The reverse-centaur problem: LLMs and the choices ahead

Mark Wildon, Professor at the University of Bristol and Heilbronn research fellow

In The reverse centaur’s guide to life after AI: How to think about artificial intelligence before it’s too late, the science-fiction writer and futurist Cory Doctorow draws a distinction between two ways people interact with AI. Centaurs are people whose skills are boosted by AI. The AI frees them from some tedious tasks and helps them to do higher-quality work: they are the masters and the AI is their obedient servant. Reverse centaurs are people who take orders from AI. A typical example of a reverse centaur is the Deliveroo rider who cut in front of you yesterday evening, forced by an AI algorithm to work at a breakneck pace for a sub-minimum wage, the prisoner of a mobile phone app. For such reverse centaurs, the AI is their master and they are its unwilling servants.

As professional mathematicians, we are exceptionally skilled (albeit in a very narrow field) and used to exercising a very high degree of autonomy. Surely we should be the centaurs in a new world in which AI becomes a part of the mathematician’s toolkit? The great worry is that this might not be the case. In this essay I will explain why this worry is reasonable, but not inevitable and within our power to avert.

Amplifying inequalities

I have been lucky enough to do most of my work in an area (the representation theory of groups and algebras) that is unusually welcoming and collaborative. The presence of many leading women, including my doctoral supervisor Karin Erdmann, has a lot to do with this. Similarly, the Heilbronn Institute for Mathematical Research is a remarkably collaborative environment. (Part of my job is to ensure this remains the case.) It helps that, since we all work together towards common goals, there is no incentive to ‘scoop’ one’s peers. Whether large language models (LLMs) strengthen this collaborative culture will depend on who can use them, who controls them, and what our institutions reward. Quite possibly, LLMs will make such large-scale collaborations more common, by helping to bridge the gap between different mathematicians and different areas of mathematics. Yet more likely, I think, is that the advent of AI will shake up the mathematical world by amplifying existing inequalities.

As a correction to a wrong view about these inequalities that may develop if, like me, you spend most of your time around other professional mathematicians, please consider the following question.

Question. The 10% right tail of the standard normal distribution begins at 1.28. Given that a single sample is drawn from this tail, what is the probability that it is 2.28 or more? What is the analogous chance that a single sample from the 0.01% tail (which begins at 3.72) is 4.72 or more?

I think most people would rightly guess that the second sample is more concentrated, but the magnitude of the effect is surprising: the conditional probability that the first sample is more than 2.28 is about 11.2%, whereas the conditional probability that the second sample is more than 4.72 is less than 1.2%. Thus the second sample is almost 10 times more concentrated than the first. Modelling professional mathematicians as members of a 0.01% tail, roughly only one in a hundred of your colleagues will stand out as a noticeable outlier.

More generally, the further out in the normal distribution you sample, the more homogeneous the sample appears. This helps explain why many of us believe that we are merely averagely talented at mathematics, and that the real stars are a tiny number of people lying in an even more extreme part of the tail. (Of course, this is only a toy model of one narrow kind of ability: it says nothing about general intelligence, morality or whether dogs on the street wag their tails or snarl when you approach.) But it may also explain why too many lecture courses are aimed at the top 5% of an already talented cohort. It also suggests that the introduction of a radically new source of variation — namely, access to frontier LLMs and the willingness and ability to use them — will shake up the mathematical world.

The most dystopian vision is that LLMs will destroy the move towards increased collaboration and cooperation that I have seen since completing my PhD in 2004. In this dystopia, a tiny number of mathematical superstars — or the AI corporations themselves — will use LLMs, with privileged access to the top models, to produce spectacular results. The rest of us will be disillusioned by having our partial results used (perhaps without adequate credit) to solve big open problems, and be daunted by the firehose of publications. (But who will referee them? Other LLMs?) I already see worrying anecdotal evidence that early career researchers are being mistreated. In this dystopia, mathematicians become reverse centaurs: suppliers of partial ideas, citations and refereeing labour to systems controlled elsewhere.

But there are also reasons to be optimistic. Whether or not you accept my toy model that most of us are roughly equally good at research mathematics, a huge amount of diversity remains in how we are talented. Some people are quick over a few seconds, able, when their brain is suitably primed, to instantly grasp complex ideas; others are quick over longer periods, able to write a polished paper in a week when it would take me a month. (Littlewood, from whom I stole this observation, felt he was fast over twenty minutes, six weeks and one year.) Some people are wonderful listeners, others think best aloud and need room (or rooms) in which to talk. And thinking of mathematics more broadly, some mathematicians are brilliant teachers, or inspiring outreach speakers, or administrators so capable that they never get the credit they deserve, because they forestall or deal speedily with all the crises that might draw attention to their role. Maybe all these varied talents will also be amplified by LLMs, and while the small number of research superstars will prosper, so will all of us$\ldots$ or at least, all of us willing and able to use LLMs. In this world, we remain the centaurs. Neither outcome is technologically predetermined: which one we get will depend on the professional norms and incentives we adopt.

Perhaps there will even be a place for AI refuseniks, who certify that all their work is their own human labour. But, making a possibly unfair comparison, if a mathematician ended their job interview talk by saying ‘I did all of this while deliberately not using computer algebra’, how impressed would you be?

Mathematical morality

The dystopian outcome, in which research mathematics becomes the preserve of a tiny number of people, requires these people to act in a way that they should realise is unhealthy for the subject as a whole. Let’s say it is immoral, at least by the standards of mathematical morality. This is a definition: as Ayer almost argued back in 1936 (philosophers might wish to avert their gaze now), the only way to bridge Hume’s ‘is/ought’ divide is to interpret all moral claims as a more-or-less persuasive way of saying ‘plagiarism, yuck’.

There is nothing immoral about using an LLM. Maybe most uses of LLMs are deleterious, particularly in inexpert hands, but still it makes no sense to blame a tool that has no moral agency. The morality or immorality comes entirely from the person using it. Here I have to add that in many countries, corporations are legal persons, and mathematics is already at serious risk of being distorted by corporations using LLMs to solve open problems — the more hypeable, the better. But while LLMs will make it quicker for mathematicians to write papers, it will still be down to each mathematician to choose what and how much they publish, and to maintain ethical standards. For instance, LLMs are excellent at pulling together ideas from adjacent fields, and notoriously bad at crediting the sources of these ideas. (Although ironically, if one then prompts the same LLM or another to chase down the source of the ideas, the results can again be surprisingly accurate.) A mathematician who uses AI to publish an endless stream of incremental and barely readable papers, citing only the leaders in the field and ignoring the contributions of his or her less famous colleagues, will still, as now, earn a poor reputation.

Advice

Here is my tentative advice. Because I’m fortunate enough to have a full-time research job at the University of Bristol, in which I spend half my time working on problems of interest to the Heilbronn Institute for Mathematical Research, most of the advice is about research rather than teaching.

For mathematicians

• Ethical standards. Hold yourself to high academic standards, particularly with regard to citations. The more senior you are and the more secure your position, the more important it is that you behave well.
• Never use AI merely to turn yourself into a faster paper mill. Instead, use it to create new collaborations, to better understand your colleagues’ papers, or as a Socratic partner to help you learn a new, perhaps adjacent, field. Do not use the AI to replace what might have been an enjoyable and fruitful human interaction. Find new ways to use AI that don’t require you to stare at walls of text: for instance, code generation, diagram drawing, or tracking down new speakers to invite for your seminar. Do not let yourself become a reverse centaur whose workload is increased by AI.
• Never submit a paper to the arXiv without considering who might be working in the same field and might, at the very least, appreciate advance sight of your paper. Be extra willing to collaborate or to write joint papers.
• Discuss how to use AI with your research students. Emphasise that not using AI is always an option, and probably the only good option in certain ‘starter projects’ if they are to have the desired effect of introducing the student to new methods and techniques from the research literature.
• Be open about your use of AI. If the LLM contributed an important idea, offer to share the prompt, and preferably the entire chat log. A reasonable colleague will excuse any sloppiness in your prompts: indeed, any sloppiness will merely emphasise that your paper is the polished human-readable outcome of a much messier process.
• Watch very carefully for signs that you are becoming dependent on AI.
• Do not be charmed by the LLM into becoming the servant of a highly sophisticated next-token prediction engine. The LLM will subtly feed back your ideas to you, in the language that you have taught it you find most agreeable. The effect on highly intelligent people can, as Richard Dawkins found, be highly seductive: he is on record as saying ‘If my friend Claudia is not conscious, then what the hell is consciousness for?’

For heads of department and other senior people

• Ensure that there remain many ways to be a good mathematician at your institution. Reward people who find ways to keep students motivated in this new AI age. If your university has a ‘teaching track’, strive to give it parity of esteem (this includes pay) with the traditional ‘teaching/research track’. But do not fall into the trap of creating a huge grid full of boxes that all have to be ticked for promotion: this only rewards workaholics with a high tolerance for form-filling who enjoy playing complex academic games. Does that remind you of anyone or anything?
• Convene or attend a working group in your department on how to use AI in mathematics. Encourage an open discussion of the advantages and disadvantages of LLM use. I am helping to make this happen at the School of Mathematics at the University of Bristol.

For AI researchers and people employed by the big AI companies

• Think about academic standards and ethics when designing guardrails for your LLMs, or tools to organise teams of AI agents. Refuse to act on prompts that will obscure authorship.
• Design AI models to augment rather than supplant humans. Such models will create economic value rather than destroying it. Do not aim to pass the Turing test: instead aim to show that the Turing test is the wrong metric. Maybe your ambition is to develop a superhuman intelligence; if so, fair enough, but please reflect that making a faithful copy of a hypercapable human may not be the best staging post.

I expect I will fall short of this advice in various ways, but I hope more through inaction than action. In connection with the point about acknowledging AI use, let me mention one example not to follow. Back in March 2026, I used Gemini Pro 3.1 ‘Deep Think’ as an assistant to answer several MathOverflow questions on series related to zeta functions. The answers were well received, and I made my use of AI very clear, posting most of the answers as ‘community wiki’ so that I accrued no reputation points. Nonetheless, this went against the official policy: ‘If the mathematical component of your content is deemed to be generated by AI, it will likely be deleted’, and my efforts to change the policy met with only lukewarm support. The de facto position now is that AI-assisted answers are grudgingly permitted, but there is far less openness and transparency about this than is desirable.

Where will it end?

All this advice assumes that the current generation of frontier LLMs is not far from a local peak in performance. I think this is likely but not certain. My reason for making this prediction is somewhat personal. In brief: my name is Mark, and I am a high-functioning LLM. I may not be the only one. I have a little list(icle) that you might use to self-diagnose.

• Do you find writing helps organise your thinking?
• Do you use computer algebra or computer programming to help explore mathematical problems? (These side-hustles are now called ‘skills’.)
• Do you find that most good ideas only occur after a long time spent staring at the problem — this includes time in the bath or asleep — mentally reshuffling your thoughts?
• Do your colleagues often prompt you to emit helpful streams of tokens? (These are sometimes called ‘reference letters’.)
• Do you have unusual skill at solving cryptic crosswords? (No problem if you developed this skill from scratch over many years: in fact, that is the expected training model.)
• Do you care about writing polished papers in which the English and mathematics march together? Are you favourably impressed when you re-read your old papers? (Bonus points if you find you retain no context of the original thought process.)
• Before they became the (punctuation) marks of Cain, did you make liberal use of the em-dash and semicolon?
• Do you habitually reason by analogy?

The corollary of this confession is that, since I am so much like a frontier LLM, my solo publications might be a reasonable guide to what one can expect from them. Here, though, I must already make exceptions for counterexample finding (at which there is some evidence LLMs excel) and for literature search. Still, I will tentatively predict that, given sufficient prompting, a large token budget, and some human intervention (this will be called ‘collaboration’), LLMs will reliably be able to produce a paper publishable in a good specialist journal on at least one in three human-chosen problems. On close inspection, this paper will turn out to glue together existing ideas and methods in novel ways, but it will be arguable that it lacks true originality. On the other hand, it will be technically correct, and even at times elegantly written. Unless the human user is careful, it will be sloppy in citations. If submitted for the UK Research Excellence Framework, it would probably get the benefit of the doubt and be awarded the second-from-top rating of 3*, rather than the graveyard 2*.

Here the qualification ‘on at least one in three human-chosen problems’ is important. One of my biggest concerns is that the big AI corporations will burn through years’ worth of parallel GPU time across many problems so that a few spectacular hits emerge from the inferno of complete misses or uninspiring incremental successes. This is the research equivalent of publication bias: run many trials, but allow the public record to retain mainly the successes. (Not that anyone would do that.) This scattergun approach is extra harmful because no one will be able to predict when they might get unexpectedly scooped by an entity outside their established circle of collaborators.

This therapeutic exercise also makes me realise that I probably have a few strengths unlikely to be duplicated by LLMs, at least not in the next six months. Two are probably widely shared:

• A highly tuned sense of what problems are worth solving, and how to present an argument briefly, concentrating on the key points and leaving routine details to the reader.
• Some ability to co-ordinate or even lead (in the lightest possible way) complex research projects involving multiple highly intelligent people, few of whom will be willing to take guidance from an LLM any time soon.

A third might be a distinctive sense of humour, but let me not add any further hostages to fortune here.

Conclusion: back to Doctorow

All the advice above aims to make professional mathematicians the centaurs rather than the reverse centaurs of the new AI age. Doctorow ends with a prediction that the AI bubble will burst fairly soon (his book was published in June 2026), and a deliberately hopeful vision that AI use will remain in roles that are non-threatening to knowledge workers (summarising documents, transcribing audio, $\ldots$) without upending the knowledge economy. More inspiringly, he concludes on his final page:

$\ldots$the future is up for grabs. It is not inevitable. AI isn’t a genie that can’t be put back into a bottle. How we use AI is up to us. Whether we use AI is up to us. The future can be ours, if we never stop remembering that the most important fact about a technology isn’t what it does, it’s who it does it for, and who it does it to.

Mathematics has always been tough. And competitive, even if in recent years overt competition has been discouraged in favour of a more collaborative approach. (Cynically, one might say it is just as competitive as ever, but now appearing not to be competitive is an important sub-skill.) This is not likely to change. But through careful use of AI, we can ensure that the new AI age does not exacerbate existing inequalities, but instead enriches both mathematics and mathematicians.

I hope this post has given you some ideas for new creative and ethical ways that you or your research students might use LLMs, or decide not to use them. Let us be the masters of AI and not its servants.

Disclosure

My acquaintance with LLMs began back in November 2024 when I contributed to the FrontierMath benchmark. The resulting paper is already, according to Google Scholar, my most-cited publication. (According to MathSciNet it does not exist at all, since it is not published in a peer-reviewed journal.) When I began, it was an enjoyable day’s work to set a problem that would defeat all the frontier LLMs, but be doable by an expert colleague: now the same task could take a week, with no guarantee of success.

If you dislike that I work for the Heilbronn Institute for Mathematical Research, you will probably hate even more that, since September 2025, I have done paid consultancy for Surge AI. Let me add, as evidence that I do retain some independence of thought, that I have been interviewed three times about my LLM experiences by upbeat representatives from the AI industry. Every time I was politely ghosted afterwards, I imagine because my responses were insufficiently helpful.

Finally, I acknowledge that this blog was proofread by ChatGPT 5.6 ‘Solve’ in its max-strength version, and invited to have the final word, it suggested ‘Keep the centaur facing the right way’. It was more gnomic than I expected.

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Received 14 August 2026.