The scope of the conversation:
Tyler calls Terence Tao arguably the greatest mathematician of his generation: a Fields Medal winner who has published serious work across harmonic analysis, partial differential equations, number theory, probability, and compressed sensing, among many other fields. His new book, Six Math Essentials, is the occasion for this conversation, but Tyler spends much of it asking something more personal: not what the proofs say, but what they feel like. Tao’s answers reveal a mathematician who treats beauty as a real signal, finds Gödel’s incompleteness theorem genuinely unsettling even decades later, and is remarkably candid about failing both Fortran and quantum mechanics as an undergraduate.
Tyler opens by asking Terence what it feels like to contemplate Euler’s identity, quantum mechanics, the Riemann hypothesis, and Gödel’s incompleteness theorem, before moving into why mathematical history gets stripped out of classrooms, whether abstract math is a lottery ticket or a unified whole, string theory ’s beauty versus its lack of confirmation, why real analysis arrived so late, why zero took until the 7th century to be invented, game theory, the Black-Scholes model, Bostrom’s simulation argument, and Newcomb’s paradox, math education and the case for teaching more statistics, Tao’s view that AI has pushed mathematics from an era of “proof scarcity ”into “proof abundance,” what that means for the research paper, why video talks may become more valuable than papers themselves, Tao’s favorite Hong Kong action movies, his unconventional path through an Australian university at age 13, his most unusual work habit, his verdict on the Collatz conjecture, and more.
Some general comments, proof vs. understanding, Gödel, limits:
Cowen begins by asking Tao how he feels about various results in math, including the Rieman Hypothesis, which concerns prime numbers. Tao notes that it’s not been proven, but we’re working on it:
We are making progress. In fact, just two days ago, there was a significant advance on the Riemann hypothesis. We are still quite a ways from the full resolution, but now there are new ideas. Unfortunately, the ideas were generated by AI, and we don’t yet understand them. This is part of the new era now. We are getting answers without understanding, which is a very strange era to be in.
Historically, the only way to make a progress on a difficult mathematical problem was to understand it, and you could communicate these insights to others. Now we have these machines that can give us answers, but we have to really work much harder than normal to extract out the useful insights from them. It is a strange era. We now have these new tools, and hopefully we will have lots of mathematicians exploring both old methods and new methods, and we will find something. It will probably require a new idea which we haven’t seen before. We know the limits of many of the existing techniques.
What interests me about that mark is the way it bears on the use of AI in exploring the “proof space” of mathematics.
Tyler then asks him about Gödel’s “incompleteness theorem, does it frustrate you?”
Yes. Mathematics is going through a crisis right now. There was a similar crisis 100 years ago. David Hilbert, who was one of the leading mathematicians in the early 20th century, had this great dream finding that every single open problem in mathematics should be solved or unsolved. We should systematically try to answer all the questions in our subject. Here’s a famous quote in German: “Wir müssen wissen, und wir werden wissen,” “we must know, and we will know.” Now, this was the rallying cry for mathematics in 1900.
Gödel proved this very, very striking result that no matter how you set out the rules of mathematics, there will always be statements that cannot be proven or disproven from those rules alone. That there is no fixed set of laws of logic that can answer every single question. They will always be incomplete. This was very traumatic for mathematicians. It said that Hilbert’s program would never be finished in some sense. Over time, I think, we acclimatized to it.
In practice, many of the results that we do care about, we can prove or disprove. The specific statements that Gödel showed which were not provable or decidable are not the ones that we encounter in real life. Sometimes there are questions that you can’t answer using the usual axioms or rules that you’re allowed to use normally, but sometimes we add new rules. There are other foundations of mathematics which can prove more things than the standard ones. Sometimes we do play with different rules, like when you play chess. There are some standard rules of chess, but sometimes we also consider other variants.
We’ ve moved to a more pluralistic view of mathematics, where there isn’t just one true mathematics and one universal true and false facts.
During my first or second year at Johns Hopkins a friend gave me a copy of Ernest Nagel and James R. Newman, Gödel's Proof (1958), which was the first attempt to explain Gödel's results to a general audience. FWIW, I was struck by how very odd those undecidable statements were.
Tao goes on about this pluralism:
This is also changing our philosophical foundations of what does it mean to prove something. What does it mean to understand something? As I said, these two statements used to be almost synonymous, and now they have been decoupled quite severely. I try to keep an open mind, which I think is part of what it means to be a good scientist. You don’t have a dogmatic view of what science is or isn’t. You update it based on events and new data.
That is to say, understanding and proving are now understood to be different. Computers can prove statements but we don’t necessarily understand those proofs.
On the importance of mathematics on the context of STEM, which Tao views as an ecosystem:
....mathematics is the soil. It’s at the foundations. It doesn’t generate the fruits, the things that are most directly useful for humans, but it sets the stage. Of course, it gives us all the formulas and recipes that other scientists use. It also sets limits. Most other sciences or engineering, they tell you what’s possible. You can build a rocket, and it flies, and now you’ ve proven that it is possible for a rocket to fly.
If you want to show that something’s impossible, that there’s a limit, for example, if you want to communicate using a wireless spectrum, what is the maximum amount of bits you can stream from A to B? There’s a branch of mathematics, information theory, that tells you this, and it gives you limits. There’s a maximum theoretical limit, and you can’t discover this by engineering.
And that's why Miriam Yevick's work is important, it provides boundaries. See her 1975 paper, Holographic or fourier logic.
AI and math
COWEN: How long do you think it will be before AIs create the equivalent of new Millennium Problems for mathematics? Is that two years away, 20 years away? Interesting ones.
TAO: It’s extremely uncertain times now. I think a lot of assumptions about how mathematics operates is going to change a lot. There’s so many changes, but in one sentence, we have moved from an era of proof scarcity to proof abundance.
It used to be that proofs were really hard and valuable, and so we would create these open problems as these goals, these landmarks, and we would march toward these goals very slowly and deliberately, but we’d pick up all these insights and techniques and learn all these things along the way. We have these Millennium Prize Problems, and we have these targets that guided us, but they ’re relatively few in number. There’s only seven Millennium Prize Problems. That was the way things worked in a proof-scarcity era.
I think going forward, the idea of focusing on individual problems, that currency has basically been hyperinflated. We now have machines that can just brute-force, basically solve massive numbers of the old problems that we used to use as landmarks, and we have to really switch to some sort of math 2.0. I think instead of dealing with individual problems, we may start creating huge datasets of thousands or millions of problems and try to sweep them out. With this AI, I can handle 30 percent of these problems. Using this idea that this human contributed, I can solve this 10 percent, but not pick out individual problems per se anymore as the goals, but work at a much higher scale.
I think that it will take a lot of cultural shift. We are not used to mass production of mathematics. We are still, in many ways, in a pre-industrial economy, math academia. We are forced, almost at gunpoint right now, to switch to an industrial or post-industrial economy, and this will take quite a bit of cultural change, but I think that’s what’s going to happen. Individual problems, even like Millennium-level, may disappear, and we will just have lots and lots of big datasets of problems.
COWEN: On net, you’re happy? Unhappy?
TAO: That’s not the right word. It’s very complicated. I’m excited by the possibility of completely new types of mathematics that didn’t exist before, but I’m also actually quite horrified by the disruption, which I think was much more than necessary.
Of course, every revolution, like the Industrial Revolution, of course, is a famous example, it creates job displacement and so forth. I think there was some medium between status quo and transition to a new paradigm where you had more of the benefits of transition and fewer of the disruptions. If the Industrial Revolution happened within one month, it would have been much more traumatic, entire careers cut short with no ability to transition halfway through.
COWEN: It would have been better overall, right? Like our standard of living, people might have had in the 1950s.
TAO: Yes, but a lot might have been destroyed permanently. There are still pre-industrial ways of handcrafting certain objects or doing things in a very old-school way. There are communities that keep these more artisanal types of craft alive, and they have cultural value. There was enough time in the revolution to preserve these culturally significant artifacts. It is possible that there could be some major disaster where we lose our power grid or something, and some of these antiquated skills actually become relevant again. Actually, from a strategic point of view, there’s actually some value in preserving what you might think of as obsolete skills.
Sometimes the historical insights of how an obsolete practice developed can inform the development of a modern skill. We talked about the importance of history and mathematics. It’s true. History is important everywhere. If you rush the transition too much and you just focus on just getting your productivity up or hitting some other benchmark, you may lose some of this less tangible value that you may only see after the fact what you’ ve lost. It’s like everything. If you want to do urban development, you need to tear down old buildings to create new ones, but some buildings have actual historical value, and there is a way to develop sustainably.
Later:
I also think that we will also work at much larger scales than papers. We will see entire databases which are like the equivalent of thousands of papers networked together in a single spreadsheet, which you can query and you can update in real time, and you can get all these visualizations, which you just could not do with the old system. Or every time you wanted a new way to visualize the literature, someone has to write a new paper, and you have to wait a year before you get a different view. You should be able to see an entire field.
I think people will have to move to more collaborative, big math projects. Instead of a small number of people working on individual papers and then contributing atomically to the literature, the literature itself would be a much more networked, living, collaborative entity where both AIs and humans can collaborate. I think this is the only way forward.
We’ ve seen in recent months that brute competition between humans and AIs is becoming very counterproductive. People are just racing to dump out a paper three minutes before someone else with an AI is generating almost exactly the same paper, and everyone is cutting corners and making the papers harder and harder to read just because they want the priority. Priority used to be a very good way to organize our field and encourage people to produce results in a timely fashion. The metric has become way too over-optimized now, and we need to switch, as you say, to beyond thinking at the scale of a single paper.
Bonus: A footnote concerning theology:
In the middle of things, Tao remarks, “One amusing thing I saw on the internet recently is that the fact that we can now create AIs that appear smarter than us has theological implications as to how intelligent our creator was compared to ourselves.” Compare this with St. Augustine on memory, Confessions, Book Ten, Chapter VIII:
Great is this power of memory, exceedingly great, O my God–a large and boundless inner hall! Who has plumbed the depths of it? Yet it is a power of my mind, and it belongs to my nature. But I do not myself grasp all that I am. Thus the mind is far too narrow to contain itself. But where can that part of it be which it does not contain? Is it outside and not in itself? How can it be, then, that the mind cannot grasp itself?
A great marvel rises in me; astonishment seizes me. Men go forth to marvel at the heights of mountains and the huge waves of the sea, the broad flow of the rivers, the vastness of the ocean, the orbits of the stars, and yet they neglect to marvel at themselves. Nor do they wonder how it is that, when I spoke of all these things, I was not looking at them with my eyes–and yet I could not have spoken about them had it not been that I was actually seeing within, in my memory, those mountains and waves and rivers and stars which I have seen, and that ocean which I believe in–and with the same vast spaces between them as when I saw them outside me. But when I saw them outside me, I did not take them into me by seeing them; and the things themselves are not inside me, but only their images. And yet I knew through which physical sense each experience had made an impression on me.
You might also want to read a recent post, which comments on that passage: Pope Leo and St. Augustine discuss the mind and A.I. with Kurt Gödel.
