YouSaid · the spoken record

Edward Frenkel

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257
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2023-04-10
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2023-04-10
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  1. “Without proof. An example would be 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 the subject familiar 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 perfectly even table. And so Euclidean geometry is about various geometric figures on the plane, specifically lines, triangles, circles, things like that. So, what's an example of an axiom? An example of an axiom is that if you have two points which are not distinct, two points on the plane, then there is a unique line which passes through them. Now kind of sounds reasonable, but this is an example of an axiom.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  2. “And what is this about? It's really about limitation, inherent limitations of mathematical reasoning, way of producing mathematical theorems, the way we do it. So to set the stage, how do we actually do mathematics? So 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 can be presented as what is called the formal system. And at the core of the formal system is a system of axioms or postulas. These are the statements which are taken for granted.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  3. “So Kurd Gyodl 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. Neumann, Hermann Weil, and so on. But one interesting quote that I like in this regard 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 Geodel in the evening. So, in other words, Einstein thought that G ⁇ del was the smart one. Okay, so his most important contribution was to incompleteness theorems. The first incompleteness theorem and the second incompleteness theorem.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  4. “Most of the time you fail, which we also love because then it sets us up for that moment of bliss when we succeed, right?”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  5. “And then if you're interested. Try to find the confirmation of those goofbumps. In science or whatever, you find interesting.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  6. “Or when it's not, it doesn't come from 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 strife to me that I observe today in our society and a 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.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  7. “And that's what we for now exactly. So on the one side, there is this impulse of finding the theory. And then there is another impulse to escape from what has already been known. So in other words, like in my basic example is one impulse to say everything is 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. And both are essential when it's done with conviction, when it's done with passion, when it's not like, you know, gratuitous.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  8. “But I will be like, uh huh, uh-huh, but I wouldn't say anything. But suddenly I find this moving. I find it moving. 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 are passionate about it. And this is what's real. So ultimately, love is not... Neither in lash language models nor in something mystical. It's exactly in this moment of passion. And I would. Even go as far as saying that in this moment when you're describing 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 theories that we can come up with.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  9. “It has 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, if you just told me what you just described, I would be like, you know, being polite, I wouldn't contradict you since you're the host anyway, right?”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  10. “Yeah, I'm kind of worried. Recovery. But In old days, I was More on the side of everything is computational, or everything can be explained by science and whatever, you know. Would dismiss and disregard the intuitive or imaginative things. So then I had a flip that suddenly I started feeling it and started seeing it and so on. So then the pendulums had swung in the opposite direction. Then I was Arguing that somehow that was real, that imagination was intuitive, imaginative was real. And discounting what you just described. And I would argue with people saying no, no, this is not real, this is all imitation game and so on. But you see that What's new now, the new advert, okay, is the 2.0, 3.0. He is the one who is seeking balance, who is not, who is”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  11. “I agree with you. I hear you when you speak about it. Speak with To me, that is proof that 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. So I used to be, okay, kind of confession, okay?”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  12. “How interesting that artists See that so long ago. You know, of course, Blade Runner was based on a novel by Philip K. Dick, the Android's Dream of Elect That guy was a genius, you kn”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  13. “Or he would turn off his Siri program. Get out of his house, go to a local bar. Strike a conversation with a human being. Although you might say by then some of those might be Androids And we don't even have a good test”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  14. “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 Partnership. So the human being rears its head anyway.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  15. “And the guy is heartbroken But see, so here's my analysis of this. Okay, it's like a couch therapist, okay? The guy did not have the courage to go out in the real world and to meet a woman and to get a girlfriend and so on. Through no fault of his own, perhaps, because you 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 series 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”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  16. “For other people, but he doesn't have a partner, he's lonely, and then he gets this sort of enhanced version of Siri with the voice of Carol Johansen, which is a very sexy voice, you know. Obviously, she's a great actress. And then at first it looks like a fantastic arrangement. He confides in her, she tells him things. 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”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  17. “Would be helpful, would be consoling, would feel, would make them happy or sad or whatever, you know, would kind of strike the nerve. For some, it won't. And I agree with you that in principle, there is no one to judge this. 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 or something.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  18. “The reverberations of the novels I have read because I have learned or movies I have seen. 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. Which there is no answer. And that's where the subjective has an important role. For someone, that type of interaction.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  19. “And with you, I would also say that you could try to, for instance, say that the origin of that. Overfed to it, for instance However, then you can retort. But what if what I consider my subjective unique feelings are also?”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  20. “But more importantly, for me, Apollo is like math and Dionysus is like love. So Dionysus and Apollo in modern version is love and math.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  21. “That's responsible for intuition, imagination, love. Dionysus was also God of wine. So it's also that side of... Human that makes Think and engage in revelry and drink wine and spend time with friends and love and enjoy it. So Nietzsche advocated this point of view that those were two complementary sides of every human and as well as society. The purpose of human life Find balance between them.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  22. “But none of them, we mentioned Niels Bohr and Newton. And Albert Einstein. So that's where the conundrum is. How do you find a balance between the two? Point I'm trying to make, and this is what I feel. Know, if you ask me what I find most important thing today, like what makes me excited and enthusiastic and passionate. This idea So, you know Nietzsche had this book, wrote this book in the 19th century called The Birth of Tragedy. And he presented this. Theory, which I think is kind of very useful. These two sides of a human. One that comes from God Apollo. And that's our left brain, so to speak, proverbal left brain. So 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 Dionysus, Dionysus.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  23. “Great, you can play with it, and you can use that. It gives you certain lens through which to see the world in a particular way, which could be beneficial for you 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 that we kind of lost this other side of their teachings. We took their numbers and they were like, 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 another side to it, which is kind of to us now. Say, oh, that's mystical, but.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  24. “For them, the universe was this infinite symphony in which every being humans, animals, as well as the earth and other celestial bodies were moving in harmony, like different nodes of different instruments in a symphony. And so they applied the same reasoning to the cosmological model as they applied to 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. They always knew that there is more. And they knew also that every pattern that you detect. Finite, but the world is infinite. They actually accept it infinity. They believe that infinity is real. And if you discern a pattern,”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  25. “They did not quite have 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. Around the central fire or hearth, they called it earth. But still, what a departure from the dogma. From the knowledge 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 movement of celestial bodies was like music. In fact, we call it Musical Universalis or Music of the Spheres.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  26. “Were imbued with the divine. And I cannot say that I think we lost it. At least I have lost it. I look at numbers and I don't really see that. The divine that they clearly did. And so whilst you know, how else would you explain? So in other words, divine is, of course, is a term which is, you know, it's bitloaded. 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 it this way. They were able to divine, okay, strike that. That the planets were not revolving, the silent 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 predecess”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  27. “Yes. So that's how I see it. Actually, 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 learned that I actually totally misunderstood Pythagoras and Pythagoreans, that they were much deeper. Then I thought Because most of us remember Pythagoras 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 famous perfect fifth. Halves for the G for the compared to the frequency of DOA or C. But actually they were much more interesting. So for them, numbers were not just clerical devices, you know, not kind of thing that you would use in accounting only.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  28. “People predicted that, you know, the age of Aquarius, the excess 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 2,000 years. So there is a traditional way of measuring that by this. Here as you know the ages. So the previous one is called the Age of Pisces because of the constellation of Pisces that it points to, so to speak, you know. And now it's, you know, as in the famous musical hair, they said the age of Aquarius is upon us. So the different people dated differently, but somewhere around the time where we are finding ourselves, how interesting, right, is all this strife and all the... Difficulties the world is experiencing. This might actually be the transition to something more harmonious. Wouldn't it be nice?”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  29. “So, in other words, what is it about poetry? What is it about art? Why are we so drawn to that? Why are we so captivated by those forms of, by, they are not intellectual necessarily. When you look at a painting that you like, when you listen to music that you love, you get lost in it. You get absorbed in it. It can make you cry. It can make you laugh. It can make you remember something. It can make you feel more confident. It can make you feel sad or happy and so on. What is this all about? Is it really just some play between some kind of like cellophone play or some neurons hitting one each other? Is it really that only? Maybe.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  30. “But if you think it's weird, and I agree with you as a recovering addict to knowledge, Know, but I am liking it more and more because there's so much freedom in it. And like Niels Bohr said, you know, I quoted that earlier, 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 things in quantum physics are so complicated. The only way you can speak of them is in poetry.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  31. “The enchanted gardens of platonic reality, you know, so where all this fruit grows, you know, and then we might, it gives you this sort of reminding sense of an explorer. And someone may be stuck in some provincial town in Russia, for instance, but have the sense of Magellan, you know, of traveling around the world. It's just not in the world that we usually think of. So 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, you cannot deny that. It's human beings who have so far discovered new mathematics. And I do not deny the possibility that computer programs will be able to discover new mathematics. But so far it's been humans. So whatever it is, whether it's discovered or invented, it is a human activity.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  32. “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 a paradoxical, the question whether mathematics is invented or discovered, whether there is this world of pure forms and so on. Is another paradoxical question, which doesn't have a simple answer, like whether electron is a particle wave. One point of view, yes, it's true. And just the fact that so many mathematicians today actually subscribe to this idea gives it a certain credibility because that's what we feel. We do feel that we dive into that mindscape, so to speak. But the very structured mindscape where I wrote in 11 math that”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  33. “Think now. I think now. I understand why I liked it because I think that I was very dissatisfied with what we call the real world, the world around me, the cruelty, the injustice of it. And I went through certain experiences as a kid, which made me love mathematics even more as this place where I could be safe and control.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  34. “So let me tell you something. My views are evolved. When I wrote Love in Math, when I wrote my book, Was squarely on the side of mathematics discovered. Does it mean usually mathematicians or others who have this idea or belief are called Platonists in honor of the great philosopher Plato who talked about this absolute perfect forms? 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 or how neat, very neat. Also, makes you feel connected to something divine.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  35. “So, there is a big conundrum as to whether Mathemax is invented or discovered. And mathematicians are divided on this. Nobody knows.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  36. “Is that place? He was like blaying. 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.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  37. “So, in other words, yes, we resolved this one mystery that we understood that square root of negative one is real, is meaningful. We build a theory to service those, to describe those numbers. Did we find the theory of everything? No, because we then invited other mysteries because we pulled the veil, so to speak, or we pushed the frontier and then new things get illuminated, which we couldn't see before. That's how I see the process of discovering mathematics. It's an endless, limitless pursuit.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  38. “Discovered by an Irish mathematician Hamilton in the 19th century. And then in the eight-dimensional, there is something similar called Actonians, and that's about it. So how interesting the structures exist in dimension one, two, four, and eight, which are all powers of Square is 4 to the third power is 8. That's one of the bigger mysteries in mathematics why it is so. So here's a hint of What's missing on our high school curriculum, the kind of fascinating. Mysteries”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  39. “If it's two dimensional, which is like geometrically, it's just like this tabletop 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. What about three-dimensional space? Is it possible to also define some operation of addition and multiplication on it so that these operations would satisfy the properties that we are 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 plane, the two-dimensional case. And by the way, next question would be, what about four-dimensional? In four-dimensional space, again, you can, and you get what's called quaternions.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  40. “Knowledge, right? And now obviously for 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 an open shooting game. 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 this one dimensional, we have a bona fide numerical system called real numbers.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  41. “World, the fact that you can actually add them up by adding together the real parts and imaginary parts, that's easy. But there is also a formula for the product for the multiplication, which uses the fact that square root of minus 1 squared is minus 1. And the amazing thing is that that product, that multiplication, satisfies the same rules, the same properties that are usual paration of multiplication for real numbers. For instance, there is an inverse for non-zero number that you can find. Like number five has an inverse 1 over five, but one plus i also has an inverse, for instance, you know.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  42. “But then also numbers like square root of 2 or pi. We plot them as points on real line. So we draw this is one of the kind of perennial concepts even in our very poor math curriculum at school. But now imagine that instead of one line, you have one axis, you have a second axis. And so numbers now have two coordinates, x and y. And you associate to this point with coordinates x and y number x, which is real number, plus y times square root of negative 1. This is a graphical geometrical representation of complex numbers, which is not mysterious at all. 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.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  43. “It is something almost unbearable. It's almost, I feel that the large language model, a computer model trying to do that would just explode. And yet a human mathematician was able to find the courage and the inspiration to say, you know what, what is wrong? Why are we so adamant that these things don't exist? That's just our past knowledge. It's based on what our past knowledge is and knowledge is limited. What if we make the next step? Today, for us mathematicians, complex 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, 2 and so on, right? All fractions like one half or three halves or four over 3.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  44. “Deal with it, but I am going to accept it and see what happens. And in fact, at the end of the calculation, this weird numbers got cancelled, it kind of cancelled out. And the formula appeared square root of negative 17, and it's negation. So they kind of conveniently gave the right answer, which is not involved those numbers. So he was like, okay. What does it mean? Mental tortures. So you see, from the point of view of the thinking mind”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  45. “So he wrote a book which is called Ars 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. Cool, we study solutions of quadratic equations, equations of degree two. 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 quartic equations, for which would start with x cubed or x to the power of 4 as opposed to x squared. And in the process of solving this equations, he came up with square root of a negative number, specifically square root of minus 17. And he wrote that I have to forgo some mental tortures.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  46. “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 made money apparently by compiling astrological readings for the elite of his era. Is 16th century, as one does a gambler All around interesting guy. I'm sure we would have an interesting conversation with him Geralamo Cardano. He also invented what's called Cardan Schaft, which is an essential component of a car. Cardana Vival, we say in Russian.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  47. “Something that is essential or meaningful because if all the information that you get 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 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. So checkmate, you know, it's over. Square root of negative 1 doesn't exist. Yet, we know that these numbers make sense. They called complex numbers. And in fact, quantum mechanics is based on complex numbers. They are essential and indispensable for quantum mechanics. Could one discover that? So to me, that sounds like”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  48. “Whereas 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. 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. No, it's always felt as a jump, as a leap. And I have actually been studying various examples in history of mathematics, of some fundamental discoveries, like discovery of 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 of negative one.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  49. “I am supposedly like this smart, intelligent person. I am afraid to say it and own it myself. I have to find 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. He actually said that. There are some quotes which are attributed to him, which he never said, but this he did. I investigated. 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. And he's explained, for knowledge is always limited.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source

  50. “Because we haven't been taught properly. So, actually, tools are there. For instance, I think, well, here we have to Have to say my conviction is that everybody knows. The heart of hearts, everybody knows that there is that, there is something. There is something. Mysterious, and in fact, you know, somehow immediately I feel that the impulse to quote somebody on this, because as if my own opinion doesn't go.”

    2023-04-10 · Lex Fridman Podcast · #370 – Edward Frenkel: Reality is a Paradox – Mathematics, Physics, Truth & Love · IDENTIFIED FROM THE TRANSCRIPT · source