Numerous posts on Walled Culture have lamented the continuing failure to realise all the benefits from open access. Even when there is a move to open access, it is frequently to gold open access, which requires a payment from institutions when one of their academics publishes a paper with a journal. On the plus side, anyone can access that paper as soon as it is published; but on the downside, any money saved by libraries on subscriptions is needed to the cover the “article processing charges” (APCs) demanded by gold open access titles. The academic publishers have cleverly managed to subvert open access by embracing it on their own terms. They get to claim that they are supporting open access, as now required by many funding bodies, while ensuring that their high profit margins are maintained.
The chief reason this has happened is that academics feel they need to publish in journals with a high “impact factor” – the titles that are referenced the most in other academic papers – and these generally offer gold open access, if anything. Academics claim they must publish there because hiring and promotion decisions use an academic’s publishing record as a proxy for their scholarly worth. In general, only well-established academics, who are already widely-respected in their field and have tenure, or at least a secure position, are brave enough to resist this tendency. They can publish in titles that do not require APCs, which may not be as high profile as gold open access journals, but which are truer to the aspirations of the open access movement.
Repeated attempts to dismantle this system based on gold open access, which brings only some of the potential benefits of open access, have failed. However, in a completely unexpected turn of events, recent results in the field of artificial intelligence may finally provide the long-desired breakthrough, as least in the field of mathematics, and probably more widely.
A few weeks ago, OpenAI announced that an “internal OpenAI system” – that is, an advanced model that has not been released publicly – had made a significant advance in the field of fluid mechanics, which is “the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them”. The main mathematical framework for fluids is encapsulated in the Navier–Stokes equations. They are widely and successfully used in real-life applications, but there are still many open questions about them. OpenAI announced that its internal model had resolved the question “whether smooth three-dimensional fluid motion can break down”, one which had remained unanswered for some 90 years. The importance – and difficulty – of resolving that issue is reflected in the fact that it was one of the Millennium Prize Problems:
The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. Each of the problems has a prize of one million US dollars for the first correct solution, pledged by the Clay Mathematics Institute.
There is some controversy about who exactly should get credit for the new solution of the Navier-Stokes problem, but there is no doubt about its huge significance for mathematicians and mathematics. That flows in part from the forces OpenAI used to obtain its result. According to the company, some 10,000 AI agents worked together for about 88 hours, on a massive scale:
Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens. In the process of resolving the Navier–Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens.
This is not mathematics as we know it. Nor is the end result, which is a 166-page proof that even the world’s top mathematicians struggle to grasp fully. But we know that the result is in fact valid, since its logical correctness has been checked using what is called Lean formalisation. That an AI model was able to solve – apparently without much guidance – this long-standing problem suggests that many others will now follow. Indeed one of the top mathematicians playing a prominent role in the debate about the future shape of mathematics in the age of AI, Terence Tao, has written:
we will transition from an era of proof scarcity to an era of proof abundance. Most of our institutions — journals, priority conventions, hiring and promotion criteria, prizes, the very notion of a research program — were designed under the assumption of scarcity, and it should not surprise us if they behave poorly under abundance.
He also notes that advent of high-level AI mathematics will exacerbate many of today’s problems in the world of academic publishing. There will be a flood of complex, verified but barely comprehensible proofs that journals will struggle to process because of their volume and their sheer opacity. Another leading mathematician, Scott Aaronson, thinks that AI will take over the task of proving theorems almost completely:
in whatever years I have left, I don’t expect that I’ll ever again prove a theorem because I’m actually needed to prove it. If I do, it will only be for my or others’ enjoyment or edification.
Assuming there is indeed a shift to using AI to prove theorems, there will be little point in publishing them in high-profile journals in order to gain academic kudos. First, because if anything deserves credit it is the AI model, and secondly because anyone could use the same or a different AI model to obtain the result independently and then publish it elsewhere. This will lead to advanced AI proofs becoming commonplace and being routinely announced and posted online for anyone to access freely, since publishing the result in a journal would offer no extra value once the proof’s validity had been established.
To begin with, top mathematicians will probably offer their own explanations and ruminations on the implications of complex and inscrutable AI proofs. There might be unique value in those, and so people will publish those as a token of academic achievement, and hiring committees can use them to judge candidates. But this will be a temporary phase. Already AI models provide explanations of AI proofs; they are not very helpful currently, but considering the pace at which AI is progressing it is likely that in the near future they will become good enough to replace the human academics here too. Again, any sufficiently advanced AI model can be used for this explanatory role, making formal publication in journals pointless.
If both the solutions and in-depth explanations of them can be carried out by AI models, the key question becomes: what, then, will today’s highly-trained mathematicians do? This is naturally the subject of extremely lively debate in the mathematical community. Some of the thoughts are quite gloomy. The mathematician Max Weinreich wrote a paper entitled “The crisis of AI-generated mathematics”. Another mathematician, Daniel Litt, wrote of “The End of Mathematics”. But Litt has also written a more positive follow-up post entitled “A beginning for mathematics”, which offers a way forward for mathematicians in the AI age. He calls for the emphasis to shift to the human and social aspects of mathematics — the very things lacking from even the most impressive AI proofs:
The allocative aspects of our job (hiring, graduate admissions, etc.) are in dire need of reform if we want to retain human mathematical expertise. Broadly speaking I think we should focus on rewarding skill in the parts of our jobs that cannot be automated: the internal (e.g. understanding mathematics) and social-relational parts, and operationalizations that hew as closely to those aspects of the profession as possible. For example, talks and sustained mathematical discussion now demonstrate understanding much better than papers. Once AI systems improve at exposition and “digestion,” this will be even more the case.
What’s interesting here is that these are precisely the things that the open access community have been calling for – rewarding all aspects of the academic’s role, not just the extremely narrow one of publishing papers in a few leading journals.
It is still early days – OpenAI’s announcement of the solution to the Millennium Prize problem is only a few weeks back – but there is an emerging consensus among mathematicians that mathematics will never be the same again. AI proofs of many outstanding problems will become quicker and cheaper, which means that it would be wilful not to accept them. As the ability of AI models to explain and “digest” these complex proofs improve, that will necessarily encourage mathematicians to place more emphasis on the human side of their activities.
That, in turn, will mean that academic bodies responsible for hiring and promotion will need to de-emphasise – perhaps ignore completely – today’s obsession with publications in high-impact titles. And once that is gone, there will be no pressure for academics to publish in “prestigious” gold open access titles, since prestige will increasing come from other, broader activities – giving lectures, providing mentorship to younger colleagues etc. This will give them greater freedom to choose whatever kind of open access they want, including green and diamond open access, for whatever publications that still make sense in this new world of AI-assisted mathematics.
These changes are already taking place in the world of mathematics, which is the first to be impacted by the new capabilities of powerful AI models. It is not yet clear how quickly other fields will be affected. Academic fields closest to mathematics – physics, chemistry, engineering etc. – are likely to be the first to see similar dramatic breakthrough results from AI models. Moreover, if mathematics successfully shifts away from today’s emphasis on publications towards a broader appreciation of skills, it may be that other fields will recognise the benefits of doing so and start following suit – with or without pressure from AI.
Featured image by OpenAI.