So people started to realize there are other geometries.

There's geometries in which the first 4 of Euclid's postulates are true and the 5th are not.

It's kind of like when you grow up and you're like, what if mom and dad don't know everything?

It is exactly right.

Like Euclid, right?

Like, who's your daddy?

Euclid.

Like, that was the answer for like 1,000 years of mathematics, right?

Exactly.

Jordan Ellenberg, welcome to Particles of Thought.

Thanks for having me on.

Yeah, man, so you are our first mathematician.

That is a big responsibility.

It is, it is, I've been waiting for you.

I'm here to represent 3,000 years of mathematical progress.

There you go, so all right, I'm glad you brought that up.

So here is a question that I have posed to people for the last 20 years or so, because I had a realization as a young scientist, right?

And I realized that, man, I think that how we portray math and think of math is completely wrong.

And so I started asking people this question.

And the question is, in one word, what is math about?

Okay, so the most common answer I get, not from people like you, but just regular people, what do you think it is?

I bet they say numbers.

Absolutely, absolutely.

Yeah, yeah.

And us overeducated folks, we say things like reasoning or logic or something like, right?

But that's the whole point, is that math is more than numbers.

And even media sort of reinforces that idea.

One of my favorite recent series was Foundation, right?

The Asimov series.

Oh yeah.

And so the dude Hari Seldon does all these calculations of the future.

And so when he and the young woman are talking about it, they're like, the numbers say this, the numbers say that.

And I'm like, I can guarantee you that that calculation is not expressed in numbers.

So what is math research?

What is math?

I only get one word?

No, no, no, no, you get to say what it really is.

Yeah.

Because I was going to say, that's too hard.

And it's always a tough— boy, you already opened up so many things.

I want to talk about whether you were like a Foundation nerd as a kid.

Were you like an Asimov guy?

I didn't— no, I knew he existed.

I read the hell out of those books.

OK, we can talk about that too.

And also, I always have a challenge with that question, because you're right.

People do think numbers— math is about numbers.

I am a number theorist.

Like, that is the kind of math that I personally do.

I am a guy who thinks about numbers and then I have to, when I say that's what I do, they're like, isn't that what everybody does?

Math does, whereas it's only one small part.

You should tell them to go talk to a physicist.

Right, exactly.

I mean, so if I were gonna be snotty and give a one-word answer, I would be like, well, everything.

That's what it's about, it's about everything.

That's good, that's good.

I mean, it is like the structure of the universe that we live in.

But I would say, I'll tell you another answer that people like to give in my community, the mathematicians like to say, which I also don't like.

They'll say like, well, math is about patterns.

It's like the study of patterns.

Some people would say that.

I think that's also insufficient.

I mean, to me, I look at the whole world, the universe, everything that we do, and there's some kind of mathematical structure to all of it, whether it has to do with quantity, like numbers, whether it has to do with like geometry, like the shape of things and the way things are related to each other, whether it has to do with like probability, the world of chance, like somehow every aspect of human existence over the years we found some mathematical undergirding to it.

Now that doesn't mean, here's, okay, now we're gonna come back to Hari Seldon, ready?

That does not mean that it's just math.

I think that's the big mistake that they make.

If they hear me say that, they're like, are you saying that we live in this like cold, like depressing world where like all there is is like— You're starting to sound like, who is it that came with the Harmony of the Spheres, Pythagoras?

World of Forms.

Yeah, but it's not supposed to be depressing, right?

It's supposed to, I mean, it's supposed to give you more to think about rather than less.

So I would never say the world is just math.

It's just that that's like underneath and enriching like all the stuff that we see.

So tell me this then, right?

So I was a scientist that built things, right?

And the thing about that kind of research is it is expensive.

And so I decided, hey, I'd like to do something that is not quite so expensive but is very relevant to the time.

So I started working on big data, which includes machine learning, which now, you know, people call AI in many ways, right?

And it's not cheap anymore.

And it's not, right?

It's not cheap anymore.

I used to do it on my, you know what I did?

I networked computers on the floor in my building to make a pseudo-cluster and ran it at night.

But anyway, the question is, is that, you know, when we think of AI, we think of language, but at its heart it's math, right?

Absolutely, it's math.

I mean, math and engineering and like statistics, but yeah.

Well, the code, yeah.

Yeah, fundamentally there's like a mathematical process, which is, it's actually kind of amazing how simple it is underneath.

I mean, it's kind of, I mean, it's kind of amazing that it works as well as it does.

Tell me more.

'Cause fundamentally what every single, large language model we have is built on is the existence of some huge body of text.

Yeah.

And what you are trying to figure out is, well, there's some function, right?

That's what we talk about in math all the time, functions, right?.

We say, well, there's some function and the input is some string of words or some string of symbols or whatever it is.

And the output is, what comes next?

That's what we do as we produce languages.

We're sitting here talking, we sort of like said something, and then I'm like, and then somewhere maybe unconsciously we're like, oh, now I gotta produce like the next word of my utterance.

And that function, you know, it's not gonna be like a button on your calculator, right?

It's not gonna be like "this plus this minus this," and then you get the next word.

It's like some immensely complicated function, and ready now in the next like, 30 seconds, I'm gonna tell you how every single AI system, like, ever built is trained and worked.

It's basically saying, okay, I try some function.

I test it against some set of trillions of words, like every piece of text ever uttered on the internet, and see how well it does at predicting the next word.

And it probably does pretty badly, right?

If I started with a pretty random function.

So let me, I'll just tweak it a little bit.

I'll take one of the parameters, one of the pieces of the function, move it a little bit this way or that.

And if that makes it a little better, I stick with that.

And if it makes it a little worse, I don't do that.

And then I do that again, millions and billions of times.

Trial and error, just being like, does this make it a little bit better?

Does this make it a little bit better?

Until at the end of the process, and this is kind of the miracle, you have something that actually works pretty well.

And what I think that one thing about it is that nobody knows why that works.

Oh boy.

In some sense it shouldn't work.

Which it sounds like is two things that are combined, right?

Back when I was, say, modeling something physically, I wanted what I calculated to match the real thing in the world.

So I used a minimizer, right?

So I used, it was the downhill simplex method if anybody cares.

But basically it's like, okay, I have this calculation that depends on some physical parameters.

I wanna know what those physical parameters are., but I have no way of directly measuring them.

What I can measure is the light that's emitted, all right?

And so my, I wanna make sure that whatever mathematical expression I have for the light that is emitted, when I calculate the light that is emitted, I just take the difference of those two things.

And when that difference gets to zero, they are the same, right?

And so that's what it sounds like you just did.

Exactly, that's what you were doing.

And you were tweaking things.

You were like, oh, that distance, that difference, I wanna get it smaller and smaller and smaller.

You were tweaking it and messing with it and screwing around until you got it down to zero.

And literally, the way a contemporary model is trained is not that different from that.

It's not that different from that.

So it's really built on simple parts that came together to make something that looks magical.

It's what we call gradient descent, for those following along at home.

Gradient descent.

Right, the gradient tells you which direction makes the program work a little bit better, and then descent, Means you're trying to get something down towards zero.

You're trying to get your failure rate down towards zero.

Oh, gradient descent.

Love that.

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So that sounds like geometry, right?

If you're looking at a difference, you're making this difference smaller and smaller.

So you can think of that in terms of like, one of the things I used to calculate was coronal loops.

And so I would make the length, you know, I had a measurement of length, so let me like make sure the lengths match what I calculate to what's in real life, and then compare that to the radiation.

So geometrically though, when you talk mathematics and geometry, things can get pretty different from the physical world, right?

You could have more dimensions than 3.

You could have dimensions that aren't spatial or temporal.

Yes.

Oh, I thought all the cool physicists had those extra dimensions too though.

Isn't that what the string theorists have?

Like, this is not— String theory ain't as cool as it used to be, my friend.

What?

I'm behind.

I'm behind.

I'm like, I'm like, it's, I'm like the guy wearing like '90s jeans or something.

Like talking about string theory.

Yeah.

That's the one that has the cool math.

String theory's gone?

I did not know that.

Man.

String theorists will say it's not gone, right?

Okay.

But the question is, is it giving us reality?

And the answer is no, right?

Its dark matter candidates haven't shown up.

It's predicted signatures in the colliders like the LHC haven't shown up.

So it's like, oh, maybe it's not that, right?

But this is the same question we're talking about, right?

You have a theory, and then in the end, you have reality principle, right?

The test of the theory is— Is the reality principle.

You're like, okay, now we can compare what it predicts with what we actually observe.

So tell me this then.

So this might not be answerable, but what we all know now is these AIs, when they produce language, they have a style.

"It's not this, it's that," right?

So how is it that, is that really reflective of how humans talk or write?

Because if it is, wouldn't that mean we wouldn't recognize it?

Or does it like have weights?

It's like, "okay, if I'm gonna say something," like for example, right now and at all times, there are phrases that people use that just become a part of the nomenclature.

Like, make it make sense, right?

Which I hate.

Make it make sense, right?

It comes in and it goes away.

So, but a large language model, I'm guessing if you have this giant database, It covers time, right?

All text that may exist.

So it might not be relevant to the current moment.

So how does this math generate style?

That's a deep question and it depends how deep we want to go.

But what I would say is that the description I just gave, it's kind of like step one of making a large language model.

That's kind of looking at, you know, all the text that's ever been typed into the internet that any— that the company has access to.

And so that produces something that maybe is a little bit like you would consider it like the average of like all, human language.

You know what I mean?

A modern language model that you talk to through a chatbot window, that has undergone a process called fine-tuning, like a very extensive process where they are trying to give it a very specific vibe, the vibe of the helpful assistant.

So that's, that style that you're talking about,— it actually does not come from the sort of just from the process I just told you where it's like, let's try to match reality, everything that's ever been said.

There, they're really working very hard because think about all the crazy things people say on the internet.

You don't want your chatbot to talk to you like that.

You don't want it to talk to you like the Reddit comments, right?

You want to talk to you in this very specific style, in this very specific vibe.

You better respect me.

Exactly.

Act like, no matter what I say to you.

Yeah.

But yeah, I mean, I think part of the math that underlies, you know, we talked about, we talked about sort of like learning some function.

And one of the things that's going on is inside every large language model is what's called an embedding.

You talked about higher dimensional spaces.

Right.

So every single word or what's called a token, which might be a part of a word, but we can just say word, doesn't matter.

You know, every single word is represented by a point in a very high dimensional space, maybe a few hundred or a thousand— Dimensions?

Yes.

Wow.

Right.

So not a 3-dimensional space.

And that's actually, it's, that's an old idea actually.

I mean, you can go back to the '50s and see psychologists making maps of adjectives.

Wow.

Where they'll be like, let's find, I mean, of course it was a map that was published in like a paper on paper.

So it was a 2-dimensional map.

And it's actually kind of amazing to like look at what they were able to do quantitatively and be like, which words, which emotions are close to each other, like which adjectives are like close to each other in like the way that humans see them.

But they didn't do it in this kind of big data, there's no internet, right?

What they would do is they would just like bring in people, volunteers for experiments, and they would like ask them on paper, okay, how similar are these two words?

They would do surveys and they would exhaustively compile that data and sort of try to find the best way to locate those words in space that matched people's judgments about what words were predicted.

So it sounds like the conclusion here is AI needs people.

Like you can't just build a, you can't just throw a bunch of words at it and then it actually works the way a human interface, it wouldn't interface with humans the right way, right?

Well, that's a complicated and somewhat controversial question.

Like fine-tuning, right?

That requires a human.

Fine-tuning, right.

Reinforcement with human feedback.

That's the sort of term of art.

So I would say, yes, as of now, they use a lot interaction with human beings and like what causes the humans to find the response helpful.

And that information is like taken in and like used to like, well, let's tweak it in the direction that people like and tweak it away from the direction that causes the user to like cuss us out in the chatbot.

Right.

So I guess I wonder this then.

So, you know, language evolves.

Yeah.

So is it possible then to come up with an equation, a differential equation that shows how language evolves in time and actually predict, like could you have predicted, yeah, in the '80s, we're gonna start using the word diss.

I was thinking about that 'cause I was asking my 15-year-old daughter to explain the phrase "get after it" to me the other day.

I was like, "does get after it mean the same thing as go for it?"

And she was like, "kind of, kind of."

Well, first she was like, "what's go for it?"

But then anyway, but.

So people have tried.

That's a great question.

People have really tried to be like, let's set up as— so again, for the viewers, differential equations are basically the part of math that studies motion and change.

So they are incredibly important.

Like, basically all of physics is described by sort of some intricate complex of differential equations that say, like, "if a thing is moving in this way, subject to this force, then this is what will happen next."

Every single statement like that, there's a differential equation that describes it.

And so people, of course, have tried to apply that outside the physical world, like to the mental world, like language.

Wow.

I'm gonna be honest with you and say I've never been convinced by anybody I've seen try to do that for linguistic development.

I don't think we could have foreseen that people would be saying get after it in 2026.

I don't think that was a predictable— Exactly.

That, that human unknown element.

Right.

So speaking of functions, so I, and this whole like predicting the future.

So there is a function that shows up in physics a lot in thermodynamics that I hear the phrase in your area, but not quite the, uh, I don't understand the full context.

And that is the problem of what we call the random walk.

So I'll give you the physics context.

So how we define it, right?

So random walk is, suppose, let's just take location.

You're in a location and you have an equal probability to take a step in any direction.

Every direction has the same probability.

So the context we have is, you're a photon of light that has been created in the core of the sun.

How long does it take you to exit the surface?

If you just went straight out, it would be 1 second, but it's not because you go like a fraction of a millimeter, right?

Some tiny distance, 10 to the minus something meters, and you're reabsorbed and then re-emitted in any random direction.

So it literally takes something like tens of thousands, if not hundreds of thousands of years to exit because of that random walk.

So how does random walk manifest in the world of math and AI?

First of all, I'm gonna answer your question, but like, if we could talk about physics for one second.

Yeah.

You just told me something I did not know and my mind is kind of blown.

'Cause in my mind, as a mathematician who doesn't know that much physics, I thought light just did move along in a straight line.

You're saying the photons like bounce off stuff?

Like are they— Yeah, they're like, when you look at a metal, right?

The reason why it looks shiny is 'cause there's free electrons.

So free charged particles interact very strongly with light.

Once they're in a bond, they ignore most light except for that which has resonances with the energy levels.

But when they're free, they interact with all wavelengths of light.

So when you're looking at a metal, it has all those free electrons in the surface.

So it's absorbing light of all wavelengths and then reabsorbing it.

And that's why it appears shiny, right?

So when you're inside the sun, everything is a free charged particle.

So you travel basically no distance before you encounter another charged particle and you're absorbed, right?

It might vibrate more, you know, more energetically then it re-emits it, right?

So it just wanders around inside the sun for a long time before escaping?

I wrote a ton about random walks, and I wish I'd known that.

I would've put that in too.

That would've been cool, right?

That's a perfect example.

I should've learned physics.

But yes, I mean, absolutely, that principle is absolutely fundament— Is it identical?

Fundamental.

I mean, what you just said, it's one of those things that is, there's kind of an amazing story where this concept is invented, again and again by independent people in different countries, like just at the very beginning of the 20th century, one of them being our guy Einstein, of course.

I mean, he's, remember at this point, we don't really know about molecules, that's like a theory.

People are like, maybe inside matter, there's some particles that are way too small for our microscopes to see, and maybe that's what's causing what's called Brownian motion, you know, the sort of like the jitter that you see of like a particle suspended in a fluid.

Right.

Um, and exactly what Einstein did was he was to say, like, it's just like we were talking about with, like, you have a theory, and then you— but there's actually something you can observe.

You can't observe the theory itself, but you can observe its consequences, and you can see if they match.

Einstein asked this very same question.

If the little chunk of stuff on the drop of fluid in the microscope, if the reason it were moving around were because it was being bombarded by tiny molecules that we can't see, like slamming into it at high speed from random directions.

He— one could ask the same question you asked.

How long does it take that photon moving around in random directions, how long does it take to get to the surface of the sun?

How rapidly does it end up moving?

But language too.

I mean, so this, you know, there's a fellow called Claude Shannon who kind of— now we're sort of jumping forward to like the 1940s, and he kind of creates information theory.

But one of the things he does in this sort of paper where he like invents so much stuff, is he talks about a random walk model for generating language.

He's like, what if there was some rule where, let's say you look at like the last 2 or 3 letters in your text, what if there was some rule for like figuring out the next letter?

Not deterministically, of course, because if you like, if you have a letter like the letters A-N, you could be talking about an ant or you could be and, right?

There's different things that could be.

Or angry.

Or angry, right?

There's a lot of things that could be, but it's probably not gonna be B, right?

Can we think of a word that's A-N-B?

Okay, I can't think of one off the top.

There probably is one somewhere, but like, but probably not that, right?

That is in some sense what a large language model is doing, developing a probabilistic law for we see the text so far and what's the probability distribution on what comes next?

It's not a deterministic law, it's not like a rule.

It's a probability distribution.

But it's a probability distribution.

And that you can already see in 1948 in Shannon's paper.

And you can see this kind of artificial text that he produces that sounds something like English.

It's not English, right?

He's had access to like, his data was very small by the modern standards.

But you can very visibly see that it's picking up features of the English language already by this kind of method.

So if AI is just math, what are humans bringing to the table when we engage with it, right?

I mean, I think as somebody, and I should say, this is something I've worked on on my own I mean, the dominant part of my work is like very traditional pencil and paper and mathematics.

But I've also like worked on this question of like, how do we leverage these new tools to generate interesting mathematics?

For me, I work— Oh, interesting.

For me, I work on examples.

I don't know if people know this, but like one of the things we love in mathematics is an interesting example.

Okay.

And so I think like example making is a super interesting AI application, and that's what I've worked on.

Tell me more.

I don't quite understand the context.

Like what kind of examples?

So these are use cases for AI or?

Oh no, so let me, I mean, so for instance, if we're studying geometry, there's the famous Platonic solids, like Plato knew about these, right?

The sort of regular polyhedra, or as I like to call them, the D&D dice.

I don't know if you were a D&D player.

No, I never played.

But there's the shape of the dice, right?

There's the cube, there's the tetrahedron, which is like the 4-sided die.

There's like, There's like an icosahedron, I don't even— there's one behind you, I don't even know if it's in the shot, but there's like— I'm looking at an icosahedron right now, like behind you, this sort of 20-sided figure.

So there's just these— there's exactly 5 solid figures that are the so-called regular polyhedra that are like wonderful examples of like geometric symmetry and regularity.

And so from examples like that, like we learn a lot.

You ask like, what can symmetry look like?

So, but interesting examples are hard to find and And AI sort of gives us like really interesting novel ways to sort of search space, maybe ways that we wouldn't think of.

So here's the question about AI then.

So as a researcher, do you think AI will ever get to the point where it can make new discoveries?

I, to be a little bit salty about it, I think that was true 30 years ago.

Really?

Like what did it do?

We have always had computational tools that help us discover things in mathematics.

Well, that's not— the idea is, Here's a bunch of data, tell me something about this physical.

Not, oh, I'm gonna classify my data this way, or I'm gonna find groups, right?

I mean, do you think, would you say a telescope makes its own discoveries about what's out in the cosmos?

No.

Okay.

Yeah.

I think at this moment, It's a tool that's in your hand, a very capable tool, but a tool.

Who knows what the future will bring?

But it appears to think, it appears to be able to create.

But at our direction.

Yeah, yeah.

Well, that's the point.

Will it ever get to the point where it doesn't need so much of our direction?

If you can give it a really broad statement and then it can go in and, you know, for example, we can say something like, oh, you know, here's an unsolved problem we've had in physics.

We don't understand, this is something the Higgs, right?

We don't understand the origin of mass.

Can you tell us what the origin of mass is?

And then it goes and predicts the Higgs field.

But we'd be asking the question.

Yeah, we are asking the question.

Okay, then we're doing it.

Is that how you see it?

That's how I see it.

Okay, so for it to be autonomous, it needs to pose its own questions and be able to answer them.

I mean, that would be a start.

That would be.

Well, because both of them are skills of a researcher, right?

You have to ask good questions and you have to have the ability to go and get the answers to these good questions that you posed, right?

So in that two-step process, you called that the beginning.

But I say like, you know, that's kind of the magic of it, right?

You know?

I think the whole history of technology and science is that we reclassify what counts as science, right?

I mean, as we develop technology that enables us to do things that before required human effort, we stop thinking of those things as science.

We start thinking of them as computation or instrumentation or engineering or something like that.

And that frees us up to do new things that we now call science.

That's sort of been the whole procession of things for like, most of human history.

Okay, so there's not gonna be a case in the future then where two AIs are sitting here and asking, hey, can you get anything creative out of those humans now that we've taken over?

Yeah, and you and me are gonna be like lying over there with our protein tubes, like feeding our, you know what I mean?

Or maybe just our heads.

They'll probably just save our heads.

Heads in a jar, right?

So here's a question then.

Can AI screw up our science?

Can it slow us down instead of speeding us up?

Can it send us in bad directions?

That could happen.

That's one dark future.

It's not what I think is the center of the probability distribution.

It's not what I think is most likely, but yes, you can imagine a world in which in some perverse sense it becomes like too easy to solve certain kinds of problems and we sort of, you know, people like what's easy.

Then we sort of forget about the hard problems because it's like, well, let's just, let's just do the stuff that— let's just do the accessible stuff.

It's like GPS, like I've lost my mental map because of GPS.

I no longer have a mental map of my town.

Yeah, you gotta fight it, man.

You gotta turn it off.

I just somehow can't make myself believe that we're just headed for like a lazy-brained future.

I feel like people are not like that.

I feel like everything I know about the way human beings are is that we will find stuff to think deeply about and care about and be, passionate.

It's gotta be that way, right?

I don't want to be corny here.

Yeah.

No, no.

I mean, you're that kind of guy.

I'm that way.

I'm driven to curiosity.

And I think children definitely are.

Yeah.

And so I feel like it— how you nurture children matters to some degree, right?

I mean, I think that's part of the reason I think that way is I'm lucky enough to like my working environment.

I'm around like 18, 19, 20-year-olds every day.

And you cannot be pessimistic about the future if you're around these kids who are like so excited to learn and like so excited to grow.

When I asked you in one word, what is math about?

You said everything.

That's right.

I stand by that.

You stand by that.

So you are known for illuminating the math in our everyday lives.

So especially where it comes to geometry.

So bring some of that to our audience right now.

Where is geometry and math everywhere?

So, I mean, the thing is, when I say geometry, I mean, it's, you know, it comes from the Greek, like measuring the earth, right?

It's like the general science of like, measure, like where things are, what is their shape, what do they look like, how are they related to each other.

So I mean it very broadly when I say geometry, I'm not just talking about triangles, although I love a good triangle.

Yeah, I mean, triangles are a key.

Who doesn't?

But I think any context in which we're thinking about— so maybe measure is a key word.

Any context in which we talk about how close or how far things are from each other, we're fundamentally thinking geometrically.

I mean, even in our family life, right, you talk about a close relative or a distant relative, there is a geometry of the family where you're like, you know, how many times, yeah, the family tree, and actually a tree is a wonderful geometric figure, right?

There's a whole world of the geometry of trees, and I don't mean biological trees.

So let's get this right.

So geometry isn't just shapes, it's relationships and shapes and structure and— Yeah, but I think all those things are related together, and then if you were to like pin somebody down and say, "Well, what do you even mean by a shape?

Like, what do you mean when you say a triangle has a certain shape?"

You would be talking about the relationship between the points of the triangle, right?

Like, how far are they one from the other?

Like, that sort of, you know, I think it was, there was a painter, one of the Cubist painters, Georges Braque, who said like, "The goal is not to paint things, it's to paint the relations between things."

I think that was a very geometric insight.

I mean, they were called Cubists, those people, right?

So you knew they liked their geometry.

So where else, like when we talk about distances and concepts like that, that's a straightforward application of geometry.

But when I got, for example, to studying quantum mechanics, we started talking about this thing called Hilbert spaces.

And I was confused as heck.

To start out, right?

Because we, when we take math in school, we learn specific context, X-Y-Z, right?

Or X-Y-Zed.

And you know, if I want to, you know, one of the most scary concepts for a physics major, if you don't have a good math upbringing, is vectors, right?

And then you come to find out like, oh, any list of numbers is a vector in some space.

So I give you sets of 5 numbers.

That is a vector in a 5-dimensional space, right?

So when we start talking about things like AI, talking about things like words, we can use those same concepts of distance and geometry, right?

Yeah, and actually, I mean, this is one of those things that when we teach math, there's stuff that, I think one of the big challenges for us in teaching math, teaching physics, whatever it may be, any kind of science, is that there's concepts that are so natural to us that we just say them, I'm gonna slightly call you out for doing this, say them as if they were obvious.

Which are not obvious at all.

So here's what you just said, I'm gonna repeat back to you.

You were like, "well yeah, like any list of numbers you could think of it as like a vector, as a geometric thing."

That's not obvious, that's René Descartes, right before Descartes.

Yeah, I know it's not obvious.

So this is like, this is an incredibly deep— It wasn't obvious to me, yeah.

This is this incredibly deep fact that I mean, I think the way we usually, I think the way we most commonly see it in our daily lives is just like the notion of longitude and latitude, right.

That on the one hand, it's a point somewhere on the planet Earth, right?

It's a geometric location, it's a geometric thing, but it's also described by these two numbers.

And this facility with going back and forth between something numerical, something algebraic, like a pair of numbers and a point in space, is this incredibly deep insight that changed geometry that Descartes brought to us.

And I think, you know, you can say like, but wait, which is it?

Is it a list of numbers or is it a point in space?

The answer is it's both, right?

It's always both.

And in some sense, really understanding things, I mean, I think physics is like this too, right?

I mean, like, is it, is light a wave or a particle?

Like, it's both.

And if you, if you spend your life being like, well, I'm trying to figure out which one it is, like you've lost the game, right?

You're not actually trying to understand light at that point.

You're trying to answer a question that doesn't have an answer.

That's right.

And I think geometry is like that too.

If you say like, well, is it a list of numbers or is it a position in space?

It is both of those things and the ability to kind of flick back and forth between those points of view is like so much what geometry is.

And just as you say, I'm gonna let you talk in a second, I'm just like, I have a long answer to this question.

I love listening to this answer.

What's great is that of course many things that are not in the physical world, like thinking about, you know, thinking of the space of words, the space of ideas, the space of computer programs, whatever, ask us to think in higher dimensions than 3.

And that's, I think, a challenge for a lot of people.

They'll say like, well, how am I supposed to like think in like higher than 3 dimensions?

Right.

Maybe with some effort you could be like, OK, time's a dimension.

I can kind of think of like 4 dimensions maybe.

That's about as far as our physical intuition goes.

And the beauty of this numerical approach is like, well, you can think of a list of 2 numbers that's the longitude and the latitude.

You can think of a list of 3 numbers that tells you maybe a point on the surface and a height above the ground, a point in 3-dimensional space, it's not that much harder to think of a list of 10 numbers than it is to think of a list of 3 numbers.

So this numerical point of view, that unlocks your ability to take geometry into these higher dimensions.

And then the incredible miracle that's such a wonderful thing about math is that so many of our geometric notions still work.

When you're thinking about these abstract— When you're thinking about these high-dimensional spaces, and like, boy, I'd love to talk a little about quantum mechanics, but Hilbert spaces are infinite dimensional.

I don't know if people know that you can have infinitely dimensional spaces, it's crazy.

But even still, I mean, one of my favorite facts about this is like, you know, you may study, there may be two things you're interested in, and you may be interested in how they're correlated.

I don't know, we could just make something up, like stock prices and like the Baltimore Orioles' record in a given year, to say like two things I care about.

What a random team.

What a random team.

That might be the team I've been suffering with for the last 40 years.

Okay, so maybe, so maybe you have a question about like how those things are correlated.

And so for each one of those, you'd have like a list of numbers, right?

You'd have the Dow Jones Index at the end of each year and you would have the Orioles record at the end of each year.

And you could compute what's called a correlation coefficient.

Meat and potatoes of statistics.

We wanna understand whether two variables tend to kind of vary in the same way or vary in opposite ways.

That correlation coefficient is an angle.

It's the same thing as we do in geometry.

When two variables are correlated, it's because some angle in high-dimensional space is very acute.

It's like two lines pointing in almost the same direction.

Oh, wow.

When they're anti-correlated, it's like an obtuse angle.

They're pointing in almost opposite directions.

I see.

And when they're uncorrelated, when two things are completely independent of each other, that's a right angle.

And it's kind of amazing that like these notions of things like angles that we think are so tied to like Euclid's geometry in the plane, they make sense.

And actual real space.

Yeah, and I think you would, but let me put it this way, let me try to convince you why you should believe that's true.

Because it doesn't bother you that you can pick the triangle up off the page and put it in 3 dimensions, you're like, yeah, the notion of angle still makes sense in 3 dimensions.

I think you should say, your intuition should be, "Well, if it makes sense in 2 and it makes sense in 3, why wouldn't it make sense in 4?

Why wouldn't it make sense in 10?

Why wouldn't it make sense in a Hilbert space with infinite dimensions?"

Which it definitely does.

So I can really remember the very first time I was introduced to these weird spaces and understood them.

And it wasn't in my quantum mechanics class, 'cause I didn't understand it yet.

No?

Okay.

I didn't understand it yet, right?

But it was Michio Kaku's book, Hyperspace, where he talked about these higher dimensions, right?

But he gave very— just like Albert Einstein, right?

You give these good thought problems that he didn't necessarily invent, but, you know, they're really good at getting the idea across.

And so for me, I felt like I opened up a whole new part of my brain in which to wonder.

So these kind of thoughts that you're talking about that you mathematicians do aren't just everyday intuition.

They are— they're very different.

Have you ever— have you ever read the book Flatland?

That's what Michio talked about in Hyperspace.

He introduced us to the book Flatland.

'Cause I wanna riff on that for a little bit, 'cause it's such an amazing story.

So Flatland is a book, it's from 1888, I think.

It's a 19th century book, it's really old.

And I just have to say that the organization that I lead, the Astro Society, was founded in 1889, but continue.

Okay, probably inspired by Flatland.

And it's this wonderful fable, it's by a guy who actually mostly wrote religious texts, Edwin Abbott.

But he wrote this book, and it's the story of a square.

The main character is a square.

He lives in Flatland.

He lives in a world of geometric figures where it's sort of a little bit of like a satire of his Victorian British society.

There's like different social classes where depending on how many sides you have as a polygon, like that's sort of your social status.

I used to be called a square.

But anyway.

We used to, I will say our high school math team, the Hell's Angles, as we were called.

No way.

We would come into, are you guys ready for some true '80s mathiana, we would come into our math meets and our team captain would come in with a giant boombox on his shoulder playing Hip to Be Square by Huey Lewis and the News.

That was our big entrance to intimidate the mathematical opposition.

So for those of you who did not live through the '80s, I promise that was cool at the time.

And for those of you who are scriptwriters.

Yeah, okay.

But let me tell you this other thing about, geometry, and you say let's go into these other realms and dimensions.

One realm of dimension I want to go into is ghosts.

Okay, that was not what I thought you were going to say.

I know you didn't.

I know you didn't, because here's the thing: a place is haunted, all right?

And here's what I'm getting at.

When I started doing geometry in physics, one of the things that they stressed to us was, bro, there is no grid of background in spacetime.

You choose your origin, and you set up your coordinate system based on whatever's convenient, right?

So when I think about, okay, that place is haunted, but here's the thing, the Earth is moving through space and the sun is moving through space.

So you're only in one place once in your entire life and existence.

That house is never in the same place ever, right?

So how can a place, you know, you left that place back in space somewhere, right?

You know, so how can that place be haunted, but in a particular geometry that is centered on Earth's center at a pole or something like that, it makes sense, right?

So all spaces aren't equivalent.

We have these abstract ones and we have ones that are called non-Euclidean.

So I got that from Michio's book as well.

So Euclid, right?

He's— we're talking Library of Alexandria, ancient Egypt.

He writes his book of geometry that dominates until, you know, we get something different, right?

How do you define Euclidean and non-Euclidean geometry?

Because we're gonna talk about that.

Yeah, so I mean, right, so Euclid, he writes his compendium, and you know, we don't know very much about the historical Euclid.

I mean, we know he lived in North Africa, and at the city of Alexandria, we sort of know about when.

We think that probably what he wrote, at least some of it was a compendium of mathematics that was known to the people of his time.

I think people don't really think he invented it all.

It was more like, invented a lot of it, and then a lot of it was like kind of an encyclopedia of like, what do people know about geometry in this point in time?

Let's put it all in one book.

In this very systematic way.

But of course what was special about Euclid was this axiomatic method that was really new.

That he's like, let us start from these basic principles of geometry.

Like a point has these properties, a line has these properties, and they have these properties in relation to each other.

He writes down these 5 axioms, and his goal was to show that everything could be derived from that starting point.

And that became a model, not just for mathematics, but for like all kinds of knowledge.

That was like considered the gold standard of like how you made knowledge that you could absolutely rely on.

So you see like, you know, you see like Spinoza, like the philosopher, there's this kind of amazing book called The Ethics where he's like, let me write, you know, what God is like, how to believe in God, how to behave, but it's all in the form of definitions and theorems and proofs.

He's imitating Euclid.

And there's— And by the way, I just have to say this, folks, if you're listening to this, I don't know much philosophy, but I used to listen to the podcast Philosophize This.

So that's how I know who you're talking about.

Oh, okay.

And by the way, I mean, it was, I should say, I mean, he's now a famous philosopher, but he eventually got excommunicated by both the Jews and the Christians.

I mean, he was not that popular.

That's an amazing feat actually to be like, rejected by everyone.

We're not buying this whole systematic God you're creating here.

Well, 'cause part of it was like, you know, he was like, "well look, this is so much like geometry, like how do you know God exists?

Well, it's like the same way we know like a triangle exists, 'cause like ideal form."

And then everybody else was like, "we don't want a God that's like a triangle, like we want a God that's gonna like smite our enemies, like a triangle doesn't do that," you know, nobody likes it.

It does have a sharp— That's a good point, right, is it acute enough?

Okay, off Spinoza.

I mean, even— Non-Euclidean geometry.

So I'll tell you where non-Euclidean geometry, so— Yeah, we're getting there.

We're gonna do it.

Yeah, yeah, yeah.

Well, so I'll tell you for me, okay?

The growth.

So first I learned Pythagorean theorem, a squared plus b squared equals c squared.

Love it.

Then, you know, they give me the Euclidean plane and they're like, x squared plus y squared.

If I have a line of length s, s squared is x squared plus y squared.

They're like, you can do this in 3 dimensions, x squared plus y squared plus z squared, right?

Right.

Then I get to Einstein.

He's like, well, actually, there's not just 3 dimensions in space.

Time is also a dimension, just like space.

So now we have to add this ct, you know, speed of light times time all squared, and it has the opposite sign from space.

And it's like, whoa, that's amazing.

And then I take cosmology, and they're like, well, that, that space part in that 4-dimensional spacetime vector, it's expanding with time.

So you got to multiply that space part by this thing called the scale factor, which is a function of time, because you don't just have an origin and a grid.

The grid is growing, right?

Oh, see, I didn't know this part.

I know it up to Einstein.

Everything.

I knew everything you said up to Einstein, but I did not know this last part.

And then the final part was, well, guess what?

Spacetime may have curvature.

It's not just growing.

It may have curvature.

So now we have to add a curvature constant K. So I'm like, bro, that was a lot.

That was a lot.

But that's day one of cosmology, right?

Robertson-Walker metric.

Yeah.

But what does that reflect?

I mean, it reflects the fact that, I mean, so here's what happens in history, that like, you have this Euclidean system, people study it for centuries, for millennia, right?

This is what geometry is, like every school child needs to do it.

But there's something about it that bothers people.

There's these 5 axioms, and there's 4 really simple ones that everybody agrees upon.

And then there's this fifth one that people do not like.

What is that one?

They don't like it, and it's called— it has many equivalent forms, but it's often called the parallel postulate.

And I'm gonna like draw a little picture of it with my hands.

You're all gonna see what it says.

It says if you have a line, here's the line, everybody sees it, great— and you have a point that's not on that line, yeah, then there's exactly one line through this point that's parallel to the other line that I just drew.

Okay.

And if you can visualize that, you can probably be like, yeah, that seems true.

But it's also like a little bit complicated.

It's more complicated than the other ones, which are things like, well, two lines intersect at exactly one point.

That's like very straightforward.

You're like, yeah, of course.

Or if they're not the same line, if they're not parallel, they intersect at exactly one point.

You're like, oh yeah, of course that's true.

This parallel postulate is a little more janky.

And I think for years people were like, "wouldn't the system be better if we didn't need that?"

If we could prove the fifth one from the other four, then we wouldn't need to have this like extra hypothesis that kind of makes the whole theory like a little bit less elegant.

And this was a problem for hundreds and hundreds of years, and people like tried to do it.

And then something kind of incredible happens in the 19th century.

Something unlocks in people's minds, and there's like a lot of discussion about like, you know, what it was about like the history of the world and people saying that maybe they will do this, then they were like, well, what if it doesn't follow from the other 4?

Like, what if it's just like its own thing?

And indeed, what if there's some alternate universe version of geometry where that's not even true?

Oh?

And this was a revelation because, because again, you got to remember we're in a cultural world where that is truth.

Right.

Like, Euclid is truth.

You're not— if you wake up and one morning, you're like, yeah, but what if like Euclid weren't true?

Right.

Like you can't, you can't do that, right?

That was, I mean, and yet people started to— It's kinda like when we, it's kinda like when you grow up and you're like, what if mom and dad don't know everything?

It is exactly what you're saying, like Euclid, right?

Like who's your daddy?

Euclid.

Like that was the answer for like 1,000 years of mathematics, right?

Exactly.

So people started to realize there are other geometries.

There's geometries in which the first 4 of Euclid's postulates are true and the 5th are not.

And, and János Bolyai, who's one of the first people to do this, a Hungarian mathematician, he's in 1820— it's an amazing story actually because his father had worked on this problem all his life with complete failure, Farkas Bolyai, and then János the son sort of starts working on it and there's all these impassioned letters.

The father's like, don't do this, like don't work on this, you'll ruin your life.

But of course mathematicians, once we kind of get our, something gets its hooks into us, right?

It's very hard to give up.

So once he, so he figures it out and he writes to his father this amazing letter where he's like, out of nothing I have created a strange new universe.

Wow.

And it's true because for us, you gotta understand, people are like, isn't it weird being in pure math and doing this thing that's completely outside the world.

For us, that is the world.

All right, so let's get back to now Euclidean math.

Oh, yeah, yeah.

And our friend Einstein, or as people who know how to say it right say Ein-shtein, which I just learned in the last 2 years.

But yeah, so Einstein is looking for a way to solve his general relativity problem after doing special relativity.

And lo and behold, hey, here's a math someone invented that just happens to work, Riemannian mathematics.

Exactly.

So how often does that happen?

And math is invented for one purpose, then it turns out, holy cow, this is exactly what we need.

And I remember reading about Ramanujan at some point did something like that.

Was it the interface with string theory?

I mean, the stuff Ramanujan did, like, yes, it touches string theory, it touches like modern number theory and cryptography, like it touches all kinds of things.

And I think that's, but I think that is because, you know, the world is mathematical.

So I don't think it's coincidence, right?

I think that, oh, I have another good answer to your question, ready?

I thought of another good answer to your one word— Do it.

Question.

What is mathematics in one word?

Maybe you could say it is about the possible.

What do I mean by that?

I mean that why was this non-Euclidean geometry, why was Riemann's work, why was it sitting there at the ready for Einstein to use when he understood what the— because, you know, physicists are like, what is the geometry of the universe?

Like, how is it shaped?

Mathematicians are like, what are all possible geometries?

Like, what is conceivable?

Like, what is possible?

Not just what is, but what could be.

And it's our nature to kind of try as much as we can to think about everything that could be, everything that logic doesn't forbid.

But even mathematical techniques, right?

So the, like, you know, two examples are Maxwell discovering electromagnetic radiation is, you know, light is electromagnetic radiation when he decides, excuse me, listeners and watchers, I'm going to decouple this set of differential equations.

And then you have the same thing with Fermi discovering antimatter.

He's like, I'm going to linearize this set of differential equations, right?

Two opposite pathways.

But basically they were like, here's some mathematical thing I'm gonna do the mathematical thing that you do with this type of mathematical object.

I'm just gonna do the, I'm just gonna turn the crank.

And when I turned the crank, what came out told me something new and profound about nature.

And it was just a mathematical operation, just like plus or times or division, right?

But, you know, a little more complex.

So in that way, you know, the problem is, is that sometimes you do that and it gives you something, but that something isn't manifested in the physical world.

And so our job for us as physicists is to determine the thing that math told us, is that the real world or not?

But then people take it to the next step and they're like, oh, maybe it's describing the universe in another dimension.

They're all true, you know?

So— I'm a little bit like that.

You're saying it in a comic tone of voice, but I have to admit that I'm a little bit like that, that if the math demands it, I'm like, some version of it must be.

Well, Einstein said something like that, right?

That if God, God had to invent general relativity.

So I guess it goes to that classic question.

I'll tell you my physics story, 'cause I know much less physics than you.

I stopped after freshman year.

But I remember taking electricity and magnetism and getting to the point where, and now I, please correct me if I flub this, but where like once you accept relativity, then if you believe in electricity, like magnetism follows, like there has to be magnetism.

And if you believe in magnetism, then electricity follows.

Like those are two sides of the same coin.

And that was like one of the most, mind-blowing and beautiful things I'd ever seen.

And I'm gonna be honest with you, this is why I'm a mathematician, not a physicist.

I learned that and I was like, great, now I believe in magnetism.

I don't have to see a single experiment, don't care.

I'm like, obviously this is correct.

Like, this could not be wrong.

Yeah, you're definitely a mathematician.

I was very bad at labs.

This math is too compelling.

I got the lowest score in the labs.

I could not do them.

But I mean, but right at that moment I was like, "oh, now I get why people do physics."

But of course it was the math I was responding to, but it's incredible.

So it's like there's so much in it, right?

There's so much hiding in there.

We love that feeling of, I mean I guess they're both kind of incredible.

I mean of course it's incredible to like go out into the natural world and observe things and see something that nobody's ever seen before, whether it's because you're a biologist using an electron microscope or like using a space telescope and sort of seeing things in the cosmos.

That's amazing, but it's also amazing this feeling, and this is how I felt when learning, and you can probably tell me I forgot who did this, this electricity magnetism thing.

Is that Maxwell, or is that like, I don't even know.

Oh, that's Maxwell, he's the theorist who put it all together, yeah.

So, but that feeling of like, oh, not only is this how it is, it couldn't possibly be any other way.

Oh yeah.

That's incredibly satisfying, and I think that's the kind of spiritual benefit, if I'm allowed to use words like that, that mathematics has to offer.

So I guess what this is saying is that, you know, this math pre-exists our use for it, it applies everywhere in the universe.

So there's a classic question, is math discovered or invented?

So what I like, what I like to say is it is something that we invent by discovering it and we sort of discover it by inventing it.

That's my non-answer to that question.

Like that's, like that's how we discover things.

We sort of like, invent something and then having invented it, we're like, oh, that feels like it was always there.

Yeah, there is something to that, right?

There is something to that.

It's kind of— it's almost like the rapper Notorious B.I.G., Biggie Smalls.

Another thing is not, not what I expected you were going to say, but like, let's go with that.

Yeah.

But when you hear him for the first time, he sounds so familiar, right?

It's not like— it's almost as if he was inevitable, right?

That is a great analogy, actually, because that's right.

That is what it's like.

On the one hand, there's some piece of knowledge that no human being had until a person comes up with it.

And yet, having come up with it, you feel, you feel, you feel more like you're seeing something that was always there.

So it sounds like math can be applied to like so many different, virtually everything.

But what changes is uncertainties, right?

You're not able to necessarily predict things 100% every time.

Yeah, and that's the thing.

I mean, I think if you were like, what is the biggest misconception people have about mathematics and what it is and mathematicians and who we are?

You know, it's not that we just think about numbers, or it's not that we're like cold automatic machines.

It's that what mathematics is about is certainty.

And it's funny because of course we do prove things.

So that is, that is part of what we do.

But if mathematics were only about certainty, it wouldn't do so well at describing the world, which after all is kind of made of uncertainty, right?

We don't in fact live in a world where things proceed exactly deterministically and everything is predictable 100%.

We live in the real world.

And math very much incorporates that.

Well, you know, Wikipedia introduced one idea to me that I use.

There's only one so far.

Only one?

Okay.

Only one so far, right?

And that is, is when you look up the branches of science, it says, okay, 'cause I, for example, what I'm about to say, this is related.

I would go home to Mississippi where I'm from, and you know, people, there were some people who were sort of like anti-establishment, and they were also like anti-science for that reason.

So I'd come home and they'd be like, "Hakeem, You know, man can't know everything."

And then they'd give me an example.

"My uncle, the doctor gave him 2 seconds to live.

That was 8,000 years ago.

He's still kicking," right?

And then I realized, like, what?

How do I respond to this?

I was like, oh, I know the response.

And I would say to them, "bro, science ain't science."

And what do I mean by that?

Uncertainties matter in physics.

We can make a prediction with our math.

And then we go make the measurement, in say, quantum mechanics.

They agree to 1 part in a trillion.

But if a doctor gives you a prognosis, right, you could die the next second or spontaneously heal.

So there's this big uncertainty.

So here's what Wikipedia introduced to me.

So it said that if you look at the branches of science, first you have the formal sciences, mathematics, some computation, logic.

They have no uncertainty.

Then you get to the physical sciences where physics has the smallest uncertainty, then chemistry has a bigger one, then the life sciences starts to get really big.

Then you get to the social sciences where the uncertainties just blow up, right?

But you're saying— so there's two things.

There's the mathematics of uncertainty and there's the uncertainty of mathematics.

So because of Wikipedia, I was like, there is no uncertainty of mathematics.

You say different.

I— now that I know it's on Wikipedia, I can go home and edit it right after this podcast.

I'm going to go change.

I'm going to go put it in.

There is uncertainty in mathematics.

There you go.

There you go—All right.

Our contribution to the world.

—By the editors.

Yeah, so it's a big topic.

And one thing I want to say, I mean, I want to emphasize that probably for most of the history of mathematics, that was the way we thought of it, that mathematics was the study of absolutely certain knowledge that was incontrovertible.

And I think that's one reason why, you know, you probably know we now have like a whole theory of probability.

We talk about probability and statistics like all the time.

But those are new fields.

Like mathematics went along for thousands of years without any notion of probability theory.

That comes, I think, in the 17th century, I think is when you sort of see the first glimmers of it with Pascal and Fermat.

And like, and it was a little bit like, wait, this is not what math is supposed to be about.

Math is supposed to be about what's certain.

But I think people started to understand that there's no natural phenomena that don't have a mathematical aspect.

Uncertainty is an extremely natural phenomenon.

So that's one aspect, that we have a theory of probability and that underlies the things we see.

I mean, it certainly underlies all the developments in artificial intelligence, for instance, where people used to think that to make a machine be able to produce something like human language, the way that was gonna work is that we were gonna figure out the rules of language, just like we know the rules of geometry, right, and the laws of physics.

We were going to figure out the rules of language, so that there would be something like a program which applied those rules and rigorously figured out like what was an English sentence and what was not.

And I think people worked very hard at that project.

And in the end, it was completely superseded by this more probabilistic approach where we're just going to say, we're going to accept that language is this fuzzy, probabilistic, fundamentally uncertain process, and we're going to try to kind of capture and channel that uncertainty as best we can, but the systems themselves are fundamentally probabilistic.

So one of the issues that we have in physics along that same line is, you know, in the quantum realm, things are probabilistic.

But then you get to a realm above the quantum realm, what I call the middle realm, and they become deterministic.

And that fuzzy interface between those two realms is not well understood at all.

How do you go from the probabilistic realm into the deterministic realm?

Yeah, and it's such a deep question, and I think it has an analogy.

It's not exactly the same math, but I think there's an analogy in the social sciences where, of course, psychology is incredibly inexact, right?

I mean, we can't really predict people's behavior well at all.

But in aggregate, sociology is different from psychology, right?

I mean, there are things we can, predict that are quite regular about, on average, how a million people are gonna behave.

And I think that's kind of analogous to like what happens in the quantum state when you look at like one particle in its state or like one small group of particles.

That's, you have to use quantum mechanics, like you can't do it any other way.

But maybe things look more deterministic when you're like, let me look at sort of a chunk of matter that's a lot larger than that.

And here again, that's always, 'cause I don't actually know physics, I'm faking it, so you gotta give me a look if I say it's I'm saying something that's completely wrong, but that's my impression.

No, that's kinda how it is, yeah, yeah.

So in terms of, so when we make measurements in physics, how many decimal places we can get, we sorta call that the precision of it.

But then we also have to say, when we make that measurement, there is an uncertainty.

So the classic example I give to my students is, so let's say you're gonna measure the width of the table, all right?

And you're gonna use one of those brown wooden rulers.

It has a thickness to it.

So the angle at which you look is going to change your answer.

That's a systematic uncertainty.

So you're like, okay, I'm going to move to a little thin metal ruler.

Okay, that reduces the uncertainty.

But then I'm like, okay, I'm going to put mirrors on each end.

I'm going to bounce a laser and I'm going to use a laser wavelength to determine it.

And what you find is that over the course of the day, the table is actually growing and shrinking based on the temperature in the room.

Right?

Oh wow, you got to think about that.

Yeah, there's a fundamental limit that I'm up against.

So one manifestation of that, I've heard, have not confirmed, is that that's why they removed the decimal point in Olympic swimming.

Right.

It used to go to like thousandths or something of a second.

But they realized that because of the lanes are actually changing their length, you know, so you can't give that extra decimal place because it could be due to the lane getting a little shorter because it got a little colder or something.

Right.

So we talk about precision and uncertainty all the time.

How would you clear up for our audiences the differences between precision and uncertainty?

Well, I think I didn't know this story about the Olympic swimming, but I fully approve of it, right?

Because I think that in some sense we have a lot of push-button ability to compute, right?

We can sort of put something in our spreadsheet and be like, you know, tell me this measurement.

Like, tell me like how much money I have.

That's a good example.

Like, let's say you're like looking at your own bank account.

Like, yes, there is some amount of money, like down to the penny.

That you have in like your whatever, like your investment portfolio or whatever.

But maybe you shouldn't actually measure because it's changing every second, right?

Like, why would you actually ever need to know?

Maybe, maybe, uh, maybe, uh, all of your, uh, financial spreadsheets should like only have like a certain number of decimal places to it.

Um, so I think in some sense, maybe what's the— maybe something that is like a mathematical kind of question is like, how much precision should you state things with?

I mean, it feels like, in other words, you might think, oh, mathematically, isn't more precision always better?

No, not if the precision is spurious.

Not if it's like measuring something you actually haven't measured and it's just like an ephemeral— And quite often it adds money, right?

So if you're making electronics, you can get a resistor or capacitor that's accurate, it's so precise, but that costs a lot more than the less precise one, and you might not need all that, right?

So here's another thing that I've come across.

People will say, well, "science has proven," or, you know, "science has proven this."

And I go, "hey, science doesn't prove, science asks, math proves."

So what does that mean when you prove something mathematically?

Does that mean this is 100% absolutely true?

Is it related to that Euclid thing?

Yes, and that is, it goes back to Euclid and it is still what we do.

So I, I don't wanna— I talk about uncertainty a lot, but I don't wanna obviously take away from this very special property that math, that when you prove a theorem, you have proven that it follows from whatever the hypotheses are, right?

You sort of have some set of starting facts and you're like, if these facts are true, then these other facts follow by logical deduction.

They must be true.

They must be, they must be true.

And that is— Have we ever been fooled—emotionally— —where one of those hypotheses, those setup statements weren't true and we thought they were, and then you later throw it out?

Well, here's what I would say.

I would say I wouldn't consider that being fooled because it is still the case that given those hypotheses, the conclusion would follow.

If the hypotheses turned out to be wrong, that's no problem for the theorem.

Like, that's— the argument is still correct.

It's just that it doesn't have the desired input.

That's like saying if you have like a recipe for something, if you don't have one of the ingredients, it's not a problem with the recipe.

The recipe is still good, you just don't happen to have one of the things you need to make the casserole you were gonna make.

Yeah, yeah, yeah.

But once you have non-Euclidean geometry, you start to realize math is not so much about what is definitely true, it's about what definitely follows from the axioms you start with.

You start with different axioms, you get different truths.

And that was this revolution that I think really starts with non-Euclidean geometry.

And now in modern mathematics, I think that is how we think of things.

There's a lot of— I'll give you an example.

One thing we love to talk about in mathematics is infinite series.

Right.

I hated that.

Was it Calculus 2?

It is in Calculus 2.

I taught that course many times, and it is the hardest part of that class, but I love teaching it because it's hard, not because you have to be like really careful or because it's kind of like you have to do a lot of steps.

It's hard because it's conceptually hard.

Yeah, and you know what I hate?

People, and my son is one of these people, they look at something and go, "Well, Dad, that diverges."

I'm like, "How do you know?"

Right?

Or, "That converges."

Okay, a famous conundrum that people love to talk about, and I've seen people like literally almost come to blows over this, is this question of if I write down like, 0.9999999, like forever, is that equal to 1?

Yes.

Or is it less than 1?

I don't know if you have a take.

My take is yes, it's 1.

That's my take too.

I've fought with my daughter over this, not with our fists, like many, many, many, many times.

My daughter's response to this is, this is why no one likes mathematicians.

So we use that word axiom.

What is an axiom?

An axiom is a starting point, right?

And it's something, but again, I think the meaning of the word has changed in some sense.

I think in Euclid's time and for many centuries afterwards, an axiom meant something that you write down and you simply have to accept it.

You cannot deny it.

So it's not a known truth necessarily.

It is something that you accept as true as a part of this hypothesis before proving something.

Well, no, I think that's the modern view.

And I think the older view is it is something is true and no thinking person can deny it.

I often think, you know, in our founding documents where we say like we hold these truths to be self-evident, that is a very mathematical statement.

And by the way, you know, our founders were like weird nerds.

Like that's who they were.

They liked math, right?

Benjamin Franklin, like Thomas Jefferson, these guys loved math.

So I truly believe, I cannot actually find historical documentation to this, but I believe that that is written because those guys liked math and they were like, we want this to be like Euclid, we want to start from these truths that are self-evident that you simply cannot— Do all mathematicians have that perspective?

'Cause I'd never heard that before.

That's pretty amazing.

Well, but that's the thing.

I think it is not the modern perspective.

I think now we see things a little bit like what you just said, that an axiom is a hypothesis, and we're interested in, you may start from one set of axioms and get Euclid's geometry.

You may start from a different set of axioms and get various non-Euclidean geometries, and you want that flexibility, and partly, because, you know, as we know post-Einstein, we have discovered that the world doesn't always match like the first set of axioms that we might come up with.

We want to have that flexibility of having different set of axioms so that we are prepared, we are studying the art of the possible, we are ready for whatever nature brings to us.

But I think one thing I really like to make sure people know is that mathematics is a human activity and it involves human judgments.

Like, okay, I'll give you an example.

Richard Dedekind, who is in some sense one of the creators of the contemporary way, late 19th century, of how we think about numbers.

His famous book, I'm not gonna be able to say it correctly in German, so I'm just gonna say the English title, is "What Are Numbers and What Should They Be?"

Okay.

I think most people don't think that mathematicians ever use the word "should."

Exactly.

But we do.

Why would you?

We do because we're human beings and we have human values and we choose what math to think about, what we think is interesting, what set of axioms are good for the purposes.

We do that all as human beings.

How much do you play?

Like, do you ever take axioms and say, what if, you know, let me just accept the set of truths that are kind of different because they're different.

All the time.

All the time.

You're like, it's like, it's like a kid playing with like a, like a, some kind of toy and being like, well, what if I loosen this screw?

Like what ha— what happens?

Yeah.

What if I loosen these screws and like turn it on?

Does it just like fall apart or does like something interesting happen?

Does this toy do something it didn't do before?

We are tinkerers.

Right.

In math.

And we are players.

Yeah.

So one of the things I asked my graduate students to do, and it panned out for them.

So when I went to Silicon Valley from doing a degree in astrophysics, I was able to get patents that just, was let me take what I know in astrophysics and apply to semiconductor manufacturing.

Plasmas are emitting light.

I know how to say what the matter is doing from that.

And I told my graduate students like, hey, we're going to look for this process, types of processes on the sun, but also pay attention to adjacent areas.

So you come up with these, you play with these axioms and you come up with these different mathematics that may or may not apply.

Do you guys pay attention to adjacent areas to see if it may have an actual real-world application, or do you stick to the mathematical realm?

No, absolutely, and that is one of the chief ways.

When we say like, which set of axioms is a good one?

How do we make these human value judgments about like what to study and like which hypotheses are good ones?

One of the big ways is feedback from the natural world.

I mean, historically, that's one of the reasons that math and physics have been so tightly tied together, because there are lots of geometries that we could study.

And I feel like the various, you know, the theory of Riemannian manifolds is a very good one.

And one of the reasons is that that's what spacetime is actually like.

Right.

So nature gives us Riemann metrics, so we should study them.

So speaking of uncertainty, you're working on a book about that.

Yeah.

A new book coming out May 2027, and it's called Don't Be Too Sure.

Oh boy.

So as you can tell, it's a pro-uncertainty book.

Like I'm in favor of uncertainty.

That's wild coming from a mathematician.

I'm thinking axioms and proofs, and you're saying it's uncertainty.

Exactly, so I mean, I think it's a little bit counter to what you might expect coming from somebody like me, but like, I think in some sense, you know, people are very allergic to uncertainty, like often with good reason.

But I think people have this natural desire to like live in a world where they don't have to deal with the unexpected.

And in certain ways, of course that's natural, right?

Of course we don't wanna be uncertain about like where we're gonna get our next meal, or like whether our house is gonna fall down, or whether the drugs we're taking are safe, or anything like that.

There's a lot of reasons that you wanna get rid of certain kinds of uncertainty.

But I think to the extent, I think we're in a moment in time where people are like really trying to expel all uncertainty from their life to kind of like algorithmize everything or to be like— I think of it— look, I don't want to be like a middle-aged man about this, but I feel like, you know, the whole world of like how people like find love today, right?

They're like, I want to be able to order it like a DoorDash.

I want to know everything about this person in advance.

And for, you know, for guys our age, we're like, that's not how it works.

Like, you have to— you have to— You have to do some courting.

And you have to accept that like there's a lot you're not going to know.

Exactly.

Yeah.

So I— My book is not about dating.

Just to, just to be clear.

Well, what you just brought up made me think about— there are uncertain certainties.

As an example, the big Hurricane Katrina.

We know it's coming, but we don't know when.

The big earthquake on the West Coast.

We know it's coming, but we don't know when.

So, you know, so what does that mean?

Let's not spend the money to protect ourselves from that yet.

Because we have tomorrow to do it, right?

We didn't— we knew the levees in New Orleans needed to be fixed, right?

We knew that they need to be upgraded.

The city had sank, so they weren't as tall as they thought they were, for example.

So is the idea of certain uncertain certainties play a role in there at all?

Yeah.

And that's such a good example because the fact that we don't know exactly when or where the hurricane is going to come, and the fact that, you know, maybe it won't come at all, that doesn't mean it's like a bad idea to protect yourself against it, right?

So I mean, there's sort of two failure modes, right?

There's the failure mode of, I'm not certain this is gonna happen, so I'm gonna assume it is gonna happen, and just like never leave my house, or I'm uncertain about whether this is gonna happen, so I'm just gonna assume it's not gonna happen, and it's fine, and not take any steps to protect myself.

Those are both bad, and they both come from the same impulse of saying, like, I just have decided I know the answer to what is in the future, and what's gonna happen.

And I think, you know, people come to a mathematician being like, I want the answers.

I wanna know numerically, like, you know, how long am I gonna live?

Like, when's the hurricane gonna come?

All these questions.

And on the one hand, you know, I'm here to say, the world's not like that.

The world's made of uncertainty.

Right.

—But then what I hope is that we can also say, but that doesn't mean you just live life shrugging all the time and being like, YOLO, like I have no idea.

There are ways to think about the uncertain world that math brings you.

So, one thing that happened to me as a physicist is that once I began to have an understanding of the subatomic nature of the world around me, it made the world around me look different, right?

You know, I see materials differently.

Your knowledge of probabilities, uncertainties, mathematics, does it make you live and see the world differently?

Live in a different world, see a different world than before you had this knowledge?

I mean, none of us are perfect, right?

So it's like hard to take your own advice.

It's easy— it's easier to give the advice and hard to take it.

So I'm not gonna say, I mean, I, you know, I'm still, I can't believe I'm admitting this on TV, but I'm still the guy who like leaves my house and then I'm like, I'm like, wait, did I lock the door?

I gotta like go back and check it again.

So I'm not gonna say I'm immune to that feeling of like, I can't take the uncertainty.

But you know, the way I think about it is if I try to think about this philosophically and like what I know as a mathematician, I'm like, you're never sure you locked the door.

Like you're never actually sure.

The question is, can you tolerate the uncertainty?

You gotta take a picture of it as you leave.

Don't tempt me, man.

Don't give me new things to check.

I'm gonna put cameras around.

I'm gonna be a Waymo.

Is that what it's called?

The car that self-drives with all these sensors?

Yeah, so I can be certain about everything.

But that is a big one, right?

Did I turn off the heating, the curling iron?

I can't, before I left the house.

So what are the things that mathematicians are uncertain about today?

Like where is the future?

Where are the frontiers right now in this effort?

Well, that's the thing.

The life of a mathematician, and I mean, if I'm honest, I mean, some of the answers to those questions are like not so exciting for TV because they're sort of like the things at the frontiers of mathematics are like these kind of like weird questions about like, you know, numbers and high-dimensional shapes and like stuff that's pretty abstruse.

But I mean, I guess the message I would want to send is just that choosing a life in mathematics is choosing a life of ignorance in some sense, in that we are always focused on what don't we know.

As a researcher generally, right?

Yeah, and I think, but I think that's kind of beautiful in a way, that we're in this tiny circle of light, and everything outside of it is dark, and you spend your whole life just trying to make that circle a little bit bigger.

I think there's something kind of beautiful about that.

Yeah, no, I love it, man.

I love being a researcher, I love uncovering new knowledge, I love the spark when you give someone some new knowledge, like we did to each other in this conversation.

Yeah.

Right, hopefully for our listeners.

And that's what I like, I mean, I gotta say, writing these books, I always learn so much, and I always, you dutifully write an outline about this is what I'm gonna say, And then you just like learn stuff as you go and you're like, oh, I never, I gotta write about that.

I never knew.

So then it always, so then what I turn in always looks like really different from my original.

Exactly.

Kind of like our conversation today.

You never know where it's gonna go and you just gotta believe that.

Yeah.

I think this is true in teaching too, that I mean, you gotta believe that if you're excited about learning something, then that excitement is gonna come through.

Yeah, absolutely.

That people are gonna wanna hear it.

Well, I was excited about this conversation, Jordan.

Me too.

And you know, I wanted to get a, a mathematician in here for a long time to prove the superiority of physicists.

I hope you're not like, never again, no more mathematicians.

This is it.

No, I mean, I had a good mathematics buddy that you happen to know, Bill Minicozzi, who's now— Shout out to Bill Minicozzi, wherever you are.

Great basketball player, Bill.

Holla at me.

Great differential geometer.

Yeah.

And, you know, Bill was great to talk math with.

You know, I've had some general relativity buddies that were great to to talk math with, and I'm happy to share this with the world.

So thank you for coming, sir.

Thank you.

Yes.

This was great.