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AI, Mathematics, and map building
It is obviously a tumultuous time in the field. And, I am merely an onlooker since I am certainly not a mainstream theorem-proving mathematician. There are a few talks/discussions/reports that I have watched over the past months that I have found provocative and that speak well on the topic.
A recent forum where the panel members are organised from left to right according to their stance on AI, from AI-leaning to, I suppose, whatever the opposite might be.
A recent talk by Terry Tao at the ICM.
I do find talks by Tim Gowers on this topic somehow always on point or at least worth hearing out.
MIT released this report that I really did read page-to-page and found very useful.
I myself feel only capable of asking some questions that I think are not unique or even my own:
How should we teach young mathematical scientists and scientists? I heard Geordie Williamson say something like "to oscillate between AI-heavy weeks and AI-zero weeks", which seems to literally find a middle ground.
How should we communicate with each other? By this, I include whether and how we should use AI in writing the words we put out into the world. I think we could, at this point, be declaring our use of AI in the words we put out there. For example, in writing this post, I have only used the auto-correct feature of my free version of Grammarly. And since I am too cheap to pay, I am only able to click on the red underlined words.
In instances where a target in applied sciences is determined, let's say to test human safety and efficacy of an AI platform developed drug, it seems like a powerful thing to develop is a benchmarked computational human model. The difficulty here of course, lies in the fact that you run a trial, designed as it is these days, I imagine, to account for the differential response arising from the standing genetic and phenotypic variation in a human population. So what we really need isn't a human computational model to test new candidates on, but one that somehow captures variation. Another way to say what I am trying to say is that we really do need to appreciate that the idea of clinical trials must evolve. They are simply too expensive now, relative to the ease with which we can generate candidates. In my view, what is rate-limiting in the advance of AI-aided human health is how we evolve clinical trials.
In the instances where the results aren't directly impacting something like human health, where, much like in the history of medicine, we were OK with the thing "working" without understanding it, we simply must understand it. Thinking of this other asymptotic, I am reminded of David Gross's comments about the geometry of knowledge. What he suggests is that, contrasting the onion theory of knowledge where we think there are some central elemental facts that science/truth-seeking is attempting to find, he thinks that the model is more inside out. We are growing our knowledge out into an expanse of the unknown, rather than getting into the core of the truth. In this geometry, knowing that something is true becomes necessarily related to why it's true, how it is true, what else is true/false, and where we can go from here. In this sense, adopting David Gross's geometry here, what AI assisted mathematics and science will lead to I think is an incredibly fractal boundary between the known and the unknown. By this I simply mean that the boundary of knowledge will now become increasingly "rough". But here is David's point. the surface area of this growing mass of knowledge is the frontier. Its where our understanding meets our ignorance. We want this interface to be smooth, not rough, because we want the surface area to be least. So one very important goal here is to ask what plays the role of an effective surface tension in this space to unruffle all the ruffled parts? Imagine a knowledge space where instead of surface tension connecting thin necks of knowledge it did the opposite. Now we have these disconnected blobs of stuff, which we call physics, chemistry, biology, mathematics, etc. And if unification is a goal that we might have in science then this surface tension is crucial. It connects knowledge space up.
Somewhat related to this is Peter Scholze's comments on what actually happens to mathematics over generations. How are young mathematicians able to contribute at the frontiers of knowledge despite the thousands of years of "stuff they have to get on top of"? Scholze says it's because what is happening in mathematics is this constant demand and desire (is it a human one or one demanded by the geometry of this space itself?) to constantly compactify. To say things more simply and compactly. To have increasingly better definitions that now allow you to get onto your budget airline seat with only a carry-on bag instead of a checked bag. This revisiting and synthesis of things already known is an enormous part of what we do in science. And this, I think, isn't happening at the frontiers of knowledge, which is ruffled and constantly dynamic, but all the way back here where things are apparently well understood.
So what was this piece of writing really about? Couldn't it "all" be automated? Maybe, I don't know. But that isn't the point. The point is to begin to identify all that is happening on all the timescales of human truth-seeking activities. I think there is more we are doing than we even know. And perhaps before we ask whether it can all be automated it is worth reflecting on what it "all" is...
