The Dark Night of Mathematics

What are we really doing?

I 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:

If The Library of Babel existed, would authors continue to write books?

What if the library of Babel was being constructed before our eyes and there was some mechanism for separating the masterpieces from the random strings of text, and all the masterpieces were dropping as fast as publishers could scoop them up? Would authors stop writing then? What about if the very second someone began composing a story in their private word processor, the demonic Master Librarian read their mind and completed their story in one million ways, then used some oracle to pluck the best for publishing. What then? The answer is probably the same no matter how bad I make it. The author still writes. But why the hell would we do this? Why would we force the author to endure this nightmare?

What about this: What if we told the author that they would never write again. They are forbidden from creating original works to express themselves. However, they are still permitted to comment on writing, interpret it, share their taste. They are still valued for their appraisal, presentation, understanding and appreciation of creative writing. They just can’t write creatively anymore. They can go on as an enthusiastic spectator. Do you think they’d snap?

Perhaps I shouldn’t tell you this, but my aim is to be open: These developments have triggered some deranged thoughts in me. I have wondered if it is the express goal of these companies to make me kill myself. Am I alone in this paranoia? If we loosed a powerful demon in the machine, what would that look like? Would it consume lots of power, and gleefully imitate us, and tell us anything we wanted to hear? Would it fuel our delusions, and generate unspeakable images and give us (for a price of course) anything we desired?

If a mathematician made a deal with the devil, what do you think they would ask for?

The story of human discovery and the triumph of the human spirit will soon be excised from this discipline. The Dinitz-Garg-Goemans counterexample was the most egregious demonstration. It revealed that the process of prompting novel proofs will be as auraless as ordering doordash. Watch as magic and mystery evaporate. Watch as the sun sets on our heroic age. Is there not something evil in the act of blocking all future generations of mathematicians from the experience of discovery? Forget about accuracy or even attribution. Something fundamental to the experience of mathematics is being taken.

You are a helpless onlooker. Before you a channel through which humans have accessed the ineffable and sacred for thousands of years is being sealed for eternity.

None of this may come to pass. I am not interested in forecasting and speculating. I am giving you only this: The impact of a worst case scenario on the human heart. That is my futile outpouring, my dark night of the soul. Thank you for reading it. I have named my suffering and maybe I have named yours. I invite any and all responses to this piece. If you feel as I do, please express it. If my words provoked a wash of sadistic elation within you, then let everyone see you. Leave nothing unsaid.

There is nothing I can do. There may be nothing you can do. I have no prescriptions, policy recommendations, or coherent call to action. I just want to be honest and open about my emotional and spiritual response. I want to feel seen. I want folks like me to feel seen. I need the architects of our new mathematical paradigm to look me in the eye and acknowledge our shared humanity and soul before they deliver the coup de grâce. I need, most of all, for us to understand what we are really doing.