The Ironies of Higher Arithmetic
The "abc conjecture," first proposed in 1985, asserts a surprising connection between the addition and multiplication of whole numbers. (The name comes from that amiable equation, a + b = c.) It appears to be one of the deepest and most far-reaching unresolved conjectures in mathematics, intimately tied up with Roth's theorem, the Mordell conjecture, and the generalized Szpiro conjecture.
Three years ago, Shinichi Mochizuki of the University of Kyoto claimed to have proved the abc conjecture—potentially a stunning advance in higher mathematics. But is the proof sound? No one has a clue.
Near the end of last year, some of the world's leading experts on number theory convened in Oxford to sort abc out. They failed.
Mochizuki's would-be proof of the abc conjecture uses a formalism he calls "inter-universal Teichmüller theory" (IUT), which features highly symmetric algebraic structures dubbed "Frobenioids." At first the problem was that no one (except, we must suppose, its creator) could understand this new and transcendently abstract formalism. Nor could anyone see how it might bear on abc.
By the time of the Oxford gathering, however, three mathematicians—two of them colleagues of Mochizuki's at Kyoto, the third from Purdue in the U.S.—had come to see the light. But when they in turn tried to explain IUT and Frobenioids, their peers had no idea what they were talking about—"indigestible," one of the participants called their lectures.
In principle, checking a proof in mathematics shouldn't require any intelligence or insight. It's something a machine could do. In practice, though, a mathematician never writes out the sort of austerely detailed "formal" proof that a computer might check. Life is too short. Instead, she (lady-friend of mine) offers a more or less elaborate argument that such a formal proof exists—an argument that, she hopes, will persuade her peers.
With IUT and abc, this business of persuasion has got off to a shaky start. So far, converts to the church of Mochizuki seem incapable of sharing their newfound enlightenment with the uninitiated. The abc conjecture remains a conjecture, not a theorem. That might change this summer, when number theorists plan to reconvene, this time in Kyoto, to struggle anew with the alleged proof.
So what's the news? It's that mathematics—which, in my cynical moods, I tend to regard as little more than a giant tautology, one that would be as boring to a trans-human intelligence as tic-tac-toe is to us—is really something weirder, messier, more fallible, and far more noble.
And that "Frobenioids" is available as a name for a Brooklyn indie band.
