What’s the Future for Pure Math Research in the Age of AI?

Stephen Wolfram (0:05)

Hello, everyone. Well, I thought I would talk about something I just posted entitled What's the Future for Pure Math Research in the Age of AI? So let me tell you what I said, and then we can have some discussion about it.

Headlines on History. The headlines keep coming. Such and such an AI system has solved such and such a math problem. More and more, I'm hearing people say, maybe we don't need people doing math research anymore. Maybe we should just delegate it to more and more powerful AIs.

Well, I must admit that I'm getting a bit impatient with some of what's being said, because it seems to me too often to involve sort of fundamental misunderstandings of what math is really about, and also frankly, what AI is about.

Perhaps I personally have a unique piece of history that informs this, because after all, back in 1988, when we first introduced Mathematica, there were also some of the same kinds of talk about math being taken over by and made pointless. Of course, that's not how it worked out at all. Instead, Mathematica, now Wolfram Language, just raised the level of math that can be done, and over the years has led to all sorts of important new math.

If working out symbolic integrals was what one thinks doing math is really about, then yes, Mathematica has essentially replaced it. But while that kind of problem-solving is what's needed in many applications of math, it's not the core of what math itself and its pure form is about.

The enterprise of pure mathematics is an old one, crucially entwined with the history of civilization. From the time of Plato and nuclear, pure mathematics was the defining example of a place where abstract rational thought could build an ever larger structure. And over the centuries, pure mathematics has come to be the single largest intellectual edifice that our civilization has built. It's not been without its pathologies and limitations. And even among those building the edifice, one runs into misunderstanding about what's important, how things should be done, and so on. Is mathematics fundamentally about producing proofs by whatever means necessary? Is mathematics always ultimately justified by its applications? Is there some inevitable book of right answers that it's the goal of mathematics to discover?

Again, I suppose I have some personal history in all of this, because my efforts in basic science have led me to ask questions about the foundations of many things, including mathematics. So I've studied questions like the space of all possible mathematicses, and what the limiting structure of the network of all theorems might be, and notably, what the role of humans is in defining the thing we call mathematics.

The value of modern AI.

I'll talk later about the general character and value of pure mathematics, but before that, let's address the issue of the moment, the role of modern AI. The first thing to say is that it's unquestionably useful, sometimes very useful. For me, its greatest use in mathematical pursuits has been its ability in effect to thematically mine the knowledge base of human mathematics. Starting back in the 1970s, being able to do keyword searches with the scientific literature was a crucial enabler of quite a bit of the research that I did. And now with modern AI, one can do so much more. Besides, because somewhere inside those LLMs, in a way that we don't yet scientifically understand, there's what amounts to a representation of raw ideas, gleaned from all those millions of papers and books about mathematics. And at its best, it's not just about retrieving things, it's also about making connections, of being able to see that this result here can be put together with that result there to come up with a surprising and useful conclusion. Humans routinely do that too, but they tend to have only read hundreds of papers. AIs have effectively read millions, and it's cheap for them in effect to try out lots and lots of possible combinations.

So how can one expect to just come on expect to just launch an AI off and have it come back with great math? As we'll talk about, great math is more than anything else defined by the questions it asks. Yes, the AI can successfully automate things that humans would normally have had to do themselves before. But as we'll talk about, at the core of pure mathematics is human imagination that guides what questions to ask. It's worth understanding the difference between what modern AI does and what pure computation does. Modern AI is first and foremost a way of leveraging the existing corpus of human knowledge. Computation is at its most powerful and open-ended way to generate things that are fundamentally and irreducibly new. Start from some rule or axiom and just repeatedly run the computation of applying it and my all-time favorite phenomenon of computational irreducibility guarantees that you'll go on getting fresh new results that can't be reached except by doing all those computational steps.