Mathematics as collective rewiring

Preamble

My brother-in-law (BIL from now on) is a software developer, a job that has apparently been rebranded as “AI engineer”. Last August he read about OpenAI’s Ten Advances and reached out on WhatsApp to ask, bluntly: “hey, is your profession being automated? Because that’s what it looks like from outside.” I replied with something about Thurston’s view that the product of mathematics is clarity and understanding, not theorems by themselves. To which BIL replied:

Yes, but the reality is that mathematics is funded for its potential applications, not for the sake of understanding.

I froze and left him on read. I felt I knew what to answer, but the words would not come. Thurston’s writings, and many similar texts circulating these days, are written from a perspective internal to mathematics and do not address why human understanding of mathematics should matter to society, beyond being a means towards “applications”. And from many conversations I have had since then, I have realised that this is not at all clear to many mathematicians either.

This text is an attempt to find the words to explain this, first to myself and then to others. But it is not (only) an answer to BIL: my intended audience is mathematicians troubled by this doubt.

What is an application of mathematics?

First, a clarification: “mathematics being funded” is not a decision taken by a king with one reason in mind. It is the product of many decisions, by many people, across countries and centuries; nobody can say what the reason is. So the question should rather be: what are the benefits of mathematics to society that paved the way to this state of affairs? The most common answer is “its applications”. But what does that mean exactly?

A common interpretation runs as follows. The product of mathematicians’ work is a giant library of results, each one calling others, like a computer program, all the way down to the axioms. The library exists to be at the disposal of scientists and engineers who need a specific result: a PDE to predict the weather, a statistical estimate. On this view most of the library will never be used, since most results will never lie on a path from the axioms to an application. But there are surprises: number theory is used in cryptography—mathematicians love to say this. We cannot know which results will be used in advance, so we do the whole thing and time will tell.

On this premise, if AI gets good enough, scientists and engineers can ask it directly for the mathematics they need, and ask it to apply that mathematics too: human understanding might become redundant. The corollary for mathematicians and mathematics teachers is straightforward: we thank you for your contribution to the scientific project and wish you success in your future endeavours. P.S. Buy a suit, nobody will hire you dressed like that.

The supply of results for such “sharp” applications is certainly one important role of mathematics. Is it the main one? On closer inspection, the picture looks incomplete at best.

• Why has mathematics existed, and been valued, for millennia before its alliance with modern science and technology in the seventeenth century?
• If sharp applications were the whole story, the highest form of success for a piece of mathematics would be ending up on the path to one. But when we celebrate the achievements of the past, we rarely do so by listing the applied leaves that sprang from them. Mathematicians tend to be grounded people. If this were the whole of mathematics’ importance, it would have emerged as the main criterion of quality, and our mathematical taste would reflect it.

The unnoticed discovery of the wheel

A concrete example first. When people tell you how much they liked a concert, or how full the room was, they may hold their hand up high or down low. They know this will trigger in you a cognitive mechanism that they have and that everyone has: call it the “intuitive real line”. I do not mean any formal mathematical object. I mean the thing in your head that makes it self-evident that any quantity can be represented as a length. Billions of people share it today, including people with little or no education, and we count on others having it.

It is worth pausing here, because its transparency hides how non-trivial it is. The intuitive real line is a tiny step away from the mechanisms we need to read the graph of a function or to use Cartesian coordinates. Those are widespread too, but not on a comparable scale, and high-school teachers fight hard to install them.

Now I would like you to imagine a world in which people do not have the intuitive real line. Did Lucy have it? Euclid probably did, but did Alexander the Great? A farmer in Descartes’ time?

Cognitive scientists have studied a simpler version of this question, the “number line”: the habit of mapping natural numbers onto positions along a segment. In Western children, the mapping appears before school age. But Núñez, Cooperrider and Wassmann studied the Yupno of Papua New Guinea [2]. In short, the study shows that unschooled adults—despite having number concepts and a well-established counting system—do not resonate with the number line at all.

I am venturing into a field where I have no competence. But this strongly suggests that the intuitive real line is not an innate, built-in mechanism of our brains. It is a legacy, transmitted through culture. And one thing is certain: nobody “discovered” it. It resonated many times in the minds of people who were exploring new ways of thinking, probably with far more sophisticated goals in mind. Through a painstakingly slow process, it took shape and spread. Those people were the mathematicians of their time.

Mathematics as a collective rewiring mechanism

Let’s agree on words

It is not easy to talk about the inner and social experience of doing mathematics. Depending on the meaning we project onto words like “clarity”, “understanding”, “idea” or “intuition”, the same sentence can carry a deep message or read as an empty tautology. This is, I think, the main difficulty in spreading Thurston’s message. So I will take a more down-to-earth approach and speak about brain rewiring. I am no neuroscientist and this is not a research paper: I am choosing clarity over correctness.

Neuroplasticity rewires your brain whenever you internalise something (a concept, a complex task, a way of organising input) so deeply that it becomes intuitive. The real phenomenon is global and complex, but as a useful simplification I will speak of wiring patterns attached to particular functions. You have a wiring pattern for riding a bike, one for “boat”, one that can play “Jolene” in your head, and one for the intuitive real line. We build our most important ones as babies.

Since humans are social animals, it matters enormously that some of these patterns be shared. I like to use the word resonance for this, in a sense that should resonate with mathematicians: resonance is what happens in a group of people when a common input triggers a similar cognitive response in all of them. For instance, “fire!”. If the response can be observed, we can take it as evidence that they have all built a wiring pattern with the same function.

Resonance on shared patterns is fundamental to cooperation. We constantly count on certain patterns existing in others, and we try to make them resonate. Consistent resonance in a large group of strangers is, basically, culture. The creation and spreading of these shared patterns must start somewhere.

We mathematicians build new wiring patterns all the time. They let us regard certain things as evident, feel mathematical objects as real, and manipulate them in our minds. Bessis’ book [1] proposes this as the essence of the experience of doing mathematics.

Let me ask the mathematicians among my readers a question. (Sorry, BIL, I will do this often, but I am sure you can still follow.) What happens in your head when you read this?

Let X be the vector space of all real-valued functions on [0, 1] with finitely many discontinuity points.

My bet is that you immediately take hold of this object. You feel it; it is no less real than “boat”. And you probably do not need to check formally that it is a vector space: you look at it and it is self-evident. These are signs that you do not simply “know” what vector spaces and continuous functions are: your brain has rewired to manipulate them. The experience is vivid to us and hard to describe to others. But we can write a proof as a certificate that it happened. And this ability serves as hard evidence that we resonate on shared “vector spaces and continuity” patterns. Not so different from “fire!”.

This is the meaning I project onto Thurston’s clarity and understanding.

The social function of mathematics

Time to state the main claim on the role of mathematics in society:

Some of these wiring patterns spill out of the community, in concentric circles, allowing large groups of people to resonate, and so to think and cooperate, in ways that were not available before. In other words, mathematics is one of the forces that make culture—scientific culture and general culture—evolve.

It is important to stress that this is an emergent phenomenon. Individually, mathematicians chase difficult theorems and problems. But our brains cannot craft and hold a complex proof in raw form. They need to rewire deeply, creating sharp, specialised patterns that treat large components of the proof (new objects, arguments) intuitively, as single units. Other mathematicians then acquire some of those patterns and reuse them in other proofs, and so the patterns start to spread. That process gradually simplifies the patterns, refines them, and merges them. At a larger scale, simpler and much wider patterns start recurring, shared by many more people. Those are the ones that usually spill out. They can rarely be traced to individuals. Only a handful of mathematicians of all time have had a visible impact on the global direction of the rewiring process. The others shape it collectively, and no single one of them is essential. This idea is uncomfortable for mathematicians, to the point of being controversial.

The intuitive real line is an example of a wiring pattern that emerged from mathematical thinking and now pervades general culture. Hugo Duminil-Copin discussed others of the same kind: zero, the equals sign, negative numbers, infinity. We underestimate the value of these jewels because of one of the strangest powers of our brains: once they have rewired, there is no way back—just as you cannot unlearn how to walk, or how to read.

An intermediate-scale example would be “linear algebra”—not the collection of definitions and results, but the patterns that all mathematicians share, which make those definitions natural and those results evident. They have left traces in millions of people, directly through undergraduate courses and indirectly through all the technology and concepts that rely on them. They change how those people see the world and cooperate, whether or not they still remember how to solve a linear system. Their impact should not be reduced to a list of “technical applications of linear algebra”.

The list could go on indefinitely. Some patterns resonate strongly in every mathematician and hardly at all outside mathematics. Many more resonate only in one specific community of mathematicians. For example (the details do not matter):

In smooth dynamics, exponential contraction guarantees that continuous objects are automatically Hölder continuous.

I am quoting this from memory from a Facebook post by David Fisher, who was asking his friends for concrete results illustrating the principle to use in a class. It shows two things. First, wiring patterns live in our heads and do not always correspond to anything in our external output of definitions, questions, theorems and proofs. There is nothing well defined in that sentence, yet David knew his friends would resonate with it, and they did. Second, his goal was to rewire his students; choosing which piece of the external output to present in class was subordinate to that. I find this a beautiful lesson in what mathematicians are really after.

What sets mathematics apart

I am not claiming that mathematics is the only mechanism of collective rewiring. Every collective human activity needs to install shared wiring patterns. But that does not make them all rewiring games: most often the rewiring is a means towards a concrete objective, not the objective itself. Mathematics is one of the activities in which exploring new wiring patterns is the goal.

It is not the only one either. Many of the arts, music, philosophy and parts of the humanities have a very similar social function. Many mathematicians feel that mathematics has something in common with them, but this is usually dismissed as a romantic view of their field. Perhaps this is what they really have in common?

What singles mathematics out, however, is the mechanism we use to make sure that the patterns we create can be transmitted, and to verify that they resonate in others. That mechanism is proof.

Compare this with music. People who practise both tend to agree that the neuroplasticity involved feels very similar. Musicians use conscious effort and practice to build patterns which, once installed, let them play and manipulate music in their heads instinctively; these patterns are shared by a community, they shape styles, and they spill out to become cultural glue. But music fine-tunes and verifies its patterns concretely: musicians play together, crowds dance or clap in rhythm. The price is that what musicians build stays anchored to the physical world. I honestly do not know how scholars in philosophy or the humanities make sure that the patterns they build are transmitted and resonate in others and in future generations. But PhDs lasting more than a decade suggest that it is an extraordinarily difficult exercise.

The Greeks conceived of mathematics as an essential component of philosophy. Euclid invented modern proof as a solution to the philosophers’ struggle to resonate. It provides something analogous to physical reality, against which what happens inside our heads can be tested. The fact that we can all consistently produce correct proofs of certain statements is a certificate that the resonance between our wiring patterns is strong. Thanks to proof, mathematics enjoys the best of both worlds: we can keep building new patterns freely out of old ones, with no anchor in the physical world, and still make sure that what we build can be transmitted and will resonate strongly in others. This combination lets people imagine together things that are both genuinely new and very precise, and it is what makes mathematical thinking so well suited to modern science and technology.

We regularly think something is evident and then discover, when trying to write the proof, that our intuition was wrong. The pattern we were using was imperfect, so we fine-tune it. I could continue my vector space example with:

Let Y be the subspace of X of all functions with an even number of discontinuities.

Yes, I know, it is not a subspace. But some readers will have overlooked that and found it evident that it was. Confronted with the task of writing the proof, they discover that their mental wiring pattern was not sharp enough, and fine-tune it. This happens to every mathematician all the time, and it is essential to how we operate. As students, we fine-tune our patterns through exercises. As researchers, we do it through the dumb questions in informal discussions, the emails to experts, and the less novel parts of our papers that rest on well-established patterns. Those parts are not useless: they refine, solidify and spread the patterns in circulation, and keep the resonance strong.

A Platonist would say that the fine-tuning process is adjusting an imperfect intuition to the truth. Maybe. But what is falsifiable is that you are realigning your intuition with that of others, using a common reference. So perhaps proofs are not the objective: they are an auxiliary, intermediate step in a human-to-human process. And for a pattern to spill out of mathematics into general culture, it must become so clear and so mature that testing it against proof is no longer necessary.

When we struggle to prove, or to understand in depth, a difficult theorem, we build and fine-tune certain patterns. Full resonance of these patterns shows up in skills such as being able, with some struggle, to reconstruct the proof years later, to export it to other settings, to find alternative routes, and to answer many surrounding questions. Reaching it is a slow process, though much faster when we have in-person access to someone who already has those patterns: our brains seem far better at copying another person’s wiring when that person is in the room. We experience the most intense resonance with close collaborators, with whom we have been rewiring in the same way: after a few words of gibberish and a potato drawn on the blackboard, everyone can go home and write a long proof that anyone else would struggle to follow.

It is worth rereading (or reading, if you haven’t!) Thurston’s famous MathOverflow post with the rewiring process in mind. I have read it several times in the last few months, and I always find new nuances in his words. If the meaning you project onto it has changed a little, then this essay has not been useless.

The intuitive real line of tomorrow?

It may seem hard to imagine that today’s mathematics could produce anything comparable to the intuitive real line for tomorrow’s world. What could it be? No individual can grasp the whole picture, and my own perspective is far too narrow for me to answer. But let me bet on a horse, to show that the idea is not unrealistic: the probabilist’s rewiring.

Once again, I do not mean probability theory as a set of results. I mean the way of thinking that emerges from it. On paper, probability theory rests on measure theory; the two are formally indistinguishable. But the wiring patterns needed to think in them are very different. Probabilists have found a sharp rewiring that lets them see and manipulate random objects in their minds, and makes natural a whole range of arguments that look like magic tricks when viewed through the lens of measure theory. The ability to think of random objects might be compared to the millennia-old human ability to think intuitively about plane figures, except that it is very recent for our species.

It is still spreading within mathematics. The status of probability theory has changed rapidly, and many other areas have started to absorb probabilistic aspects. When that happens, many mathematicians in the area at first keep translating everything into measure-theoretic or counting arguments, and the probabilistic rewiring takes over gradually. In my own field, group theory, this has been perceptible over the decade I have been working in it. (I do not resonate with the probabilist’s rewiring nearly as much as I would like. But I can tell, from people I have crossed paths with, how far it can go.)

This process is still ongoing within mathematics. But one day, the fundamental ability to think of random objects—in some simplified form—might spread well beyond it.

An answer to BIL

Now I can try to answer my brother-in-law. First of all, yes, AI is becoming more efficient than humans at many mathematical tasks—not all of them yet, but the process does not seem to be slowing down. And yes, in modern times mathematics is largely funded for its potential applications. But application does not only mean the use of a specific theory for something concrete. Those sharp applications exist and matter, but above all, mathematics is a slow and constant force acting on culture, and that is what made many of those applications imaginable in the first place.

It is remarkable that number theory has creative applications in cryptography. But saying that number theory is useful because it is used in cryptography is like saying that thermodynamics is useful because your car has air conditioning. The slow transformation of collective thinking that begins in mathematics is why money (first physical and then immaterial), the negative numbers in your bank account, computers and the internet could be imagined at all, and on a large scale. Then they were built and combined into your online banking app, which—yes—does use number theory through cryptography. To my colleagues: please think carefully about this before using this example to justify your profession.

So, can mathematics be automated? The point, BIL, is that AI can only produce external output: definitions, questions, theorems and proofs. So far it is much better at the last two; some of us are clutching at that straw, but there is no reason to believe it will hold. Some of its output may be directly applicable to the real world. But the rewiring process happens, by definition, in humans. At the moment AI-generated output appears, no brain has been rewired. Nobody has a new way of thinking to transmit to others. Without further human effort, AI output is useless, no matter how famous or how old the problem was. The rewiring process can never be automated.

It could, in theory, be shut down by cutting all funding, on the grounds that all thinking can now be delegated to AI, that we therefore no longer need a motor for the evolution of collective human thinking, and that states could save a few peanuts. I do not think that will happen. Even in the most naive version of the AI-optimist paradise, where machines imagine and build great things on our behalf, it remains valuable for humans to keep their own collective rewiring alive and to go on imagining things themselves. And people will realise this.

But the most concrete risk is that the rewiring process could slow down, or even atrophy. Let me explain this with an example. Take the probabilist’s rewiring discussed earlier, and imagine a world in which measure theory predates AI but probability theory was developed entirely in the age of AI, with humans “understanding and digesting” proofs produced by machines. The problem is that these words are vague. You can “understand” many proofs in probability by thinking in terms of measure theory; it is only when you are confronted with the task of finding new proofs that you see the difference. As a result, many of the humans doing the digesting might never rewire as probabilists did. They might go on thinking in measure-theoretic terms and attribute the remarkable tricks to “AI superiority”. We would have all the theorems of probability theory much sooner, at a serious cost to human cultural evolution.

Finding a way to avoid this is, dear BIL, the challenge we mathematicians are facing now.

Pleas to other mathematicians

I want to conclude with a few pleas to other mathematicians. I really do not think of mathematicians as split in two opposite camps: the two categories below should and do overlap largely.

If you are writing LLM-assisted papers

In mathematics, “importance” is largely recursive: a result is important because our predecessors considered it so, or because it leads to something else that is. The notion is anchored in a past without LLMs, and we cannot simply import it without rethinking it.

I am sure you have solid reasons for letting LLMs contribute non-trivially to your papers, and that you judge the benefits to outweigh the costs and risks, which many agree are non-zero: fuzzier understanding, overproduction of papers, and so on. But that judgement is invisible to everyone else. So please, and I am begging here, make your cost-benefit analysis explicit in concrete cases: an appendix, a note on your webpage, anything. It can greatly help the debate.

If the main benefit is “it is an important result”, please try to explain, in a non-circular way, how the result itself can benefit others—not merely as a proxy for the understanding that produced it, or as a way of fulfilling the moral duty to explore the mathematical world faster. In many cases, I am sure, this can be done.

(Personal note: I have never used LLMs to produce results, but I am considering some cautious tests, and I have committed to doing this exercise if I do.)

If you are worried and wonder what to do

Then here is my plea: stop crying at night, throw away the whiskey bottle, and do something now.

“Yes, but what can I do? I have no power.” Codes of conduct like ours are not enforced from above, least of all in communities with no chain of command. They emerge in groups of people who share strong goals and values, after a great deal of argument about how best to serve them. And if the goals and values are truly shared, this always happens. So here are some places to start.

• Help clarify the goals and values, and widen the group of people who share them. Talk to your colleagues, write posts like this one, and put serious time into refining your opinion. Many of us already are doing this.
• Start thinking now about the paper inflation problem. The number of papers on the arXiv is already exploding; your papers this semester might be worth about as much as an Argentine austral in 1989. Soon nobody will read them, we will not know whether even their authors understood them, and we will not be able to use them in hiring. I hear of committees planning to ignore everything that appeared after the summer of 2026. That may work for a year, but what then? Evaluation is not a game whose rules we can rewrite at will: it is an estimate of a person’s potential to contribute something good, and rules not aligned with that are useless. Sit down with colleagues and work on concrete solutions—new output formats, new forms of peer review, new hiring procedures—with a start-up mindset; then advertise them. I intend to propose this at my institute. Pretend that if your start-up will be bought, you will become rich. You won’t, and the odds that any proposal, yours or ours, is implemented at scale are ridiculously low. But if everyone tries, something will emerge.
• Change the narrative about mathematics in outreach to students. Some already in the system may take fright and leave, but the bigger problem is who will enrol after years in which we deemed that a good way of advertising our activity to the general public was to put a million-dollar reward on some problems, now all over the news. Nobody in their right mind. Start presenting mathematics instead as a collective motor of cultural evolution, and explain that human mathematics has become an outpost of resistance against the total atrophy of collective human thinking. That argument will resonate strongly with bright young people.

Post-scriptum

For the record, BIL read this. He got it despite the math references, and we had some very interesting conversations. If your BIL asks you the same questions, you can try to send him this text.