Duminil-Copin on AI and Mathematics

Hugo Duminil-Copin recently wrote a very nice piece on AI and mathematics, where he gives us a delightful image:

A mathematical question is much more than a theorem waiting to be proved; it is a source of momentum. First and foremost, it is a lighthouse in the night: it illuminates and guides mathematicians in their wanderings, both esthetic [sic] and scientific…

Arguing that we ought to “let the lighthouses built for us shine a little longer,” he reminds us:

What is at stake is not only how we choose problems, but the culture and ethic that shape our profession. The social consequences for our community, especially for the younger generation, are devastating.

And later, recalling Thurston’s On Proof and Progress in Mathematics (1994), he writes:

Let us recall that our mission as mathematicians is not limited to producing theorems, we must before anything produce understanding. We remain the first link of the chain turning very abstract notions mastered by few experts into concepts that become part of the culture of next generations… Ideas need time to emerge, to be refined, tested against one another, understood by multiple mathematicians. That is the price we must pay for ideas to mature in our minds and ultimately become genuine intuitions. Once properly mastered, these intuitions can be passed to other mathematicians, then teachers, and kids. This is exactly what happened with 0, the = sign, negative numbers, infinity, fractals, etc. And this process is ongoing with more recent concepts.

I am reminded, in this regard, of the following remarks by Wittgenstein in Culture and Value (1980):

Our civilization is characterized by the word ‘progress’. Progress is its form rather than making progress being one of its features. Typically it constructs. It is occupied with building an ever more complicated structure. And even clarity is sought only as a means to this end, not as an end in itself. For me on the contrary clarity, perspicuity are valuable in themselves. I am not interested in constructing a building, so much as in having a perspicuous view of the foundations of possible buildings.