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 behavior depends 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 that doesn't rely on physical events-and whether they're inside or outside your skull doesn't make them any less real."
"Fair enough," I conceded warily. "But that doesn't mean-"
"And maybe Andrew Wiles's brain-and body, and notepaper-comprised the first physical system whose behavior depended on the theorem being true or false. But I don't think human actions have any special role . . . and if some swarm of quarks had done the same thing blindly, fifteen billion years before-executed some purely random interaction that just happened to test the conjecture in some way-then those quarks would have constructed FLT long before Wiles. We'll never know."
I opened my mouth to complain that no swarm of quarks could have tested the infinite number of cases encompassed by the theorem-but I caught myself just in time. That was true-but it hadn't stopped Wiles. A finite sequence of logical steps linked the axioms of number theory-which included some simple generalities about all numbers-to Fermat's own sweeping assertion. And if a mathematician could test those logical steps by manipulating a finite number of physical objects for a finite amount of time-whether they were pencil marks on paper, or neurotransmitters in his or her brain-then all kinds of physical systems could, in theory, mimic the structure of the proof . . . with or without any awareness of what it was they were "proving."
I leant back on the bench and mimed tearing out hair. "If I wasn't a die-hard Platonist before, you're forcing me into it! Fermat's Last Theorem didn't need to be proved by anyone-or stumbled on by any random swarm of quarks. If it's true, it was always true. Everything implied by a given set of axioms is logically connected to them, timelessly, eternally . . . even if the links couldn't be traced by people-or quarks-in the lifetime of the universe."
Alison was having none of this; every mention of timeless and eternal truths brought a faint smile to the corner of her mouth, as if I was affirming my belief in Santa Claus. She said, "So who, or what, pushed the consequences of 'There exists an entity zero' and 'Every X has a successor et cetera, all the way to Fermat's Last Theorem and beyond, before the universe had a chance to test out any of it?"
I stood my ground. "What's joined by logic is just . . . joined. Nothing has to happen-consequences don't have to be pushed' into existence by anyone, or anything. Or do you imagine that the first events after the Big Bang, the first wild jitters of the quark-gluon-plasma, stopped to fill in all the logical gaps? You think the quarks reasoned: well, so far we've done A and B and C-but now we must do D, because D would be logically inconsistent with the other mathematics we've 'invented' so far . . . even if it would take a five-hundred-thousand-page proof to spell out the inconsistency?"
Alison thought it over. ''No. But what if event D took place, regardless? What if the mathematics it implied was logically inconsistent with the rest-but it went ahead and happened anyway . . . because the universe was too young to have computed the fact that there was any discrepancy?"
I must have sat and stared at her, open-mouthed, for about ten seconds. Given the orthodoxies we'd spent the last two-and-a-half years absorbing, this was a seriously outrageous statement.
"You're claiming that . . . mathematics might be strewn with primordial defects in consistency? Like space might be strewn with cosmic strings?"
"Exactly." She stared back at me, feigning nonchalance. "If space-time doesn't join up with itself smoothly, everywhere . . . why should mathematical logic?"
I almost choked. "Where do I begin? What happens-now-when some physical system tries to link theorems across the defect? If theorem D has been rendered 'true' by some over-eager quarks, what happens when we program a computer to disprove it? When the software goes through all the logical steps that link A, B, and C-which the quarks have also made true-the dreaded not-D . . . does it succeed, or doesn't it?"
Alison sidestepped the question. "Suppose they're both true: D and not-D. Sounds like the end of mathematics, doesn't it? The whole system falls apart, instantly. From D and not-D together you can prove anything you like: one equals zero, day equals night. But that's just the boring old-fart Platonist view-where logic travels faster than light, and computation takes no time at all. People live with omega-inconsistent theories, don't they?"
Omega-inconsistent number theories were non-standard versions of arithmetic, based on axioms that "almost" contradicted each other-their saving grace being that the contradictions could only show up in "infinitely long proofs" (which were formally disallowed, quite apart from being physically impossible). That was perfectly respectable modern mathematics-but Alison seemed prepared to replace "infinitely long" with just plain "long"-as if the difference hardly mattered, in practice.
I said, "Let me get this straight. What you're talking about is taking ordinary arithmetic-no weird counter-intuitive axioms, just the stuff every ten-year-old knows is true-and proving that it's inconsistent, in a finite number of steps?"
She nodded blithely. "Finite, but large. So the contradiction would rarely have any physical manifestation-it would be 'computationally distant' from everyday calculations, and everyday physical events. I mean . . . one cosmic string, somewhere out there, doesn't destroy the universe, does it? It does no harm to anyone."
I laughed drily. "So long as you don't get too close. So long as you don't tow it back to the solar system and let it twitch around slicing up planets."
"Exactly."
I glanced at my watch. "Time to come down to Earth, I think. You know we're meeting Ju1Ia and Ramesh-?"
Alison sighed theatrically. "I know, I know. And this would bore them witless, poor things-so the subject's closed, I promise." She added wickedly, "Humanities students are so myopic."
We set off across the tranquil leafy campus. Alison kept her word, and we walked in silence; carrying on the argument up to the last minute would have made it even harder to avoid the topic once we were in polite company.
Half-way to the cafeteria, though, I couldn't help myself.
"If someone ever did program a computer to follow a chain of inferences across the defect . . . what do you claim would actually happen? When the end result of all those simple, trustworthy logical steps finally popped up on the screen-which group of primordial quarks would win the battle? And please don't tell me that the whole computer just conveniently vanishes."
