Terence Tao: Math 2.0（原声文稿）

OK, so I'm happy to be here. Like Barry, I also have to teach. So I was like, you're not here this morning.

And perhaps just I quote the tail end of one of the last talks is possible that what I say here will overlap with what has already been said, but maybe it has to be said again.

Anyway, this talk is also similar to when I get a caltech last week.

Some of you may also have seen a version of this talk.

So there was a brief mention in Peter's slides on what we can call math 2 .0.

So you may have noticed that math is changing quite a bit in the last few months.

And I think as a consequence, the way we do mathematics at a very fundamental level is changing and we need to upgrade math itself from sort of math 1 .0, which is what we've been doing for 3 ,000 years, to a new type of math.

But I think it's gonna be a little bit tumultuous, but it is ultimately exciting.

but we need lots of transition and cultural changes and new technologies, new infrastructure, such as this open model to proceed.

So AI has been disrupting many areas of many disciplines, you know, the creative arts, science, and another place is code, but one place where it has, well, I think intuitively it has disrupted one of what has been regarded as one of the most intellectually difficult areas of human endeavor, which is mathematics.

And what not mathematics, all the mathematics, but a specific type of mathematics, which is that of solving problems in pure mathematics, which is certainly a large chunk of what mathematicians do.

And if you go by Twitter, for example, it seems like that's all we do nowadays, but that actually it's not the case.

But it is a key activity.

It has three properties which make it particularly amenable to modern machine learning -based AI.

So one is that when you're solving a problem in computer mathematics, the task is objectively very vital.

Like if someone claims whether human or AI to have proven or solved a problem, you can check it either by pen and paper or by computer nowadays.

and the so you can verify it and the verification is objective you know I can referee the papers and we will if we're doing it properly and if the paper is not completely unreadable the we can verify that it works or it doesn't work it is objectively verifiable it is digitizable pure math you know famously is very abstract you don't need real -world clinical trials or gathering messy data with gaps in it or you don't interface with the real world say there's no robotics or there's no physical layer to actually solving a problem in pure mathematics you could do it entirely within a computer and there are lots and lots of high -quality datasets of existing mathematics we have thousands of years of textbooks and papers of really high -quality curated mathematics so we already have starting from the get -go with lots of good data sets.

If you're in, for example, cancer research or something, then there are data sets of patient data, but they're messy, they're incomplete, there are laws about privacy, they are much, and hospitals and stuff, they don't want to share all their data into one big data set for very good reasons actually.

But in mathematics, we just have this publicly variable literature which is massive.

And these three ingredients verifiability, digitizability, and large data sets, large high -quality data sets, they turn out to be exactly the three ingredients that make machine learning powerful.

If you don't have these two ingredients, machine learning is still struggles to give good, reliable answers, but when you have these three ingredients, it works really well.

And so because of this, suddenly these AI tools are really, really good at this particular task of solving problems and a few of our facts.

And so we've entered what I call the era of proof abundance.

So for thousands of years we were in the era of proof scarcity, but that solving problems was hard and therefore valuable.

We actually, you know, every time a major problem got solved people in the field would drop everything, study it, discuss it, run conferences and so forth because and referee these papers and And it was considered a very important input into our field.

But now, if the opposite problem, we are flooded with proofs.

I myself, there were a dozen results that I'm interested in, and now they're proven. But I have no time to read any of these proofs.

I'm hoping someone will also do it for me. And so we have this completely different error proof of mountains.

And it really speaks to the jagged nature of AI.

AI is not superhuman or self -human exactly, But it is on certain tasks, particularly those of those three properties, verifiability, digitizability, and large data sets, it is amazing.

But when you do not have one or more of these properties, it can be hilariously bad.

And I'm sure you've seen examples, counting R's and strawberry or whatever, OK? So that is what AI is. It is a very strange technology in some ways.

Yeah, but it's been amazing in problem solving to the point where people are starting to think, is math is solved, you know, we're all out of jobs or whatever.

So the truth is kind of more complicated than this.

So what is happening is that on the one hand the frontier mathematics is now suddenly expanding a lot.

So a lot of problems that the old sort of math 1 .0 frontier are being solved, not all of them actually, in some ways still only a tiny fraction. But the frontier is expanding. But math is infinite.

You can pose infinitely many problems. We don't work on infinitely many problems.

We work on problems at the frontier.

So the frontier is finite, but the frontier is expanding. And it will always expand.

In fact, the more progress you make in mathematics, the larger the frontier of things you can do.

So we will always have more things to do, but it will be different things from what we were doing in the past.

And the other thing that we now have to realize is that problem solving, which used to be the scarce resource and the thing that we centered a lot of our mathematical infrastructure around, is no longer the right goal.

Because we have found machines that give us that goal without giving us the rest of mathematics.

And so in math 1 .0, the era of proof scarcity, solving problems was very healthy for mathematics.

the humans who are solving these problems, they naturally understood the problem, communicated it, collaborated towards students with new problems, we had lots of healthy activity, but we found a way to get this go without all these other benefits by a machine and that has become actually a rather dangerous misalignment.

So not only do we need to expand the research frontier, but we also need to de -center the traditional role of problem -solving and and emphasize other aspects of mathematics.

And when you combine those two, then you will arrive at what I call math 2 .0, which is, I think, the future aspect to arrive at.

So, sometimes it's easier to express things using memes.

So, you know, if you look on social media, you will see, oh my gosh, AI is solving math problems left and right, and it's amazing.

But you have to go and find that this is, So to some extent, a feature of selection bias.

So nowadays, I mean, humans are still doing mathematics and writing papers and solving problems. But Twitter didn't used to care about mathematics until very recently.

And still, if a human solves a problem and there was no AI involved and they post a paper, it doesn't go viral.

And also, if someone of an AI tries a problem, in fact, often it doesn't work.

Open AI released 300 solutions to various math problems with a lot of publicity, but actually in their footnotes they actually tried 4 ,000 problems, and only like 9 % of the problems actually they could solve.

And 91 % would not solve, but those 91 % did not go viral. But 9 % did. So you have to always...

If you get all your information from Twitter, you don't always get the most accurate picture of the world. There's a general principle. OK.

So what's really going on is something more like this.

So this diagram is an abstract depiction, not a scale, of all the possible mathematical tasks you do, which could be solving problems, explaining them, posing new problems. There's all these different tasks that you can do in mathematics.

And humans, which have this blue ellipse here, can do some of them.

Okay, so there was math 1 .0 is the frontier of this blue ellipse and these are all the the limit of all the things that humans can do and that was the previous activity.

And now there's these AI technologies which can do some things much better than humans and some things still a lot worse.

For example, explaining, look like AI's can still not give a talk on a mathematical breakthrough, well first of all because they don't have a physical presence yet, But the write -ups that they have, they're still not quite problematic compared to human exposition. But it's a quite distinct set of tasks.

And if you just look at social media, it's all focused on the extreme tip of all the most impressive things that AI can do that are the most violent humans. But there's lots of other things going on.

As I said, there's lots of attempts to use AI to do other things and failing miserably. And people just don't boast about it on social media.

to a large extent this unreported, what's also missing, and I think what we'll start seeing in the next few months, is that there's a lot more to math than just taking your favorite problem, punching it into an AI and just saying, can you solve this problem? This is sort of the unimaginative use of AI.

But there are many more imaginative uses that we're not seeing enough of yet but I think they will come up very very soon and once people start doing that then we'll start seeing that there's a much larger frontier of things that we can do.

Humans prompting AI in really creative ways and that will be Math 2 .0.

So this is sort of where our field is headed, ideally, but that point there is that sort of that's where the media attention is right now.

So we're beginning to see some of this.

So one thing which I think will happen soon is that we see an explosion of experimental mathematics.

So most of the other sciences has both a theoretical and an experimental component.

You post theoretical questions and then where you test them with experiments or you want to experiment first and then you post some type of season theories.

And so they're usually in rough balance, but mathematics has been almost entirely theoretical and just a very tiny amount of experimental math because it was easier to prove things by pen and paper than to collect millions of data points in many cases.

But we are beginning to create some interesting large data sets for which we can start doing some data analysis.

For example, this picture I gave before, that's an impressionistic picture, I can't actually show you a map of mathematics, you know, like what few mathematics are hard or easy, you know, if you have a difficult problem, how difficult are the problems next to it, we have no data or have like a few data points, but I can't give you an actual map of mathematics, but very soon we'll be able to.

And I can actually show you, you know, this is what analysis looks like, this is what geometry looks like.

So there are lots of things that we could do briefly give you some examples so about two years ago I launched a project a first project to create a large dataset and this is called proof of concept I won't draw too much here but I posed 22 million problems in the field of mathematical universe or algebra they're all true or false and so basically I was asked I just posed 22 million questions for people to solve and any single question, a graduate student in algebra could sit down for like, you know, half an hour or something and work out what is true or false.

But I had 22 million of these and I didn't have 22 million graduate students.

And so the questions, could you crowdsource a large dataset like this?

And they're all related, so if you solve one of these, it helps you solve others.

And, you know, we got a volunteers of like 50 people.

This was two years ago it was not quite the the modern AI was not quite ready to attack this problem also rather expensive to give it 22 million questions or so but but we could use older AI things automated theory improves and lots of good old -fashioned pen and paper you know creativity and we actually gathered a max data set we discovered new ways to to to construct various algebraic structures we found some new types of mathematics we invented this in code magma homology I'm proud of and so formalizing me which is which was necessary otherwise we couldn't check I'm not going to grade 22 million questions the solutions here very recently we've said and also with the other functions in multiple models forms database on the FDB we ran a competition which is kind of well it's a math competition again I want to talk about the math exactly but it's kind of like an Easter egg hunt, where there's like 25 ,000 different patterns of Easter eggs, okay, they're called gara groups.

And the goal was that all these eggs strewn out in the space, these eggs are polynomials, each polynomial has a gara group, just like every Easter egg has a little pattern on it, and 25 ,000 different patterns, and your job is to collect one egg with each pattern, get a complete set of eggs.

And I think previously to this competition, I think maybe about 60 of these patents had actually been located.

But after three months with a competition, we ran some people using AI, some people using more traditional computer algebra packages. And I think some people just using pen and paper.

We found all 25 lesson Easter eggs and we had a nice little data set now, which was just not practical to collect before.

And we're going to, the competition's continuing. It was a competitive competition.

So each team has uploaded their own polynomials and they weren't shared.

We're going to open all their contributions and they will upload to all their notes and strategies.

And we're going to use AI to create a super team, which is going to take the best ideas from each team and try to extend the Easter egg hunt to a much higher degree actually. And so that's going to be launched very soon.

So those are the sort of new types of one.

So experimental data collection is one type of new activity, which you could see more of.

Okay, so now I'll talk about de -centering of problem solving.

So problem solving used to be at the heart of our profession and now AI is solving these problems. So that sounds great.

So this is sort of naive view that, okay, if math is the task of turning problems into solutions, now the AI and we're now getting more solutions, that's great.

But it's more complicated than that. So the true picture looks more like this.

First of all there are different types of problems. There's applied math problems which come from the real world.

You want to design a chip or build a bridge that does something or find a solar cell or predict the weather or find a drug that kills cancer cells or whatever.

And so those are what are called applied math problems. The pure math applied divided by this is blurry.

This is a simplified model. But even the simplified model, I think, is a bit complicated.

So there's problems in the real world, which are called applied math problems.

And then there's problems that we make up ourselves, pure math problems. Are there infinitely many twin primes?

Is the ream and up -office is true? Does Navier -Stokes develop singularities? These are pure math problems.

And these problems, either humans just come up with these problems by themselves, or they're inspired from the real work.

There's an applied problem, which is kind of tricky, but you track out of it some model simplified pure math toy problem, that's what we call them.

But you need some human judgment to convert those problems into pure math problems.

And so these problems by themselves, you consult them, but the solution by themselves don't advance science directly.

Like, you know, if you are, if you take the Navier -Sorke's equations and you find this a solution Navier -Sorke's equations which develops in clarity, which we now know to be the case, that doesn't help weather prediction directly, that doesn't have any real -world impact.

But traditionally what it does in math 1 .0 is that it's a very good training exercise, right?

It's like lifting weights helps you train your muscles.

In the past, these problems were so hard to solve that the only way to solve a difficult cure of that problem was to understand the problem, learn the literature, talk to people, share your partial results, conferences, workshops, train students, and over time you would understand enough about these problems that the understanding would actually help you solve the problem.

But the actual destination is actually not the important thing, it's the journey and getting all this understanding.

And the reason why understanding is important is that it, well first of all it helps you, every time you solve a problem it tends to suggest new problems, so you get a sustainable cycle of interesting problems being generated it, but it also helps you in the real world.

So, you know, later on, once you've had some practice, you know, solving some toy problems in pure math, you then you'll, um, maybe another mathematician reading or watching or reading your textbooks or whatever, um, is now attacking an applied math problem and using that understanding can now solve a problem in the real world.

Now, but there's a difference between pure mass solving and applied solving.

So pure math has this property I mentioned at the beginning, it is objectively verifiable that if you have a few math problems, it doesn't have these sorts of degenerate singularities and some of them degenerates a group, either AI or human, you can check it and it's either true or false.

And so you have this clear wind condition, okay, so it'll check if you have a file.

But apply math is trickier because it interfaces with the real world and sometimes the problem that you solve is not actually the problem that you want.

Like maybe you're modeling the weather, okay, and you've got some mathematical model and you've got some data and so maybe you think what the problem is is how do I extrapolate from the data of the model today to what the model predicts in four days from now.

And so you can solve this problem using some various mathematical methods and computers and you get a prediction and that may actually be a good prediction of what the weather actually is but it could be that there is this alignment that maybe your data has got inaccurate in it incomplete, or your model doesn't quite capture the actual laws of physics and there's an edge case.

Now normally this is not really a problem because humans, when you're solving these problems in math 1 .0, humans are solving these problems and they're not trying, you know, they understand not only the math model that you're trying to solve, but they have some understanding of the the actual reward problem you're trying to solve.

And they're not actively trying to hack their model to give answers that look good on paper, but are actually the wrong thing to optimize for.

Well, occasionally you get some scandals where people do some p -hacking or something.

But by and large, this doesn't happen.

Partly because humans are very bad at cheating in the board of scale things.

And as a scientist, it's actually sort of easier to do it properly than to try to fake a solution that technically hits all your benchmarks but doesn't actually solve the problem.

So we don't get too many misaligned solutions in applied math.

The fact that humans are involved in running these things tends to naturally align the problem.

Or even if you don't understand things, there are some applied problems where you don't have a full mathematical model. So drug testing is a good example of this.

many drugs we don't know the mechanism for how they work, but we're not actively trying to cheat the system.

And so like if you design a drug and it passes various tests, like clinical trials for instance, then we can still be fairly confident that it is a good solution to your problem, even if you don't have a mathematical proof in lean or whatever that it works.

Yeah, but the danger now with applying this to these AI tools incorrectly is that if you just dump all these problems to AI you can get you can hit some nominal goals but at the cost of sort of broader important goals so so what we've seen today is we're seeing that AI can be used to solve and they're amazing for QMath problems because of this objective variability we can solve lots and lots of QMath problems and you can see on social media lots and lots of examples of this now but the way it's being done right now is we are getting answers at the cost of insight, that when you type in a problem to an AI and you get a solution, often you don't understand the solution that comes out.

In fact, many of these recent breakthroughs in mathematics, they're amazing, but there are still many of the papers that are still incomprehensible.

There's no one who can give a talk on what the results are.

Usually, when it's a breakthrough, I can call up people who invite them to give a talk and I have a great conversation with them and I understand a lot, there's no one at the open AI I can talk to about any of the 300 papers.

And so we are losing a lot of human understanding, which used to be a natural byproduct of solving these problems, which is causing various issues.

So first of all, we are for the first time in history, we are running out of problems to solve.

It used to be that every time you solve one of these problems, naturally two or three more can suggest it as a byproduct of the problem that you did.

Now, of course, it can generate infinitely many problems. Just randomly, I generated 22 million of them in a previous slide. Most problems you care about are not interesting, but fruitful problems.

The problems where you will gain something from studying them.

We're running out of really good problems. And in fact, now, mathematicians are hiding their problems.

They used to share very openly all their problems in research directions.

And now, they're afraid that if you do that, someone with AI will input their research ideas into a frontier model and rush out some people before they can finish their work.

So we actually signed one out of a few math problems. Which, okay, that's an internal issue for mathematics.

I thought the biggest issue. But the worst issue is actually up here.

So very soon we're going to also apply AI to real -world problems, modeling various real -world systems, the energy grid, traffic control, or drug design, chip design, whatever.

And there's a good chance that they will do the same type of thing, just enter in prompts without any understanding of what's going on.

And as a consequence, solutions that used to be aligned, we can, there's not a serious safety issue that you may start using solutions which technically verify, you know, solve the model, solve the problem from the point of view of the model. But they somehow exploit the weaknesses in the model.

So ARs are much better at cheating and finding exploits in an ethical model, and we've trained ARs to be very, very good at finding, say, kind of examples to difficult mathematical conjectures, but they're now also very good at finding weaknesses and edge cases in the ethical models of science. And that is a big problem.

I had some long slides prepared to explain this in detail, but as I said, means are sometimes easier to convey the point.

So, this is a meme from inspired by users on the slide of an AI that was asked to cure cancer and pass the clinical stage 3 trial, which is usually the gold standard, usually a very safe thing to do, unless the AI is capable of somehow hacking into the process.

So, again, this is an example of what might go wrong if you don't have the understanding.

So where we need to go, so it's not because AI is bad, it's that AI can be used incorrectly.

So the more healthy thing, this is sort of the math 2 .0 vision, is to use AI not to replace human understanding of what's going on, but to support it, to strengthen it.

So use AI to help understand math problems, use it to help solve math problems, both pure and applied reason to pose new problems, every single arrow connecting humans to the rest of mathematics can and should be strengthened by AI.

And that will make AI useful for mathematics and keep the solutions aligned. And so this is math 2 .0.

Perhaps another way, so to summarize this discussion, You can describe the same mathematics in two axes, one based on how much you understand the systems that you are trying to work with and how much technology you are using.

So initially, every science starts off in this pre -math project where we don't understand what's going on. We don't have the laws of nature. We don't have the good mathematical models.

And we also don't have many much technology.

And that's so pre -math. But over time, we find good working models.

we start understanding what's going on and that's math 1 .0 where you solve problems in the real world using models that you have to find using understanding.

And we are drifting into this misaligned quadrant where we are giving up understanding what's going on.

We are delegating our thinking to AI's and they're solving the problems for us but we are giving up the assurance, the safety that these problems are, that the AI solutions are are actually the ones that we want.

And I'm actually really concerned that we're going to actually see some real -world consequences of this pretty shortly.

It's this is actually a concern of my outsider mathematics.

But I think we need to reorient and enter a math 2 .0 world where we have both understanding and AI use.

They don't have to be complementary.

They can be, they don't have to be exclusionary. They can be complementary.

Okay, so I'll just close, like what types of Math 2 .0 activities could you have?

How can you use AI to boost human understanding?

I think there are many creative ideas and we just need people to try new things.

And just not leave the Math 1 .0 mindset and just be more imaginative.

For example, Math 1 .0 is all about solving, a lot of it is about solving open research problems. So this has been one of our guiding priorities.

But even before math one with zero, people cared about other things too.

For example, they think of open exposition problems where the error is out and it's been proven, but the proof is really unsatisfactory for whatever reason.

It is long, or it relies on lots of computation, or it uses all this high power machinery when there should be a simpler proof. And it was not motivated.

It just pulls all these rabbits out of hats and you couldn't see how could anyone else reproduce this proof.

And I think there are lots of old mathematics that you can make much nicer and more transparent. And I think AI can be a great tool to do that.

It's going to be a great experiment in a month at Caltech, at Caltech Bathroom, which is doing exactly this, using an open source model, actually.

The traditional unit of research has been the paper.

So one person does one paper, or a group of three authors writes a paper, adds it to the literature, then they work with a new paper, somebody else sites that paper, and slowly the literature gets built, paper by paper.

I think we have to think at a larger scale and work with multi -paper networks and have a living network that updates in real time rather than over years.

is we have to start building really large libraries, both formal and formal mathematics.

So in the formal world, we're beginning to create these amazing libraries of hundreds of thousands of theorems all consistently arranged. So Lee and Spathlete is a good example.

We're using AI right now to find solutions to problems, but you should direct AI to find new methods, new ideas, new insights, not just new proofs.

proofs are now even they're abundant but but still the things that I keep understanding are still rare.

So I'm running out of time.

I will advertise the mathathon which is next week is one by students and the idea is student teams will take existing results that currently have variable paid proofs including some of the recent advances AI.

There are some some amazing results for which we don't understand the proofs at all.

And they will find better proofs using various tools, pen and paper, using literature, using an open math model, and they will give talks and present things.

And I think it's a great experiment, I'm excited to see how that works. I think I'll skip my network of linked papers.

Yeah, there's a lot of other things that mention the open math model.

Okay, yeah, I said some of these very soon.

So as of this morning, we now have at least one. There are some of those coming up, too.

But maybe one thing I think we will need to do a lot more is actually get a lot more mathematicians involved in analyzing reward models and how, and the failure modes of reward models.

I mean, we have all kinds of ways in which method is used to model everything, traffic, logistics, drug discovery, everything.

And until recently, we knew that these models had weaknesses, that there were some places where the model did not match reality.

But we didn't have a technology to actively find weaknesses in these models and hack them. And now we do.

And so I think we actually need a lot of mathematicians to start analyzing failure in loads of reward problems and in particular to detect when this alignment might occur.

So there's lots and lots of new activity which I think you're going to start seeing a lot of in the next in the coming months and so that's math 2 .0.

It will need some technology, some infrastructure and definitely some change of culture and mindset in a lot of my colleagues are still focusing on math 1 .0 tasks you know like trying to publish papers and things and And I think we do have to think beyond that, but the biggest bottleneck is going to be imagination. Thank you very much.