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 of this story,
I thought, hmm, how interesting.
So what a coincidence that this came to him
when he was whistling.
But in fact, it's not.
This is how it works, in some sense,
that you have to prepare for it,
but then it happens when you stop thinking, actually.
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,
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 the Lex Friedman Podcast.
To support it, please check out our sponsors
in the description.
And now, dear friends, here's Edward Frenkel.
You open your book, Love and Math, with a question,
how does one become a mathematician?
There are many ways that this can happen.
Let me tell you how it happened to me.
So how did it happen to you?
So first of all, I grew up in the Soviet Union.
In a small town near Moscow called Kolomna.
And I was a smart kid in school,
but mathematics was probably my least favorite subject.
Not because I couldn't do it.
I was 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 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 and teachers.
It's a provincial town.
Imagine, it's like 117 kilometers from Moscow,
which would be something like 70 miles, I guess.
You do the math.
I like how you remember the number exactly.
Yeah, it's not funny how we remember numbers.
So his name was Evgeny Evgenyevich 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, with the patronymic names.
But this was all real, this was all happening.
So my mom, one day, by chance, met Evgeny Evgenyevich
and told him about me, that I was this bright kid
and interested in physics.
And he said, oh, I wanna meet him.
I'm going to convert him into math.
And my mom was like, nah, math, he doesn't like mathematics.
So they said, okay, let's see what they can do.
So I went to see him, so I'm about 15,
and a bit arrogant, I would say, like average teenager.
So he says to me, so I hear that you are interested
in physics and elementary particles.
I said, yeah, sure.
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 one
of the greatest discoveries in theoretical physics
in the 60s, that those particles were not elementary,
but in fact had the smaller parts.
And he said, oh, so then you probably know
the presentation theory of the group SU3.
I said, like, SU what?
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 look like.
But how did physicists come up with these ideas?
How were they able to theorize them?
And so I remembered, 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 them.
It's somewhere in the middle.
And there I see the diagrams that I saw in popular books.
But in popular books, there was no explanation.
And now I see all these weird symbols and equations.
It's clear that it is explained in there.
Oh my God.
He said, you think what they teach you at school
is mathematics?
It's like, no, this is real mathematics.
So I was instantly converted.
That you understand the underpinnings of physical reality,
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
were those diagrams.
But for that story that a proton consists of three quarks
and the 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.
If you can just linger on it,
what drew you to quantum mechanics?
Is there some romantic notion
of understanding the universe?
What is interesting to you?
Is it the puzzle of it
or is it like the philosophical thing?
Now I am looking back.
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 revealed that that 15-year-old
is still in there somewhere.
Well, I think it is a compliment.
Some of the joy.
He's probably still here now, yes.
Yeah, in some way.
Yeah, I think it was a joy of discovery
and the joy of going deeper into the root,
to the deepest structures of the 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.
It's interesting.
I don't think there are so few, to be honest,
because I think I find a lot of people
are actually interested.
If you talk to 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're,
mathematician, okay, so that's a separate story.
A lot of people, I think, have been traumatized
by their experience in their math classes.
We can talk about it later.
But then they ask me what kind of research I do,
and I mentioned that I work on the interface
of math and quantum physics, and their eyes light up.
It's like, oh, quantum physics, or Einstein's relativity.
Oh, I am really curious about it.
I watch this podcast, or I watch that podcast,
and I've learned this, and it's like,
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.
Mathematicians are behind.
We're starting to catch up a little bit,
have been starting last 10 years, but we're still behind.
But I think people are curious.
Science is still very much something
that people want to learn, because that's the best way
we know to establish some sort of objective reality,
whatever that might be.
To figure out this whole puzzle,
to figure out the secrets that the universe holds.
Things that we can agree on.
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 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 upbringing,
religion, 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 two plus two equals five
in George Orwell's 1984.
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 two plus two equals five,
and would demand their citizens to repeat that.
The night is still young.
Has not happened yet, okay?
Yeah, but it does feel like math and physics
are both sneaking up to a deep truth
from slightly different angles,
and you stand at the crossroads
or at the intersection of the two.
It's interesting to ask, what do you think is the difference
between physics and mathematics,
in the way physics and mathematics look at the world?
There is actually an essential difference,
which is that physicists are interested
in describing this universe, okay?
Mathematicians are interested in describing
all possible mathematical universes,
of which, 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 three spatial dimensions
and one time dimension, right?
So what I mean is that-
Allegedly.
Allegedly.
Observed, but that we can observe today, right?
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 two dimensions
because you can have two coordinate axes,
X and Y, but then there is also the third one
to describe the space of this room.
And then there's a time dimension.
So realistic theories of physics
have to be about spaces of three dimensions
or space-time, so four dimensions.
But mathematically, we are just as interested
in theories in 10 space-time dimensions
or 11 or 25 or whatever, or infinite dimensional spaces.
So that's the difference.
On the other hand, I have to give it to the physicists.
We don't have the same satisfaction that they have
of having their theories confirmed by an experiment.
We don't get to play with big machines like LHC in Geneva,
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.
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 I kind of envy a little bit
my friends' physicists, that they get to experience
these big toys 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 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 Tegmark,
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.
But at the core, is it just mathematics?
At the core of, you know,
I would say mathematics is one half of the core.
So the book is called Love and Math, okay?
So these are the two pillars in my view.
In other words, you can't cover everything by math.
So mathematics gives you tools,
it gives you kind of a clear vision.
But mathematics by itself is not enough
for one 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 he 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,
the 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.
You know, 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 shaped as a 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,
the 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 an array 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 special spatial dimensions,
the way parallels and meridians come together.
In a small scale, they look like perpendicular lines,
but if you zoom out, you see that the space,
I think, 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 one time, imagine a sphere
which has one of the meridians correspond to time
and the parallelists correspond to space.
I can't imagine it,
but I can write a mathematical formula
expressing that curvature.
And in fact, that curvature is responsible
for the force of gravity attraction
between the sort of simplest instantiation of it,
attraction between two planets
or between two human beings, for that matter.
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.
And the fact that you can start now.
Up to now, there are people who claim
that they possess other ways of experiencing it.
So it truly can visualize messing with time.
Well, messing with time,
but not necessarily messing with time
because one point of view is that,
you know, I think, who was it?
I think William Blake,
who wrote that eternity loves time production.
So one point of view is that it is eternity,
which is fundamental, where time stands still,
which our mind conceptualizes as the time.
So, but in fact, you know, it's not something mystical.
If you think about 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, two hours have passed.
And it felt like a couple of seconds.
When you are absorbed, when you're in love,
when you are passionate about something,
when you're creating something,
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, you're just there.
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
making, taking those actions.
But what if all of that is just an illusion,
and a nice narrative we tell ourselves?
That's a really difficult thing to internalize.
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.
But I think we like to play with time.
I think we like to play with these experiences,
with all the drama of it, with all the memories,
with all the tribulations.
I think that's-
We love it, we love it, otherwise we wouldn't be doing it.
I think that's, or Earth loves it.
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 is defined, not just humans.
And I'm glad you mentioned that
because what I find fascinating
is that the greatest scientists are on record
saying that when they were making their discoveries,
they felt like children.
So Isaac Newton said to myself,
I only appeared as a child playing on the seashore.
And every once in a while, finding a prettier pebble
or a prettier shell whilst, I think he said something
like the infinite ocean of knowledge was lying before me.
Alexander Grothendieck, who probably
it 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, 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, right?
So who said every child is an artist.
The question is how to preserve that as we grow up.
Do you struggle with that?
You're one 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 towers.
Yeah.
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?
Absolutely.
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 could maintain 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're 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,
we get punished by parents, and so on.
And don't get respect when you act childlike
in your thinking, when you are fearless
and looking like an idiot.
That's right.
Because there's a hierarchy in society.
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 of,
you don't think in these terms.
You just play, you're just playing,
and you are open to possibilities,
to these infinite possibilities
that this world presents to us.
So how do we, I'm not saying that education system
should not be also kind of taming that a little bit.
Obviously, the goal is balance,
that acquiring knowledge so that we can be more mature
and more discerning, more discriminating
in terms of our approach to the world,
in terms of our connections to the world
and people and so on.
But how 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.
That's one 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 this 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.
If we take that point of view,
then what I just said kind of doesn't fit
because obviously a child has not been fed
any training data as far as we know.
Yet they're perfectly capable of distinguishing
between cats and dogs, for instance, and stuff like that.
But much more than that, they're also capable of that
wide-eyed sort of perspective.
So can it really be captured, that perspective,
that sense of awe, can it really be captured
by computation alone?
I don't know the answer.
So I'm not sort of trying to present
a particular point of view.
I'm just trying to question any theory
that starts out by saying life is this
or consciousness is this.
Because when you look more closely,
you recognize that there are some other things at play
which do not quite fit the narrative.
And it's hard to know where they come from.
It's also possible that the evolutionary process
that's created is the very, it is computation.
And the child is actually not a blank slate,
but the result of one of the most incredible
several billion year old computations
that had explored all kinds of aspect of life on Earth.
Of war and love and terror and ambition
and violence and invention, all of that,
from the bacteria to today.
So that young child is not a blank slate.
They're actually hold within them the knowledge
of several billions of years.
Right, the question is whether as a child
you carry that in the form of the kind
of computational algorithms that we are aware today.
You see, what strikes me as unlikely is that,
how should I put it?
How interesting that, you are a computer scientist
and there are other people,
I have studied computer science, so I know a little bit.
And so it's tempting to say, oh, the whole world
is computer science or is based,
can be explained by computer science.
Why?
Because it makes me feel good.
Because I have mastered it, I have learned it.
My ego is very happy and people come to me
and they look up to me and they revere me.
Kind of like priests in old days
when religion was paramount,
when you would tend to explain things
in theological, religious terms.
Today, science has progressed.
There are fewer people who kind of buy
into official religion.
So we have this urge, I suppose, to explain
and to know and to dissect and to analyze
and to conceptualize, which is a wonderful quality
that we have and we should definitely pursue that.
But I find it a little bit unlikely
that the universe is just exactly what I have learned
and not something that I don't know.
You see?
Well, there's a lot of interesting aspects
to the current large language models
that one perspective of it, I think,
speaks to the love and math that you talk to,
which is they're trained on human data from the internet.
So at its best, a large language model, like GPT-4,
captures the magic of the human condition
on its full display, its full complexity.
And since it's mimicking, it's trying to compress
all the weirdness of humans, of all the debates
and discussions, the perspectives,
all the different ways that people approach
solving different problems, all of that compressed.
So we live, we're each individual ants.
We only have like, we have a family,
we interact with a few little ants.
And here comes AI that's able to summarize,
like a TLDR, report of humanity.
And that's the beauty of it.
So I embrace it.
What I wonder- I'm very impressed by it.
I wonder if it can be very impressive,
meaning way more impressive in being able to fake
or simulate or emulate a human.
Fake, I'm glad you mentioned that
because that's just, it seems to be the mantra.
It's just worth it.
Fake it till you make it, isn't it?
Isn't that what we all do though?
No, well, yes, we do that.
But we also do other things.
We can be truly in love.
We can be truly inspired when it is not fake.
I do believe, call me romantic, okay?
But I do believe, and this is a very good,
I'm glad you're putting it in these terms
because I've had conversations like that,
that yeah, fake it till you make it.
But that's like, that's what humans do.
Yes, we do that, but not all the time.
So, and that is debatable because also I speak
from my own experience.
And that's where the first person perspective comes in,
the subjective view.
I cannot prove to you, for instance, or anyone else
that there are certain moments in my life
where I am genuine, I am pure, so to speak,
when it's not faking it.
But I do have a tremendous certainty of it.
And that's a subjective certainty.
Now, I am, as a scientist, I'm also trained
to give more credibility to objective arguments
that there are things that can be reproduced,
things that I can demonstrate, that I can show.
But as I get older, so we say,
as I get more mature, so hopefully, you know,
I'm starting to question why I am not giving
as much credibility to my subjective understanding
of the world, the kind of the first person perspective,
when actually modern science has already sold on that.
You know, quantum mechanics has shown unambiguously
that the observer is always involved in the observation.
Likewise, Gödel's incompleteness theorems, to me,
show how essential is the observer
of a mathematical theory.
For one thing, that's the one who chooses the axioms.
And we can talk about this in more detail.
Likewise, Einstein's relativity,
where time is relative to the observer, for instance.
That's brilliant.
You're just describing all of these different scales,
the observer, what the observer means.
Science of 19th century, from modern perspective,
and I don't want to offend anybody,
had the delusion that somehow you could analyze the world
being completely detached from it.
We now know, after the landmark achievements
of the first half of the 20th century,
that this is nonsense.
That is simply not true.
And this has been experimentally proved time and time again.
So to me, I'm thinking maybe it's a hint
that I should take my first person perspective seriously
as well, and not just rely on kind of objective phenomena,
things that can be proved in a traditional,
sort of objective way, by setting up an experiment
that can be repeated many times.
Maybe I fall in love, the deepest love of my life perhaps,
perhaps hasn't happened yet.
Perhaps I will fall in love, but it's unique.
It's a unique event.
You can't reproduce it necessarily, you see.
So in that sense, you see how these things
are closely connected.
I think that if we are declaring from the outset
that all there is to life is computation
in the form of neural networks or something like this,
however sophisticated they might be,
I think we are from the outset denying to ourselves
the possibility that yes, there is a side of me
which is not faking it.
Yes, there is a side of me which cannot be captured
by logic and reason.
And you know what another great scientist said,
bless Pascal, he said, the heart has its reasons
of which the reason knows nothing.
And then he also said, the last step of reason
is to grasp that there are infinitely many things
beyond reason.
How interesting.
This was not a theologian.
This was not a priest.
This was not a spiritual guru.
It was a hardcore scientist who actually developed,
I think, one of the very first calculators.
How interesting that this guy also was able
to impart on us that wisdom.
Now, you can always say that's not the case.
But why should we from the outset exclude this possibility
that there is something to what he was saying?
That is my question.
I'm not taking sides.
What I'm trying to do is to shake a little bit the debate
because most mathematicians that I know
and computer scientists even more so,
they're kind of already sold on this.
It reminds me of this famous Lord Kelvin's quote
from the end of 19th century.
There's some debate whether he actually said that,
but never let a good story stand in the way of truth.
He said, physics is basically finished.
All that remains is more precise measurement.
So I find a lot of my colleagues are happy to say,
yeah, everything's finished.
We already got it.
Maybe a little tweaks in our large language models.
So now here's my question.
I'm kind of playing devil's advocate a little bit
because I don't see the other side,
so quote unquote, represented that much.
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 being such.
It's confirmation bias, if you will.
This also reminds me, I think a good analogy
is a friend of mine, Philippe Koshin told me
that in France, there is this literary movement
which is called Oulipo, O-U-L-I-P-O.
And it's a bunch of writers and mathematicians
who create works of literature
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 Georges Perec,
which is a 300-page novel in French
which never uses the letter E,
which is the most widely used letter
of the French language.
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 experiment and I applaud it.
But it's one thing to do that
and to kind of show his gamesmanship, if you will,
and his proclivity and his ability as a writer.
But it's another thing, if at the end of writing this book,
when you finish the book,
he would say letter E actually doesn't exist.
And he tried to convince us that in fact,
French language does not have that letter
simply because he was able to go so far without using it.
So self-imposed limitation, that's how I see it.
And I wonder why we should do that.
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, it's a personal question for me
because I am addicted to knowledge myself.
Hi, my name is Edward.
And I'm a knowledge addict, okay?
I'm being serious, I'm not being facetious.
Up until very recently, maybe a couple of years ago,
I simply did not feel comfortable
if I could not say, give an answer, explanation.
It's like, oh, there has to be some explanation.
And I tried to frantically search for it.
Just for somebody like me and her left brainiac,
and that's kind of typical for a scientist.
For a mathematician.
It is incredibly hard just to allow the possibility
that it's a mystery and not to feel the urge
to get the answer.
It is incredibly hard, but it's possible.
And it is liberating, it's recovering,
it's recovering, I think, to knowledge.
Let me say what you gain from it.
For instance, I understand the value of paradoxes.
I appreciate paradoxes more.
And to use another philosopher, Søren Kierkegaard,
the Danish philosopher said,
I think or without paradox is like a lover without passion.
A paltry mediocrity.
Ooh, that's a good line.
Right?
And you know, Niels Bohr said in Simmervane,
the great Danish also.
Something about Danes.
I think it all started with Hamlet.
He said, the opposite of a simple truth is a falsity,
but the opposite of a great truth is another great truth.
In other words, things are not black and white, you know?
They're not, and I would even venture to say
the most interesting things in life are like that.
The ones which are ambiguous is an electron,
a particle, or a wave.
It depends how you set up an experiment.
It will reveal itself as this or that,
depending on how you set up an experiment.
This bottle, if you project it down onto the table,
you will see more or less a square.
If you project it onto a wall,
you will see a different shape.
A naive question would be, is it this or that?
Because we understand that it's neither.
But both projections reveal something.
They reveal different sides of it.
A paradox is like that.
It's only paradoxical if we are confined
in a particular vision,
if we are wedded to a particular point of view.
It's a harbinger, if you will,
of a possibility of seeing things in a more,
as they are, more sophisticated than we thought before.
That's such a difficult idea for science to grapple with.
I don't know how there's so many ways to describe this,
but you could say maybe that the subjective experience
of the world from an observer is actually fundamental.
But we know that.
Our best physical theories tell us that unambiguously.
In quantum mechanics, actually,
Heisenberg, I think, captured it the best
when he said, what we observe is not reality itself,
but reality subjected to our method of questioning.
When I talk about electrons, for instance,
so that there is a very specific way
in which this is realized.
There is a so-called double slit experiment.
So for those who don't know,
you have a screen and you have an emitter
from which you shoot electrons.
And in between, you put another screen
which has two vertical slits parallel to each other.
If we were shooting tennis balls,
each ball would go through one slit or another
and then hit the screen behind this or that slit.
So you would have, let's say they're colored, they're painted.
So there'll be bumps or spots of paint behind this or that.
But that's not what happens when we shoot electrons.
We see an interference pattern
as if we were actually sending a wave
so that each electron, it seems like each electron
goes through both slits at once
and then has the audacity to interfere with itself
where at some points, two crests would amplify
and at some points, a crest and a trough
would cancel each other.
Yet, so that suggests, okay, so the electron is a wave.
Not so fast because if you put a detector
behind one of the slits and you say,
I'm going to capture you.
I'm going to find out which slit you went through.
The pattern will change and it will look like the particles.
So that's a very concrete realization of the idea
that depending on how we set up an experiment,
we will see different results.
And the problem is that our psyche, I feel,
kind of is lagging behind,
in part because maybe our scientists
are not doing such a great job.
So I take responsibility for this,
why haven't I explained this properly?
I tried in a bunch of talks and so on,
so now I'm talking about this again.
Our psyche kind of is lagging behind.
Even though our science has progressed so much
from the certainty and the determinism and all of that,
of the 19th century,
our psyche is somehow still attached to those ideas,
the ideas of causality or this naive determinism
that the world is a bunch of billiard balls
beating each other, driven by some blind forces.
That's not at all like it is.
And we've known this for about a hundred years, at least.
And you call this self-imposed limitation.
It is a self-imposed limitation when we pretend that,
for instance, that these naive ideas of 19th century physics
are still valid and then start applying them to our lives
and then also derive conclusions from it.
For instance, people say, there is no free will.
Why?
Oh, because the world is just a bunch of billiard balls.
Where is the free will?
But excuse me, didn't you get the memo
that this has been debunked thoroughly
by the so-called quantum mechanics,
which is our best scientific theory?
This is not some kind of bullshit
or some kind of concoction of a madman.
This is our scientific theory,
which has been confirmed by experiment.
So we should pay attention to that.
But of course, it's not just self-imposed limitation.
Unfortunately, in this case,
there is a big issue of education.
So a lot of people are not aware of it
through no fault of their own
because they were never properly taught that
because our system is broken, education system is broken,
especially in math.
And then our, so where do we get information?
You get information from our scientists
who actually write popular books and so on,
which is a great thing that they do.
But a lot of scientists somehow,
when it comes to explaining the laws of physics,
they're doing a fantastic job
talking about this phenomenon, for instance,
double slit experiment and things like that.
But then, interviewed by Science Magazine
about free will and so on,
they revert back to 19th century physics
as if those developments actually never happened.
So to me, this is single most important sort of issue
in our popular science.
The idea that somehow there is this world out there,
but it has nothing to do with me.
So I can revel in the intricacies of these particles
and their interactions, but completely ignore
what implications this has for my own relationship
to physical reality, to my own life.
Because it's kind of scary, I guess.
But also, what are the tools with which we can
talk about the observer, the subjective view on reality?
What are the tools with which we could talk about,
rigorously talk about free will and consciousness?
What are the tools of mathematics that allow that?
I don't think we have those tools.
Because we haven't been taught properly.
So actually, tools are there.
For instance, I think, well, here we have to,
I have to say, my conviction is that everybody knows.
In the heart of hearts, everybody knows that there is that.
There is something in a football.
There is something mysterious.
And in fact, you know, somehow, immediately,
I feel that, you know, the impulse to quote somebody
on this, because as if my own opinion doesn't count.
Yeah, there's a long dead expert that has said it.
Even Einstein said that, you know, so like,
how, see, look at me.
I am supposedly like this smart, intelligent person.
I am afraid to say it and own it myself.
I have to find confirmation.
I have to find an authority who agrees with me.
And in fact, it's not so difficult to find,
because Albert Einstein literally said
the most important thing in life is the mysterious.
Okay, he actually said that.
There are some quotes which are attributed to him,
which he never said, but this he did.
I investigated, okay?
So, but more importantly, you know,
how do you feel about it?
I think that everybody knows.
But in other words, he also said Einstein,
imagination is more important than knowledge, okay?
And he explained, for knowledge is always limited.
Whereas imagination embraces the entire world,
giving birth to evolution.
It is strictly speaking a real factor
in scientific research, he says.
And he says, I am enough of an artist
to follow my intuition and imagination.
That's Albert Einstein again.
So, and I feel the same way, to be honest.
If I think about my own mathematical research,
it's never linear.
It's never like, give me more data,
give me more data, give me more data, boom.
The glass is full, and then I come up with a discovery.
No, it's always felt as a jump, as a leap.
And I have actually been studying various examples
in history of mathematics of some fundamental discoveries,
like discovery of complex numbers,
like square root of negative one.
I wonder if a large language model
could actually ever come up with the idea
that square root of negative one
is something that is essential or meaningful.
Because if all the information that you get,
that all the knowledge that had been accumulated
up to that point, tells you that you cannot
have a square root of a negative number, why?
Because if you had such a square root,
we know that if 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 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,
astronomer, astrologer, he made money apparently
by compiling astrological sort of readings
for the elite of his era.
This is 16th century, as one does, a gambler.
All around interesting guy.
I'm sure we would have an interesting conversation with him.
Girolamo Cardano.
He also invented what's called Cardan Shaft,
so which is an essential component of a car.
Cardano Vival, we say in Russian.
So he wrote a book, which is called Ars Magna,
which is a great art of algebra.
And he was writing solutions
for the cubic and quartic equations.
This is something that is familiar because it's cool.
We study solutions of quadratic equations,
equations of degree two.
So you have ax squared plus bx plus c equals zero.
And there is a formula which solves it using radicals,
using square roots.
And Cardano was trying to find a similar formula
for the cubic and quartic equations,
for which would start with x cubed or x to the power of four
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 minus 17.
And he wrote that I have to forego some mental tortures
to deal with it, but I am going to accept it
and see what happens.
And in fact, at the end of the calculation,
this weird numbers got canceled, kind of canceled out.
And the formula appeared square root of negative 17
and its negation.
So they kind of conveniently gave the right answer,
which does not involve those numbers.
So he was like, okay, what does it mean, mental tortures?
So you see, from the point of view of the thinking mind,
it is something almost unbearable.
It's almost I feel that a language model,
a computer running a large language model,
trying to do that would just explode.
And yet a human mathematician was able to find the courage
and inspiration to say, you know what, what is wrong?
Why are we so adamant that these things don't exist?
That's just our past knowledge.
It's based on what our past knowledge is
and knowledge is limited.
What if we make the next step?
Today, for us, mathematicians,
complex numbers that we call them,
are not at all mysterious.
The idea is simply that you plot real numbers,
that is to say all the whole numbers,
like zero, one, and so on, two, and so on, right?
All fractions like one half or three halves or four over three
but then also numbers like square root of two or pi,
we plot them as points on the real line.
So we draw, this is one of the kind of perennial concepts
even in our very poor math curriculum at school.
But now imagine that instead of one line,
you have one axis, you have a second axis.
And so 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 negative one.
This is a graphical, geometrical representation
of complex numbers, which is not mysterious at all.
Now, it took another 200 or 300 years
for mathematicians to figure that out.
But initially, it looked like a completely crazy idea.
So all it is, all complex number is,
is just an expansion.
Real number, two real numbers.
Yeah, it's just two.
The real part and the imaginary part.
It's just an expansion of your view
of the mathematical world.
The fact that you can add them up
by adding together the real parts
and the imaginary parts, that's easy.
But there is also a formula for the product,
for the multiplication, which uses the fact
that square root of minus one squared is minus one.
And the amazing thing is that that product,
that multiplication satisfies the same rules,
the same properties that are usual
operation of multiplication for real numbers.
For instance, there is an inverse for every non-zero number
that you can find.
Like number five has an inverse, one over five.
But one plus I also has an inverse, for instance.
That was always there in the mathematical universe,
but we humans didn't know it.
And here comes along this guy
who engages in the mental torture,
who takes a leap off the cliff of comfort,
of like mathematical comfort.
Established knowledge.
Established knowledge.
Right.
And now, obviously, for each sort of fruitful leap like that
there probably were thousands of like things
which went nowhere.
I'm not saying that every leap, you know,
it's an open shooting game.
Because, for example, you can try to do the same
with three-dimensional space.
So you have coordinates X, Y, and Z.
And you can say, oh, if it's one-dimensional,
we have a bona fide numerical system called real numbers.
If it's two-dimensional, which is like, you know,
geometrically, it's just like this tabletop
extended to infinity in all directions.
These are complex numbers,
and we can define addition and multiplication,
and they will satisfy the same properties
as real numbers that we're used to.
What about three-dimensional space?
Is it possible to also define some operation
of addition and multiplication on it
so that these operations would satisfy the properties
that we're used to?
And the answer is no.
You can define addition, but you can't define multiplication
for which there would be an inverse, for instance.
So there is something special about the plane,
the two-dimensional case.
And by the way, next question would be
what about four-dimensional?
In four-dimensional space, again, you can,
and you get what's called quaternions,
discovered by an Irish mathematician, Hamilton,
in the 19th century.
And then in the eight-dimensional,
there is something similar called actonions,
and that's about it.
So how interesting.
The structures exist in dimension one, two, four, and eight,
which are all powers of two.
Two squared is four, two to the third power is eight.
That's one of the bigger mysteries in mathematics,
why it is so.
So that's a hint.
That's a hint of what's missing in our high school curriculum
the kind of fascinating mysteries, yes.
The appreciation of the mysterious.
So in other words, yes, we resolved this one mystery
that we understood that square root of negative one
is real, is meaningful.
We build a theory to service those now,
to describe those numbers.
Did we find the theory of everything?
No, because we then invited other mysteries,
because we pulled the veil, so to speak,
or we pushed the frontier, and then new things
come get illuminated, which we couldn't see before.
That's how I see the process of discovering mathematics.
It's an endless, limitless pursuit.
Can you comment on what you think this human capability
of imagination that Einstein spoke about,
of the artist following their intuition
in this big Alice in Wonderland world of imagination,
what is it?
You visit there sometimes.
What does it feel like?
Yeah, what does it feel like?
What is it?
What is that place? It feels like playing,
but I think all of us are engaged in that kind of play.
When we do what we love, I think it always feels the same.
But it's not real, right?
So you're describing a feeling,
but that place you go to in the imagination,
it's bigger than the real world.
So there is a big conundrum
as to whether mathematics is invented or discovered,
and mathematicians are divided on this.
Nobody knows.
Where do you bet your money on financially?
The next advice, investment advice.
So let me tell you something.
My views have evolved, 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 idea,
what I believe, are called Platonists
in honor of the great philosopher Plato,
who talked about these absolute perfect forms.
So for me, about 10 years ago,
the world of mathematics was this world of pure forms,
this beautiful pure forms,
which existed outside of space and time,
but I was able to connect to it through my mind.
And as it were, kind of dive into it
and bring treasures back into this world,
into this space and time.
That's how I viewed the process of mathematical discovery.
How nice, how neat, very neat.
That's the picture.
Also makes you feel connected to something divine.
It allows you the sense of escape
from the cruelty and injustice of this world,
which I now recognize.
And the divine world of forms is stable, reliable.
There's something stable.
And in that world, everything is clear cut.
It's either true or false.
How nice, huh?
It's very nice.
The biggest illusion of all.
Allegedly.
I think now, I think now.
I understand why I liked it,
because I think that I was very dissatisfied
with what we call the real world, the world around me.
The cruelty, the injustice of it.
And I went through certain experiences as a kid,
which made me love mathematics even more
as this place where I could be safe and in control.
made you see the human world
as lesser than the mathematical world,
as more limited than the mathematical world.
Yes, yes.
And I think that,
I think that it's still missing the mark in some sense,
because in fact, what I now think,
it's a paradoxical question,
the question whether mathematics is invented to discover it,
whether there is this world of pure forms and so on,
is another paradoxical question,
which doesn't have a simple answer.
Like whether electron is a particle or a wave.
From one point of view, yes, it's true.
And just the fact that so many mathematicians today
actually subscribe to this idea
gives it a certain credibility, because that's what we feel.
We do feel that we dive into that mindscape, so to speak,
but the very structured mindscape,
where I wrote in the Love and Math
that the enchanted gardens of platonic reality,
where all this fruit grows,
it gives you this sort of romantic sense of an explorer.
And someone may be stuck in some provincial town
in Russia, for instance,
but have the sense of Magellan,
of traveling around the world.
It's just not in the world that we usually think of.
So it's one point of view,
but the other point of view is that,
yes, it is a human process.
Of course it is.
I mean, you cannot deny that.
It's human beings who have so far
discovered new mathematics.
And I do not deny the possibility that computer programs
will be able to discover new mathematics,
but so far it's been humans.
So whatever it is, whether it's discovered or invented,
it is a human activity.
Well, the possibility that paradoxes
are actually fundamental to reality
and really, really internalizing that,
that we exist in a world of not forms, but of paradoxes.
Bingo.
And so it's like what I said-
Weird world.
But if you think, it's weird,
and I agree with you as a recovering addict to knowledge,
but I am liking it more and more
because there's so much freedom in it.
And like Niels Bohr said, I quoted that earlier,
the opposite of a great truth is another great truth.
He's pointing out to this fact that,
you know, and he also said that some things
in quantum physics are so complicated,
the only way you can speak of them is in poetry.
So in other words, what is it about poetry?
What is it about art?
Why are we so drawn to that?
Why are we so captivated by those forms of,
they are not intellectual necessarily.
They are not, when you look at a painting that you like,
when you listen to music that you love,
you get lost in it, you get absorbed in it.
It can make you cry, it can make you laugh,
it can make you remember something,
it can make you feel more confident,
it can make you feel sad or happy and so on.
What is this all about?
Is it really just some play between,
some kind of like cellophone play
or some neurons hitting on each other?
Is it really that only?
Maybe.
It could be both.
I'm just worried about kids these days
that might live in a world of paradoxes.
If there's no God, everything is possible.
And yet just, they'll have a little too much fun.
And we have to put a constraint to the fun.
But have you looked at the world lately?
I haven't checked in in a while.
You think it's perfect, the way it is now?
The world without paradoxes?
The world in which we believe that every question
can be answered as yes or no, that it is this or that.
And if you disagree with me, you are my enemy?
Wouldn't that be interesting if this 21st century
is a transition into seeing the world
as a world of paradoxes?
I'm telling you, people predicted that.
The Age of Aquarius, the axis of the Earth
is rotating relative to the plane
in which the Earth goes around the Sun.
And the period of this revolution is around 2,000 years.
So there is a traditional way of measuring that
by this eras, the ages.
So the previous one is called the Age of Pisces
because of the constellation of Pisces
that it points to, so to speak.
And now, as in the famous musical Hair,
they said the Age of Aquarius is upon us.
So the different people, they did differently,
but somewhere around the time
where we are finding ourselves.
How interesting, right, is all the strife and all them?
Difficulties the world is experiencing.
This might actually be the transition to something
more harmonious, wouldn't it be nice?
It's also interesting that people from long ago
are able to predict certain things.
And it's almost like from long ago,
and you've talked about this with Pythagoras,
that it seems that they had a deep sense of truth
that sort of permeates all of this, even now.
So it's not just a linear trajectory
of an expanding knowledge.
There's a deep truth that permeates the whole thing.
Yes, so that's how I see it.
Actually, I gave a talk about Pythagoras and Pythagoreans
just a few weeks ago at the Commonwealth Club of California
in San Francisco.
And because of that, I did a kind of a deep dive
into the subject.
And I learned that I actually totally misunderstood
Pythagoras and Pythagoreans,
that they were much deeper than I thought.
Because most of us remember Pythagoras
from the Pythagoras theorem about the right triangles.
We also know that Pythagoreans were instrumental
in introducing the tuning system for the musical scale,
the famous perfect fifth,
three halves for the G, for the soul,
compared to the frequency of DAW or C.
But actually, they were much more interesting.
So for them, numbers were not just clerical devices,
not kind of thing that you would use in accounting only.
They were imbued with the divine.
And I cannot say that I think we lost it.
At least I have lost it.
I look at numbers, and I don't really see that.
The divine.
The divine, that they clearly did.
And so why else, how else would you explain?
So in other words, divine is of course is a term
which is 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,
to intuit that the planets were not revolving,
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.
They did not quite have the Copernicus model
with the Sun in the middle.
They had what they call the central fire in the middle.
And all the planets and the Sun were revolving around,
around the central fire, or hearth, they called it hearth.
So, but still, what a departure from the dogma,
from the knowledge of the era
that the Earth was at the center.
How could they come up with this idea?
The reason was, in my opinion, that for them,
the movement of celestial bodies was like music.
In fact, we call it music universalis,
or music of the spheres.
For them, the universe was this infinite symphony
in which every being, humans, animals,
as well as the Earth and other celestial bodies,
were moving in harmony,
like different notes of different instruments in a symphony.
And so they applied the same reasoning
to the cosmological model
as they applied to their model of music.
And from that perspective,
they could see things deeper than their contemporaries.
You see?
So in other words, they saw mathematics as a tool,
but that tool was not limited to itself.
They always knew that there is more.
And they knew also that every pattern that you detect
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
the world in a particular way,
which could be beneficial for you to learn more and so on.
But they never had the illusion that that was the final word,
that they always knew that it's not the whole thing.
So there is more.
There are more sophisticated patterns
that could be discovered using mathematics or otherwise.
And I think that what happened was
we kind of lost this other side of their teachings.
We took their numbers and their idea
that you could use mathematics to discern patterns
and to find regularities
and to explain things about the world.
We took that and we ran with it.
And we kind of dropped the other idea
that in fact there is another side to it,
which is kind of, to us now, we say, oh, that's mystical.
But what does it mean mystical
if it is something that helps you to make great discoveries?
And the interesting thing is that the people
who are in touch with the mystical among us
are often seen as mad, and many of them are,
most of them are, but not all of them.
But not all of them.
We mentioned Niels Bohr and Newton and Albert Einstein.
So that's where the conundrum is.
How do you find the balance between the two?
So the point I'm trying to make,
and this is what I feel,
if you ask me what I find most important thing today,
like what makes me excited and enthusiastic and passionate
is this idea of balance.
So Nietzsche had this book, wrote this book
in the 19th century called The Birth of Tragedy.
And he presented this theory,
which I think is kind of very useful,
of these two sides of a human.
One that comes from God Apollo,
and that's our left brain, so to speak,
proverbial left brain.
So it's everything has to do with,
that has to do with logic and reason and analyzing,
dissecting, conceptualizing.
And the other side, which comes from God Dionysus, Dionysus.
And that's responsible for intuition, imagination, love.
Dionysus was also God of wine.
So it's also that side of human that makes us sing
and engage in revelry and drink wine
and spend time with friends and laugh and enjoy it.
So Nietzsche advocated this point of view
that those were two complementary sides of every human
and as well as society.
And the purpose of human life
is to find balance between them.
So math is both discovered and invented.
We should be okay with having both ideas in our head
and living the balance.
More importantly, for me, Apollo is like math
and Dionysus is like love.
So Dionysus and Apollo in modern version is love and math.
I return to that question we had about GPT
and the language models.
And I think about this quite a lot,
which is when the model, and we know what it's 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 result, 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.
And not only that, the model will say
that it's deeply in love with us.
And who are we to say it is not?
I think there's, it's the same imperative
that you described in the scientific mind
that wants to throw away the subjective.
That same imperative wants to throw away the feelings
that AI might have.
And I'm very careful to not ignore
when an AI system says it's lonely, it's afraid,
it doesn't want to die, it misses you, it loves you.
I am with you.
I would also say that you could try to,
you could, for instance, say that the origin of that
is the romantic novels that were fed to it, for instance.
Yes.
However, you could also, then you can retort.
But what if my, what I consider my subjective,
unique feelings are also...
Novels, you were fed.
The reverberations of the novels I have read
because I have learned, or movies I have seen.
Because that's the purpose of movies,
what kind of to teach us how to express ourselves,
how to feel, maybe even, one could argue that.
Some people have argued that.
I agree that this is,
there is no obvious answer to this.
But see, that's exactly my point.
That is an example of something which is paradoxical,
for which there is no answer.
And that's where the subjective has an important role.
For someone, that type of interaction
would be helpful, would be consoling,
would feel, would make them happy or sad or whatever,
would kind of strike the nerve.
For some, it won't.
And I agree with you that in principle,
there is no one to judge this.
This is where subjective is paramount.
But remember, a lot of this has been anticipated
by artists.
The great movie, Her.
There you have this guy who is this lonely,
he kind of writes letters or something.
The romantic letters, yeah.
The romantic letter for other people.
But he doesn't have a partner.
He's lonely.
And then he gets this sort of enhanced version of Siri
with the voice of Scarlett Johansson,
which is a very sexy voice, you know?
Obviously, she's a great actress.
So, and then at first,
it looks like a fantastic arrangement.
He confides in her, she tells him things,
she makes him happy and so on,
until he finds out that she has a relationship,
quote, unquote, if you can call it that,
with 10,000 other people.
Not two others, not three others.
Yeah, like 10,000.
Because it has the computing capability.
So yes, definitely.
Oh, it certainly makes sense.
It's a good explanation.
And the guy is heartbroken.
But see, so here's my analysis of this, okay?
It's like a couch therapist, okay?
The guy did not have the courage to go out
in the real world and to meet a woman
and to get a girlfriend and so on.
Through no fault of his own, perhaps,
because you may have had some experiences
which made him withdrawn and closed and so on.
And a lot of us are like this.
I had periods like that myself.
Definitely can sympathize and relate.
However, part of the joy of having
this Siri-like relationship for him, one could say,
was the absence of that fear that she would abandon him,
which prevented him from initiating
a relationship with a human being.
And yet, it turns out that he could be betrayed,
quote unquote, that she could be unfaithful to him,
quote unquote, anyway.
So then, that means that it did not resolve
the underlying fear, having that relationship.
So in other words, that human element of the relationship
still found its way into the seemingly sterilized,
protected partnership.
So the human being rears its head anyway.
And I think the lesson there is that the system
in the movie Her actually gave him a lesson
that even AI could betray you, even AI can leave you,
even AI can be unfaithful to you.
And I would argue that the next AI he meets
will be one he actually falls in deep love with
because he knows the possibility of betrayal is there,
the possibility of death is there,
the possibility of infidelity is there,
because we need that possibility to truly feel good.
Or he would turn off his Siri program
and get out of his house, go to a local bar,
and strike a conversation with a human being.
Although you might say by then,
some of those might be Androids.
So who knows?
And we don't even have a good test
to know the difference between one or the other.
And that was predicted by another great movie.
Yeah.
Right?
The Blade Runner.
The Blade Runner.
How interesting that artists could see that so long ago.
Of course, Blade Runner was based on a novel
by Philip K. Dick, The Androids Dream of Electric Ship.
That guy was a genius, you know?
It's somehow that artists have their eyes open
to bigger reality. How is it
that they anticipate?
Is it also a large language model
that they're using for that?
An even larger one.
Even larger.
I hesitate to dismiss the magic in large language models.
A lot of the work I've done is in robotics,
and the robotics community generally doesn't notice
the magic of feeling.
I've been working a lot with quadruped recently,
Legged Robots with Four Legs,
and the feelings I feel when I see,
I'm programming the thing,
but when the thing is excited to see me,
or shows with its physical movement
that it's excited to see me,
I cannot dismiss the feeling I feel
as not somehow fundamental to what it means
to program robots.
And I don't want to dismiss that.
Please don't, please don't.
The robotics community often doesn't gender robots.
They really try to work hard
to not anthropomorphize the robots,
which is good for technical development
of how to do control, how to do perception.
But when the final thing is alive and moving,
and it does whatever,
like I've been doing a lot of butt wiggling,
it can wiggle his body,
it can turn around and look up excited.
That's not just, I know how it's programmed,
but the feeling I feel, that's something.
I don't know what that is.
I agree, I agree with you.
I hear you when you speak about it,
you speak with passion.
And that's, to me, that is proof
that it is magical, you see.
So don't, I would say, don't dismiss that.
Don't discard that.
On the contrary, I think magic is everywhere.
So I used to be, okay, kind of confession, okay?
Yeah, you already confessed to quite a few addictions.
Yeah, I'm kind of, yes, I'm kind of worried.
Recovering from many.
But, you know, in the old days,
I was more on the side of everything is computational
or everything can be explained by science and whatever.
I would dismiss and disregard
the intuitive or imaginative things.
So then I had to flip.
Then suddenly I started feeling it
and started seeing it and so on.
But so then the pendulums had swung
in the opposite direction.
Then I was arguing that somehow that was real,
that imagination was intuitive,
imaginative was real,
and discounting what you just described.
And I would argue with people saying,
no, no, this is not real.
This is all imitation game and so on.
But you see that what's new now,
the new Edward, okay, is the 2.0, 3.0,
is the one who is seeking balance,
who has suddenly become aware that no matter
which one-sided, lopsided point of view you take,
you're limiting yourself.
So whereas even a couple of years ago,
if you told me what you just described,
I would be like, being polite,
I wouldn't contradict you
since you're the host anyway, right?
But I would be like, uh-huh, uh-huh,
but I wouldn't say anything.
But suddenly I find this moving.
I find it moving.
Honestly, I'm not being facetious.
I find it moving.
And I almost feel like I can see it through your eyes
because the way you describe so vividly
and you're passionate about it.
And this is what's real.
So ultimately, love is neither in language models
nor in something mystical.
It's exactly in this moment of passion.
And I would even go as far as saying that in this moment
when you're describing it,
there was a connection of sorts
so that I could feel your passion for it.
And in this moment, something else comes up
which is far beyond any theories that we can come up with.
And that's what we, for now, exactly.
So on the one side,
there is this impulse of finding a theory.
And then there is another impulse to escape
from what has already been known.
So in other words, like in my basic example
is one impulse to say everything is a real number,
square root of negative one doesn't exist.
But another impulse is I'm going to be this naughty child
who is not afraid to be an idiot.
And I will say square root of negative 15 is real.
And both are essential when it's done with conviction,
when it's done with passion,
when it's not like, you know, meh, you know, gratuitous.
Or when it's not, it doesn't come from self-limiting,
but comes from this sense of this is how I am.
This is how I feel.
It is real.
That's where the progress is.
That's where creativity is.
And that's where I would even say a real connection is.
Because the strife to me that I observe today
in our society and the society level
and the level of humans and so on,
it comes from not seeing the other person actually
and being caught up in a very specific
conceptual bubble, you see.
And the way out of it is not to refine the bubble,
but just break out of it.
A good guide out of the bubble is a childlike passion.
Discovering that and following it.
Goosebumps.
Yeah, following the goosebumps.
Not the rigor of science, but the magic of goosebumps.
And then, if you're interested,
try to find a confirmation of those goosebumps
in science or whatever you find interesting.
And most of the time you'll fail.
And most of the time you fail, which we also love,
because then it sets us up for that moment of bliss
when we succeed, right?
Exactly.
Quick pause.
Bathumbreich, you mentioned
Godel's The Completeness Theorem.
Can you talk a little bit about it?
What is it as you understand it?
Did it break mathematics?
Maybe another question is,
what are the limits of mathematics?
What is mathematics from the perspective
of Godel's The Completeness Theorem?
Well, yes.
How much time do you have?
We talked about time previously, so it's all different.
Time is an illusion, right?
So we agreed.
So Kurt Godel was a great Austrian mathematician
and logician.
He moved to the United States before Second World War
and worked at the Institute for Advanced Study in Princeton,
where he was a colleague of Einstein
and other great scientists, von Neumann,
Hermann Weyl, and so on.
But one interesting quote that I like in this regard
is that Einstein said that, at some point,
he said that the only reason he came to the Institute
was that he would have the privilege
of walking back home with Godel in the evening.
So in other words, Einstein thought
that Godel was the smart one, okay?
So his most important contribution
was his two incompleteness theorems,
the first incompleteness theorem
and the second incompleteness theorem.
And what is this about?
It's really about inherent limitations
of mathematical reasoning,
the way of producing mathematical theorems,
the way we do it.
So to set the stage, how do we actually do mathematics?
So we know that, we discussed that,
say, physics is based on mathematics,
and you could say chemistry is based on physics,
biology based on chemistry.
Okay, so it comes to mathematics.
What is mathematics based on?
Well, mathematics is based on axioms.
So any field of mathematics can be presented
as what is called the formal system.
And at the core of the formal system
is a system of axioms or postulates.
These are the statements which are taken for granted.
Given without proof.
Without proof.
An example would be,
so one of the very first formal systems
was Euclidean geometry, developed by Euclid
in his famous book, Elements, about 2,200 years ago.
And it's about, well, it's a subject familiar from school,
because we study it.
But what it's really about
is about the geometry of the plane.
And the plane, by plane, I mean just this tabletop
extended to infinity in all directions.
Kind of a perfect plane, a perfectly even table.
And so Euclidean geometry is about
very geometric figures on the plane,
specifically lines, triangles, circles, things like that.
So what's an example of an axiom?
An example of an axiom is that if you have two points,
which are distinct, two points on the plane,
then there is a unique line which passes through them.
Now, it kind of sounds reasonable,
but this is an example of an axiom.
In mathematics, you have to have a seed, so to speak.
You have to start with something.
And you have to choose certain postulates or statements
which you simply take for granted,
which do not require proof.
Usually there are ones which kind of intuitively clear
to you, but in any case, you cannot have any mathematics
without choosing those axioms.
And you refer to those as the observer
because they're kind of subjective.
The observer comes in the process of choosing the axioms.
Who chooses the axioms?
The turtles that it's all sitting on top of.
As Alan was, you know, like to say,
who is watching the water?
And so in mathematics,
but you see mathematicians are so clever.
It's really kind of like a little kind of a game of mirrors
that we often like to say, and I used to say that,
that mathematics is objective.
It's really the only objective science.
But that's because we hide this fact
in the basement is based on axioms.
And the fact that there is no unique choice,
that there are many choices.
And so Euclidean geometry is actually a good illustration
of this because Euclid had five axioms.
Four of them were kind of obvious,
like the one I just mentioned.
And the fifth, which came to be known famously
as the fifth postulate, was that if you have a line
and you have a point outside of this line,
there is a unique line passing through that point,
which is parallel to the first line,
meaning that doesn't intersect it.
And Euclid himself was uncomfortable about this
because he felt that it was kind of a, you know,
that he takes for granted something that is not obvious.
And for many centuries after that,
mathematicians were trying to derive this axiom
from other axioms, which were more obvious in some sense,
and they failed.
And it was only almost 2,000 years later
that mathematicians realized that you can't,
not only you cannot derive,
but you can actually replace it with its opposite.
And you will still get a bona fide consistent,
not self-contradictory,
which is called non-Euclidean geometry,
which of course sounds very complicated, but it's not.
Think of a sphere, just the surface of a basketball
or the surface of the earth, I know, idealized.
The analogs, so you have points,
you have analogs of lines, which are meridians, right?
Every two meridians intersect,
unlike parallel lines on a flat space.
There is also so-called hyperbolic plane,
where there are infinitely many lines
which do not intersect.
So every possibility can be realized.
There are different flavors.
This is a good illustration of what a formal system is.
You start with a set of axioms,
those statements that you take for granted,
and this is where you have a choice.
And by making different choices,
you actually create different mathematics.
After that, there are rules of inference, logical rules,
such as if A is true and A implies B, then B is true.
Most of them were actually introduced already by Aristotle,
even before Euclid.
And then it runs as follows.
You have the axioms, which are accepted as true statements.
Then you have a way to produce new statements
by using the rules of logical inference from the axioms.
Every statement you obtain, you call a theorem,
and you kind of add it to the collection of true statements.
And then the question is, how far can you go?
How many statements can you prove this way?
Of course, you want the system to be non-trivial
in the sense that you don't prove everything.
Because if you prove everything,
it would mean that it's self-contradictory,
that you prove a statement A and it's negation.
So that's kind of useless.
It has to be discriminating enough
so that it doesn't prove contradictory statements.
So there is already a question
of that mathematician called consistency.
It has to be consistent in the sense
that it is not self-contradictory.
And then the idea that was basically prevalent
in the world of mathematics
by the beginning of the 20th century
was that in principle,
all of mathematics could be derived this way.
We just have to find the correct system of axioms.
And then everything you ever need
could be produced by this procedure,
which is really algorithmic procedure,
which actually could be run on a computer.
Now, think about it.
What is special about this process?
In this process,
you are just manipulating symbols, basically.
You're going from one statement to another
without really understanding the meaning of it.
So it's an ideal playground for a computer program.
It's a purely syntactic process
where there are some rules,
some rigid rules of passing from one statement
to the next.
Most mathematicians believed that this way,
you can produce all true statements.
And if this were true,
it would give a lot of credibility to the thesis
that everything in life is computational,
or life is computation.
Because then at least mathematics is computational
because then it can be programmed.
And the computer, after sufficient time,
depending on its capacity,
would produce every true statement.
So Gödel's first incompleteness theorem
says that that's not the case.
And it not just says it,
but it proves it at the highest level of rigor
that is available in mathematics.
That is to say within another formal system
that he was operating in.
So more precisely, what he proved was that
if you have a sufficiently sophisticated formal system,
that is to say that you can talk about numbers,
whole numbers in it,
that you have whole numbers, one, two, three, four,
you have formalized the operation of addition
and multiplication within the system.
If it is consistent, that is to say,
if it's not completely useless,
then there will be a true statement in it
which cannot be derived by this linear syntactic process
of proving theorems from axioms.
It's really incredible.
So this was a revolution, 1931,
revolution in logic, revolution in mathematics.
And we're still feeling the tremors of this discovery.
And at a similar time, the computer is being born.
The actual engineering of the computational system
is being born, which is ironic.
Turing was, Alan Turing, who is considered
as the father of modern computing, right?
So he actually did something very similar.
So he had this halting problem.
He proved that halting problem
cannot be solved algorithmically,
that you cannot, out of all computer programs,
roughly speaking, you cannot have an algorithm
of choosing out of all possible computer programs
which ones are meaningful, which ones will halt.
Very depressing results all across the table.
Or, on the contrary, life affirming.
Depends on your point of view.
Because everything is full of paradoxes.
So that means, so you're right, it's depressing
if we are sold on a certain idea from the outset
and then suddenly this doesn't pan out.
But, okay, to which I retort,
what if he proved that actually everything can be proved?
So then what?
What is left to do if you're a mathematician?
So that would be depressing to me.
And here there is an opportunity to do something new,
to discover something new,
which maybe a computer will not be able to.
Again, with a caveat,
according to our current understanding,
maybe some new ideas will be brought into the subject.
And the meaning of the word computation,
like now we think of computation in a particular framework,
Turing machines or Church thesis and stuff like that.
But what if in the future another genius
like Alan Turing will come and propose something else?
The theory will evolve the way we went
from Newton's gravity to Einstein's gravity.
Maybe in the framework of that concept,
some other things will become possible.
So it's not,
to me, it's kind of like not so much about
deciding once and for all how it is or how it should be,
but kind of like accepting it as an open-ended process.
I think that's much more valuable in some sense
than deciding things one way or another.
I wonder, I don't know if you think
or know much about cellular automata
and the idea of emergence.
I often return to Game of Life and just look at the thing.
Amazing, right?
And wonder.
The kind of things they can do with such a small tools.
That from simple rules, a distributed system
can create complex behavior.
And it makes you wonder that maybe
the thing we'll call computation is simple
at the base layer, but when you start looking
at greater and greater layers of abstraction,
you zoom out with blurry vision.
Maybe after a few drinks, you start to see
something that's much, much, much more complicated
and interesting and beautiful than the original rules
that our scientific intuition says
cannot possibly produce complexity and beauty.
I don't know if anyone has a good answer,
a good model of why stuff emerges.
Why complexity emerges from a lot of simple things.
It's a why question, I suppose, not a,
but every why question will eventually
have a rigorous answer.
Not necessarily.
We could have an approximate answer
which still eludes something.
Like quantum mechanics.
99%, maybe.
We would be able to describe it with 99% certainty
or 99% accuracy.
And then maybe in 100 years or next year,
somebody will come up with a different point of view
which suddenly will change our perspective.
To this point, I want to say also one thing
that I find fascinating, speaking of paradoxes and so on.
Do you remember how everybody was freaking out
about this blue dress?
And was it blue or was it black?
I think yellow and white or black and blue.
It almost broke Twitter, I remember that, that night.
So there are many examples like that
where you can perceive things differently
and there is no way of saying which is correct
and which is not.
For instance, you got this, the vase, the Rubens vase,
where you have, from one perspective, it's a vase.
From another perspective, it's two faces.
Then there is this duck rabbit picture
where you can Google it.
If somebody doesn't know, they can Google it and find it.
It's very easy.
Actually, Ludwig Wittgenstein devoted several pages
to duck rabbit in his book and so on.
There are many others.
There are like squares where you can see,
a square you can see from different perspective,
this way, that way, and so on.
So when we talk about neural networks,
we're talking about training data and stuff
so that you have some pictures, for example,
that you feed to your program
and you try to find the most optimal neural network
which would be able to decide which one is it,
so the dog or a cat or whatever.
But sometimes it doesn't have a definite answer.
So what do you do then?
So actually, it's a question I actually don't know.
Has modern AI even come to appreciate this question?
That actually sometimes you can have a picture
on which you cannot say what it is in it.
From one perspective, it's a rabbit.
From another perspective, it's a duck.
How are you supposed to train?
If you have a neural network which is supposed
to distinguish between ducks and rabbits,
how is it going to process this, you see?
Well, so the trivial trick it does is to say
there's this X probability that it's a duck
and this probability that it's a rabbit.
Well, that's a good approach.
But also, I would say there is no given percentages.
For instance, actually, at some point,
I was really curious about it and I looked.
And for each picture of this nature,
and there are a bunch of them you can easily find online,
my mind immediately interprets it in a particular way.
But because I know that other people
could see it differently, I would then strain my mind
and strain my eyes and stare at it
and try to see it in a different way.
And sometimes I could see it right away
and then I could go back and forth between the two.
And sometimes it took me a while for some pictures.
So in that sense, even if these probabilities exist,
they are subjective.
Some people immediately see it this way.
Some people may see it that way.
And I think that nobody knows, not psychologists,
not neuroscientists, not philosophers, what to make of it.
The best answer, of course, as a scientific mind,
even though I say, no, don't look for interpretation,
give some place for mysticism or mystery, right?
I say that, but of course I want a theory.
I want an explanation.
So the best explanation I find
is from Niels Bohr's Complementarity Principle.
So it is like particle and wave,
that there are different ways to look at it.
And when you look at it in a particular way,
another side will be obscured.
Think about it like the other side of the moon, you know?
So like we are observing the moon from one side
and then we don't see the other side.
There is a complementary perspective
where we see the other side,
but not the side we normally see.
But the moon is the same, it's still there.
It's our limitations of being able to grasp the whole.
That's complementarity.
And we know that from quantum mechanics
that our physical reality is like that,
rather than being certain,
rather than being one way or another.
And we should just, as a small aside,
in terms of neural networks, mention that
at the end of the day, there's humans,
it's built on top of humans.
Or with ChatGPT, that it's using
reinforcement learning by human feedback.
We're actually using a set of humans to teach the networks.
And that's the thing that people don't often talk about
because, or I sometimes think about,
that those humans all have a life story.
Each human that annotated data,
that fed data to the network,
or did the RLHF, they have a life story.
They grew up, they have biases.
There's some things that they like,
there's some things they don't like,
which can kind of appear under the radar screen.
They may not be aware that they are exercising those biases.
That's the point.
What you brought up is a very important issue here.
Not so much issue, but it's not a bug.
It's a feature, in my opinion,
that implicit in the discussion of the question,
is thinking computational and so on,
is the idea that our conscious awareness
covers everything within our psyche.
And we just know that that's not the case.
We have, all of us have observed other people
who have had sort of destructive tendencies.
So obviously, they did things destructive for themselves.
And many of us have observed ourselves doing that
as part of human nature, right?
So, and there is great research in analytic psychology
and in the past hundred years,
strongly suggesting, if not proving,
the existence of what Carl Jung called
the personal unconscious and also collective unconscious,
the kind of circle of ideas
which are under the radar screen,
which lead us to some strong emotions
and inspire us to act in certain ways,
even if we cannot really understand.
So if we accept that,
then the proposition that somehow everything
can still be covered by our actions,
which are totally kind of neutral
and totally righteous and totally conscious,
that it becomes really tenuous.
Let me ask you some tricky questions.
Uh-oh.
In terms of how big they are.
In terms of how, you know,
they become difficult because of how much
of a romantic you are.
What to you is the most beautiful idea in mathematics?
Another one we can ask is,
what is the most beautiful equation in mathematics?
Well, I mean.
I may have just broken your brain.
Because what your brain is doing
is walking down a long memory lane of beautiful experiences.
Well, you see, in mathematics,
we have this idea that we have an idea of a set, right?
So we have a collection of things.
For instance, you know, the set of tables,
the set of chairs, and so on, or set of microphones.
But it could be a set of numbers.
Could be a set of ideas.
Could be a set of formulas, mathematical equations.
And then we have the notion of an ordered set.
Ordered, like the set in which there is order.
Which means that for every two members of the set,
we will say which one is better than the other,
or greater than the other.
For instance, all numbers are ordered.
Five is greater than three, five is less than seven,
and so on.
But not all sets are ordered.
So the set of beautiful theorems is not what,
beautiful equations is not ordered.
So in other words, there are many best equations.
And so Richard Feynman chose one,
which I think one of the best,
is that if you take e, the base of natural logarithm,
to the power pi i, so you have pi,
you have e in it, the base of natural logarithm,
you have pi i, which is square root of negative one,
then the result is negative one.
So that's up there, for sure,
in the pantheon of beautiful formulas, you know?
That I think pretty much every mathematician would agree.
I don't know what my favorite one is.
I'm just lingering on that one, Euler's identity.
What makes it beautiful?
Just a few symbols together.
I mean, part of it is actually just trying to define
what is beautiful about mathematics
that is laid in there in this particular equation
that is somehow revealed when the human eye looks at it.
Why is it beautiful, do you think?
Pi, i.
There is an element of surprise in it.
How is it possible?
We always think of pi as the ratio
between the circumference of a circle and its diameter.
Here, we are taking some number to the power pi,
not even pi, mind you,
but pi multiplied by square root of negative one.
Ooh.
Surely, this is something completely incomprehensible,
and yet the result is negative one, you see?
And if you take e to the power two pi i,
you get one, actually one.
So I would guess that that's, but in other words,
the initial reaction is just that of a surprise, I guess.
I guess for anyone who first comes across.
That these three folks, four folks got together,
it reminds me of the idea that Hitler, Stalin,
Trotsky, and Freud were all in Vienna
in some early, at the beginning of the 20th century.
And Wittgenstein was classmate of Hitler, you know this.
I did not know this, no.
So there, it makes you, you can imagine a situation
where they're all sitting at a bar together at some point,
not knowing it, but they somehow,
it all made sense in space-time to be located there.
And that's what this feels like, some kind of intersection.
– Intersection, yes.
But I would say that after the initial shock,
you look at the proof of this equation,
and it actually does make sense.
And actually, it is nothing but the statement
that the circumference of the circle is.
And in fact, in this case, it's the circumference
of a semicircle is equal to pi,
and that's where it comes from.
– In the end, the truth is simple.
– In the end, the truth is simple.
Not necessarily easy, but simple.
– So I mentioned to you offline that I desperately,
in trying to figure out the optimal,
in an order to set questions to ask you,
texted Eric Weinstein asking for what questions
he can ask you, and he said that you are definitively
one of the greatest living mathematicians,
so don't screw this up.
But he did give me a few questions.
So he asked to ask you what are the most
shockingly passionate, this isn't Eric's language,
what are the most shockingly passionate
mathematical structures, and he gave a list of four for him,
but he said he really wanted your list.
Okay, let me say that.
Shockingly passionate mathematical structures.
– Shocking.
– Is there something you can,
is there something that jumps to mind?
– Sure.
I'm here to shock, obviously.
So first of all, Eric Weinstein is a very dear friend,
I have to say, and I really, really, really appreciate
and love him, he's just like my brother,
so it's interesting to have a question posed by him.
– Maybe if we can linger for a moment,
what do you think is special about Eric Weinstein
for what you know of his work and his mind?
– The way he sort of straddles
so many different disciplines.
He's like a Renaissance man.
There are very few people like that at any given moment,
let alone the 21st century,
where information has become so huge
that it's almost physically impossible
to be able to keep track of things,
and yet he does, and he has his own unique vision
and unique point of view, and he has integrity,
which is almost impossible.
I can't think of so many people who possess those qualities,
almost no one.
– And also the ability, in some sense,
to embody the balance that you talked about
of both the rigor of mathematics and the imagination.
– Humanity, also, I would say.
We talk about imagination as a kind of a counterpoint
to knowledge or logic.
But just basic humanity, just compassion,
just being able to...
Because every destructive, I would say,
like every destructive society,
be it Germany, under Hitler, or Soviet Union,
under Stalin, and so on,
was based on some kind of what was considered
unassailable truths.
So it's kind of a conceptual system,
if you think about it, right?
There is a beautiful episode of this series
by Jacob Bronowski,
where he talks about, he filmed it in Auschwitz,
talking about the certainty that what led the Nazis
to be killing people wholesale
was almost a mathematical idea.
And they just basically bought into this idea
and checked out their humanity at the door.
So I would say that antidote to this type of thing
is not necessarily even imagination
in a kind of elevated sense
that we have been discussing today
that is exemplified by our greatest scientists
and philosophers.
But just basic humanity,
basic human, basic common sense
of just knowing that it's just not right.
And I don't care what my ideology tells me,
but I'm just not going to do it.
So that, I think, is kind of missing a little bit
in today's society.
People get a lot too caught up in the ideology,
in certain conceptual frameworks.
So societies that lose that basic human compassion,
that basic humanity around the trouble.
Oh, very much so.
Not only society, like a human being.
And Eric is one of the people, I agree with you,
keeps that flame of human.
I trust that he will not do something that's not human,
that's not right.
I just feel, you know, like there's some people,
you just kind of feel that they won't cross that line.
And that's a huge thing today.
Because I have to say, looking back,
definitely I have not hurt people personally,
but I could be mean, for instance.
I could be harsh.
And now I see it as a sign of weakness,
as a sign of insecurity.
I saw your interview with Ray Kurzweil the other day.
Beautiful, I was really moved by it.
But you know, at some point I was like,
I looked at him as sort of like Dr. Evil.
I'm kind of ashamed of it now,
but like, you know, I'm kind of clean.
And I would, you know, because, well, why?
Because I needed an adversary in my mind,
because I projected onto him kind of the fears that I had,
that AI will conquer us and so on.
And this was rooted in my kind of awakening moment,
in a sense, kind of a moment where I suddenly started
to see the other side.
So, but I wasn't sure yet, you see?
You had to feel it.
So I had to have a fight about it.
Yeah, you had to actually have the projection.
I had to, so it was not in,
I believe that it was not in me already.
So I had to throw it onto somebody.
And that's not balanced yet.
So balance is when you recognize that it's you, actually.
And I had this moment actually, it was so amazing.
Like I would give this mean,
I would talk about AI and the dangers,
and he would always be my foil, you know?
I would put like a sinister photograph of him in the slide.
And I was like, look at this guy.
He wants to put nanobots into your brain.
And he's also like high-end top executive at Google
and so on.
So I would create this whole narrative.
And then something happened where I was giving a lecture,
this is 2015, in Aspen, Aspen Ideas Festival,
which is a wonderful festival.
So keynote speech, actually.
And I was doing my usual stick.
And then suddenly I said,
I came up to that, there was a big screen,
and there was a picture of him there.
And I came up to the screen,
and I kind of touched it with my hand.
And I said, but I don't want to pick on Mr. Kuzma
because he's me.
I had this revelation that I'm actually fighting
with myself, with my own fears.
And then I learned about his father,
that his father died when he was young.
And that he is, in fact, he's very, to his credit,
he's very sincere and upfront about it.
Self-disclosure, I think it's very essential, by the way,
in all this discussion, like what really motivates you?
He said it publicly many times,
even as early as 2015, I could find this information,
that he wanted to reunite with his father in the cloud.
And suddenly I saw him, not as a caricature
that exemplified all my fears, but as a human being,
a child longing for his father, grieving for his father.
So suddenly it became a story, a love story.
And so in other words, I've seen it in myself,
this capacity to project my own fears
and then fight with other people over something.
That actually was my own.
And as soon as I got to this point of seeing him,
and then my next lecture, actually, I talked about it,
about him in this way.
And I said, look, it's a love story.
And he is actually,
and it's not how I would want to reunite with my father.
But like you said, if I am consistent,
I have to allow the possibility that different people
perceive things differently.
And so for him, that's his imagination.
So you know how, who is this?
Voltaire, I think, is ascribed to Voltaire.
It's like, I disagree with you, but I will fight to death
for you to have the right to say it.
So now that I feel like my position is more like,
I disagree with him, that this is the way to approach death
and to approach the death of our loved ones
and how we miss them and how we,
that sense of loneliness and inability to interact directly.
That's not something that is nice with me,
but I think it can also be called imagination
from his perspective.
And look, motivated by that, how much he has brought,
how many interesting inventions,
like his musical inventions, for instance,
naturally because his father was a composer,
a music composer and a conductor.
So in other words, in the bigger scheme of things,
even if I think he's misguided,
still, I can't deny that it's a certain leap of faith
from his perspective to try to say that this is the way
we can all connect to our loved ones.
And because it is sincere, and I see it now as sincere,
and in fact, in your interview,
you really teased it out of him,
and I was really moved by it, I have to say.
It's like, he has mellowed a little bit too, I said.
You know, it was really, really sweet
when he talked about his father.
And I can relate, you know, my father died four years ago,
and I can relate what a heartbreak,
I was much older than Ray was when his father died,
but I can relate to this longing and that grief, you know.
And when somebody is sincere and he opens his cards
and says, this is why, this is why I want to do it,
because I want to recreate my father
and I want to be able to talk to him this way,
then we have a serious, then we understand, you know.
The opposite of it would be not disclosing
and just pretending that this is how it's supposed to be
in scientific terms.
So he was replacing the real emotion
that come from the heart by some kind of a theory
which comes from the mind.
And this is where we can go astray,
because then we become captives of frameworks
and conceptual systems,
which may not be beneficial to our society.
In tough times, we need the people
that have not lost their way in the ideologies.
We need the people who are still in touch with their heart.
And you mentioned this with Eric, it's certainly true.
I disagree with him on a lot of stuff,
but I feel like when the world is burning down,
Eric is one of the people that you can still count on
to have a heart.
I've talked a lot over the past year
about the war in Ukraine
and the possibility of nuclear war.
And it feels like he's one of the people I would call first
if, God forbid, something like a nuclear war would begin,
because you look for people with a heart,
no matter their ideas.
That's right.
It takes courage and it takes a certain self-awareness,
I would say.
And which brings me,
I think the crucial is that which was inscribed
on the temple of Apollo and Delphi.
There was a statement, know thyself, know yourself.
Like, who am I?
Ultimately, it goes down to this and all these debates.
And the point is that I used to be, like I said,
pessimistic at some point,
and I was scared even of where development of AI was going.
This is about 2014, 2015.
And now I'm much more,
so for instance, after I saw Ray Kurzweil as a human being,
after I could relate to him and sympathize with him,
suddenly I stopped seeing him in the news.
Like before that, I would always see him in the news saying,
we're going to put nanobots in your brain,
da, da, da, by the year 2030, whatever, you know?
And then we upload you by the 21st.
And I would be like, no, you know, the story terrible.
Suddenly I didn't see him anymore.
I had to, you know?
So now it makes me question, who was creating the trouble?
What was all within me?
Was it him who was creating, stirring the trouble?
Or was it my mind, you see?
And so as I become, as I became self-aware,
suddenly other possibilities opened.
And suddenly that conflict, which by the way,
if I kept giving this nasty, you know, talks about him,
one day I suppose we'd have a debate.
And so you have this, one person stays this and that.
And what I learned is that it's a never ending conflict.
This conflict just does not end.
But there is an alternative, there is a better way,
which is to realize that it is you arguing with yourself.
Now, if you want to continue arguing with yourself,
continue as long as you need.
Just be careful not to destroy too many things,
you know, in the process.
But there is an option of actually dropping it,
of actually dropping it.
This is so, I was so surprised by this.
Yeah, it's discovering in yourself the capacity,
the human capacity for compassion.
And you understand that he has a perspective,
he is operating in the space of imagination,
a human being like you.
And we're all in this kind of together
trying to figure this out.
But we're on the same boat, ultimately.
And also it's like, with realizing how much
I have screwed up, you know, comes this humility also.
So like, I find it extremely hard now
to like really lash out at somebody.
And to say like, you're horrible, whatever.
Because immediately the question is,
who am I to criticize, you know?
So is there another way to have a dialogue?
Is there a way to, you know, speaking,
you know, since we talked about the innocence of a child
and how much it drives discovery in science and so on,
you know, I remember, I think I heard of Adyashanti
who gave this nice example.
He's like, when you're a kid, you know,
you go and you play with your friends
and then you fight with another kid.
And you're just like, I hate you.
I don't want to see you again.
And you just go home like after half an hour.
Okay, what are you going to do?
You want to play?
So you come out, it's like, hey, you want to play?
You don't talk about what happened.
You don't rehash this, you know, just keep going.
And sometimes I think we are on the verge maybe
of learning that.
Because I think that if we are,
if we continue to push each of us, our set of ideas
and like ideologies and like, you know,
what matters to us and so on, like, yeah,
no, no, what matters to you?
But like, there are other ways to approach other people.
There are other ways you can find point of contact.
Speaking of which, mathematics, mathematical formulas
are universal, represent universal knowledge.
Two plus two is four, whether you vote for this guy
or that guy in the election, you know,
how about that as a point of contact of commonality,
you know, and nobody can patent those formulas.
Did you know that?
There is a Supreme Court decision
that mathematical formulas cannot be patented.
Like Einstein could not patent E equals MC squared.
It doesn't belong to him because if the formula is correct,
then it belongs to everyone.
So what do you think of that all too tricky question?
And if you want, I can deeply bias your answer
by giving the list of four that Eric provided.
Oh no, let me give mine.
I cannot see, by the way, what you have.
But I can guess some of them.
So I'm going to try to do something different from him.
So I already mentioned one,
which is that you have one-dimensional numerical system,
which is real numbers.
You have two-dimensional, which is complex numbers.
You have four-dimensional and it's probably connected
to what he wrote because it has to do
with some homotopic groups of spheres and stuff like that.
Then of course, one I love, okay,
one plus two plus three plus four plus five plus six
and so on.
Does it make any sense, the sum?
You probably heard about this one.
It became very popular at some point.
One plus two plus three, I did a video for Numberphile,
the YouTube channel about it maybe 10 years ago.
So one plus two plus three plus four plus five
ostensibly diverges, goes to infinity
because you get a bigger and bigger number.
And yet, there is a way to make sense of it
in which it comes up to minus one over 12.
How fascinating.
First of all, the answer is not even a positive number
and it's not an integer.
It's not a whole number.
It's minus one over 12.
So sometimes people ask me, what is your favorite number?
And it's a kind of a joke.
I say minus one over 12.
It's actually 42.
So your favorite number is not in order to set.
So what else?
What else?
So Langlands program, of course.
I have to mention that.
And we'll explore that in depth.
Do you want to know what Eric said?
Sure.
Sphere aversion, Boy's surface, hop vibration.
Hop vibration, okay.
And pi one of SO three.
Okay, oh yes.
So that's the famous cup trick, you know?
Okay, look.
So this is how it works.
No tricks.
No tricks.
No magic.
It is magical, okay?
But not because I'm tricking you.
So you start with a bottle like this or a cup
and you start twisting it.
And the same time you twist your arm.
Then you come, so this is actually going to rotate
at 360 degrees, the full turn.
Then you say, okay, I won't be able to do another turn
because then my arm would really get twisted.
I'll have to go see a dog.
Yet, if I do it second time, it untwists.
This is the pi one of SO three Eric is talking about.
So there is something where the first motion is not trivial,
but if you double down on it,
you come back to the initial position.
It's very closely connected to the fact
that we have elementary particles of two types,
bosons and fermions.
So bosons are, for example, photons
or carriers of other forces or the Higgs boson.
It is called a boson for a reason, because it is a boson.
In honor of Indian mathematician, Bose, B-O-S-E,
and Einstein.
So these particles obey what's called
Bose-Einstein statistics.
But then there are other particles called fermions
in honor of Enrico Fermi, Italian born mathematician
who worked in the US.
And they follow what's called Dirac-Fermi statistics.
And those are electrons and constituents of matter,
electrons, protons, neutrons, and so on.
And they have a certain duplicity, if you will.
And that duplicity is rooted mathematically
in this little experiment that I have just done.
So I imagine, speaking of imagination, okay?
So I'm just kind of riffing on this.
Imagine a world in which this will not be shocking
or like, in this case, it's not even shocking
because I haven't really explained the details
because I can't do it in two minutes.
I indicated what this is all about and so on.
But imagine a world in which this is not foreign
to most people, that most people have seen it before.
They're not afraid to approach this type of questions
because, you know, we talked a little bit
about mass education, but I really believe
that a lot of people in our society,
and it is not only in the United States,
but throughout the world,
a lot of people have been traumatized.
It's really PTSD.
That's why people, when they see mass microformula
or even how they need to calculate tip on the bill,
they're terrified because it brings up those memories
when they were kids and being called to blackboard
and solve a problem.
You can't solve a problem and unscrupulous teacher says,
you're an idiot, sit down and you feel ashamed and lowly
and that stays with you.
And so I think that, unfortunately, that's where we are,
but one can dream.
And so my dream is that one day we'll be able
to overcome this and actually all of these treasures
of mathematics will become widely available
or at least people will know where to find them
and they will not be afraid of going there and looking.
And I think this will help because like I said,
for one thing, it gives you a sense of belonging.
It gives you, it kind of is an antidote
to the kind of alienation and separation
that we feel today oftentimes
because of ideological divide, sectarian strife
and all kinds of things like that.
Because then once you see there's like a critical mass
of this beauty that kind of like dawns on you is like,
my God, this is what we all have in common.
You mentioned Langlands program.
We have to talk about it.
Sure.
At the core of your book and your work
is the Langlands program.
Can you describe what it is?
Sure, so Langlands is a mathematician.
It's a name of a mathematician, Robert Langlands.
Canadian born, still alive.
He was a professor at the Institute for Advanced Study
that we talked about where Einstein and Gödel
and other great scientists have worked.
In fact, he used to occupy the office of Albert Einstein
at the Institute for Advanced Study.
So he, in the late sixties,
he came up with a set of ideas
which captivated a lot of mathematicians,
several generations of mathematicians by now,
which came to be known as the Langlands program.
And what it is about
is connecting different fields of mathematics,
which seem to be far away from each other.
For example, number theory,
which as the name suggests, deals with numbers
and various equations with, you know,
like X squared plus Y squared equals one.
And on the other side, harmonic analysis,
something that any music lover can appreciate
because the sound of a symphony can be kind of decomposed
into sounds of different instruments.
And each of those sounds can be represented by a wave
like this, like a sine function.
Those are the harmonics.
They actually, the period of harmonic periods
of different notes are different.
They correspond to different notes
and different instruments, different semitones, if you will.
But they all combine together into something special,
which cannot be reduced to any one of those.
So mathematically, it's the idea
that you can decompose a signal into,
as a collection, as a simultaneous oscillation
of several elementary signals.
That's called harmonic analysis.
So what Langlands found
is that some really difficult questions in number theory
can be translated into much more easily tractable questions
in harmonic analysis.
That was his initial idea.
But what happened next surprised everybody,
that the kind of patterns that he was able to observe,
the kind of regularities that he was able to observe,
which were quite surprising,
were subsequently found in other areas of mathematics,
for example, in geometry and eventually in quantum physics.
So in fact, Ed Witten,
who is kind of a dean of modern theoretical physicists,
a professor at the Institute for Advanced Study as well,
got interested in this subject.
I describe in my book how it happened.
And he was instrumental in bridging the gap
between these patterns found in physics and in geometry,
finding kind of a super stratum, if you will,
or kind of a way to connect these two things,
kind of a bridge between these two fields.
So I subsequently collaborate with Witten on this,
and this has been one of the major themes of my research.
I always found it interesting to connect things,
to unite things.
When I was younger, I couldn't understand why,
but I was always interested in when,
not in working in specific field,
but kind of cutting across fields.
And then I would discover that, for instance,
I took some people who know what happens in this field,
but don't know what happens in their field, or conversely.
And then I would find it imperative
to go out and explain to them,
to the different sides, what this is all about,
so that more people are aware of this hidden structure,
so these hidden parallels, if you will.
So that has been sort of a theme in my research.
And so I guess now I kind of understand more
why it's kind of a balance,
like what we talked about earlier.
So can you elucidate a little bit how,
what are the mathematical tools that allow you
to connect these different continents of mathematics?
Is there something you can convert it towards
that Langlands was able to find
and you were able to explore further?
I would say what it suggests
is that there is some hidden principles,
which we still don't understand.
My view is that we still don't know why,
that we can prove some instances of this,
correspondences and connections.
But we still don't know the real underlying reasons,
which means that there is a certain layer
beneath the surface that we see now.
So the way I see it now is like this,
that there is something three-dimensional like this bottle,
but what we are seeing is this projection onto the table
and the projection onto a wall.
And then we can map things from one projection to another
and you say, oh my God, that's incredible.
But the real explanation is that both of them
are projections of the same thing.
And that we haven't found yet,
but that's what I want to find.
So that's what motivates me, I would say.
From number theory to geometry to quantum physics.
So there is this one thing which has different projections,
except it's not just a table and a wall,
but there are like many different walls, if you will.
So what is the philosophical implication
that there is commonalities like that
across these very disparate fields?
It means that what we believe
are the fundamental elements of mathematics
are not fundamental, there is something beyond.
It's like we previously thought that atoms were indivisible.
Then we found out that there is a nucleus and electrons
and the nucleus consists of protons and neutrons.
Then we thought, okay, protons and neutrons
must be elementary.
Now we know they consist of quarks.
So it's about kind of finding the quarks of mathematics.
And of course, beyond that, there's maybe even more.
Which was my initial motivation
to study mathematics, by the way, right?
Quarks was the first time you fell in love
with understanding the nature of reality.
What was it like working with Ed Witten,
who many people say is one of the smartest humans in history
or at least mathematical physicists in history?
Yes, fascinating, I enjoyed it very much.
I also felt they have to keep up.
So we wrote this long paper in 2007
and we collaborated for about a year.
I have known him before and we talked before
and I've seen him since and we talked,
but it's very different to just meet somebody
at conferences and have a conversation
as opposed to actually working on a project together.
So he's very, very, very serious, very focused.
This is one thing which I have to say.
I was really struck by this.
Why is he considered to be such a powerful intellect
by many other powerful intellects?
He has had this unique vision of the subject.
He was able to connect different things,
especially find connections
between quantum physics and mathematics,
almost unparalleled.
I don't think anyone comes close in some sense
in the last 50 years to him
in terms of finding just consistently time after time,
breaking ground, new ground, new ground.
So he would basically, one way one could describe it is
he would take some idea in physics
and then find an interpretation of it in mathematics
and then say, distill it, present it in mathematical terms
and tell mathematicians, this should be like that.
Kind of like one plus two plus three plus four
is minus one over 12.
And mathematics should be like, no way.
And then it would pan out and mathematicians would then,
like a whole industry would be created
of groups of mathematicians trying to prove his conjectures
and his ideas and he would always be proven right
So in other words, being able to glean
some mathematical truths from physical theories,
that's one side.
On the other hand, conversely,
applying sophisticated mathematics,
he's probably the physicist who kind of
could learn mathematics the fastest, I don't think.
Some younger physicists maybe could come close,
but it's still for them a long way to go
to be comparable to Witten,
to take some of the most sophisticated mathematics
and not learn it to the point where
it becomes a practitioner of the subject, practically,
and then use it to gain some new insights
on the physics side.
Now, of course, the thing is that the theory is that
physics, one could say, is in a sort of a crisis
in some sense because of a current gap
between the sophisticated theories,
which came from applying sophisticated mathematics
and the actual universe.
So we have theories, for instance,
which describe 10-dimensional worlds,
10-dimensional space-time coming from string theory
and things like that, but we don't know yet how to apply it
to understanding our universe.
A lot of progress has been made,
but it's kind of at a kind of an impasse right now.
And at the same time, our most realistic theories,
most advanced theories of the four-dimensional universe
are in contradiction with each other.
The standard model describing the three known forces
of nature, the electromagnetic, strong and weak,
with great accuracy, and Einstein's relativity,
which describes the fourth, called gravity.
Everybody above a certain age knows that one.
So these two theories are in contradiction at the moment.
And string theory was one of the,
the promise of string theory was that
it would unify those two.
And so far it has not happened.
So we are kind of at a very interesting place right now.
And I think that new ideas perhaps are needed
and I wouldn't be surprised if Witten
is one of those people who come up with those ideas.
Well, he has been one of the people
that added a lot of ideas under the flag of string theory.
What do you think about this theory?
What do you think is beautiful about it, string theory?
Well, first of all, kind of,
you remember we talked about Pythagoreans
and how for Pythagoreans, the whole world was this symphony
where you have this different vibrations of all the humans.
Every human is a vibration, every animal,
every being, every tree, and every celestial body and so on.
So string theory is kind of like that
because in string theory, there is this fundamental object,
which is a vibrating string.
And all particles are in a sense supposed
to be different modulations or vibrations of that string.
So that by itself is already interesting
that you kind of describe this diversity
of various particles and interactions between them
using one guiding principle in some sense.
But also just the mathematical things that come out of it,
it looks impossible to satisfy various constraints
and then there is sort of like a unique way to do it.
So that's sort of the, every time that happens
when you have some system, like over-determined system.
Let's suppose you have to do like five interviews
in one day and you wake up in the morning
and you're like, that's impossible
because then so many things have to align.
For instance, let's suppose you have to go
from one place to another.
So then you have a commute and then who knows,
maybe there is a traffic jam and stuff like that.
And now suppose that it all works seamlessly
and there were like a bunch of places
where it could have gone hopelessly wrong and it didn't.
And then in the evening, you're like, wow, it worked.
That's beautiful, right?
That's kind of like great luck, we would say.
But in science, this happens sometimes
that you have this theory which is not supposed to work
because there are so many seemingly contradictory demands
on it and yet there is a sweet spot
where they balance each other.
So string theory is kind of like this.
The unfortunate aspect of it is that it balances itself
in 10 dimensions and not in four.
So maybe there is another universe somewhere.
But see, as a mathematician, for me,
all spaces are created equal.
10 dimensional, four dimensional.
So mathematicians love string theory
because it has given us so much food for thought.
But do you think it's a correct or a incorrect theory
for understanding this reality?
So it might be a theory that explains
some 10 dimensional reality in some other universe,
but is it potentially, what do you think are the odds?
Again, financial advice, if you were to bet.
What do you think are the odds that it gets us closer
to understanding this reality?
Well, in the form that it is now, that seems unlikely.
But it could well be that based on these ideas
with some modifications, with some essential new elements,
it could work out.
So I would say right now it doesn't look so good,
like from the point of view from what we know.
But maybe somebody will come and introduce
like square root of negative one.
I mean, they already introduced,
but I mean kind of like as a metaphor.
Maybe somebody will come and say, what if we do this?
It looks crazy.
You know, speaking of Niels Bohr,
he had this famous quote that he said to somebody,
there is no doubt that your theory is crazy.
The question is whether it's crazy enough
to describe reality.
So that's where we are kind of, you know.
Speaking of crazy and crazy enough,
let me ask for therapy, for advice, for wisdom,
in returning to Eric Weinstein
and maybe give some guidance, understanding his view
on his attempt at theory of everything
that he calls geometric unity,
that he told me that you may have some inkling
of an understanding of.
If you were to describe this theory
to aliens that visited Earth,
how would you do it?
Or you could try if it was just me visiting Earth.
How would you describe it?
What is your best understanding of it?
He shared with me some of it
when I was in New York at Columbia, like 11 years ago.
We actually spent a lot of time where he explained to me
and I found it beautiful.
He has a very original idea at the core of it,
where you have this, instead of four-dimensional,
instead of 10-dimensional, he has 14-dimensional space.
And I thought it was really original.
And this exactly goes to the point I made earlier
that we need new ideas.
I feel that without some fundamental new idea,
we won't be able to get closer
to understanding our universe.
Now, I have a problem with the whole idea
of theory of everything.
I don't believe that one exists,
nor that we should aim to construct one.
And I think it's really, not to offend anybody,
but it's ultimately a fault of education system
of physicists.
Like in mathematics, we're not brought up,
we're not educated as mathematicians with the idea
that one day we will come up with the theory of everything.
Even though, as a joke, I said that Langlands program
is mathematical theory of everything.
But I made it kind of a tongue in cheek.
But isn't it a little bit kind of that?
It's not really, because first of all,
it doesn't cover all fields of mathematics.
And it covers specific phenomena.
But isn't it spiritually striving
towards the same platonic form of a theory of everything?
Like connecting fields?
Connecting, but connecting doesn't mean
that it covers everything.
So you could connect two things,
and then you have infinitely many other things
which are outside of the purview of this connection.
That's how it is in mathematics, I feel.
And I would venture to say that most mathematicians
look at it this way.
There is no idea that somehow,
I think it's actually impossible
because we're not talking about such a thing
as like one universe.
We're talking about all possible universes
of all possible dimensions and so on.
It is just not feasible to have a unified,
unify everything in one equation.
Now, physicists, on the other hand,
have been brought up, educated for decades with this idea.
And to me, and I am not sure I should say that,
but I feel like it's kind of an ultimate ego trip.
So that I have come up with the unified.
I have found the unitary theory of everything.
It's me, and my name will be on it.
I think a lot of physicists get educated this way,
especially men, take it seriously.
And I've seen that happen,
and I think it is counterproductive.
I think that, a lot of people agree,
that this debate is kind of,
I feel like it's kind of settled.
I hear it less and less,
but I disagree with the whole premise.
So you, it's interesting,
because both are interesting points you made,
which is you don't think a theory of everything exists,
and you don't think the pursuit
of a theory of everything is good.
So I think you spoke to the second thing,
which is basically that the pursuit of a theory
of everything becomes like a drug to the human ego.
That's right, so it's a huge motivating factor.
I don't deny that, but I feel that there are better ways
to motivate people than like that, than this way, okay?
So I would say, for instance, if one,
because then it's not a game of winner takes all,
in some sense.
And in fairness, when physicists say theory of everything,
grand unified theory, they mean something very specific,
which is unifying the standard model
and Einstein's relativity theory,
which is the theory of gravity.
So they don't necessarily,
a lot of physicists may say these words,
but they don't really mean them.
I think it's important to realize that,
that in my opinion, that's not productive
and it's not feasible anyway.
So having said that, there are some theories
are better than others, obviously.
So for instance, Eric's theory has, as far as I understand,
does have a certain way of producing
some of the elementary particles that we see,
and as well as the force of gravity.
So it does have that promise.
I feel that, at least from the place
where I had seen it about 10 years ago,
it still required a lot of work to get to the point
of actually saying that it does work,
because there are a lot of elements.
It's a huge enterprise to have a theory,
because you have, just to describe the field,
sort of the building blocks of the theory,
it's already a tremendous undertaking.
And he's trying to do it for curved spaces
in great generality, which is what makes it so unique
and so beautiful.
But then, on top of that, there are all this issue
of quantization, of actually describing them
as quantum field theory.
And the quantum field theory, even as a language,
as a framework, is currently incomplete, in my opinion.
And not only my opinion, it's like everybody agrees on that,
that it's a collection of tricks, so to speak.
It's a collection of tools.
It's a toolbox, but it is not a consistent way
of rigorous theory, like number theory in mathematics.
Physicists have still been able to derive predictions
from it and inform them to great accuracy.
But the underpinnings, it doesn't have
the real rigorous foundation from mathematical perspective.
So in that sense, even if, in that framework,
a new theory could lead to a new explanation
of some phenomena, it would still be incomplete,
in a sense, because it wouldn't be
mathematical rigorous, you see what I mean?
Because the whole framework is not yet on a firm foundation.
So it's not consistent.
Why is it that the universe should have?
So that's to your first point.
Do you think the universe has a beautiful clean?
When you show up and meet God
and there's one equation on the board
and the two of you just chuckle,
do you think such equation exists?
Yeah, there are such equation, for instance,
let's say I am interested in a particular question, right?
So in the language program, so moving away from physics,
so let's talk about math.
So in the context of language program,
I have recently developed with my co-authors,
Etengof and Kashdan, a kind of a new strand,
a new flavor of the language program, if you will.
But so far, it's a vision, it's a set of conjectures,
which we have proved in some cases,
but not in full generality.
So yes, I would like to use your framework,
meet the creator, and ask her,
what is the explanation of this?
And it may well be that she will answer
in a way that I will just burst out laughing.
It's like, how could we not see it, you see?
So that I totally see.
But I don't see one equation governing,
one equation governing them all.
Not one equation to govern them all,
but it does seem that such equations exist,
where she will tell you something
and you look back and say, how could I not see it?
It seems like the truth at the end of the day is simple,
that we're seeking, especially through mathematics.
It seems somehow simple.
The nature of reality, the thing that governs it
seems to be simple.
I wonder why that is.
And I also wonder if it's not totally incorrect,
and we're just craving the simplicity.
And then mixing into the whole conversation
about how much the observer that craves simplicity
is part of the answer.
It's a whole big giant mess.
Or a whole big beautiful painting or symphony.
You said of Eric Weinstein that,
I find it remarkable that Eric was able to come up
with such beautiful and original ideas,
even though he has been out of academia for so long,
doing wonderful things in other areas,
such as economics and finance.
I'd like to use that kind of quote
as just a question to you about different places
where people of your level can operate.
So inside academia and outside.
What is the difference of doing mathematics
inside academia and outside?
Not even mathematics,
but developing beautiful original ideas.
Where's the place that your imagination can flourish most?
So the limitations of academia is,
there's a community of people that
take a set of axioms as gospel,
so it's harder to take that leap into the unknown.
But it's also, the nice thing about academia
is some of the most brilliant people in the world are there.
And it's that community,
both the competition and the collaboration is there.
Yeah.
I wonder if there's something you could say
sort of to further about this world,
that people might not be familiar with.
But I think you gave a very good description.
I'm not sure I could improve on it,
because I don't have an overarching theory of academia.
I definitely have been part of it,
and I'm grateful because it gives you
a great sense of security,
which comes with its own downside too,
because you kind of get a little disconnected
from the real world,
because you get tenure, so you feel financially secure.
They don't pay you that much, so to speak,
relatively speaking.
It's comfortable, but it's not that much.
But you can't be fired.
So there is something about this,
which I definitely have benefit from it.
People are not even aware what it's like to live outside,
where you don't have this type of security.
On the other hand, that also means
that we're lacking certain skills
that real people in the real world have developed
out of necessity to deal with that sort of insecurity.
So it kind of always cuts both ways.
With one hand, it gives,
and with the other hand, it takes away.
It's a very interesting setup.
And also, on the one hand,
we are all supposed to be the truth seekers.
But in reality, of course, it is a human activity,
and it is a human community
with all kinds of good, bad, and ugly things that happen,
a lot of them under the radar screen, so to speak.
But maybe there is something to it.
There are definitely people who are upholding
kind of the old tradition, definitely.
And that is inspiring.
And I aspire to be one of those, do my best.
So whether this is a system that will stay or should stay,
I don't know.
I really don't know.
That's really fascinating.
Yeah, it's fascinating what, especially with it,
just to introduce the bit of AI poison into the mix,
as that changes the nature of education, perhaps, as well,
what the role of the university is
in the next 10, 20, 50, 100 years, I wonder.
I wonder.
And I wonder that, how did it make sense
that Einstein was working, after attempting, I believe,
to be a university professor?
It was in the patent office.
Yeah, as a patent clerk.
But I have to say, these days,
the science has become so much faster.
It is really hard to do it being outside.
Now, Eric is unique in this way.
Even though he did go to great undergraduate and graduate
schools, and then worked for a while in academia,
there are very few examples like that.
There's Yutong Zhang, who proved an important conjecture
in numbers theory about 10 years ago,
and is now, as I understand, is a professor
at UC Santa Barbara.
He worked outside of academia and was able to make
tremendous advance on his own.
This case is exceedingly rare.
In part, because academia is trying to protect its turf
and it's creating the prohibitive cost of an outsider,
that is true.
But there is also something about how much concentration.
In mathematics, I don't think people who are not
in the field understand what kind of focus and concentration
actually doing mathematics at the top level these days
requires, because we're not talking about something
that is more or less good.
It is something which is unassailable.
It's finding this treasure at the bottom of the ocean
without the alkaline, without oxygen.
And that's why people go crazy sometimes.
There is a reason for that.
Let me ask you about that, just to linger on that,
the amount of concentration required.
Cal Newport wrote a book called Deep Work.
He's a theoretical computer scientist.
He took quite seriously the task of allocating the hours
in the day for that kind of deep thinking.
And then the mathematicians is theoretical
computer scientists on steroids.
So for your own life and what you've observed,
let me ask the big question, how to think,
how to think deeply, how to find the mental,
psychological, pragmatic space to really sit there
and think deeply, how do you do it?
In the moments you remember where you really
deeply thought, was it an accident, was it deliberate?
No, it's deliberate because, first of all,
my first years as a mathematician, I worked every day,
weekends, holidays, doesn't matter.
I didn't even question that.
So I would feel something's missing if I took a day off.
And so it was just a kind of a sustained effort.
The point is that still the process is nonlinear
to go back to what we discussed earlier,
that in other words, the way I see it is you just,
you are making an effort to bring all the information
into focus, what you believe is correct.
And you're playing with different ways of connecting things.
But it is a total miracle when suddenly
there is insight strikes.
It is not something that, in my experience,
could be predicted or even anticipated or brought closer.
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, hmm, how interesting.
So what a coincidence that this came to him
when he was whistling.
But in fact, it's not.
This is how it works in some sense,
that you have to prepare for it,
but then it happens when you stop thinking, actually.
The moment of discovery is the moment when thinking stops.
And you kind of almost become that truth
that you're seeking.
But you cannot do it by will in some sense.
It's kind of like, you know how in the Eastern tradition
they have this concept of Satori,
like in Buddhism, in Zen Buddhism,
Satori, which is enlightenment.
And so there are various reports of Buddhist monks
or Buddhist pastors who have experienced Satori.
But they say you can't do it by will.
You cannot make it happen.
If anything, you have to relax to let it come to you.
It's kind of like that.
It's kind of like that.
So I think that what matters,
but you say how to think.
The point is that we're talking about
such an esoteric area.
Mathematics is really esoteric area.
It's a really strange subject
where you try to fit everything
in this very, very stringent set of rules.
To obey those rules.
Isn't it basically the pure, the hardest manifestation
of a puzzle that we're all solving
in different other disciplines,
but this is the hardest puzzle.
Yes or no, because there is just a different,
for instance, there is a different criterion
for what constitutes progress.
For instance, physics, a lot of arguments they make,
they are not rigorous from mathematical perspective.
It is kind of an intuitive argument.
We think it is like this.
And this is acceptable in the subject for a good reason.
And so there is some play.
It's more like human activity, day-to-day activity.
For instance, if you and I discuss something,
you have an idea, and I have an idea,
and we argue about it, and something seems more plausible.
Something seems less plausible.
And so we may decide to take this point of view
or that point of view as a provisional point of view
and go with it.
In mathematics, it doesn't work this way.
You either prove it or you don't.
And oftentimes, you get to the point
where there is this much you need to prove,
and it just wouldn't come to you.
And you just don't see it.
And it can go on for months.
Super frustrating.
But without it, it is nothing, kind of.
I would love to hear your opinion
to the degree that you know it
of the proof of Fermat's Last Theorem by Andrew Wiles,
which seems to have this element perhaps for years.
To the degree that you know,
perhaps can you explain Fermat's Last Theorem
and what your thoughts are in the process
that Andrew Wiles took that seemed to,
at least from my sort of romantic perspective,
seem to be very lonely.
Yes, it's a lonely profession.
And hopeless, and sort of the,
you put it really nicely because it feels like
there's a lot of moments where you feel like you're close.
You feel like 99% is done.
And there is this one stubborn thing
which just does not compute, it doesn't happen.
And you're trying to find that, push for this last link.
And it could take, and nobody knows
how long it's going to take.
Would it be useful to maybe try to explain
Fermat's Last Theorem?
Sure, it's easy to do.
I am an optimist, I'm an optimist.
I think, I always think that everything can be explained.
Even though I say that not everything can be explained.
But in mathematics, within this particular framework,
I think that I always feel optimistic
when people ask me to explain something.
I always start with the assumption
that they will understand, you know?
So let's try.
Fermat's Last Theorem, one of the jewels
sort of mathematics of all time.
A beautiful story also behind it.
Pierre Fermat, a great French mathematician
who lived in the beginning of,
mostly worked at the beginning of 17th century.
And he actually has to his credit
a number of important contributions.
But the most famous is called Fermat's Last Theorem,
or Fermat's Great Theorem.
And the reason why it became so famous
is in part because he actually claimed
to have proved it himself.
And he did it on the margin of a book that he was reading,
which was actually an important book by Diophantus
about equations with coefficients in whole numbers.
And he wrote on the margin, literally,
this equation, you know, this problem,
which I will explain in a moment,
I have solved it, I have found a proof,
but this margin is too small to contain it.
At some point, I was giving a public talk about this,
and as a joke, I made a tweet in which I wrote
that I have proved this theorem,
but 280 characters are not enough,
and it kind of cuts me in mid-sentence.
So this was 17th century Twitter-style proof, okay?
But a lot of mathematicians took it seriously
because he had great credibility.
He did make some major contributions.
And the search was on.
So for 350 years, about 350 years,
it remained unproved with many people trying and failing
until in 1994, no, in 1993, Andrew Wiles announced,
a mathematician from Princeton University,
announced the proof, and it was very exciting
because he was one of the top number theorists in the world.
And unfortunately, about a year later, a gap was found.
So this is exactly what we were talking about earlier.
You have 99% of the proof.
This one little thing does not quite connect,
and this nullifies the whole thing,
even though, well, you could say
there are some interesting ideas,
but it's not the same as actually having a proof.
So he apparently was really frustrated,
and he was really, a lot of people thought
that it's going to be another 100 years or whatever.
And then luckily, he was able to enlist,
with the help, assistance of his former student,
also great number theorist, Richard Taylor,
they were able to do that 1%, so to speak.
Well, some people might say it may be not one,
but 5% or whatever, but it definitely
was an important ingredient, but it was not,
he had a sort of like a big new set of ideas,
and this one thing didn't pan out.
They were able to close it with Taylor,
and it finally was published,
and I think it was accepted and refereed in 95,
and is believed to be correct.
Now, what he proved actually was not Fermat's theorem.
It's not a theorem itself, but a certain statement,
which is called Shimura-Taniyama-Wei conjecture,
named after three mathematicians,
two Japanese mathematicians,
and one French-born mathematician who worked
also at the Institute for Advanced Study in Princeton.
And it was my colleague at UC Berkeley, Ken Ribet,
who in the 80s connected the two problems.
So this is how it often works in mathematics.
You want to prove statement A.
Instead, you prove that A is equivalent to B.
So after that, if you can prove B,
this would automatically imply that A is correct.
So this is what happened here.
A was Fermat's last theorem,
B was Shimura-Taniyama-Wei conjecture,
and that's what Andrew Wiles and Richard Taylor really proved.
So it requires, to get to Fermat's last theorem,
it requires that bridge, which was established
by my colleague Ken Ribet at UC Berkeley.
So now, what is the statement of Fermat's last theorem?
Let me start with Pythagoras.
Since we already talked about it,
let me start with Pythagoras' Theorem,
which describes the right triangles.
So what is a right triangle?
It's a triangle in which one of the angles is 90 degrees,
like this.
So it has three sides.
The longer side is called hypotenuse,
and then there's two other sides.
So if we denote the lengths of hypotenuse by z
and the two other sides x and y,
then z squared is equal to x squared plus y squared.
So that's the equation,
or x squared plus y squared equals z squared.
And it turns out that this equation
has solutions in natural numbers,
actually infinitely many solutions in natural numbers.
For example, if you take x equals three, y equals four,
and z equals five, then they solve this equation,
because three squared is nine, four squared is 16.
Nine plus 16, 25.
And that's five squared.
So x squared plus y squared equals z squared
is solved by x equals three, y equals four, z equals five.
And there are many other solutions of that nature.
And we should say that natural numbers
are whole numbers that are non-negative.
That's right.
One, two, three, four, five, six, and so on.
Now, what's Fermat's last theorem?
Fermat asked, what about what will happen
if we replace squares by cubes, for example?
So x cubed plus y cubed equals z cubed.
Are there any solutions in,
what do you call, natural numbers?
It turns out there are none.
What about fourth powers?
Again, none.
Well, it seems like none, right?
So that was the statement.
So this theorem says that the equation
x cubed plus y cubed equals z cubed
has no solutions in natural numbers.
Remember, natural means positive whole numbers.
So of course there is a trivial solution, zero, zero, zero.
So this works, but you need all of them to be positive.
X to the fourth plus y to the fourth equals z to the fourth
also has no solutions.
X to the fifth plus y to the fifth equals z to the fifth,
no solutions.
So you kind of see the trend.
X to the n plus y to the n equals z to the n.
If n is greater than two,
has no solutions in natural numbers.
That is a statement of Fermat's last theorem.
Deceptively simple as far as famous theorems are concerned.
You don't need to know anything beyond standard arithmetic,
addition and multiplication of natural numbers.
That's why a lot of people,
both specialists and amateurs, tried to prove it.
Because it's so easy, it's certainly so easy to formulate.
So in fact, I think Fermat proved the case of cubes.
I think he did actually prove some elsewhere,
the case of cubes.
But so it remained like fourth,
there are infinitely many cases, right?
You have to, even if you prove it for cubes
and for fourth power and fifth,
then still there is six, seven, and so on.
There are infinitely many cases
in which it has to be proved.
And so you see, the separately simple result
took 350 years to prove.
And you know, but in a sense, it's like mathematicians,
you know, you would think mathematics
is such a sterile profession.
Everybody's so serious, you know,
almost like we're all wearing like lab coats
and like take an elevator to every tower.
And however, look at all this drama.
Look at all this drama.
It's like we also like drama.
We also have narratives.
We also have our myths.
Here is a guy, he's a 16th century mathematician
or 17th century mathematician who leaves a note
on the margin and motivates others to find the proof.
Then how many hearts were broken
that they believe that they found the proof.
And then later it was realized
that the proof was incorrect and so on
and brings us to modern day and one last attempt
and reviles who is very serious
and respected and esteemed mathematician
announces the proof only to be faced
with the same reality of his hopes dashed, seemingly dashed
and like there is a mistake, it doesn't work.
And then to be able to recover a year later,
how much drama in this one story?
It's amazing, but from what you understand,
from what you know, what was the process for him
that is similar perhaps to your own life
of walking along with a problem for months?
Not years.
Yes, so he worked, he has given interviews
about it afterwards.
So we know that he described his process
that number one, he did not want to tell anybody
because he was afraid that people find out
that he's working on it.
Because he was such a top level mathematician,
people would guess that he has some idea,
that there is some idea.
So, you know, if you just know that somebody has an idea,
this already gives you a great boost of confidence, right?
So he didn't want people to have that information.
So he didn't tell anybody that he was working on it,
number one.
It's been lonely.
Number two, he worked on it for seven years,
if I remember correctly, by himself.
And then he thought he had it and he was elated,
obviously he was very happy.
And he announced that at the conference,
I think it was in Cambridge University
or Oxford University in the UK in 1993, I believe.
So, you know, this is really interesting
because all of us, all mathematicians can relate to this
because I remember very well my first problem,
how I solved my first problem.
I describe it in Love and Math in my book.
So it was, how old was I?
I was 18 years old.
I was a student in Moscow and I just lucked out
that I was introduced to this great mathematician
since I was not studying at Moscow University
because of antisemitism in the Soviet Union.
So I was in this technical school,
but I was lucky that I had a mathematician
who took me under his wing and Dmitry Fuchs,
who actually later came to the US
and he's still a professor at UC Davis, actually.
Not so far from me.
So he gave me this problem and it was rather technical,
so I will not try to describe it,
but I do remember how much effort and all that excitement,
but also kind of a fear.
What if I don't have what it takes, you know?
I lost sleep, so this was one consequence of this.
For the first time in my life,
I had trouble falling asleep
and this actually stayed for a couple of years afterwards.
So then it was kind of like a wake-up call
that I should be, take care of myself,
not work too late and so on.
So that was sort of like that experience.
And I was lucky that I was able to find a solution,
number one, within two months maybe.
And it was surprising and it was beautiful.
The answer was in terms of something
which seemed to be from a different world,
from a different area of mathematics.
So I was very happy.
But I do remember this moment when suddenly you see that.
Just like, in this case, it was literally,
I had to compile these diagrams
with what mathematicians call cohomology groups
and spectral sequences and manually calculate some numbers
and trying to discern some system in it.
And suddenly I saw that, how they all were governed
by this one force, so to speak, one pattern.
And that was absolutely, wow.
So it's like.
I mean, what was it?
So you're sitting there at a desk.
Actually, I lived in a town outside of Moscow.
So I would take a train to Moscow.
So it's what we call in Russian, electric,
this electric train, which was super slow.
It took more than two hours to cover that distance.
And I think that the crucial insight came when I was in this.
And I had to contain myself so I don't start screaming
because there were other passengers.
In the car.
So I was sitting there and staring at this paper.
So you know what I remember, that's what came to me.
I have something now which nobody else in the world has.
I have a proof of, first of all, it was not just a proof.
Like in the case of Fermi, the statement is already made.
That's why it's called conjecture.
You make a statement, you don't have a proof yet.
Then you try to prove it.
In my case, I did not know what the answer would be.
There was a type of question where the answer was unknown.
So I had to find the answer and prove it.
And the answer was very nice.
So nobody knew, as far as I could tell, nobody knew
because my teacher told me that he explored
all the literature and this was not known.
Suddenly I felt that I was in possession of this.
Now, it was a little thing.
It was not cure for cancer.
It was not a large language model.
But it was something undeniably real, meaningful.
And it was mine, kind of.
I had it, nobody else.
I had not published it yet.
I didn't even tell, hadn't even told anybody.
And it is a very strange feeling to have that.
Were you worried that this treasure could be stolen?
Not at the time, not at the time.
So later on, there were situations where I was exposed
to those type of experiences.
But at that time, I didn't think of that.
I was still this starry-eyed kid
who was just obsessed with mathematics,
with this beauty and discovering those beautiful facts,
beautiful results.
So I didn't think about,
I didn't even think that it could be possible
that somebody could steal it or whatever.
I just wanted to share it with my teacher
as soon as possible.
And he understood quickly.
And he's like, yeah, good job.
Is there something you can give color to the drama,
Eric Weinstein has spoken about some of the challenges,
some of the triumphs and challenges of his time at Harvard.
So is there something to that drama
of people stealing each other's ideas
or not allocating credit enough?
Oh, sure, yes.
All of that and creating psychological stressors
because of that.
Happens all the time, yes, unfortunately.
On young minds and so on.
Could have a very bad effect.
Is that just the way of life or is this?
I think we can definitely do better.
And I think the first step is to kind of admit
that we are not 100% seekers of truth,
that we are human beings and all the good and bad
and the ugly qualities can be present.
And to have some kind of dialogue in my subject,
in mathematics, this has not happened yet.
There have been some famous cases
where people have been accused, which had been resolved
or partially resolved or unresolved and everybody knows it,
but there isn't a systematic effort as far as I can tell
of really trying to create some rules, some ethics rules.
This is fair game, this is not fair game.
So that as a community, we strive to get better.
I think that for most people,
it's more like keeping your head in the sand
and kind of pretending that it doesn't happen
or it happens some isolated incidents.
Well, my experience is not like that at all.
I think it does happen much more often than it should.
That's my opinion.
So there's the pool of academia is fascinating.
One of the reasons I really love it
is you have young minds with fresh ideas
and that same innocence you had
with when you first on the train
have that brilliant breakthrough.
And then you throw that in together
with senior exceptional world-class scientists
who have, first of all, are getting older.
Second of all, maybe they have partaken
in the drug of fame and money and status and recognition.
So that starts to a little bit corrupt
all of our human minds.
And you throw that mix in together.
Yeah.
Mostly without rules.
And it's beautiful because that's where the ideas of old
contend with the new wild-eyed crazy ideas
and they clash and there's a tension
and there's a dance to it.
But then there's the old human corruption
that can take advantage of the young minds.
It's unclear what to do with that.
I mean, part of that is just the way of life
and there's tragedies.
And oftentimes when you look at who wins the Nobel Prize,
it's also tragic because sometimes so many minds are,
the trajectory to the breakthrough idea
involves so many different minds, young and old.
Yeah, I mean, you're right.
I think it's like everything else.
The path is to more self-awareness
and it's like owning up your own stuff
and not blaming other people,
not projecting onto other people,
but taking responsibility.
And that's true for everything.
And the problem here, unique problem for mathematics,
I would say physicists and chemists are better.
They actually have better sort of ethical rules and so on,
especially biologists.
Because I think in part it's because
there is much more money involved
because they have to get grants and so on.
So for them, the question of priority,
who discovered what first is much more serious
because there's really some serious money.
Mathematics, who cares?
You know that Fermat's Last Theorem was proved?
Did Andrew Wiles become a millionaire?
No.
As far as I know.
I think he got a prize.
He won a prize, but those prizes are not.
I think that one was a big prize,
but in general, there's not going to be.
I think he won the Nobel Prize eventually,
which is about a million dollars.
So, but, you know, sometimes I joke about this,
that this is the hardest way to win a million dollars.
So, you know, but amongst mathematicians,
I think the trouble is that we are so insulated
from society because it's such a pure subject.
It draws in very specific psychological types.
And I can speak about myself.
I did not realize it at the time,
but later on, I definitely saw,
I mentioned some of it earlier,
that for me, mathematics was a refuge
from the cruelty of the life I experienced,
from discrimination that I experienced
when I applied to Moscow University at 16,
being failed at the exam and stuff like that,
which I describe in my book as well.
But that was my way.
I was like, I don't trust this world.
I don't want to deal with it.
I want to hide in this platonic reality of pure forms.
This is where I know how to operate.
I love this.
And I couldn't be bothered in some sense.
For a while, up to a point,
as I was getting older and more mature,
I was becoming more and more interested in other things.
But I think that's one of the reasons.
And one of the reasons why I wrote Love and Math
was precisely to break that cycle,
that it's the quiet guy in the corner that goes into math
and not the flamboyant jock or DJ.
I wanted to show how beautiful the subject is,
to attract this new blood
so that different psychological types
and more women would join.
Because then they would have students who would look at them
and whom they will inspire.
And then it would be, instead of a vicious circle,
it would be a virtuous circle.
And I have to say, I think it's happening,
not because of my efforts alone.
Obviously, there are many other mathematicians
who are around the same time, started to put more effort.
Because see, if the old stereotype of mathematician,
you're so enclosed, you're not interested
in even exposing the beauty of your subject to other people.
And then it becomes this vicious circle.
But this one day, not one day,
all the time I meet the students who say,
your book is the reason why I chose math as my major.
And I am proud, especially when it's women who tell me this.
And they are cool, they are DJs at the same time,
and they are social and they have friends
and they go out and so on.
You see, so then they carry the torch
because then they will be more likely
to share this beauty with others,
to attract more students and so on.
So I think this dumbness is broken.
So now you'll have more influx.
And once we have people who are more able to connect
at the personal level,
that's when we also become more self-aware
as a community, I think.
And that's when we might be,
we should be able to have a chance to improve
in terms of our ethical rules and stuff like that.
So let me return to our friend Eric Weinstein
for a question that I would ask anyway.
But let me, let's have a non-Russian
ask the Russian question.
Ask him about the Russian concepts of friendship,
science, gender, and love versus the American.
You can, you can, so there is a deep romanticism
that you have that runs through your book, Love and Math.
Is part of that something you've picked up
from the Russian culture?
What can you speak to that fueled both your fascination
with math and your fascination, no,
your prioritization of the human experience of love?
Good question.
I definitely, there is some influence
of the Russian culture, Russian literature, perhaps.
But also, there's so many things.
How do we develop certain sensibilities?
Why do we care about this and not that?
Why do I care, for instance, about, like you said,
about this romantic ideal, so to speak, of mathematics?
That's certainly not something that is automatic.
Some people care about it, some people don't.
And I'm not saying it is superior or inferior.
It's just how my composition,
my psychological composition is like that.
And it's an interesting question.
What is the cause of it?
So I think that we cannot really know,
but there are some aspects of it, of course.
The experiences, life experiences are upbringing, family.
Like I was surrounded by love by my parents on the one hand,
but on the other side,
perhaps they were a little overprotective of me.
So I was kind of like too much taken care of.
So then when I had developed some sensitivity,
but I was kind of not ready for the challenges.
Of the real world.
So then that struggle and then being lost
and then being able to overcome and to learn.
And then if you don't lose, you don't appreciate maybe.
But sometimes when we lose something and then regain it,
then we cherish it, we appreciate it,
and then it becomes something important.
Also various difficulties, the upsetting experiences,
or one could say traumatic experiences.
Growing up in the Soviet Union,
that was not a walk in the park.
There were a lot of issues there that I had to go through.
And then it doesn't break you, makes you stronger.
But in my case, what happened was that,
for some of it, it took me 30 years
to really come to terms with it
and to really understand what happened.
It gave me this motivation
to strive to become a mathematician,
which maybe I wouldn't have otherwise.
It charged me, supercharged me.
I'm talking about, for instance,
the experience with exam at Moscow University.
Can you take me through that experience?
So this is 1984, we spoke about Orwell earlier.
And I was applying to Moscow University,
mathematics department, it's called Mehmat.
Which is like, for people who don't know, the place.
It was the only place to study pure mathematics
in Moscow, period.
But also considered to be one of the great places on Earth.
And it's like a huge building,
this monolith of a building of Moscow University.
Because as I said, a year earlier,
Evgeny Evgeny, which converted me into math,
capitalizing on my love for quantum physics.
And so I spent a whole year studying with him
and I was already kind of at the level of,
in some subjects, a level of early graduate studies.
So it seemed like it would be a breeze
to get into Moscow University.
But in fact, little did I know that there was
a policy of antisemitism where students like me
would be failed by special examiners.
Mostly during the oral exam with mathematics,
but occasionally it would be written tests and stuff.
Now, my father is Jewish by blood.
It was not religious.
His family was not religious.
My mom is Russian.
But since my last name was my father's name,
so it was very easy to read what my nationality was.
And so can you imagine there were special people
who would screen up applicants,
who would put aside the files of the undesirables.
There would be special examiners
who were actually professors at this university
who would be designated as those who would take the exam
from those undesirables.
It's almost comical when you look back now.
And also questions of why.
There was no reason other than just hatred of the other.
That's how I see it.
It's just you give a little bit more color.
So because you mentioned nationality,
it's a little quirk that perhaps gives an insight
to the bigger system,
that the nationality listed on your birth certificate
when you're Jewish is Jewish,
and when you're non-Jewish is as Russian.
For me, it was Russian.
So first of all, in the inner part,
everybody has an internal passport.
And there you have first name, patronymic name,
last name, date of birth, so these are four.
And the fifth colon is nationality,
which comes from the nationality of the parents and so on.
In my case, it was written Russian
because my mom was Russian.
But it didn't save me.
Because that was my dad's last name.
And so anyway, this was the toughest experience
that I had up until that point.
And there was these two people who came into the room
where I was the only undesirable.
All other kids were being questioned by other examiners,
but they told me that we cannot question you.
We are waiting for special examiners.
So I was like, uh-oh, something is afoot.
And so these two guys came, and for four hours, basically,
and were asking me questions
which were not in the program and so on.
But anyway, I was a kid, I was 16 years old.
I tried to answer the best I can, but it was a setup.
It's been documented since then.
There are even lists of problems
that were given to undesirables in those days.
In my year, no Jewish applicants, as far as I know,
Jewish, by this metric, were accepted.
So then I had to go to this...
There was one school, technical school in Moscow,
which was the Institute for Oil and Gas Exploration,
which had an applied mathematics program.
And that's where me and many of the kids
who were not accepted to Moscow University ended up.
And so, but the point is, so...
And then I was so motivated by this,
because I wanted to show those guys, you know,
that within five years, less than five years,
I got a letter from the president of Harvard University
inviting me as a visiting professor to Harvard.
I was 21, I was barely 21,
because I already did some research in the meantime.
That's how motivated I was, you know?
But the interesting aspect of it is that
for the longest time afterwards,
I was telling myself a story that nothing really happened.
It wasn't so bad.
Okay, so I was failed,
but I knew that I was going to succeed.
It was 30 years later that I finally got to meet that boy,
that 16-year-old that I neglected this time.
And I realized that he died, that it was a crushing blow.
The innocence?
Not just the innocence, because there was no way,
it looked like there was no way
I could become a mathematician,
because if they don't accept me there, it's over.
I didn't know that I could actually find
this thriving applied math program,
and then eventually somebody would take me under his wing
and so on, and then could move to the United States.
That was not in the realm of possibilities.
There was nothing to look forward to.
It was clear that it's over.
I cannot do what I love.
And so when I finally connected to that boy, oh my God,
that was a totally different experience.
All the pain and all the trauma came to the surface,
and it was a kind of tsunami.
I wasn't sure I would survive this.
It was so hard.
And what happened was I was invited
to give a talk about this in New York.
It was kind of a spoken word event about science,
but personal experiences related to science.
This was almost a year after my book came out.
In my book, one of the first chapters
is the chapter about this experience.
But what I realize now is that I wrote it
from the third-person perspective.
I knew the facts, but I was not emotionally connected
to that experience.
However, since I wanted to write the book
and to connect to my readers, I allowed the boy to write it.
So a lot of people were touched by it,
and people would say, wow, that chapter,
it really got a lot of resonance.
It was translated into other language
even before the book was published.
I was surprised by this because I didn't know yet.
So the adult Edward was not yet in touch,
but the book gave the outlet to the child.
And that kind of started the process.
So finally, almost a year later,
I'm in New York at this event.
And the night before, I'm in my hotel room,
and I was like, okay, what am I going to talk about tomorrow?
And I take a piece of paper just to,
my usual preparation for things.
And then suddenly I have this vision
that I will walk up to the microphone tomorrow
and I will just start crying.
And I was like, by that time, I already had an insight
that it's possible to have that kind of a splitting,
kind of dissociation.
I was, but things were happening quickly.
There was someone in my life who explained to me this idea
that some things are under the radar of awareness,
but they may still influence you.
And a lot of that could be connected
to some experiences in your childhood.
So I was kind of ready for it from different angles.
But I was so surprised because I was like,
what is there to remember?
I know, I know everything.
So then my inner voice says, all right then,
you have nothing to worry about.
Go tomorrow and you will speak about this.
And if you start crying, it's not a problem.
I was like, no, I don't want to cry in front of people.
I want to find out what it is, what happened.
And I sat on my bed, closed my eyes and it came.
So it's hard to describe.
So this is what, and the sheer energy of it
and how much effort it took to suppress it actually
for all these years, how much effort it took
to build that, I want to say in Russian,
that hardcore around myself so that,
and the thing later I realized there were moments
when it could come out.
And for instance, I developed this fear of public speaking,
all kinds of little things that I now feel were connected.
So anyway, I saw what happened now
through the eyes of that child.
I saw how difficult it was, how crushed he was.
And it looked completely hopeless.
And I felt like, what's the point of living now for me,
now that I know how cruel this world is,
which I didn't realize before because I prefer
to wear this pink, the rose colored glasses.
But then something happened, it's so strange.
It's like you feel that inside of you
there is this dead child and it is incredibly sad.
I mean, it's like, I can't even describe it,
but suddenly he comes alive.
And suddenly it's like, oh, he's here.
And I had a little talk with him and I said,
look, I know, and now I want to thank you.
I'm so sorry that I neglected you for so long.
I didn't know, thank you for doing this.
And it's almost like, I felt like the image came to mind
is like a fallen soldier, like you leave a fallen soldier
on the battlefield, a wounded soldier,
and then you come back to take him with you.
And I said, but look, look what we have done.
Look at us now.
It was not in vain.
We are doing okay.
And it's kind of almost just like holding, holding,
holding that child and that sense of who I am, you know,
and feeling it.
So the next day I went to the microphone
and I let him speak for the first time about his experience
in his own voice.
It was incredible.
People were crying and afterwards came up to me
and started sharing stories and so on.
Because it is a story, it's a universal story.
It's archetypal story.
It's the story of rejection and being treated unfairly.
We all know it.
And I think it's so important to realize
that it's possible to revisit those moments.
It's possible to reconnect to our little ones.
It's possible to bring them back and we are better for it
because this changed my life, this experience.
Then suddenly it's like a floodgates.
There were many other things that came.
That's when I became interested in the dimensions
of imagination and intuition and so on.
Because suddenly I realized that I was deprived
of that possibility of looking at the world
through the eyes of a child
because that child was frozen in time.
I was not connected to him, but suddenly he's with me.
And he's like, almost like opens his doors and says,
look at, look at this, look at this.
So.
If I could ask you about, there's a difficult idea here.
There's a tension.
I've interacted with a few folks in my personal life
and in general that have lived through this experience
of unfairness and cruelty in the world as young people.
And what wisdom do you draw from the action you took
of not acknowledging that you were a victim to cruelty,
but instead just working your ass off, working harder?
And then the flip side of that is you eventually
reconnecting with the cruelty that you experienced.
Because if you did that early on.
I was not ready for it.
It is a defense mechanism.
I could have come, you know, there were kids
who commit suicide after this experience.
I could have come in suicide because it's too much.
And it is well-known afterwards, of course,
I became aware of all the literature
about childhood trauma and so on.
And I have been speaking publicly about it since then too.
And so, you know, it is well-known issue
and well-known kind of a universal phenomenon.
I think that, interestingly enough,
even though now I see a lot of discussion of it,
now that my eyes are open, but somehow before,
I didn't see it, which also shows you
how our confirmation bias, kind of like
how we screen ourselves, how we turn the blind eye
to things which do not confirm our views
or for which we are not yet ready.
And by the way, nobody should push to do it too soon.
I developed certain strengths.
I was confident.
I was strong to withstand this.
And if I weren't, who knows how it could turn out.
So it is a very subtle kind of alchemical process,
which I don't think there is a recipe, there's a formula.
The reason I'm talking about this
is just to share this experience,
because I think that the only thing we can do in this,
in some sense, is to share with each other,
because then we can find, for instance,
if somebody shared with me, it would naturally lead me
maybe to get closer to that kind of understanding.
It's really just personal stories.
It's not, obviously there is a component
where professionals could be involved,
professional therapists and so on.
In my case, it somehow happened miraculously.
Well, I did have support,
but not from professional therapists,
but from like dear friends.
So I did have, I had somebody at the time
who basically held my hand through this experience.
Yes, it was invaluable and it could not be done otherwise.
So I think it's very common.
And here's the thing.
I would not do it in any other way.
When I reconnected and I saw all the horrors and so on,
but I also was able to see that my examiners
were victims of their own situation,
that they became, they fell for this bogus theories,
or maybe it was more of an issue
of career advancement or something.
And I also realized they must've suffered as well,
because they must've had some kind of consciousness
about it, that acting in this way
towards sort of basically kids, you know?
So it wasn't pretty from their point of view.
So I could forgive them.
And I could like also appreciate
what a boost of energy it gave me.
If I was accepted and I was just,
where I was a first year student,
I would live in a dorm because I was in,
you know, I'd be probably partying and drinking
and who knows what,
maybe I wouldn't even become a mathematician.
But this focused me like a laser
without me even thinking about it.
It just happened.
I didn't care about anything but doing mathematics.
And it paid off, you know?
It changed my life.
So was it good or bad?
Paradox.
Seems like life is full of those.
You said you lost your father four years ago.
Yeah.
What have you learned about life from your dad?
That's another big one.
Yeah, because I was very close to him.
And it was tough, it was tough.
And I was not, I was sort of not ready for it,
because up until that point,
I lived pretending that death does not exist.
When my grandparents died, I was already in the US.
So it was very convenient and I couldn't go back.
So I grieved, but it kind of was a bit abstract for me.
I didn't see their dead bodies, you know?
I didn't bury them and so on.
So I waited till the latter morning.
So the first death in my life was my father,
like of really close loved ones.
And I was absolutely devastated.
You know, he was such an amazing creature,
such an amazing human being.
He was the kindest, the smartest, the most funny,
just really funny and just really fun to be with, you know?
This is what I miss, obviously.
I mean, I would just love to hang out with him.
So, and then suddenly he's not there.
So it was tough, but it kind of changed my perspective.
You miss him?
I miss him tremendously.
I miss him tremendously, but in a way,
I learned that he never left me in some sense.
I mean, it sounds so, ugh.
Words are so, you know, like they are,
they cannot express in words this, what I'm trying to say.
But-
Do you carry him with you?
Yeah, and in some sense I always did.
And I saw that, that it's always been,
it was really, we were one in some sense, you know?
Like we were, but there was this experience
of two people being together,
and that I missed tremendously.
But he gave me so much.
And, you know, let me tell you one aspect of it,
for instance, when he was a kid,
his father was sent to Gulag on bogus pretenses, right?
So he, when he was 16, he applied to a university.
He wanted to become a theoretical physicist.
By the way, my love for theoretical physics
was to a large extent because of that.
And he was not accepted, even though he was brilliant,
because he was the son of the enemy of the people.
And he kind of broke him, this experience,
that he didn't care when he was, you know,
he went to technical school and he didn't really care.
That's my take on it.
And then he ended up in this little provincial town,
and he thought he would escape from it as soon as possible.
And then he met my mother, and I fell in love.
And so I am sort of the product of that, you know?
But then what I learned is that
because he was not able to overcome
that specific experience, it fell to me to do it.
And if I didn't, my son or my daughter would have.
I think that that was one of the things I learned,
that it was not by chance, that about the same age,
for slightly different reasons,
I was subjected to the same kind of unfairness and cruelty.
And in some ways, I feel like I did it for him also,
because he was also always so proud of me and so happy.
And I had this tremendous gift, twice.
I was invited by American Mathematical Society
to give these big lectures, twice.
It was in 2012 and in 2018.
And both times, they were in Boston.
It could be anywhere in the US.
Both times, it was in Boston,
walking distance from my parents' place.
So he could be there, and my mom as well.
And that was such a gift, that he was beaming,
you know, like seeing me on the stage, like, ah.
So, you know?
Now that he's no longer here, and it's just you.
Well, I still have my mom, I still have my sister.
Yeah, but as a man, there's some aspect that it's,
that it does hit you hard.
Are you afraid of your own death?
Do you think about your death?
Are you afraid of it?
I have a certain conceptual view of life and death today,
which is informed by my experiences.
In particular, going through my father's death.
And that is something which cannot be conceptualized,
like that experience, like he cannot give it to somebody.
One thing I will say is that I felt that what it was,
it was actually love totally exposed, like naked.
And you try to throw, it is so acute.
So facing that love is incredibly painful
because it's so intense.
When a person is alive, we have conversation, we have wars,
we have some actions, we have some stuff that's going on,
and it puts a filter.
So we rarely actually feel love
in this totally, completely pure, unadulterated state.
But when a person dies, it's there.
And it's staring at you, and no matter what you do,
you cannot run away.
Like, I tried to, it's like, almost I felt like
I want to throw a blanket over it.
It burns, like immediately, like boom, gone.
It's there.
Live through it.
And I kept saying to myself, just live through it,
live through it.
And that's how you know, also learn what is love,
for example, what is it really?
What is love?
What is life also?
Because I was completely, I had no idea.
And then you kind of learn that, okay,
so maybe it's not quite,
there's more to it.
There is more to it.
There's more to this experience
than what can be put in a concept or in a sentence,
maybe poetry or music can do some justice to it.
But if so, then my own life has that component,
has that dimension,
which is beyond anything I can say about it.
And even though I love playing this role, I love it.
And it kind of makes me feel different
about all kinds of difficulties that arise,
because it's almost like I want to enjoy it,
because that's what being human is.
It's being terrified.
It's being frustrated.
It's being self-loathing sometimes.
It's not knowing, but also being joyful
and just like, ah, let's just enjoy it.
All of it, that's why you came here for, in some sense.
It's like not trying to run away from things,
but kind of trying to just live through them and appreciate.
So the biggest thing is gratitude, in some sense.
It's just gratitude, so thank you for letting me play.
Gratitude for every single moment, even if it's dark,
even if it's a loss.
Yes, and that's why I'm so,
people around me, they all say there is total doom and gloom,
and the world is ending.
And I'm like, first of all, that's how you see it, okay?
That's not the only point of view.
But also, even if it is, that's your challenge.
What are you going to do about it?
Stop complaining about other people.
Do something yourself.
How can you make it a better world?
And I think all of that starts
with just a gratitude for the moment,
to be able to play this game.
Yeah, how beautiful it is.
What, we've talked about love, but let me ask,
what role does love play in this whole game,
in the human condition?
It's like the glue, you know, it's like for me,
it's like the,
it's, and it's not because people say love
is like for a human being, like a romantic love,
which is a huge component of it, obviously,
because it's so, so beautiful
to be able to express it in this way.
But it could be love for what you do,
for your passion for something, you know?
And, or love for your friends, for instance,
or love, it doesn't have to be.
And so, in some sense, that's what it's all about.
And ultimately, because living without love,
it's kind of bland, boring, and so.
And I don't think it's possible for science
to explain exactly what it is.
You can do evolutionary biology perspective,
you can talk about some kind of sociology perspective,
psychology perspective, but the experience,
the intensity, where you forget,
where time, where reminded, becomes an illusion,
and everything just freezes.
And then, it's kind of beautiful and painful
to hear you say that, when you've experienced love,
the deepest is when you lost it.
Yes, but in a sense, you can say that
you could not have one without the other.
I could not have that deep connection with my father,
like really on so, so many levels.
If there weren't a moment, that's how I see it.
And I'm not trying to say that's how everybody should see.
For instance, I respect Ray Kurzweil.
I respect and I feel, and I almost like,
I feel good bumps right now.
I feel that desire to reconnect,
even if it is in the form of a computer program,
let's be honest about it.
I find it to be very moving.
I find it very moving.
And I understand because he actually didn't have a chance
to spend much time, I think he was 16 or 17 when he was,
he was a teenager when his dad died.
I was lucky because my father died.
I was much older.
I've had so many moments with him.
But that's not my thing.
Like, I think it is a feature, it's not a bug,
and it sounds crazy.
Like, I would love, I would give anything
to have him or he right now.
Right now, everything I have, I give it away.
Right now, where do I sign?
Just see him for one hour.
I promise you, I will.
But I also know that then I'll still lose him,
or I will die, or whatever, that thing.
So why is it so worse to just hold on to it?
Why?
Why are we holding on to this?
And I am the first sucker, I'm not the first one to hold on.
But I'm questioning it now.
Like, is there another way to approach life
where you just, you know how Buddha,
it's like, just let it go.
Enjoy, and let it go.
Enjoy, and let it go.
Is it possible?
Except the paradox of it.
Well, ask me in a couple of years, I will report.
But I think that, but to my mathematical mind,
it sounds like a very interesting idea, to be honest.
Because to me, the idea of holding
sounds like an impasse.
Because no matter, in all my experience,
and if you look in history,
every time somebody's holding,
you know it's how they said in the matrix,
whatever has a beginning has an end.
It's like, you cannot go around it.
If you have a beginning, you will have an end.
So then, might as well just enjoy it
and not worry too much about extending it longer.
That's how I see it now.
But maybe tomorrow will be something else.
You know?
Yeah, the rollercoaster of life, the paradox of life.
Edward, you're an incredible human being.
I've been a fan for a long time.
Thank you for writing Love and Math.
Thank you, thank you for being who you are,
being both one of the greatest living mathematicians
and still childlike wanderer of the,
exploring how this whole world works,
the nature of the universe.
And thank you so much for speaking with me today.
This is amazing.
It's been a pleasure.
Thank you.
Thanks for listening to this conversation
with Edward Frankel.
To support this podcast,
please check out our sponsors in the description.
And now, let me leave you with some words
from Sofia Kovalevskaya, a Russian mathematician.
It is impossible to be a mathematician
without being a poet in the soul.
Thank you for listening and hope to see you next time.
Thank you.