Graph Theory and \(\mathfrak{sl}_2\)

December 15, 2016

How many unlabeled graphs are there with \(n\) vertices and \(k\) edges? (More below the fold.)

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February 21, 2026

Mathematics in the Library of Babel

February 21, 2026

Mathematics isn't only about saying true things. It's about asking the right questions, being confused, stumbling about, getting distracted, being wrong, recognizing when you're wrong, being stuck. Mostly being stuck. It's about clinging to a giant edifice and feeling it out until you understand some tiny piece of it. It's about finding meaning in and intuition for the texture of an object which, at first, can only be apprehended by bashing your skull into it until it imprints on your forehead. Then trying to convey some of that insight to someone else, and watching as they find their own way to it.

I started trying to get LLMs to do math in July 2020, through the game "AI Dungeon," one of the earliest applications powered by GPT-3. I first got GPT-3 to produce a correct proof (of Fermat's Little Theorem) in April 2022. At the time I did not think they would become useful for math research in the near term.

This changed when the first reasoning models were released: on February 1, 2025, I wrote that the model o3-mini-high “clearly has passed the threshold of genuine usefulness” for research, while still making many, many mistakes. Since then, the models have improved, and ChatGPT 5.2 Pro (released in December 2025) can regularly provide reasonable proofs of lemmas that I would characterize as “involved but routine for experts,” though it still makes many errors. And I have been using Codex, OpenAI's coding/computer use agent, for scientific computing tasks I would not have considered attempting a few months ago.

In public comments, I've tried to credit successes while pushing back against hype. I've talked a lot about "slop" papers on arXiv. I have worried that we are polluting the scientific commons with incorrect mathematics whose errors are enormously difficult to detect. I've tried to focus on the present. In this essay I'll talk about the future.

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arXiv in trouble?

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