Edward Frenkel is a mathematician at UC Berkeley working on the interface of mathematics and quantum physics. He is the author of Love and Math: The Heart of Hidden Reality. Please support this podcast by checking out our sponsors:
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EPISODE LINKS:
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OUTLINE:
0:00 - Introduction
1:10 - Mathematics in the Soviet Union
11:20 - Nature of reality
22:39 - Scientific discoveries
36:00 - Observing reality
52:12 - Complex numbers
1:00:58 - Imagination
1:08:49 - Pythagoreanism
1:16:44 - AI and love
1:29:22 - Gödel's Incompleteness Theorems
1:49:48 - Beauty in mathematics
1:54:17 - Eric Weinstein
2:16:12 - Langlands Program
2:22:51 - Edward Witten
2:25:57 - String theory
2:31:25 - Theory of everything
2:40:18 - Mathematics in academia
2:45:45 - How to think
2:51:31 - Fermat's Last Theorem
3:06:22 - Eric Weinstein and Harvard
3:13:47 - Antisemitism
3:34:00 - Mortality
3:41:58 - Love

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There is a famous story about Einstein that he used to go think, think, think, and then go for a walk, and he would whistle sometimes.

So I remember the first time I heard this story, I thought, how interesting.

So what a coincidence that this came to him when he was whistling.

This is how it works in some sense, that you have to prepare for it, but then it happens when you stop thinking, actually.

So the moment of discovery is the moment when thinking stops and you kind of almost become that truth that you're seeking.

The following is a conversation with Edward Frenkel, 1 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.

To support it, please check out our sponsors in the description.

And now, dear friends, here's Edward Frankl.

You open your book, Love and Math, with a question, how does 1 become a mathematician?

There are many ways that this can happen.

So first of all, I grew up in the Soviet Union in a small town near Moscow called Kolomna.

And I was a smart kid, you know, in school, but mathematics was probably my least favorite subject.

I was, you know, a straight A student, and I could do all the problems easily, but I thought it was incredibly boring.

And since the only math I knew was what was presented at school, I thought that was it.

And I was like, what kind of boring subject is this?

So what I really liked was physics, and especially quantum physics.

So I was buying, I would go to a bookstore and buy popular books about elementary particles and atoms and things like that, and read them, devour them.

And so my dream was to become a theoretical physicist and to delve into this finer structure of the universe.

So then something happened when I was 15 years old.

It turns out that a friend of my parents was a mathematician who was a professor at the local college.

It was a small college preparing educators, teachers.

It's a provincial town, imagine it's like 117 kilometers from Moscow which would be something like 70 miles, I guess.

I like how you remember the number exactly.

Yeah, isn't it funny how we remember numbers?

So his name was Yevgeny Yevgenyevich Petrov.

And if this doesn't remind you of the great works of Russian literature, then you haven't read them.

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

But this was all real, this was all happening.

So my mom 1 day by chance met Evgeny Evgenievich and told him about me, that I was a bright kid and interested in physics.

I'm going to convert him into math." And my mom's like, nah, math, he doesn't like mathematics.

So I was like, okay, let's see what I can do.

And a bit arrogant, I would say, you know, like average teenager.

So he says to me, so I hear that you are interested in physics and elementary particles.

And he said, for example, do you know about quarks?

And I said, yes, of course I know about quarks.

Quarks are the constituents of particles like protons and neutrons, and it was 1 of the greatest discoveries in theoretical physics in the 60s that those particles were not elementary, but in fact had the smaller parts.

And he said, oh, so then you probably know representation theory of the group SU3.

So in fact, I wanted to know 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 looked like.

But How did physicists come up with these ideas?

And so I remember, like it was yesterday, so he pulls out a book, and it's kind of like a Bible, like a substantial book, and he opens it somewhere in the middle, and there I see the diagrams that I saw in popular books, but in popular books there was no explanation.

And now I see all these weird symbols and equations.

It's clear that it is explained in there.

He said, you think what they teach you at school is mathematics?

That to understand the underpinnings of physical reality, you have to understand what SU3 is.

You have to learn what are groups, what is group SU3, what are representations of SU3.

There was a coherent and beautiful, I could appreciate the beauty even though I could not understand heads and tails of it.

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

Well, in retrospect, I think what I was really craving was a deeper understanding.

And up to that point, the deepest that I could see was for those diagrams, but for that story that a proton consists of 3 quarks and a neutron consists of 3 quarks and they're called up and down and so on.

But I didn't know that there was actually underneath, beneath the surface, there was this mathematical theory.

If you can just link around it, what drew you to quantum mechanics?

Is there some romantic notion of understanding the universe.

Is it the puzzle of it or is it like the philosophical thing?

So whatever I say about Edward at 15 is colored by all my experiences that happen in the meantime, my current views and so on.

For the people who may not know you, I think your book and your presentations kind of reveal that that 15-year-old is still in there somewhere.

Yeah, I think it was a joy of discovery and the joy of going deeper into the, kind of the, to the root, to the deepest structures of universe, the secrets, the secrets.

And we may not discover all of them, we may not be able to understand, but we're going to try and go as far and as deep as we can.

I think that's what was the motivating factor in this.

Yeah, there's this mystery, there's this dark room, and there's a few of these mathematical physicists that are able to shine a flashlight briefly into there.

We'll talk about it, but it also kind of makes me sad that there's so few of your kind that have the flashlight to look into the room.

I don't think there are so few, to be honest, because I think, I find a lot of people are actually interested.

If you talk to people, you know, like some people you wouldn't expect to be interested in this, from all walks of life, from people of all kinds of professions.

I tell them I'm a mathematician, and they, oh, a mathematician, okay, so that's a separate story.

A lot of people, I think, have been traumatized by their experience in their math classes.

But then they ask me what kind of research I do, and I mentioned that I work on the interface of math and quantum physics, and their eyes light up.

It's like, oh, quantum physics, or like Einstein's relativity, or I'm really curious about it, I watch this podcast or I watch that podcast, and I've learned this, what do you think about that?

So I actually find that actually physicists are doing a great job educating the public, so to speak, in terms of popular books and videos and so on.

We are starting to catch up a little bit, have been starting last 10 years, but we are still behind.

Science is still very much something that people want to learn because that's our kind of, the best way we know to establish some sort of objective reality, whatever that might be.

Yeah, to figure out this whole puzzle, to figure out the secrets that the universe holds.

Things that we can agree on, kind of, you know?

Even though for me, at this point, I always make an argument that our physical theories always change, they get updated.

So you had Newton's theory of gravity, then Einstein's theory, you know, superseded it.

But in mathematics, it seems that theories don't change.

Pythagoras' theorem has been the same for the last 2, 500 years.

X squared plus y squared equals z squared.

We don't expect that next year, suddenly it will be z cubed.

And so that to me is actually even more, hints even more at how much we are connected to each other.

Because Pythagoras theory, if you think about it, or any other mathematical theorem, means the same thing to anyone in the world today, regardless of their cultural, you know, bringing religion, you know, ideas, ideology, gender, whatever, nationality, race, whatever, right?

And it has meant the same to everyone everywhere, and most likely will mean the same.

So that's, to me, kind of an antidote to the kind of divisiveness that we sometimes observe these days, where it seems that we can't agree on anything.

To the political complexity of 2 plus 2 equals 5 in George Orwell's

I was in the Soviet Union in 1984, and so in many ways I see that it was prescient, the novel was prescient, but we still have not found a dictator who would actually say 2 plus 2 equals 5 and would demand their citizens to repeat that.

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 2.

It's interesting to ask, what do you think is the difference between physics and mathematics and the way physics and mathematics look at the world?

There is actually an essential difference, which is that physicists are interested in describing this universe, okay?

Mathematicians are interested in describing all possible mathematical universes, of which, you know, in some of our work, I still consider myself more of a mathematician than a physicist, my first love for physics notwithstanding.

Mathematicians are, in a way, we have more diversity, if you might say.

So we are accepting, for instance, our universe has 3 spatial dimensions and 1 time dimension.

So of course there are theories where there are some hidden dimensions as well.

Well, let's just say observed dimensions.

So this tabletop has 2 dimensions because you can have 2 axis, 2 coordinate axis, X and Y, but then there is also a third 1 to describe the space of this room.

So realistic theories of physics have to be about spaces of 3 dimensions, or space-time, so 4 dimensions.

But Mathematically, we are just as interested in theories in 10-space-time dimensions, or 11, or 25, or whatever, or infinite-dimensional spaces, you know?

On the other hand, I have to give it to the physicists, we don't have the same satisfaction that they have of having their theories confirmed by an experiment.

We don't get to play with big machines like LHC in Geneva, a large hadron collider that recently discovered the Higgs boson and some other things.

For us, it's all like a mental exercise in some sense.

And That's our way of confirming, experimental confirmation, if you will.

But I think I kind of envy a little bit my friends physicists that they get to experience this sort of, these big toys, you know, and play with them.

But it does seem that sometimes, as you've spoken about, abstract mathematical concepts map to reality, and it seems to happen quite a bit.

That's right, so mathematics is, underpins physics, obviously, it's a language.

The book of nature, as Galileo famously said, is written in the language of mathematics.

And the letters in it are the circles, triangles, and squares, and those who don't know the language, I'm paraphrasing, are left to wander in a dark labyrinth.

That's a famous quote from Galileo, which is very true and has become even more true more recently in theoretical physics in the most sort of far out parts of the theoretical physics that have to do with elementary particles as well as the structure of the cosmos at the large scale.

What do you think of Max Tegmarker who wrote the book Mathematical Universe?

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

Meaning the ones that deeply understand the way the universe works.

At the core of, you know, I would say mathematics is 1 half of the core.

In other words, you can't cover everything by math.

So mathematics gives you tools, it gives you a kind of a clear vision.

But mathematics by itself is not enough for 1 to have a harmonious and balanced life.

So I am suspicious of any theory that declares that everything is mathematics.

So math can generate things that are beautiful, but it can't explain why it's beautiful.

Math, you could say, is a way to discern patterns, to find regularities in the universe, and both physical and mental universe.

Mathematics explores the mind as much as it explores the physical world around us, and it helps us to find those patterns, which makes our perception more sophisticated, our ability to perceive things such as beauty, and it sharpens our ability to see beauty, to understand beauty.

So our world becomes more complex from thinking that Earth is flat, we go to realizing that it is round, that it's the shape of the sphere, so that we can actually travel around the Earth, so there isn't a place where we hit the end, so to speak.

And then, proceeding in the same vein, then Einstein's general relativity theory tells us that our space-time is not flat either.

This is much harder to imagine, a bent three-dimensional or four-dimensional space or four-dimensional space-time, because this idea that the space around us is flat is so deeply entrenched.

And yet we know from this theory and from the experiments that have confirmed it, that a ray of light bends around a star as if being attracted by the force of gravity.

But in fact, the force of gravity is the bending.

It's just that it's not only the bending of the space, it's also the bending of space-time.

There is a curvature, not only between spatial dimensions, the way parallels and meridians come together.

In the small scale, they look like perpendicular lines, but if you zoom out, you see that the space, are they curving the space?

They are sort of the tracks along which the space gets curved.

That would be the curvature of spatial dimensions.

But in fact, now throw in time, And 1 time, imagine a sphere which has, 1 of the meridians correspond to time, and the parallels correspond to space.

I can't imagine it, but I can write mathematical formula expressing that curvature.

And that's, in fact, that curvature is responsible for the force of gravity attraction between, the sort of simplest instantiation of it, attraction between 2 planets, or between 2 human beings.

Yeah, the time, bending time, it's not very nice what that theory did to time, because it feels like the marching of time forward is fundamental to our human experience.

The arrow of time marching forward nicely seems to be the only way we can understand the universe.

There are people who claim that they can, that they possess other ways of experiencing it.

So truly can visualize messing with time.

Well, messing with time, but not necessarily messing with time, because 1 point of view is that, you know, I think, who was it?

I think William Blake, who wrote that eternity loves time production.

So 1 point of view is that it is eternity which is fundamental, where time stands still, which our mind conceptualizes as the time.

But in fact, it's not something mystical.

If you think about it, when you're really absorbed in something, time does stand still.

And then you look at the clock and it's like, oh my God, 2 hours have passed.

When you are absorbed, when you're in love, when you are passionate about something, when you're creating something, we lose ourselves and we lose the sense of time and space for that matter, you see?

So there is only that which is happening, that creative process.

So I think that this is familiar to all of us.

And we may be actually the closest to the truth at that moment, to the true natural reality.

So yes, so then there is a point of view that this is where we are, we are who we are, at our sort of fundamental, at our fundamental level.

And after that, the mind comes in and tries to conceptualize it.

It's like, oh, because I was writing something.

I was writing a book, I was painting this painting, or maybe I was watching this painting and got totally absorbed in it, or I fell in love with this person, that's what happened.

But in the moment when it's happening, you're not thinking about it.

Yeah, we construct narratives around the set of memories that seem to have happened in sequence, or at least that's the way we tell ourselves that.

And we also have a bunch of weird human things, like consciousness and the experience of free will, that we chose a set of actions as the time unrolled forward.

And we are intelligent, conscious agents taking those actions, but what if all of that is just an illusion?

And imagine that to make it really catch 22.

Imagine that our minds are set up in such a way that they can't approach the world or experience otherwise.

So in other words, to understand, to see that from a more kind of all-encompassing point of view, we have to step out of the mind.

Well, I wonder what's the more honest way to look at things.

I think we like to play with these experiences, with all the drama of it, with all the memories, with all the tribulations.

The evolutionary process somehow loves it.

Whatever this thing that's being created here on Earth, it seems to like to create, like to allow its children to play with certain truths that they hold, the subjective truths that are useful for the competition or whatever, this dance that we call life, broadly defined, not just humans.

And you know, I'm glad you mentioned that because what I find fascinating is that the greatest scientists are on record saying that when they were making their discoveries, they felt like children.

So Isaac Newton said to himself, I only appeared as a child playing on the seashore and every once in a while finding a prettier pebble or a prettier shell, whilst I think he said something like the infinite ocean of knowledge was lying before me.

Alexander Grothendieck, who probably was the greatest mathematician of the second half of the 20th century, the French mathematician, Alexander Grothendieck, wrote that discovery is a privilege of a child, the child who is not afraid to be wrong once again, to look like an idiot, you know, to try this and that, I'm paraphrasing, and go through trial and error.

That is for them, in other words, for them, that innocence of a child who is not afraid, who has not yet been told that it cannot be done, okay, that was essential to scientific pursuit, to scientific discovery.

And now also compared to Pablo Picasso, a great artist, so who said every child is an artist.

The question is how to preserve that as we grow up.

You're 1 of the most respected mathematicians in the world.

You're Berkeley, you're like, there's a statue, you're supposed to be very like, you know.

Ivory tower, Sometimes I joke, I say, I take an elevator to the top of the ivory tower every day, yeah.

And you're supposed to speak like royalty.

Do you struggle to 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?

I mean, that's part of being human, because when we grow up, I mean, 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, that spontaneity, you know, and kind of looking at the world through the eyes of a child.

But it's difficult because, you know, you go through education system, And for many of us, it's not especially helpful for maintaining that connection.

That we kind of like, we'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 in a way that doesn't fit sort of the pattern.

We get punished by the teachers, get punished by parents, and so on.

And don't get respect when you act childlike in your thinking, when you are fearless and looking like an idiot.

Nobody wants to look like an idiot, you know?

Once you start growing up, or you think you're growing up.

In the beginning, you don't even think in these terms.

And you are open to possibilities, to these infinite possibilities that this world presents to us.

So how do we, I'm not saying that education system should not be also kind of taming that a little bit.

Obviously, the goal is balance, that acquiring knowledge so that we can be more mature and more discerning, more discriminating in terms of our approach to the world, in terms of our connections to the world and people and so on.

But how do we do that while also preserving that innocence of a child?

And my guess is that there is no formula for this.

Every life, every human being is 1 particular answer to how do we find balance?

That's 1 imperfect approximation, approximate solution to the problem.

But we can look up to the great ones who have credentials, in the sense that they have shown and they have proved that they have done something that other humans appreciate, our civilization appreciates, say, Isaac Newton or Alexander Grotendieck or Pablo Picasso.

So they have established their right to speak about these matters.

And we cannot dismiss them as mere madmen.

They say, okay, well, if the same thing was said by somebody who never achieved anything in their field of endeavor, It would be easy for us to dismiss it.

But when it comes from someone like Isaac Newton, we take notice.

So I think there's something important that they teach us.

And especially today, in this age of AI, of course there's a big elephant in the room always, which is called AI, right?

And so I know that you are an expert in the subject, and we are living now in this very interesting times of new AI systems coming online pretty much every couple of weeks.

So I kind of, to me, that whole debate about what is it, what is artificial intelligence, where is it going, what should we do about it, needs an influx of this type of considerations that we've just been talking about.

That, for instance, the idea that inspiration, creativity, doesn't come from accumulation of knowledge, because obviously a child has not yet accumulated knowledge.

And yet, the great ones are on record saying that a child has a capacity to create.

And an adult credits the inner child for this capacity to create as an adult.

You see, that's kind of weird if we take the point of view that everything is computation, everything is accumulation of knowledge, that just bigger and bigger data sets, finer and finer neural networks, and then we will be able to replicate human consciousness.

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