This article is written in response to the announcement of OpenAI’s “Sol” model making 10 mathematical “breakthroughs”. These events have been well covered elsewhere and I don’t intend to cover them in any detail. Instead I intend to focus on the likely trajectory of developments in Mathematics, based on what we have seen in Programming and Computer Science.
I would recommend readers the excellent essays by Kirwin Hampshire and Tamsin Chu who share similar feelings of horror and revulsion at this development.
I should also make clear that the concerns outlined here mostly cover Generative AI, LLMs, and not the narrower forms of AI that serve specific purposes for useful human ends. I would have the same concerns, however, about any future form of technology that takes the process of discovery entirely out of human hands.
This is a long essay, so here is a brief summary of the conclusions on mathematical practice:-
LLMs will claim a number of easy victories in maths, maybe even a surprising number.
These will mostly come in the form of disproofs and bound improvements.
Mathematical practice will become sloppified.
The institutions of mathematics will be overwhelmed and engagement, in the long run, will drop overall.
We will fail to train sufficient new mathematicians to replace the ones who currently check machine proofs.
Corporations Are Not Your Friends
Corporations are not your friends. They do not care about you. They do not care about your research project, they do not care about whether you have a happy and fulfilling life and they do not care about Mathematics – except as it helps them to enact their primary purpose.
Corporations exist to make money for shareholders. That’s all. During the Fordist Era there were perhaps some crumbs thrown to workers to convince them that they were part of an ascendant middle class, but those times are gone now, and the mask is well and truly pulled back.
Even as Toy Story 5 becomes Pixar’s highest grossing film of all time, the company lays off 15% of its staff. Even as banks rake in record profits off of the back of sustained high interest rates, they announce multiple rounds of lay-offs. Even the “spoiled” workers in Big Tech now feel the cold chill of a labour market where labour is no longer a desirable commodity.
Corporations exist to make money for shareholders, and workers (including you) are an unwelcome speed bump in that process.
There is no money to be made in proving theorems. Even the $1m prize attached to each of the Clay Millennium Problems is chump change to companies that think nothing on spending $800bn on a technology that has yet to generate a tangible economic return.
The Uberisation of Maths
Why then, would OpenAI, Anthropic or any other AI company care about solving maths problems?
For much of the general public, pure mathematics is synonymous with intelligence. Proving hard maths problems, the reasoning surely goes, will convince an increasingly sceptical public that these machines are truly thinking and that we are therefore at the age of “AGI” or “Within the Singularity” or whatever marketing buzz phrase the oligarchs of AI are favouring this week.
Sam Altman has openly admitted that he favours the idea of Intelligence-As-A-Service – a commodified model of thought, where researchers (and normal folk) have a problem and then pay for the thought needed to resolve it by, the second. Simply think of a maths problem (or ask Sol to think of one for you) pour in compute, and wait for the answer. You can also ask it to write up the discovery in LaTeX and formalise it in Lean. There’s absolutely no need for you to do anything. For a price, humans can be fully relieved of the onerous burden of thinking.
The model has been seen before. Uber spent billions of VC cash putting small taxi firms out of business, before raising costs and taking full advantage of their new monopolies.
I doubt that any mathematicians actually welcome this outcome, but I suspect many are unaware of the trap that has been sprung. Anyone acting without the awareness of its potential is doing a grave disservice to one of the foundational sources of human knowledge and understanding.
“We have To Check Everything” – The Reverse Centaur
I remember being invited to the office of an AI vendor back in late 2023 (I was there at the request of my employer). They showed us how we could think of a little function, ask the AI and it would produce a function that more or less did what we asked. Very cute. Reality however, has been less kind.
Software developers and engineers complain of sloppy, buggy code, produced in enormous volumes that requires detailed checking by humans. AI start-ups boast of how many lines of code are already in their repos – probably one of the worst measure of code reliability and stability. Data Scientists at Meta complain of being turned into glorified data labellers for AI, and quit for less well paid work to escape the monotony. Meanwhile, software developers complain that their once magnificent skills are rotting.
Cory Doctorow has labelled this situation the “Reverse Centaur”. Far from being AI assisted centaurs, galloping into a future of achievement and wonder, instead, humans find themselves ridden by unfeeling beasts, and overwhelmed by their new chores.
Weird glitches
At every generation we are told that the reliability of AI is now solved, and yet the reliability issues continue. The Cloudflare debacle(s), plural. AWS outages. Windows 11 as a farce (again, multiple, serious failures). BlueSky developers proudly announcing their dependence on AI just as their website suffers major, hours long outages.
This list is not exhaustive, and the link to AI is difficult to prove definitively, but in an age in which we are told that ‘coding is a solved problem’ these incidents are indicative that all is not well in the AI Coding Utopia.
As of today, Anthropic’s main Claude Code Repo has 14,678 open issues and the company continues to advertise to hire more software engineers.
‘Solved’ indeed.
The Costs (Programming)
AI costs are becoming serious issue. OpenAI, Anthropic and all of the Mag 7 companies are setting cash on fire every day. Even servicing their $200 pcm users is not profitable.
Anthropic recently claimed to have had their first profitable quarter. It is a lie. Another accounting trick caused by XAI leasing compute infrastructure to them, and paying for it at a later date.
The pressure to increase costs is real and at the same time, companies are realising that it can never be economical to replace developers at the prices that AI companies want to charge.
Even ‘lovely’ open source, open weights modellers DeepSeek, are announcing that they will need to increase costs significantly.
The result of this will be botched attempts to reverse course and switch to cheaper models – what Ed Zitron has called “island hopping” as institutions attempt to survive the drowning effect of suffocating platform costs, much like the players in a game of “Escape From Atlantis”.
The recent exodus from GitHub Copilot shows exactly how this can unfold and their reversal on cost rises is instructive. The fact remains that these companies are spending at unprecedented rates and costs will have to rise if profit is ever to be achieved.
Again – the only purpose of a corporation is to make money for shareholders.
The Likely Trajectory – Lessons From Computer Science
If the above examples from the programming world – and bear in mind that AI companies largely seem to see mathematics as a coding challenge – are at all indicative, I see the following trajectory for AI in mathematics.
In the early months and years a fairly large number of open conjectures and hard to calculate limits will be disproved or will be improved upon. It will seem like AI is making great progress in mathematics.
Over time it will become apparent that the search space for mathematical problems is more limited than first suggested, and rumours of very high failure rates will begin to spill out – indeed, we have already heard unverifiable rumours of 5% success rates for the OpenAI Sol team, and that work is being guided by seriously heavyweight mathematicians, up to and including Field’s Medallists. Work guided by amateurs and early career mathematicians will fail much more often.
Terence Tao has said that he finds machine generated proofs ‘weird’ in that they will spend a whole page deriving a well known and simple result, and then almost skip over the key findings of a paper. Therefore they will require constant supervision.
Despite this – Universities, under political and corporate pressure will divert scarce budgets from academic staff and grad students, towards AI subscriptions.
Journals will quickly be overwhelmed with proofslop, conferences will be overwhelmed with slop submissions, and graduate students, much like the poor software engineers, will increasingly be expected to triage noisy garbage as part of their training to become a mathematician.
The overwhelmed University system will fail to train enough first rate mathematicians to keep up with the deluge of machine proofs, many will leave the field in any case rather than deal with endless proofslop.
The Costs (Mathematics)
One leaked rumour from AWS talks of a project to match author names to product listings. $1.8m dollars spent, >860% over budget. Little to no cost oversight from management. Never went into production.
There are many such stories – Klarna, Salesforce, even Microsoft have all gone full throttle on AI code spending only to yank the leash hard as costs get way ahead of productivity returns and issues mount.
Pressure on AI costs is going nowhere and restricted University budgets are going to start screaming at a far earlier adoption stage then corporate budgets did. Pure mathematics has effectively zero financial return. Institutions will fail to gain any prestige from machine produced proofs as proofs become just another slop commodity.
The Deskilling
Students and staff alike will deskill as cognitive atrophy caused by constant shortcut taking takes its toll. Everyone using these tools will insist that they are not tempted by the whispering of The One Ring, and that they are able to resist its demonic power. They will tell themselves they are only using it for minor parts of their work, and that of course they still understand the broad thrust of mathematical development.
It will be a lie.
All of the currently available evidence tells us that deskilling is inevitable, when friction is removed.
It will happen just as it is happening in software development.
The Sloppification of Everything
Why does AI slop seem like such a threat to so many areas of our culture? The writer Brian Phillips wrestles with this topic here. He argues that our culture is drowning in ‘stuff’. Rumours of of six generations’ worth of clothing sitting in warehouses, never to be worn. Fast food everywhere, even as half of our food goes straight to landfill. Entertainers rebranding themselves as “Content Providers”.
At one time, all of us would have shared the same cultural touchpoints, as nations or even internationally.
Now it is unlikely that you even share particularly many within your own family, and AI providers want to make this situation even more atomised.
We do not lack for stuff. Everything from clothing, to cars, to TV shows, to ephemeral lines of code is produced in super-abundance on its short journey from creation to landfill. Our personal preferences are catered for in ever more fine grained detail and all of it, all of it, is hyper-mediated mush.
In catering to our every impulse, the companies involved destroy everything that made these IPs meaningful. Try blue instead of red!
As Freddie De Boer argues (excellently) here, a big part of the reason we cannot measure or see big boosts in productivity from AI because it is simply producing more of things that were already abundant.
Producing more and more lines of code does not seems to be producing more and more usable applications, in fact, user satisfaction with apps is now declining, as shown in this well known chart from John Burn-Murdoch at the Financial Times.
I am afraid I see maths research going the same way in the age of AI.
More atomised. Sloppier. Less satisfactory. Less engagement overall.
The medium will decide the form of the content – not the users.
Why Even be Human? And Why Do Maths?
For me and many others the pursuit of mathematical knowledge is one of the purest form of being human that I can experience. Others feel the same from art and writing, or crafts, all of which have been severely impacted by AI.
In these cases, consumers are often roundly rejecting the outputs of AI slop art and writing generators, even as companies try ever harder to shove them down our throats.
Mathematics has no such moat. A proof is s proof. Yes, the AI might write a bad proof that reads weirdly, but as long as it remains logically valid, as long as the Lean code compiles, the proof stands. The Leiden Declaration is a well meaning but very mealy mouthed attempt to warn of the dangers of AI stealing all of the low hanging fruit from the mathematical orchard, or of AI distorting mathematical research towards these easy wins. Only Peter Scholze seems to get the real danger here and states bluntly:-
“Just like I do not want my children to be educated by AI, I am pondering my mathematical ideas without use of AI, and generally avoid reading AI-generated text as best as I can.”
I think he is to be commended in this, and I hope he is supported in resisting the pressure to force AI into his work.
Kirwin Hampshire captures much of my own feelings, when he talks of mathematics as an encounter with the sublime and the ineffable. We risk now foreclosing on this possibility for all future generations with a devil’s bargain that offers a radically different form of ‘progress’ from all progress so far.
Terence Tao has argued (correctly) that we are not terribly far from a world in which machines can spit out proofs that no human can understand. What will it mean to know that the machines have proven the Riemann Hypothesis, or that they have found smooth solutions for the Navier-Stokes equations in R3, but that no human will ever understand them?
I suspect that for most people, mathematicians included, that this will be about as meaningful as being told that they have been solved by the Centaurs of the Alpha Centauri star system, and that having the existence of solutions to all major problems “spoiled” in such a fashion, will greatly diminish the enjoyment of mathematics pursuant to such problems.
The “Enhanced” Games
The Enhanced Games were a joke. Sponsored by tech oligarchs, including the reptilian Peter Thiel, the idea was that chemically ‘improved’ athletes would mangle world records from ‘ordinary’ athletics and show the world that the marriage of synthetic chemistry, with human biology would take us to new plateaus of human achievement. A new Olympics for an age of scientifically enhanced centaurs.
The reality was quite different.
Unenhanced athletes won multiple events, only one record was broken – under extremely dubious circumstances – and only by a tiny margin. Exhibition events where audiences were promised that records would be shattered, turned into embarrassing farces.
It turns out, that there really is no substitute for hard training.
Humans brains (and, to an extent bodies) find it very easy to be lazy. To put off that workout until a receding tomorrow. To trust that big sum in the restaurant to the cashier, even if they might have made a mistake.
But in order to get good at anything hard training is what is required and there is no other way. Will humans continue to try hard when everything, from your first fractions homework, up to discrete Fourier transforms can be handled by the same easy, friendly chat interface?
The Gilded Cage
I read recently that AI excels at helping B grade students produce A grade work, while becoming C grade students [please help me attribute this properly if you can - SH]
In one recent study, the vast majority of students (83%) were not able to quote anything from an essay they has recently finished using AI. Students obtain higher scores using AI, but remember less. Students give up sooner once they get used to using AI, and this effect sets in very fast.
It is hard to blame any individual mathematician for wanting to use AI to boost their output. In a ‘publish or perish’ society, and with the fear that they will lose out to others who may be openly or secretly using AI, the pressure they feel is real, and significant.
But they are likely setting themselves a trap, a gilded cage where easy results give way to dependence and erosion of the fundamental qualities needed to do novel mathematical work – depth, persistence, and a well worked mind capable of chewing through tough problems.
Academics and Solidarity
I very much appreciate and understand the call from Chu for mathematicians to stop working with AI companies, but I see virtually no chance of it happening unless forced by external circumstances (such as extreme prices rises, or destruction of data centres as a result of war)
Even during the 1950s and 60s heyday of the organised working classes, academics were notoriously fickle on supporting the class struggles of their fellow workers. This situation has not improved even as academic working conditions and jobs have come under sustained attack in the Neoliberal Era.
We face the prospect of a Prisoner’s Dilemma Game with many tens of thousands of players, with the rewards for choosing the ‘defect’ option being very great, and cooperation being very hard to arrange.
Maybe I am wrong, I fear I am not.
What Hope do Human Minds Have?
The temptation here will be to end on an optimistic note, to say that we can all come together and resist the worst effects of the inexorable tide of slop, but I am not optimistic, and I did not write this essay to give false hope.
I suspect that low quality output will swamp mathematics. I expect that mathematicians will be proletarianised. I suspect that mathematics, overall, will suffer.
But I will invite you to consider the costs to yourself of participating in such a system and the effect on your own mathematical understanding.
Costs of AI are going to rise dramatically – maybe Prometheus should steal as much fire as he can from the Gods today before the vultures of corporate finance come to peck at his liver tomorrow! However, I expect that Generative AI will survive in some form for the foreseeable future.
Perhaps, in an age of imperial and environmental collapse, people will come to regret relying on these Faustian knowledge machines, and maybe sooner than they expect.
Generative AI has failed to live up to its promise in computer science and programming, it will fail to live up to its breathless promises in mathematics, I sincerely hope that it does not prevent you from living up to your own promise as a scholar and member of a vibrant community of mathematicians.
So repackage it and sell it at a premium to the originator. Aren’t you smart. What your LLM learned was us, is us and becomes trough us. Steal and sell, that is a tactic. I fret what will happen if this mindset prevails while AI becomes more powerful.
Sell the river in bottles to the ones who live in the river.
Finalmente alguém escreveu com a raiva, o ódio, a energia, a indignação necessária sobre esse assunto. Estou cansado do bom tom, do burocrático, do meio termo, do apaziguador. Essas empresas estão explodindo todas as nossas fontes de sentido e de valor, e ninguém parece se importar.