What does the end of mathematics look like?

Updated on: August 2, 2025

Word count: 1485

As a prelude to what is to follow, I must say that while I am not a professional mathematician (I have a masters degree in theoretical physics and work in the software world), I do enjoy reading the occasional textbook or review paper, and wandering through its pages in a type of reverie, like walking through a glade looking at flowers. My wife has opinions about this - I suppose most of us who compare textbooks and theorems to forests and flowers are not so well glued to the more mundane aspects of life (such as remembering to feed ourselves, or to tame our appearance so as not to startle strangers). My distance from a professional mathematical setting gives me the space to indulge a type of speculation which might invite scorn from the more hard-headed type of academic, and it is to this freedom that I owe this essay.

I was recently contacted by someone who wants to pay people to, as far as I can determine, write Lean/mathlib proofs to train an LLM, or some other type of machine learning model. This was interesting for several reasons - one is that I don’t actually know Lean/mathlib (a fact I hope to remedy at some point), but also because it brought to the fore a topic I have been mulling over for a while. What would it mean if the thing that some people intuitively believe to be important about mathematics, namely that it is part of, and an expression of, the human desire to understand our world and ourselves (and perhaps thereby improve these things) becomes completely subsumed by the machinery of capitalism?

To expand: what if the practice of mathematics becomes completely determined by the diktats of a vast capitalist machinery of proprietary machine learning models churning out proof after proof, and theory after theory, conjured from the aether of all possible true statements? What happens when the actual act of mathematical thought, beyond but including even that which constitutes official mathematical ‘research’, becomes completely and utterly commodified?

I do not, to be clear, believe this is an imminent event; indeed such an event may be far in the future, or never occur (for more level-headed views on the current state of applying machine learning methods to mathematical research I direct the reader to ‘Machine-Assisted Proof’ by Terence Tao and ‘Can AI do maths yet? Thoughts from a mathematician.’). But I think it should at least be clear to those who pay attention that such a future no longer seems completely far-fetched. And this is, or at least should be, reason for introspection and perhaps alarm on the part of people who care about thinking as a normative value, rather than as, for instance, something you do for money or to solve concrete ‘real world’ problems.

If you believe that research-level mathematics exists primarily to enable the posing and proof/disproof of theorems that are somehow ‘important’, or the development of new algorithms and methods that can be applied to achieve some end in the physical world, the rise of machine learning models that can (dis-)prove, let us say, the Riemann hypothesis, might excite you. But if you also harbour a certain type of belief about researchers making fundamental contributions to our understanding that is meaningful on an ultimately human scale; or about the importance to the individual of attaining the vistas attained by previous mathematical explorers who have left us guideposts to follow - that is to say, to aid in understanding and intellectual growth - you may be less enthused.

The most dystopian version of this that I can imagine would look like the ascendancy of inscrutable models that have completely surpassed our level of understanding, that can only with extreme effort be made to render these results in a manner comprehensible to even top mathematicians, and that are all owned by private companies or governments. The outputs of such machines would themselves perhaps be parsed and analysed by other machines, that would exist to score these models in terms of ‘what the humans would like or care about’; and perhaps these in turn would dictate which research programs are worthy of pursuit, and would, for example, carefully plan which types of data centers must be built where, and with what technology, to make progress with some problem or set of problems. This is arguably already the case in a metaphorical sense - the decisions aren’t literally coming from machines, but due to ‘market forces’ (i.e. intense competition for funding and prestige, and the perverse incentives of the current academic system) the situation is analogous. The motto ‘all large-scale opaque processes eventually resemble malevolent machines’ could almost be a dictum of the modern era (of course the modern age also contains many marvels and much that is good, although the ability to notice such things is in short supply, except perhaps in small children as yet unjaded by cynicism and social media).

We could also imagine a new type of priest class developing - descendants of today’s data engineers, scientific software developers, and chip designers, who exist to increase the expanse of this computational wonderland, at the behest of the capitalist class and beholden only to the mythical ‘shareholder value’. That we are all shareholders in the common inheritance of humanity will be increasingly forgotten, even as it today, but only more and completely so. They will be the keepers of secret grimoires of intellectual property they are prohibited from sharing, under threat of unemployment, or merely being sued and shamed into poverty.

It could be that I am completely misrepresenting the situation; perhaps advanced ML research models will be more analogous to improvements in climbing gear, aiding the development of mountaineering as a sport, than an intrusion of corporate control into our minds. But I believe this is to fundamentally misunderstand both the type of results that machines can in principle produce, and the actual stated goals of the incredibly wealthy people funding the development of these things. In the image I have sketched, machines will not merely work as aids to the human mind, although they will do that too. By the wholesale generation of entire universes of proof arrangements they may conceivably replace the central role that the human mind, albeit already aided with computers, has hitherto played in its historical development; organisation of results in a coherent whole may be accomplished much faster, and hence more economically, than mere humans could achieve. Perhaps a few eccentric people could hang around and try to present the results in a more ‘humane’ way; but doubtless models would be trained for this too. One might have officials, as in ‘Brave New World’, who are breezily aware of the old way of doing things, but consider it quaint and quite outmoded by the needs of a modern society.

The picture I have sketched above may seem ridiculous, and is intended more as a call to introspection than a prophecy. I still think it bears thinking about, simply because mathematics is often conceived to be the closest to an application of ‘pure reason’ that we can obtain. Because of this it occupies a special place in our shared cultural reality, and serves in turn to orientate us in the world, and to inform our intellectual norms. As the creative, as opposed to merely rote computational, mathematical faculty (that the difference between the two is one of scale and speed is still not adequately appreciated), becomes increasingly performed by machines, we will inevitable be forced to rethink what role mathematics plays in our world, decide what it is actually for, and be confronted in a much more direct and visceral way with how expendable our minds are to capital and the people who control it. It should go without saying that this also has direct bearing upon what we conceive of as the goal of mathematical education for children, and indeed education itself.

If something like the preceding were to occur, not much might appear to have changed to an ordinary person going about their day; but a subtle shift would have occurred. I fear it is happening already; our attention has been captured (due in part to regulatory capture by tech companies) in a way that makes it difficult for many of us to think constructively outside our work environments (and sometimes not even then).

I imagine and hope for a full turning of the circle, where we must come to terms again with the relation between mathematics and art, the role they play in our mental and emotional lives, and the purpose of our human societies. I hope the end result will be a new flourishing of the human spirit, where we will shrug off some of the chains by which we have trapped ourselves. But this is far from certain, and many incentives are driving in a diametrically opposed direction.