It is the morning, and I’m working on a paper; it’s one I’ve had going for a few years, and suitably unfashionable that I’m not too worried about getting auto-scooped by a large-language model. But the work is complicated, it’s technical, and the main argument keeps coming apart at the seams; I tweak a definition and fix up one part, and then another lemma doesn’t quite match up. But I understand what is going on intuitively, and I’m confident that it should work out in the end. I tell myself that an LLM could not do such a thing, and I do not believe it. I realise that my latest attempt at a patch falls foul of a trivial obstruction (but I realise this only after writing several pages) and think about how to get around it. I flip between this and the work that I really should be doing, which is writing an application to another postdoc. It feels like I always should be filling in an application. The form requires that I explain what impact my research has had, what I will work on for the next few years, and what I will aim to achieve. What will I achieve in the future? This has always to be difficult to answer, but I at least used to be sure that I would have problems to work on. I write about how I will work on percolation and how I will work on the Komlós conjecture. I put in a paragraph about how I will also think about how to best present ideas and focus on writing good exposition. Even I am not convinced by it.
In the evening, I read. There are blog posts and forums and discussions and an outpouring of words from mathematicians unsure about what the future will bring. I read and hope to find the reassurance that mathematicians will still be necessary to understand and weave narratives. If the historians suddenly had access to a biography of every person who had ever lived, would that be the end of the study of history? Surely not. What if there was a library containing every proof of every question or conjecture that had ever been written? What if they were already well-written, with good exposition? What would we do except write more of the same? A mathematician is a machine for turning coffee into theorems, Erdős said. I think that was on the wall of a classroom when I was at school. I realise that I never really internalised any argument for why mathematics was useful; I was aware of them, of course. I believed them, I still do. But I chose to do mathematics because I loved the process, I loved solving problems. I would become a machine for turning coffee into theorems. That it had a use was a convenient justification.
The next day, I must finish the application; it is due the day after. My research plan needs some polish and some more references, and so I search for a paper I know and stumble upon a reddit thread with four comments: the Komlós conjecture is solved. Ten minutes later, I see that it has been known for three weeks (an eternity in the current world) that there is no percolation at criticality. So I will re-write the proposal and try to click submit before the cutting edge can cut its way onwards. What will you achieve in the future, the form asks me. A mathematician is a machine for turning coffee into funding applications.
I am running two LLMs at once; one on the cloud and one local. The remote frontier model I pay for is checking the manuscript which I was working on yesterday, and I know it has a flaw. I tell the AI, find the flaw; find the flaw and fix it. Produce a Lean verification that the proof works. Do not stop until you have it. I know why this proof should go through, but there have been so many problems and patches and trips back to the drawing board that I’m starting to doubt myself; the certainty would be welcome. The offline model can run on the feeble integrated GPU of my mid-range desktop computer. I ask it to prove a statement, a research problem, and give it a substantial hint; a frontier model would solve this in a heartbeat (I check; it does). The model thinks and chugs away and the fans on my PC whir into life. A few minutes later, it produces a response, crawling out one word at a time, while the frontier model in a data centre who-knows-where rattles through layers of protocols and hammers out lines of Lean by the thousand. The offline model seems happy, it has produced a proof, and its response is full of exclamation marks. I am stunned at first: the proof is only five lines long (and the paper I have written on this spent several pages on this problem). But then I read it and remember, this is an offline model, it is weak. It has failed on some basic algebra, and the trivial induction it has attempted is nonsense. I point out the error and it trundles through some simple rearrangements and concludes – yes! There is an error! My proof was wrong! Would you like me to correct it? I smile at its innocence (how easy it is to anthropomorphise, however hard I try not to). It has no hope of proving anything; it is like the distant past of last year. Will an open-weights offline model be hammering out formalised proofs in 2027? The frontier model pings to tell me it is finished, and the result is formalised, though it has made some significant changes. It summarises the changes in a fifteen-page pdf, which I open and, to my relief, find that the write up is horrible, barely readable. I still have a purpose. Until the next version comes out, at least.
It is the evening again, and I read more. I read about how the goal of mathematics is not solving problems but developing understanding, that there is no point in rushing for priority. I read these articles and scheme about how, now that application is submitted, I’ll have more time to have a good crack at that big problem I’m trying to solve. Soon, AI models will be superhuman across all problem-solving, I read online. I’ll have to be quick in my work, then, before the machines get there. It takes a long time before I see the contradiction in my own thinking. The goal of mathematics is not just to solve a problem and move on, I read.
But if this is not and never was the real goal, then why did we act so much like it was? What have I spent my whole postgrad career (be it only four years long) trying to do? Was it writing good exposition and deepening human understanding? Perhaps I managed to do some of that on the way. A mathematician is a machine for turning coffee into theorems, Erdős said. I read about how the goal of mathematics should not be to solve big problems and move on, and still I plan how I will do exactly this. Would my solution be more insightful, easier to read? Maybe it would, for now.
How deeply I have internalised this aim, to solve problems! But who gets awards, and who gets recognition? I cannot imagine how one could look at mathematics from the outside and not conclude that the goal is to solve big problems, how one could see us argue otherwise and perceive it as anything but us trying to cope. A mathematician is a machine for turning coffee into theorems. So I will try to change. Perhaps I will write an exposition article and make existing proofs clearer. Perhaps that is better mathematics. Perhaps, soon, I will even believe it.
Received 17 September 2026.
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