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

Wolfram, Stephen

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

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

September 28, 2026

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 observers of the ruliad. And this leads to certain “general laws of mathematics”, such as the very fact that higher-level mathematics is possible—at least for observers like us. But other aspects of the sampling we can see as historical accidents: choices that the mathematical community made to explore one direction rather than another. And once again it’s important to recognize that it’s inevitable that one has to make such choices.

In a sense the ruliad is too big for it to be otherwise: our finite minds can’t span all directions, so we have to choose just some. And then we have to summarize what we find in terms of some limited set of concepts—out of which we can then build our mathematical narratives.

It’s very much like with human language. Out of all possibilities we choose certain concepts to represent, say by words. And these concepts are how we “encapsulate thoughts” to be able to communicate them in finite ways, and be able to fit them in our minds.

At a raw computational level it’s straightforward to do what amounts to mathematical ruliology and start axiomatically generating huge numbers of new theorems that have never been seen before. But with overwhelming probability this will lead us only to “alien mathematics” that doesn’t usefully connect with mathematics as we know it—or, for that matter, with anything that we can view as a higher-level concept at all.

Generating Math with AI

But, OK, let’s say we’re using modern AI. Can’t it operate directly with higher-level mathematical concepts? Certainly LLMs manage to deal with human language. And indeed, just as LLMs successfully learn the general patterns for putting together words in human language, so also they can learn the general patterns for putting together constructs in mathematics.

But generating useful mathematics is a much more exacting activity than generating language. A written story can’t really be “wrong”; a written piece of mathematics certainly can be. Of course, it helps a lot when the LLM can call on Wolfram Language—as humans do—to reliably compute things. (And, yes, everyone should spend the few seconds it takes to connect their AIs to our Wolfram MCP system!) But a serious piece of math research will typically involve fitting together many parts, doing many steps in an argument, etc. And the issue is that the fundamentally statistical nature of how an LLM works inevitably makes it progressively less likely that, as things get more complicated, they will work out correctly. And it can become something of a game to try to find cases where the LLM happens to do the right thing.

But how can one even tell? It’s a frustrating feature of modern times that someone like me gets sent many AI-generated documents every day that have the “statistical texture” of math papers, but that one at least expects have a very low probability of being meaningfully correct (and that were, one assumes, mostly made by people who didn’t particularly understand math themselves, but told their AIs to “make math” for them). So, yes, the LLM can in effect work in terms of human-level mathematical concepts, but with the same kinds of uncertainties and imprecisions that affect human thinking.

The Power and Challenge of Formalization

But what about turning those human-level constructs into something precise and formal—something more at an axiomatic level? And, yes, I’ve been involved in thinking about such things for several decades. There’s been quite a bit of excitement of late in the idea of autoformalization: take “human-level math”, and automatically turn it into something formal and axiomatic, that can for example be verified by a proof assistant system. At first it sounds like a good idea. But there’s a problem. Yes, the proof assistant might verify a proof. But was it a proof of what you thought it was a proof of? The weak link is being sure that the formalization of your human-level math is correctly capturing what you were trying to express.

I’ve had the experience quite a few times now: I try to autoformalize something, and an AI will tell me “I did it; look, the proof checks out!” But, actually, in some sense it cheated: instead of formalizing what I intended, it found a (sometimes very squirrely) way to interpret what I asked so that it could successfully prove it. It’s often very hard to tell, though, that this is what happened—not least because the formalized versions of things (say as expressed in popular proof assistant systems) tend to be very low level, very verbose and very hard for us humans to understand.

So what can one do? Well, I actually think there’s something very good one can do, that’s centrally based on the tower of ideas and technology that we’ve been developing for so long in the Wolfram Language. Our goal with the Wolfram Language has always been to create a language—or in effect a notation—that can represent in precise computational terms things we humans think about. It’s not like a programming language where one’s just representing raw constructs and operations in a computer; instead it’s based on representing actual things in the world, and actual concepts we think about. And also, it’s not just a way of telling a computer what to do; a bit like a vast generalization of mathematical notation, it’s a language that one can read, and think in.

In the past couple of years a powerful general workflow has emerged: tell an AI what you want, then have the AI write Wolfram Language to give a precise representation of what it thinks you mean. If it’s properly done, this representation should be something you as a human can understand—then check and modify as desired.

A New High-Level Language for Pure Mathematics

Over the past four decades we’ve steadily grown the Wolfram Language to cover a great many domains, including most of what one can think of as today’s “applicable mathematics”. And indeed what the Wolfram Language does has proved extremely useful in pure math research. But in the past—notwithstanding our efforts a bit more than a decade ago—we’ve never quite been able to justify building broad capabilities specifically for pure math research. But thanks in part to the general advance of the Wolfram Language platform, and in part to the changing economics of development associated with AI, we are in the middle of a large effort to extend the design of Wolfram Language to encompass the constructs of pure math research, from sheaves to Lie groups to Clifford algebras.

Our goal is to use our methods and long experience in computational language design to make pure math broadly computational—in a way that both humans and AIs can use. And, yes, we can then at long last expect to have papers where underneath every mathematical statement is a precise and unambiguous computational version, that one can systematically execute and process computationally.

But also we’ll have a real target for autoformalization—in effect a nexus of human mathematics, AI and computation. So let’s say we have a statement comprehensibly formalized in our pure-math-extended Wolfram Language. Often what we’ll in practice want to do with it is to compute a result from it. But sometimes—following the traditions of pure math—we’ll just want to abstractly prove it.

Automated Theorem Proving

So can we do that automatically? Well, there’s automated theorem proving, many types of which are, for example, built into the Wolfram Language. Sometimes automated theorem proving is in effect implicit (like, “Solve showed this polynomial has no real roots”). Sometimes (like with FindEquationalProof) it can be more explicit, with a precise—and easily verifiable—symbolic representation of all steps in a proof.

One might imagine this would be something broadly powerful for pure math. But at least for the past many decades it hasn’t been. A large part of the reason is that it’s been hard to break typical research-level pure math down to a sufficiently granular axiomatic level to make it amenable to such methods. But another part has been that—thanks to the phenomenon of computational irreducibility, and undecidability—proofs can be unreachably long.

And indeed I think it’s fair to say that in the whole history of automated theorem proving there’s actually only one example of something that can even plausibly be considered a real mathematical result that wasn’t already believed to be true, and that was found for the first time with automated theorem proving. It’s something I did in 2000—to find the minimal axiom system for Boolean algebra. But there was another problem here: the proof is very long, very low level, and very alien. And in the past 26 years it’s never been possible—with AI or otherwise—to find any human-level version of it. In effect, so far as we can tell, it’s a kind of prong in the metamathematical universe, unconnected to existing human-level mathematical concepts.

Of course, we can just use the result of the proof—as a kind of “black-box piece of mathematics”. But insofar as our objective in mathematics is to provide human-level mathematical narrative this represents a gap in the narrative.

But let’s say instead you take a proof that was already known in the mathematical literature, and you somehow formalize it—whether by AI or by direct human effort. If it’s a complicated proof, then this is certainly an achievement. And if you’re sure that you actually formalized the right thing, then it’s a good check that the proof really is correct. But one shouldn’t expect to get new understanding, or to surface new reusable components. And then there’s the question of actual mathematical content. Yes, a formal result has been established—but for example what underlying axiom system did it use (and, no, it’s probably in the end not something as familiar as set theory)?

(It’s worth commenting that formalized proofs can themsleves become objects of study. For example, with many proofs of a given theorem, one can start asking about the geometry and topology of proof space. And with proofs of many theorems one can start doing “empirical metamathematics” to understand their fundamental relations.)

Problem Solving in Math

As we’ll discuss, much of the most important progress in pure mathematics comes from the invention of new concepts and structures (often encapsulated in new definitions). But there’s still a vast amount to do operating with existing concepts and existing structures. Sometimes the goal is to “compute an answer”. Sometimes it’s to “find a proof”. Sometimes it’s to “find an example or counterexample”. And sometimes it’s to “figure out what’s true”. In all of these types of cases there’s plenty of powerful algorithmic computation that can be used (e.g. in Wolfram Language) to get results, often very directly and quickly.

AIs can certainly use such computation (say through Wolfram MCP). But how can modern AI directly contribute? One can imagine just straightforwardly asking it “chatbot style” to generate results. And often, yes, it will come up with a result. But how can one tell if it’s correct? Sometimes the only way is in effect to formalize the whole process of getting it, as we discussed above. But sometimes we can just take the result and do a computation on it to check if it’s correct. For example, if the AI claims to solve an integral, we can just differentiate its solution and see if we get back something equivalent to the integrand. Or if the AI comes up with a counterexample to something, then we can just do a computation to check whether it is in fact a counterexample.

We don’t know much about what’s “going on inside” when an LLM comes up with a result. But it’s probably not a bad model to think of it as a process of fitting together digested pieces from the millions of mathematical papers on which it’s been trained. In the end, though, what it does may actually amount to something that’s algorithmically quite simple. The AI has the advantage of great breadth. And in my experience it’s rather common to find that it’s come up with something that would be quite simple—or even obvious—if one just happened to look in the right place. But having found it, it can then be implemented as a piece of pure computation (e.g. in Wolfram Language), that doesn’t need any AI around.

When humans do math and solve problems an important part of the process tends to be the use of intuition. But what actually is intuition? At some level it seems to be a kind of “I know how this will go” procedural pattern matching. And—speaking of intuition—my intuition is that at least in its basic form this isn’t a capability that’s fundamentally out of reach for LLMs.

But one of the challenges is that even to solve a problem stated in terms of existing mathematical concepts it’s common to have to “break out of the system” and invent new concepts—a classic example being the need for complex numbers in finding even real solutions to cubic equations. At some level this is yet another manifestation of computational irreducibility, and the way it limits any given pocket of computational reducibility. But its effect is that it forces what might seem like “pure problem solving” to actually often involve the invention of new concepts and structures.

Open-Ended Math

So what about what one can think of as open-ended math? In ruliology it’s very common to just go out into the computational universe and “look for interesting things”. In math that’s traditionally not something that’s common. Of course, one can certainly in principle imagine, say, systematically enumerating possible theorems based on some set of axioms, and then asking: which of these are “interesting”? Operationally one might ask “Which of these theorems would one mention if one was writing a book or paper?”

Years ago I looked at the very simple case of possible theorems of Boolean algebra and found empirically that there was a criterion for which of them are typically given names in textbooks of logic: it’s basically ones that can’t be proved from theorems earlier in a lexicographically ordered list of possible theorems—or, in some sense, the simplest theorems that “give new information”.

In more complicated cases one might imagine that an AI trained on millions of examples of theorems that people chose to publish in the literature of mathematics might be able to channel human criteria for interestingness. But in my experience this doesn’t really work. And I think the reason is that in all but the simplest cases interestingness is not something that can be determined theorem by theorem: rather, it requires one to map out the whole historical development of a particular area of mathematics. And, yes, then one can ask what aspects of the arc of that development are somehow inevitable, and what are just historical accidents.

I’ve asked a somewhat similar question recently for biological evolution. And in that case I’ve discovered that there do seem to be some (fundamentally computational) general principles. For example, adaptive evolution according to a computationally limited fitness function quite generally seems to lead to the presence of modular structures (analogous to organelles, organs, etc.) Those structures don’t “come with an explanation of what they are”. But they do in effect represent “raw pockets of computational reducibility”.

In the development of pure mathematics one thing that’s clearly critical is the identification of new structures and concepts. Historically such structures and concepts have always been introduced by humans, and have in effect “come with explanations”. But could one for example imagine new structures and concepts that are automatically found, say by an AI? And could one then build new mathematics with these?

New Mathematical Concepts

Those two questions turn out to be somewhat separate. At least at a simple level, it’s fairly straightforward (say by looking at autoencoders) to identify in neural nets particular patterns of activation that one can think of as summarizing pockets of computational reducibility, or in effect “representing new concepts”. But insofar as mathematics is about defining mathematical narratives for us, this isn’t immediately helpful.

Yes, the AI can in effect identify zillions of new concepts. But our human minds can only deal with limited numbers of them. It’s very much like with natural language. We can imagine an AI looking at patterns of human discourse and figuring out zillions of new words that could be introduced to summarize them. But we humans seem to only be able to know about a limited number of distinct words (typically a few tens of thousands).

So what about in mathematics? An AI could in principle come up with a zillion definitions of new mathematical concepts. But something built from them will inevitably seem about as alien as something built “ruliologically” from raw axiomatic-level mathematics. The building blocks might be more “human level”. But they’re not familiar to us humans, and so we can’t readily think in terms of them.

There are of course new words introduced into human languages, and new concepts introduced into mathematics. But it’s a gradual process, that in practice tends to involve a certain “societal consensus”. (And indeed in developing the design of the Wolfram Language over the past four decades, I’ve quite explicitly limited the rate at which new concepts are introduced, to make sure to let people keep up.)

The Goals of Math

But, OK, so there are lots of possible new concepts and new structures—all corresponding to pockets of computational reducibility—that are “out there in the metamathematical universe”. But in developing mathematics suitable for us humans, we can only ever pursue a tiny fraction of them.

In other words, it’s inevitable that the development of mathematics has to make choices about where to go. It can’t meaningfully “pursue all possibilities”. It has to have definite goals.

But where do goals for pure mathematics come from? In technology development, goals tend to be set quite directly by what people find useful. In natural science, by what one observes and has tools to interpret in the natural world. But in pure mathematics, there don’t at first seem to be such obvious “external” ways to define goals—making the math that’s been done seem more determined by the internal choices of the mathematical community and the leaders within it.

But actually if one looks at the historical arc of human mathematics there are at least definite trends to be seen. The journey of mathematics began with basic formalizations of things like numbers and space that somehow reflect everyday human perception of the world. And although a century or so ago mathematics seemed intent on trying to “generalize to all possibilities”, most of the strongest trajectories of mathematical development over the past century have actually stayed much closer to what humans can intuitively understand and visualize.

There is still plenty of freedom in where to go in math. But to be meaningful to us humans we somehow have to have a certain familiarity with whatever choices are made. Yes, an AI could, say, pick a direction at random, perhaps informed by the observed statistics of existing human choices. But there’s then no reason for humans to care about it; it’s not something that relates to our actual experience of mathematics.

At a more practical level, some part of the doing of math is about figuring out how to achieve goals that have been set—and this is something we can imagine AI doing. But what’s ultimately more important is the setting of the goals in the first place. And almost by definition, these must come from “outside the system”. Or, in particular, from us.

It’s a notable observation that the most significant reported successes for AI in math so far tend to come from some of the most skilled human mathematicians. And in some sense we should not be surprised—because it’s the setting of goals (or, in effect, knowing the questions to ask) that is the most important and unique part. Indeed, it’s often the case that once one really knows the question to ask, one’s already done the lion’s share of the work to answer it.

But just where do goals for pure mathematics in practice come from? There tend to be two basic sources. One is what amounts to problem solving; the other to what amounts to creative building. Problem solving is in a sense more cut and dried, measurable, and amenable to AI. Take a problem that’s been defined in the past, and worked on for a while. Then the goal is to solve it. And if one succeeds, there is a clear way to say one has done something (“an Erdős problem was solved!”, etc.), even if it sometimes seems more like a sporting achievement, with at best muddy ultimate intellectual content.

The Aesthetics of Math

But then there’s what one might think of as “creative building”: coming up with new structures and new concepts for mathematics. At some ultimate metamathematical level, these structures and concepts must already “be there”, as pockets of computational reducibility. So then one can think of the goal as being to identify particular ones, and bring them into the “lexicon” with which one talks about mathematics.

But of all the possibilities, which ones does one want? One could imagine various criteria. But mostly they come down to wanting to create abstractions that best unify, summarize and simplify disparate things. And put this way, one might think this would be an objective that could be automated, say by AI. But there’s a crucial wrinkle. One might imagine one could define something as being “simpler” if it’s shorter to state. But how long something is to state depends on the language it’s stated in. So again there’s nothing immediately absolute; it’s something that inevitably depends on a whole history of interconnected choices.

Often people describe the choices as aesthetic ones. What makes a structure more elegant, more perfect, etc.? Ultimately these are things grounded in human preference and human experience, and extended by what one can describe as human initiative. But does one actually need a human to make the choices? Or could one instead just have some kind of generative AI trained on existing human choices? The issue is that to usefully create new building blocks for mathematics one needs more than just to invent new concepts and structures; they also have to be knitted into the fabric of mathematical culture.

It’s like introducing a new word in a human language. For the word to be useful, its meaning has to be widely enough known for it to routinely stand for a particular concept. And spreading this meaning is inevitably at some level a social process. And so it is for new concepts and structures in mathematics. Insofar as mathematics is about developing mathematical narratives for human minds, concepts and structures have to be shared if they’re going to usefully be used as building blocks. It’s a bit like the issue in a typical automated proof. If one can say that a particular step is from “X’s theorem”, that provides a kind of conceptual anchor. But if every step is in some sense bespoke, one ends up with something that in effect seems alien.

So where does this leave AI in math? It’s similar to AI in many other places. Properly define a goal and AI can be very helpful. Without a goal, it doesn’t know where to go. Exploring possibilities in a ruliological way—and in effect doing experimental mathematics—can bring one lots of surprising new results. But in a sense they’re “born alien”, not immediately connected to what we normally think of as mathematics.

Why Do Pure Math Anyway?

But, OK, so AI can help in extending the enterprise that we can call pure mathematics. But it’s still inevitably going to take human effort and human initiative. So why do it?

A common argument is that it’ll eventually end up being useful for something “practical”. Often the notion is that pure math somehow manages to lay down certain “beacons” that science and technology will eventually reach. But I think the true picture is rather the reverse. The math that we create provides ways of thinking. And it’s when we have these ways of thinking that we end up creating science and technology that makes use of them. It’s not that there’s a convergence of what’s been done in math with what’s been discovered in science. It’s that the math we know guides what we end up looking for in science.

Sometimes there’s a notion that the things we discover in science are somehow unique and inevitable. But thinking in terms of the ruliad we realize that there are actually an infinite number of different slices of computational reducibility, each in effect defining their own take on science. And over and over again historically what seems to happen is that first a conceptual framework is developed—often from math—and only then does it become clear that there’s an associated regularity one can look at, that shows the way to some piece of science. In other words, it’s not a mysterious convergence between pure math and science; it’s that the science is developed because the pure math exists.

In a sense this makes the enterprise of pure mathematics seem less speculative. It’s not that we’re doing a piece of pure math in the hope that one day an application for it will be found. Rather, it’s that doing the piece of pure math develops a way of thinking (and in effect reveals a pocket of computational reducibility) that gives one the chance to come up with the application.

Could a piece of pure math just turn out to be “fundamentally useless”? In some sense the answer is no. Because any time there’s a piece of pure math that’s successfully been done, it must correspond to some pocket of computational reducibility. And any pocket of computational reducibility will be associated with regularities which are inevitably the “raw material” for some form of science and some form of technology.

The catch, though, is that those might be very alien forms of science and technology, far from anything we’ve so far considered. It’s not that the math can’t give us science and technology; it’s just that the arc of scientific and technological development may not yet have reached the point where we care about what it gives.

Back in antiquity the great contribution of math to our way of thinking was the idea of deductive reasoning. In more recent times came a multitude of ideas around continuity, real numbers, etc. Then, a bit more than a century ago came the idea of abstract functions and then the idea of computation. Once reached, this idea might seem almost obvious. But historically it took a surprisingly tall tower of abstract mathematical thought to reach it. And, yes, the general notion of computation is of great practical importance. But—as I have explored over the course of many decades—it’s also foundational for the general way we think about things.

One of the notable features of pure math is that it involves a certain style of structured thinking that has long been seen as important in education. In some ways I think that computation—and particularly computational language, with its way of describing the world—provides a powerful alternative for learning structured thinking. But the foundational study of computation is much younger than mathematics, and much more immediately subject to the limitations of computational irreducibility. And the result is that modern pure mathematics has developed a much taller tower of structured thinking and formalization—and indeed, its tower is taller than any other field.

It’s a big investment to learn that tower. And few people make it. Those that do have a variety of motivations. For some, it’s a matter of challenge. For others, more a matter of the “scenic view from the tower”. In some ways the automation provided by AI makes the challenge less appealing—though, as in something like chess, there can still be fulfillment in pure human achievement. But when it comes to the scenic view, the fulfillment is in the human aesthetic experience, which is quite unrelated to automation and AI.

Half a millennium ago, there were mathematical challenges set up as spectator sports. Today the challenge aspect of mathematics is not as successful at garnering public interest. Indeed, nothing about the process of doing mathematics is typically exposed to the public at all, except to a tiny extent through things like personality-based movies.

(And, yes, one might wonder if there’s any way the doing of mathematics could provide any form of public entertainment. I can report that I myself have done a fair number of successful live mathematical computer experiments. And during the pandemic I livestreamed a great many hours of highly technical working sessions about mathematical physics—which garnered a surprising number of views.)

So what about math as an essentially aesthetic activity? Yes, it’s fulfilling and enriching for the person in the middle of doing it. But what does society as a whole get out of it? Much as some tiny fraction of art is public art, so one might wonder if there can be public math. Occasionally there will be public visual or sculptural math, or perhaps architecture or music that embodies math. But realistically it’s not math at the level of the top of the tower of pure math.

One thing to understand about the upper reaches of pure mathematics is that to a surprising extent it’s been passed down as an oral tradition and a chain of human connections. Perhaps this is partly a consequence of the structure of academia and academic publishing. But the fact remains that to maintain the flame of pure mathematics seems to require a certain community of individuals continually and actively pursuing it, and talking about how it’s done. In other words, even if their personal motivations are quite internal, their collective effort has the long-term external value of allowing the progress of pure mathematics to continue.

One might ask if at some point pure mathematics will somehow be “finished”. And to this we can definitively answer that it will not. Computational irreducibility ensures that there will always be an infinite sequence of mathematical facts—and surprises—to be discovered. And there will also be an infinite number of pockets of computational reducibility, representing new mathematical concepts and structures.

Pure mathematics stands as the single largest intellectual edifice built by our civilization. And out of it have come some of the most impactful ideas in human intellectual history—like formalization, abstraction and computation. In modern times, there are some accelerators for pure mathematics. Practical computation has been one. AI is another. Experimental mathematics in the style of ruliology is yet another, still fairly undeveloped. And I’d like to think that the language for pure mathematics that we’re building will—particularly when combined with AI—be still another.

The work of pure mathematics is long and hard (yes, a consequence of computational irreducibility). But over the course of millennia its importance to the core intellectual development of our civilization has been very great. And there is no doubt that it has more to give. And that even after the issues of today are long forgotten pure mathematics will continue to shine as a beacon of human achievement. So, yes, there’s every reason to expect a bright future—now with some additional help from AI—for that most rarefied of human pursuits: research in pure mathematics.

Note

Some of the ideas in this piece I discussed over the course of more than three decades with my wife Elise Cawley, who sadly died shortly before the piece was written, and who would no doubt have had many valuable and incisive insights about it, now lost forever.

Posted in: Artificial Intelligence, Mathematics