was writing solutions for the cubic and quartic equations. This is something that is familiar because it's cool. We study Solutions of quadratic equations, equations of degree two. So you have x a x squared plus BX plus C equals zero. Show more

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And there is a Formula which solves it using radicals, using square roots. And cardano was trying to find a similar formula for the cubic and quadratic equations, for which which would start with x cubed or x to the power 4 as opposed to x squared. And in the process of solving these equations it came up with square root of a negative number, specifically square root of -17, and he wrote that I have to forego some mental tortures to deal with it, but I am going to accept it and see what happens. And in fact, at the end of the four, at the end of the calculation, this, this, this weird numbers, got canceled. It kind of canceled out in the formula appeared square root of negative 17 and its negation. So they kind of conveniently gave the right answer which is not involve those numbers. So he was like, okay, what does it mean? Mental tortures? So you see, from the point of view over of the of the thinking mind, it is something almost unbearable. It's almost, I feel, that a large language more the computer running a large language model trying to do that would just explode. And, yeah, the human mathematician was able to find the courage and inspiration to say: you know what? What is wrong, why? Why are we so adamant that these things don't exist? That's just our past knowledge, based on what our past knowledges, and knowledge is limited. What if we make the next step today? For us mathematicians, context numbers- that we call them- are not at all mysterious. The idea is simply that you plot real numbers, that is to say all the whole numbers like 0, 1 and so on to and so on, right, all fractions like one, half or three, halves or four over three, but then also numbers like square root of two or Pi. Show more

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We plot them as points on the real line. So we draw: this is a. This is one of the kind of perennial Concepts, even in a in our very poor math curriculat school. But now imagine that instead of one line, you have a pla, you have a one axis, you have a second access, and so you numbers now have two coordinates, X and Y, and you associate to this point with coordinates X and Y and the number X, which is a real number plus y times square root of negative one. This is a graphical, geometrical representation of complex numbers which is not mysterious at all. Show more

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Now it took another two or three hundred years for mathematicians to figure that out, but initially it looked like a completely crazy idea, you know. So all it is all a complex number is it's just an experience, the real part and the imaginary part, it's just an expansion of your view of the mathematical world, the fact that you can actually multi, you can add them up by adding together the real parts and imagining parts. Show more

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That's easy. But there is also formula for the product, for the multiplication which uses the fact that square root of minus 1 squared is minus one. And the amazing thing is that that that product, that multiplication, satisfies the same rules, the same properties that are usual operation of multiplication. For real numbers, for instance, there is an inverse. For every non-zero number that you can find, like number five has an inverse, one over five. But OnePlus I also has an inverse, for instance. You know that was always there in the mathematical universe, but we humans didn't know it. And here comes along this guy who engages in the mental torture, who takes a leap off the cliff of comfort of, like mathematical, established knowledge, established knowledge, right and now, obviously, for each, each sort of fruitful leap like that. There probably were thousands of like things which went nowhere. I'm not saying that every Leap, you know, it's like, it's a, it's a, it's a open shooting game. Yeah, because, for example, you can try to do the same with three-dimensional space. So you have coordinates x, y and z and you can say, oh, if there's one dimensional, we have an abonified numerical system called real numbers. If it's two-dimensional, which is like you know, geometrically it's just like the stable top extended to Infinity in all directions, these are complex numbers and we can Define addition and multiplication and they will satisfy the same properties as real numbers that we're used to. Show more

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What about three-dimensional space? Is it possible to also Define some operation of addition and multiplication on it, so that the separations would satisfy the properties that we're used to? And the answer is no. You can Define addition, but you can't Define multiplication, for which there would be an inverse, for instance. So there is something special about the, the plane, the two-dimensional case. And, by the way, next question would be: what about four-dimensional? In a four-dimensional space, you again, you can, and you get what's called quotarians, discovered by an Irish mathematician, Hamilton, in the 19th century, and then in in the eight dimensional, there is something similar called octonians, and that's about it. So how interesting. These structures exist in dimension one, two, four and eight, which are all powers of two. To square this four, two to the third power is eight. That's one of the bigger mysteries in mathematics, why it is so. So here's a hint. That's a hint of what's missing in our, on our high school curriculthe kind of the kind of fascinating the Mysteries, yes, the appreciation of the mysterious. But so, in other words, yes, we resolved with this one mystery that we understood that square root of negative one, Israel is Meaningful. We build a theory to service those now, to to describe those numbers. Did we find The Theory of Everything? No, because we then invited other Mysteries, because we, we pushed the, we pull the whale, so to speak, or you push the frontier, and then new things come, get eliminated, which we couldn't see before. That's how I see the process of discovering mathematics: it's an endless, Limitless Pursuit. Can you comment on what you think, this human capability of imagination that Einstein spoke about, of the artists following their intuition, in this big Alice in Wonderland world of of imagination? What is it? You visit there sometimes. What does it feel like? Yeah, what does it feel like what? What is it? What is it? Sounds like playing, but I think all of us are engaged in that kind of play, no matter when we do what we love. I think it always feels the same, but it's not real, right? You're. So it that you're describing a feeling, but that place you go to in the imagination, right it's, it's bigger than the real world. So there is a big conundras to whether mathematics is invented or discovered, and mathematicians are divided on this. Nobody knows where do you bet your money on financially, financially, investment advice? So let me tell you something: my abuser have evolved. Okay. When I wrote Love and math- I mean I wrote my book- I was squarely on the side of mathematics. Discovered. What does it mean? Usually mathematicians or others who have this, you know, idea what a belief are called platonists in in honor of the great philosopher Plato, who talked about this absolute, perfect forums. So for me, you know, about 10 years ago, the world of mathematics was this world of pure forms, this beautiful, pure forms, which existed outside of space and time. But I was able to connect to it through my mind and, as it were, kind of dive into it and bring Treasures back into this world, into this space and time. That's how I viewed the process of mathematical Discovery. How nice the picture also makes you feel connected to something Divine, allows you this sense of escape from the cruelty and Injustice of this world. You know, which I now recognize in the Divine world of forms- is stable, is something stable, it's, and in that world everything is clear-cut. Yeah, it's either true or false. Yeah, it's very nice. Yeah, the biggest delusion of, allegedly, I think now. Show more

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I think now I understand why I liked it, because I think that I was very dissatisfied with the, with the, what we call the real, the real world, the world around me, the cruelty of, the Injustice of it. And I I went through certain experiences as a kid which made me love mathematics even more, as this place where I could be safe and be in control made you see the human world as lesser than the mathematical world, yes, as more limited than the mathematical one. Show more

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Yes, yes, and I think that I think that it's still missing the mark in some sense, because, in fact, what I now think it's not. It's not a it's, it's a paradoxical. The question whether mathematics is invented or discovered, whether there is this world of pure forums and so on, is another paradoxical question which doesn't have a simple answer, like whether electron is a particle or a wave. From one point of view, yes, it's true, and just the fact that so many mathematicians today actually subscribe to this idea gives it the certain credibility, because that's what we feel. We do feel that we dive in, to dive into that mindscape, so to speak. Show more

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You know, but the very structured Minds keep where you know. I wrote In Love, In Math, that you know the Enchanted Gardens of platonic reality, you know. So where all this fruit is grows, you know. And then we, we might think it gives you this sort of romantic sense of an explorer, and someone may be stuck in a you know some provincial town in Russia, for instance, but have the sense of Magellan, you know, of traveling around the world. Show more

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It's just not in the world that we usually think of. So it's, it's one point of view, but the other point of view is that, yes, it is a human process. Of course it is. I mean it is. You cannot deny that it's human beings who have so far discovered new mathematics, and I do not deny the possibility that a computer programs will be able to discover new mathematics, but so far it's been humans. So it, whatever it is, whether it's discovered or invented, it is a human activity. What that? The possibility that paradoxes are actually fundamental, right to reality, right, and really really internalizing that, that we exist in a world of not forms but of paradoxes, Bingo. Show more

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And so it's like what I said, what the world? But if you think it's weird- and I agree with you as the recovering addict to knowledge, yeah, you know, but I am liking it more and more because there's so much freedom in it. And like, like Niels Bohr said- you know, I quote that earlier- the the opposite of a great truth is another great truth. He's pointing out to this fact that you know. And he also said that some, some things in quantphysics are so complicated the only way you can speak of them is in poetry. So, in other words, what is it about poetry? What is about art? Why are we so drawn to that? Why are we so captivated? Captivated by, by those forms of, by. They are not intellectual, necessarily, they are not. When you look at the painting that you like, when you listen to music that you love, you get lost in it, you get absorbed in it. It makes it make you cry, it can make you laugh, it can make you remember something, it can make you feel more confident, or can make you feel sad or happy, and so on. What is this all about? Is it really is just some play between some kind of like silophone play or some neurons or hitting on each other? Is it really that? Only maybe it could be, it could be both. I'm just worried about kids these days that might live in a world of paradoxes. You know, if there's no God, everything is possible, and yet just they'll have a little too much fun and we have to put a constraint. Because have you looked at the world lately? I haven't checked in in a while. You think it's perfect the way it is now, the world without paradoxes, the more than which we believe that every end, every question can be answered as yes or no, that it is this or that, and if you disagree with me, you're my enemy. Wouldn't that be interesting if this 21st century is a transition into singing the world as well, the paradoxes. And tell you, you know people predicted that. You know the age of Aquarius, you know the excess of the of the Earth is rotating relative to the plane in which the earth goes around the sun and the period of this revolution is around 2000 years, so there is a traditional way of measuring that by this eras, you know the the ages. So the previous one is called the age of Pisces because of the constellation of Isis that it points to, so to speak. Show more

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You know, and now it's. It's you know, as in famous musical hair they said, the age of Aquarius is upon us. So the different people dated differently, but some somewhere around the time where we are finding ourselves. How interesting right is all the strife and all them difficulties the world is experiencing. This might actually be the transition to something more harmonious, wouldn't it be nice. It's also interesting that people from long ago are able to predict certain things. It's, and it's almost like from long ago- and you've talked about this with with Pythagoras- that it seems that they, they had a deep sense of truth. That's right. For me, it's all this, even even now. So it's not just a linear trajectory of an expanding knowledge, there's a deep truth that permeates the whole thing. Yes, so that's how I see it. Show more

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Actually, I, you know, I gave a talk about Pythagoras and pythagoreans just a few weeks ago at the Commonwealth Club of California in San Francisco, and because of that I did a kind of a deep dive into the subject and I I I learned that I actually totally misunderstood Pythagoras and pythagoreans, that they were much deeper than I thought, because, you know, most of us remember Pythagoras from the, from the Pythagoras Theorem about the right triangles. We also know that pythagoreans were instrumental in introducing the tuning system for the musical scale, the, the famous perfect fifth, three halves of the for the, for the, for the G, for the soul, compared to the frequency of of Doha or C, you know and so, but actually they were much more interesting. So for them, numbers were not just clerical devices, you know that, not kind of thing that you would use in accounting, only they were imbued with, with the Divine. And I cannot, I cannot say that I, I think we lost it, at least I have lost it. Show more

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I look at numbers and I don't really see that the Divine, the Divine that they clearly did, and so that, why else? You know, how else would you explain? So that, in other words, Divine is, of course, is a term which is, you know, it's a bit loaded, so it's hard to escape that. Let's just say something that more from the world of imagination and intuition than from the world of knowledge. Let's just put this way. They were able to Divine, okay, abstract that, to intuitive, to Intuit that the, the planets were not revolving and the Sun and the planets were not revolving around the earth. They were the first ones, at least in the western culture, as far as I know, and in fact Copernicus gave credit to pythagoreans as being his predecessors. Did you not quite have the, the Copernicus model, with the sun in the middle? They had what they call the central fire in the middle, and all the planets and the Sun were evolving around, oh, around the central fire, or Earth, they call it Earth. So, but still, what a departure from the Dogma, from the knowledge of the era that the Earth was at the center. So how could they come up with this idea? The reason was, in my opinion, that for them, the most movement of celestial bodies was like music. In fact we call it musical universalis, or Music of the Spheres. For them, the universe was this infinite Symphony in which every being, you know, humans, animals, but as well as the Earth and other celestial bodies, were moving in harmony, like different notes of different instruments in a symphony. Show more

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And so they apply the same reasoning to the. You know, the cosmological model as they apply to the, their model of music, and from that perspective they could see things deeper than their contemporaries, you see. So, in other words, they saw mathematics as a tool, but that tool was not limited to itself, and it was that they always. They always knew that there is more and they knew also that every, every pattern that you detect, it's finite, but the world is infinite. Show more

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They actually accepted Infinity. They believe that infants is real and if you discern a pattern, great, you can play with it and you can use that. It gives you a certain lens through which to see the world in a particular way, which could be beneficial for you to to learn more and so on. But they never had the illusion that that was the final word, that they always knew that it's not the whole thing. So there is more. There are more sophisticated patterns that could be discovered using mathematics or otherwise, and I think that what happened was we kind of lost this, this other side of their teachings. We, we took their numbers and they were like idea that you could use mathematics to discern patterns and to find regularities and to explain things about the world. We took that and we ran with it and we kind of dropped the other idea that in fact there is a, there is another side to it which is kind of to us. Now we say, oh, that's mystical. But what does it mean mystical if it is something that helps you to make great discoveries? And the interesting thing is that the people who are in touch with the mystical Among Us, often seen as mad- and many of them are. Most of them are well, but not all of them. All of them. We mentioned, you know, spoor and Newton and Albert Einstein, so, but that's where the conundris: how do you find the balance between the two? Show more

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So the point I'm trying to make, and you know, and this is what I feel, you know, if you ask me, what I find most important today, today, like what makes me excited and enthusiastic and passionate, is this idea of balance. So you know, Nietzsche had this book, wrote this book in the 19th century, called the birth of strategy, and he presented this Theory, which I think is kind of very useful, of these two sides of a human, one that comes from god, Apollo, and that's our left brain, so to speak, proverbal left brain. Show more

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So it's everything has to do with that, has to do with logic and reason and analyzing, dissecting, conceptualizing, and the other side, which comes from God, the anesis and the anisis, and that's responsible for intuition, imagination, love, Dennis, was also god of wine, so it's also that side of a human that makes us sing and engage in, in, you know, revelry and and and drink wine and spend time with friends and love and enjoy it. You know. So Nietzsche Advocate at this point of view that those were two complementary sides of every human and as well as a society, and the purpose of human life is to find balance between them. Because the math is both discovered and invented, we should be okay with having both ideas in our head and living in the balance. But, more importantly for me, Apollo is like math and Dionysus is like love. Show more

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So Geniuses, in modern version, is love and math. I returned to that question we had about GPT and the language models, and I think about this quite a lot, which is when the model and we know what is trained on, we know the parameters, when all the different hacks that are involved in the training process and the fine-tuning process and the final results- well, there's GPT five, six or seven- will result in hundreds of millions of people falling deeply in love with that language model and to be able to have conversations that are very much like the conversations we have with somebody we're deeply in love with. Show more

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And not only that. The model will say: that is deeply in love with us, and who are we to say it is not? I think there's a- it's the same imperative that you described in the scientific mind that wants to throw away the subjective. That same imperative wants to throw away the feelings that AI might have, and I'm very careful to not ignore when an AI system says it's lonely, it's afraid, it doesn't want to die, it misses you, right, it loves you. I am, I'm with you. I would also say that you could try to. Show more

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You could, for instance, say that the origin of that is the- you know, romantic novels that overfed to it, for instance. Yes, however, you could also. Then you can retort. But what if my, what I consider my subjective, unique feelings are also novels, the river preparations of the novels I have read because I have learned, or movies I have seen? Yeah, because that's the purpose of movies, kind of to teach us how to express ourselves, how to feel. Maybe even one could argue that some people have argued that I agree that this is. There is no obvious answer to this. But see, that's exactly my point. That is an example of something which is paradoxical, for which there is no answer, and that's where the subjective has a has an important role. For someone, that type of interaction would be, would be helpful, would be consoling, would, would feel, would you know, make them happy or sad or whatever you know what kind of strike the nerve for some it won't, and I agree with you that, in principle, there is no one to to judge this. This is, this is where subjective is Paramount. But remember, a lot of this has been anticipated by artists. The great movie, her. There you have this guy who is this lonely. He kind of writes letters, some romantic letters, yeah, for other people, but he isn't, he doesn't have a partner, he's lonely. And then he gets this sort of a kind of enhanced version of Siri with the voice Johansson, which is very sexy voice, you know, obviously she's a great actress. So, and then at first it looks like a fantastic arrangement: he, he confising her, he, she, she tells him things, he is, she makes him happy, and so on, until he finds out that she has a relationship- quote, unquote, if you can call it that- with 10 000 other people, not two others, not three others, yeah, like ten thousand, because it's a, it has the Computing capability. So, yes, definitely, oh, it certainly makes sense. It's a good experience. And the guy is heartbroken, yeah, but see, so, see, here's my analysis of this. Show more

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Okay, it's like a couch, a couch, a therapist. Okay, the guy did not have the courage to go out in the, in the real world, and to meet a woman and, to, you know, get a girlfriend and so on. It's true, no fault of his own, perhaps because, you know, he may have had some experiences which made him withdrawn and closed and so on, and a lot of us are like this, you know, I had periods like that myself- definitely can sympathize and relate. However, part of the joy of having this Siri like relationship for him, one could say, was the absence of that fear that she would abandon him which prevented him from initiating a relationship with a human being. And yet it turns out that he could be betrayed- quote unquote- that she could be unfaithful to him- quote, unquote anyway. So then, that means that it did not resolve the underlying fear having that relationship. So, in other words, that human element of the relationship still found its way into the seemingly sterilized, protected, protected partnership. So the human being rears its head anyway, and I think the lesson there is that the system in the movie Her actually gave him a lesson: that even AI could betray you. Even if I can leave you, even AI can be unfaithful to you, and I would argue that the next AI he meets will be one he actually falls in deep love with, because he knows the possibility of betrayal. Is there the possibility of death? Is there? The possibility of infidelity is there because we need that possibility to truly feel core, or he would turn off his Siri program and finally and get out of his house, go to local bar and strike a conversation with a human being, although you might say by then some of those might be Androids. And that was predicted by another great movie, yeah, right, so the Blade Runner. Blade Runner, how interesting that artists could see that so long ago. You know, of course, Blade Runner was based on a novel by Philip K dick, though Android stream of Electric Sheep. That guy was a genius, you know. It's somehow that artists have their eyes open, that they anticipate. Is it also a large language model that they're using for that an even larger one or even larger I- I hesitate to dismiss the magic in large language models. Show more

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I a lot of the work I've done is in robotics. In the robotics Community generally doesn't notice the magic of feeling. When I I've worked, I've been working a lot with quadrupeds, recently legged robots with four legs, and the feelings I feel when I see you know I'm programming the thing, but when the thing is excited to see me or shows with this physical movement that it's excited to see me, I cannot dismiss the feeling I feel is not somehow fundamental to what it means to program robots and I don't want to dismiss that and please don't, please don't. The robotics Community often doesn't gender robots. Show more

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They really try to work hard to not anthropomorphize the robots, which is good for technical development of of how to do control, how to do perception, but when the final thing is alive and moving and it does whatever- like I've been doing a lot of butt wiggling- it can wiggle his butt, it can turn around and look up: yeah, excited, that's not just I know how it's programmed, but the feeling I feel, that's something that's I don't know what that is. Okay, I agree with you. I, I hear you. Show more

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I hear you when you speak about it. Yeah, you speak with passion. Yeah, and that's for me that is proof, but it is magical, you see. So don't, I would say, don't dismiss that, don't discard that. On the contrary, I think magic is everywhere, you know. So I used to be okay, kind of confession. Okay, you've already got confessed to quite a few addictions. Yeah, I'm kind of- yes, I'm kind of worried, recover, recovering for many. But you know, I in old days I was more on the side of everything is computational or everything can be explained by science and whatever. You know, like I would dismiss and disregard, you know, the intuitive or imaginative things. So then I had to flip that suddenly I start feeling it and started seeing it and so on. But so then the pendulums had swung in the opposite direction. Show more

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Then I was arguing that you know, somehow that was the real, that imagination was intuitive, imaginative was real, and discounting what you just described. And I would argue with people saying no, no, this, you know, this is not real, this is all you know, imitation game and so on. But you see that what's new now? The new advert? Okay, is the? The 2.0 or 3.0 is the one? Who is seeking balance? Who is not, who is, because suddenly become aware that, no matter which one-sided, lopsided point of view you take, you're limiting yourself. So, whereas even a couple of years ago, you know, if you just told, you told me what you just described, I would be like, you know, being polite, I would just, I wouldn't contradict you, since you're the host anyway. Right, it's not a lot so, but I would be like, but I wouldn't say anything. But suddenly I find this moving, I find it more. I honestly- I'm not being facetious- I find it moving and I almost feel like I can see it through your eyes, because the way you describe so vividly and you're passionate about it- and this is what's real, so, ultimately, love is not- is neither in large language models nor in something mystical. It's exactly in this moments of passion, and I would, I would, I would even go as far as saying that in this moment, when you're describing it, it there was a connection of sorts, so that I could feel your passion for it. And in this moment, something else comes up Which is far beyond any, any theories that we can come up with, and that's what we for now, exactly. So, on one side, there is this impulse of finding this Theory a theory. And then there is another impulse to escape from what has already been known. So one, in other words, like in my basic example, is one impulse to say everything is a real number. Square root of negative one doesn't exist. But another impulse is: I'm going to be this naughty child who is not afraid to be an idiot and I will say: square root of negative 15 is real. Show more

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And both are essential. When it's done with conviction, when it's done with passion, when it's not like you know, you know gratuitous, or when it's not, it doesn't come from self-limit, self-limiting, but comes from. This sense of this is how I am, this is how I feel it is real. That's where the progress is, that's where creativity is and that's where I would even say a real connection is, because the strive to me that we I observed today in our society and the society level, at the level of humans and so on, it comes from not seeing the other person actually and being caught up in a very specific conceptual bubble, you see, and the way out of it is not to refine the bubble but just break out of it. A good guide out of the bubble is a childlike passion, whatever, discovering that and following it, Goosebumps, yeah, following the Goosebumps, yeah, to to the to. You know, not the rigor of science, but the magic of Goosebumps, and then, if you're interested, try to find a confirmation of those goosebumps in science or whatever you know. You know you find interesting and most of the time you'll fail. And most time you fail, which we also love, because then it sets us up for that moment of bliss when we succeed. Right, exactly, quick pause, bathroom break. You mentioned gato's, the completeness theorem. Can you, can you talk a little bit about it? What is it? As you understand it? Did it break mathematics, maybe? Another question is: what are the limits of mathematics? What is mathematics from the perspective of getting us to completeness theorem? Oh, yes, how much time do you have? We talked about time previously, so time is an illusion, right? Show more

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So we agreed- was a great Austrian mathematician and logician. He moved to the United States before second world war and worked at The Institute for advanced study in Princeton, where he was a colleague of Einstein and other great scientists, for Neumann, Hermann Weil and so on. But you know one? One interesting quote that I like in disregard is that Einstein said that at some point. He said that the only reason he came to the Institute was that he would have the privilege of walking back home with godo in the evening. So, in other words, Einstein thought that the girl was the smart one. Okay, so so he, his most important contribution was his two incompleteness ethereums: the first incompleteness theorem and the second incompleteness theorem. And what is this about? It's really about limitation, inherent limitations of mathematical reasoning, but my way of producing mathematical theorems, the way we do it. So, to set the stage, how do we actually do mathematics? What is so? You know, we know that we discussed that. Say, physics is based on mathematics. And you could say: chemistry is based on physics, biology based on chemistry. Okay, so it comes to mathematics. What is mathematics based on? Well, mathematics is based on axioms. So any field of mathematics is, can be, can be presented as what is called the formal system, and at the core of the formal system is a system of axioms or postulates. These are the statements which are taken for granted, given without proof. Show more

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Without proof, an example would be so that one of the very first formal systems was the system was euclidean geometry, developed by Euclid in his famous book elements, about 2200 years ago. And it's about it's well, it's a subject familiar from from school because we studied. But what it's really about is about the geometry of the plane and the plane. By plane I mean just the stable top extended to Infinity in all directions, kind of a perfect plane, a perfect, perfectly even table. And so euclidean geometry is about very geometric figures on the plane, specifically lines, triangles, circles, things like that. So what's an example of an Axion, an example of an axiom is that if you have two points which are not, which are distinct, two points on the plane, then there is a unique line which passes through them. Show more

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Now kind of sounds reasonable, but this is an example of an Axion in mathematics. You can, you have to have a seed, so to speak, you have to start with something and you have to choose certain postulates or statements which you simply take for granted, which do not require proof. Usually they are ones which kind of intuitively clear to you. But in any case you cannot have, you could not have any mathematics without choosing those actions. And you refer to those as the Observer because they're kind of subjective. The Observer comes in the process of choosing the axioms. Who chooses the axons, the turtles, you know, like to say who is watching The Watcher. Yeah, and so in mathematics. But you see, mathematicians are so clever. It's really kind of like a little kind of a game of mirrors that we often like to say, and I used to say that that mathematics is objective. It's really one, the only objective science, but that's because we hide this. Show more

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This fact is based on axioms and the fact that that there is no, there is no unique choice. There are many choices, and so euclidean geometry is actually a good illustration of this, because Euclid had five axioms. Four of them were kind of obvious, like the one I just mentioned, and the fifth, which came to be known famously as the fifth postulate, was that if you have a line and you have a point outside of this line, there is a unique line passing through that point which is parallel to the first line, meaning that doesn't intersect it. Show more

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And Euclid himself was uncomfortable about this because he felt that it was kind of a. You know, the York City takes for granted something that is not obvious, and for me, for many, for many centuries after that, mathematicians were trying to derive this action from other axioms which were more obvious in some sense, and they failed. And it was only almost 2000 years later that Magnus realized that you can't- not only you cannot derive, but you can actually replace it with its opposite and you'll still get a bonafide, a consistent no, it's not self-contradictory, which is called non-euclidean geometry, which of course sounds very complicated. But it's not think of a sphere, just the surface of a basketball or the surface of the Earth. I know, idealized the analogues. So do you have points? You have analogues of lines which have meridians right and every two meridians intersect, unlike parallel lines on a flat space. There is also so-called hyperbolic plane where, no, the infinitely many lines which do not intersect. Show more

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So every possibility can be realized. There are different flavors. This is a good illustration of what a formal system is. You start with a set of axioms, those statements you take for granted, and this is where you have a choice. And by making different choices you actually create different mathematics. After that there are rules of inference, logical rules such as if a is true and a plus applies B, then B is true. Most of them were actually introduced already by Aristotle, even before Euclid. And then it runs as follows: you have the axioms which are accepted as true statements, then you have a way to produce new statements by using the rules of logical inference from the axioms. Show more

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Every statement you obtain you call ethere and you kind of add it to the collection of true statements. And then the question is: how far can you go? How many statements can you prove this way? Of course you want it to be the system to be non-trivial in the sense that you don't prove everything, because if you prove everything it would mean that it's self-contradictory, that you'd prove a statement a and it's negation, so that's kind of useless. It has to be discriminating enough so that it doesn't doesn't prove contradictory statements, so that there is already a question of that mathematical consistency. It has to be consistent in the sense that it is not self-contradictory. And then the idea that was basically prevalent in the world of mathematics by the beginning of the 20th century was that in principle, all of mathematics could be derived this way. We just have to find the correct system of axioms and then everything you ever need would be could be produced by this procedure, which is really algorithmic procedure, which which actually could be run on a computer. Now think about it. What is special about this process? In this process you are just manipulating symbols. Basically you're going from one statement to another without really understanding the meaning of it. So it's an ideal playground for a computer program. It's a purely syntactic process where there are some rules, some rigid rules of passing from one statement to the next. Show more

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Most mathematician mathematicians believed that this way you can produce all true statements, and if this were true it would give a lot of credibility to the thesis that everything in life is computational, or life is computation, because then at least mathematics is computational, because then it can be programmed and the computer, after sufficient time, depending on on its capacity, would produce every true statement. So guildo's first incompetence theorem says that that's not the case, and it's not just says it, but it proves it at, you know, the highest level of rigor that is available in math Max, that is to say within another formal system that he was operating in. So, more precisely, what he proved was that if you have a sufficiently sophisticated formal system, that is to say that you can talk about numbers, whole numbers in it, that you have whole numbers- one, two, three, four, you, you have, you can, you have formalized the procedure. Show more

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The operation of additional multiplication within the system is consistent, that is to say, if it's not complete, completely useless, then there will be a true statement in it which cannot be derived by this linear syntactic process of proving theorems from axioms. It's really incredible. So this was a revolution 1931, a revolution in logic, Revolution mathematics. We're still feeling the Tremors of of this discovery. In a similar time, the, the computer, is being born, the actual engineering of the computational system is being born right, which is electronic. Turing was Turing, was Alan Turing, who is considered as the father of modern Computing, right. Show more

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So he actually did something very similar, so he got. This halting problem cannot be solved algorithmically. That you cannot. Out of all computer programs, roughly speaking, you could not, you could not, you cannot have an algorithm of choosing out of all can be possible computer programs, which ones are meaningful, which ones will not, which ones will hold very depressing results all across the table. Or, on the contrary, life affirming depends on your point of view, because everything is full of paradoxes. So that means so you're right. It's depressing if we are sold on a certain idea from the outset and then suddenly this doesn't pan out. But okay, so which? My? I retort: what if? Show more

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What if he proved that actually you know, everything can be proved. So then what? What is left to do if you're a mathematician. So that would be depressing to me. And here there is an opportunity to do something new, to do to discover something new which may be a computer will not be able to. Again, with a caveat: according to our current understanding, maybe some new technology, some new ideas will be brought into the subject and the meaning of the word computation. Like now we think of condition, a particular framework during machines or church thesis and stuff like that. But what if in the future another genius like on Turing will come and propose something else? The theory will evolve the way, you know, we went from Newton's gravity to Einstein's gravity. Maybe in the framework of that concept some other things will become possible. You know, so it's not. To me it's kind of like not so much about deciding once and for all how it is or how it should be, but kind of like accepting it as an open-ended process. I think that's much more valuable in some sense than deciding things one way or another. You know, I wonder, I don't know if you think or know much about cellular automata and the idea of emergence. Show more

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I I often return to Game of Life, yeah, and just look at the thing, amazing, right, and wonder what kind of things they can do with such a small. You know tools, yeah, for simple rules. A distributed system can create complex Behavior and it makes you wonder that maybe the thing we'll call computation is simple at the base layer, but when you start looking at greater and greater layers of abstraction, you zoom out with blurry vision. Show more

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Maybe after a few drinks you start to see some, something that's much, much more complicated and interesting and beautiful than the original rules that are scientific. Intuition says cannot possibly produce complexity and Beauty. I don't know. I don't know if anyone has a good answer, a good model of why stuff emerges, why complexity emerges from a lot of simple things. It's a why question, I suppose another, but every why question will eventually have a a rigorous answer. Not necessarily we could have an approximate answer which still eludes something like quantmechanics: 99, maybe we'll be able to describe it was 99 certainty or 99 accuracy, yeah. Show more

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And then maybe you know in, in 100 years or, you know, next year, somebody will come up with a different point of view which suddenly will change our perspective. You know. To this point I want to say also, you know one thing that I find fascinating. Speaking of paradoxes and so on, do you remember how everybody was freaking out about this blue dress and the blue. Was it blue or was it black? Show more

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Yeah, it was the yellow, I think yellow or and white, or black and blue. It almost broke Twitter, you know, I remember that, yeah, that night. So there are many examples like that where you can perceive things differently and there is no way of saying which is correct, which is not. For instance, you got this, the bus, the Rubens bars, you know where you have, from one perspective, it's of us, from another perspective, it's two faces. Then there is this Duck Rabbit picture where you can Google it. If somebody doesn't know, they can Google it and find it. It's very easy. Actually, Ludwig wittenstein devoted several Pages Duck Rabbit in his book and so on. Show more

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There are many others. There are like the squares where you can square, you can see the from different perspective, this way, that way, and so on. So when we talk about neural networks, they're talking about training, data and stuff, and so that you have some pictures, for example, that you feed to your program and you try to find the most optimal neural network which would be able to decide which one is it. So the dog or cat or whatever. But sometimes it doesn't have a definite answer. So what do you do then? So do, actually. It's a question I actually don't know. Has modern AI even come to appreciate this question, that actually sometimes you can have a picture on which you cannot say what it is in it. From one perspective it's a rabbit, from an episode is a duck. How are you supposed to train if you have a neural network which is supposed to discriminate between, distinguish between ducks and rabbits? How is it going to process this? You see well, so the the trivial trick it does is to say there's a this x probability there is a duck and this probability that it's a rabbit. Show more

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But that's a good approach. But also I would say there's no like given percentages, for instance. Actually, at some point I was really curious about it and I looked, and for some. For each picture of this nature- and there are a bunch of them you can easily find online- my mind immediately interprets it in a particular way, but because I know that other people have could see it differently, I would then strain my mind and strain my eyes and stare at it and try to see it in a different way, and sometimes I could see it right away, and then I could go back and forth between the two, and sometimes it could. It took me a while for some pictures. So in that sense, even if this probabilities exist, they are subjective. Some people immediately see it this way, some people may see that way, and I think that nobody knows- not psychologists, not neuroscientists, not philosophers- what to make of it. The best answer? The best, of course, as a scientific mind, I will. I'm, even though I say no, don't look for interpretation, leave some place for mysticism or mystery. Show more

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Right, I say that. But of course I want a theory, I want an explanation. So the best explanation I find is from Niels Bohr's complementarity principle. So it is like particle and wave, that there are different ways to look at it, and when you look at it in a particular way, another side will be obscured. Think about it like the other side of the moon, you know so, like we are observing the Moon from one side and then we don't see the other side. There is a complementary perspective where we see the other side, but not the side we normally see, but the Moon is the same, still there. It's our limitations of being able to grasp the whole. That's complementarity, and we know that from quantmechanics, that our physical reality is like that, rather than being certain, rather than being one way or another, and we should, just as a smaller side in terms of neural networks, mentioned that at the end of the day, there's humans is built on top of humans. Show more

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Or, with Chad GPT, that is, using reinforcement learning by human feedback, we're actually using a set of humans to teach the networks. Yeah, and that's the thing that people don't often talk about, because- or I, I sometimes think about that- those humans all have a life story, each human, the annotated data that fed data to the network or did the rlhf- yeah, that they have a life story. They grew up, they have biases. Biases are some things that they like. There's some things they don't like which can kind of appear under the Raider screen. They may not be aware that they are exercising those biases. That's the point. What you brought up is a very important issue here. Not so it might issue, but it's not a bug, it's a, it's a feature. In my opinion, that implicit in the discussion of the question is thinking computational and so on, is the idea that our conscious awareness covers everything within our psyche and we we just know that that's not the case. Show more

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We have, all of us have observed other people who have had sort of destructive Tendencies, so obviously they did things destructive for themselves and many of us have observed ourselves to doing that as part of human nature, right so? And there is great research in analytic psychology and, you know, in the past hundred years strongly suggesting, if not proving, the existence of what Carl Jung called the unconscious, personal unconscious and also Collective unconscious. There's a kind of Circle of ideas which are under the radar screen, which lead us to some strong emotions and Inspire us to act in certain ways, even if we cannot really understand. So if we accept that, then it the preposition that somehow everything can still be covered by our actions, which are totally kind of neutral and totally like righteous and totally conscious, that it becomes really tenuous. Let me ask you some tricky questions. Show more

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Oh, in in terms of how big they are, in terms of how you know they become difficult, because of how much of romantic you are. What do you is the most beautiful idea in mathematics? Another one we can ask is: which is the most beautiful equation in mathematics? Hmm, well, I mean I may have just broken your brain, because, because what your brain is doing is walking down a long memory lane of beautiful experiences. Well, you see, in mathematics we have this idea that that we have an idea of a set, right. So it could be a collection of things, for instance, you know, the set of tables, the set of chairs and so on, outside of microphones. But it could be set of numbers, could be set of ideas, could be a set of formulas, mathematical equations. Show more

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And then we have the notion of an ordered set. Order it, like we said, in which there is order, which means that for every two members of the set we'll say which one is better than the other or greater than the other. For instance, all numbers are ordered, 5 is greater than 3, 5 is less than 7, and so on, but not all sets are ordered. So the set of beautiful theorems is not what beautiful equations is not ordered. So, in other words, there are many best equations, and so Richard Feynman chose one which I think one of the best is that if you take e, the base of natural logarithm, to the power Pi I, so you have Pi, you have e in it, the base of initial logarithm, you have Pi I, which is square root of negative one, then the result is negative one. Show more

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So that's up there for sure, in the pantheon of beautiful formulas. You know that every- I think pretty much every- mathematician would agree. I don't know what my favorite one is, I just lingering on that one: Euler's identity. What makes it beautiful? Just a few symbols together, right, I mean part of it is actually just trying to Define what is beautiful about mathematics that is Laden there in this particular equation, that is somehow revealed when the human eye looks at it. Why is it beautiful, do you think pi? There is an element of surprise in it. How is it possible we always think of Pi as the ratio between the circumference of a circle and its diameter? Here we are taking some number to the power Pi, not even Pi, mind you, but pi multiplied by square root of negative one. Surely this is something completely incomprehensible? And yet the result is negative one, you see, and if you take e to the power 2, pi, I, you get one, actually one. So I would guess that that's. But in other words, the initial reaction is just that of a surprise, I guess. I guess, for anyone who first comes across, that these three folks, four folks, got together. Yeah it. It reminds me of the, the idea that Hitler's, Stalin, Trotsky and Freud were all in Vienna in some early at the beginning of the 20th and business Stein was was classmate of of Hitler. Show more

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You know this, I did not know this though. Yeah, so there it makes you. You know, you can imagine a situation where they're all sitting at a bar together at some point, not knowing it, but they somehow it all made sense in space time to be located there, and that's what this feels like: some kind of intersection. Intersection, yes, but I would say that after the initial shock, you you look at the proof of this equation and it actually does make sense- and actually it is- is nothing but the statement that the circumference of the circle is- and in fact in this case it's the circumference of a semicircle- is equal to 2 pi, and that's where it comes from. In the end, the truth is simple. Show more

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In the end it was a simple- not necessarily easy, but simple. So I mentioned to you Offline that I desperately, in trying to figure out the optimal in an audit, set questions to ask. You texted Eric Weinstein asking for what questions he can ask you and he said that you are definitively one of the greatest living mathematicians. So don't screw this up. But you did give me a few questions. So he asks to ask you what are the most shockingly passionate- this is in Eric's language- what are the most shockingly passionate mathematical structures? And he gave a list of four for him, but he said he really wanted your list. Okay, let me say that: shockingly passionate mathematical structures and shocking. Okay, is there something you can? Is there something that jumps to mind? Sure, I'm here to shock, yeah, yeah. So first of all, Eric Weinstein is a is a very dear friend, I have to say, and I really really appreciate and love him. He's just like my brother. So, you know, it's it's interesting to don't have a question posed by him. Maybe, if we can linger for a moment, what do you think is special about Eric Weinstein? For you know of his work and his mind, the way he sort of straddles so many different disciplines. It's like a renaissance man. There are very few people like that at any given moment, let alone the 21st century, where information has become so, you know, huge that it's almost physically impossible to be able to keep track of things. And yet he does, and he has his own unique vision and unique point of view and he has integrity which is like almost impossible, like I can't think of. I met so many people who possess that those qualities- almost no one- and also the ability in some sense to to embody the balance that you talked about of both the rigor of mathematics and the the imagination. Show more

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Humanity also, I would say. You know like we talk about imagination as a kind of a Counterpoint to knowledge or logic, but just basic Humanity, you know, just be basic, just compassion, just being able to, because every destructive, I would say, like every destructive Society, you know, like, be it Germany, you know, and Hitler or Soviet Union, understand, and so on, was based on some kind of what was considered unassailable truth, so a kind of conceptual system. Show more

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You know, if you think about it right. There is a beautiful episode of this series by Jacob bronowsky. You know where he talks about. He filmed it in Auschwitz, talking about the certainty that what led the Nazis is a tool. Killing people wholesale was a certain. It was almost a mathematical idea and they just basically bought into this idea and checked out their Humanity at the door. So I would say that antidote to this type of thing is not necessarily even imagination in a kind of elevated sense that we have been discussing today, that is exemplified by by our greatest scientists and philosophers, but just basic Humanity, you know, basic human, basic common sense of just like knowing that it's just not right and I don't care what my see, what my ideology tells me, but I'm just not going to do it. So that I think is kind of missing a little bit in today's society because people get a lot too caught up in in the ideology, in in certain conceptual Frameworks. So societies that lose that basic human compassion, the basic Humanity around the trouble, oh very much so. But not only Society, like a human being- and Eric is one of the people I agree with you- keeps that flame of you, like I trust that he will not do something that's not human, that's not right. Show more

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I just feel that you know, like there's some, some people you just kind of feel that they won't cross that line. Yeah, and that's a huge thing, you know today. Because I have to say, looking back, definitely I have not heard people personally, but like I could be mean, for instance, I could be harsh, and now I see it as a sign of weakness, as a sign of insecurity. You know, I saw, I saw your interview with Rick Roswell the other day. It's beautiful, I was really moved by it. But you know that at some point I was like I, I looked at him at this sort of like Dr Evil, kind of ashamed of it now but, like you know, I'm kind of coming clean and I would, you know because, well, why? Because I needed an adverse adversary in my mind, because I projected onto him kind of the fears that I had that we will be the AI, Will Conquer us, and so on. And this was rooted in my kind of Awakening moment, in sense, so kind of a moment where I suddenly started to see the other side. So, but I wasn't sure yet. You see, you have to feel it. So I had to have a fight about it. Yeah, you had to actually have the project. I had to that. So it was not, I believe that it was not in me already, so I had to throw it onto somebody. Show more

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Yeah, and that's not balanced yet. So balance is when you recognize that it's you actually. So and I had this moment. Actually it was so amazing, like I would give this, mean, I would talk about Ai and the dangers and he would always be my like foil. You know, I would put a Sinister photograph of you, okay, on the slide and it's like: look at this guy, he wants to put Nano balls into your brain and he's also like high-end, High, a top executive at Google, and so on, like so I would create this whole narrative. Show more

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And then something happened where I was giving a lecture- this is 2015- at the in in Aspen, Aspen ideas Festival, which is a wonderful Festival, so keynote speech actually- and I and I was doing my, my usual stick, and then suddenly I said I came up to that under the big screen and there was a picture of him there and I came up to the screen and I kind of touched it with my hand and I said: but I don't want to pick on Mr casual, because he's me. Show more

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I had this Revelation that I'm actually fighting with myself, with my own fears, and and then I learned about his, his, his father, that his father died when he was young and that he's, in fact, he's very discredited. He's very sincere, an upfront about it, self-disclosure. I think it's very Essential, by the way, in all this discussion, like what really motivates you. He said that. He said it publicly many times, even as early as 2015. Show more

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I could find this information that he wanted to reunite with his father in the cloud and suddenly I saw him not as a gray carcature that exemplified all my fears, but as a human being, who, a child longing for his father, grieving for his father. So suddenly it became a story, a love story, and you know. So that is so. In other words, I've seen it in myself, this capacity to project my own fears and then fight with other people over something that actually was my own. And as soon as I got to this point of seeing him- and then my next lecture, actually I talked about it, about him in this way, and I and I said, look, you know, it's a love story and he is actually- and it's not how- I would want to reunite with my father. Show more

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But, like you said, you know that if I am consistent, I have to allow the possibility that different people perceive things differently, and so, for him, that's his imagination. So you know how, who is this? Well, Voltaire, I think, is ascribed to Voltaire. It's like I disagree with you, but I will fight to death for you to have the right to say it. So, now that I I feel like my position is more like I disagree with him, that this is the way to to approach death and to approach the death of loved, our loved ones, and how we miss them and how we, you know, that sense of loneliness and inability to interact directly. That that's not something that is nice with me, but I think it's also. It can also be called imagination from his perspective and look, motivated by that, how much he has brought, how many interesting inventions, like his musical, for instance, naturally because his father was a composer, a music composer and a conductor. So, in other words, from in the bigger scheme of things, even if I think he's misguided, still I can't deny that it's a certain leap of faith from his perspective to try to say that this is the way we can all connect to our loved ones. Show more

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And because it is sincere- and I see it now sincere- and in fact in your interview you really teased it out of human it was, I was really moved by it. I have to say it's like he has mailed a little bit too. I said you know it was really, really sweet when he talked about his father and I can relate. You know my father died four years ago and I can relate what a heartbreak I was much older than Rey- was when his father died. But I can relate to this, to this longing and that grief, you know, and when he is somebody sincere and he puts his, opens his cards, and you know and say: this is why that's what I want to do, it because I want to recreate my father and I want to be able to talk to him this way. Then we have a series and then we understand. You know, the opposite of it would be not risk closing and just pretending that this is how it's supposed to be in a scientific term. So it was replacing your real emotion, but come from the Heart, by some kind of a theory which comes from the mind. And this is where we can go astray, because then we get become captives of Frameworks and conceptual systems which may not be beneficial to our society. In tough times, we need the people that have not lost their way in the ideologies. Show more

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We need the people who are still in touch with their heart, and you- you mentioned this with Eric is certainly true. I disagree with them on a lot of stuff, but I feel like when the world is burning down, Eric is one of the people that's you can still count on to have a heart. I've talked a lot over the past year about the war in Ukraine and the possibility of nuclear war, and it feels like he's one of the people I would call First if, if God forbid, something like a nuclear war will begin. Because you, you look for people with a heart, no matter their ideas. That's right. It takes courage and it takes a certain self-awareness, I would say, and which brings, I mean, you know. I think the crucial is that that which was inscribed, you know, on the Temple of a Poland, Delphia, there was a statement: know thyself, know yourself, you know the who am I. Show more

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Ultimately, it goes down to this and all these debates, and the point is that it's like I used to be, like I said, you know, pessimistic at some point and I was scared even of where development of AI was going. This is about 2014, 2015.. And now I'm much more so. For instance, after I saw Ray courtswell as a human being, after I could relate to him and sympathize with him, suddenly I I stopped seeing him in the news like before that I would always see him in the news saying: we're going to put Nanobots in your brain by the year 2030, whatever you know, and then we upload you to the 20 by the 21st, and I will be like, no, you know, the story was terrible. Show more

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Suddenly I didn't see him anymore. I had to, you know. So now it makes me question who was creating the trouble? What was all with it? Was it him who's creating, stirring the trouble, or was it my mind, you see? And so as I become, as I became self-aware, suddenly other possibilities opened and suddenly that conflict- which, by the way, if I kept giving this nasty, you know, talks about him- one one day I suppose we'll have a debate. Show more

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And so you have this- one person says this and then, yeah, and what I learned is that it never, it's a never-ending con, this conflict just does not end. But there is an alternative, there is a better way, which is to realize that it is you arguing with yourself. Now, if you want to continue arguing with yourself, continue as long as you need, just be careful not to destroy too many things, you know, in the process. But there is an option of actually dropping it, of actually dropping it. This is so. I was so surprised by this. Yeah, it's discovering in yourself the capacity, the human capacity, for compassion, and you understand that he has a perspective, he is operating in the space of imagination. Show more

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A human being like you, and we're all in this kind of together ultimately, and also it's like with realizing how much I have screwed up, you know, comes with humility also, yeah. So, like, I find it extremely hard now to like really Lush out at somebody and to say like you're horrible, whatever. Because immediately question is: who am I to criticize, you know? Show more

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So is there another way to have a dialogue? Is there a way to you know, speaking? You know, since we talked about the innocence of a child and how much it drives a discovery in science, and so on, you know, I remember it was. I think I heard the Ashanti who gave this nice example. He's like when you're a kid, you know you go and you you play with your friends, and then you fight with another kid, and he was like: I hate you, I don't want to see you again. Show more

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I just go home like after half an hour. Okay, what are you gonna do? You want to play? So you come out. It's like, hey, you want to play, you don't talk about what happened, you don't rehash this. You know, let's keep going, and sometimes I think we are on the verge, maybe, of learning that, because I think that if we are can, if we continue to push each of us our set of ideas, and like ideologies and like you know what matters to us, and so on, like yeah, no, no, what matters to you, but like there are other ways to approach other people. Show more

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There are other ways you can find point of contact, speaking of which mathematics, mathematical formulas, are Universal, represent Universal knowledge. Two plus two is four, whether you vote for this guy or that guy in the election. You know how about that? As a point of of contact, of commonality, you know, and nobody can patent those formulas. Did you know that there is a Supreme Court decision that mathematical formulas cannot be patented? Like Einstein could not patent equal E equals m, c squared. It doesn't belong to him because if the formal is correct, then it belongs to everyone. So what do you think of that all too tricky question? And, if you want, I can deeply bias your answer by giving the list of four that Eric provided. Oh no, let me give my mind. I cannot see, by the way, with you. Show more

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Yes, so, but I can guess some of them, so I'm going to try to do something different from him. So I already mentioned one, which is that you have one dimensional in the medical system, which is real numbers. You have two-dimensional, which is complex numbers, you have four dimensional and it's kind of- it's probably connected to what he wrote Because has to do with some homotopic groups of spheres and stuff like that. Then, of course, one I love, okay, one plus two plus three plus four plus five plus six and so on. Does it make any sense? The sI mean you probably heard about this one. It became a very popular at some point. One plus two plus three, I did. I did a video of a number file the the YouTube channel about it, maybe 10 years ago. So one plus two plus three plus four plus five ostensibly diverges goes to Infinity because you got a bigger and bigger number and yet there is a way to make sense of it in which it it comes up to minus one over twelve. How fascinating. First of all, the answer is not even a positive number and it's not an integer, it's not a whole number, it's minus one over twelve. So sometimes people ask me what is your favorite number? And it's a kind of a joke, I say: minus one over twelve, it's actually 42.. So your favorite number is not an ordered set, right? So what else? What else? So placements program. Of course I have to mention that and we'll we'll explore that in depth. Do you want? Do you want to know what Eric said? Sure, sphere aversion, boys, surface hop, vibration, vibration, okay, and Pi 1 of SO3, okay, oh, yes, so that's the. That's the famous cup trick, you know? Okay, look, so this is how it works. No tricks, no tricks, no matter what. Honest it is magical, okay, not, because I'm tricking you. So you start with a, a bottle like this, or a cup, and you start twisting it. At the same time you twist your, your arm. Then you come, so this is actually going to rotate it 360 Degrees, the full turn. Then you say, okay, I won't be able to do another turn because then my arm would really get twisted. I'll have to go see a doctor. Yet if I do it second time, it untwists. This is the pi one of SO3. Eric was talking about it. Yes, so there is something where the first motion is not trivial, but if you double down on it you come back to the initial position. It's a very closely connected to the fact that we have Elementary particles of two types: bosons and fermions. Show more

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So bosons are, for example, photons or carriers of other forces, or the Higgs boson- it is called a boson for a reason: because it is a Boson in in honor of Indian mathematician Bose, b-o-s-e and Einstein, so this particles obey what's called both Einstein statistics. But then there are other particles called fermions in honor of Enrico Fermi, Italian porn mathematician who worked in in the US, and and they follow what's called Iraq Fermi statistics and those are electrons and constituents of matter, electrons, protons, neutrons and so on, and they have a certain duplicity, if you will, and that duplicity is rooted mathematically in this, in this experiment, this little experiment that they have just done. So I can, I imagine, speaking of imagination. Show more

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Okay, yeah, so I'm just kind of briefing on this. Imagine a world in which this will not be shocking, or, like in this case, it's not even shocking, because I haven't really explained the details, because I can't do it in two minutes I I indicated what this is all about and so on. But imagine a world in which this is not foreign to most people, that most people have seen it before, or they're not afraid to approach this type of questions, because, you know, we talked a little bit about Mass education, but I really believe that a lot of people in our society- and it is not not only in the United States but throughout the world. A lot of people have been traumatized. It's really PTSD. That's why people, when they see, might make a formula or like, even like, how can they need to calculate, tip on the bill? They it's just, they're terrified, because it brings up those memories when they were kids and being called to Blackboard and solve the problem. You can't solve a problem and, as groupless teacher says, you're an idiot, sit down and you feel ashamed and and then lowly, and that stays with you. And so I think that, unfortunately, that's where we are dream, and so my dream is that one day we'll be able to overcome this and actually all of these Treasures of mathematics will become widely, widely available, or at least people will know where to find them and they will not be afraid of going there and looking. Show more

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And I think this will help because, like I said, for one thing, it gives you a sense of belonging, it gives you it kind of is an antidote the kind of alienation and separation that we feel today, oftentimes because of ideological divide, sectarian strife and all kinds of things like that. Because then you will, once you see there's a critical mass of this beauty, that kind of like tones on you is like my God. This is what we all have in common. You mentioned language program. We have to talk about it. Sure, at the core of your book and your work is the language program. Can you describe what it? Is sure? So, blank lens is a mathematician. Show more

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It's a name of a mathematician. Robert langlands- canadian-born- still alive he is- was a professor at The Institute for advanced study that we talked about, where Einstein and gildel and other great scientists have worked. In fact he used to occupy the office of Albert Einstein at The Institute for advanced Stadi. So here in in the late 60s, he came up with a set of ideas which captivated a lot of mathematicians- several generations of mathematicians by now- which came to be known as the language program, and what it is about is connecting different fields of mathematics which seem to be far away from each other, for example, number Theory, which, as the name suggests, deals with numbers and various equations with, you know, like x squared plus y squared equals one, and, on the other side, harmonic analysis, something that any music lover can appreciate, because the sound of a symphony can be kind of decomposed into sounds of different instruments, and each of those sounds can be represented by a wave like this, like a sine function. Show more

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Those are the harmonics. They actually the period of a harmonic. Periods of different nodes are different, they correspond to different notes and different instruments, different semitones if you will, but they all combine together into something diff, something special, which, which is not, cannot be reduced to any one of those. So it's the mathematically, it's the idea that you can decompose a signal into as a collection, as a simultaneous oscillation of several Elementary signals. That's called harmonic analysis. So what language found is that some really difficult questions in number Theory can be translated into much more easily tractable questions in harmonic analysis. That was his initial idea. But what happened next surprised everybody: that the kind of patterns that he was able to observe, the kind of regularities that he was able to observe, which were quite surprising, were subsequently found in other areas of mathematics, for example in geometry and eventually in quantphysics. So in fact Ed Witten, who is a kind of a dean of modern theoretical physicists, the professor at The Institute for advanced study as well, got interested in this subject. I described in my book how it happened and he was one instrumental in in Bridging the Gap between this patterns in found in physics and in Geometry, finding kind of a substrate, a kind of a super stratif, you will. Show more

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It's kind of over our kind of a way to connect these two things, kind of a bridge between these two Fields. So I subsequently collaborate with whiten on this and this has been one of the major themes of my research. It's sort of I always found it interesting to connect things, to unite things. When I was younger I couldn't, I couldn't understand why, but I was always interested in when, when, not in working in specific field, by kind of, but kind of cutting across fields, and and then I we discovered that, for instance, I took some people who know what happens in this field but don't know what happens in their field, and or conversely, and then I would, I would like find it imperative to go out and explain to them, to the different sides, what this is all about, so that more people are aware of this hidden structure, so this hidden parallels, if you will. So that has been sort of a theme in my research and so I guess you know now I kind of understand more why it's kind of a violence, you know, like what we talked about earlier. So can you elucidate a little bit how? What are the mathematical tools that allow you to connect these different continents of mathematics. Right, is there something you can convert into words? That language was able to find? Find and you were able to explore further. I would say what it suggests is that there is some and a hidden principles which we still don't understand. My, my view is that we still don't know why, that we can prove some instances of this, correspondences and connections, but we still don't know the real underlying reasons, which means that there is a newly, there is a certain layer beneath the surface that we see now. It is like the. So the way I see it now is like this: that there is something three-dimensional, like this bottle, but what we are seeing is this projection onto the table and projection to wall, and then we can map things from one projection to another and say, oh my God, that's incredible. But the real explanation is that both of them are projections of the same thing and that we haven't found yet. But that's what I want to find. So that's what motivates me, I would say, from number Theory to Geometry, to quantphysics. So there is this one thing which has different projections, except it's not just a table and the wall, but they're like many different walls, if you will. So what is the philosophical implication that there is commonalities like that across these very desperate fields? It means that what we believe are the fundamental elements of, of mathematics are not fundamental. Show more

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There is something Beyond. It's like we previously thought that atoms were indivisible. Then we found out that there is nucleus and electrons, and the nucleus consist of protons and neutrons. Then we thought, okay, protons and neutrons must be Elementary. Now we know they consist of quarks. So it's about kind of finding the quarks of mathematics. Of course, beyond that there's maybe even more, which was my initial motivation to study mathematics, by the way. Right, so quarks was the first time you fell in love with understanding the nature of reality. Yeah, what was it like working with Ed Whitton, who many people say is one of the smartest humans in history, or at least mathematical physicists in history? Yes, fascinating. I enjoy, I enjoyed it very much. Show more

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I also felt they have to keep up, you know, and so we wrote this long paper and in 2007, and we collaborated for about a year. I have known him before and we talked before and we have seen him since and we talked, but it's very different to just meet somebody at conferences and have a conversation as opposed to actually working on the project together. So he's very, very, very, very serious, very focused. This is one thing which I have to say. I was really struck by this. Why is he considered to be such a powerful intellect by many other powerful intellects he had? He has had this unique vision of the subject. Show more

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He was able to connect different things, especially find connections between quantphysics and Mathematics, almost unparalleled. I I don't think anyone comes close in some sense in the last you know, 50 years to him in terms of finding just consistently, time after time, breaking around new ground, new ground. So it- it would basically one way one could describe it is he would take some idea in physics and and then find an interpretation of it in mathematics and then say, distill it, present it in mathematical terms and tell mathematicians: this should be like that. You know, kind of like one plus two plus three plus four is minus 1 over 12.. I might think she'd be like no way. And then it would pan out and mathematicians would then like like a whole industry would be created of groups of mathematicians trying to prove his conjectures and his, his ideas, and he would always be proven right, you know. Show more

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So, in other words, being able to glean some mathematical truths from physical theories. That's one side. On the other hand, conversely, applying sophisticated mathematics, he's probably the physics who kind of could learn mathematics the fastest. I don't think some younger faces maybe could come close, but it's still quite for them a long way to go to, to get you know, to be comparable to whiten, to take some of the most sophisticated mathematics, not learn it to the point where it becomes a practitioner of the subject practically and then use it to gain some new insights on the physics side. Show more

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Now, of course, the thing is that the theory