To grieve, or not to grieve?

Recent events in the field of AI for mathematics have shown us beyond all reasonable doubt that the field is currently undergoing a rapid transformation, unlike anything that we have ever seen before. Language models are solving hard problems which humans could not, and mathematicians are reacting very differently to this news. I personally am extremely excited about the future of our field. However it is becoming clear to me that my views are not shared by everyone in the community: indeed, many of my colleagues seem to be upset. So let me start by outlining my understanding of their positions.

The AI pessimists.

Elisabeth Kübler-Ross’ model of the process of grief is now well-known: the five stages are denial, anger, bargaining, depression and acceptance. The first four of these are currently very well-represented within the mathematical community, and perhaps give some weight to the claim that many mathematicians are currently grieving what they perceive as a loss of something much beloved to them. However there will also be some mathematicians whose viewpoints will fall into the following categories and who will probably strongly argue that what they are feeling is not grief at all. We are not all grieving. But many of us are in one of the stages of grief.

Denial

The Association for Human Mathematics prominently states that its goal is to protect mathematics as a human endeavor against what it calls the “threat of artificial intelligence”. Members commit to not publishing AI-generated outputs, and there is an “AI-free” option to completely renounce the use of AI in a research capacity and commit to finding proofs and exploring examples without using these tools at all. From talking to its members, my understanding is that there are multiple motivations for joining this association, ranging from serious concerns about what AI is doing to mathematics, to serious concerns about what AI is doing to the planet or what it might do to humanity. A prominent member of the association is Fields Medallist Peter Scholze, who in the recent Heidelberg Laureate Forum panel on AI and mathematics said that he will not use AI and furthermore he will “die on that hill, and be some public figure that dies on that hill”. You can listen to his reasons here.

Anger

The proofs and prompts website contains posts which represent a wide variety of opinions, and it is not difficult to find posts by people who are angry. One prominent example is Vlad Lazic’s post here. Lazic has outlined his views in more detail here and that site contains several links to pieces written by other mathematicians who have similar feelings.

Bargaining

The Advisory Group on Mathematics and Artificial Intelligence are an independent committee of senior mathematicians (including three Fields Medallists) who on Tuesday released recommendations in response to the news that OpenAI is sitting on “a large number of significant results in mathematics that they report have been produced by their internal model”. One quote from the first paragraph of the recommendation is that the group “do not endorse this practice, and we ask [the frontier AI labs] to stop testing advanced mathematical problems on proprietary models.” They go on to ask the labs to release results “responsibly” and suggest various courses of actions that the labs could consider going forwards, such as “to provide support, including funding, for the development of human understanding of the AI mathematical output that they release.” I’ll come back to these “significant results”, and also the phrase “human understanding”, later.

Depression

If you are a mathematician then you almost certainly already know at least one person who feels like this. A faculty member I know who works in fluids told me that the Navier–Stokes news was “extremely depressing”. A post-doc I know told me that they were considering leaving mathematical research because of what it was about to become. A PhD student I know told me that they were stuck on a lemma in their research and ChatGPT one-shotted it and it made them wonder what the point of it all was.

Interlude: why humans do mathematics.

I should first make it clear that the above list is certainly not intended to be a criticism of any of these positions; I hope I have given each one a fair treatment. Several people whose opinions I value highly are represented in the above list, and recently I have been trying to listen hard to what my possibly-grieving friends and colleagues are concerned about. But it is still the case that I wake up every morning feeling excited about the future of my field. Before I explain why, let me discuss the contentious question of why we do mathematics.

Thurston’s view of mathematics

The fact that AI can now prove hard theorems has quickly led humans to a discussion about why humans do mathematics at all. I would imagine that a few years ago, many mathematicians would have been happy to agree that the mathematics community gives out our biggest prizes and awards to the people who prove the hardest theorems. However, now AI is proving hard theorems, we seem to be spending a lot of time explaining that in fact we are giving these awards and prizes to the people whose ideas are giving us the deepest understanding of our field. It is also worth noting that at this point in time the deepest AI-generated mathematical proofs have been typically accompanied by a poorly-written pdf or no pdf at all, and perhaps an end-to-end formalization in Lean, giving super-human confidence in the correctness of the proof but little clue as to where the ideas came from, or even where the new ideas are. Again I defer to Scholze, who here points out that the Elkies–Klagsbrun construction of a rank 29 elliptic curve over the rationals came with a geometric explanation, but the new rank 30 and rank 31 elliptic curves came with equations and nothing else. “We didn’t learn anything really”.

The oft-cited justification of the “it’s not about the theorem tally, it’s about the human understanding” framing of our field is Thurston’s 1994 essay “On proof and progress in mathematics”, and indeed Tao’s blog now carries a quote from this essay as the splash at the top. As one can imagine, this framing of the “point” of mathematics has been met with some cynicism on social media, and even claims that mathematicians are moving the goalposts in a desperate attempt to survive. However this cynicism is not really justified; experts in the field have known perfectly well for decades or more that the prize-winning results give us not only a chunky new theorem to add to the pile, but also a big dollop of new understanding to go on top. To choose an example from my own field, the Wiles–Taylor–Wiles proof of Fermat’s Last Theorem simultaneously resolved a silly little 350-year old puzzle about positive integers with no applications, and gave us genuinely profound new insights into the Langlands Program, the ramifications of which are still being felt 30 years on (see Frank Calegari’s 2022 ICM talk for plenty of evidence for this claim).

Other views of mathematics

But Thurston’s view is not the only view in town. I re-read Hardy’s “A mathematician’s apology” this week. I first attempted to read it as an undergraduate and back then it struck me as extremely pompous and sexist; I know that the world was a different place in 1940, but I did not even make it to the end before giving up. This time I got there, and my understanding of Hardy’s viewpoint is that it is completely different to Thurston’s. Hardy is doing mathematics because he views it as art (so presumably he would be angry about AI doing mathematics for the same reasons that many artists are angry about AI doing art). He is scathing about those who “explain” (take a look at the first paragraph of the book) and frank when he tells us that one of his main motivations to prove theorems is so that he is remembered after his death. This is a view very different to Thurston’s, but also very different to my own.

Let me next discuss why promoting Thurston’s view of mathematics (which is what many of us are doing right now) may ultimately backfire on our community. AI has solved a Millennium problem and this has shown us where AI is in mathematics. But I believe that many people in our community are still vastly underestimating how fast AI is moving, perhaps because they only just started paying attention to it. If we use “understanding” as a justification for the continued existence of mathematics as a subject worth studying, then where exactly do we retreat to when in 1 year’s time AI is not only proving theorems, but also doing a perfectly good job of explaining them to humans?

I remember when I first read Thurston’s essay, and how it struck me as being rather at odds with what personally motivates me to do mathematics. I have recently been diagnosed with autism (my few friends all think that this is hilarious, having known this for decades) and I wonder whether this is one of the reasons that I hold my position. Thurston’s essay was a response to an earlier paper by Jaffe and Quinn, who argue that a mathematical idea really only becomes valid or useful once it is anchored to the ground by a rigorous proof, and they criticise some authors (including Thurston) for having important insights and then not taking the time to write down full details. My own area (the Langlands philosophy) is full of arguments which seem to be “known to the experts” without a publicly-available write-up, and this situation was what caused me to become disillusioned with the field and ultimately withdraw from curiosity-driven research completely and switch to mathematical formalization. You cannot fool Lean; one learns this very early on. Lean will not accept proof by authority or proof by intimidation; it doesn’t work like that. I feel safe with Lean.

“Understanding” mathematics.

As a PhD student of Richard Taylor in the early 1990s, I quickly understood that the statement of the theorem which I would attempt to prove in my thesis relied on a construction of Deligne attaching Galois representations to modular forms. I suggested to Taylor that I first read Deligne’s proof before continuing, and he was quick to shoot down this idea, pointing out that my funding was for 3 years only and I simply did not have enough time to get on top of all of the relevant literature at this point in my career. I never did find the time to read Deligne’s construction, or the proof of the Langlands–Tunnell theorem which was crucial in Wiles’ work, or many of the other things which I needed in my thesis and subsequent work. So do I “understand” my own work? What exactly do we even mean by “human understanding of mathematics”?

I think that whilst mathematicians are now beginning to agree that mathematics is all about human understanding, I am not entirely convinced that they will agree on what it means to understand high-level mathematics. Of course we all know it means to understand undergraduate-level results; we have taught the courses and checked everything carefully. I am able to explain all of the undergraduate algebra courses which I have ever lectured, right down to the axioms of set theory and also right down to the axioms of type theory. I understand the material in a visceral way.

But do I “understand” the proof of Fermat’s Last Theorem, given that earlier this year I gave 22 hours worth of lectures on the topic? I know that Langlands–Tunnell is crucial for Wiles but I know very little about the details of what goes into it. What if there had been a mistake discovered in the process of its formalization? Can one “understand” mathematics which is wrong? Mathematics which is incomplete? I personally can not, and I personally am happy to not understand some things. I’m in it for the theorem tally. In stark contrast to Thurston, I believe that understanding is a slightly nebulous concept, whereas total number of theorems correctly proved is something which we can measure. I am well aware that I seem to be in a minority here. But I wanted to share these views anyway.

I personally call for OpenAI to simply dump their collection of theorems upon the world so that we can see exactly what they claim to have done. The tidying up can come later, and it will come later (although it will certainly come sooner if it is funded by OpenAI); in every case other than Navier-Stokes we have seen humans reverse-engineering AI-generated mathematics, and both Navier-Stokes and the new currently-secret theorems in OpenAI’s possession will be no different: if humans want to understand, we will understand. Right now humans may well be spending their time trying to prove theorems which are already proved by a tech company. Where is the logic in that? These humans have a right to know what is known. I signed this open letter urging OpenAI to simply go public with their data, because I think it is the optimal approach, given the situation we find ourselves in. If you agree then please sign it too. Sure the tech companies could make our life easier. But what is happening now is akin to censorship.

An AI optimist.

Let me now explain why I am an optimist. Time will tell us how well my opinions age.

In a 2020 piece in the Notices of the AMS, I asked the following question: “If one human had an understanding of all of modern pure mathematics simultaneously, how much further would they immediately be able to see?” Six years later we are beginning to understand the answer to this question. Machines have ingested the mathematics on the internet and are able to manipulate this data in a coherent way. The Erdős unit distance disproof came about because a machine happened to be an expert both in discrete geometry and class field theory; one rarely finds humans who are simultaneously experts in both. Machines are answering questions now which one could easily believe that humans, were they to be left alone for a few more years in the right groups, could also have answered. It’s just happening a whole lot more quickly.

Above I expressed some scepticism about mathematics being all about understanding — both the sentiment itself, and the risks that one is taking by reducing mathematics to this. My personal answer to the question of what mathematics is all about is that it’s about proving theorems (problem solving), developing tools (theory building) and seeing where to go next (conjecture formulation).

Right now we have seen a lot of evidence that computers are good at problem solving, but far less evidence of their abilities at either theory building or conjecture formulation. This puts a hard limitation on how far today’s machines will go. Time will tell us how much better tomorrow’s machines will be at these skills, and there surely will be progress, but we are yet to see anything decisive. Will there be sufficient growth in these key areas before external pressures such as cost become too great for progress to be worthwhile?

To those who dismiss the observation that machines can’t do X today with the reply that “humans just don’t understand what exponential growth looks like”, I invite you to consider the following facts. Current progress is indeed exponential, and may well continue to be exponential for a while, however we cannot expect exponential progress to continue indefinitely (exactly because of what exponential growth looks like), and furthermore mathematics is infinite which beats exponential hands down; those that don’t believe this just don’t understand what infinity looks like.

I thus believe that in the future we will reach a new “natural boundary” in mathematics, beyond (and perhaps way beyond) where we are now, but where machines are going to get stuck and where it is not viable to expend any more resources to make the next big leap. I am aware that I might be wrong. I am an optimist, so I am expecting future machines to be awesome; but I simply cannot see how they can get to infinity with finite resources, so they must stop somewhere. I believe that the optimal thing to do (at least from my personal perspective) is to let the machines loose, see what happens, and then begin the journey to where they have stopped. Things are currently moving fast. They cannot move fast forever. But if we get on board now then they will take us to extraordinary new places. And after we have arrived, the new adventure will begin.

This is why I have also signed this letter. I know that some readers will find the ideas in it abhorrent. But I cannot fail to be excited by the powerful new tools which we have. Wir müssen wissen — wir werden wissen!