Colby Cosh: The mechanical monster storming through an ancient priesthood

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Oct. 6 will be remembered in mathematics as the day the bomb went off. OpenAI, the San Francisco-based “public benefit corporation” that has been making impressive mathematical advances every few weeks for a while now, unleashed a flood of 377 new machine-generated papers, each of which contains a new result. Literally hundreds of outstanding theoretical conjectures of longstanding interest have been resolved, found true or false, without human intervention.

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Some of these have resisted human solution for decades. (The one known as “Hilbert’s tenth problem” has been partially open since 1900.) No active subfield of higher mathematics is left untouched; the mechanical monster has stormed through the corridors of number theory, algebraic geometry, combinatorics, topology and computing science. And while the degrees of claimed rigour in the new papers vary slightly, the typical reaction of most expert human inquirers into the results they are capable of understanding has been, “Gee, this looks pretty solid,” often with a side of, “I didn’t think of trying that solving strategy back in 2003 when I spent 18 months on this problem.”

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As you already know if you are close to any professional mathematician, OpenAI’s earlier accomplishments have already led to total moral crisis in the world of mathematics. None of the questions that OpenAI’s software has answered for us are easily explainable to a layman without some prior education. That certainly includes me; I’ve spent a weirdly large amount of time “studying” higher math and listening to lectures in a superficial way over the last few years, so I have a journalistic, half-comprehending sense of what mathematicians value and what sorts of questions absorb them. I don’t see a paper in this barrage that I could discuss intelligently and specifically, with you or with anyone else.

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The big picture is that mathematicians are a privileged caste that civilizations, western and non-western alike, have sustained for thousands of years in a state of sacralized purity. There are specialists in “applied” mathematics, but in general, no mathematician is ever expected to create anything economically or practically “useful” to anyone but another mathematician. It is the fostering of the whole tribe, with a necessary stellar core of persons engaged in pure thought, that sometimes delivers technological value to the rest of us and justifies its existence.

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Long ago a priesthood embarked on tackling questions about number and geometric magnitude and how they’re related. These questions proved unexpectedly fertile and complex. Notions of dimension and probability entered the picture. Concepts of number began to multiply confusingly. The fundamental binary of “true” versus “false” began to be joined by other dichotomies: infinite versus finite, countably infinite versus uncountably infinite, algebraic versus non-algebraic, computable versus noncomputable, decidable versus undecidable.