Author’s note: This essay is chiefly hosted on Proofs and Prompts, a new initiative spearheaded by four mathematicians (Raphael Appenzeller, Francesco Fournier-Facio, Simon Machado, and Anand Rao Tadipatri). Proofs and Prompts is a public forum for mathematicians to discuss AI and mathematics. I encourage you to read the following essay there, so that we can start having communal conversations rather than spreading the discourse across disparate personal blogs. I have turned off Substack comments as a result, since there is a comments section on Proofs and Prompts; use the comments there instead. You should subscribe to Proofs and Prompts by clicking Subscribe in the lower right-hand corner.

In the following essay, I would like to give a reasonable defense of what I call the AI dissenter viewpoint. I have argued along broadly similar lines in two previous essays.1 My views are continually evolving and I recognize that a no-AI future for mathematics is unlikely to succeed, but I think it would be intellectually dishonest of me to water down the argument and propose some nominally more pragmatic scheme. So let me push the Overton window a little bit further and offer my honest thoughts.

The thrust of my argument is the following: the introduction of increasingly capable LLMs has a disastrous effect on the social context and professional environment through which human mathematical understanding is produced. Moreover, we are not alone in the crisis we now face, and we should expect LLMs to affect a range of white-collar and scientific professions in the long-term. As citizens of a broader society, we have moral obligations to avoid collaborating with AI companies and improving their models with the aid of our extensive mathematical training; in fact, we should take an explicitly adversarial position to these companies and the future they are creating. When we use LLMs to generate new proofs of mathematical theorems, we are directly benefitting AI companies, who stand to financially and politically benefit. The more important2 the theorem, the worse the impact. For moral reasons alone, we should not use LLMs to generate new proofs, even if increasing scientific progress might otherwise be a good use case for AI tools.

We should not embrace the use of LLMs to maximally increase mathematical progress, but instead safeguard a social context which preserves and increases human mathematical understanding. Human mathematicians must extensively cooperate to establish reasonable professional norms around AI use in this new regime. In the near-term, we should preserve the current model as much as possible until a consensus coalesces; that means an ongoing moratorium on asking LLMs to prove novel mathematical theorems. Do not collaborate with AI companies, do not defect, and do not pollute the commons with LLM-generated mathematics. Since non-defectors are obviously punished by the existence of defectors, cooperation and coordination is the only path forward. In every regime, the current model of mathematics will clearly substantially change. (For instance, I believe it is likely that the “ownership model” of mathematics will collapse, and so we should redesign our incentives to preserve human understanding of mathematics.)

The current regime: Humans competing with LLMs

Right now, in the ongoing ownership model of mathematics, we have created a situation where human mathematicians are now directly competing with LLMs. So we need to redesign our incentives, fast. Humans who are doing mathematics without AI use are spending labour and time; humans prompting LLMs are spending money on tokens.

Concern: LLM capabilities are quite spiky; their capabilities not only vary field-by-field, they vary problem-by-problem, such that an LLM may be capable of proving Hard Theorem A, yet spit out nonsense when asked to prove Easy Theorem B. Moreover, different models have varying capabilities, and some mathematicians have access to frontier models while others do not. As a result, vast inequities result for mathematicians in different research niches and with differing access to models.

Concern: Mathematicians are directly disincentivized to communicate and disseminate their ideas, proof sketches, and new theories with other people, because they can be scooped by LLM users. People are incentivized to either quickly publish lower-quality work in an LLM-accelerated environment (slop mathematics), or to fully flesh out a theory entirely alone and then drop a 100-page mathematical monograph. Long-term, high-quality projects are now significantly more risky.

Concern: Various actors (AI companies, social media users, mathematicians) will pollute the commons in the near future by quickly producing and generating mathematical work using LLMs and releasing it without refining, disseminating, communicating, and understanding that work carefully. In other words, they will produce slop mathematics. This is already evident in the appalling phenomenon of important mathematical counterexamples being dumped on social media by users who clearly have no intention of ever producing an ArXiv preprint or publication; instead, the mathematical community steps in to pick up the pieces.3 In short, we should expect that all the worst aspects of the publish-and-perish model (the production of ‘slop’ mathematics, underverified proofs, etc.) will be exacerbated in the near-term LLM regime.

Concern: Early-career mathematicians will leave en masse in the next 5 years. They will respond to a field-wide sense of uncertainty and fear if reasonable professional norms do not coalesce quickly. Many of these mathematicians already face suboptimal working conditions of severe financial precarity.

Concern: Mathematicians with serious ethical concerns about AI companies are directly penalized in the near term; this fuels discrimination in our field. In other words, a person’s ideological lens or political sensibilities could unacceptably curtail their career opportunities.

Broadly: My sense is that using LLMs contributes to a social ecosystem in which human mathematical understanding is broadly degraded, disincentivized, and punished. The degree to which human mathematical understanding is preserved is largely due to individual actors consciously or pro-socially moderating their LLM use.

Arguments against LLM uptake in the mathematical profession

I would like to present three different arguments for avoiding using LLMs in our profession. I should add that clearly, LLMs are complex enough tools as to create a moral spectrum of responsible use; but I believe the conclusion of the three arguments below is that in our profession, the less we use them, the better.

The social context lens (the preservation of mathematical understanding)

The introduction of highly capable LLMs imperils human understanding of mathematics, because human understanding of mathematics happens in a social context. We should expect a serious loss of understanding even if we only allow LLMs to generate and verify proofs, while we digest these proofs as an active and living community.4 Let me offer some arguments.

Argument 1: When humans generate mathematics, it is useful for mathematical understanding. We understand mathematics more when we do it ourselves. If we increasingly delegate the generation of mathematical proofs to LLMs, that already has a drastically negative impact on our understanding.5 Of course we spend vast amounts of time understanding other people’s work, but frequently we do so with the aim of eventually writing original mathematical work ourselves.6

Argument 2: People need to believe they add value to our field. If LLMs are pervasively used to generate research mathematics and eventually become more capable at generating new proofs than human mathematicians, then many people will not choose to work in mathematics. Long term, many people are not attracted to working on solved problems. Does reading AI-generated mathematics full-time sound like a profession or a hobby?

Argument 3: When we outsource the work of generating research mathematics to LLMs, we degrade the working conditions of our field. The labour of doing original mathematical work is creative, important, and enjoyable. Is refereeing papers your favourite part of your job? What about papers written by LLMs?

These are serious concerns which I cannot quiet even if we wholeheartedly pivot to incentivizing the communication and digestion of proofs (which I believe we should: see below). I think if humans increasingly relinquish the work of generating autonomous proofs, that is an incalculable loss to our understanding and our profession.

The labour lens

AI companies have extracted value from the scientific community’s labour by using our papers, textbooks, and research output as training input for frontier models. Simultaneously, they exert pressures that in the near-term will result in mathematicians losing their jobs and our working conditions deteriorating. How is that fair? To state the obvious: AI companies have not turned to our field because they are benignly interested in increasing the scientific progress of our field. They use their models to prove theorems because when they announce a nonsofic group exists, they increase shareholder value and establish market dominance.

AI companies are attempting to create models which outperform humans at almost all economic tasks7. If they succeed, almost all of human society stands to be affected. This is so unbelievably inequitable that it is poised to be one of the greatest wealth transfers in human history.

It is incumbent upon us to organize in light of this and to hold the line. We must defend our material interests and our livelihoods, particularly when AI companies used our corpus of work to create these highly capable models. Many of us produced scientific work with the understanding that it would become the collective intellectual property of humanity; we did not produce it anticipating that it would become fodder for private interests and the creation of models which disrupt the production of exactly that scientific work.8

The ethical lens

In mathematics, we have a tendency to treat AI like it is some massive exogenous force which came out of thin air to disrupt our profession. But highly capable LLMs were not produced ex nihilo; they were produced by an industry with intense financial interests, appalling professional norms around safety, and executives with so little message discipline they say things like this in public:

Weirdly enough, if you think that this moment is, I don’t necessarily believe this, but a lot of people would say we’re living through this kind of eclipse of the human intellect where we’re in the final days of humans being the primary actors on this planet, um, and that soon machines will rise. […] It’s a little bit like, it’s, in that sense, it’s a very beautiful time period to live through because in a Dionysian way, there’s a lot of ugliness about it, but there’s a beauty in the ugliness of when a star dies, it grows super big into the red giant, right? And it’s like that, where you, as you watch this final flowering of humanity and the birthing of the machine intelligence, it’s like you see this greatness in human effort.9

Right now, thousands of people are attempting to build artificial general intelligence which outperforms humans at all cognitive tasks. Is that a good idea? Is that democratic? Set aside for a moment empirical claims about whether or not AI companies succeed at their stated goals. Should they even be trying?

Here is my sense of the situation. It is scientifically irresponsible to initiate a scientific project which, if it succeeds, leads to disastrous consequences for human society. Why should we not spend our time doing frontier human genetic engineering or creating super-contagious pathogens? We could make huge scientific progress in those domains. The reason why is that if we succeeded, we would have done something deeply unethical.

In light of that, we have moral obligations to avoid collaborating with AI companies, and in fact to resist them as much as possible. When we use LLMs to generate novel mathematical proofs, AI companies benefit. When we offer our expertise to help mathematically benchmark models, we feed AI hype.10 When we collaborate with AI companies, we are aiding and abetting the Manhattan Project of our time.11 The labour of mathematicians is critical for helping AI capabilities improve; it’s past time for a boycott. I believe that we are desperately in need of a global AI pause. We can either lend our support to the movement for an AI pause, or continue to manufacture consent for AI companies.

Of course, many mathematicians are not directly building LLMs or working for AI companies, and we may console ourselves that we are just downstream beneficiaries of this irresponsible scientific project. Maybe we can just use LLMs to resolve our own scientific curiosities and finally make some progress on our favourite pet conjecture. After all, we tell ourselves, scientific progress is morally neutral. Whether humans or machines generate proofs is morally neutral. But this is clearly not true. The more important the theorem, the more these companies stand to benefit. The more we use LLMs, the more social consent we manufacture for this dangerous and rogue industry.

The fact that increased AI capabilities are broadly bad for mathematicians is a corollary of a more general proposition: increased AI capabilities are bad for everyone. So we can’t use LLMs, even if nominally a proof is a proof, whether it comes from a human or a machine.

Rebuttals to the view above

I now address some rebuttals to the view above.

LLMs are not that good at mathematics yet. LLMs may be good at [y] thing that was never really the essence of math (e.g. problem-solving, finding counterexamples to conjectures, tedious computations of technical lemmas), but it will not replace [x] aspect of our mathematical work which is actually the essence of mathematics (building theory, teaching younger mathematicians, clearly expositing mathematical proofs).

Unconvincing. What happens if this intermediate regime does not last? What happens when [x] aspect of our career can be performed by LLMs too?

Of course, the one kind of labour that LLMs cannot do for us, definitionally, is the human digestion of mathematical proofs. But the human digestion of mathematical proofs occurs in a social context. See the social context argument above.

Human mathematical understanding will just become much higher-level. LLMs will function similarly to calculators and computers.

Low-level details are also key to understanding things deeply. Moreover, LLMs are substantially different from calculators and computers in their capabilities. See the social context argument above.

Mathematicians are paid to produce theorems and it is unfair to taxpayers and the public to not maximally accelerate mathematical progress with LLMs.

I think this is a terrible argument. The main way the public benefits from government-funded math research is from the ancillary benefits of having a large, public research mathematics community. I think it is pretty easy to argue that human mathematicians are more useful to the public than LLMs. Mathematicians can explain mathematics to the public (in the classroom, on Youtube, in popular science magazines). They teach undergraduates across scientific disciplines foundational tools like calculus and linear algebra. They educate and train graduate students on problem-solving skills which transfer to industry and academia. The public largely doesn’t care about the production of new mathematical theorems; the main benefits they receive are indirect benefits like those above.

LLMs may prove to be cheaper at producing novel mathematical theorems than human mathematicians, but the indirect benefits the public receives will only decrease if we replace mathematicians with LLMs.

If the theorem economy is artificial12, then there is no harm in simply producing as many new theorems as possible with LLMs in the interest of satisfying our mathematical understanding.

Theorems may have little market value (in plainer terms: the majority of mathematical work is useless). However, that does not entail that it is consequence-free to produce as many theorems as possible with LLMs. See: the social context and ethical anti-AI arguments above.

If we institute some kind of professional ban on using LLMs, that actually decreases and hinders human mathematical understanding (possibly to an enormous degree). It is incoherent to defend human mathematical understanding while asking mathematicians to desist from LLM use, and to inhibit scientific progress, which is nominally the goal of a research community. In other words, scientific progress is one of the positive use cases of artificial intelligence.

I think this is one of the most interesting and compelling rebuttals to the arguments I have outlined above. It is by far the strongest.

One possible rebuttal is to reiterate the social context argument. Already, a vanishingly small fraction of human society is interested in learning the proofs of famous theorems. If LLMs become more capable than humans at generating mathematics, I think many people will lose interest in mathematics. If our profession is hollowed from the inside out and mathematics becomes a hobbyist endeavour, then it’s not really the frontier that matters anyway. We can know; we won’t know.

But I think the more intellectually honest thing is to concede the point. Yes, we won’t make the most mathematical progress possible without the use of LLMs. But we have compelling reasons to refrain from using them anyway. See: the ethical anti-AI argument above.

People can just lie and use LLMs anyway. Mathematics is a competitive environment and actors who don’t use LLMs will be punished. A worldwide total ban on AI use is unlikely to succeed. There are morally grey methods of AI use (literature searches, learning classical mathematics, etc.) which would undermine the efficacy of an outright ban. Even if we control the use of LLMs in some countries, other countries may not abide by such agreements. In short, the AI dissenter view is unrealistic and unlikely to succeed.

I concede all of these points. I expect that the actual course of events does not land us in the AI dissenter’s ideal world, and at best in some intermediate regime.

Nevertheless, I would like to shift the Overton window to a more radical place by being as intellectually honest as possible. From a pragmatic perspective, I think if we don’t have the intellectual courage to articulate ‘naive’ points of view, then we have no chance of them ever succeeding. From a moral perspective, I think the AI dissenter viewpoint is just correct. To state a fairly obvious observation in moral philosophy, there is no reason to believe the morality of a given action is at all related to the facility of performing that action. But actually, I think we still have very good options for protecting our field from AI companies; see the radical proposals below. 13

If LLMs outcompete humans at research mathematics, then we should just allow LLMs to largely monopolize the work of generating research mathematics (that is, producing proofs). The job of a mathematician could instead be about digesting and disseminating frontier AI-generated mathematics. The work of our profession will be teaching each other math, educating undergraduates, service work, and community.

I actually find this very interesting. Broadly, I think the mathematical community is coalescing around redesigning the incentives in this direction, and I support this initiative. These sorts of labour are valuable. The work of it assembles into the labour of a real profession. I think it is an interesting rebuttal which addresses concerns about both labour and the social context by which human mathematical understanding is produced.

There are two issues with this proposal to my mind. In this theoretical regime, the work of generating original research mathematics now plays a marginal career in our careers. That is an enormous loss.

The other is that this course of action does not address the ethical anti-AI arguments. Mathematicians will still have aided and abetted AI companies in improving AI capabilities. When novel mathematical theorems are proved by us with extensive AI use, then AI companies stand to gain enormously, in terms of both stock value and sociocultural sway. Every time you use AI to prove a new “big” theorem, you are helping AI companies (even if the ownership model is long dead).

Some radical proposals

I now suggest some proposals which can help address the AI crisis in mathematics. Critically, supporting these proposals does not require too much anti-AI buy-in; I believe these proposals could be supported by a coalition of mathematicians.

1. Move away from the ownership model and the current theorem economy. Redesign the incentives to encourage human mathematical understanding. Incentivize work like: mathematical communication; education; and textbook writing, instead of just the production of original mathematical work.

   1. Even those of us who are attached to the human generation of mathematics should support this initiative, because this initiative directly disincentivizes defection. It curbs the creation of slop mathematics. People will be much less motivated to use LLMs to thoughtlessly produce novel mathematical theorems if they aren’t rewarded for it. Already, too many theorems were proven for the community to be able to reasonably digest them.

2. Limit the number of papers that an individual is allowed to output per year. This is an immediate way to punish actors who want to flood ArXiv with LLM slop, and to encourage higher-quality work. This essay by Mario Pasquato expands on this proposition at greater length.

3. Determine the leverage we have over AI companies and exercise it. What can we do to oppose this industry? Options could include: a field-wide boycott on collaborating with AI companies; deliberately polluting the commons with bad and faulty mathematics to wreck models; restricting frontier mathematics from being accessed as training data by AI companies; demanding direct remuneration from AI companies; building coalitions and organizing politically; writing public essays and speaking to the media.

4. Organize at your own university. I reiterate here urgent proposals from mathematician Max Weinreich:

   Every college mathematics department needs a committee, formal or not, to regularly discuss and make recommendations for AI use. You can start this committee. Time is too short to coordinate a national project from the top down or to wait for someone else to do it. Departments hold the keys to ensuring that human mathematicians may at least persist as a minority in the mathematical world in many ways. Departments can lead by establishing anti-AI policies for student work, by reserving hire lines for mathematicians who eschew AI, by valuing AI-free papers more highly in tenure promotion, and by increasing resources for talks, seminars, and conference attendance. [emphasis mine]

5. At an individual level: avoid using LLMs to prove novel mathematical theorems. Don’t defect and don’t pollute the commons. Every time you do this, you directly benefit AI companies and you marginalize the production of human-generated mathematics. You behave unethically. If you use LLMs, limit their use to applications which don’t directly compete with humans. In all cases, act conscientiously and pro-socially, and don’t engage in mathematical arbitrage.

Thoughts on cooperation and defection

Clearly, it is extremely difficult for humans to cooperate on a large scale even when it is in their best interests to do so (see: all of human history). But any path forward requires extensive cooperation. Everything from here on out is game theory. People can make locally rational decisions to globally disastrous effect. If you don’t make your own views known in a public forum, then sympathetic actors are unable to coordinate with you. Within mathematics, we are obviously deeply divided as a community on our views about how to proceed in an era of increasingly capable LLMs, and every person will have to make concessions.

I would like to push back against this manufactured consensus that we all have to start using AI tools or risk falling behind. I have been told it is exceedingly naive to resist these tools and to ask people to cooperate and not defect. Probably this is true. But I think it’s also exceedingly naive to embrace the use of LLMs in mathematics and expect good long-term outcomes for our field. I do not find the people whose argument is to submit to entropy practically convincing. I do not find even thoughtful and nuanced AI proponents morally convincing. I find AI companies morally despicable and I continue to dissent.

All of us know that LLMs are certainly not better than the mathematical community in aggregate, yet. What happens when LLM capabilities at mathematics dramatically increase? An entire industry exists which will only meet its financial targets if it succeeds at exactly that. Thus, we will be on the back foot if we have not already built a coalition.

It takes a certain kind of person to form a union. It takes a certain kind of community to cooperate en masse. Our working conditions involve relentless competition; thus, from the outset, it seems unlikely that we are that community.14 But I’m not so sure. Only time will tell.

Acknowledgements. I am grateful for conversations with Zachary Glaser and Merrick (Dongming) Hua. All of the views above are solely my own and do not reflect those of anyone else.

Further reading:

• Statement on LLMs, by Jonny Evans.

• The crisis of AI-generated mathematics, by Max Weinreich.

• What will become of us? by Mario Pasquato. This essay outlines a proposal to limit the number of scientific publications a person can output each year.

1. Do not collaborate with or work for AI companies. This includes unpaid labour, part-time scientific collaborations, verifying AI-written proofs, etc.

2. A moratorium on using LLMs to prove novel mathematical theorems, until mathematicians have decided on a course of action to preserve the mathematical community. We must respond to this crisis now. We must build a new ecosystem which preserves human mathematical activity in an era of increasingly capable artificial intelligence systems. It is vital that we secure our future in collaboration with governments, universities, and non-profits. In the years to come, we should anticipate a lot of damage: an exodus of early-career researchers and potentially huge swings in funding. It is time to discuss the future we would like to live in.

I am aware that the second prescription is extremely controversial, and indeed many will see it as logistically untenable in the long-term. I do not disagree with this. The introduction of LLMs will create a prisoner’s dilemma environment in which a large number of actors are likely to defect. For this reason, it appears to me that we have no choice but to move away from what I call the “ownership model of mathematics” and the current incentive structures which determine our mathematical ecosystem. There were already serious issues with this model. Nevertheless, I believe in the near-term, we must protect and preserve the ownership model as much as possible until we have secured a future for the mathematical community. We must not defect.

Moving away from the ownership model does not necessitate that we embrace the use of AI in mathematics to the greatest extent possible. In fact, I believe we should have serious qualms about AI companies and the future that they are hurtling all of us towards. I find the prospect of an artificial intelligence that outperforms humans on all cognitive tasks (or, more weakly, all mathematical tasks) alarming. We are clearly not yet in this regime with respect to math, and we may never be, but we need to muster a response to the worst-case scenario.

For that reason, we must cooperate now. Artificial intelligence is increasingly capable of automating cognitive labour. Huge fractions of human society are directly threatened as a result, and mathematicians are far from the only professionals in this position. However, I believe that the mathematical community is uniquely well-positioned to respond to this crisis. We must hold the line and provide an exemplar of how a field can cogently and collectively respond to the advent of increasingly capable artificial intelligence systems. Below, I offer some calls to action for the greater mathematical community.

A note on Ludditism: Calling for collective action is not the same thing as asking that our field indefinitely abstain from using AI to aid in mathematical discovery. I am sure that many mathematicians want to use LLMs to enhance our mathematical understanding, and this is perfectly reasonable. Before we wholesale embrace the use of AI in mathematics, we must secure our future and protect human mathematical understanding.

What is the ownership model of mathematics?

Previously, the incentive structure for doing mathematical research was based on what I will call the “ownership model” of mathematics.1 To concisely summarize this state of affairs:

• In this model, a mathematician proves a theorem (usually by combining existing work in the literature with new ideas) and then claims attribution for it. The community then judges whether this theorem is important and interesting. The more “interesting” theorems a mathematician proves, the more likely she is to be hired by a good university with generally better working conditions (a lower teaching load, more remuneration, etc), invited to disseminate her mathematical ideas at conferences, and generally rewarded with career accolades.

• In particular, one might notice that this model heavily incentivizes proving as many theorems as possible, and devalues other contributions to human mathematical understanding, such as mathematical communication, dissemination of ideas, and teaching of undergraduates and graduate students. Sometimes the funding pressures of this model lead to a surfeit of “slop mathematics”, where people release underwhelming, poorly-written, or underverified papers so that they can maximize their number of publications.

• The current ecosystem is one of scarcity and competition: there are not enough professorships for everyone who would like to be a mathematics professor. Industry typically does not compensate people to do pure math research. Academia is therefore the only option for someone who would like to do pure math research. Thus, the number of interesting theorems you prove determines not just the success of your career but whether you will have a career at all.

• I have always had serious qualms about the process of stratification by which we decide who gets hired where, and I believe there are serious ways in which this model is detrimental to human understanding of mathematics.

When a large number of mathematicians are simultaneously using LLMs, then this model is untenable.

• First, let us state the obvious. With AI-assisted mathematics, it becomes extremely difficult to determine who can claim attribution for a result: the LLM or the human mathematician. Consider the following scenarios:

  • Scenario 1: A mathematician needs to internalize Artin-Wedderburn theory quickly for a research problem. This is a classical area of mathematics which was worked out by human beings in the previous century. The mathematician holds long Socratic conversations with an LLM and quickly learns the theory.

  • Scenario 2: A mathematician can formulate a true statement, and knows that it should follow roughly by Propositions 1, 2, and 3 in the literature, but cannot find the right references and is struggling to fill in the details of the argument. After the mathematician gives the proof sketch to an LLM, the model synthesizes the existing literature, fills in all missing details, and returns a complete proof which goes along the lines of the original proof sketch.

  • Scenario 3: A mathematician is the process of proving Main Theorem through a long and complicated argument which involves 100 pages of write-up and many intermediate propositions. After working on the problem, the mathematician becomes stuck on Technical Lemma 42, a result which on its own is not of independent interest but is necessary for the proof of Main Theorem. The mathematician has no clue on how to prove Technical Lemma 42 but has a posteriori reasons for believing that is true. The mathematician asks the LLM to prove Technical Lemma 42, and the LLM returns a complete proof of Technical Lemma 42 with no other input from the mathematician.

  • Scenario 4: A mathematician would like to resolve Famous Open Conjecture. They ask the LLM for ideas on high-level proof strategies, and the LLM returns a promising high-level proof strategy, although many technical details have not been proven and need to be verified. The mathematician then independently work to verify these technical details and completes the proof.

  • Scenario 5: A human mathematician proves Big Theorem through a convex combination of Scenarios 1, 2, 3, and 4, using the LLM as a regular collaborator throughout the mathematical process.

  • Scenario 6: A mathematician prompts an LLM the following: “Solve Famous Open Conjecture.” After 2 hours of autonomously thinking about the problem, the LLM completely resolves Famous Open Conjecture.

• Exercise: In the scenarios above, how much credit should be attributed to the human mathematician, and how much should be attributed to the LLM?

• In the future, we may face the following scenarios:

  • Scenario 6: An LLM autonomously conjectures an interesting mathematical theorem and proves it.

  • Scenario 7: An LLM autonomously develops a new subfield of math with useful and important applications, and human mathematicians endeavour to learn it.

  • Scenario 8: An LLM proves a new result after synthesizing a large body of AI-written mathematics.

What about mathematicians who abstain from using LLMs to do research mathematics? Unfortunately, I think their prospects are dismal in the current “Wild West” era, where access to LLMs coexists with the ownership model.

• Human mathematicians who do not use LLMs will be outcompeted by human mathematicians who do use LLMs, in the near term. Before new professional norms around attribution are cemented, mathematicians who collaborate with LLMs as much as possible stand to gain in the ownership model and will be able to engage in mathematical arbitrage. Consider the following.

  • Scenario: You are a human mathematician on the job market, attempting to prove Hard Theorem without AI assistance, and you are close to the finish line after a year of work. Simultaneously, another mathematician proves Hard Theorem in about 2 weeks using AI assistance to some degree (see Scenarios 1-5 above). They receive enough credit for the proof of Hard Theorem to be hired. Without Hard Theorem on your résumé, you are rejected by universities, your postdoc contract expires, and you are forced to leave math after a period of unemployment.

• If we continue in the current Wild West ecosystem, in which individual actors make highly personal choices about their use of LLMs, mathematicians with ethical qualms about using LLMs will be heavily penalized.

In my previous blogpost, I strongly urged that mathematicians take a collective and cooperative approach to the crisis we face. I believe that the use of any technology is a choice. Nevertheless, I think the following proposition is likely true.

Proposition. The ownership model will collapse even if mathematicians attempt to cooperate to protect it. The existence of highly capable LLMs introduces a prisoner’s dilemma for all mathematicians, and there will be many defectors.

Argument. Even if a vast majority of mathematicians would prefer to preserve the ownership model of mathematics, there will be huge pressures on this model.

1. In the ownership model, mathematics stand to gain by proving the largest number of important theorems possible, and LLMs are useful tools for doing so. Thus, mathematicians are highly incentivized to use LLMs covertly and take credit for the results. In other words, there will be many defectors. At best, a culture of surveillance and deception will take over our community.

2. Many mathematicians already have strong reservations about the values implicit in the ownership model. It will not prove a compelling enough philosophy to stand the external pressures that it will face.

3. Many mathematicians would like to embrace the use of LLMs to do mathematical work, for a multitude of reasons: accelerated mathematical progress; a preference for the resulting working conditions; curiosity about the mathematical world. They will be an outspoken fraction of our community, and many actors from industry and government will be sympathetic to them. If the ownership model continues, the mathematical community will be fractured along ideological lines.

Corollaries.

1. In the long-term, the current model of trying to get [x] theorem published in [y] journal and speaking at it about in [z] conference will not survive.

2. We will no longer be able to instrumentalize proving theorems for career success or make hiring decisions based on a human being’s autonomous theorem-proving ability.

3. The current funding model for mathematics is greatly jeopardized.

Some comments are in order. I am not trying to position myself as a heroic defender of the ownership model of mathematics or the publish-or-perish environment of academia more broadly. We should remember that the ownership model of doing mathematics is a very recent historical development and has only been in place the last 80-100 years. However, I believe in the importance of human understanding of mathematics, and it is clear that the system above at least incentivizes that understanding in so far as it is necessary to produce research mathematics. Will the model which succeeds it have such clear-cut incentives?

How good will LLMs become at mathematics?

Right now, we live in an intermediate regime, where LLMs are extremely useful for pushing mathematical progress forward, but human-led mathematical activity is the still the source of vast majority of novel mathematical theorems. LLMs are capable of doing novel and interesting mathematics, but they are not as good as the mathematical community, in aggregate. I certainly do not want to join AI companies in overstating current frontier model capabilities, as impressive as they are. (As an example, the recent proof that nonsofic groups exist heavily used preexisting work of Kun and Thom.)

It is possible that this intermediate regime proceeds indefinitely, and LLMs do not continue to linearly improve at proving theorems. It may be true that human mathematicians remain instrumental for proving mathematical theorems for an indefinite period of time. In this scenario, human mathematicians will continue to prove mathematical theorems, although increasingly with assistance from AI. They will remain employed by universities for nominally the same reasons—teaching, service work, and the production of original mathematical research. In fact, mathematicians may be able to point to their dramatically increased productivity as an argument for the prolongation of their profession. In all regimes, only humans with extensive mathematical training will be able to understand the output of AI-generated mathematics.

However, we should not allow the existence of our careers to be dictated by the state of AI capabilities, and I am afraid that the scenario above will not last for long. We must instead secure our future now and prepare for the following worst-case scenario:

Scenario A: Artificial intelligence outperforms humans on all mathematical tasks.

Imagine a world in which artificial intelligence is capable of proving novel mathematical theorems without any human guidance whatsoever. Moreover, the less human guidance the LLM uses, the faster and better it is at proving new, interesting theorems. Imagine a world in which artificial intelligence is capable of educating human beings about mathematical content better than mathematicians are; it can quickly curate bespoke textbooks, hold long Socratic conversations with students, and educate human beings about the most recent AI-generated mathematics. Imagine a world in which artificial intelligence has exceptional mathematical taste, and is more than capable of asking interesting questions and determining the future of a mathematical field. Imagine a world in which mathematics is no longer a collective human endeavour but instead an AI-led endeavour in which humans take the backseat.

You may notice something about this scenario: Human mathematicians will lose all bargaining power when all aspects of our labour can be automated. When I read optimistic visions of the future outlined by prominent mathematicians in the scenario above, I often find them breathtakingly naive. Will we really just sit around, going to our conferences and disseminating the latest AI-generated mathematics to each other? Will human mathematical understanding be preserved in the scenario above? I am doubtful that selling math as a humanist pursuit will go over well. We live in a broadly anti-intellectual society. We have maintained our lifestyles of intellectual freedom, service work, and mentorship because we nominally produce original mathematical content and mathematical understanding in others. When our labour is replaceable, we should fear being replaced.

Let us all think outside of ourselves for a second. A world in which artificial intelligence outperforms humans on all mathematical tasks is not so very far from a world in which artificial intelligence outperforms humans on all cognitive tasks. I believe that we should be very frightened of this prospect, and that AI companies should probably not be building technology towards this end. Nevertheless, they are, and our entire society is throwing its resources and compute towards this endeavour. In light of that, let us imagine we are in Scenario B.

Scenario B: Artificial intelligence outperforms humans on all cognitive tasks.

What would we do if such a technology existed? This is clearly a dangerous scenario. One would hope we would not cede important decisionmaking power to such a technology. We would heavily regulate this technology and prevent actors from accessing it. We might pass inter-country agreements to avoid the proliferation of such a technology. We would affirm that the existence of a technology is not an obligation to use it.

Mathematical solidarity and holding the line

Today, mathematicians are on the bleeding edge of a cross-societal phenomenon: the automation of cognitive labour. AI has already disrupted other highly-skilled industries like animation and software engineering. It has degraded the working conditions of these fields and contributed to increased unemployment. Now we are in the crosshairs.

I believe mathematicians are uniquely well-positioned to face this challenge. We are highly educated and literate; we have a distinct subculture which gives us cultural unity; we face broadly similar working conditions and we work for the same employers. We do not have ethical obligations to use AI to accelerate mathematical progress as quickly as possible. We have extraordinary professional latitude and intellectual freedom compared to other careers. We are knowledge workers whose cognitive labour is directly threatened by AI, and we can cooperate with each other in light of this. It is our duty to preserve and increase human mathematical understanding.

I believe the greatest threat to our solidarity is self-imposed. In reading visions of the future articulated by other prominent mathematicians, I am struck by the degree to which we are afraid to selfishly defend our material interests. Well, we live fantastic lives of intellectual freedom, service work, and mentorship; we provide assistance to theorists and experimentalists in other scientific fields; we carry on an intellectual tradition which is thousands of years old. If AI makes those things obsolete or performs aspects of our labour better than us, shouldn’t we just let it? No. Stop capitulating in advance to market forces and start defending human values that you think are actually important.

I think that we have a moral obligation to resist the world that AI companies are creating. Perhaps your own material self-interests are not enough to motivate you; fair enough. What about the material self-interests of society at large? While LLMs undoubtedly have the potential to be a powerful scientific tool, I believe that we are on shaky ground. It is my opinion that AI companies are creating a future that most people do not want to live in. We can provide an exemplar of how a community can preserve itself against the automation of cognitive labour. It is incumbent on us to set an example for other industries by pre-emptively acting to safeguard our working conditions and livelihoods. We have an obligation to hold the line, not only for ourselves but for others.

Calls to action for the mathematical community

1. We need to hold a large professional meeting within the next two to three months, featuring a coalition of mathematicians from various subfields, to outline possible scenarios for the future and determine the scenario we would like to live in. We must continue litigating the details in the months and years to come. It will be professionally irresponsible if such a meeting does not materialize. The Leiden working group is a start. We have to continue.

2. Professional mathematical organizations should begin to hire mathematical advocates and emissaries, who will represent the interests of mathematicians to governments, nonprofit, and industries. This should include not only prominent senior-career mathematicians but early-career representatives whose livelihoods are most in jeopardy.

3. We must build class solidarity and stop collaborating with AI companies for the foreseeable future, until we have safeguarded our material interests and that of our larger society. We must not defect.

Setting societal precedents

We will lose bargaining power if AI technologies are equally capable of doing the mathematical labour we do—unless we set societal precedents about the kinds of labour we allow AI to do. The time to set such precedents is now, in the intermediate regime where LLMs are still unable to replace our labour.

Here are some precedents I could imagine a broad coalition of mathematicians supporting (or at minimum, discussing in the professional meeting suggested above):

1. We should not allow AI to educate our children and junior researchers in mathematics, even if it is capable of doing so. We must prevent universities from outsourcing the labour of teaching and education to LLMs.

2. We should not allow AI to decide the direction of mathematical research or the future of mathematical fields, even if it is capable of doing so.

3. We should maintain mathematics as a collective enterprise, even if it becomes more efficient to individually learn about mathematics from AI. We should continue disseminating and expositing mathematics to each other.

4. We need to secure our livelihoods. We need to convince universities and non-profits to maintain a large community of living, human, employed mathematicians who are paid to talk to each other, do mathematics, and control our field, even if originality no longer determines the quality of a piece of mathematics.

5. We must draw clear professional norms about how mathematicians are allowed to use LLMs, and abide by them.

   1. For instance, for prompts on the level of “Solve Famous Open Conjecture” with no other human assistance, I would prefer that such prompts are initiated on the basis of a collective decision of a subfield of mathematicians.

6. We must remember that the use of a technology is a choice and that we are not ethically obligated to maximize the use of LLMs for the end of mathematical progress (as measured, e.g., by the number of true theorems with Lean-typed proofs in existence.)

A sombre note

Over the last year, I have been haunted by dreams of an impending future. In this future, the frontiers of mathematical knowledge are pushed ever forward while our universities are hollowed from the inside out. Undergraduate students across scientific fields fail to master basic concepts. Difficult theorems are proven with increasingly little input from humans; in light of this, governments and industries funnel money towards buying tokens rather than hiring expensive human mathematicians. Young, talented mathematicians leave the field en masse and are pushed towards careers in finance, the military, and increasingly, AI companies. The professional mathematicians who remain fail to convince governments and industries to fund a large, active community to disseminate and pass on mathematical knowledge. After all, mathematicians already lost all bargaining power when almost all aspects of their labour could be capably performed by LLMs. The community struggles to attract and retain talented people to work on and understand the proofs of important theorems. Cognitive labour across many industries is increasingly automated, and political and financial power concentrates in the hands of a few AI companies. In general, human beings increasingly offload cognition onto LLMs, and far fewer people pursue intellectual professions as more of them are effectively replaced by AI. Humans now chiefly work in the relational sector, providing hospitality, therapy, artisanal goods, and personal services to each other. Mathematics is now a hobbyist endeavour which is done by a handful of atomized individuals, and most people don’t know very much about it. The AI industry remains poorly regulated; after all, superpowers spent the early 21st century competing to build the most powerful models possible, and they were well-aware that regulations would only slow the arms race down. Dangerous actors control frontier models and artificial intelligence is given increasing amounts of decisionmaking power, to catastrophic effect.

Is this a touch hysterical? I’m not sure. Certainly, we must not allow this future to come to pass.

About the author:

I am a rising 2nd-year PhD student at the California Institute of Technology. All the views espoused in the essay above are my own. I am 24 years old and have spent the last seven years of my life studying mathematics.

Let me begin with a few technical notes. I first learned about the problem of nonsofic groups in undergrad; it is one of Gromov’s famous open problems, who in 1999 essentially asked if every group was sofic (though the reader may notice he did not use the word sofic in that paper, which was coined by Weiss a year later). For the last 15 years, the mathematical community widely believed that nonsofic groups existed, but it is quite hard to show that a given group is nonsofic. This was known to be a very important problem in group theory, with real implications; sofic groups are known to satisfy interesting properties, such as Gottschalk’s conjecture and Kaplansky’s direct finiteness conjecture. As recently as March of 2026, I went to a conference at UT Austin which was solely about sofic groups and the Connes embedding problem, although as a percolation theorist I have never worked directly on this problem or even adjacent problems myself.1

Nine other results were announced by OpenAI in the same press release, including new bounds on sphere packing and a superexponential lower bound on multicolour triangular Ramsey numbers. Of all the facts accompanying this press release, the following stood out the most to me: the cost of these results was roughly $2,000 USD of tokens at Sol API rates.

In lieu of continuing with the technical details of this quite impressive result, let me now speak as a human being. This is a polemic, not a press release. This ruined my morning. I know that others feel the same way; I have spent today crashing out and learning that top mathematicians I know are crashing out. We in the mathematical community are all grappling with the implications of these highly effective large language models. What does this mean for our field? How should the practice of mathematics continue in light of increasing capable artificial intelligence systems? How should we interact with AI companies? Should we partner with them and work for them? Should we use these technologies to accelerate mathematical progress as quickly as possible and initiate scientific breakthroughs? What role do human mathematicians have to play in a world where artificial intelligence is increasingly capable of autonomously proving important theorems?

Many prominent mathematicians have articulated visions of what the future could hold, such as Terence Tao, Tim Gowers, Jacob Tsimerman, Daniel Litt, and others. Some, like Terence Tao, predict that we may have to adapt to an “abundance” environment where mathematical insights are plentiful rather than rare. Others, like Jacob Tsimerman, are broadly pessimistic about the implications of artificial intelligence for mathematics. Let me also draw attention to the Leiden Declaration on Artificial Intelligence, which offers a good number of intelligent and unobjectionable statements, and simultaneously assert that it does not go far enough.

I have talked to mathematicians who believe that in the future, we will simply be like artists; is this so different from what we usually do, which is already useless? One talented PhD student told me that while photography may have the obliterated the industry of portraiture when it made the production of images extremely cheap, at least photography still exists. Of course, artists and photographers are famously well-respected and paid in this society of ours.

I am sure that many obsolete careers of yesteryear were quite wonderful. Perhaps it was a great gig being the court astrologer. Pick up Silas Marner and you will realize that weaving used to be a well-paid and respected profession; while “Luddite” may be a pejorative today, the textile workers who destroyed mechanized looms were not behaving so irrationally from an economic standpoint.

I believe that today, mathematicians may be in the same position as these bygone professions. While I believe mathematical activity will continue and that some version of the profession of “mathematician” will continue, I think that LLMs pose an existential threat to the mathematical community and the values of that community. I believe our lives and livelihoods are important. I believe in chasing fantastic scientific and mathematical progress with the help of LLMs, we risk imperiling our institutions, our professional norms, and human mathematical understanding. Moreover, I believe that we are capitulating and surrendering in advance to AI companies that are broadly building a world that we do not want to live in. Mathematicians are not the only victims of these disruptive technologies, and we are far from the most important ones. There is a reason why AI companies have turned their attention to our field. Every new press release of this type enriches the shareholders of AI companies.

I am not a Luddite. I think that it is fine if human mathematicians decide that we want to use LLMs to accelerate scientific progress and introduce new ideas to mathematical fields which have been stuck on the same conjectures for a long time. However, it needs to come from us. We need to be making these decisions collectively. We cannot preemptively lie down and surrender control to AI companies. Their success hinges on our continued collaboration: our technical knowledge, our careful verification, our write-ups. Whether we offer that collaboration is a choice.

The existence of a technology is not an obligation to use it. The rapid pace of AI progress is both impressive and alarming, but it alone does not dictate the uptake of that technology. There is no denying that AI companies have made spectacular scientific advancements: this means we are in the domain of politics now. Remember: As human beings, we have the power to write laws and enforce social rules. We instill professional norms in our mentees and colleagues; we ostracize those who behave in distasteful and predatory ways. We regulate dangerous and disruptive industries; we enforce legislation. We organize, form unions, and socially punish “scabbing” so that we maintain our collective bargaining power.

The use of a technology is always a choice.

The values of the current mathematical community

Let me first attempt to articulate some of the values of the current mathematical community and highlight how the use of artificial intelligence is in tension with those values.

1. Humility and correct attribution of work to others. As mathematicians, we generally spend the majority of our time understanding and digesting the work of others. As a result, honestly attributing mathematical ideas to other people is an important value in our community. In fact, sometimes a mathematician may write down a completely original proof and call it “folklore” or post it as an expository note, because they believe that the techniques were not essentially original and that the community at large owns this proof. Mathematical work is carefully-attributed collective property. Unlike many other scientific fields, in multi-authored works it is our norm to list names in alphabetical order rather than by who “contributed” the most or “proved” the most.

2. Accountability over mathematical work and admitting mathematical mistakes. Suppose that two mathematicians are having an argument, and one believes that (P) is true, while the other believes (Not P) is true. Both may have very convincing mathematical arguments at hand, but at the end of the day, only one of them can be correct. This results in a strong social norm in our community of admitting to mathematical errors. I have seen a person who has been arguing for 60 minutes instantaneously admit they were wrong as soon as they are given a valid argument. Similarly, when we write papers, we take full ownership of the mathematical content therein, and we expect a high level of rigour in mathematical arguments. If an error or gap is found in a proof, it is fixed or the paper is retracted.

3. A culture of open-access and knowledge-sharing. We upload our preprints to ArXiv, an open-access archive, to speed up knowledge sharing and avoid creating financial barriers (even if we ultimately publish in for-profit journals). We go to conferences to disseminate knowledge to each other. We teach other people what we know rather than hoard techniques which can be used to prove lots of results. We are far from the dueling Italian mathematicians of the 16th century; we learn techniques to solve problems so that we may educate other people how to solve the same problems.

Let me now say a word on how these values are threatened by the use of AI.

1. Large language models frequently misattribute theorems or altogether forgo citing the works of other authors. When synthesizing the vast domain of (human-authored) mathematical works they are trained upon, they often do not give adequate credit.

   1. This is generally quite convenient from a market-valuation perspective for AI companies, who stand to be enriched by an increase in “hype” and downplaying the achievements of the relevant mathematical field which helped lead to a result.

   2. For example, the proof of the existence of a nonsofic group heavily uses work of Andreas Thom and Gabor Kun, but details like these are generally de-emphasized in press write-ups.

2. Large language models frequently write nonsense proofs and double down on errors. In doing so, they waste human attention span, they confuse experts and non-experts, and they lead to human beings propagating these nonsense arguments further. This undermines our values of accountability and admitting to our mistakes.

3. Large language models threaten our culture of openness and knowledge-sharing, though perhaps at a second-order level. I believe that human mathematicians are increasingly incentivized to be more secretive about their mathematical ideas, since otherwise they risk being “scooped” by large language models. This is especially true when LLMs are regularly trained on large open-source databases.

   1. Let me give an illustrative example from my own life. I am currently working on the proof of a very interesting result with a tractable proof strategy: the sort that involves combining (x) technique from this paper with (y) observation from the classical theory, in novel and substantive ways. If I told a human about this proof idea, I could get an interesting mathematical idea across, and I would not be afraid of them writing up this result: it would take them about 3-6 months of work to do so, and about 70 pages of write-up. (More saliently, it would be viewed as in poor mathematical taste to do this, especially to an early-career researcher.) However, if a frontier model was told about this proof strategy, then perhaps after about $200 tokens it could write up the same result in a vastly shortened timescale.

   2. The way in which LLMs threaten open source movements is well-articulated in the following essay by Sean Boots:

      The process for creating and improving generative AI tools (scraping vast quantities of information without compensating the original creators) incentivizes restricting rather than sharing information. It also damages open source movements and collaborative human efforts (like Wikipedia) that society as a whole benefits from.

Why we do mathematics and the goals of doing mathematics

Why do we do mathematics and what are the goals of doing mathematics? We prove theorems and write papers for many reasons: appreciation of mathematical beauty; professional competition; a stable livelihood; career prestige; the joyful experience of problem-solving. As with all basic sciences, we are motivated by the pure desire to understand the world around us, including our mathematical world. The acquisition of this understanding necessitates proving and learning new theorems and lemmas.

In light of this, why are so many mathematicians reacting negatively to the use of AI in mathematics? Should we not want to understand as much of the mathematical world as possible? Studies show that humans enjoy artistic works less after being told they were generated by AI; clearly, humans have a pro-human bias. I have meditated a long time on my personal negative reactions to the use of AI in mathematics. After considering, I believe that it is rational that I have serious qualms about the use of AI in mathematics.

To me, the primary goal of mathematical progress is human understanding of mathematics. In this, I take essentially the same perspective as Thurston in his seminal essay, On Proof and Progress in Mathematics. I do not believe the sole aim of mathematics is to prove the most theorems or lemmas possible; it is to enrich human understanding of the mathematical world. While the use of AI has the potential to be a serious tool in the pursuit of that endeavour, I believe it poses a grave threat to the current practice of mathematics. In particular, what alarms me most about the use of AI in math is that I believe it has the potential to disrupt, mitigate, and harm human mathematical understanding. I believe it will push people away from mathematics and into other careers; I believe that access to an increasingly-powerful oracle will paradoxically disincentivize people from doing the work of mathematical understanding; and I believe that the median person (say, the person who takes a couple of courses in math in high school or university) will learn strictly less mathematics.

How the use of artificial intelligence risks disrupting mathematical understanding

Large language models are already disrupting mathematical understanding. Let me now give some examples, including some obvious ones.

1. Undergraduate students in scientific fields have a generally weaker mathematical understanding, because they do not solve problems autonomously but instead use LLMs. Those of who teach undergraduates know that free access to AI is generally to the detriment of undergraduate education (although of course, talented undergraduates can probably learn more with LLMs). When I was a master’s student in 2024-2025, students would often come to my office hours with nonsense proofs from AI. Now that AI capabilities have improved tremendously in the last year, many universities are observing an uptick in students performing extremely poorly on tests and in-person exams while obtaining perfect grades on homework, some of which are written with zero human input.

   1. This is likely very bad. Although research mathematicians are disincentivized to prioritize it, I believe undergraduate education is extremely important. Let us state the obvious: most people, including those with math degrees or substantial mathematical education, will never prove an original mathematical result in their lives. They work through exercises and learn or rederive proofs of known theorems; they discover the crown jewels of 19th-century and 20th-century mathematics. It is important that they do this, because learning mathematics is a valuable human activity. However, with access to an oracle which can basically answer any question you want, including frustrating homework questions, it is tempting and tractable to complete one’s mathematics course without actually learning mathematical content or how to prove lemmas and theorems.

2. LLMs have the potential to disrupt the pipeline through which we produce mathematical researchers. At the moment, they are most capable of doing mathematical problems that are pedagogically useful for PhD students to do.

   1. How do we produce research mathematicians? In fact, it is incredibly difficult. We often give graduate students quite tractable problems that involve reading a proof in the literature and extending it to a slightly more general case, or using a technique from one field in another slightly different context. It is through problem-solving and autonomously writing proofs like this that people internalize the techniques of their field and learn how to write original mathematical content. However, if artificial intelligence is used to write up proofs of this level, this raises the minimum level of a “publishable” result to something much higher, which is potentially out of reach for early-career researchers. The same concern is raised by Gowers in this blogpost, although he points out that if the bar instead becomes “proving something in collaboration with LLMs that LLMs cannot manage on their own”, this may actually be substantially easier.

3. In response to AI capabilities improving, talented, young people will leave mathematics because they believe they do not add value, and this will impoverish the mathematical community. This will be particularly true in problem-solving fields. AI companies will take advantage of this situation: they will hire these people and lure them to their safety departments to solve the problems that they are actively creating.

   1. I believe this may pose the gravest long-term threat to the mathematical community. Many mathematicians, including early-career researchers, react to new announcements of AI-led results with despair. This does not seem wholly irrational to me: we have been culturally conditioned to believe that proving theorems is how we add value to the world and to the mathematical community, and it increasingly seems that artificial intelligence technologies are autonomously capable of doing the same. If people believe that they do not add value in this career, they will choose different careers. Our profession, which already loses talented people every year because it demands so much personal sacrifice, will hemorrhage more people. Our human mathematical understanding, as measured by the number of people who actively understand and do research-level mathematics, will shrink further.

   2. Imagine the following scenario, where human beings had access to an oracle whom it could ask arbitrarily hard mathematical questions and receive a correct answer. As humans, would we understand more mathematics, or less? Would we talk to each other more about mathematics, or less? I believe the answer is less. This is clearly a defect of human psychology—nevertheless, we must grapple with the implications. To quote Tsimerman:

      1. “There are many people whose primary enjoyment of math comes through problem solving in one of its incarnations. If that disappears, that is not a trivial issue and many of them might not want to do it anymore (even if there were some way to proceed).”

Are we ethically obligated to use artificial intelligence?

It seems obvious to me that if mathematicians do not use artificial intelligence to the fullest extent possible, then mathematical progress will proceed at a strictly slower rate than if we did. Does this have any ethical ramifications? Are we obligated to use artificial intelligence to pursue mathematical progress at the fastest rate possible?

Fortunately, I believe the answer is no. Of course, some of us are working on mathematics that are directly relevant to other scientific fields or building new technologies. (For instance, a theorist working on quantum error-correcting codes may write an algorithm which makes building quantum computers more tractable.) If these technologies dramatically improved human life, perhaps we are obligated to accelerate progress in those areas as much as possible. However, the work of the vast majority of pure mathematicians does not directly impact other human beings (aside from other mathematicians).

I am inspired in this regard by the following philosophical analysis by my friend Marcel Goh, a PhD student at McGill University. Consider the following scenarios:

1. You are a doctor that [sees] many patients on a daily basis, and there is an AI tool that helps you diagnose twice as many patients in a day, and the diagnoses are accurate 50% more often.

2. You are a low-level employee at an office job, paid for eight hours of work a day, five days a week. Your work is not a matter of life and death, but produces value to some interested parties. Implicit in this hourly wage structure is that you’re expected to do as much work as you reasonably can every day. You have access to an AI tool which allows you to produce twice as much value at the same effort cost.

3. You are a digital artist commissioned to preparing some visual art that will go on some organisation’s website. You have the opportunity to use generative AI tools to produce in a day what would have taken a week in the past.

In Scenario 1, Marcel argues that it is unethical not to use AI. (I concur with him.) In the other scenarios, it is much less clear. Like Marcel, I believe mathematicians land in the latter two scenarios. I do not believe that, in the majority of cases, we are ethically obligated to use artificial intelligence to speed up mathematical progress. Of course, if we want to, we may do so anyway.

If the impact of AI on mathematics is dictated by AI capabilities, we will not succeed

I believe that in a world where AI-led mathematics directly competes with human-led mathematics, AI-led mathematics will ultimately surpass us. If the competition is “Who is better at proving theorems: humans or artificial intelligence systems?” I believe that humans will lose, possibly in the very short term (within the next 2-5 years). This seems fairly obvious to me, though I know some experts will disagree. Just think: every person who wants to say something interesting about symmetric spaces has to learn semisimple Lie theory, and every person who exists is born knowing nothing about it. Moreover, each person lives a finite life, perhaps only eighty to a hundred years, and when we die we cannot contribute anything to the mathematical literature. But a large language model can be trained on Lie theory and only has to learn it “once”.

Thus, if we want human-led mathematical activity to continue, we must avoid a scenario where large language models directly compete with humans. We must convince governments and nonprofits to continue employing mathematicians and those who do mathematical work, even if LLMs are more capable of autonomously proving and verifying theorems. We must find ways to incentivize humans to continue doing mathematics.

Some mathematicians may react to this state of affairs as follows: we should give up on protecting or preserving human-led mathematical activity. Perhaps the future career of the mathematician is that of a prompt engineer, and not an autonomous researcher. Perhaps we will simply read and digest AI-created output rather than autonomously prove theorems. Perhaps mathematics will evolve into a recreational hobby; well, chess engines are better than ever, but simultaneously more laypeople are interested in chess than ever.

I don’t find this scenario particularly appealing, although that may reveal a lack of imagination on my part. I also fear it is unlikely that many people will continue learning mathematics to the level at which they could digest (AI-written) research mathematics in this ecosystem.

On artificial intelligence companies

Cui bono? Of course, someone is enriching themselves from the latest AI advances in mathematical progress and the cultural power which comes attached. How should we feel about the companies which are building these scientifically impressive technologies? What about the immediate impacts of these technologies? To state the obvious, LLMs consume vast quantities of electricity; lead to the creation of widely unpopular data centres; introduce algorithmic opportunities for bias and discrimination; and are increasingly harnessed for warfare. The issue of aligning them with human incentives is a serious one that remains unsolved; these technologies come laden with catastrophic risks. As I write this, a populist anti-AI backlash is already brewing.

Like many of us, I myself have used LLMs in the last year. Much like the internet, I find these LLMs extremely fun to use, and I think they are helpful for deciphering mathematical content and learning mathematics. I am sure they will only get better and make fewer mistakes. I am sure that using LLMs will make human mathematics more productive. Should I continue using them or desist from doing so? What career consequences will I face if I don’t use them?

The main objection I keep coming back to is that they are creating a society and a future that I broadly do not want to live in. They threaten the community of people that I personally care about. I feel I owe a debt to my mentors and friends to preserve our community. If that involves me desisting from using LLMs, I would be more than happy to never use them again. The problem is that our mathematical community will be wrecked anyway if we do not cooperate.

A call to arms and some prescriptions

As time goes on, the use and uptake of these technologies by all industries will only become more normalized. But as mathematicians, we are in a special position compared to many other industries. We do not have heavy profit incentives which force us to use these technologies. We have an extraordinary amount of professional freedom and latitude compared to many other careers.2 We by and large do not have ethical obligations to use AI.

Moreover, these AI companies need us. They need us to validate their proofs; to certify their write-ups; to use their free subscriptions; to work for their companies. They need us to imbue their products with soft power and cultural prestige. We are just one of many industries that they are prepared to disrupt. I am sure that many users are out there prompting an LLM: “Do a breakthrough! Do a breakthrough!” This strategy is unlikely to lead to success without serious mathematical training; indeed, even if a breakthrough resulted, the reader is unlikely to be able to understand the proof. As of this moment, mathematicians are still required for the production of AI-assisted mathematics.

In light of that, here are some prescriptions for the near-term:

1. Do not work for AI companies. This includes unpaid labour. Do not intern there; do not seek employment there; do not verify proofs for them. Take an explicitly hostile stance towards this industry.

   1. Work in AI safety, but judiciously. It is of critical importance that more people work in AI safety. However, ensure that you do not help legitimize and hasten the production of disruptive models; take an adversarial stance, and do work that reflects your values. I am skeptical that the net impact of the AI safety teams of companies like OpenAI and Anthropic is a good one.

2. Refuse all gifts and partnerships from AI companies. Do not take free or subsidized subscriptions; do not engage in “scientific partnerships” in which they pay you some heavy chunk of money in order to extract prestige from your scientific project.

3. Use LLMs for mathematics judiciously, or desist from using them entirely. Include chat logs with LLMs in mathematical papers.

   1. My personal stance at the moment is that, in the current climate, it is unethical to ask a large language model to prove a new result or theorem. Until we establish firmer professional norms about what counts a contribution, we should avoid engaging in "arbitrage” by using LLMs to quickly prove results.

   2. In particular, asking LLMs to prove new theorems can lead to unintentional scooping of human mathematicians who are concurrently proving these results. I am already aware of cases like this. Consider the following two scenarios:

      • Scenario 1: Mathematician A and Mathematician B are both trying to prove on Statement X. Mathematician B finishes proving Statement X before Mathematician A and publishes the result.

      • Scenario 2: Mathematician A and Mathematician B are both trying to prove statement X. Mathematician B asks an LLM to prove Statement X, verifies the proof, and publishes the result before Mathematician A finishes proving statement X.

   3. I am not sure why if I can articulate a coherent moral theory that makes Scenario 1 morally permissible and Scenario 2 morally impermissible, but I nevertheless think we should attempt to avoid Scenario 2 as much as possible in the near future.

   4. I personally find using LLMs to learn classical mathematics a morally grey area, although I am willing to revise my opinion on this. Others, like my friend Marcel, prefer to abstain from using LLMs entirely.

What could an ethical model of using AI in mathematics look like?

Imagine the following scenario. Field A has spent the last 80 years trying to prove Famous Open Conjecture. Mathematicians in Field A would like to make some amount of progress on this conjecture and inject new and diverse ideas into Field A. 60% percent of mathematicians in Field A would prefer to use LLMs to attempt to attack Famous Open Conjecture. In light of this majority opinion, they proceed to use LLMs to collaboratively solve Famous Open Conjecture, which introduces many interesting new definitions and conjectures into Field A.

A priori, I see nothing wrong with the model of doing mathematics outlined above. In fact, it is not so dissimilar to the funding model in fields like astronomy, where people put large amounts of money together to build and use high-technology telescopes. For some Famous Open Conjectures, I might be in the 40% who would prefer not to use LLMs, but I am not the only mathematician.

What is important to me is that these decisions are collectively made by mathematicians. I would like us to maintain a choice in how we do mathematics. I am angry about the exploitation of our field by AI companies and the work that other mathematicians have done to collaborate with those companies.

A dose of reality

I am broadly cynical about the future (for many, many reasons). I think it is likely that no matter what we do, the way in which mathematics is conducted will change forever. A technology, once introduced, cannot be put back in Pandora’s Box. However, I would encourage others in the mathematical community to act now to safeguard our values and preserve the mathematical community. To build a future we would like to live in, we cannot be passive and preemptively surrender.

Right now, we are living in the Wild West; these technologies have been freely introduced to the public. As a result, individual actors are using them in a variety of ways. Morally confusing scenarios are already playing out across mathematics every day. Each month heralds a new result, and we should expect many big conjectures to fall in the next year.

I urge that we quickly act to instill cultural and professional norms that safeguard the practice of mathematics. To me, this involves desisting from using LLMs to prove new results, and I wish that as a community we could agree on doing this for the time being. Of course, OpenAI and similar companies will use their models to prove new results anyway. In light of that, we must accrue political power and we must campaign for regulations against these companies which directly threaten our livelihoods and those of many other people.

About the author:

I am a rising 2nd-year PhD student at the California Institute of Technology. All the views espoused in the essay above are my own. I am 24 years old and have spent the last seven years of my life studying mathematics.

Further reading:

• Mathematics without Mathematicians, by Fernando Borretti.

• Existential Risk from AI: An Exposition for Mathematicians, by Xiaoyu He.

• The Dark Night of Mathematics, by Kirwin Hampshire.