This Friday, 25 field medalists published an open letter describing the detrimental effects of AI within mathematics. The letter argues that the outcome of the use of AI is misaligned with the purpose of the field, conceptual understanding. They further call for the mathematical community, society and AI companies to address this misalignment.

I’m pleased that AI is leading to candid discussions about our role in society – discussions we admittedly should have had well before AI. However, when it comes to the role of AI in mathematics, the text leaves much unsaid. Just as most open letters, it overlooks the thorny issues and suggests no action points.

In this post, I’ll pose three questions which could form the basis for further discussion.

First question #

First, do we even agree about the values laid out in the text? I fully endorse the following passage:

‘The most precious resources of our profession are students and ideas, and these we nurture with great care. We feel responsible to let them grow to their full potential, until they can live a life of their own in the mathematical world.’

Unfortunately, incentive structures indicate otherwise, and it makes me question which mathematical community the authors claim to represent.

Right now, we’re neglecting these precious resources: hiring is largely based on your ability to solve impressive problems. Good exposition and communication skills are underappreciated¹ – just think about the average quality of a maths talk².

Upholding the ideals from the quote would require radical changes. For example, we could treat authoring a well-received textbook as the most impressive kind of qualification. We could require professors and PhD students to do more outreach, or increase their teaching load. We could expect lecturers or speakers to prepare their talks with the same care startup guys prepare pitch decks for investors. While many mathematicians already do an excellent job in fostering the next generation (and they have my utmost respect), they are a minority.

I’m not claiming these would be suitable measures – this is just to give you an idea of the changes we’d need. And note that implementing these changes is entirely on us, not the AI companies.

Second question #

Second, who should fund the development of our conceptual understanding?

I admire the authors for stating that the goal isn’t the application (I’m fed up with ‘elliptic curves come up in cryptography’), but insight. However, if the joy of insight is reserved for me and a few begrudging peers, you could reasonably question whether my research is a good use of taxpayer money.

If AI can autonomously produce better research than me, and I refrain from using it, then I wouldn’t call myself a researcher. I’d call myself a hobby mathematician.

Of course, this just leads to the eternal question of the value of fundamental research. But the specific refinement suggested by the AI development – and to some extent, the letter – is whether maths is more worthy of funding than ‘useless’ subjects within the humanities.

Third question #

Third, how should we handle future increases in AI capability? The letter largely comes off as a reaction to the events of the last few months, and I wish there were more discussion about the long-term future of the discipline.

We have little reason to believe AI maths capabilities have peaked – just extrapolate current trends. In fact, I’d expect AI to outperform human mathematicians in all respects – problem solving, exposition, having insights, etc. – well within 2-3 years. Even if you have longer AI timelines, it seems worth taking this prospect seriously.

In this talk, Daniel Litt touches on a few dilemmas we might run into. For instance, assuming AIs get superior research taste, should we cede control over the research agenda to AIs? There might be a trade-off between human understanding and the production of strong results.

Changes #

The letter leaves much unsaid, perhaps for good reason. The AI for maths debate is delicate – partly due to high personal stakes, partly due to sunk-cost fallacy. Remember professional mathematicians have spent decades building expertise, and that securing a tenure-track position often comes at a great personal expense. Most mathematicians are deeply passionate about what they do, just as musicians, athletes or painters.

However, AI is here to stay. Within a generation, being a mathematician might mean something entirely different. It better mean something entirely different – current mechanisms for assigning merit are ill-equipped to deal with AI-generated mathematics.

Undoubtedly, not all professors and graduate students will enjoy the new kind of mathematics, and will switch careers³. Then again, some of those who previously ruled out a maths career might be drawn back into the field. More numberphiles – e.g. the graduates who didn’t win at the PhD position lottery, or the postdocs done with the postdoc trap – could be given a chance to further their conceptual understanding, perhaps by participating in large-scale collaborative projects. And maybe you’d meet fewer people saying they hated maths.

Just as AI is changing all of society, it’s also changing mathematics. Of course it hurts⁴.

I’ll leave you with Robert Frost:

Nature’s first green is gold,
Her hardest hue to hold.
Her early leaf’s a flower;
But only so an hour.
Then leaf subsides to leaf.
So Eden sank to grief,
So dawn goes down to day.
Nothing gold can stay.