Into the Impossible With Brian Keating - Love & Math: Edward Frenkel (#281)

Episode Date: December 21, 2022

Edward Frenkel’s latest book Love and Math, a New York Times bestseller, was named one of the Best Books of the year by both Amazon and iBooks, and won the Euler Book Prize from the Mathematical Association of America. The book reveals a side of math seldom seen, suffused with all the beauty and elegance of a work of art. Mathematics, he writes, directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars. Love and Math is also about accessing a new way of thinking, which empowers us to better understand the world and our place in it. It is an invitation to discover the hidden magic universe of mathematics. Edward Frenkel is Russian born and overcame a discriminatory educational system to become one of the twenty-first century’s leading mathematicians. He is a professor of mathematics at the University of California, Berkeley, which he joined in 1997 after being on the faculty at Harvard University. He is a member of the American Academy of Arts and Sciences, a Fellow of the American Mathematical Society, and the winner of the Hermann Weyl Prize in mathematical physics. Frenkel has authored 3 books and over 90 scholarly articles in academic journals and is an electronic music aficionado. Frenkel’s research is 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. twitter.com/edfrenkel www.edwardfrenkel.com www.youtube.com/@edfrenkel Connect with Professor Keating: 🏄‍♂️ Twitter: https://twitter.com/DrBrianKeating 📸 Instagram: https://instagram.com/DrBrianKeating  🔔 Subscribe https://www.youtube.com/DrBrianKeating?sub_confirmation=1 📝 Join my mailing list; just click here http://briankeating.com/list ✍️ Detailed Blog posts here: https://briankeating.com/blog.php 🎙️ Listen on audio-only platforms: https://briankeating.com/podcast Subscribe to the Jordan Harbinger Show for amazing content from Apple’s best podcast of 2018! https://www.jordanharbinger.com/podcasts  Can you do me a favor? Please leave a rating and review of my Podcast:  🎧 On Apple devices, click here, https://apple.co/39UaHlB scroll down to the ratings and leave a 5 star rating and review The INTO THE IMPOSSIBLE Podcast. 🎙️On Spotify it’s here: https://open.spotify.com/show/2G3PRMUhxGQkyQzLiiCqlf?si=8656119458df4555 🎧 On Audible it’s here : https://www.audible.com/pd/Into-the-Impossible-With-Brian-Keating-Podcast/B08K56PXJX?action_code=ASSGB149080119000H&share_location=pdp&shareTest=TestShar Other ways to rate here: https://briankeating.com/podcast Support the podcast on Patreon https://www.patreon.com/drbriankeating  or become a Member on YouTube- https://www.youtube.com/channel/UCmXH_moPhfkqCk6S3b9RWuw/join Learn more about your ad choices. Visit megaphone.fm/adchoices

So where we go astray is when we say there is nothing else but computation, there is nothing else but thinking and so on. The great ones knew that the hype has its reasons of which the reason knows nothing. And I think that we have to take some wisdom from that and take it seriously. And then, you know, that's the balance. And that's the balance. Welcome, friends, to another wonderful episode of the Into the Impossible podcast featuring yours truly Brian Keating, the Chancellor's Professor of Physics at UC San Diego, and also the associate director of the Arthur C. Clark Center for Human Imagination,

and today we take a deep dive into the imagination with none other than friend and colleague in the University of California, Professor Edward Frankel of UC Berkeley. He is a renowned mathematician, he is a filmmaker, and he is an author of one of the New York Times most popular books ever written on math called Love and Math, The Heart of Hidden Reality, Best Seltieth,

book reviewed so splendidly all over the place. We talk about many things ranging from so-called artificial intelligence and the challenge it provides. Will future mathematicians be artificial or natural? Talk about quantum weirdness, giants in mathematics, and of course love and math. What does it mean to him? And last but not least, his research is a mathematician. We're normally here talking with physical scientists, whether they be theoretical physicist or astronomers that observe the universe or experimentalists like me who build things. But today we're with a mathematician and not the first one. We've had on many mathematicians, Stephen Wolffron, Jim Simons, and also Stephen Stroggatz. And it's a real pleasure to talk to mathematicians whenever I can.

My father was a mathematician and it's kind of baked into my blood, although I don't have the aptitude of an Ed Frankel. Nevertheless, it's quite fascinating to hear about his approach to the so-called Lang Landon's program. We'll find out what that means, why it was so a and what he did to help solve it. In the meantime, I ask you to subscribe to the YouTube channel in which I broadcast these videos. That's Dr. Brian Keating. You can also find me and Ed on Twitter. I'm Dr. Brian Keating.

He is Ed Frankel. And this is a real delight. And so addition to following us on Twitter, YouTube, you can also go to my mailing list, which is briankeating.com slash list, and sign up. Who knows? You may win a piece of space schmutz, some genuine and meteorite material from the early solar system before there was love or math. Here's some jiggling around in my office right now.

So that's at brightgating.com slash list and last but not least, please do subscribe and leave a rating of the podcast. Wherever you're listening to this on iTunes slash formerly known as iTunes, it's now known as Apple Podcast. Or on Spotify, you can leave ratings reviews, even audible.com. It got 11 reviews on Audible of the podcast. Or you can also find my book and S books and many, many other cool things. And I often love to read reviews.

that people leave. And some are more kind of detailed than others. But I want to get one that I received recently from Jayman's 88 on Apple Podcasts Awesome Learning Experience. Professor Keating, wonderful discussions with very interesting guests on a wide variety of topics.

Highly recommended. So I thank you. I hope you will continue to kind of honor me with those reviews and ratings and subscriptions and shares and all the stuff you do. It's all free. It's all good fun. And now sit back and relax.

And there's no homework. Don't worry about it. This will not be on the test. but this will test the limits of your imagination as we go deep into the heart of hidden reality with Professor Edward Frankel of UC Berkeley. Let's go. Any sufficiently advanced technology is indistinguishable from magic. It is a great pleasure to be talking with a renowned scientist, mathematician, thinker, philosopher, actor, producer, author. He wears many hats. And he's actually coming to us from where, you know, where I'd

got a start in a certain sense. My father was a math PhD at UC Berkeley and worked very closely with Jim Simons. And that's how they met and that's how the Simon's Observatory came to be. But we're not here. We're not here to talk about my favorite subject, which is me. We're here to talk with Edward Frankel. Edward, how are you today up north? Doing great, Brian. Good to be here with you. And thank you for this generous introduction. Yes, yes, it's all true. So your legend has been known to me for many, many years. So we have mutual friends like Eric Weinstein and Stefan Alexander.

And I started to become aware of you originally back in 2015, 2016, when I started to think about writing a book. And I came across your wonderful book called Love and Math, the Heart of Hidden Reality. And it was really shocking to me that a math book, even though I'm the son of a mathematician, that a math book could be so fascinating, and so touching and personal. So I want to do the thing you're never, ever supposed to do, which is to judge a book by its cover. That's verboten. You're not supposed to do that. But we're going to do it.

So I want you to explain, hold up the book, explain the title, the subtitle, and what that beautiful image is meant to represent. Well, you know, I can take credit for the title, but I cannot take credit for the cover. for the cover art. It was really a stroke of genius by my publisher basic books. So the cover depicts a fragment of the famous painting by Van Gogh, Steynard. And they chose it. And I was just like, wow, this is great. Now, this was not totally random, because in the book, I do say that Van Gog was my favorite painter when I was a kid, So I was exposed to his art early on. I remember my mother took me to a museum,

the Pushkin Museum, art museum in Moscow when I was maybe 10 or 11 years old. And I was absolutely fascinated with Van Gogh's paintings. They had a very good collection of Impressionists in general. But somehow I was just drawn to Van Gogh. I was just like standing in front of those paintings and just totally mesmerized, you know. And so that affinity, that love for Van Gogh kind of, continued. I discovered other artists and so on, but it's still up there for me, one of my absolute favorite artists. So I guess maybe the artist who designed the cover for the book

read that passage or something. Also, it's about, what is it about this story night? It's about sort of looking at the world, at the universe and appreciating its beauty and asking questions, where does this light come from? What does it mean? What does it all mean? And I think all of us have a memory of an experience of that nature when we were as kids, maybe when the air was not as polluted,

especially for people of our generation. Speak for yourself up north up there. Crystal clear down here. Oh, really? Well, good for you. But, you know, I remember with the kid, we went on a trip, you know, like a hiking, like, you know, with backpacking trip was my with my classmates.

I must have been 14 or something. I remember we were somewhere in a village in the middle of Russia and I saw that the night that started night sky. Unbelievable. It was like, wow. It was just like really made an enormous impression on me. So I guess that cover maybe communicates some of that also. That sort of like unspoiled, innocence.

sense of beauty and mystery of the universe, which I guess I try to convey in my book as well. Now, the title is kind of interesting because people kept asking me, why did I choose this title, love and that? And I didn't know when I was writing. Sometimes, well, you're an author, so you know that sometimes you only later come to appreciate and understand what you write. Sometimes you just try the kind of, you know, people say sometimes channel or something. I don't want to go on you, but it is true in my experience that I rereading the book

years later, I was like, I wrote this. Really? I didn't. It's like, now I can see the meaning of it, which I could not possibly consciously appreciate at the time that I was writing.

So, you know, when people ask me, after the book was published, What does it mean? What does the title mean? I would make some kind of jokes, I would say, you know, love and math have a lot in common. It's easy at first, but then it gets awfully complicated, something like that.

But later on, I kind of appreciate what it means to me. It is like, it is a new iteration on the eternal theme that we have as humans. this sort of balance between yin and young, masculine and feminine, the sun and moon, Apollo and Dionysus, the left brain and the right brain, logic and intuition. So to me, love and matter present like these two pillars of humanity,

you know, on which we kind of, on which our life is based. And sort of our life, to me, is kind of unfolds in this sort of infinite spectrum between those two polarities. And in a way, the older I get, the more I kind of become interested in the question of how do we find balance between the two? How do we find in our personal lives, but also in our life as a society, you know?

So that's what this title represents to me. Wow. And you mentioned the subtitle is the heart of hidden reality, which is similar in some sense to a book by Brian Green, who's not been a guest on the podcast, but he assures me he will be on someday. So in the context of string theory, so that kind of harkened to me,

this quote by my hero, as my listeners are probably long suffering from hearing about, but Galileo, Galilei, I was actually just in his prison slash house a couple of weeks ago in Archetri, Italy. And Galileo said something beautiful. the book of nature is written in the language of mathematics. And he considered himself a natural philosopher.

That's what we used to call physicists. And I want to quote a different physicist mathematician, Eugene Vigner, who gave a lecture at New York University in 1960, called the unreasonable effectiveness of mathematics in the natural sciences. And it sparked a debate that continues to this day. How could it be that mathematical concepts purely of the human mind, and we'll talk about the great debate as math invented or discovered. But for example, he mentioned in the statistical distribution of populations,

which we have for voting and all sorts of aggregations, appears the ratio of the circle's circumference to its diameter. What the hell is that doing in there? So how do you react? Or the sum of inverse squares of forced to vintage just like one plus one over four, plus one over nine, plus one over 16, and so on. You want to do it yourself. Thumbs up to pi squared over four.

Right. Par squared over six. Over six. You're just a math. My father of blessed memory, he used to say, don't ask me to calculate a tip on a bill because I'm a mathematician. Mathematicians are the worst with the two. You know, is the worst.

So never try to divide the bill with the math. No, never do it. So what does that quote mean to you? Why is math so effective in physics and in the heart of hidden reality making it illuminated? I quote, I use this quote on page two of my book, okay? Yes. So, yes, you're asking, so you're the right question.

What does it mean? Well, there's so many layers to this, you know. But our Western society is built on this idea, which is very powerful idea of analyzing things and ordering things. And actually, I would trace it. back much earlier than Galileo to my hero Pythagoras, it was a great

great mathematician philosopher. So he said all things known are numbers, right? So there is this impulse to represent things, to analyze things, to put them as part of a system. And ultimately it means

actually representing things by numbers and representing the various connections that we observe between things as equations and as formulas and so on, right? So Galileo, so then for a while that thread, that theme,

which to me signified sort of the birth of Western civilization. We're talking about 2,500 years ago, give or take. For a while, that thread was kind of, went kind of, not silent, but kind of quiet. And it was picked up by a renaissance and people that followed, especially Galileo, and that eventually led us to Descartes and to the Industrial Revolution, and here we are, you know.

So the book of nature is written in the language of mathematics. And then he says, for those who don't, and the letters, the in it are circles and triangles and genetic figures like that, and those who, I'm paraphrasing those who are not aware of them, are bound to wander in a dark library. So in other words, it gives us tremendous capacity to innovate, to evolve, to progress as a species, what I might say, you know, as a humanity as species,

to create all this technological wonders and so on. Also brings us a lot of grief. So, you know, sometimes I say, we have built these beautiful cathedrals, but also bombs. to destroy those cathedrals. So in that sense, it's always like, you know,

in one motion, the mathematics gives and takes away. And that's where our wisdom needs to come forward and to help us to discern, to understand, to see the limitations of knowledge, also the dangers of logical reasoning run amok

when it's not supported by love, by connection, by compassion, you know. And so I think this is where we find ourselves a few centuries after Galileo, 2500 years after Pythagoras, in a way re-evaluating these words, these maxims that the instructions that they have given us.

I think we have taken them maybe sometimes a little too literally, and maybe time has come for us to show more maturity and more wisdom in appreciating that Mike's is not everything. It gives us a lot. It is the language of nature. But there are other aspects of nature that are just as important. Does it make sense?

Yeah. No, it does. And actually, it seems to me that we fit the language of nature to the needs at hand. For example, Galileo was very close to getting the inverse square law, but he couldn't quite get it. He got this law of squared times from dropping things down, but then we remember, he didn't even have clocks back. I mean, they're really, we're dealing with very primitive physical entities. And if we fast forward, you know, 400 years from Galileo to Feynman, Feynman then refines the statement and says calculus is the language God speaks. And, you know,

these are very breathless things, but, you know, great physicists, great mathematicians. but do you feel like we find the math that we need for the time? In other words, could we find, sometimes they hear, oh, string theory is the math of the 22nd century that fell into the 21st century? Well, no. Here's the first thing that comes to mind, okay, when you're asking this. And again, there's so many layers to this.

I hope we'll talk about that. Here's the first thing that comes to mind. Is don't you find fascinating how people usually speak about, so it used to be, one of the ways, If people say God is things they didn't know, but they don't understand. They say it's mysterious. That was sort of the paradigm in the years past. But what I find more often today is if people speak of what is God,

is what they have mastered, what they have understood, or what they feel they have understood. So I have understood calculus, so calculus is God. It comes from God. And that's the essence of things. That's the core essence of that. Oh, I understood things, theory. Great.

This must be the grand unified theory of everything. Because I have understood it now. So what else could be? Right. So I'm like, don't you find it a little bit strange? Yeah. What you have learned. Right.

You know what I mean? It's like, coincidence or not? Or like, should you be a little bit, should we have a little more humility to say, just because I've learned this doesn't mean that I already know. everything or I'm close to knowing everything. And in fact, if you look in history, every time somebody went out and said,

we have now mastered it. This must be the mess from God, and that's basically all there is. They are put to shame. So the famous quote from Lord Kelvin from around 19 countries, when he said, the edifice of physics is basically finished. And all these little problems.

The Michelson morely explains, about the speed of light, which of course we know led to Einstein's relativity. That's right. And the radiation of the black body, which led to quantum mechanics. Okay, just the two little things. Otherwise, we understand everything. So I am a student of history. I would like to be a student of history and say,

what it shows to me is that every time I have this impulse, and it's a very natural impulse, nothing wrong with it, to say, I got it, I should hold myself back, and I should not allow myself this type of hubris to say that that's it. That's what it is. So, yes, maybe it is part of the story. It is a part of some design or depends on your sensibility, to what extent you want to, how friendly you are with a mysterious set of speed, and how much you want to emphasize it.

But I would like to believe and hope, actually, that I don't know. Most of the things I still don't know. And in fact, why I say hope, because I think life would be boring if, in fact, everything was known. Or we were close to be to the point where everything is known. So, no, we are not close. And that's good because that means that there is so much more to discover. There's going to be so much more joy going forward of learning more things and saying Eureka all over again, you see. So that's the first thing that comes to mind.

But on the other hand, mathematics to me is inexhaustible, so sort of continuing the same line. So in other words, whereas physicists sometimes, in my opinion, no criticism intended, succumb to this idea that they are

chasing this ultimate theory, that somehow if ultimate theory is possible. And sometimes they say, okay, we'll come up eventually with a theory where all the main equations can be written on a T-shirt. It sounds like a great, you know, motivator. But I think mathematics is different.

So if you're a student of mathematics, even though my first love, as I explained in my book, was physics. I really wanted to be a theoretical physicist and study quantum physics. But due to very circumstances, I actually went in a slightly different direction. I went to mathematics. And then eventually I came back and I have collaborated with various brilliant physicists. And so many ways I'm a mathematical physicist. But my upbringing is as a mathematician. When you study as a mathematician, you never hear this from your teachers, you know, from your mentors.

We are chasing the final theory. No, because there is no such idea that there is a final theory. Mathematics is inexhaustible. It's limitless. It's infinite. So the progress in mathematics is infinite and eternal, and you always continue to discover new things. And so for each era, for each of focus, if you will. It's own mathematics. It's own emphasis, you know, on the particular area of mathematics.

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you know, kind of gratifying or beautiful things in mathematics that eventually in hindsight turnout to have applicability in physics, I can't think of anything more kind of surprising or spectacular, than the fact that we have these classical commutation relations between position and momentum in classical mechanics, in Lagrange's theory, back in the 17th hundreds. And you take that theory and you just add in the square root of negative one and a little constant called H-bar, and you get the quantum commutation relate, which don't commute, and so the Poisson bracket are the... The key point is non-comitivity, though, right? So there was the brackets that those guys wrote.

They were plason bracket. Yes. You have a commutative algebra, but you have a certain operation on the commuting quantities. There are still numbers. They're still, or numerical, they could be functions, but valued in numbers. And numbers commute. 3 times 5 is 15, and so is 5 times 3. But in fact, I think it's kind of an accident.

What happened is just because we use numbers, like whole numbers or rational numbers or real numbers so much. And they commute that it's natural for us to believe that the world is commutative. And the greatest, one of the greatest, what to say, breakthrough or sort of like discoveries in quantum physics was that the world is actually non-commutative. And as a mathematician, if you are a professional mathematician, you know that most algebraic structure are non-commutative, which means that AB is not equal to B.

And of course, I usually explain it by saying, you know, if you put socks on and then shoes, it's not the same as the other way around. Or a carriage before the horse or after the horse. These are two different things. So that's non-comitivity. And the point is at the very basic fundamental level, our universe is non-commutative. So that was the big discovery. That was a big jump, the big insight. Those guys in classical physics, they still thought that everything is commutative.

Classical physics is commutative. But there is a germ on quantitativeity, because you know that to write down the equations, in a Hamiltonian form, it is useful to have this additional structure, which is called possible post-on bracket. And like you said, you know, you can think of passing from classical physics to quantum physics, what we might as should quantizing possible bracket, where instead of commuting quantities, you suddenly now have non-commuting quantities, such as one. momentum and the coordinate.

And this non-comitativity is really at the core of all the paradoxes and all this weird and strange behavior in the subatomic world. You see. But you're right that mathematicians already anticipated in some sense, even though they could not possibly imagine that it could be taken this far. That's where you have to appreciate physicists. You have to appreciate people like Heisenberg, who actually, discovered this non-comitivity without even knowing mathematics involved, which had been

constructed, had been theorized a hundred years earlier. What I'm talking about this theory of matrices. So matrices, when you multiply them, they also don't commute. And so Heisenberg just driven by experimental results and trying to build a new theory, which would explain them. Right. And he was in this famous story. It was an island where there were no books, no mathematicians.

which may actually be a good thing. No department chairs. No department chairs, no committees to, you know, but also the point is he actually invented matrices. Yeah. How cool is that?

So there are some fundamental structures in mathematics which are inevitable and which are brought upon us, eventually by physics. Same with complex numbers, by the way, square root of negative one that you mentioned, was theorized by mathematicians, you know,

in the 16th century, but then make their, appearance in quantum mechanics, the Scheringer equation and other things. So they are also woven into the fabric of the universe. And my conjecture that you'll probably laugh at is, what if you had all these things? You had, well, I mean, as we just said,

Bamboli, I think it was, convented the complex algebra and the rules of it. He is a contemporary of Galileo. Cardana, Cardano. A Cardano, but... Gerolama Cardano. At least he wrote, I mean, there were several people who talked about,

but he wrote it in his book. Right. He introduced complex numbers there. Yeah, and then Descartes and, of course, Euler and all sorts of other people got into it. But the important thing to realize, they were contemporaries with Galileo, not with, not with Heisenberg. So what I wonder is, you know, this is my controversial proposal for the podcast. So could you look at all the mathematical structures that are known to exist and say, let's pick the Langlan's program. Say, there must be some physical instantiation of that.

I mean, you've called it a grand unified theory. We don't have to get into that. But as a little tongue in cheek. I know, I know. But still. Yes, I did. I did. Yeah, if you'll humor me, what if you looked at all the,

just put all the mathematics in front of. Good question. Yeah. So could you discover new physics that we don't know about now? Okay. Okay. So again, several things. You are, you're inspiring me to think in civil direction.

So first of all, I would say, my, you may be, will be right that this example, first example being square root of negative 1 and complex numbers being theorized by mathematicians just abstracted in order to, but not completely abstracting, there was the reason

to try to find formulas for solutions of quartic equations where this square roots naturally appear. So then they pop up in quantum physics and they are fundamental to quantum physics as we now realize. And then Quaternians and Hamilton Quaternians, then Poisson brackets and stuff like that.

And so non-contactivity and so on. So that might lead one to speculate, to conjecture, that maybe all of mathematics somehow will find its proper place in the natural world and the world around us. It is good out. May well be true. I know at the moment, I am inclined to think otherwise

simply because mathematics is just so enormous. Yeah. And so there are certain, there parts of it which seem to have no connection whatsoever. But it may well be my own limitation and my own prejudice, you see. So because immediately what I'm starting to think are things like Hilbert space, and then, oh, wait, that's an essential ingredient of quantum physics already. Okay, so it hasn't been found its place. Okay, okay, bad example.

So the added numbers. So there are these counterparts of real numbers, which are called the heidiatic numbers, which are, give us a different generalization of rational numbers, the fractions of integers. I was going to say they haven't found any place in physics. But I just remember that a couple of months ago, I was actually looking at some articles by a physicist who are trying to feed them in some models of quantum mechanics also. So they actually may well be. relevant. So I would say this is one of those cases where I would say, I don't know.

Very good question. May well be true, but maybe not. So because in the end also it's possible that Math Max is just a kind of has a certain not necessarily focused on the or pegged

to the physical reality because it is a kind of an activity of the mind, right? So in a way one could also say that it's like a event diagram so where you have a an overlap between the two, but the two subjects kind of like developed on separate paths. Right, they may not. And that was something else I was going to say, but anyways.

I mean, the other thing I want to run by you because, you know, we don't have that much time, but, but, you know, I can't resist talking about, you know, in both string theory and in inflationary cosmology, which is what I study experimentally, there's a notion of a multiverse In string theory, it's typically called a landscape where you have all these different instantiation of vacuum energy level. And then we often, and then there's the multiverse of many universes. And if you know one thing about it, typically in both scenarios, there's a claim that there'll be different laws of physics. There'll be different laws of physics in multiverse, you know, universe number is 65,012 versus 65,11, you know, whatever. Or in the string landscape, you know, where there's an infinite distribution of vacuum level.

My question to you is, could it be that in some of those universes, the laws of math are different? In other words, why should it be that just changing the vacuum energy density changes the speed of light, C, or Newton's capital G, or the electronic charge or fine structure? Why couldn't it change the ratio of the circumference to the diameter of a circle? Why couldn't it change modus tollens? could you see a scenario where a multiverse of mathematical universe, not in Max Tagmark's conception, but really that you'd have different versions of mathematics in different universes. And maybe we could use that to rule out such a fantasy. Well, I think, again, I don't know.

But I'm inclined to think that that's not possible. That mathematics is universal in some sense, so that even if there are many universes, In fact, I used to do this thing. I would sometimes start give talks, and I would talk about this universality of mathematics, and I say, what if we meet aliens, which I guess now has become much closer to the realm of possibilities. What if we meet aliens and we start talking to them?

Is it possible that they actually have different mathematics? And I like to illustrate it with one example, where people say, for instance, one way that I heard that people would articulate the possibilities that they would have different mathematics. They would say there would be the following argument. For us, for many of us, mathematics starts with numbers,

with natural numbers, one, two, three, four, five. Why? Because we see multiplicity around us. You know, we have, we see many people. We have seen many trees. You know, we have, you know, we eat strawberries and there's several strawberries,

several of them. So it's natural for us to count. But what if you have a kind of a civilization or which only has one entity, which one conscious entity, like in the movie Solaris or the book by Stanislav Lam,

Solaris, you know, where it is a planet, which is conscious. And it doesn't have any other, there are no other Solaris in its world. Right. So then it's not, it's not natural for Solaris type intelligence,

which is one, which is unique. To think in terms of numbers, so they would do different type of mathematics. So it's not even the question of mathematics contradicting two different mathematics structures contradicting each other, but just sort of developing it from a different place. And to this, and I think this is a very interesting point where actually one could articulate that the two, it's possible that they would start from a different place, but they would still meet with us. And the way I illustrate it is by saying that numbers can be discovered, Salaris intelligence could discover numbers through what mathematicians call homotopy groups. So to explain this, imagine, I wish I had a floss handy.

So imagine just taking a floss and wrapping it around your finger. Okay? So you can wrap the floss around your finger once, twice, three times, four times, and so on. This is how a finger could discover, given a floss, could discover natural numbers, even if there were no other fingers around. by wrapping things you can wrap things around several times and actually it's much better because

this wrapping in some sense does more justice to numbers because I'm an experimentalist I have some floss yeah I have it in my so you're saying a homotopy group is homotopy class is the number of windings it's number of windings

and so but actually see is much better so number one this windings you can really see that they are equal whereas if you have a bowl of strawberries no two strawberries are exactly the same. So actually, you may have a hard time convincing a child that they should be counted in the same progression, in the same process, you see.

But when you wrap things around, it does look like that. It's just winding. It's the same. And you will also realize negative numbers, not only positive numbers, because you can also wind things in the opposite direction. And I've never seen minus five strawberries, you see.

So in other words, likewise, you can also wrap a sphere, onto itself. It is harder to find a probe for that. But mathematicians have a theory of what's called homotopy groups, where it's sort of go one dimension higher than the previous

example, and you can also wrap a sphere onto itself and you can wrap it any positive number of times or any negative number of times. So an advanced Solaris intelligence would would naturally be led to this type of abstraction of wrapping its own surface onto itself

and discover numbers as a winding number, as a number counting how many times it wraps on itself. To me, this example illustrates the unity of mathematics that the same concept can arise from different fields of mathematics. In this case, from topology slash geometry and from number theory proper, where you're actually just counting things, But the concept of numbers is sort of the intersection of this, and there are many other ways to get to the natural numbers. It's like you were saying earlier about Pye, how interesting that Pye appears as a ratio of a... Well, that was Vigner. Yeah, I can't take credit. That was Vigner, so not me, Eugene Vigner. Okay, well... I'd love to take credit for Vigna.

And you can take, and you can get also Pye from formulas for voting for statistical distributions. And also, as I mentioned, although I mispronounced the answer, the sum of inverse squares is pi squared over six. You see, so pi appears in this variety of ways. And likewise, every mathematical concept. So this actually brings us to this idea of language program. So I think for me personally, I've always been interested in this idea of unity, of how you can get the same things from different parts of mathematics. And so language program is actually about that. Yeah, could you do the favor of, you know, it's kind of like having, you know, having a rock superstar, like having Mick Jagger here and say, you don't have to sing satisfaction, you know, don't worry about that.

So, come on, you're the, you're the foremost master of lines. It's funny, it's funny you, it's funny, you mentioned that because I just had a conversation about exactly this song. It was a friend of mine, with an artist. And he's saying, you know, can you imagine Mick Jagger? he's done it so many times and he's still doing it like how can you possibly do it so many times and I said but he explains to you that he can't get no satisfaction

he gets the satisfaction when he's singing this song that's the whole point that's in it that's right I said it I didn't know where it came from it was kind of an insight which I just had in that moment so you know it's kind of like this I get satisfaction by explaining what the language program is about

like Nick Jen Yeah, right. But thank you for the comparison. So, well, it's exactly like that. So it's like you get natural numbers or whole numbers in two different ways, from two different fields of mathematics. So I would call this kind of a baby version, an example, prototype, or unifying different parts of mathematics, number theory and topology. Or the topology would reference this general idea of things like winding,

and wrapping things on itself and so on. But in Langan's program, the original connection was between the number theory and what's called harmonic analysis, which is harmonic analysis, basically you can appreciate it by thinking about the sound of a symphony, playing a piece of music, as being a kind of a superposition of different notes of different instruments. So a note played by a single instrument is what we call a harmonic. So it's mathematically represented by sine or cosine wave. But they have different frequencies. And when you put them next, if you combine them, you can actually create sounds, very sophisticated sounds, which are much more spisky than the original sign and cosine functions, right?

So that's the idea of harmonic analysis. And so what Langlands imagined or what he saw in some manifestations, in situation, is that some difficult problems of number-term. where you're basically counting things, like counting solutions of some complicated equations, polynomial equations, could be recast as a problem in harmonic analysis, where the answer could be found in a much more straightforward thing. So kind of much more easily solvable problems. So that was the original formulation.

And you could say that the way I talk about it is like he established a bridge between two continents of mathematics. number theory and harmonic analysis. And there is a pattern. So it's not just a random connection, but there's a particular pattern where certain structures arise. And then similar structures also appeared in other areas of mathematics like geometry

and representation theory. But then also in quantum physics. So in quantum physics, there is this idea of electromagnetic duality where amongst all the equations in vacuum, they're just invariant and they're exchanging electric and magnetic fields, even though they have,

have very different manifestations in the physical world. And if we try to see if that holds in the quantum world, we would have to accept the existence of what called Dirac monopholes, particles which carry magnetic charge. Haven't been observed yet, as far as I know. Right. Now, I have a video about that. I'll link to it above here.

We did a video about the Valentine's Day event. In 1982, collaborator at our friend at Stanford, Blas Cabrera, claimed he detected the monopole. Because Dirac said you only need one monopole in the whole universe to explain the value of the quanta of the electric charge. Why electric charge is quantized. It's such a beautiful mathematical argument. Or Dirac was a genius and a very beautiful mathematical arguments. But so interestingly enough, electromagnetism is what's called a gauge theory. So where the group governing this gauge theory is called U1.

It's a circle group, the group of rotations of a circle. But we know that there are also gauge theories. responding to other groups, for instance, the weak interaction, weak force is described by Gage theory, in which the group is called USU2, so it's a group of two by two matrices, the matrix is determined one, and so on, and the strong interactions, SU3, and so on. And so, for instance, wondered in the 70s whether there is an analog of electromagnetic duality for those gauge theories, and they found to their astonishment that if such a duality existed, And now I have to say that these are not quite realistic models because they are what's called supersymmetric models with maximal supersymmetry in four dimensions.

So don't quite correspond to the real world, but not too far. Kind of like microformulations are very similar to the realistic models. What they discovered is that if electromagnetic duality holds for such models, then it would not be between the model and itself, but between the model and another gauge theory in which the group is another group, which actually turned out to be what? we call in Mathesian Langlan's dual group. And that's one of the biggest mysteries of the language correspondence is that the symmetry group gets replaced by this other group. And then suddenly physicists find the same phenomenon in the electromagnetic duality. So that's a kind of astonishing, you know, surprise, you know, that first of all, it sort of cuts to several things that we have discussed, the connection between mathematical discoveries and discoveries and physics,

but also this unity, there's some phenomena which appear in many different branches of mathematics and physics. And to me, that signifies that there are higher levels of understanding, higher levels. So mathematicians, you know, in 100 years or 200 years, we'll just look at things differently.

They will see, you know, in a way, what we perceive now at different fields of mathematics is just like different projections of this much more multidimensional subject. It's like, you know, if I take this cup and project it onto the floor, I will see a disc, but if I project it onto a wall, I will see a rectangle. So it's like, now that theory, manifestations of the cup would be like projection onto the floor.

And we see them in a certain way. And then, you know, manifestations in geometry or harmonic analysis or math, microphysics would be projections onto the wall. But in fact, the subject itself isn't so much richer. And when we discover, oh, it's actually a cup. It's actually a tea cup. And what we were looking at before were the projections. There is projections of it. That's why this mysterious connections, they are not longer mysterious when we see them manifested inside the cup.

Yes. Excellent. Well, before we turn to the existential questions, I have one more quote from Vigner. You said, mathematics is the science of skillful operations with concepts and rules invented just for this purpose. the principal emphasis is on the invention of concepts. Seems to me he's definitively answering the eternal question of whether or not mathematics is invented or discovered. I've asked Stephen Stroggatz. I've asked Jim Simons. I want to ask you, is mathematics invented or is it discovered? You said this place was steps from the water.

We just haven't found the steps yet. How much did we save? Enough. Enough to get lost. Or you could book a stay with Hilton. Welcome to your oceanfront room. Just steps from the water. The Hilton sale is on now.

Book on Hilton.com or the Hilton app and save up to 20% to get the stay you expected. When you want savings, not surprises. It matters where you stay. Hilton, for the stay. Yes. Well, I knew you were going to ask this. You know, I used to really think deep and hard about this.

But my understanding, my feeling, or my intuition about it, because obviously there's no proof either way. Right. Maybe there will be one day, but we don't have proof today or either way. So it's more of a speculation, it's more like intuition. My intuition of it has evolved over the years. I used to be squarely a Platonist. And when I say Platonist, in this context, it means that you believe that my micro objects live somewhere outside of space and time.

So in love and math, I have, you know, I sort of waxed romantic about this. Enchanted gardens, you know, number theory, where Maurice Galois went and brought us the flowers, you know, these Gala groups and so. So I was really seduced by this idea, which I thought was a very romantic idea, that there is something outside of this world. And Mathematicals, when you do mathematics as a professional, you get exposed to so many things, which unfortunately our math education, it doesn't let most people to see. And it really looks so incredibly dazzling and fascinating that it's, and not linked to the physical world at our current understanding, right?

It is very natural. And I would say most mathematicians are closest to platonists. They wouldn't tell you maybe directly. Maybe the whisper, if you're all friends with them. But they will not tell you directly, because it sounds like it's like a bad form to be like a mystic, you know, to say something like that. But most people believe that that is the case, that there is something weird about Math Max.

It is not all of it. It's not true that all of it comes from the physical world around us. There is something beneath the surface, okay? And I was very firmly in that camp. But what I think now is a little bit different. So what I think is that actually it's one of those cases where there is not an objective answer. It depends on how you look at it. So, for instance, and to give you an analogy, this happens in quantum physics all the time.

For instance, you ask, is an electron a particle or a wave? And the answer is, it's neither and both in some sense. In other words, if you set up an experiment in one way, it will manifest itself as a particle. If you set up an experiment in another way, and what I'm saying is something very well known and one can Google it and find out exactly what I mean, that doubles the experiment. If you set it up in one way, the electrons will behave as particles. if you set it up with a detector behind the sleeve or so on. If you set up in a different way, it will behave as a wave.

So what is it? A particle or a wave? Is this cup a disk or a rectangle? Neither. It is something else. So there is something else. It's beyond those concepts. And what we absorb is that Heisenberg said beautifully is not nature itself,

but nature exposed to our method of questioning. And that was the nature which we have put in a particular context of our experimental protocol, if you will. And this experimental protocol can be changed, and then nature will expose other sides. And sometimes the sides seem contradictory to us at this moment of evolution, where these things seem to us as contradictory in nature. But in fact, for nature, is much more mysterious than we know at the moment, in my opinion, that from the point of view of nature, higher point, which we have not yet reached,

there is no contradiction. Just like there is no contradiction in the fact that the cup projects onto a disk in one way, projects onto a rectangle and another way. So, same for me today about the question if mathematics is invented or discovered. It's like electron, if it's a particle or a wave. It depends on how you ask. It depends on from which angle you look. It is like boris complementarity principle.

The two properties seem to be complementary. But I think it's both. In other words, there are some elements in mathematics which were discovered. But if you look at it in a certain way, if you look another way, they are invented by humans. but who are humans anyway? Who are we? Who are you, Brian? Do you know who you are?

Depends who you ask. Exactly. The point is, in the year, to me, it actually leads to, and this is where I'm not being facetious, I'm actually being serious. These questions are useful. This question about, is mathematics discovered or invented and so on? They're useful, just like a question, is a human being a robot, a thinking machine? So there is nothing but a sequence of zeros and ones?

Or is a human being, collection of particles and nothing else. And I know that many of our colleagues are very brilliant ones who subscribe to these ideas. But what I'm, I'm not trying to say they're correct or they're not correct, right or wrong. But what it reveals is what they think about who they are. So all questions, ultimate, all questions of that nature. We can call it metaphysical questions. We can call it plasothical questions or whatever. They ultimately, we have to appreciate that they are, There is a review when one speaks about it. A review is that, in fact, they all lead to one question,

like all roads lead to Rome. And that question is, who am I? Who am I? If I think that I am a sequence of zero and ones, then I will believe that the world is a computer simulation on somebody's computer and so on. And then I will believe that something else about what the mathematics is discovered or invented.

And if I think of something else, then I will, so you see that I mean. So this discussion, are very important. But I think sometimes we have to also make the next step and actually ask the question directly. Which, my friend Nassim Taleb,

who by the way, speaking of the cover of my book. Oh, yeah. Black Swan himself, yes. He wrote, he gave me a beautiful blur that if you are not a mathematician, this book might make you want to become one. So Nassim

has this thing, which he calls skin in the game, okay? So it's It's all fun at games when you just talk about things abstractly, which we are used to in some ways as scientists. We pretend that we're talking about the physical world, but these electrons are weird and so on. But nothing has nothing to do with me, you know? So that's like no skin in the game. You're just, you're removing yourself from your world.

And quantum actually, to me, shows that you can. That's the whole point. That's what Heisenberg meant when he said what we are observing is not nature itself, but nature exposed to our way, method of questioning it or method of observation. You are always involved as an observer. Quantum physics proves it, okay? So that's skin in the game. So let's have skin in the game and let's ask those questions too.

And I know you're asking me now, so that's why I'm giving you the answer. And maybe on that last topic before we wrap up with the existential questions, when we hear in physics that there's only a few years left before we have artificial intelligent physicist, artificial Galileo, Galileo, A-I-O, or artificial Einstein, A-I-A-E. But I always like to point out that good old Albert Einstein, he had a very famous quote on his most famous discovery, which was that, which he called the happiest thought of his life,

which was that if an observer was in free fall, they would experience no gravitational acceleration. Equivalence, equivalence, but it's... Exactly. But I want to ask you, Edward, how can an artificial intelligent computer, A, have a happiest thought? What does that even mean? And B, how could they relate to the physical, visceral sensation of falling? It seems impossible to me.

But I want to ask you in the context of mathematics. Is there mathematics? I know that there's obviously lots of chess and so forth. But I'm actually not as interested in whether or not computers can beat humans at chess. I want to know, or shakmati, all right? But I want to know, can a computer invent a game like chess? I don't seem to feel like that's possible, but I want to know what you think. Right.

That's exactly the kind of thing I'm talking about. And that's one of the questions which leads to the question of who am I, right? So that who am I? Do I believe that everything in my life is programmed? In other words, can be ascribed to intelligence, to logical thinking, or do I believe that there are things in my life which go beyond logic? You know, bless Pascal, no less, was a very smart guy. and he was a great scientist and mathematician.

He said, the heart has its reasons of which the reason knows nothing. The heart has its reasons of which the reason knows nothing. And all the great ones, they knew about it. That's how they felt. And you could say, oh, Pascal was a long time ago,

so he was a religious man, perhaps. Einstein is on record saying the most important thing in science is a mysterious. And I'm paraphrasing, the one who doesn't see it has their eyes closed, you know. So they're not really, doing their job in some sense. Not was mentioned, Niels Bohr, Heisenberg, I have already

quoted, and so on. So this guys knew that there are other dimensions in life, other than thinking. Call it intuition, imagination, which, by the way, Einstein said, imagination is more important than knowledge and so on. Except if you go to the dentist, Edward, you know, you go to the dentist, you want him to have some knowledge or her to have some knowledge, not... But here's this creative thing on your teeth here. Here's what Edward Franklin adds to this discourse, okay? So here's a thought experiment. Let's suppose Sometimes I use it

When I'm asking Christian, so maybe I'll close with that. Let's suppose that AI today, official intelligence, whatever you call it, computers, whatever, you know, neural networks,

which by the way, oh my God, neural networks solve every problem. Really? And it's based on 19th century mathematics. What about the 20th century mathematics?

What about the 21st century mathematics? Doesn't sound a little odd that none of that is useful? Come on. But let's suppose that neural networks, AI, machine learning, whatever you even call it, today can cover 99% of all human experience. Let's suppose, ask yourself this.

Excuse me. Ask yourself this. What is more important to you, this 99% or the one remaining? Right. Yes. Ask yourself this. And, well, where I stand on this, probably is clear from what I said earlier. Yes.

To me, the most interesting part in life is that which is not captured. Right. Now, to get to that, you may actually have to do a lot of computation, but there is a moment of or there is a moment of inside. There is a moment of when you reach the peak and see the beautiful valley, which is opening in front of you, does it, is it covered by 99% or 1% and so on. So I believe that we will progress and we will maybe cover more.

And in ways which we cannot now imagine, computation, logic, reason, is one of the basic major human impulses. And wonderful, look how much great stuff we can do within. And obviously, in the professional mathematician, that's what I do for a living. Where we go straight is when we say there is nothing to life but computation, there is nothing but thinking. And so we should just do more thinking and then we will solve our problems. Well, first of all, the history of a civilization shows that's not the case. Because even if you're so thinking, maybe other people don't.

I'm not subscribing to that, number one. But is everything in your life based on thinking, really? You really always make decisions based on thinking, well, congratulations. I'm glad you do, but I have some serious doubts about it. And if you are honest with yourself, maybe you will remember a moment or two, when your emotions took over, when your intuition over the overriding your thinking and so on. So where we go astray is when we say there is nothing else but computation, there is nothing else but thinking and so on.

The great ones knew that the heart has its reasons of which the reason knows nothing. And I think that we have to take some wisdom from that and take it seriously. And then, you know, that's the balance. And that's the balance. Yeah. Between love and not. That's the balance. Between love and not.

Right there. I want to ask you one rapid question and one, and then one longer question just to finish up for a minute response. But the one single word answer, I'm going to ask you now, Edward, you ready? Love or math? choose one. And then in the next breath, I say, don't choose one over the other. Because we go astray also if we go too far on the other side. I've seen people do that too.

And I don't want to do that. You know, I have a while. You can see it on the Internet, folks. You can see a hunky mathematician named Edward Franklin. So what I am, what my practice is in some sense, or what my goal invite is to find the balance between the two. I think today I spoke more about the dangers of being imbalanced or going too far on the side of math. But I have to say that the same is true.

If you go too far on the other side, also. And sometimes, you know, we play so we can go. Sometimes we go ahead of ourselves, go too far. But we have to remember and come back. And I think that's where we are at our best as human beings, as individuals and as a society. when we are both because you know we're best when we use both hands not just one but both you see and so think of this as love and think of this is mad and so you know bond them together when we do both you know well yeah i can't think of anyone else who's more superlative at doing that and communicating

it and doing the most far-reaching research that the mind can even grasp i mean just to go through some of your research and preparation. It's just so fascinating. Edward, I hope we can meet in person. We've never met in person. We're in the same university system. Next time I come up to Berkeley, I will stop by to see you, my friend. For now, I want to thank you so much for going into the impossible. Thank you. There's a lot of fun. Any sufficiently advanced technology is indistinguishable from magic. Well, I hope you enjoyed that interview with Edward Frankel, Professor Edward Frankel, of UC Berkeley, the Bears. Bears, not those bears, not those

Chicago Bears. If you like the show, please leave a quick rating wherever you're listening to this. Spotify for 308 reviews and Apple were over 700 total. Please leave a little asterism, a small constellation of stars. It really means a lot to me. And I'll see you over at Brian Keating.com slash list to join my mailing list and Dr. Brian Keating

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