Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, the problem is that I used arguments from probability theory.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's always this exceptional event.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in probability, we have these low, large numbers, which tells you things like if you play a casino with a game at a casino with a losing expectation, over time, you are guaranteed, almost surely, with probability,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
probably probably as close to 100 as you wish you're guaranteed to lose money but there's always this exceptional outlier like it is mathematically possible that even in the game is the odds are not in your favor you could just keep winning slightly more often than you lose very much like how in navier stokes it could be you know most of the time your waves can disperse there could be just one outlier choice of initial conditions that would lead you to blow up
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there could be one outlier choice of a special number that you stick in that shoots off to infinity while all other numbers crash to earth, crash to one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In fact, there's some mathematicians, Alex Kontorovich, for instance, who've proposed that actually these Caldatz iterations are like these similar automata.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you look at what happened in binary, they do actually look a little bit like
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
like these Game of Life-type patterns.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And in an analogy to how the Game of Life can create these massive self-applicating objects and so forth, possibly you could create some sort of heavier-than-air flying machine, a number which is actually encoding this machine, whose job it is to encode, is to create a version of itself which is larger.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
heavier-than-air machine encoded in a number that flies forever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So Conway, in fact, worked on this problem as well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Oh, wow.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So Conway, so similar, in fact, that was one of my inspirations for the Navi Stokes project, that Conway studied generalizations of the collapse problem where instead of
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
multiplying by three and adding one or dividing by two, you have more complicated branching rules.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But instead of having two cases, maybe you have 17 cases and then you go up and down.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And he showed that once your iteration gets complicated enough, you can actually encode Turing machines and you can actually make these problems undecidable and do things like this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In fact, he invented a programming language for these kind of fractional linear transformations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He called it FactRat as a play on FortRat.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And he showed that you can program... It was too incomplete.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You could...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You could make a program that if your number you insert in was encoded as a prime, it would sink to zero.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It would go down, otherwise it would go up, and things like that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the general class of problems is really as complicated as all the mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So yeah, that is literally the million-dollar question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, if you want to do it, not statistically, but you really want 100% of all inputs for the Earth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so what might be feasible is statistically 99%, you know, go to one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But like everything, you know, that looks hard.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So this is what distinguishes mathematicians from pretty much everybody else.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Riemann is up there.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
P equals NP is a good one because that's a meta problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you solve that in the positive sense that you can find a P equals NP algorithm, then potentially this solves a lot of other problems as well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If the Riemann hypothesis is disproven, that would be a big mental shock to the number theorists.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If something holds 99.99% of the time, that's good enough for most things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it would have follow-on effects for cryptography.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Because a lot of cryptography uses number theory, uses number theory constructions involving primes and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it relies very much on the intuition that number theories have built over many, many years of what operations involving primes behave randomly and what ones don't.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And in particular, encryption methods are designed to turn
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
text with information on it into text which is indistinguishable from random noise.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And hence, we believe to be almost impossible to crack, at least mathematically.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if something as core to our beliefs as the Riemann hypothesis is wrong, it means that there are actual patterns at the primes that we are not aware of.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And if there's one, there's probably going to be more.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And suddenly a lot of our crypto systems are in doubt.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You want it to be random.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So more broadly, I'm just looking for more tools, more ways to show that things are random.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
How do you prove a conspiracy doesn't happen?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But mathematicians are one of the few people who really care about whether 100%, really 100% of all
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, there's various scenarios.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, there's one where it is technically possible, but in fact, it's never actually implementable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The evidence is sort of slightly pushing in favor of no, that probably P is not equal to NP.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Certainly more on the no than on the yes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The funny thing about Pico's MP is that we have also a lot more obstructions than we do for almost any other problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So while there's evidence, we also have a lot of results ruling out many, many types of approaches to the problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is the one thing that the computer scientists are actually very good at.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's actually saying that certain approaches cannot work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
No-go theorems.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It could be undecidable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We don't know.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
situations are covered by, yeah, so most fluid, most of the time, water does not blow up, but could you design a very special initial state that does this?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, he's somewhat of an outlier, even among mathematicians who tend to have somewhat idealistic views.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I've never met him.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think I'd be interested to meet him one day, but I'd never have the chance.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I know people who've met him, but he's always had strong views about certain things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it's not like he was completely isolated from the math community.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, he would give talks and write papers and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But at some point, he just decided not to engage with the rest of the community.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He was disillusioned or something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I don't know.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And he decided to peace out and collect mushrooms in St.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Petersburg or something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that's fine.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can do that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, that's another sort of flip side.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, a lot of problems that we solve, some of them do have practical application, and that's great.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if you stop thinking about a problem, so he hasn't published since in this field, but that's fine.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's many, many other people who've done so as well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I guess one thing I didn't realize initially with the Fields Medal is that it sort of makes you part of the establishment.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Most career mathematicians just focus on publishing the next paper, maybe getting promoted to one rank, and starting a few projects, maybe taking some students or something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But then suddenly people want your opinion on things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You have to think a little bit about things that you might just so foolishly say because you know no one's going to listen to you.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's more important now.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Is it constraining to you?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I have a lot less free time than I had previously.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, mostly by choice.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, obviously I have the option to sort of decline.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I decline a lot of things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I could decline even more.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or I could acquire a reputation of being so unreliable that people don't even ask anymore.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I love the different algorithms here.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is great.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's always an option.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But, you know, there are things that are...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, I don't spend as much time as I do as a postdoc, you know, just working on one problem at a time or fooling around.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I still do that a little bit.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But yeah, as you advance in your career, the more soft skills, so math somehow front-loads all the technical skills to the early stages of your career.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So yeah, as a postdoc, as a publisher or parish, you're
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You're incentivized to basically focus on proving very technical theorems, so prove yourself as well as prove the theorems.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But then as you get more senior, you have to start mentoring and giving interviews and trying to shape direction of the field both research-wise and sometimes you have to do various administrative things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it's kind of the right social contract because you need to work in the trenches to see what can help mathematicians.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so it has practical importance.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So this Clay-Price problem concerns what's called the incompressible Navier-Stokes, which governs things like water.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some recognition is necessarily important.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But yeah, it's...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's also important to not let these things take over your life and only be concerned about getting the next big award or whatever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Again, you see these people try to only solve the really big math problems and not work on things that are less sexy, if you wish, but actually still interesting and instructive.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
As you say, the way the human mind works, we understand things better when they're attached to humans, and also if they're attached to a small number of humans.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's something called the compressible Navier-Stokes, which covers things like air.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The way our human mind is wired, we can comprehend
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
the relationship between 10 or 20 people.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But once you get beyond 100 people, there's a limit.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think there's a name for it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Beyond which, it just becomes the other.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you have to simplify the whole mass.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
99.9% of humanity becomes the other.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And often these models are incorrect, and this causes all kinds of problems.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, yeah, so to humanize a subject, you know, if you identify a small number of people and say, you know, these are representative people of a subject, role models, for example, that has some role, but it can also be, yeah, too much of it can be harmful because it's,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that's particularly important for weather prediction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'll be the first to say that my own career path is not that of a typical mathematician.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Weather prediction, it does a lot of computation of fluid dynamics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I had a very accelerated education.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I skipped a lot of classes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think I always had very fortunate mentoring opportunities, and I think I was at the right place at the right time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Just because someone doesn't have my knowledge
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
trajectory, it doesn't mean that they can't be good mathematicians.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They can be good mathematicians in a very different style, and we need people of a different style.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Sometimes too much focus is given on the person who took the last step to complete
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A lot of it is actually just trying to solve the Navier-Stokes equations as best they can.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
a project in mathematics or elsewhere that's really taken centuries or decades with lots and lots of previous work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But that's a story that's difficult to tell if you're not an expert, because it's easier to just say one person did this one thing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It makes for a much simpler history.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Also gathering a lot of data so that they can initialize the equation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so I was a graduate student at the time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I vaguely remember there was press attention and everything.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Um, we all had the same, um, uh, we had pigeon holes in the same mail room, you know, so we all particular mail and like suddenly Andrew Wiles's mailbox exploded to be overflowing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a lot of moving parts.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's a good, that's a good metric.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, so yeah, we all talked about it at tea and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, we didn't understand, most of us didn't understand the proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We understand sort of high-level details.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In fact, there's an ongoing project to formalize it in Lean.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Kevin buzzed it, I think.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's a very important problem practically.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's really difficult to formalize now.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I guess you're right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The objects that they use, you can define them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They've been defined in Lean.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Just defining what they are can be done.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Why is it difficult to prove general things?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's really not trivial, but it's been done.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a lot of really basic facts about
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
these objects that have taken decades to prove in all these different math papers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so lots of these have to be formalized as well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Kevin Buzzard's goal, actually, he has a five-year grant to formalize Fermat's last theorem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And his aim is that he doesn't think he will be able to get all the way down to the basic axioms
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But he wants to formalize it to the point where the only things that he needs to rely on as black boxes are things that were known by 1980 to number theorists at the time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then some other work would have to be done to get from there.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's a different area of mathematics than the type of mathematics I'm used to.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In analysis, which is kind of my area, the objects we study are kind of much closer to the ground.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I study things like prime numbers and
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
and functions, and things that are within the scope of a high school math education to at least define.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But then there's this very advanced algebraic side of number theory where people have been building structures upon structures for quite a while.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a very sturdy structure.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
At the base, at least, it's extremely well developed with textbooks and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The short answer is Maxwell's demon.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it does get to the point where if you haven't taken these years of study and you want to ask about what is going on at level six of this tower, you have to spend quite a bit of time before they can even get to the point where you can see something you recognize.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So Maxwell's demon is a concept in thermodynamics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, that is a romantic, yeah, so it kind of fits with sort of the romantic image I think people have of mathematicians to the extent they think of them at all as these kind of eccentric, you know, wizards or something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So that certainly kind of fits.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
accentuated that perspective.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, I mean, it's, it is a great achievement.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
His style of solving problems is so different from my own.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like if you have a box of two gases, oxygen and nitrogen, and maybe you start with all the oxygen on one side and nitrogen on the other side, but there's no barrier between them, right?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But, which is great.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, we need people like that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Can you speak to it?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like what, in terms of like the, you like the collaborative.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I like moving on from a problem if it's giving too much difficulty.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Got it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But you need the people who have the tenacity and the fearlessness.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, I've collaborated with people
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
People like that where I want to give up because the first approach that we tried didn't work and the second one didn't approach.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But they're convinced and they have the third, fourth, and the fifth of which works.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I have to use my words.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, I didn't think this was going to work, but yes, you were right all along.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, it works for me.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, I mean, there are also people who are very productive and they focus very deeply on, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think everyone has to find their own workflow.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like, one thing which is a shame in mathematics is that we have mathematics, there's sort of a one-size-fits-all approach to teaching mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And, you know, so we have a certain curriculum and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Then they will mix.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, you know, maybe, like, if you do math competitions or something, you get a slightly different experience.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think many people, they don't find their native math language until very late, or usually too late, so they stop doing mathematics, and they have a bad experience with a teacher who's trying to teach them one way to do mathematics, and they don't like it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And they should stay mixed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
My theory is that evolution has not given us a math center of a brain directly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We have a vision center and a language center and some other centers which evolution has honed, but we don't have an innate sense of mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But our other centers are sophisticated enough that we can repurpose other areas of our brain to do mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's no reason why they should unmix.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So some people have figured out how to use the visual center to do mathematics, and so they think very visually when they do mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some people have repurposed their language center, and they think very symbolically.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some people, if they are very competitive and they like gaming, there's a part of your brain that's very good at solving puzzles and games, and that can be repurposed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But in principle, because of all the collisions between them, there could be some sort of weird conspiracy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
When I talk to other mathematicians, I can tell that they're using some other different styles of thinking than I am.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, not disjoint, but they may prefer visual.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I don't actually prefer visual so much.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I need also visual aids myself.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Mathematics provides a common language, so we can still talk to each other even if we are thinking in different ways.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But you can tell there's a difference...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
set of subsystems being used in the thinking process.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They take different paths.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They're very quick at things that I struggle with and vice versa.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And yet they still get to the same goal.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's beautiful.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like maybe there's a microscopic demon called Maxwell's demon that will, every time an oxygen and nitrogen atom collide, they will bounce off in such a way that the oxygen sort of drifts onto one side and the nitrogen goes to the other.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And yeah, but I mean, the way we educate, unless you have like a personalized tutor or something, I mean, education sort of just by initial scale has to be mass produced.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, you have to teach the 30 kids, you know, they have 30 different styles.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can't, you can't teach 30 different ways.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, it's a tricky problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
One nice thing is that there are now lots of sources for mathematical enrichment outside the classroom.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, in my day, there were already math competitions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And, you know, there are also popular math books in the library.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Now you have YouTube.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There are forums devoted to solving math puzzles.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And math shows up in other places.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For example, there are hobbyists who play poker for fun.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And they, for very specific reasons, are interested in very specific probability questions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's a community of amateur probabilists in poker.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In chess, in baseball, I mean, there's math all over the place.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I'm hoping actually with these new sort of tools for Lean and so forth, that actually we can incorporate the broader public into math research projects.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like this is almost, it doesn't happen at all currently.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in the sciences, there's some scope for citizen science, like astronomers, the amateurs who discover comets, and there's biologists, there are people who identify butterflies and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you could have an extremely improbable configuration emerge.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there are a small number of activities where amateur mathematicians can discover new primes and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But previously, because we had to verify every single contribution, most mathematical research projects, it would not help to have...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
input from the general public.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In fact, it would just be time-consuming because just error-checking and everything.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But one thing about these formalization projects is that they are bringing together more, bringing in more people.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I'm sure there are high school students who've already contributed to some of these formalizing projects, who've contributed to MathLib.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You don't need to be a PhD holder to just work on one atomic thing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
which we never see, and statistically it's extremely unlikely, but mathematically it's possible that this can happen, and we can't rule it out.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can execute code and you can get results.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, you can print out the world pretty quickly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Um,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If programming was taught as an almost entirely theoretical subject, where you just taught the computer science, the theory of functions and routines and so forth, and outside of some very specialized homework assignments, you're not actually programmed on the weekend for fun, that would be considered as hard as math.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So as I said, there are communities of non-matheticians where they're deploying math for some very specific purpose, like optimizing their poker game.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And for them, then math becomes fun for them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's a tough, tough, tough question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a lot of certainty now in the world.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There was this period after the war where, at least in the West, if you came from a good demographic, there was a very stable path to a good career.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You go to college, you get an education, you pick one profession and you stick to it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's becoming much more a thing of the past.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I think you just have to be adaptable and flexible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think people will have to get skills that are transferable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, like learning one specific programming language or one specific subject of mathematics or something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this is a situation that shows up a lot in mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That itself is not a super transferable skill, but sort of knowing how to reason with abstract concepts or how to problem solve when things go wrong.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
These are things which I think we will still need
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Even as our tools get better, you'll be working with AI and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A basic example is the digits of pi, 3.14159 and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A lot of mathematicians.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
First of all, I've always been interested in new ways to do mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I feel like a lot of the ways we do things right now are inefficient.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Me and my colleagues, we spend a lot of time doing very routine computations or doing things that other mathematicians would instantly know how to do, and we don't know how to do them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Why can't we search and get a quick response?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's why I've always been interested in exploring new workflows.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
About four or five years ago, I was on a committee where we had to ask for ideas for interesting workshops to run at a math institute.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
At the time, Peter Schultzer had just formalized one of his new theorems, and there were some other developments in computer-assisted proof that looked quite interesting.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The digits look like they have no pattern, and we believe they have no pattern.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I said, oh, we should run a workshop on this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This would be a good idea.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then I was a bit too enthusiastic about this idea, so I got voluntold to actually run it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I did with a bunch of other people, Kevin Buzzer and Jordan Ellenberg and a bunch of other people.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it was a nice success.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We brought together a bunch of mathematicians and computer scientists and other people, and we got up to speed on the state of the art.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
On the long term, you should see as many 1s and 2s and 3s as 4s and 5s and 6s.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it was really interesting developments that most mathematicians didn't know was going on.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Lots of nice proofs of concept, just hints of what was going to happen.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This was just before ChatGPT, but even then there was one talk about language models and the potential capability of those in the future.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So that got me excited about the subject.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I started giving talks about this is something we should, more of us should start looking at now that I've arranged to run this conference.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then ChatGPT came out and like suddenly AI was everywhere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so I got interviewed a lot about this topic.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And in particular, the interaction between AI and formal proof of existence.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There should be no preference in the digits of pi to favor, let's say, 7 over 8.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I said, yeah, they should be combined.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is a perfect synergy to happen here.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And at some point I realized that I have to actually do not just talk the talk, but walk the walk.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, like, you know, I don't work in machine learning and I don't work in proof formalization.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's a limit to how much I can just rely on authority and saying, you know, I'm a, I'm a, I'm a Warner mathematician.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Just trust me.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, when I say that this is going to change mathematics and I'm not doing it any, and I don't do any of it myself.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I felt like I had to actually justify it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A lot of what I get into, actually, I don't quite see in advice as how much time I'm going to spend on it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it's only after I'm sort of waist deep in a project that I realize by that point I'm committed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But maybe there is some demon in the digits of pi that every time you compute more and more digits, it biases one digit to another.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think mathematics is so huge these days that nobody knows all of modern mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Inevitably, we make mistakes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can't cover up your mistakes with just bravado, because people will ask for your proofs.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you don't have the proofs, you don't have the proofs.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I love math.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it does keep us honest.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it's not a perfect panacea, but I think we do have more of a culture of admitting error because we're forced to all the time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Big, ridiculous question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So first of all, as I mentioned before, there's some time dependence.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
On the day.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, if you plot cumulatively over time, for example, Euclid is one of the contenders.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then maybe some unnamed anonymous mathematicians before that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Whoever came up with the concept of numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, Hilbert spaces.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We have lots of things that are named after him, of course.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Just the arrangement of mathematics and just the introduction of certain concepts.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, 23 problems have been extremely influential.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is a conspiracy that should not happen.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, I mean, there's this bystander effect everywhere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If no one says you should do X, everyone just mills around waiting for somebody else to do something, and nothing gets done.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's no reason it should happen, but there's no way to prove it with our current technology.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
One thing that actually you have to teach undergraduates in mathematics is that you should always try something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you see...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
a lot of paralysis in an undergraduate trying a math problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If they recognize that there's a certain technique that can be applied, they will try it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But there are problems for which they see none of their standard techniques obviously applies.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the common reaction is then just paralysis.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I don't know what to do.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think there's a quote from The Simpsons, I've tried nothing and I'm all out of ideas.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, you know, like the next step then is to try anything, like no matter how stupid, and in fact, almost the stupid of the better, which, you know, I think we're just almost guaranteed to fail, but the way it fails is going to be instructive.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like it fails because you're not at all taking into account this hypothesis.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Oh, this hypothesis must be useful.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's a clue.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, so getting back to Navier-Stokes, a fluid has a certain amount of energy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Psychology is really important.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You talk to athletes like marathon runners and so forth, and they talk about what's the most important thing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Is it the training regimen or the diet?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So much of it is psychology.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Just tricking yourself to think that the problem is feasible so that you're motivated to do it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And because the fluid is in motion, the energy gets transported around.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Is there something our human mind will never be able to comprehend?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, as a mathematician, I mean, it's a reduction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There must be some large number that you can't understand.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That was the first thing that came to mind.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And water is also viscous.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, okay.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, how much augmentation are you willing?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For example, if I didn't even have pen and paper, if I had no technology whatsoever, so I'm not allowed blackboard, pen and paper.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You're already much more limited than you would be.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Incredibly limited.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Even language, the English language is a technology.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if the energy is spread out over many different locations, the natural viscosity of the fluid will just damp out the energy and it will go to zero.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's one that's been very internalized.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yes, yeah, yes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So humanity, plural, has much more intelligence, in principle, on its good days than the individual humans put together.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It can all have less.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, but...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so the mathematical community, plural, is an incredibly super intelligent entity that no single human mathematician can come close to replicating.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You see it a little bit on these question analysis sites.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's Math Overflow, which is the math version of Stack Overflow.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And sometimes you get very quick responses to very difficult questions from the community.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it's a pleasure to watch, actually, as an expert.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think the younger generation is always really creative and enthusiastic and inventive.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this is what happens in...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a pleasure working with young students.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The progress of science tells us that the problems that used to be really difficult can become trivial to solve.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it was like navigation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Just knowing where you were on the planet was this horrendous problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
People died or lost fortunes because they couldn't navigate.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
when we actually experiment with water.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we have devices in our pockets that do this automatically for us.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a completely solved problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So things that seem unfeasible for us now could be maybe just homework exercises for them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You splash around, there's some turbulence and waves and so forth, but eventually it settles down and the lower the amplitude, the smaller the velocity, the more calm it gets.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, well, already plenty has happened.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you could go back in time and talk to your...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
teenage self or something, you know what I mean?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Just the internet and our AI, I mean, again, they're beginning to be internalized and say, yeah, of course an AI can understand our voice and give reasonable, you know, slightly incorrect answers to any question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But yeah, this was mind-blowing even two years ago.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, when the infrastructure and the culture is healthy, the community of humans can be so much more intelligent and mature and rational than the individuals within it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But potentially there is also a demon that keeps pushing
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
the energy of the fluid into a smaller and smaller scale.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it will move faster and faster.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And at faster speeds, the effect of viscosity is relatively less.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so it could happen that it creates some sort of what's called a self-similar blob scenario where, you know,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
the energy of the fluid starts off at some large scale, and then it all sort of transfers its energy into a smaller region of the fluid, which then, at a much faster rate, moves into an even smaller region, and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And each time it does this, it takes maybe half as long as the previous one, and then you could actually
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
converge to all the energy concentrating in one point in a finite amount of time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That scenario is called finite-time blow-up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In practice, this doesn't happen.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Water is what's called turbulent.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It is true that if you have a big eddy of water, it will tend to break up into smaller eddies.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it won't transfer all the energy from one big eddy into one small eddy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It will transfer into maybe three or four.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then those ones split up into maybe three or four small eddies of their own.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so the energy gets dispersed to the point where the viscosity can then keep the thing under control.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if it can somehow concentrate all the energy
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
keep it all together and do it fast enough that the viscous effects don't have enough time to calm everything down, then this blob can occur.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there were papers who had claimed that, oh, you just need to take into account conservation of energy and just carefully use the viscosity and you can keep everything under control for not just Navier-Stokes, but for many, many types of equations like this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so in the past, there have been many attempts to try to obtain what's called global regularity for Navier-Stokes, which is the opposite of finite time blow-up, that velocity stays smooth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it all failed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There was always some sign error or some subtle mistake, and it couldn't be salvaged.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So what I was interested in doing was trying to explain why we were not able to disprove finite time blow-up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I couldn't do it for the actual equations of fluids, which were too complicated.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if I could average the equations of motion of the Navier-Stokes, basically, if I could turn off certain types of ways in which water interacts and only keep the ones that I want.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in particular, if there's a fluid and it could transfer its energy from a large eddy into this small eddy or this other small eddy, I would turn off the energy channel that would transfer energy to this one and direct it only into this smaller eddy while still preserving the law of conservation of energy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you're trying to make a blow-up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I basically engineer a blow-up by changing the rules of physics, which is one thing that mathematicians are allowed to do.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We can change the equation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
How does that help you get closer to the proof of something?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it provides what's called an obstruction in mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So what I did was that basically if I turned off certain parts of the equation, which usually when you turn off certain interactions, make it less nonlinear, it makes it more regular and less likely to blow up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But I found that by turning off a very well-designed set of interactions, I could force all the energy to blow in finite time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So what that means is that if you wanted to prove
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
global regularity for Navier-Stokes, for the actual equation, you must use some feature of the true equation which my artificial equation does not satisfy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it rules out certain approaches.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The thing about math is it's not just about taking a technique that is going to work and applying it, but you need to not take the techniques that don't work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And for the problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it's only after a lot of experience that you realize there's no way that these methods are going to work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So having these counterexamples for nearby problems kind of rules out, it saves you a lot of time because you're not wasting energy on things that you now know cannot possibly ever work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the key phenomenon that my technique exploits is what's called supercriticality.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in partial differential equations, often these equations are like a tug of war between different forces.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in Navier-Stokes, there's the dissipation force coming from viscosity, and it's very well understood.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's linear.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It calms things down.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If viscosity was all there was, then nothing bad would ever happen.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But there's also transport, that energy in one location of space can get transported because the fluid is in motion to other locations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that's a nonlinear effect, and that causes all the problems.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there are these two competing terms in the Navier-Stokes equation, the dissipation term and the transport term.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If the dissipation term dominates, if it's large, then basically you get regularity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And if the transport term dominates, then we don't know what's going on.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a very nonlinear situation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's unpredictable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's turbulent.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So sometimes these forces are in balance at small scales, but not in balance at large scales or vice versa.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So Navier-Stokes is what's called supercritical.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So at smaller and smaller scales,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
the transport terms are much stronger than the viscosity terms.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the viscosity terms are the things that calm things down.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so this is why the problem is hard.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In two dimensions, so the Soviet mathematician Ladislav Skaya, she, in the 60s, showed in two dimensions there was no blow-up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And as you mentioned, the Navier-Stokes equation is what's called critical.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The effect of transport and the effect of viscosity are about the same strength, even at very, very small scales.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we have a lot of technology to handle critical and also subcritical equations and improve regularity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But for supercritical equations, it was not clear what was going on.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
and i did a lot of work and then there's been a lot of follow-up showing that for many other types of super critical equations you can create all kinds of blow-up examples once the non-linear effects dominate the linear effects at small scales you can have all kinds of bad things happen so this is sort of one of the main insights of this this line of work is that super criticality versus criticality and sub-criticality this this makes a big difference i mean that's a key qualitative feature that distinguishes
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some equations were being sort of nice and predictable and, you know, like planetary motion.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, there's certain equations that you can predict for millions of years or thousands at least.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Again, it's not really a problem, but there's a reason why we can't predict the weather past two weeks into the future because it's a super critical equation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Lots of really strange things are going on at very fine scales.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, and if non-linearity is somehow more and more featured and interesting at small scales.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's many equations that are non-linear, but in many equations you can approximate things by the bulk.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For example, planetary motion, if you want to understand the orbit of the Moon or Mars or something, you don't really need the microstructure of the seismology of the Moon or exactly how the mass is distributed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can almost approximate these planets by point masses.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Just the aggregate behavior is important.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if you want to model a fluid, like the weather, you can't just say in Los Angeles, the temperature is this, the wind speed is this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For supercritical equations, the finite scale information is really important.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Can you describe this idea?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So this came out of this work of constructing this average equation that blew up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So one...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
As part of how I had to do this, there's sort of this naive way to do it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You just keep pushing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Every time you get energy at one scale, you push it immediately to the next scale as fast as possible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is sort of the naive way to force blow up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It turns out in five and higher dimensions, this works.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But in three dimensions, there was this funny phenomenon that I discovered, that if you keep...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
change the laws of physics, you just always keep trying to push the energy into smaller and smaller scales.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
What happens is that the energy starts getting spread out into many scales at once.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you have energy at one scale, you're pushing it into the next scale, and then
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
as soon as it enters that scale, you also push it to the next scale, but there's still some energy left over from the previous scale.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You're trying to do everything at once.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this spreads out the energy too much.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then it turns out that it makes it vulnerable for viscosity to come in and actually just damp out everything.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it turns out this directed motion doesn't actually work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There was a separate paper by some other authors that actually showed this in three dimensions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So what I needed was to program a delay.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So kind of like airlocks.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I needed an equation which would start with a fluid...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
doing something at one scale, it would push its energy into the next scale, but it would stay there until all the energy from the larger scale got transferred.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And only after you pushed all the energy in, then you sort of open the next gate and then you push that in as well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So by doing that, the energy inches forward scale by scale in such a way that it's always localized at one scale at a time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then it can resist the effects of viscosity because it's not dispersed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in order to make that happen,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
yeah, I had to construct a rather complicated non-linearity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Um, and it was basically like, um, you know, like it was constructed like an electronic circuit.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I, I actually thank my wife for this because she was trained as a electrical engineer.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Um, and, um, you know, she talked about, um,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He had to design circuits and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And if you want a circuit that does a certain thing, like maybe have a light that flashes on and then turns off and then on and then off, you can build it from more primitive components, capacitors and resistors and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you have to build a diagram.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And these diagrams, you can sort of follow it with your eyeballs and say, oh yeah, the current will build up here and then it will stop and then it will do that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I knew how to build the analog of basic electronic components, like resistors and capacitors and so forth, and I would stack them together in such a way that I would create something that would open one gate, and then there would be a clock, and then once the clock hits a certain threshold, it would close it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It was kind of a Rube Goldberg-type machine, but described mathematically.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this ended up working.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So what I realized is that if you could pull the same thing off for the actual equations, so if...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The equations of water support a computation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can imagine a steampunk, but it's really a waterpunk type of thing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Modern computers are electronic.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They're powered by electrons passing through very tiny wires and interacting with other electrons and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Instead of electrons, you can imagine these pulses of water moving at a certain velocity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Maybe there are two different configurations corresponding to a bit being up or down.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
probably that if you had two of these moving bodies of water collide, they would come out with some new configuration, which would be something like an AND gate or OR gate.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The output would depend in a very predictable way on the inputs.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you could chain these together and maybe create a Turing machine
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then you have computers, which are made completely out of water.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And if you have computers, then maybe you can do robotics, hydraulics and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so you could create some machine, which is basically a fluid analog of what's called a von Neumann machine.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So von Neumann proposed, if you want to colonize Mars, the sheer cost of transporting people and machines to Mars is just ridiculous.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if you could transport one machine to Mars,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this machine had the ability to mine the planet, create some more materials, smelt them, and build more copies of the same machine.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Then you could colonize the whole planet over time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if you could build a fluid machine,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so it's a fluid robot.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And what it would do, its purpose in life, it's programmed so that it would create a smaller version of itself in some sort of cold state.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It wouldn't start just yet.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Once it's ready, the big robot, Configured Water, would transfer all its energy into the smaller configuration and then power down.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then it'd clean itself up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then what's left is this newer state, which would then turn on and do the same thing, but smaller and faster.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then the equation has a certain scaling symmetry.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Once you do that, it can just keep iterating.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So this, in principle, would create a blur for the actual Navier-Stokes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this is what I managed to accomplish for this average Navier-Stokes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it provided this sort of roadmap to solve the problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Now, this is...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
a pipe dream because there are so many things that are missing for this to actually be a reality.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I can't create these basic logic gates.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I don't have these special configurations of water.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, there's candidates that include vortex rings that might possibly work, but also, you know, analog computing is really nasty compared to digital computing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, because there's always errors.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You have to do a lot of error correction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
along the way.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I don't know how to completely power down the big machine so that it doesn't interfere with the running of the smaller machine.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But everything in principle can happen.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It doesn't contradict any of the laws of physics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's sort of evidence that this thing is possible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There are other groups who are
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
now pursuing ways to make Navisworks blow up, which are nowhere near as ridiculously complicated as this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They actually are pursuing much closer to the direct self-similar model, which can, it doesn't quite work as is, but there could be some simpler scheme than what I just described to make this work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, that's a big leap.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's precedent.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, so the thing about mathematics is that it's really good at spotting connections between what you might think of as completely different things
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
problems.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if the mathematical form is the same, you can draw a connection.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's a lot of work previously on what's called cellular automata, the most famous of which is Conway's Game of Life.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's this infinite discrete grid, and at any given time, the grid is either occupied by a cell or it's empty.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's a very simple rule that tells you how these cells evolve.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So sometimes cells live and sometimes they die.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And when I was a student, it was a very popular screensaver to actually just have these animations going.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And they look very chaotic.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In fact, they look a little bit like turbulent flow sometimes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But at some point, people discovered more and more interesting structures within this game of life.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, for example, they discovered this thing called a glider.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So a glider is a very tiny configuration of like four or five cells.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
which evolves and it just moves at a certain direction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's like this vortex rings.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is an analogy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The Game of Life is a discrete equation and the fluid Navier-Stokes is a continuous equation, but mathematically, they have some similar features.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so all the time people discovered more and more interesting things that you could build within the game of life.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The game of life is a very simple system.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It only has like three or four rules to do it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But you can design all kinds of interesting configurations inside it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's something called a glider gun that does nothing but spit out gliders one at a time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then after a lot of effort, people managed to create AND gates and OR gates for gliders.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like there's this massive, ridiculous structure, which if you have a stream of gliders coming in here and a stream of gliders coming in here, then you may produce extreme gliders coming out.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If both of the streams have gliders, then there'll be an output stream.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if only one of them does, then nothing comes out.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So they could build something like that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And once you could build these basic gates, then just from software engineering, you can build almost anything.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can build a Turing machine.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it's like an enormous steampunk type things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They look ridiculous, but then people also generated self-replicating objects in the game of life.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A massive machine, a binomial machine, which over a huge period of time, and it always looked like glider guns inside doing these very steampunk calculations, it would create another version of itself,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
which could replicate.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's so incredible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A lot of this was like community crowdsourced by like amateur mathematicians, actually.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I knew about that work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so that is part of what inspired me to propose the same thing with Navier-Stokes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Now it was a much, as I said, analog is much worse than digital.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like it's going to be,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can't just directly take the constructions from the game of life and plunk them in.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But again, it shows it's possible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
the thing is you can get this emergent very complicated structures but only with very carefully prepared initial conditions yeah so so these these these glider guns and and gates and and software machines if you just plunk down randomly some cells and you when they get left you'll not see any of these um and that's the analogous situation of navier-stokes again you know that that with with typical initial conditions you will not have any of this weird computation going on um but
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
basically through engineering, you know, by specially designing things in a very special way, you can make clever constructions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is a recurring challenge in mathematics that I call the dichotomy between structure and randomness.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That most objects that you can generate in mathematics are random.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They look like random, like the digits of pi.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
we believe is a good example but there's a very small number of things that have patterns but now you can prove something as a pattern by just constructing you know like if something has a simple pattern and you have a proof that it does something like repeat itself every so often you can do that but
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you can prove that most sequences of digits have no pattern.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if you just pick digits randomly, there's something called low-large numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It tells you you're going to get as many ones as twos in the long run.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But we have a lot fewer tools to, if I give you a specific pattern, like the digits of pi,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
How can I show that this doesn't have some weird pattern to it?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some other work that I spend a lot of time on is to prove what are called structure theorems or inverse theorems that give tests for when something is very structured.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So some functions are what's called additive.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like if you have a function that has natural numbers, they're natural numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So maybe, you know, two maps to four, three maps to six, and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some functions are what's called additive, which means that if you add
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you add two inputs together, the output gets added as well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For example, a multiplying by a constant.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you multiply a number by 10, if you multiply a plus b by 10, that's the same as multiplying a by 10 and b by 10 and then adding them together.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So some functions are additive.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some functions are kind of additive, but not completely additive.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So for example, if I take a number n, I multiply by the square root of 2, and I take the integer part of that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So 10 by square root of 2 is like 14 point something, so 10 up to 14.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
20 went up to 28.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in that case, additivity is true then, so 10 plus 10 is 20, and 14 plus 14 is 28.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But because of this rounding, sometimes there's round-off errors, and sometimes when you add A plus B, this function doesn't quite give you the sum of the two individual outputs, but the sum plus or minus one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's almost additive, but not quite additive.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's a lot of useful results in mathematics, and I've worked a lot on developing things like this, to the effect that if a function exhibits some structure like this, then there's a reason for why it's true, and the reason is because there's some other nearby function which is actually completely structured, which is explaining this sort of partial pattern that you have.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you have these inverse theorems, it creates this dichotomy that either
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
the objects that you study either have no structure at all, or they are somehow related to something that is structured.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And in either case, you can make progress.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A good example of this is that there's this old theorem in mathematics called Szemeredi's theorem, proven in the 1970s.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It concerns trying to find a certain type of pattern in a set of numbers that the patterns have made progression.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Things like 3, 5, and 7, or 10, 15, and 20.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And Andrea Zamorelli proved that any set of numbers that are sufficiently big, what's called positive density, has arithmetic progressions in it of any length you wish.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, for example, the odd numbers have a density of one-half,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
and they contain rhythmic progressions of any length.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in that case, it's obvious because the odd numbers are really, really structured.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I can just take 11, 13, 15, 17.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I can easily find rhythmic progressions in that set.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But Zermattislim also applies to random sets.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If I take the set of odd numbers and I flip a coin for each number, and I only keep
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
the numbers which for which i got a heads okay so i just flip coins i just randomly take out half the numbers i keep one half so that's a set that has no no patterns at all but just from random fluctuations you will still get a lot of um of ethnic progressions in that set can you prove that there's arithmetic progressions of arbitrary length within a random yes um have you heard of the infinite monkey theorem
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Usually, mathematicians give boring names to theorems, but occasionally they give colorful names.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each typewriter, they type out text randomly, almost surely one of them is going to generate the entire square root of Hamlet, or any other finite string of text.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It will just take some time, quite a lot of time, actually.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if you have an infinite number, then it happens.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So basically the thing is that if you take an infinite string of digits or whatever, eventually any finite pattern you wish will emerge.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It may take a long time, but it will eventually happen.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In particular, ethnic progressions of any length will eventually happen
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, but you need an extremely long random sequence for this to happen.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
How are we humans supposed to deal with infinity?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, you can think of infinity as just an abstraction of
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
a finite number for which you do not have a bound for.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So nothing in real life is truly infinite.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But you can ask yourself questions like, what if I had as much money as I wanted?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or what if I could go as fast as I wanted?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And a way in which mathematicians formalize that is, mathematics has found a formalism to idealize, instead of something being extremely large or extremely small, to actually be exactly infinite or zero.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And often the mathematics becomes a lot cleaner when you do that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, in physics, we joke about assuming spherical cows.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, like real-world problems have got all kinds of real-world effects, but you can idealize, send some things to infinity, send some things to zero.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the mathematics becomes a lot simpler to work with there.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so there's a lot of pitfalls.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We spend a lot of time in undergraduate math classes teaching analysis, and analysis is often about how to take limits.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So for example, A plus B is always B plus A. So when you have a finite number of terms and you add them, you can swap them and there's no problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But when you have an infinite number of terms, they're these sort of show games you can play, where you can have a series which converges to one value, but you rearrange it, and it suddenly converges to another value.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so you can make mistakes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You have to know what you're doing when you allow infinity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You have to introduce these epsilons and deltas, and there's a certain type of way of reasoning that helps you avoid mistakes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In more recent years, people have started taking results that are true in infinite limits and what's called finitizing them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you know that something's true eventually, but you don't know when.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Now give me a rate.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, so if I don't have an infinite number of monkeys, but a large finite number of monkeys, how long do I have to wait for Hamlet to come out?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that's a more quantitative question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this is something that you can attack by purely finite methods, and you can use your finite intuition.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And in this case, it turns out to be exponential in the length of the text that you're trying to generate.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so this is why you never see the monkeys create Hamlet.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can maybe see them create a four-letter word,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
but nothing that big.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so I personally find once you finalize an infinite statement, it does become much more intuitive and it's no longer so weird.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The downside is that the finalized proofs are just much, much messier.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the infinite ones are found first, usually, like decades earlier.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then later on, people finalize them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I think science in general is interaction between three things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's the real world.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's what we observe of the real world, our observations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
and then our mental models as to how we think the world works.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we can't directly access reality.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
All we have are the observations, which are incomplete, and they have errors.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there are many, many cases where we want to know, for example, what is the weather like tomorrow, and we don't yet have the observation that we'd like to predict.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then we have these simplified models, sometimes making unrealistic assumptions, you know, spherical cow type things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Those are the mathematical models.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Mathematics is concerned with the models.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Science collects the observations and it proposes the models that might explain these observations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
What mathematics does is we stay within the model and we ask, what are the consequences of that model?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
What observations, what predictions would the model make of future observations?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or past observations, does it fit observed data?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's definitely a symbiosis.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I guess mathematics is unusual among other disciplines.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We start from hypotheses, like the axioms of a model, and ask what conclusions come up from that model.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In almost any other discipline, you start with the conclusions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I want to do this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I want to build a bridge.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I want to make money.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I want to do this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then you find the path to get there.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
a lot there's a lot less sort of speculation about suppose I did this what would happen you know planning and modeling speculative fiction maybe is one other place but that's about it actually most of the things we do in life is conclusions driven including physics and science I mean they want to know where is this asteroid going to go what is the weather going to be tomorrow but mathematics also has this other direction of going from the axioms
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
What do you think?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, you need both, top-down and bottom-up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, it's a really interaction between all these things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So over time, the observations and the theory and the modeling should both get better.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
closer to reality.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But initially, and this is always the case, they're always far apart to begin with.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But you need one to figure out where to push the other.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if your model is predicting anomalies that are not predicted by an experiment, that tells experimenters where to look to find more data to refine the models.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it goes back and forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Within mathematics itself, there's also a theory and experimental component.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's just that until very recently, theory has dominated almost completely.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
99% of mathematics is theoretical mathematics, and there's a very tiny amount of experimental mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, people do do it, you know, like if they want to study prime numbers or whatever, they can just generate large data sets.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So once we had the computers, we began to do it a little bit.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Although even before, well, like Gauss, for example, he discovered, he conjectured the most basic theorem in number theory, which is called the prime number theorem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
which predicts how many primes, up to a million, up to a trillion.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's not an obvious question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And basically what he did was that he computed, I mean, mostly by himself, but also hired human computers, people whose professional job it was to do arithmetic, to compute the first 100,000 primes or something, and made tables and made a prediction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That was an early example of experimental mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Theoretical mathematics was just much more successful because doing complicated mathematical computations was just not feasible until very recently.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Even though we have powerful computers, only some mathematical things can be explored numerically.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's something called the combinatorial explosion.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you want to study, for example, Zermatt's theorem, you want to study all possible subsets of numbers 1 to 1,000.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's only 1,000 numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
How bad could it be?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It turns out the number of different subsets of 1 to 1,000 is 2 to the power of 1,000.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
which is way bigger than any computer can currently enumerate.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In fact, any computer ever will ever enumerate.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There are certain math problems that very quickly become just intractable to attack by direct brute force computation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Chess is another famous example.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The number of chess positions, we can't get a computer to fully explore.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But now we have AI.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We have tools to explore this space, not with 100% guarantees of success, but with experiment.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We can empirically solve chess now.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For example, we have very, very good AIs that don't explore every single position in the game tree, but they have found some very good approximation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And people are using actually these chess engines to do experimental chess.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They're revisiting old chess theories about, oh, this type of opening, this is a good type of move, this is not.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And they can use these chess engines to actually refine, in some cases overturn, conventional wisdom about chess.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I do hope that mathematics will have a larger experimental component in the future, perhaps powered by AI.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, there are these three ontological things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's actual reality, there's observations, and our models.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And technically they are distinct, and I think they will always be distinct.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But they can get closer over time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The process of getting closer often means that you have to discard your initial intuitions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Astronomy provides great examples.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
An initial model of the world is flat because it looks flat, and it's big.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The rest of the universe, the sun, for example, looks really tiny.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so you start off with a model, which is actually really far from reality, but it fits kind of the observations that you have.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So things look good, but over time, as you make more and more observations, bring it closer to reality, the model gets dragged along with it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so over time, we had to realize that the Earth was round,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
that it spins, it goes around the solar system, the solar system goes around the galaxy, and so on and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the universe is expanding.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Expansion is self-expanding, accelerating.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And in fact, very recently in this year or so, even the acceleration of the universe itself is evidence that it's non-constant.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the explanation behind why that is... It's catching up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's catching up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it's still, you know, the dark matter, dark energy, this kind of thing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We have a model that sort of explains, that fits the data really well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It just has a few parameters that you have to specify.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But so, you know, people say, oh, that's fudge factors.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, with enough fudge factors, you can explain anything.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But the mathematical point of the model is that you want to have fewer parameters in your model than data points in your observational set.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if you have a model with 10 parameters that explains 10 observations, that is a completely useless model.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's what's called overfitted.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But like if you have a model with two parameters and it explains a trillion observations, which is basically, so yeah, the dark matter model, I think it has like 14 parameters and it explains petabytes of data that the astronomers have.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can think of a theory, like one way to think about a physical mathematical theory is it's a compression of the universe and a data compression.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, you know, you have these petabytes of observations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You'd like to compress it to a model which you can describe in five pages and specify a certain number of parameters.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And if it can fit to reasonable accuracy,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
almost all of your observations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, the more compression that you make, the better your theory.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Einstein had a quote like that, the most incomprehensible thing about the universe is that it is comprehensible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And not just comprehensible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can do an equation like E equals MC squared.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There is actually some mathematical possible explanation for that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's this phenomenon in mathematics called universality.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So many complex systems at the macro scale are coming out of lots of tiny interactions at the macro scale.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And normally, because of the common form of explosion, you would think that the macroscale equations must be exponentially more complicated than the microscale ones.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And they are, if you want to solve them completely exactly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like, if you want to model...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
all the atoms in a box of air.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's, like, Avogadro's number is humongous, right?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a huge number of particles.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you actually have to track each one, it'll be ridiculous.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But certain laws emerge at the microscopic scale that almost don't depend on what's going on at the macroscale, or only depend on a very small number of parameters.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if you want to model a gas of, you know,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
quintillion particles in a box.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You just need to know its temperature and pressure and volume and a few parameters, like 5 or 6.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it models almost everything you need to know about these 10 to 23 or whatever particles.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we have...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We don't understand universality anywhere near as we would like mathematically, but there are much simpler toy models where we do have a good understanding of why universality occurs.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The most basic one is the central limit theorem that explains why the bell curve shows up everywhere in nature, that so many things are distributed by what's called a Gaussian distribution, a famous bell curve,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's not even a meme with this curve.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Even the meme applies broadly, the universality to the meme.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yes, you can go meta if you like.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But there are many processes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For example, you can take lots of independent random variables and average them together in various ways.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can take a simple average or more complicated average, and we can prove in various cases that these bell curves, these calciums emerge, and it is a satisfying explanation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Sometimes they don't.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if you have many different inputs and they're all correlated in some systemic way, then you can get something very far from a bow curve show up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this is also important to know when a situation fails.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So universality is not a 100% reliable thing to rely on.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The global financial crisis was a famous example of this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
People thought that mortgage defaults had this sort of Gaussian-type behavior that
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
that if you ask a population of 100,000 Americans with mortgages, ask what proportion of them will default on their mortgages, if everything was de-correlated, it would be a nice bell curve and you can manage risk with options and derivatives and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it is a very beautiful theory.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if there are systemic shocks in the economy that can push everybody to default at the same time, that's very non-Gaussian behavior.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this wasn't fully accounted for in 2008.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Now I think there's some more awareness that this systemic risk is actually a much bigger issue.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And just because the model is pretty and nice, it may not match reality.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the mathematics of working out what models do
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
is really important, but also the science of validating when the models fit reality and when they don't.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, you need both.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But mathematics can help because it can, for example, these central limit theorems, it told you that if you have certain axioms like non-correlation, that if all the inputs were not correlated to each other, then you have this Gaussian behavior, so things are fine.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It tells you where to look for weaknesses in the model.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if you have a mathematical understanding of central limit theorem and someone proposes to use these Gaussian copulars or whatever to model default risk, if you're mathematically trained, you would say, okay, but what are the systemic correlation between all your inputs?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so then you can ask the economists, how much of a risk is that?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then you can go look for that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's always this synergy between science and mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's certainly a lot of connecting threads, and a lot of the progress of mathematics can be represented by taking stories of two fields of mathematics that were previously not connected and finding connections.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
An ancient example is geometry and number theory.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in the times of the ancient Greeks, these were considered different subjects.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, mathematicians worked on both.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Euclid
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
worked both on geometry, most famously, but also on numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But they were not really considered related.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, a little bit, like, you know, you could say that this length was five times this length because you could take five copies of this length and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it wasn't until Descartes, who really realized that, who developed what we now call analytic geometry, that you can parameterize the plane, a geometric object, by two real numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Every point can be...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, geometric problems can be turned into problems about numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Today, this feels almost trivial.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's no content to this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Of course, a plane is xx and y, because that's what we teach, and it's internalized.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it was an important development that these two fields were unified.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this process has just gone on throughout mathematics over and over again.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Algebra and geometry were separated, and now we have algebraic geometry that connects them over and over again.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that's certainly the type of mathematics that I enjoy the most.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I think there's sort of different styles to being a mathematician.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think hedgehogs and foxes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A fox knows many things a little bit, but a hedgehog knows one thing very, very well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And in mathematics, there's definitely both hedgehogs and foxes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then there's people who can play both roles.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I think ideal collaboration between mathematicians involves... You need some diversity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A fox working with many hedgehogs or vice versa.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But I identify mostly as a fox, certainly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I like...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
arbitrage somehow, like learning how one field works, learning the tricks of that wheel, and then going to another field, which people don't think is related, but I can adapt the tricks.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So see the connections between the fields.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there are other mathematicians who are far deeper than I am.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They're really hedgehogs.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They know everything about one field, and they're much faster and more effective in that field, but I can give them these extra tools.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
know searching for the connections versus like very singular focus i'm much more comfortable with with the the the fox paradigm yeah so um yeah i i like looking for analogies narratives um i i spend a lot of time if there's a result i see it in one field and i like the result it's a cool result but i don't like the proof like it uses
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
types of mathematics that I'm not super familiar with.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I often try to re-prove it myself using the tools that I favor.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Often my proof is worse, but by the exercise of doing so, I can say, oh, now I can see what the other proof was trying to do.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And from that, I can get some understanding of the tools that are used in that field.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's very exploratory, doing crazy things in crazy fields and reinventing the wheel a lot.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Whereas the hedgehog style is much more scholarly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You're very knowledge-based.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You stay up to speed on all the developments in this field.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know all the history.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You have a very good understanding of exactly the strengths and weaknesses of each particular technique.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So yeah, I think you'd rely a lot more on sort of calculation than sort of trying to find narratives.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So yeah, I mean, I can do that too, but there are other people who are extremely good at that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Let's step back and...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's a good question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
When I came to graduate school in Princeton, so John Conway was there at the time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He passed away a few years ago.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So Conway just had this amazing way of thinking about all kinds of things in a way that you wouldn't normally think of.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He thought of proofs themselves as occupying some sort of space.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you want to prove something, let's say that there's infinitely many primes, there will be different proofs, but you could rank them in different axes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some proofs are elegant, some proofs are long, some proofs are elementary, and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The space of all proofs itself has some sort of shape.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He was interested in extreme points of this shape.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Out of all these proofs, what is the shortest at the expense of everything else, or the most elementary, or whatever?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He gave some examples of well-known theorems, and then he would give what he thought was the extreme proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
in these different aspects.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I just found that really eye-opening.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's not just getting a proof for a result was interesting, but once you have that proof, trying to
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
to optimize it in various ways.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That proofing itself had some craftsmanship to it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's something for my writing style that, you know, like when you do your math assignments as an undergraduate, your homework and so forth, you're sort of encouraged to just write down any proof that works and hand it in.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
As long as it gets a tick mark, you move on.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if you want your results to actually be influential and be read by people, it can't just be correct.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It should also be a pleasure to read, motivated, be adaptable to generalize to other things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's the same in many other disciplines, like coding.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a lot of analogies between math and coding.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I like analogies, if you haven't noticed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But, you know, like, you can code something spaghettical that works for a certain task, and it's quick and dirty and it works, but there's lots of good principles for writing code well so that other people can use it, build upon it, and so on, and has fewer bugs and whatever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's similar things with mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
sort of the programs or in this case the proofs but also the different languages maybe that's a different notation or whatever to use to accomplish a different task yeah you learn a lot I mean it may seem like a frivolous exercise but it can generate all these insights which if you didn't have this artificial objective to pursue you might not see
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, as I said, I mean, what I find most appealing is connections between different things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if the pi i equals minus one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So yeah, people are like, oh, this is all the fundamental constants.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, that's, I mean, that's cute.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But to me, the exponential function was to measure exponential growth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Compound interest or decay, anything which is continuously growing, continuously decreasing, growth and decay or dilation or contraction is modeled by the exponential function.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Whereas pi comes around from circles and rotation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you want to rotate a needle, for example, 180 degrees, you need to rotate by pi radians.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I, complex numbers, represents the swapping between rule and imaginary axes of a 90 degree rotation, so a change in direction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the exponential function represents growth and decay in the direction that you already are.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
When you stick an i in the exponential, instead of motion in the same direction as your current position, it's motion at right angles to your current position, so rotation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then so, e to the pi i equals minus one tells you that if you rotate for time pi, you end up at the other direction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it unifies geometry through dilation and exponential growth, or dynamics, through this act of complexification, rotation by i. So it connects together all these fields of mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
complex and the complex numbers, they're all considered almost, they're all next door neighbors in mathematics because of this identity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, it's confirmation that you have the right concepts.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So when you first study anything, you have to measure things and give them names.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And initially, sometimes because your model is, again, too far off from reality,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
you give the wrong things the best names.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you only find out later what's really important.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Physicists can do this sometimes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, but it turns out okay.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So actually, with physics, so E equals mc squared.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So one of the big things was the E. So when Aristotle first came up with his laws of motion and then Galileo and Newton and so forth,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They saw the things they could measure.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They could measure mass and acceleration and force and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Newtonian mechanics, for example, I think it was MA, was the famous Newton's second law of motion.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Those were the primary objects.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They gave them the central building in the theory.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It was only later, after people started analyzing these equations, that there always seemed to be these quantities that were conserved, in particular momentum and energy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it's not obvious that things happen in energy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's not something you can directly measure the same way you can measure mass and velocity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But over time, people realized that this was actually a really fundamental concept.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Hamilton eventually, in the 19th century, reformulated Newton's laws of physics into what's called Hamiltonian mechanics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
where the energy, which is now called the Hamiltonian, was the dominant object.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Once you know how to measure the Hamiltonian of any system, you can describe completely the dynamics, like what happens to all the states.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It really was a central actor, which was not obvious initially.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this helped, actually, this change of perspective really helped when quantum mechanics came along.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Because the early physicists who studied quantum mechanics, they had a lot of trouble trying to
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
adapt their Newtonian thinking because everything was a particle and so forth to quantum mechanics because everything was a wave.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It just looks really, really weird.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like you ask, what is the quantum vision if it was MA?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it's really, really hard to give an answer to that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it turns out that the Hamiltonian, which was so
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
secretly behind the scenes in classical mechanics, also is the key object in quantum mechanics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's also an object called Hamiltonian.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a different type of object.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's what's called an operator rather than a function.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But again, once you specify it, you specify the entire dynamics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's something called Schrodinger's equation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
that tells you exactly how quantum systems evolve once you have the Hamiltonian.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So side by side, they look completely different objects.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
One involves particles, one involves waves, and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For example, in classical mechanics, there's this thing called Noether's theorem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Every time there's a symmetry in a physical system, there is a conservation law.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the laws of physics are translation invariant.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like if I move 10 steps to the left, I experience the same laws of physics as if I was here.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that corresponds to conservation momentum.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If I turn around by some angle, again, I experience the same laws of physics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This corresponds to the conservation of angular momentum.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If I wait for 10 minutes, I still have the same laws of physics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So this time transition invariance, this corresponds to the law of conservation of energy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's this fundamental connection between symmetry and conservation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that's also true in quantum mechanics, even though the equations are completely different.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But because they're both coming from the Hamiltonian, the Hamiltonian controls everything.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Every time the Hamiltonian has a symmetry, the equations will have a conservation law.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So once you have the right language, it actually makes things a lot cleaner.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
One of the problems why we can't unify quantum mechanics and general relativity yet
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We haven't figured out what the fundamental objects are.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For example, we have to give up the notion of space and time being these almost Euclidean-type spaces.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we kind of know that at very tiny scales, there's going to be corner fluctuations, there's space-time foam.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And trying to use Cartesian coordinates X, Y, Z, it's a non-starter.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But we don't know what to replace it with.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We don't actually have the mathematical concepts.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The analog of Hamiltonian that sort of organized everything.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I believe so.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, the history of physics has been that of unification.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
much like mathematics over the years.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Electricity and magnetism were separate theories, and then Maxwell unified them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Newton unified the motions of the heavens for the motions of objects on the Earth and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it should happen.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's just that the... Again, to go back to this model of observations and theory...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Part of our problem is that physics is a victim of its own success.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Our two big theories of physics, general relativity and quantum mechanics, are so good now that together they cover 99.9% of all the observations we can make.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You have to either go to extremely insane particle accelerations or the early universe or things that are really hard to measure in order to get any deviation from either of these two theories.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
to the point where you can actually figure out how to combine them together.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But I have faith that we've been doing this for centuries.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We've made progress before.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
What often happens is that when the physicists need some theory of mathematics, there's often some precursor that the mathematicians worked out earlier.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
When Einstein started realizing that space was curved, he went to some mathematician and asked, is there some theory of curved space that mathematicians already came up with that could be useful?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He said, oh yeah, I think Riemann came up with something
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Wiemann had developed Riemannian geometry, which is precisely a theory of spaces that are curved in various general ways, which turned out to be almost exactly what was needed for Einstein's theory.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is going back to Wiemann's unreasonable effectiveness on mathematics, I think.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The theories that work well to explain the universe tend to also involve the same mathematical objects that work well to solve mathematical problems.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Ultimately, they're just both ways of organizing data in useful ways.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, that was a leading candidate for many decades.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think it's slowly falling out of fashion because it's not matching experiment.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
what that reality really is like this is why analogies are so important you know i mean so yeah the round earth is not intuitive because we're stuck on it um but you know but you you know but round objects in general we have pretty good intuition a little bit uh and we have interest about light works and so forth and like it's it's actually a good exercise to actually work out how eclipses and phases of the sun and the moon and so forth can be really easily explained by by by round earth and round moon you know um
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
models.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you can just take, you know, a basketball and a golf ball and a light source and actually do these things yourself.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the intuition is there, but you have to transfer it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So modern science is maybe, again, a victim of its own success is that
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
in order to be more accurate, it has to move further and further away from your initial intuition.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so, for someone who hasn't gone through the whole process of science education, it looks more and more suspicious because of that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, you know, we need more grounding.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, I think, I mean, you know, there are scientists who do excellent outreach, but there's lots of science things that you can do at home.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's lots of YouTube videos.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I did a YouTube video recently with Grant Sanderson.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We talked about this earlier.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
how the ancient Greeks were able to measure things like the distance to the Moon, the distance to the Earth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Using techniques that you could also replicate yourself, it doesn't all have to be like fancy space telescopes and very intimidating mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Change of perspective is really important.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You say travel broadens the mind.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is intellectual travel.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Put yourself in the mind of the ancient Greeks or some other person, some other time period.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Make hypotheses, spherical cows, whatever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Speculate.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this is what mathematicians do and some artists do, actually.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you propose axioms, then the mathematics does follow those axioms to their conclusions, and sometimes you can get quite a long way from initial hypotheses.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I have worked on some equations, there's something called the wave maps equation, or the sigma field model, which is not quite the equation of space-time gravity itself, but of certain fields that might exist on top of space-time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, Einstein's equations of relativity just describe space and time itself.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But then there's other fields that live on top of that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's the electromagnetic field, there's things called Young-Mills fields, and there's this whole hierarchy of different equations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
of which Einstein is considered one of the most nonlinear and difficult.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But relatively low on the hierarchy was this thing called the wave maps equation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's a wave which at any given point is fixed to be like on a sphere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I can think of a bunch of arrows in space and time, and yeah, so it's pointing in different directions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But they propagate like waves.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you wiggle an arrow, it will propagate and make all the arrows move kind of like sheaves of wheat in the wheat field.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I was interested in the global regularity problem again for this question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Is it possible for all the energy here to collect at a point?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the equation I considered was actually what's called a critical equation, where it's actually the behavior at all scales is roughly the same.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I was able barely to show that...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
that you couldn't actually force a scenario where all the energy concentrated at one point.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The energy had to disperse a little bit, and the moment it dispersed a little bit, it would stay regular.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This was back in 2000.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That was part of why I got interested in Navier-Stokes afterwards.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I developed some techniques to solve that problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Part of it is that this problem is really non-linear because of the curvature of the sphere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There was a certain nonlinear effect, which was a non-perturbative effect.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
When you looked at it normally, it looked larger than the linear effects of the wave equation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so it was hard to keep things under control, even when the energy was small.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But I developed what's called a gauge transformation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the equation is kind of like an evolution of heaps of wheat, and they're all bending back and forth, and so there's a lot of motion.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if you imagine stabilizing the flow by attaching little cameras at different points in space, which are trying to move in a way that captures most of the motion, and under this sort of stabilized flow, the flow becomes a lot more linear.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I discovered a way to transform the equation to reduce the amount of nonlinear effects, and then I was able to solve the equation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I found this transformation while visiting my aunt in Australia.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I was trying to understand the dynamics of all these fields, and I couldn't do it with a pen and paper.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I had none of the facility of computers to do any computer simulations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I ended up closing my eyes, being on the floor, and just imagining myself to actually be
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
this vector field, and rolling around to try to see how to change coordinates in such a way that somehow things in all directions would behave in a reasonably linear fashion.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And, yeah, my aunt walked in on me while I was doing that, and she was asking, what am I doing doing this?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's complicated, is the answer.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, yeah, and she said, okay, fine, you're a young man, I don't ask questions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A lot of pen and paper.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
One thing you pick up as a mathematician is sort of, I call it cheating strategically.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the beauty of mathematics is that you get to change the rule, change the problem, change the rules as you wish.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You don't get to do this for any other field.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like, you know, if you're an engineer and someone says, build a bridge over this river, you can't say, I want to build this bridge over here instead, or I want to build it out of paper instead of steel.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But in mathematician, you can do whatever you want.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's like trying to solve a computer game where you can, there's unlimited cheat codes available.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so, you know, you can set this, there's a dimension that's large, I'll set it to one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'd solve the one dimension problem first, or whatever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a main term and an error term.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'm going to make a spherical cow assumption.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'll assume the error term is zero.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so the way you should solve these problems is not in sort of this Iron Man mode where you make things maximally difficult.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But actually, the way you should approach any reasonable math problem is that if there are 10 things that are making your life difficult,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Find a version of the problem that turns off nine of the difficulties but only keeps one of them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And solve that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you install nine cheats.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, if you solve ten cheats, then the game is trivial.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if you solve nine cheats, you solve one problem, that teaches you how to deal with that particular difficulty.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then you turn that one off and you turn something else on, and then you solve that one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And after you know how to solve the 10 problems, 10 difficulties separately, then you have to start merging them a few at a time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
As a kid, I watched a lot of these Hong Kong action movies.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's from a culture.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And one thing is that every time there's a fight scene, you know, so maybe the hero gets swarmed by a hundred bad guy goons or whatever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it would always be choreographed so that he'd always be only fighting one person at a time, and then he would defeat that person and move on.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And because of that, he could defeat all of them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But whereas if they had fought a bit more intelligently and just swarmed the guy at once, it would make for much worse cinema, but they would win.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Are you usually pen and paper?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Are you working with computer and latex?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'm mostly pen and paper actually.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in my office I have four giant blackboards and sometimes I just have to write everything I know about the problem on the four blackboards and then sit on my couch and just sort of see the whole thing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Is it all symbols like notation or is there some drawings?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a lot of drawing and a lot of bespoke doodles that only make sense to me.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the beautiful blackboards you raise, it's a very organic thing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'm beginning to use more and more computers, partly because AI makes it much easier to do simple coding things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If I wanted to plot a function before, which is moderately complicated, some iteration or something, I'd have to...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I remember how to set up a Python program and how does a full loop work and debug it and it would take two hours and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And now I can do it in 10, 15 minutes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's much, yeah, I'm using more and more computers to do simple explorations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So Lean is a computer language, much like sort of standard languages like Python and C and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Except that in most languages, the focus is on using executable code.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Lines of code do things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They flip bits, or they make a robot move, or they deliver you text on the internet or something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So Lean is a language that can also do that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It can also be run as a standard traditional language, but it can also produce certificates.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So a software language like Python might do a computation and give you that the answer is seven.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Does the sum of three plus four is equal to seven?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But Lean can produce not just the answer, but a proof that how it got the answer of 7 as 3 plus 4 and all the steps involved.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it creates these more complicated objects, not just statements, but statements with proofs attached to them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And every line of code is just a way of piecing together previous statements to create new ones.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the idea is not new.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
These things are called proof assistants.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so they provide languages for which you can create quite complicated, intricate mathematical proofs.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And they produce these certificates that give a 100% guarantee that your arguments are correct if you trust the compiler of Lean.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But they made the compiler really small.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there are several different compilers available for the same language.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So lean, a lot of mathematicians were involved in the design of lean.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's designed so that
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
individual lines of code resemble individual lines of a mathematical argument.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You might want to introduce a variable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You might want to prove a contradiction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There are various standard things that you can do, and it's written so that ideally it should be like a one-to-one correspondence.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In practice, it isn't because Lean is like explaining a proof to an extremely pedantic colleague
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
who will point out, okay, did you really mean this?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
What happens if this is zero?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
How do you justify this?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So Lean has a lot of automation in it to try to be less annoying.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So for example, every mathematical object has to come with a type.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If I talk about x, is x a real number or a natural number or a function or something?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you write things informally,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's often in some context.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You say, you know, clearly X is equal to, let X be the sum of Y and Z, and Y and Z were already real numbers, so X should also be a real number.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So Lean can do a lot of that, but every so often it says, wait a minute, can you tell me more about what this object is, what type of object it is?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You have to think more at a philosophical level, not just sort of computations that you're doing, but sort of what each object actually is in some sense.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's using much more traditional, good old-fashioned AI.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can represent all these things as trees, and there's always algorithm to match one tree to another tree.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's actually doable to figure out if something is...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
a real number or a natural number.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, every object sort of comes with a history of where it came from and you can kind of trace it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Oh, I see.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so it's designed for reliability.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, modern AIs are not used in, it's a disjoint technology.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
People are beginning to use AIs on top of Lean.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, when a mathematician tries to program a proof in Lean, often there's a step, okay, now I want to use the fundamental calculus, say, okay, to do the next step.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, the Lean developers have built this massive project called MethLib.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
a collection of tens of thousands of useful facts about mathematical objects.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And somewhere in there is the fundamental theorem of calculus, but you need to find it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So a lot of the bottleneck now is actually lemma search.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a tool that you know is in there somewhere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
and you need to find it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so you can, there are various search engines specialized for Mathlib that you can do.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But there's now these large language models that you can say, I need the fundamental calculus at this point.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I've said, okay, for example, when I code, I have GitHub Copilot installed as a plugin to my IDE.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it scans my text and it sees what I need.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It says, you know, I might even type, okay, now I need to use the fundamental calculus, okay.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it might suggest, OK, try this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And maybe 25% of the time, it works exactly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then another 10%, 15% of the time, it doesn't quite work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it's close enough that I can say, oh, yeah, if I just change it here and here, it will work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then half the time, it gives me complete rubbish.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But people are beginning to use AIs a little bit on top, mostly on the level of basically fancy autocomplete, that you can type half of one line of a proof and it will tell you.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So right now I estimate that the effort, time and effort taken to formalize a proof is about 10 times the amount taken to write it out.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's doable, but you don't, it's annoying.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But doesn't it like kill the whole vibe of being a mathematician?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I mean, having a pedantic co-worker.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If that was the only aspect of it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But, um,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, there's some cases where it's actually more pleasant to do things formally.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there was a theorem I formalized, and there was a certain constant 12 that came out in the final statement.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so this 12 had to be carried all through the proof, and everything had to be checked.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
All these other numbers had to be consistent with this final number 12.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we wrote a paper through this theorem with this number 12.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then a few weeks later, someone said, oh, we can actually improve this 12 to an 11 by reworking some of these steps.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And when this happens with pen and paper, every time you change a parameter, you have to check line by line that every single line of your proof still works.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there can be subtle things that you didn't quite realize.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some properties on number 12 that you didn't even realize that you were taking advantage of.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So a proof can break down at a subtle place.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we had formalized the proof with this constant 12.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then when this new paper came out, we said, oh, so that took like three weeks to formalize, and like 20 people to formalize this original proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I said, oh, but now let's update the 12 to 11.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And what you can do with Lean is that you just, in your headline theorem, you change it 12 to 11, you run the compiler, and like of the thousands of lines of code you have,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
90% of them still work, and there's a couple that are light and red.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Now I can't justify these steps, but it immediately isolates which steps you need to change.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But you can skip over everything which works just fine.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And if you program things correctly, with good programming practices, most of your lines will not be red.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there'll just be a few places where you, I mean, if you don't hard code your constants, but you sort of
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You use smart tactics and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can localize the things you need to change to a very small period of time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's like within a day or two, we had updated our proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Of course, this is a very quick process.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You make a change, there are 10 things now that don't work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For each one, you make a change, and now there's five more things that don't work, but the process converges much more smoothly than with pen and paper,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so the proofs are longer, but each individual piece is easier to read.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if you take a math paper and you jump to page 27 and you look at paragraph 6 and you have a line of text of math, I often can't read it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
immediately because it assumes various definitions, which I have to go back and maybe on 10 pages earlier, this was defined.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the proof is scattered all over the place.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you basically are forced to read fairly sequentially.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's not like, say, a novel where in theory, you could open up a novel halfway through and start reading.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a lot of context
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But with a proof in Lean, if you put your cursor on a line of code, every single object there, you can hover over it and it will say what it is, where it came from, where it was justified.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can trace things back much easier than flipping through a math paper.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So one thing that Lean really enables is actually collaborating on proofs at a really atomic scale that you really couldn't do in the past.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So traditionally with pen and paper, when you want to collaborate with another mathematician,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Either you do it at a blackboard where you can really interact, but if you're doing it sort of by email or something, basically, yeah, you have to segment it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I'm going to finish section three, you do section four, but you can't really sort of work on the same thing collaboratively at the same time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But with Lean, you can be trying to formalize some portion of the proof and say, I got stuck at line 67 here.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I need to prove this thing, but it doesn't quite work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Here's the three lines of code I'm having trouble with.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But because all the context is there, someone else can say, oh, okay, I recognize what you need to do.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You need to apply this trick or this tool.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you can do extremely atomic level conversations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So because of Lean, I can collaborate effectively
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
you know, with dozens of people across the world, most of whom I don't, have never met in person.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I may not know actually even whether they're, how reliable they are in their, in the proofs they give me.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But Lean gives me a certificate of trust.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I can do, I can do trustless mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's so many interesting questions there.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's always a brainstorming process first.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Math research projects, by their nature, when you start, you don't really know how to do the problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's not like an engineering project where somehow the theory has been established for decades and its implementation is the main difficulty.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You have to figure out even what is the right path.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So this is what I said about cheating first.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's like, to go back to the bridge building analogy, first assume you have an infinite budget and unlimited amounts of workforce and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, I mean, in your undergraduate education, you learn about the really hard impossible problems, like the Riemann hypothesis, the twin-primes conjecture.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Now can you build this bridge?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Now have infinite budget but only finite workforce.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Now can you do that and so forth?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, I mean, of course, no engineer can actually do this because they have fixed requirements.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yes, there's this sort of jam sessions always at the beginning where you try all kinds of crazy things and you make all these assumptions that are unrealistic, but you plan to fix later.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you try to see if there's even some
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
skeleton of an approach that might work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then hopefully that breaks up the problem into smaller subproblems, which you don't know how to do, but then you focus on the sub ones.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And sometimes different collaborators are better at working on certain things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So one of my theorems I'm known for is a theorem, Ben Green, which is now called the Green-Tau theorem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a statement that the primes contain arithmetic progressions of any length.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it was a modification of this theorem as I'm already.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the way we collaborated was that Ben had already proven a similar result for progressions of length 3.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He showed that sets like the primes contain lots and lots of progressions of length 3, and even subsets of the primes, certain subsets do.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But his techniques only worked for length 3 progressions, they didn't work for longer progressions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But I had these techniques coming from ergodic theory, which is something that I had been playing with and I knew better than Ben at the time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so if I could justify certain randomness properties of some set relating to the primes, there's a certain technical condition which if I could
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can make problems arbitrarily difficult.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
habit if ben could supply me that this fact i could give i could conclude the theorem but i what i asked was a really difficult question in number theory which um he said there's no way we can prove this can so he said can you prove your part of the theorem using a weaker hypothesis that i have a chance to prove it and he proposed something which he could prove but it was too weak for me uh i can't use this um so there's this there's this conversation going back and forth um
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's not really a problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Different cheats too.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In fact, there's even problems that we know to be unsolvable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I want to cheat more, he wants to cheat less.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But eventually we found a property which A, he could prove, and B, I could use, and then we could prove our view.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's all kinds of dynamics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Every collaboration has some story.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
No two are the same.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
What's really interesting are the problems just on the boundary between what we can do perfectly easily and what are hopeless.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so it makes everything compatible and trustable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so currently, only a few mathematical projects can be cut up in this way.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
At the current state of the art, most of the lean activity is on formalizing proofs that have already been proven by humans.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Math paper basically is...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
a blueprint, in a sense.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It is taking a difficult statement, like a big theorem, and breaking it up into maybe a hundred little numbers, but often not all written with enough detail that each one can be sort of directly formalized.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A blueprint is like a really pedantically written version of a paper where every step is explained to as much detail as possible, and you're trying to make each step kind of self-contained, depending on only a very specific number of previous statements that have been proven.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But what are problems where existing techniques can do like 90% of the job and then you just need that remaining 10%?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
so that each node of this blueprint graph that gets generated can be tackled independently of the others.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you don't even need to know how the whole thing works.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's like a modern supply chain.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you want to create an iPhone or some other complicated object, no one person can build a single object.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But you can have specialists who just, if they're given some widgets from some other company, they can combine them together to form a slightly bigger widget.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yes, yes, yes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I told you before about the split between theoretical and experimental mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And right now, most mathematics is theoretical and only a tiny bit is experimental.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think the platform that Lean and other software tools, so GitHub and things like that, allow experimental mathematics to scale up to a much greater degree than we can do now.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So right now, if you want to...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
do any mathematical exploration of some mathematical pattern or something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You need some code to write out the pattern.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, sometimes there are some computer algebra packages that help, but often it's just one mathematician coding lots and lots of Python or whatever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And because coding is such an error-prone activity, it's not practical to allow other people to collaborate with you on writing modules for your code, because if one of the modules has a bug in it, the whole thing is unreliable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you get these bespoke...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think as a PhD student, the Kakeya problem certainly caught my eye, and it just got solved, actually.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
spaghetti code written by non-professional programmers, mathematicians.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And they're clunky and slow.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so because of that, it's hard to really mass-produce experimental results.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, but I think with Lean, I mean, so I'm already starting some projects where we are not just experimenting with data, but experimenting with proofs.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I have this project called the Equational Theories Project.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Basically, we generated about 22 million little problems in abstract algebra.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Maybe I should back up and tell you what the project is.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, so abstract algebra studies operations like multiplication and addition and their abstract properties.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, so multiplication, for example, is commutative.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
X times Y is always Y times X, at least for numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it's also associative.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
X times Y times Z is the same as X times Y times Z. So these operations obey some laws and they don't obey others.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a problem I've worked on a lot in my early research.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For example, X times X is not always equal to X. So that law is not always true.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So given any operation, it obeys some laws and not others.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so we generated about 4,000 of these possible laws of algebra that certain operations can satisfy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And our question is, which laws imply which other ones?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So for example, does commutativity imply associativity?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the answer is no, because it turns out you can describe an operation which obeys the commutative law but doesn't obey the associative law.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So by producing an example, you can show that commutativity does not imply associativity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Historically, it came from a little puzzle by the Japanese mathematician Soichi Kakeya in 1918 or so.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But some other laws do imply other laws by substitution and so forth, and you can write down some algebraic proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we look at all the pairs between these 4,000 laws, and there's about 22 million of these pairs.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And for each pair, we ask, does this law imply this law?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If so, give a proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If not, give a counterexample.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So 22 million problems, each one of which you could give to an undergraduate algebra student, and they had a decent chance of solving the problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Although there are a few, at least 22 million, there are like 100 or so that are really quite hard, but a lot are easy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The project was just to work out, to determine the entire graph, like which ones imply which other ones.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so it would not have been feasible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, the state of the art in the literature was like 15 equations and sort of how they imply.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's sort of at the limit of what a human pen and paper can do.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you need to scale it up
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you need to crowdsource, but you also need to trust all the, I mean, no one person can check 22 million of these proofs.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You need to be computerized.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so it only became possible with Lean.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We were hoping to use a lot of AI as well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the project is almost complete.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So these 22 million, all but two have been settled.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The puzzle is that you have a needle on the plane.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Wow.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, actually, and of those two, we have a pen and paper proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
of the two, and we're formalizing it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In fact, this morning I was working on finishing it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we're almost done on this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's incredible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, fantastic.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
How many people were able to get it?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
About 50, which in mathematics is considered a huge number.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a huge number.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's crazy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we're going to have a paper with 50 authors and a big appendix of who contributed what.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The beauty of these lean projects is that automatically you get all this data.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Think of it like driving on a road.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So like everything's uploaded to this GitHub and GitHub tracks who contributed what.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you could generate statistics at any later point in time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You could say, oh, this person contributed this many lines of code or whatever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, these are very crude metrics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I would definitely not want this to become part of your 10-year review or something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But I mean, I think already in enterprise computing, people do use some of these metrics frequently.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
as part of the assessment of performance of an employee.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Again, this is a direction which is a bit scary for academics to go down.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We don't like metrics so much.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You want to execute a U-turn.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You want to turn the needle around.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, I think it's interesting to study.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, I think you can do studies of whether these are better predictors.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's this problem called Goodhart's law.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If a statistic is actually used to incentivize performance, it becomes gamed, and then it is no longer a useful measure.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
but you want to do it in as little space as possible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Oh, humans, always gamed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, I know.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it's rational.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So what we've done for this project is self-report.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there are actually standard categories from the sciences of what types of contributions people give.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's this concept and validation and resources and coding and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's a standard list of 12 or so categories.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we just ask each contributor to, there's a big matrix of all the authors in all the categories, just to tick the boxes where they think that they contributed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you want to use this little area in order to turn it around.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And just give a rough idea, you know, like, oh, so you did some coding and you provided some compute, but you didn't do any of the pen and paper verification or whatever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I think that that works out.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Traditionally, mathematicians just order alphabetically by surname.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we don't have this tradition as in the sciences of lead author and second author and so forth, which we're proud of.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We make all the authors equal status, but it doesn't quite scale to this size.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So a decade ago, I was involved in these things called polymath projects.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It was the crowdsourcing mathematics, but without the lean component.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it was limited by, you needed a human moderator to actually check that all the contributions coming in were actually valid.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this was a huge bottleneck, actually.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But still, we had projects that were, you know, 10 authors or so.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But we had decided at the time...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But the needle is infinitely maneuverable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
not to try to decide who did what, but to have a single pseudonym.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we created this fictional character called DHJ Polymath in the spirit of Bobakhee.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Bobakhee is the pseudonym for a famous group of mathematicians in the 20th century.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so the paper was authored on the pseudonym.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So none of us got the author credit.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This actually turned out to be
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Not so great for a couple of reasons.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So one is that if you actually wanted to be considered for tenure or whatever, you could not use this paper as you're submitted as one of your publications because you didn't have the formal author credit.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you can imagine just spinning it around its, as the unit needle, you can spin it around its center.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But the other thing that we've recognized much later is that when people referred to these projects, they naturally referred to the most famous person who was involved in the project.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So this was Tim Gower's project, this was Terence Tao's project, and not mention the other 19 or whatever people that were involved.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we're trying something different this time around where everyone's an author, but we will have an appendix with this matrix, and we'll see how that works.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I think that gives you a disc of area, I think pi over four.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, the difficulty increases exponentially with the number of steps involved in the proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or you can do a three-point U-turn, which is what we teach people in the driving schools to do.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a combinatorial explosion.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the thing with large language models is that they make mistakes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so if a proof has got 20 steps and your large language model has a 10% failure rate at each step of going in the wrong direction, it's just extremely unlikely to actually reach the end.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Oh, yeah, it's extremely hard, actually.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Natural language, you know, it's very fault tolerant.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like, you can make a few minor grammatical errors and a speaker in the second language can get some idea of what you're saying.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But formal language, if you get one little thing wrong, the whole thing is nonsense.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Even formal to formal is very hard.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There are different incompatible prefaces and languages.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that actually takes area pi over eight.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's Lean, but also Koch and Isabel and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Even converting from a formal language to a formal language is an unsolved problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's a little bit more efficient than a rotation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
we talked earlier about things that are amazing over time become kind of normalized.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So now somehow it's, of course, geometry is a silver bullet problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's true.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's true.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it's still beautiful.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a great work that shows what's possible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so for a while, people thought that was the most efficient way to turn things around.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it's, it, it,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The approach doesn't scale currently.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Three days of Google's server time to one high school math formula, this is not a scalable prospect, especially with the exponential increase as the complexity increases.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We should mention that they got a silver medal performance
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The equivalent of.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So first of all, they took way more time than was allotted.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And they had this assistance where the humans helped by formalizing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But also they're giving us those full marks for the solution, which I guess is formally verified.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I guess that's fair.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But Bezekovic showed that, in fact, you could actually turn the needle around using as little area as you wanted.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there will be a proposal at some point to actually have an AI Math Olympiad, where at the same time as the human contestants get the actual Olympiad problems, AIs will also be given the same problems, the same time period, and the outputs will have to be graded by the same judges, which means that it will have to be written in natural language rather than formal language.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Oh, I hope that happens.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I hope this IMO happens.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I hope next one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It won't happen this IMO.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The performance is not good enough in the time period.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But there are smaller competitions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There are competitions where the answer is a number rather than a long-form proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And AI is actually a lot better at problems where there's a specific numerical answer.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Because it's easy to reinforce learning on it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, you got the right answer, you got the wrong answer.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a very clear signal.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So 0.001, there was some really fancy multi back and forth U-turn thing that you could do.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But a long-form proof either has to be formal, and then the lean can give it thumbs up, thumbs down.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
or it's informal, but then you need a human to grade it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And if you're trying to do billions of reinforcement learning runs, you can't hire enough humans to grade those.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it's already hard enough for the last language learners to do reinforcement learning on just the regular text that people get.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But now if you hire people and not just give thumbs up, thumbs down, but actually check the output mathematically, yeah, that's too expensive.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's a good question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think the nature of what mathematicians do over time has changed a lot.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, so a thousand years ago, mathematicians had to compute the date of Easter, and those really complicated calculations, you know, but it's all automated, been automated for centuries.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
that you could turn a needle around.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We don't need that anymore.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They used to do spherical trigonometry to navigate how to get from the old world to the new.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Very complicated calculations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Again, we'd been automated.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Even a lot of undergraduate mathematics, even before AI, like Wolfram Alpha, for example, it's not a language model, but it can solve a lot of undergraduate level math tasks.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And in so doing, it would pass through every intermediate direction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So on the computational side, verifying routine things like having a problem and say, here's a problem in partial differential equations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Could you solve it using any of the 20 standard techniques?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the AI will say, yes, I've tried all 20.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I hear that 100 different permutations and here's my results.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that type of thing, I think it will work very well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Type of scaling to once you solve one problem to make the AI attack 100 adjacent problems.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Is this in the two-dimensional plane?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The things that humans do still, so where the AI really struggles right now is knowing when it's made a wrong turn.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
that it can say, oh, I'm going to solve this problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'm going to split up this one into these two cases.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'm going to try this technique.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And sometimes if you're lucky, it's a simple problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is in the two-dimensional plane.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's the right technique and you solve the problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And sometimes it will get, it will have a problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It would propose an approach which is just complete nonsense.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we understand everything in two dimensions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
but it looks like a proof.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is one annoying thing about LLM-generated mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We've had human-generated mathematics that's very low quality, like submissions of people who don't have the formal training and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if a human proof is bad, you can tell it's bad pretty quickly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the next question is what happens in three dimensions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It makes really basic mistakes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But the AI-generated proofs, they can look
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
superficially flawless.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it's partly because that's what the reinforcement learning has trained them to do, to produce text that looks like what is correct, which for many applications is good enough.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the errors are often really subtle, and then when you spot them, they're really stupid.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like no human would have actually made that mistake.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So suppose the Hubble Space Telescope is a tube in space, and you want to observe every single star in the universe.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so the sense of smell.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is one thing that humans have.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's a metaphorical mathematical smell that it's not clear how to get the AI to duplicate that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Eventually, I mean, so the way...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
AlphaZero and so forth, they make progress on Go and chess and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In some sense, they have developed a sense of smell for Go and chess positions, that this position is good for white, it's good for black.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They can't enunciate why, but just having that sense of smell lets them strategize.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you want to rotate the telescope to reach every single direction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if AIs gain that ability to sort of, a sense of viability of certain proof strategies, so you can say,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'm going to try to break up this problem into two small subtasks, and they can say, oh, this looks good.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The two tasks look like they're simpler tasks than your main task, and they've still got a good chance of being true.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So this is good to try.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or, no, you've made the problem worse because each of the two subproblems is actually harder than your original problem, which is actually what normally happens if you try a random thing to try.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Normally, it's very easy to transform a problem into an even harder problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Very rarely do you transform it into a simpler problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so if they can pick up a sense of smell, then they could maybe start competing with human-level mathematicians.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And his unrealistic part, suppose that space is at a premium, which it totally is not.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You want to occupy as little volume as possible in order to rotate your needle around in order to see every single star in the sky.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think all of the above.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A lot of it is we don't know how to use these tools because it's a paradigm that it's not...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We have not had, in the past, systems that are competent enough to understand complex instructions, that can work at massive scale, but are also unreliable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a bit unreliable in subtle ways, whilst providing sufficiently good output.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's an interesting combination.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You have graduate students that you work with who are kind of like this, but not at scale.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we had previous software tools that can work at scale, but very narrow.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we have to figure out how to use... I mean, so Tim Gower, like you mentioned, he actually foresaw, like in 2000, he was envisioning what mathematics would look like in actually two and a half decades.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And he wrote in his article, like a hypothetical conversation between
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
a mathematical assistant of the future and himself, you know, he's trying to solve a problem and they would have a conversation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
How small a volume do you need to do that?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Sometimes the human would propose an idea and the AI would evaluate it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And sometimes the AI would propose an idea and sometimes a competition was required and AI would just go and say, okay, I've checked the 100 cases needed here.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or the first, you said this is true for all N, I've checked N up to 100 and it looks good so far.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or hang on, there's a problem at N equals 46.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so just a free-form conversation where you don't know in advance where things are going to go, but just based on, I think ideas are good to propose on both sides, calculations are good to propose on both sides.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so you can modify it based on Copic's construction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I've had conversations with AI where I say, okay, we're going to collaborate to solve this math problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it's a problem that I already know the solution to.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I try to prompt it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, so here's the problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I suggest using this tool.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it'll find this lovely argument using a completely different tool, which eventually goes into the weeds and say, no, no, no, try using this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it might start using this and then it'll go back to the tool that it wanted to do before.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you have to keep railroading it onto the path you want.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I could eventually force it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so if your telescope has zero thickness, then you can use as little volume as you need.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
to give the proof I wanted.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it was like herding cats.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the amount of personal effort I had to take to not just sort of prompt it, but also check its output, because a lot of what it looked like was going to work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I know there's a problem on 917.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And basically arguing with it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It was more exhausting than doing it unassisted.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But that's the current state of the art.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I wonder if there's...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I believe so.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in formalization, I mentioned before that it takes 10 times longer to formalize a proof than to write it by hand.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's a simple modification of the two-dimensional construction.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
With these modern AI tools, and also just better tooling, the lean developers are doing a great job adding more and more features and making it user-friendly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's going from 9 to 8 to 7, okay, no big deal.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But one day it will drop below 1.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that's a phase shift.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Because suddenly it makes sense
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
when you write a paper, to write it in Lean first, or through a conversation with AI, which is generally on the fly with you.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But the question is that if your telescope is not zero thickness, but just very, very thin, some thickness delta, what is the minimum volume needed to be able to see every single direction as a function of delta?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it becomes natural for journals to accept, maybe they'll offer expedite refereeing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If a paper has already been formalized in Lean, they'll just ask the referee to comment on the significance of the results and how it connects to literature, and not worry so much about the correctness, because that's been certified.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Papers are getting longer and longer in mathematics, and it gets harder and harder to get good refereeing for the really long ones, unless they're really important.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It is actually an issue which, and the formalization is coming in at just the right time for this to be.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's a virtuous cycle, okay.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, one phase shift of this type that happened in the past was the adoption of LaTeX.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So LaTeX is this typesetting language that all mathematicians use now.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in the past, people used all kinds of word processors and typewriters and whatever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But at some point, LaTeX became easier to use than all other competitors.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And people would switch within a few years.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It was just a dramatic phase shift.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, it depends on the level of collaboration.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
No, like it deserves to be, to get the Fields Medal.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like, so half and half.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Already, like, I can imagine a Fields Medal winning paper having some AI assistance in writing it, you know, just, you know, like the old complete alone is already, I use it, like it speeds up my own writing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like, you know, you can have a theorem, and you have a proof, and the proof has three cases.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I write down the proof of the first case, and the autocomplete just suggests that now here's how the proof of the second case could work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it was exactly correct.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That was great.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Saved me like five, ten minutes of typing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So as delta gets smaller, as your needle gets thinner, the volume should go down, but how fast does it go down?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But in that case, the AI system doesn't get the field's metal.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
What do you think?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I gave a prediction in print about 2026, which is now next year.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There will be
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
math collaborations with AI.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So not Fields Medal winning, but actual research level math papers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like published ideas that are in part generated by AI.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Maybe not the ideas, but at least some of the computations, the verifications.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Has that already happened?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Has it already happened?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, there are problems that were solved
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
by a complicated process, conversing with AI to propose things, and the human goes and tries it, and the contract doesn't work, but it might propose a different idea.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's hard to disentangle exactly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There are certainly math results which could only have been accomplished because there was a human mathematician and an AI involved.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the conjecture was that it goes down very, very slowly, like logarithmicly, roughly speaking.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it's hard to sort of disentangle credit.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, these tools, they do not replicate all the skills needed to do mathematics, but they can replicate sort of some non-trivial percentage of them, 30, 40%.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So they can fill in gaps.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So coding is a good example.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's annoying for me to code in Python.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'm not a professional programmer.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But with AI, the friction cost of doing it is much reduced.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it fills in that gap for me.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
AI is getting quite good at literature review.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, there's still a problem with hallucinating references that don't exist.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But this, I think, is a civil world problem if you train in the right way and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
and verify using the internet.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You should, in a few years, get to the point where you have a lemma that you need and say, has anyone proven this lemma before?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it will do basically a fancy web search AI-assisted and say, yeah, there are these six papers where something similar has happened.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, you can ask it right now and it will give you six papers of which maybe one is legitimate and relevant.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that was proved after a lot of work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
One exists but is not relevant and four are hallucinated.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It has a non-zero success rate right now, but there's so much garbage, the signal-to-noise ratio is so poor that it's...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's most helpful when you already somewhat know the literature, and you just need to be prompted to be reminded of a paper that was already subconsciously in your memory.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, that's...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So this seems like a puzzle.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, that it can sometimes do, but when it doesn't, it's buried in a list of options to which the other- They're bad.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Why is it interesting?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, yeah, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a big hump to overcome right now.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it's like self-driving cars.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The safety margin has to be really high for it to be feasible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So yeah, so there's a last mile problem with a lot of AI applications that they can develop tools that work 20%, 80% of the time, but it's still not good enough.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And in fact, even worse than good in some ways.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right, right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, this decade, I can see it like making a conjecture between two unrelated, two things that people thought was unrelated.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Oh, interesting.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, and actually has a real chance of being correct and meaningful.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For example, in wave propagation, you splash some water around, you create water waves and they travel in various directions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
No, that would be truly amazing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Current models struggle a lot.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, so a version of this is, I mean, the physicists have a dream of getting the AIs to discover new laws of physics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The dream is you just feed it all this data, and here is a new patent that we didn't see before.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But the current state of the art even struggles to discover old laws of physics from the data.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or if it does, there's a big concern of contamination, that it did it only because somewhere in its training data it already somehow knew Boyle's law or whatever law you're trying to reconstruct.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Part of it is that we don't have the right type of training data for this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For the laws of physics, we don't have a million different universes with a million different laws of nature.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A lot of what we're missing in math is actually the negative space.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we have published things of things that people have been able to prove and conjectures that ended up being verified or maybe counterexamples produced.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But we don't have data on things that were proposed and they're kind of a good thing to try.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But then people quickly realized that it was the wrong conjecture and then they said, oh, but we should actually change
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But waves exhibit both particle and wave type behavior.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
our claim to modify it in this way to actually make it more plausible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a trial and error process, which is a real integral part of human mathematical discovery, which we don't record because it's embarrassing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We make mistakes and we only like to publish our wins.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the AI has no access to this data to train on.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I sometimes joke that basically AI has to go through a grad school
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And actually, you know, go to grad courses, do the assignments, go to office hours, make mistakes, get advice on how to correct the mistakes, and learn from that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you can have what's called a wave packet, which is like a very localized wave that is localized in space and moving a certain direction in time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
All right, so it's a question about curved spaces.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Earth is a good example.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So let's say Earth, you can think of it as a 2D surface.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And just moving around the Earth, there could maybe be a torus with a hole in it, or it could have many holes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there are many different topologies a priori that a surface could have, even if you assume that it's bounded and smooth and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we have figured out how to classify surfaces.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
As a first approximation, everything is determined by something called the genus, how many holes it has.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So a sphere has genus zero, a donut has genus one, and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so if you plot it in both space and time, it occupies a region which looks like a tube.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And one way you can tell these surfaces apart, probably the sphere has, which is called simply connected.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you take any closed loop on the sphere, like a big closed loop of rope, you can contract it
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
to a point while staying on the surface.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the sphere has this property, but a torus doesn't.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you're on a torus and you take a rope that goes around, say, the outer diameter of a torus, there's no way, it can't get through the hole.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's no way to contract it to a point.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it turns out that the sphere is the only surface with this property of contractibility, up to like continuous deformations of the sphere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So things that I want to go topologically equivalent to the sphere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So Poincare asked the same question in higher dimensions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it becomes hard to visualize because,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A surface you can think of as embedded in three dimensions, but a curved free space, we don't have good intuition of 4D space to live in.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then there are also 3D spaces that can't even fit into four dimensions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so what can happen is that you can have a wave which initially is very dispersed, but it all focuses at a single point later in time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You need five or six or higher.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But anyway, mathematically, you can still pose this question that if you have a bounded three-dimensional space now, which also has this simply connected property that every loop can be contracted, can you turn it into a three-dimensional version of a sphere?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so this is the Poincaré conjecture.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Weirdly, in higher dimensions, 4 and 5, it was actually easier.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it was solved first in higher dimensions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's somehow more room to do the deformation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's easier to move things around to a sphere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But 3 was really hard.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So people tried many approaches.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's sort of commentary approaches where you chop up the surface into little
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
triangles or tetrahedra, and you just try to argue based on how the faces interact with each other.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There were algebraic approaches.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's various algebraic objects, like things called the fundamental group that you can attach to these homology and cohomology and all these very fancy tools.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They also didn't quite work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But Richard Hamilton proposed a partial differential equations approach.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the problem is that you have this object which is sort of secretly a sphere, but it's given to you in a really weird way.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So think of a ball that's been kind of crumpled up and twisted, and it's not obvious that it's a ball.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you have some sort of surface which is a deformed sphere, you could think of it as the surface of a balloon.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like you can imagine dropping a pebble into a pond and it will spread out.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You could try to inflate it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You blow it up, and naturally, as you fill it with air, the wrinkles will smooth out and it will turn into a nice round sphere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Unless, of course, it was a torus or something, in which case it would get stuck at some point.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you inflate a torus,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
there would be a point in the middle.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
When the inner ring shrinks to zero, you get a singularity and you can't blow up any further.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can't flow any further.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So he created this flow, which is now called Ricci flow.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
which is a way of taking an arbitrary surface or space and smoothing it out to make it rounder and rounder, to make it look like a sphere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But then if you time reverse that scenario, and the equations of wave motion are time reversible,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And he wanted to show that either this process would give you a sphere or it would create a singularity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I very much like how PDEs either have global regularity or finite envelope.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Basically, it's almost exactly the same thing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's all connected.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And he showed that for two-dimensional surfaces, if you started something, no singularities ever formed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You never ran into trouble, and you could flow, and it would give you a sphere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So he got a new proof of the two-dimensional result.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, these are quite sophisticated equations on par with the Einstein equations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's slightly simpler, but...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, but they were considered hard nonlinear equations to solve.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's lots of special tricks in 2D that helped.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can imagine ripples that are converging to a single point, and then a big splash occurs, maybe even a singularity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But in 3D, the problem was that this equation was actually supercritical.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's the same problem as Navier-Stokes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
As you blow up, maybe the curvature could get concentrated in finer and smaller regions, and it looked more and more nonlinear, and things just looked worse and worse.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there could be all kinds of singularities that showed up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some singularities, like there's these things called neck pinches, where the surface sort of behaves like a barbell and it pinches at a point.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Some singularities are simple enough that you can sort of see what to do next.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You just make a snip and then you can turn one surface into two and evil them separately.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But there was the prospect that some really nasty, like, knotted singularities showed up that you couldn't see how to resolve in any way, that you couldn't do any surgery to.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you need to classify all the singularities.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like, what are all the possible ways that things can go wrong?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So what Perlman did was, first of all, he made the problem, he turned the problem from a supercritical problem to a critical problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I said before about how the invention of energy, the Hamiltonian, really clarified Newtonian mechanics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So he introduced something which is now called Perlman's reduced volume and Perlman's entropy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He introduced new quantities, kind of like energy, that looked the same at every single scale and turned the problem into a critical one where the nonlinearities actually suddenly looked a lot less scary than they did before.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then he had to solve, he still had to analyze the singularities of this critical problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
and that itself was a problem similar to this wave map thing I worked on actually so on the level of difficulty of that so he managed to classify all the similarities of this problem and show how to apply surgery to each of these and through that was able to resolve the Poincare conjecture so
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's possible to do that, and geometrically what's going on is that there's always light rays.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
quite a lot of really ambitious steps and like nothing that a large language model today, for example, could.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, at best, I could imagine a model proposing this idea as one of hundreds of different things to try.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But the other 99 would be complete dead ends, but you'd only find out after months of work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He must have had some sense that this was the right track to pursue because it takes years to get them from A to B. So you've done, like you said, actually, even strictly mathematically, but
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If this wave represents light, for example, you can imagine this wave as a superposition of photons, all traveling at the speed of light.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I'm a fox, I'm not a hedgehog.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can modify the problem too.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If there's a specific thing that's blocking you, some bad case keeps showing up for which your tool doesn't work, you can just assume by fiat this bad case doesn't occur.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you do some magical thinking.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
strategically, to see if the rest of the argument goes through.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If there's multiple problems with your approach, then maybe you just give up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if this is the only problem, then everything else checks out, then it's still worth fighting.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So yeah, you have to do some forward reconnaissance sometimes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that is sometimes productive to assume like, okay, we'll figure it out eventually.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Sometimes actually it's even productive to make mistakes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So one of the, I mean, there was a project which actually we won some prizes for.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They all travel on these light rays, and they're all focusing at this one point.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We were four other people.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We were working on this PDE problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Again, actually this blow-up regularity type problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it was considered very hard.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Jean Bourguin, who was another field specialist who worked on a special case of this, but he could not solve the general case.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we worked on this problem for two months, and we thought we solved it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We had this cute argument that if everything fit, we were excited.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We were planning celebrationary to all get together and have champagne or something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we started writing it up.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And one of us, not me actually, but another co-author, said, oh, in this lemma here, we have to estimate these 13 terms that show up in this expansion.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can have a very dispersed wave focused into
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we estimated 12 of them, but in our notes, I can't find the estimation of the 13th, can you?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Can someone supply that?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I said, sure, I'll look at this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And actually, yeah, we completely omitted this term.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this term turned out to be worse than the other 12 terms put together.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In fact, we could not estimate this term.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we tried for a few more months, and all different permutations, and there was always this one term that we could not control.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so this was very frustrating.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But because we had already invested months and months of effort in this already, we stuck at this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
a very concentrated wave at one point in space and time, but then it defocuses again and it separates.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We tried increasingly desperate things and crazy things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And after two years, we found an approach that was somewhat different, but quite a bit from our initial strategy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
which actually didn't generate these problematic terms and actually solved the problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So we solved the problem after two years.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if we hadn't had that initial false dawn of nearly solving the problem, we would have given up by month two or something and worked on an easier problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If we had known it would take two years, I'm not sure we would have started the project.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Sometimes actually having the incorrect, it's like Columbus traveling in the New World, the incorrect version of the measurement of the size of the Earth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
He thought he was going to find a new trade route to India.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or at least that was how he sold it in his prospectus.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it could be that he secretly knew.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But potentially, if the conjecture had a negative solution, so what that meant is that there's a very efficient way to pack tubes pointing in different directions into a very, very narrow region of very narrow volume, then you would also be able to create waves that start out, there'll be some arrangement of waves that start out very, very dispersed,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, I think different athleticians have different levels of emotional investment in what they do.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, I think for some people it's just a job.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You know, you have a problem and if it doesn't work out, you go on the next one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so the fact that you can always move on to another problem, it reduces the emotional connection.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, there are cases, you know, so there are certain problems that are what are called mathematical diseases where we just latch on to that one problem and they spend years and years
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
thinking about nothing but that one problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And, um, you know, maybe the, the career suffers and so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But how could this big win, this will, you know, once I, once I finish this problem, I will, I will make up for all the years of, of, of lost opportunity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that's, that's, I mean, occasionally, occasionally it works, but I, um, I really don't recommend it for people without the right fortitude.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I, I've never been super invested in any one problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Um,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
One thing that helps is that we don't need to call our problems in advance.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, when we do crowd proposals, we sort of say, we will study this set of problems.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But even then, we don't promise, definitely by five years, I will supply a proof of all these things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You promise to make some progress or discover some interesting phenomena.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And maybe you don't solve the problem, but you find some related problem that you can say something new about.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that's a much more feasible task.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Again.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, I mean, the problem is like a Riemann hypothesis.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Those are so far out of reach.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You think so?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's no even viable strategy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like even if I,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
activate all the cheats that I know of in this problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's just still no way to get me to be.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think it needs a breakthrough in another area of mathematics to happen first, and for someone to recognize that it would be a useful thing to transport into this problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the natural numbers have two basic operations attached to them, addition and multiplication.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if you want to generate the natural numbers, you can do one of two things.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can just start with one and add one to itself over and over again.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
but they would concentrate not just at a single point, but there'll be a lot of concentrations in space and time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that generates you the natural numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So additively, they're very easy to generate.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
One, two, three, four, five.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or you can take the prime number, if you want to generate multiplicatively, you can take all the prime numbers, two, three, five, seven, and multiply them all together.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And together, that gives you all the natural numbers, except maybe for one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there are these two separate ways of thinking about the natural numbers from an additive point of view and a multiplicative point of view.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And separately, they're not so bad.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So any question about the natural numbers that only involves addition is relatively easy to solve.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And any question that only involves multiplication is relatively easy to solve.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But what has been frustrating is that you combine the two together.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And suddenly you get an extremely rich, I mean, we know that there are statements in number theory that are actually as undecidable.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There are certain polynomials in some number of variables.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Is there a solution in the natural numbers?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the answer depends on an undecidable statement, like whether the axioms of mathematics are consistent or not.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But even the simplest problems that combine something multiplicative, such as the primes, with something additive, such as shifting by two.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Separately, we understand both of them well, but if you ask, when you shift the prime by two, how often can you get another prime?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's been amazingly hard to relate the two.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you could create what's called a blow-up, where these waves, their amplitude becomes so great that the laws of physics that they're governed by are no longer wave equations, but something more complicated and nonlinear.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so what we've realized because of this type of research is that different patterns have different
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
levels of indestructibility.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So what makes the twin prime problem hard is that if you take all the primes in the world, you know, 3, 5, 7, 11, so forth, there are some twins in there.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
11 and 13 is a pair of twin primes, so forth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But you could easily, if you wanted to, redact the primes to get rid of these twins.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The twins, they show up, and there are infinitely many of them, but they're actually reasonably sparse.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Initially, there's quite a few, but once you go to the millions, trillions, they become rarer and rarer.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you could actually just... If someone was given access to the database of primes, you just edit out a few primes here and there, they could make the twin-prime conjecture false by just removing 0.01% of the primes or something.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Just well-chosen to do this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so you could present a...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
censored database of the primes, which passes all of the statistical tests of the primes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It obeys things like the prime number theorem and other things about the primes, but doesn't contain any twin primes anymore.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this is a real obstacle to the twin prime conjecture.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It means that any
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
proof strategy to actually find trend primes in the actual primes must fail when applied to these slightly edited primes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so it must be some very subtle, delicate feature of the primes that you can't just get from like aggregate statistical analysis.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Okay, so that's out.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
On the other hand, arithmetic progressions has turned out to be much more robust.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can take the primes, and you can eliminate 99% of the primes, actually.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In mathematical physics, we care a lot about whether certain equations in wave equations are stable or not, whether they can create these singularities.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you can take any 99% you want.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it turns out, and another thing we proved, is that you still get arithmetic progressions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Arithmetic progressions are much, you know, they're like cockroaches.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Of arbitrary length.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's crazy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it's again like, it's an infinite monkey type phenomenon.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For any fixed length of your set, you don't get arbitrary length progressions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You only get quite short progressions.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But you're saying twin prime is not an infinite monkey.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, it's a very subtle one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's still an infinite monkey phenomenon.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If the primes were really genuinely random, if the primes were generated by monkeys, then yes, in fact, the infinite monkey theorem would... Oh, but you're saying that twin prime, you can't use the same tools.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, we don't know.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We believe the primes behave like a random set.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so the reason why we care about the Trim-Half Conjecture is it's a test case for whether we can genuinely confidently say with 0% chance of error that the primes behave like a random set.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Random versions of the primes we know contain twins, at least with 100% probability, or probably tending to 100% as you go out further and further.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so the primers, we believe that they're random.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The reason why athmic progressions are indestructible is that regardless of whether your set looks random or looks structured, like periodic, in both cases, athmic progressions appear, but for different reasons.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a famous unsolved problem called the Navier-Stokes regularity problem.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this is basically all the ways in which there are many proofs of these sort of athmic progression epitheliums, and they're all proven by some sort of dichotomy where your set is either structured or random, and in both cases, you can say something, and then you put the two together.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But in twin primes, if the primes are random, then you're happy, you win.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But if the primes are structured, they can be structured in a specific way that eliminates the twins.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The Navier-Stokes equations are equations that govern the fluid flow for incompressible fluids like water.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we can't rule out that one conspiracy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the one funny thing about conspiracies is that any one conspiracy theory is really hard to disprove.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That, you know, if you believe the world is run by lizards, you say, here's some evidence that it's not run by lizards.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Well, that evidence was planted by the lizards.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You may have encountered this kind of phenomenon.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's almost no way to...
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
definitively rule out a conspiracy.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the same is true in mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
A conspiracy that is solely devoted to eliminating twin primes, you have to also infiltrate other areas of mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it could be made consistent, at least as far as we know.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But there's a weird phenomenon that you can make one conspiracy rule out other conspiracies.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The question asks, if you start with a smooth velocity field of water, can it ever concentrate so much that the velocity becomes infinite at some point?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, you know, if the world is run by lizards, it can't also be run by aliens.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right, right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So one unreasonable thing is hard to disprove, but more than one, there are tools.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, yeah, so, for example, we know there's infinitely many primes that are, no two which are, so the infinite pairs of primes which differ by at most 246, actually, is the code.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's like a bound on the... Yes, right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, like,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's twin primes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a thing called cousin primes that differ by four.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There's a thing called sexy primes that differ by six.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
What are sexy primes?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Primes that differ by six.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Got it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The name is much less exciting than the name suggests.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Got it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you can make a conspiracy rule out one of these, but once you have like 50 of them, it turns out that you can't rule out all of them at once.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It just requires too much
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
energy somehow in this conspiracy space.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's ultimately based on what's called the pigeonhole principle.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the pigeonhole principle is a statement that if you have a number of pigeons, and they all have to go into pigeonholes, and you have more pigeons than pigeonholes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
then one of the pigeon holes has to have at least two pigeons in it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there has to be two pigeons that are close together.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, for instance, if you have 100 numbers and they all range from 1 to 1,000, two of them have to be at most 10 apart.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's called a singularity.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Because you can divide up the numbers from 1 to 100 into 100 pigeon holes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Let's say we have 101 numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If we have 101 numbers, then two of them have to be a distance less than 10 apart because two of them have to belong to the same pigeon hole.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's a basic feature of a basic principle in mathematics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
We don't see that in real life.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it doesn't quite work with the primes directly because the primes get sparser and sparser as you go out, that fewer and fewer numbers are prime.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But it turns out that there's a way to assign weights to numbers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there are numbers that are kind of almost prime, but they don't have no factors at all other than themselves and one, but they have very few factors.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And it turns out that we understand almost primes a lot better than we understand primes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
If you splash around water on a bathtub, it won't explode on you.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so, for example, it was known for a long time that there were twin almost primes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This has been worked out.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So almost primes are something we can't understand.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you can actually restrict the attention to a suitable set of almost primes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And whereas the primes are very sparse
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
overall, relative to the almost primes, they actually are much less sparse.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can set up a set of almost primes where the primes have density like, say, 1%.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that gives you a shot at proving, by applying some sort of original principle, that there's pairs of primes that are just only 100 apart.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But in order to prove the twin-prime conjecture, you need to get the density of primes inside the almost primes up to a threshold of 50%.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Or have water leaving at the speed of light.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Once you get up to 50%, you will get twin primes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But unfortunately, there are barriers.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
we know that no matter what kind of good set of almost primes we pick, the density of primes can never get above 50%.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's called the parity barrier.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And I would love to find, yeah, so one of my long-term dreams is to find a way to breach that barrier.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Because it would open up not only the twin-prime conjecture, the go-back conjecture, and many other problems in number theory are currently blocked because our current techniques would require going beyond this theoretical parity barrier.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's like going past the speed of light.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But potentially, it is possible.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Oh, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, sometimes you try something and it works super well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Again, the sense of mathematical smell we talked about earlier.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You learn from experience when things are going too well.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Because there are certain difficulties that you sort of have to encounter.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I think the way a colleague might put it is that if you are on the streets of New York and you put on a blindfold and you put on a car and after some hours the blindfold is off and you're in Beijing,
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
In fact, in recent years, the consensus has drifted towards the
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, that was too easy somehow.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There was no ocean being crossed.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Even if you don't know exactly what was done, you're suspecting that something wasn't right.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, when I have nothing better to do, which is less and less time, which is I get busy with so many things these days.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But yeah, when I have free time and I'm not, and I'm too frustrated to work on my sort of real research projects, and I also don't want to do my administrative stuff or I don't want to do some errands for my family, I can play with these things for fun.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And usually you get nowhere.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You have to just say, okay, fine.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Once again, nothing happened.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I will move on.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Very occasionally, one of these problems are actually solved.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The belief that, in fact, for certain very special initial configurations of, say, water, that singularities can form.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Sometimes, as you say, you think you solved it, and then you're euphoric for maybe 15 minutes, and then you think, I should check this, because this is too easy, too good to be true, and it usually is.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Prime, I think we'll keep getting more partial results.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It doesn't need
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
at least one, this parity barrier is the biggest remaining obstacle.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
There are simpler versions of the conjecture where we are getting really close.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So I think we will, in 10 years, we will have it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
many more, much closer results.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It may not have the whole thing.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, so twin primes is somewhat close.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Riemann hypothesis, I have no, I mean, it has happened by accident, I think.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Right, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's states that are sort of viewed multiplicatively.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like for questions only involving multiplication, no addition, the primes really do behave as randomly as you could hope.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So there's a phenomenon in probability code, square root cancellation, that if you want to poll, say, America on some issue, and you ask one or two voters, and you may have sampled a bad sample, and then you get a really imprecise measurement of the
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
full average, but if you sample more and more people, the accuracy gets better and better, and the accuracy improves like the square root of the number of people you sample.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But people have not yet been able to actually establish this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if you sample a thousand people, you can get like a 2-3% margin of error.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So in the same sense, if you measure the primes in a certain applicative sense, there's a certain type of statistic you can measure, and it's called the Riemann's data function, and it fluctuates up and down.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But in some sense, as you keep averaging more and more, if you sample more and more, the fluctuations should go down as if they were random.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's a very precise way to quantify that, and the Riemann hypothesis is a very elegant way that captures this.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
As with many other ways in mathematics, we have very few tools to show that something really genuinely behaves really random.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The Clay Foundation has these seven millennium prize problems, has a million dollar prize for solving one of these problems.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this is actually not just a little bit random, but it's asking that it behaves as random as an actually random set, this square root cancellation.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And we know, because of things related to the parity problem, actually, that most of us' usual techniques cannot hope to settle this question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The proof has to come out of left field.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, but what that is, no one has any serious proposal.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's various ways, as I said, you can modify the primes a little bit and you can destroy the Riemann hypothesis.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it has to be very delicate.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You can't apply something that has huge margins of error.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It has to just barely work.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's all these pitfalls that you have to dodge very adeptly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The prime numbers is just fascinating.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
This is one of them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's a good question.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Of these seven, only one of them has been solved.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So, like, conjecturally, we have a good model of them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I mean, like, as I said, I mean, they have certain patterns, like the primes are usually odd, for instance.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But apart from these obvious patterns, they behave very randomly.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And just assuming that they behave... So, there's something called the Kramer random model of the primes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That after a certain point, primes just behave like a random set.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And there's various slight modifications to this model, but this has been a very good model.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It matches the numerics.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It tells us what to predict.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Like, I can tell you with complete certainty the Trim-Priority Context is true.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
The random model gives overwhelming odds that it's true.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
At that point, great conjecture, my parliament.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
I just can't prove it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Most of our mathematics is optimized for solving things with patterns in them.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And the primes have this anti-pattern, as do almost everything, really.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But we can't prove that.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, I guess it's not mysterious that the planets are being random because there's no reason for them to have any kind of secret pattern.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the Kakeya conjecture is not directly, directly related to the Navier-Stokes problem, but understanding it would help us understand some aspects of things like wave concentration, which would indirectly probably help us understand the Navier-Stokes problem better.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But what is mysterious is what is the mechanism that really forces the randomness to happen.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And this is just absent.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Oh, yeah.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's a problem that you can explain.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It helps with some visual aids.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So you take any natural number, like, say, 13, and you apply the following procedure to it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So if it's even, you divide it by 2.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And if it's odd, you multiply it by 3 and add 1.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So even numbers get smaller, odd numbers get bigger.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So 13 will become 40, because 13 times 3 is 39, add 1, you get 40.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's a simple process.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
For odd numbers and even numbers, they're both very easy operations.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then you put them together, it's still reasonably simple.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But then you ask what happens when you iterate it.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
You take the output that you just got and feed it back in.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So 13 becomes 40.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
40 is now even, divided by 2 is 20.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
20 is still even, divided by 2 is 10.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
5, and then 5 times 3 plus 1 is 16.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then 8, 4, 2, 1.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And then from 1 it goes 1, 4, 2, 1, 4, 2, 1.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It cycles forever.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So this sequence I just described, 13, 40, 20, 10, so both, these are also called Hailstone Sequences.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
because there's an oversimplified model of hailstorm formation, which is not actually quite correct, but it's somehow taught to high school students as a first approximation, is that a little nugget of ice gets an ice crystal.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It forms in a cloud, and it goes up and down because of the wind, and sometimes when it's cold, it acquires a bit more mass, and maybe it melts a little bit, and this process of going up and down creates this partially melted ice, which eventually becomes this hailstorm, and eventually it falls down to the earth.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the conjecture is that no matter how high you start up, like you take a number which is in the millions or billions, this process that goes up if you're odd and down if you're even, it eventually goes down to earth all the time.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Can you speak to the Navier-Stokes?
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
No matter where you start with this very simple algorithm, you end up at one.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And you might climb for a while, come down.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yeah, if you plot it, these sequences, they look like Brownian motion.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They look like the stock market.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
They just go up and down in a seemingly random pattern.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And in fact, usually that's what happens, that if you plug in a random number, you can actually prove, at least initially, that it would look like a random walk.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And that's actually a random walk with a downward drift.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
It's like if you're always gambling on a roulette at the casino with odds slightly weighted against you.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So sometimes you win, sometimes you lose.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
But over, in the long run, you lose a bit more than you win.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
And so normally your wallet will go to zero if you just keep playing over and over again.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Yes.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So the result that I proved, roughly speaking, is that statistically, like 90% of all inputs would drift down to, maybe not all the way to one, but to be much, much smaller than what you started.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
So it's like if I told you that if you go to a casino, most of the time you end up, if you keep playing for long enough, you end up with a smaller amount in your wallet than when you started.
Lex Fridman Podcast
#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
That's kind of like the result that I proved.