The n-Category Café

Egan also has a new story out called “Dark Integers”. It’s a sequel to “Luminous”, which I wrote about back in week123:

It all started out as a joke. Argument for argument’s sake. Alison and her infuriating heresies.

“A mathematical theorem,” she’d proclaimed, “only becomes true when a physical system tests it out: when the system’s behaviour depends in some way on the theorem being true or false.”

It was June 1994. We were sitting in a small paved courtyard, having just emerged from the final lecture in a one-semester course on the philosophy of mathematics — a bit of light relief from the hard grind of the real stuff. We had fifteen minutes to to kill before meeting some friends for lunch. It was a social conversation — verging on mild flirtation — nothing more. Maybe there were demented academics, lurking in dark crypts somewhere, who held views on the nature of mathematical truth which they were willing to die for. But were were twenty years old, and we knew it was all angels on the head of a pin.

I said, "Physical systems don’t create mathematics. Nothing creates mathematics - it’s timeless. All of number theory would still be exactly the same, even if the universe contained nothing but a single electron."

Alison snorted. "Yes, because even one electron, plus a space-time to put it in, needs all of quantum mechanics and all of general relativity - and all the mathematical infrastructure they entail. One particle floating in a quantum vacuum needs half the major results of group theory, functional analysis, differential geometry - "

"OK, OK! I get the point. But if that’s the case… the events in the first picosecond after the Big Bang would have `constructed’ every last mathematical truth required by any physical system, all the way to the Big Cruch. Once you’ve got the mathematics which underpins the Theory of Everything… that’s it, that’s all you ever need. End of story."

"But it’s not. To apply the Theory of Everything to a particular system, you still need all the mathematics for dealing with that system - which could include results far beyond the mathematics the TOE itself requires. I mean, fifteen billion years after the Big Bang, someone can still come along and prove, say… Fermat’s Last Theorem." Andrew Wiles at Princeton had recently announced a proof of the famous conjecture, although his work was still being scrutinised by his colleagues, and the final verdict wasn’t yet in. "Physics never needed that before."

I protested, "What do you mean, ‘before’? Fermat’s Last Theorem never has - and never will - have anything to do with any branch of physics."

Alison smiled sneakily. "No branch, no. But only because the class of physical systems whose behaviour depend on it is so ludicrously specific: the brains of mathematicians who are trying to validate the Wiles proof."

"Think about it. Once you start trying to prove a theorem, then even if the mathematics is so ‘pure’ that it has no relevance to any other object in the universe… you’ve just made it relevant to yourself. You have to choose some physical process to test the theorem - whether you use a computer, or a pen and paper… or just close your eyes and shuffle neurotransmitters. There’s no such thing as a proof which doesn’t rely on physical events, and whether they’re inside or outside your skull doesn’t make them any less real."

And this is just the beginning… the beginning of a wild tale of an inconsistency in the axioms of arithmetic — a "topological defect" left over in the fabric of mathematics, much like the cosmic strings or monopoles hypothesized by certain physicists thinking about the early universe — and the mathematicians who discover it and struggle to prevent a large corporation from exploiting it for their own nefarious purposes.

I think you can download the story “Luminous” for $1.59 here, though I’m too old-fashioned to have tried this sort of thing. The sequel, “Dark Integers”, seems right now to be available only through the magazine Asimov’s Science Fiction — the October/November 2007 issue. (Strangely, this futuristic magazine seems to be available only in the form of hand-delivered pulverized tree fragments.)