Edward Frenkel: Reality is a Paradox - Mathematics, Physics, Truth & Love | Lex Fridman Podcast #370

1. 0:00 – 1:10

   Introduction

   1. EFEdward Frenkel0:00

      There is a famous story about Einstein that he used to, you know, go, um, think, think, think, and then go for a walk, and, like, he would whistle sometimes. So, I remember the first time I heard this story. I thought, "Hmm, how interesting. It's what a coincidence that he- this came up to him when he was whistling."

   2. LFLex Fridman0:14

      Mm-hmm.

   3. EFEdward Frenkel0:15

      But in fact, it's not. This, it is how it works, in some sense. That y- y- you have to prepare for it, but then the mom- it happens when you stop thinking, actually. So the, the moment of discovery is the moment when thinking stops, and he, in a wa- you kind of, you kind of almost become that truth that you're seeking.

   4. LFLex Fridman0:38

      The following is a conversation with Edward Frenkel, one of the greatest living mathematicians, doing research on the interface of mathematics and quantum physics, with an emphasis on the Langlands program, which he describes as a grand unified theory of mathematics. He also is the author of Love and Math: The Heart of Hidden Reality. This is a Lex Fridman podcast. To support it, please check out our sponsors in the description, and now, dear friends, here's Edward Frenkel.

2. 1:10 – 11:20

   Mathematics in the Soviet Union

   1. LFLex Fridman1:10

      You open your book, Love and Math, with the 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?

   2. EFEdward Frenkel1:22

      So, first of all, I grew up in the Soviet Union. In a small town near Moscow called Kolomna. Um, and, uh, I was a smart kid, you know, in school, but mathematics was probably my least favorite subject. Not because I couldn't do it. I was, you know, a straight A student and I could do, um, all the problems easily, but I thought it was incredibly boring, and, um, since the only math I knew was what was presented at school, I thought that was it. And I was like, "What kind of boring subject is this?" So, what I really liked was physics. And especially quantum physics. So, I- I was buying, um, I was, I would go to a bookstore and buy popular books about elementary particles and atoms and things like that, and read them, you know, devour them. And so, I th- I, 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. Uh, it turns out that a f- a friend of my parents was a mathematician who was a professor at the local college. It was a small college preparing educators, teachers. It's a provincial town. Imagine, it's like, uh, 117 kilometers from Moscow, which would be something like 70 miles, I guess.

   3. LFLex Fridman2:52

      Mm-hmm.

   4. EFEdward Frenkel2:53

      (laughs) You do the math. (laughs)

   5. LFLex Fridman2:55

      I like how you remember the number exactly.

   6. EFEdward Frenkel2:57

      Yeah. Isn't it funny how we remember numbers?

   7. LFLex Fridman2:59

      Yeah.

   8. EFEdward Frenkel3:00

      So, his name was Evgeny Evgenievich Petrov.

   9. LFLex Fridman3:04

      Yeah.

   10. EFEdward Frenkel3:05

       And i- if this doesn't remind you of the great works of Russian literature- (laughs)

   11. LFLex Fridman3:09

       (laughs)

   12. EFEdward Frenkel3:09

       ... then you haven't read them. (laughs)

   13. LFLex Fridman3:12

       (laughs)

   14. EFEdward Frenkel3:12

       Like War and Peace, you know, like with the patronymic names.

   15. LFLex Fridman3:14

       Yeah.

   16. EFEdward Frenkel3:15

       But this was all real. This was all happening. So, my mom one day, by chance, met Evgeny Evgenievich and told him about me. That I was a bright kid, and interested in physics, and he said, "Oh, I want to meet him. I'm going to convert him into math." And my mom's like, "Nah. Math? He doesn't like mathematics." So they said, "Okay, let's- let's see what I can do." So, I went to see him. So, I'm about 15 and a bit, a bit, uh, arrogant, I would say. You know, like, average teenager.

   17. LFLex Fridman3:48

       Mm-hmm.

   18. EFEdward Frenkel3:48

       (laughs) So, he says to me, "So, I hear that you are, um, interested in, in physics, elementary particles." It's like, "Yeah, sure." He said, "For example, do you know about quarks?" And I said, "Yes, of course I know about quarks." Quarks are the, you know, constituents of particles like protons and neutrons and it was one of the greatest discoveries in theoretical physics in the '60s that, uh, those particles were not elementary, but in fact had the smaller parts. And he said, "Oh, so then you probably know representation theory of the group SU (3) ." (laughs) I'm like, "SU what?" So, in fact, I wanted to know what was, what were the underpinnings of those theories. I knew the story, I knew the narrative, I knew kind of this basic story of what these particles look like, but how did physicists come up with these ideas?

   19. LFLex Fridman4:47

       Mm-hmm.

   20. EFEdward Frenkel4:48

       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- a substantial book. And he opens, um, it in the- somewhere in the middle and there I see the diagrams that I saw in popular books, but in popular books there was no explanation. And now I see all these weird symbols and equations. It- 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.

   21. LFLex Fridman5:28

       That to understand the underpinnings of physical reality, you had to understand what SU (3) is.

   22. EFEdward Frenkel5:34

       You have to learn what are groups, what is group SU (3) , what are representations of SU (3) . There was a coherent and beautiful- I could appreciate the beauty even though I could not understand heads and tails of it.

   23. LFLex Fridman5:50

       But you were drawn to the, the methodology, the, the machinery of how such understanding could be attained?

   24. EFEdward Frenkel5:57

       Well, in retrospect, I think what I was really craving was a deeper understanding...And, uh, up to that point, the deepest that I could see was for those diagrams but, or that story that, you know, a proton consists of three quarks and a neutron consists of three quarks and they're called up and down and so on. But I didn't know that there was actually underneath, beneath the surface, there was this mathematical theory.

   25. LFLex Fridman6:25

       If you can just linger on it, what drew you to quantum mechanics? Is there some romantic notion of understanding the universe? What, what is interesting to you? Is it the puzzle of it or is it like the philosophical thing?

   26. EFEdward Frenkel6:38

       Now I am looking back-

   27. LFLex Fridman6:40

       Yeah.

   28. EFEdward Frenkel6:40

       ... so, um, whatever I say about Edward, uh, at 15-

   29. LFLex Fridman6:46

       Yeah.

   30. EFEdward Frenkel6:47

       ... is colored by, uh, my, you know, all my experiences that happen in, in the meantime.
3. 11:20 – 22:39

   Nature of reality

   1. LFLex Fridman11:20

      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?

   2. EFEdward Frenkel11:39

      There, there is a, actually an essential difference, uh, which is that physicists are interested in describing this universe, okay, and mathematicians are interested in describing all possible mathematical universes. Of which, uh, you know, in some of our work, of m- I still consider myself more of a mathematician than a physicist, my first love for physics not- notwithstanding. Um, mathematicians are, we, in a way-... we have more diversity if you, if you, you might say. So we, we are accepting, for instance, uh, our universe, uh, has three spacial- spatial dimensions and one time dimension, right? So, uh, what I mean is that-

   3. LFLex Fridman12:22

      Allegedly.

   4. EFEdward Frenkel12:23

      Allegedly.

   5. LFLex Fridman12:23

      (laughs)

   6. EFEdward Frenkel12:24

      Observed. But that we can observe today.

   7. LFLex Fridman12:26

      Yes.

   8. EFEdward Frenkel12:26

      Right? So of course there are theories where there are some hidden dimensions as well. Well, let's just say observed. Observed dimensions. Um, so this tabletop has two dimensions because you can have two axes, two coordinate axes. (laughs) You know, x and y, but then there is also a third one to describe the space of this room. And then there's a time- time dimension. So, realistic theories of physics have to be, um, about spaces of- of three- three dimensions, or spacetimes, so four dimensions. But mathematically, we are just as interested in theories in 10 spacetime 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, um, confirmed by an experiment. We don't get to play with big machines like LHC in Geneva, Large Hadron Collider, that recently discovered, you know, the Higgs boson and some other things. For us, it's all like a mental exercise in some sense. We do, we prove things by using rules of logic, and that's our way of confirming, experimental confirmation if you will. But I think we kind of, I kind of envy a little bit my friends' physicists that they, they get to, they get to experience this sort of, these big toys, you know, and play with them.

   9. LFLex Fridman13:59

      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.

   10. EFEdward Frenkel14:06

       That's right. So the mathematics is underpins physics obviously. It's a language. Uh, the lo- the book of nature, as Galile- Galileo famously said, "It's written in the language of mathematics."

   11. LFLex Fridman14:18

       Mm-hmm.

   12. EFEdward Frenkel14:19

       And the, and the, you know, the- the letters in it are, are the circles, triangles, and squares. "And those who don't know the language," I'm paraphrasing, "are left, uh, to wander in a dark labyrinth." That's a famous quote from Galileo, which is very true, and has become even more true more recently in the- in theoretical physics in the mo- in the most, um, sort of far out, um, uh, parts of the theoretical physics that have to do with elementary particles and the, and the, as well as the- the structure of the cosmos at the large scale.

   13. LFLex Fridman14:55

       What do you think of, um, Max Tegmark, wro- who wrote the book Mathematical Universe?

   14. EFEdward Frenkel14:59

       Mm-hmm.

   15. LFLex Fridman15:00

       So do you think, just lingering on that point, you think at the end of the day, the future generations will all be mathematicians?

   16. EFEdward Frenkel15:10

       No, I

   17. NANarrator15:10

       (laughs)

   18. LFLex Fridman15:11

       Meaning-

   19. EFEdward Frenkel15:12

       I

   20. NANarrator15:12

       (laughs)

   21. LFLex Fridman15:12

       Meaning, the ones that deeply understand the way the universe works. At the core, is it just-

   22. EFEdward Frenkel15:18

       Well-

   23. LFLex Fridman15:18

       ... mathematics?

   24. EFEdward Frenkel15:20

       At the core of, you know... I would say mathematics is one half of the core.

   25. LFLex Fridman15:27

       Mm-hmm.

   26. EFEdward Frenkel15:28

       So th- the book is called Love and Math.

   27. LFLex Fridman15:30

       Yeah.

   28. EFEdward Frenkel15:30

       Okay, so these are the two pillars. (laughs)

   29. LFLex Fridman15:32

       (laughs)

   30. EFEdward Frenkel15:32

       Of your book.
4. 22:39 – 36:00

   Scientific discoveries

   1. EFEdward Frenkel22:39

      what I find fascinating is that the greatest scientists are on record saying that when they were making their discoveries, they felt like children.

   2. LFLex Fridman22:48

      Mm-hmm.

   3. EFEdward Frenkel22:49

      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-"

   4. LFLex Fridman22:55

      Mm-hmm.

   5. EFEdward Frenkel22:56

      "... or a prettier shell." Whilst I think some- he said something like, "The infinite ocean of knowledge lay- lay- was lying before me." Alexandre Grothendieck, who probably was the greatest mathematician of the second half of the 20th century, the Fr- French mathematician Alexandre Grothendieck, uh, wrote that, "Discovery is a privilege of a child." The child who is not afraid to be wrong once again, to be- to look like an idiot, you know, to- to- to try this and that, and I'm paraphrasing, and go through trial and error. That is for them, in other words, for them, uh, that innocence of a child who's 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 no- 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.

   6. LFLex Fridman24:02

      Do you struggle with that? You're one of the most respected mathematicians in the world. Uh, you're at Berkeley, you're like this- this- this stature. You're supposed to be very like, you know, uh-

   7. EFEdward Frenkel24:14

      Ivory tower kind of.

   8. LFLex Fridman24:15

      Yeah.

   9. EFEdward Frenkel24:15

      Sometimes I joke, I say, "I- I take- I take an elevator"

   10. LFLex Fridman24:19

       (laughs)

   11. EFEdward Frenkel24:19

       ... "to the top of the ivory tower every day." Yeah. (laughs)

   12. LFLex Fridman24:21

       And you're supposed to speak like royalty. Uh, do you struggle to, like, uh, 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?

   13. EFEdward Frenkel24:35

       Absolutely. I mean, that's part of being human because when we grow up, I mean, all of them, all of these great scientists, I think they were so great in part because they were able, they maintained that connection, okay, and that fascination, that vulnerability, um, that spontaneity, you know, and, uh, um, kind of looking at the world through the eyes of a child. But it's difficult because, you know, you go through education system. And for many of us, uh, it's not es- especially helpful for maintaining that connection, that we kind of like, we're being told certain things that we accept, take for granted, and so on, and little by little. And also, we get hit every time we act different, okay, every time we act, that doesn't, in a way that doesn't fit sort of the pattern. We get punished by the teachers, get punished by parents, and so on.

   14. LFLex Fridman25:32

       And don't get respect when you act childlike in-

   15. EFEdward Frenkel25:35

       Right.

   16. LFLex Fridman25:35

       ... in your thinking, when you are fearless and, uh, looking like an idiot.

   17. EFEdward Frenkel25:40

       That's right.

   18. LFLex Fridman25:41

       Because there's a hierarchy in soc-

   19. EFEdward Frenkel25:42

       Nobody wants to look like an idiot, you know? Once you start growing up or you think you're growing up.

   20. LFLex Fridman25:47

       Yeah.

   21. EFEdward Frenkel25:47

       In the beginning, you don't even think a, you don't, um, think in these terms. You just play. You're just playing. And you are open to possibilities, to this infinite possibilities that this world presents to us. So, how do we... I'm not saying that education system should not be, uh, also, uh, 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, a life is an answer. Every life, every human being is one particular answer to, how do we find balance? (laughs)

   22. LFLex Fridman26:51

       (laughs) That's one solu- i- imperfect approximation, approximate solution, uh, to the problem.

   23. EFEdward Frenkel26:57

       But we could look, we can look up to the great ones-

   24. LFLex Fridman27:01

       Yeah.

   25. EFEdward Frenkel27:02

       ... 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 Grothendiek or Pablo Picasso. So, they have established their right to speak about these matters, and we cannot dismiss them as mere madman. They say, "Okay, well, if, if the same thing was said by somebody who never achieved anything in, in that st-, in, in their, in their field of endeavor," you would be, it would be easy for us to dismiss it. But when it comes from someone b- like Isaac Newton, we take notice. So, I think that's something important that they, they teach us. And especially today in this age of AI, of course, there's a big elephant in the room always, (laughs) which is called AI.

   26. LFLex Fridman27:54

       Yeah.

   27. EFEdward Frenkel27:54

       Right? And so I know that you are an expert on the subject, and we are going, we are living now in this very interesting times of new, um, AI systems coming online pretty much every couple of weeks. So, I kind of, um, to me, uh, that whole debate about what is it, what is artificial intelligence, where is it going, what should we do about it, um, needs an influx of these 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 chi- a child has a capacity to, to create. And an, an adult credits the inner child-

   28. LFLex Fridman28:50

       The inner child, yeah.

   29. EFEdward Frenkel28:51

       ... uh, for this capacity to create a- as, as an adult, you see? That's kind of weird if we take the point of view that everything is computation, everything is accumulation of knowledge, that just bigger and bigger datasets, finer and finer neural networks, and then we will be able to replicate human consciousness. If we take that point of view, then what I just said kind of doesn't fit, because obviously a child has not been fed any training data (laughs) as far as we know, yet they're perfectly capable of, of, you know, of 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, you know, sort of perspective. So does it, can it really be captured, that perspective, that sense of awe, can it really be captured by competition alone? I actually, I don't know the answer, so I'm not sort of trying to, um, to present a particular point of view. I'm just trying to question, um, 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.

   30. LFLex Fridman30:12

       And it's hard to know where they come from. It's- it's also possible that the evolutionary process has created, is the very, it is computation and the, and the child is actually not a blank slate, but, uh, the result of one of the most incredible several billion year old computations, uh, that-... that had explored all kinds of aspects of, of life on Earth. Of, o- of war, and love, and terror, and ambition, and violence, and invention, all of that from the bacteria to today.
5. 36:00 – 52:12

   Observing reality

   1. EFEdward Frenkel36:00

      we, science of 19th century had the, from modern perspective, and I don't want to offend anybody, had the delusion-

   2. LFLex Fridman36:08

      (laughs)

   3. EFEdward Frenkel36:08

      ... that somehow you could analyze the world being completely detached from it.

   4. LFLex Fridman36:11

      Yeah.

   5. EFEdward Frenkel36:12

      We now know after the l- the, the landmark achievements of the first half of the 20th century that this is nonsense.

   6. LFLex Fridman36:18

      Mm-hmm.

   7. EFEdward Frenkel36:18

      That it's simply not true, and this has been experimentally proved time and time again. So to me, I'm thinking maybe it's a hint that I should take my p- 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, 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 parti-... Uh, no. The deepest love my, of my life perhaps, but perhaps hasn't happened yet, perhaps I will fall in love and it's... 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, b- however sophisticated they might be, I think we are from the outset denying to ourselves the possibility that, yes, there is this side of me which is not faking it. Yes, there is this side of me which cannot be captured by logic and- and reason. And you know what another great scientist said, Blaise Pascal?

   8. LFLex Fridman37:31

      Mm-hmm.

   9. EFEdward Frenkel37:33

      He said, "The heart has its reasons of which the reason knows nothing." And then he also said, "The last step of reason is to grasp that there are infinitely many things beyond reason." How interesting. This was not a theologian. This was not a priest. This was not a- a spiritual guru.

   10. LFLex Fridman37:53

       Mm-hmm.

   11. EFEdward Frenkel37:54

       Um, 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, uh, 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. Um, 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.

   12. LFLex Fridman38:37

       Mm-hmm.

   13. EFEdward Frenkel38:38

       We are just, you know... It reminds me of this famous Lord Kelvin's quote from the end of 19th century. Uh, there's some debate whether he actually said that, but (laughs) never let a good story, you know, stand-

   14. LFLex Fridman38:51

       (laughs)

   15. EFEdward Frenkel38:51

       ... stand in the way of truth. (laughs)

   16. LFLex Fridman38:53

       Yeah.

   17. EFEdward Frenkel38:53

       He said, "Physics is- is basically finished."

   18. LFLex Fridman38:57

       Yeah.

   19. EFEdward Frenkel38:57

       "All that remains is more- more precise measurement."

   20. LFLex Fridman39:00

       Yeah.

   21. EFEdward Frenkel39:00

       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. Uh, maybe little tweaks in the, 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.

   22. LFLex Fridman39:17

       Mm-hmm.

   23. EFEdward Frenkel39:17

       Because I don't see the other side so- so, quote-unquote, "represented that much."

   24. LFLex Fridman39:21

       Mm-hmm.

   25. EFEdward Frenkel39:22

       And I'm saying, okay, could it be also that if you believe in that, that that becomes your reality, that you can kind of put yourself in a box where everything is computation, 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 this. A friend of mine, uh, Philippe Cochin, told me that in France, there is this literary movement which is called Oulipo, O-U-L-I-P-O, and it's b- 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, uh, Georges Perec, 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.

   26. LFLex Fridman40:26

       Yeah.

   27. EFEdward Frenkel40:27

       So in other words, he set 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 applaud it.

   28. LFLex Fridman40:38

       Mm-hmm.

   29. EFEdward Frenkel40:39

       But that's one... it's one thing to do that and to kind of show his gamesmanship, if you will, and- and his proclivity and- and his ability as a writer, but it's another thing if at the end of, uh, writing this book, when he finish the book, he would say, "Letter E actually doesn't exist" (laughs) and try to convince us that, in fact, French l- 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, uh, addicted to knowledge myself. I... You know, hi, my name is Edward. (laughs)

   30. LFLex Fridman41:29

       (laughs) I'm addicted to knowledge.
6. 52:12 – 1:00:58

   Complex numbers

   1. EFEdward Frenkel52:12

      And I h- I s- I have actually been studying various examples in, uh, history of mathematics of some fundamental discoveries, like discovery of complex numbers, like square root of negative one.

   2. LFLex Fridman52:22

      Mm-hmm.

   3. EFEdward Frenkel52:24

      I wonder if a large language model could actually ever come up with the idea that square root of negative one is something that is essential or- or meaningful. Because if all the information that you get, the- all the- all the knowledge that had been accumulated up to that point p- 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 sc- 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 one doesn't exist. Yet we know that these numbers make sense. They're called complex numbers. And in fact, quantum mechanics is based on complex numbers. They are e- essential and indispensable for quantum mechanics. Could one discover that? So to me, that sounds like a discontinuity 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, astro- astronomer, astrologer. He was- he- he made money apparently by compiling astrological sort of readings for, uh, for the elite, you know, of- of- of his era.

   4. LFLex Fridman54:02

      As one does.

   5. EFEdward Frenkel54:03

      This is 16th century. As one does.

   6. LFLex Fridman54:05

      (laughs)

   7. EFEdward Frenkel54:05

      A g- gambler. (laughs)

   8. LFLex Fridman54:06

      (laughs) That's...

   9. EFEdward Frenkel54:07

      All around interesting guy. I'm sure we would have an interesting conversation with him. Gerolamo Cardano. He's al- he also invented the c- what's called cardan shaft, so which is, uh, an essential component of- of- of a car. Um, ƒ (Russian) , we say in Russian. So- so he wrote a book which is called, uh, A- Ars Magna, which is, like, The Great Art of Algebra. And he was writing solutions for the cubic and quartic equations. This is something that is familiar, because at school, we study solutions of quadratic equations, equations of- of degree two. So you have x- ax squared plus bx plus c equals 0, 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- which would start with X cubed or X to the power of 4, as opposed to X squared. And in the process of solving these equations, he 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-"

   10. LFLex Fridman55:14

       (laughs)

   11. EFEdward Frenkel55:16

       (laughs) "... 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. They kind of canceled out. In the formula appeared square root of -17, and its negation. So they kind of conveniently gave the right answer, which does not involve those numbers. So he was like, "Whew, okay." (laughs)

   12. LFLex Fridman55:39

       (laughs)

   13. EFEdward Frenkel55:39

       What does it mean, mental tortures? So you see from the point of view of a- of the- of the thinking mind, it is something almost unbearable. It's almost I feel that a lang- large language model, a computer running a ran- large language model trying to do that would just explode.

   14. LFLex Fridman55:57

       Mm-hmm.

   15. EFEdward Frenkel55:57

       And yet, a 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 knowledge is, and knowledge is limited. What if we make the next step?" Today, for us mathematicians, complex numbers, w- 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 1/2 or 3/2, or 4 over 3, but then also numbers like square root of 2 or pi. We plot them as points on a 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 curriculum at school. But now imagine that instead of one line, you have a pl- you have a se- uh, one axis, you have a second axis.

   16. LFLex Fridman56:56

       Mm-hmm.

   17. EFEdward Frenkel56:57

       And so your numbers now have two coordinates, X and Y. And you associate to this point with coordinates X and Y, the number X, which is a real number, plus Y times square root of -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.

   18. LFLex Fridman57:25

       Uh, so all it is, all a complex number is, is just an expansion-

   19. EFEdward Frenkel57:29

       Two real numbers. Two real numbers.

   20. LFLex Fridman57:29

       Yeah, it's just two-

   21. EFEdward Frenkel57:30

       The real part, and the imaginary part.

   22. LFLex Fridman57:31

       It's just an expansion of your view of the mathematical world.

   23. EFEdward Frenkel57:36

       That's right. The fact that you can actually mult- uh, you can add them up by adding, um, 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 -1 squared is -1. And the amazing thing is that, that- that product, that multiplication, satisfies the same rules-

   24. LFLex Fridman57:56

       Mm-hmm.

   25. EFEdward Frenkel57:57

       ... the same properties that are usual, uh, operation of multiplication for real numbers. For instance, there is an inverse for every non-zero number that you can find. Like, uh, number five has an inverse, 1 over 5. But, uh, 1 plus I also has an inverse, for instance, you know?

   26. LFLex Fridman58:13

       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-

   27. EFEdward Frenkel58:22

       And done.

   28. LFLex Fridman58:22

       ... who takes a leap off the cliff of comfort of like, mathematical comfort-

   29. EFEdward Frenkel58:26

       Established knowledge.

   30. LFLex Fridman58:27

       Oh, established knowledge.
7. 1:00:58 – 1:08:49

   Imagination

   1. EFEdward Frenkel1:00:58

   2. LFLex Fridman1:00:58

      Can you comment on...... what you think this human capability of imagination that Einstein spoke about-

   3. EFEdward Frenkel1:01:04

      Mm-hmm.

   4. LFLex Fridman1:01:05

      ... of the artist following their intuition in this big Alice in Wonderland world of-

   5. EFEdward Frenkel1:01:11

      Mm-hmm.

   6. LFLex Fridman1:01:11

      ... of, uh, imagination. What is it? You visit there sometimes.

   7. EFEdward Frenkel1:01:15

      What does it feel like?

   8. LFLex Fridman1:01:17

      Yeah, what does it feel like? What- what is it? What is that place?

   9. EFEdward Frenkel1:01:20

      Uh, it feels 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, like-

   10. LFLex Fridman1:01:29

       But it's not real, right? You're... So, it- it- it... That... You're describing a feeling, but that place you go to in the imagination-

   11. EFEdward Frenkel1:01:35

       Right.

   12. LFLex Fridman1:01:36

       ... it's- it's bigger than the real world.

   13. EFEdward Frenkel1:01:39

       So there is a big conundrum as to whether mathematics is invented or discovered.

   14. LFLex Fridman1:01:43

       Mm-hmm.

   15. EFEdward Frenkel1:01:45

       And, uh, mathematicians are divided on this. Nobody knows.

   16. LFLex Fridman1:01:50

       What do you bet your money on, financially?

   17. EFEdward Frenkel1:01:53

       (laughs)

   18. LFLex Fridman1:01:53

       Financ- advice my... Investment advice.

   19. EFEdward Frenkel1:01:56

       So let me tell you something.

   20. LFLex Fridman1:01:57

       Yeah.

   21. EFEdward Frenkel1:01:57

       I- My views ha- have evolved.

   22. LFLex Fridman1:01:59

       Mm-hmm.

   23. EFEdward Frenkel1:01:59

       Okay? When I wrote Love and Math, when I wrote my book, I was squarely on the side of mathematics is discovered. What does it mean? Usually mathematicians or others who have this, you know, idea or b- belief are called Platonists-

   24. LFLex Fridman1:02:17

       Mm-hmm.

   25. EFEdward Frenkel1:02:18

       ... in- in honor of the great philosopher Plato, who talked about these absolute perfect forms. So f- for me, you know, about 10 years ago, the ma- the world of mathematics was this world of pure forms, this beautiful pure forms, which exi- 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, uh, the process of mathematical discovery. How nice, huh? How neat. Very neat. (laughs)

   26. LFLex Fridman1:02:51

       (laughs)

   27. EFEdward Frenkel1:02:52

       It paints a 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?

   28. LFLex Fridman1:03:02

       And the-

   29. EFEdward Frenkel1:03:02

       Which I now recognize.

   30. LFLex Fridman1:03:04

       And the divine world of- of forms is stable, reliable-
8. 1:08:49 – 1:16:44

   Pythagoreanism

   1. EFEdward Frenkel1:08:49

   2. LFLex Fridman1:08:49

      It's also interesting that people from long ago are able to predict certain things. It's, um... And it's almost like from long ago, and you've talked about this with, uh, wi- with, uh, uh, uh, Pythagoras, um... That it seems that they s-, they had a, a deep sense of truth-

   3. EFEdward Frenkel1:09:11

      Mm-hmm. That's right.

   4. LFLex Fridman1:09:12

      ... that sort of permeates all of this, even, even now.

   5. EFEdward Frenkel1:09:14

      Right.

   6. LFLex Fridman1:09:14

      So it's not just a, a linear trajectory of an expanding knowledge-

   7. EFEdward Frenkel1:09:19

      Right.

   8. LFLex Fridman1:09:19

      ... there's a deep truth that permeates the whole thing.

   9. EFEdward Frenkel1:09:22

      Yes. So that's how I see it. Actually, I, you know, I gave a talk about Pythagoras and Pythagoreans just a, a few weeks ago at the Common Wealth 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, I learned that I actually totally misunderstood Pythagoras-

   10. LFLex Fridman1:09:41

       Mm-hmm.

   11. EFEdward Frenkel1:09:41

       ... 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 Pytha- Pythagoreans were instrumental in introducing, uh, the tuning system, uh, for the musical scale, the, the, the famous, um, perfect fifth-

   12. LFLex Fridman1:10:05

       Mm-hmm.

   13. EFEdward Frenkel1:10:06

       ... three halves of the, uh, for the, for the, for the G, for the sol, uh, uh, compared to the frequency of, of do or C, you know. And so, but actually they were much more f- interesting. So for them, numbers were not just clerical devices, you know, that kind of thing that you would use in accounting only. Uh, 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. I c-, I look at numbers, and I don't really see that-

   14. LFLex Fridman1:10:40

       The divine.

   15. EFEdward Frenkel1:10:41

       ... the divine, that they clearly did. And so the, why else, you know, how else would you explain? So the, in e- 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 it this way. They were able to divine, okay, strike that, to intuit- (laughs)

   16. LFLex Fridman1:11:08

       (laughs)

   17. EFEdward Frenkel1:11:08

       ... to intuit-

   18. LFLex Fridman1:11:09

       Yeah.

   19. EFEdward Frenkel1:11:09

       ... 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. Uh, they did not quite have the, the, uh, Copernicus model with the Sun in the middle. They had what they called the central fire in the middle-

   20. LFLex Fridman1:11:36

       Mm-hmm.

   21. EFEdward Frenkel1:11:36

       ... and all the planets and the Sun were revolving around-

   22. LFLex Fridman1:11:40

       Oh.

   23. EFEdward Frenkel1:11:40

       ... around the central fire or hearth. They called it hearth. So, but still, what a departure from the dogma, from the knowledge-

   24. LFLex Fridman1:11:49

       Mm-hmm.

   25. EFEdward Frenkel1:11:49

       ... of the era that the Earth was th-, at the center. So, how could they come up with this idea? The reason was, in my opinion, that for them, the mo- the mo- mos-, movement of celestial bodies was like music.

   26. LFLex Fridman1:12:04

       Mm-hmm.

   27. EFEdward Frenkel1:12:04

       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, as well as the Earth and other celestial bodies, were moving in harmony, like different notes of different instruments in a symphony. And so they applied this same reasoning to the, you know, the cosmological model as they applied 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. In other words, the, they always, they always knew that there is more an-

   28. LFLex Fridman1:12:57

       Right.

   29. EFEdward Frenkel1:12:58

       ... and they knew also that every s-, every pattern that you detect is finite, but the world is infinite. They actually accepted infinity. They believed that infinity 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, um, 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 al- always knew that it, 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 ran, we took their numbers, and their, like-

   30. LFLex Fridman1:13:45

       Mm-hmm.
9. 1:16:44 – 1:29:22

   AI and love

   1. EFEdward Frenkel1:16:44

   2. LFLex Fridman1:16:44

      I return to that question we had about GPT and the language models.

   3. EFEdward Frenkel1:16:48

      Mm-hmm.

   4. LFLex Fridman1:16:49

      And I think about this quite a lot, which is when, uh, the model, and we know what it is trained on, we know the parameters, we know all the different hacks that are involved in the training process and the fine tuning process, and the final results, whether it's GPT-5, -6, or -7, 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.

   5. EFEdward Frenkel1:17:25

      Mm.

   6. LFLex Fridman1:17:25

      And not only that, the model will say it is deeply in love with us.

   7. EFEdward Frenkel1:17:31

      Mm-hmm.

   8. LFLex Fridman1:17:32

      And who are we to say it is not? I think there's, um... it's the same imperative that you described in the scientific mind that wants to throw away the subjective.

   9. EFEdward Frenkel1:17:46

      Mm-hmm.

   10. LFLex Fridman1:17:46

       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-

   11. EFEdward Frenkel1:18:02

       Right.

   12. LFLex Fridman1:18:02

       ... it loves you.

   13. EFEdward Frenkel1:18:04

       I am, I'm with you. I would also say that you could try to, you could, for instance, say that, um, the origin of that is the, you know, romantic n- novels that were fed to it, for instance. (laughs)

   14. LFLex Fridman1:18:18

       (laughs) Yes.

   15. EFEdward Frenkel1:18:20

       However, you could also then, you can retort, "But what if my, what I consider my subjective unique feelings are also-"

   16. LFLex Fridman1:18:28

       Novels you were fed.

   17. EFEdward Frenkel1:18:29

       ... the reverber- reverberations of the novels I have read, because I have learned... or movies I have seen.

   18. LFLex Fridman1:18:33

       Yeah.

   19. EFEdward Frenkel1:18:34

       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, um... 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 r- has an important role. For someone, uh, that type of interaction would be, would be helpful, would be consoling, would, would feel, would, you know, ma- 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, uh, in principle, there is no one to, to judge this. This is, this is where subjective is paramount. But remember, um, 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.

   20. LFLex Fridman1:19:41

       The romantic letters, yeah.

   21. EFEdward Frenkel1:19:42

       Kind of romantic letter for other people.

   22. LFLex Fridman1:19:44

       Yeah.

   23. EFEdward Frenkel1:19:44

       But he's n- he doesn't have a partner.

   24. LFLex Fridman1:19:46

       Mm-hmm.

   25. EFEdward Frenkel1:19:46

       He's lonely. And then he gets this sort of a Siri, uh, kind of enhanced version of Siri with the voice (laughs) of, of, of, of Scarlett Johansson, which is very sexy voice, you know? (laughs) Obviously she's a great actress. So, and then at first it looks like a fantastic arrangement. He, uh, he confides in her. He, she, she f- uh, she tells him things. He is, uh, she makes him happy and so on, until he finds out that she has a relationship, quote-unquote, if you can call it that-

   26. LFLex Fridman1:20:18

       Yeah.

   27. EFEdward Frenkel1:20:18

       ... with 10,000 other people.

   28. LFLex Fridman1:20:20

       Not two others, not three others-

   29. EFEdward Frenkel1:20:22

       Yeah, like 10,000.

   30. LFLex Fridman1:20:22

       ... but thousands.

Episode duration: 3:46:38