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

Headlines and History The headlines keep coming: such and such an AI system has solved such and such a math problem. And more and more I’m hearing people saying: maybe we don’t need people doing math research anymore; maybe we should just delegate it all to more and more powerful AIs. 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 fundamental misunderstandings of what math is really about—and also, frankly, what AI is about. Perhaps I have a unique piece of personal history that informs this. After all, back in 1988, when we first introduced Mathematica, there was also some of the same kind of talk about math being taken over, 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 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, in 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 Euclid 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, etc. 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 is 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 what the space of all possible mathematicses is, 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 We’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. And 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 knowledgebase of human mathematics. Starting back in the 1970s, being able to do keyword searches of 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. 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 can one expect to just launch an AI off and have it come back with great math? As we’ll discuss, 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 discuss later—at the core of pure mathematics is the 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—an 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. (For those who aren’t already familiar with it, computational irreducibility is an idea I introduced in the 1980s to capture the notion that many computational processes—even when defined by simple rules—allow no general shortcut: the only way to determine their outcome is to explicitly run each step. It’s turned out to be a phenomenon that’s quite ubiquitous in the computational universe of possible programs—and to be connected to a long sequence of foundational issues in many areas of science, as well as in philosophy, etc.) So, yes, having nothing to do with AI, computation can generate an infinite sequence of new theorems, representing an infinite sequence of new facts, etc. And among those theorems there’ll be all sorts of “originality” and, in effect, “surprise”. But there’s a catch. Those theorems are in a sense just “results plucked from the computational universe”. And as such they certainly fall under the purview of the new—and I think very important—field of ruliology on which I have spent so much effort. But are they math? What Is Math Anyway? Well, of course that depends on what math really is—which is exactly what we need to understand. In its early history, math was thought of as a way of making precise and formal statements about the world, say about arithmetic or about geometry. But by the later part of the nineteenth century higher levels of abstraction had been reached, no longer tethered to features of the world as we experience it. And there swept across mathematics an increasingly formalistic view: that ultimately math really is just the collection of theorems that can be “mechanically” (i.e. computationally) derived from certain axioms, say the axioms of set theory. Gödel’s theorem put a small dent in that picture. But even today, if pressed, many mathematicians will try to define mathematics as being the formal study of the consequences of certain chosen axioms. But while they may say that, it’s not a good description of what they actually do. The vast majority of actual pure math research does not operate at the level of axioms and mechanical derivations. Instead, it works at a much higher level, building and studying abstract structures and their interrelationships. It’s not obvious that this should be possible. It could be that the only way to get right answers in math would be to operate at the lowest level, say directly in terms of axioms. But it’s an essential—if typically unspoken—feature of pure mathematics that in practice one can reason at the level of, say, the Pythagorean theorem, without constantly having to dive down and talk, say, about the axiomatic definition of real numbers. I’ve recently argued that this happens for much the same reasons that in physics we can successfully do fluid mechanics, without always having to dive down and trace the collisions of individual molecules. Ultimately the fluid is made of all those molecules. But the point is that observers like us typically sample it only at the much more human level of overall fluid motions, etc. And so it is also with mathematics. Human mathematicians typically sample the vast metamathematical web of underlying axiomatic derivations only in overall, collective ways, in terms of “human-level” structures and concepts. Ultimately the story actually seems to be very much the same in mathematics and in physics. At the lowest level, everything is full of computational irreducibility, so that one can figure things out only by mechanically taking every computational step. But within such computational irreducibility there are always pockets of computational reducibility where it’s possible to “jump ahead” in a “higher-level” way. And it’s within these pockets of reducibility that our laws of physics—and our human-level mathematics—reside. We can see physics—or indeed natural science in general—as an effort to take the complexities of the natural world and find aspects that can be described by narratives that fit in finite human minds. Mathematics can be seen in very much the same way: as an effort to take the complexities of the metamathematical world and find aspects that can be described by narratives—now mathematical ones—that fit in finite human minds. In other words, the problem of advancing pure mathematics is at some level not so much about pushing back some raw metamathematical frontier as about finding human ways to represent what’s there. The history of human mathematics has been characterized by the building of ever taller towers of concepts—that capture ever greater abstraction. And one might suppose that this process would somehow in the end inevitably “reveal all mathematics”. But the story is more complicated. One can imagine that as an ultimate limit one could start from all possible axiom systems and generate all possible theorems. What one gets this way is a unique object that I call the ruliad—that corresponds to the entangled limit of all possible computational processes. But the issue is that finite minds—like ours—can only perceive a tiny part of the ruliad. And that means that we can have no “absolute mathematics”, only mathematics based on how we sample the ruliad. I’ve argued elsewhere that some aspects of this sampling follow inevitably from general features of the way we are as