The end of an era in mathematical research

Alonso Castillo-Ramirez, Professor and Researcher at Universidad de Guadalajara

For more than two thousand years, research in pure mathematics has retained basically the same structure. Euclid began with definitions and axioms, and from them he proved theorems through logical arguments. Essentially, we still do the same today. Of course, this does not mean that mathematical research has been static since Euclid: we now have more efficient notation, thousands of specialized journals, computers, databases, and symbolic computation software. Yet the core of the activity had remained almost untouched: a person would think for days, months, or years until they found a new idea. They would then write a proof and try to convince other mathematicians that it was correct.

Pressing a Button

New artificial intelligence models such as GPT‑5.6 Sol and Claude Fable 5 are no longer mere calculators or search engines. They can work on research problems, propose definitions, find examples and counterexamples, connect ideas from different fields, and construct original proofs. Now it is becoming possible to enter a problem, press a button, and receive a new proof that would have taken a researcher months to find.

This does not happen every time. AI systems still make mistakes, misunderstand some questions, and occasionally produce arguments that appear convincing but contain errors. Their answers must be checked carefully. Nevertheless, it is becoming increasingly common for AI-generated proofs to be correct, and this is set to change the rules of mathematical research completely.

We have all read news reports about recent AI models solving important open problems. But this revolution is not confined to headlines or to the most powerful systems operating far beyond the reach of ordinary researchers. This is something that any of us can experience firsthand simply by interacting with recent models such as GPT‑5.6 Sol or Claude Fable 5. In their combination of speed, breadth of knowledge, and ability to explore many strategies at once, these systems are beginning to surpass what any human being can do. A mathematician may still notice something the machine missed, correct one of its mistakes, or solve a particular problem that the model could not. But no person can compete with an artificial intelligence that draws on an enormous amount of information, and explores several paths simultaneously.

The End of Handcrafted Mathematics

Until now, an important part of the value of a piece of research came from the effort required to produce it. A theorem could be the result of years of work, countless failed attempts, and long conversations with colleagues. Finding the proof was a craft.

That kind of research will probably disappear as the dominant way of producing mathematics. Solving a problem without AI assistance will be like performing by hand a calculation that a computer can complete in seconds. People will still be able to do it for pleasure, training, or educational purposes, but it will no longer be the most efficient way to work at the frontier of knowledge.

These changes may be painful for may of us. Our professional identity is closely tied to our ability to solve problems. We have devoted much of our lives to developing intuition, learning techniques, and finding proofs. Now a tool has emerged that can do in minutes what would have taken us months.

But the end of handcrafted mathematics does not mean the end of mathematics. When proofs become abundant, understanding will become scarce. Someone will have to decide which questions matter, distinguish a deep idea from a superficial observation, verify that the arguments are correct, and explain why a result deserves our attention.

The role of the mathematician will change. It will no longer consist primarily of producing every step of a proof, but of reviewing, discerning, digesting, and communicating ideas generated by AI. It will also be necessary to organize thousands of results, determine which of them form a coherent theory, and translate complex proofs into concepts that human beings can understand.

In other words, AI will be able to produce mathematics much faster than we can absorb it. The bottleneck will no longer be finding new results, but understanding them.

What Chess Can Teach Us

Something similar has already happened in the world of chess. For centuries, grandmasters were the ultimate authorities over the chessboard. They discovered new strategies, prepared openings, and found combinations hidden in positions that other players could not understand. Their prestige stemmed, to a large extent, from seeing things that no other human being could see.

Today, no human plays better than the most advanced computers. And yet chess did not die. On the contrary, its community has flourished. Millions of people play online, study games, follow tournaments, and enjoy the explanations of masters and commentators. Computer programs discover extraordinary moves, but we still need a human being to explain why those moves are extraordinary.

No one stops enjoying a game because a computer could have played it better. What interests us is understanding the ideas, experiencing the tension of the position, and sharing the game with others.

Something similar may happen with mathematics. Artificial intelligence may discover most theorems, but human beings will still be able to enjoy the moment when we understand a proof, recognize an unexpected connection, or find an elegant way to explain a theory. The source of the ideas will change. Our capacity to marvel at them need not disappear.

Teaching Will Remain Human

Artificial intelligence will also transform teaching. Every student will be able to have a tutor available at any time, capable of offering explanations adapted to their level, generating examples, and answering questions without ever becoming tired.

Even so, teaching mathematics will remain a primarily human activity. We learn not only because we receive information, but because we belong to a community. We are motivated by a teacher’s enthusiasm, the recognition of our classmates, and the satisfaction of explaining something to another person. We often persevere with a difficult problem because someone believes that we can solve it.

A good teacher does more than transmit definitions and theorems. A good teacher also communicates curiosity, patience, mathematical taste, and a way of approaching the unknown. They help students tolerate frustration, formulate better questions, and discover that they are capable of understanding ideas that initially seemed inaccessible. AI will be an extraordinary tool in the classroom. But human connection will remain what gives learning its meaning.

A new industrial revolution

The industrial revolution mechanized physical production. Machines could manufacture goods faster, and on a scale that no individual artisan could match. The AI revolution is doing something similar with intellectual production. This comparison also reveals why the present transformation can feel so unsettling. Industrialization did not merely provide artisans with better tools; it challenged the economic value and social status of their skills. In the same way, AI does not simply help mathematicians calculate more quickly. It reaches into the activity many of us regard as our deepest contribution.

However, I am convinced that mathematics will not disappear, just as material production did not disappear. But its methods, institutions, and professional roles may change just as profoundly.

Note: This text was written with the assistance of AI. All the central ideas are the author’s own, and the author takes full responsibility for them.


Received 11 August 2026.

11 responses to “The end of an era in mathematical research”

  1. O L Avatar

    The steam engine changed how we travel and produce stuff, but we still needed to know where want to go and what to produce. And so it is with AI for research. The risks with AI are the commercial aspects where the machine tells us what to buy, which music to listen to and what kind of mathematics (or other intellectual activity) to pursue.

    1. Snoo Avatar
      Snoo

      The only problem is steam engine can’t think while AI can ( atleast in the future)

  2. Nilima Nigam Avatar
    Nilima Nigam

    The analogy with the Industrial Revolution is apt, but perhaps incomplete.

    There were real human costs inflicted upon millions, because governments and policies were not able to cope with the pace of change. Famines in the colonies (agricultural land was diverted to cultivate raw materials for textile mills, food continued to be exported out of Ireland to feed growing British urban centers even in the midst of the famine, etc.) It took many, many decades and many, many fatalities before the net benefits of industrialization were realized. Marx was inspired by the suffering he saw amidst industrial laborers; his response (collectivization) lead to other horrors.

    The counterfactual to consider isn’t: ‘should we have turned our back on the benefits of industrialization’?

    The question in my mind is: ‘could the Industrial Revolutions have been designed differently to mitigate the suffering that accompanied them?’ A sharper question is: ‘how many thousands of deaths were an acceptable cost for the benefits accrued?’ China industrialized, but was the Great Leap Forward the right path?

    As a mathematician, I am not competent to design public policy. But if I’m to participate centrally in ushering in a new industrial revolution, it is reasonable to ask me to contemplate subsidiary and undesirable effects of -how- as well.

    1. Steven Kelk Avatar

      I completely agree, Nilima. This point cannot be emphasized enough!

  3. Alonso Castillo-Ramirez Avatar
    Alonso Castillo-Ramirez

    This is a very good point, and I completely agree. Mitigating the human costs of this new Industrial Revolution is something that should definitely concern us. Thanks for sharing!

  4. Nicholas A Scoville Avatar

    This is an excellent and extremely honest (sorry, I know that is an AI word, haha) take. I appreciate the acknowledgement that this is a complete and total shift in the way humans do mathematics, but at the same time, the optimism and the path forward. In the same theme as one of the comments, I am hoping this will provide us with an opportunity to re-invest in humans where AI is used as a tool to contribute to human flourishing rather than acting as a replacement for humans.

    1. Alonso Castillo-Ramirez Avatar
      Alonso Castillo-Ramirez

      Thank you very much for your feedback, Nicholas! As I’ve recently heard Yuval Noah Harari emphasise, an important point that has no precedent in human history is that AI is not just a tool, but an agent that can take decisions and act autonomously to achieve a goal. Still I’m hopeful that as a society we’ll be able to overcome the challenges and flourish as you said.

  5. Gene Avatar
    Gene

    This piece has a few hallmarks of shallow thinking on this subject. The tells are: the (self-evidently poor) analogies to chess and the history of industrial technology; believing, for some reason, that there are parts of the process that AI fundamentally cannot do; and general optimism about the engagement of professionals with this entirely new way to work.

    Appeals to historical precedent simply fail. We’ve never had a tool that could think in such a general way. A tool that encroaches on cognitive labor, not just on computation but on reasoning and on problem solving, in a domain-nonspecific way, has literally no analogy in history. Nothing even comes close.

    The computerization of chess is also an incredibly poor analogy. I wonder what you think “the death of chess” would look like? What do you think would happen if instead of human beings, the WCC instead decided to put the top two chess engines against each other? How many people would care, and how deeply? What do you think that means? For what it’s worth, many chess professionals dislike using engines for analysis. At the end of the day, it’s a game that decides which human is better at chess. That is, the more interesting question is, what human is best at chess? It is definitionally immune to computerization in the way you imagine, at least until everyone on earth is more interested in which AI is best at chess. Math, on the other hand, isn’t a game about deciding which human is better at solving problems. It’s a science, and the results are (sometimes) socially and technically instrumental. I hope this makes it clear enough. More in this in a second, because it relates to the larger question of how doing math will change.

    There is also no reason to think that there are some special aspects of the mathematical process that are closed off to AI. You mention – deciding what questions to ask, what is useful, in education, etc. There is literally no reason to think LLMs can’t perform these functions. LLMs are not going to stop improving suddenly, and more and better architectures and algorithms are already on the horizon. If you catch yourself saying, “surely, an AI can’t do X,” I’d seriously urge you to wonder why you think that. You are almost certainly wrong, and if you aren’t wrong now, you will be in six months.

    So if mathematics, at least at the professional or research level, becomes “prompt engineering,” which is what it *will* become and on this I agree with you, every mathematician will be forced to confront their reason for doing what they do. No two people are identically motivated. I know many, many people enjoy the craft, of thinking deeply and of both failing and succeeding together, because both are needed. Many are motivated by authoring results and feeling the buzz of discovery and for its attribution. All of this goes away with prompting. With prompting – you are not discovering, you are off-loading explorative thinking and you become a permanent referee. Would it surprise you if many, maybe most, mathematicians will NOT enjoy that? That the AI-ification of doing math robs all of us of those elements that are most conducive to personal satisfaction? We’re being shamed thus: that we are simply narcissists who should be excited about how much more we will learn and how much more quickly we will learn. I’m excited to see what truths emerge. And I grieve for the loss of the art and the practice I care so deeply about in its sort of artisanal form.

    Of course no one can stop me from doing “traditional” mathematics but if I’m in any way accountable for my output not using LLMs now is productive suicide. Doing things the old way becomes a quaint anachronism, like many other things have become. That is a personal tragedy for anyone who feels like I do about the work.

    1. Alonso Castillo-Ramirez Avatar
      Alonso Castillo-Ramirez

      Thank you for taking the time to write such a substantial comment. I did not find convincing arguments in your text supporting the shallow thinking you mention.

      The analogy with chess certainly has its limits: chess and mathematics have different purposes. My point was much narrower: that machine superiority need not eliminate human enjoyment of an intellectual activity. In fact, your observation that we care much more about games between humans than games between engines partly reinforces my point. Human participation can remain meaningful even when machines perform the activity better. Nevertheless, I agree that this does not prove that professional mathematics will remain unchanged or equally satisfying.

      Likewise, an analogy with the Industrial Revolution does not imply that AI has an exact historical precedent. The relevant similarity is that both technologies mechanize a previously artisanal form of production, dramatically reduce its cost, and place pressure on professionals to adopt the new methods. Indeed, your description of traditional mathematics becoming “productive suicide,” and the loss of its “artisanal form” closely resembles the transformation that industrialization brought to many crafts. I have acknowledged in another comment that AI technology is unprecedented in human history because it is not a mere tool, but an agent.

      I agree that AI may eventually outperform humans not only in proof construction but also in problem selection, exposition, evaluation, and the organization of mathematical knowledge. I do not assume that these cognitive functions are fundamentally inaccessible to AI. Teaching, however, is different in an important respect. Education is not only a cognitive process; it is also an emotional and social one. Human beings have evolved to learn from and alongside other human beings, through trust, admiration, recognition, imitation, and belonging. AI may become better than human teachers at many instructional tasks, but I do not believe that it will outperform human teaching as a whole in the foreseeable future. Again, the chess example illustrates this biological and social component: we care about human games precisely because the human participants are part of what gives the activity meaning.

      Finally, I agree that the transition may constitute a genuine personal tragedy for mathematicians who love the craft of discovery. My optimism about the future abundance of mathematical knowledge is not meant to imply that every mathematician will enjoy becoming a supervisor or referee of machine-generated work. More mathematics and a profound loss for those who practise it can occur simultaneously. That tension deserves to be taken seriously.

    2. Matt Alexander Avatar
      Matt Alexander

      I think the chess analogy is quite interesting, personally: for a few years now chess has received a huge boost in popularity due to the tireless work of players like Levy Rozman, Anna Cramling, Alex and Andrea Botez, and of course (the late) Daniel Naroditsky. Danya (Daniel) in particular was very focused on dismantling the kind of “ivory tower” around chess as he perceived it.

      Nevertheless, chess has never had the popularity of other sports (barring perhaps isolated instances, like games during the height of the Cold War). Its top players are consistently underfunded, to say nothing of amateurs, or lower-rated professionals. It’s well-known within the community that you simply cannot make a living solely off of playing chess, unless you’re about the level of the top 100 world-wide. But the community is vibrant, as it’s been for over a hundred years. People play the game because it’s fun, and they become interested particularly when they feel it’s accessible.

      Mathematics has a lot of the same properties (or problems, depending on your view). In the course of questioning AI’s role and its incursion into mathematics, we should also reflect on the makeup of our community, and its accessibility to amateurs.

      The “death of chess” could only possibly arise in tandem with the death of its accessibility: the lack of funding and fame, and the general disinterest from most people on the planet isn’t enough to kill the community any more than it’s killed poetry. This has made it incredibly resistant to any kind of damaging blow by AI: it doesn’t matter if computers are good or bad at chess, people (particularly the community’s large amateur base) are interested in the game precisely because (as viewers or players) they are meaningfully participating in it.

      But if mathematics and its value only (or substantially) exist to the extent that its /professional/ community does, that seems to me a much bigger problem; it keeps math fundamentally at the whims of industry and marketability, even if AI itself were to leave no lasting damage.

  6. Marcin Kotowski Avatar

    The “this text was written with AI assistance” should be at the very beginning, not as a footnote at the end!

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