Through the looking-glass

In Suzanna Clarke’s wonderfully entertaining novel Jonathan Strange and Mr Norrell, there is an episode in which the magician Jonathan Strange travels from the Bedford coffee-house in central London to a villa in suburban Hampstead by walking into a mirror in the former and out of one in the latter. Behind the mirrors, he explains, he has discovered the King’s Roads, an immense shadowy labyrinth of bridges and corridors stretching beyond even his imagination. At the risk of reinforcing still further the mathematical magus complex, this is the incident that I was reminded of by the recent news of Shinichi Mochizuki’s claimed proof of the abc conjecture.

I’m not competent to understand, let alone to judge, Mochizuki’s work, but as far as I can make out the outlines they’re something like this. The abc conjecture, while it doesn’t have the absolutely elementary quality of, say, the Fermat–Wiles theorem, is a result that can be explained in a few minutes to anyone with a basic mathematical background. The essence of its power lies in the fact that it connects the essentially multiplicative property of primality with the simplest possible additive relationship, a + b = c; perhaps it’s not surprising, then, that if proved it would have repercussions across the field of Diophantine equations and beyond.

What is so Strange-like about Mochizuki’s approach, like so much of modern mathematics, is its indirectness. Amateurs, by and large, seek elementary proofs of elementary-looking problems. Professionals, increasingly, approach them by stepping away from the original problem, into levels of higher and higher abstraction. This was the case for the Fermat conjecture, which as most of us know by now falls out as a consequence of the Taniyama–Shimura modularity theorem famously proved by Andrew Wiles — a deep relationship between objects in the far-flung fields of topology and number theory.

This indirectness seems, as far as I can make out, to be even more vivid for Mochizuki’s proof of the abc conjecture. He appears to have constructed (discovered?) not just a new mathematical bridge, but an entire landscape of “Inter-Universal Teichmüller Theory”, itself building on the work of that supreme visionary Alexandre Grothendieck. Having found a way to enter this landscape, he has spent years exploring it — remarkably, issuing bulletins on his progress but without giving details of his discoveries — until a few days ago he walked out of the mirror with four papers setting out what he has found, including a proof of abc.

I’m reminded — though this is probably an echo of imagery rather than methodology — of Seamus Heaney’s statement in The Redress of Poetry that “If our given experience is a labyrinth, its impassability can still be countered by the poet’s imagining some equivalent of the labyrinth and presenting himself and us with a vivid experience of it.” A profound mathematician, it seems, is an imaginer and sharer of labyrinths; someone who can walk from one place to another in this world by taking a journey through that stranger world between the mirrors.

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