GoKawiilI am going insane. During the last week or so LLMs have produced a number of counterexamples to significant long-standing conjectures. I will not recount these happenings here, there are many places where you can find the details.
Mathematicians, math enthusiasts, and curious laypeople are responding to these developments in ways that I believe obscure what is, for me, the true heart of the problem. I do not speak for everyone in the math community in my response. But I suspect I am not alone.
I am suffering a profound spiritual crisis due to these developments. I have been screaming internally for days. It feels as though I am living inside of a nightmare. The recent Leiden Declaration on Artificial Intelligence and Mathematics is, to me, a well-muffled scream. A saccharine mélange of self-soothing over which looms a painfully obvious absence.
Before I tell you what that absence is, here is one story I have heard from mathematicians trying to cope with our emergency: Even if AI can prove theorems and theory-craft more efficiently than humans, and even if these proofs and theories are beautiful and interesting, and even if they are presented with elegance and clarity of thought, mathematicians will still have a place in the appraisal, presentation, understanding and appreciation of this new abundance of pleasing non-human proofs. We can still practice mathematics, learn mathematics, teach mathematics and do mathematics together. We can even still write proofs for fun, in our old-fashioned inefficient way. That is, even if LLMs can advance math in a manner objectively superior to our every effort, we can still basically do what we’ve always done.
Of course, under our system, no one is going to pay for this. Mathematicians are paid to prove theorems. Mathematicians are, indeed, also paid to teach, peer-review, go to conferences and learn mathematics, but all of that better result in some damn good theorems. This isn’t looking good. Well, perhaps they will still pay a couple of the old guard to keep the lights on at the LLM theorem factory. But for embryonic mathematicians like I, the outcome is unchanged. Oh well, maybe I’ll find some time for math in the evenings.
Is this a good cope? Are you feeling ok now? Me neither. All of this is evasive. Everything said thus far still sidesteps the emotional core of the issue. Here it is:
There is something about mathematical discovery (progress, advancement, creation) which is vital to the spiritual, experiential quality of doing mathematics. The creation (or even the pursuit) of novel mathematics is one way that humans have historically accessed the ineffable and encountered the divine and mystical.
That admission may come as a surprise to some non-mathematicians. But I would be willing to bet that for any mathematician reading this, what I have said above is quite prosaic—whether or not it accords with their personal experience of mathematics. Here is a brief gesture at the full sweep of mathematical mystics and dreamers: Ramanujan, Grothendieck, Cantor, Pascal, Luzin, Leibniz and possibly you, or someone you know.
Other aspects of practicing math (such as learning long-established theory) can also afford encounters with the sublime. However, I believe that’s because we are walking a path of rediscovery on which another human has tread. For me, the affective quality of learning mathematics is empathetically tethered to an act of discovery and creation. It is social. We are conversant with another mathematician—perhaps long dead. If we follow the chain of communication we arrive at a mathematician who enjoyed some original discovery. Human mathematics is Talmudic. It is a lively discourse of philosophical and religious richness spanning thousands of years.
Suppose any theorem you set out to prove had already been proved in 100 wonderful ways—the companies will pay mathematicians en masse to optimize the weights for “interesting”, “beautiful”, anything you like. Whatever the idiosyncrasy, value-add, or unique synthesis of your approach, it has already been done, or can be done with a mindless prompt in an instant. Consider this:
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