Cryptonomicon

'All right, Alan.'

'Won't take a minute if you will just stop interrupting.'

'But there is a locomotive too named Leibniz.'

'Is it that you don't think I give enough credit to Germans? Because I am about to mention a fellow with an umlaut.'

'Oh, would it be Herr Turing?' Rudy said slyly.

'Herr Turing comes later. I was actually thinking of Godel.'

'But he's not German! He's Austrian!'

'I'm afraid that it's all the same now, isn't it?'

'Ze Anschluss wasn't my idea, you don't have to look at me that way, I think Hitler is appalling.'

'I've heard of Godel,' Waterhouse put in helpfully. 'But could we back up just a sec?'

'Of course Lawrence.'

'Why bother? Why did Russell do it? Was there something wrong with math? I mean, two plus two equals four, right?'

Alan picked up two bottlecaps and set them down on the ground. 'Two. One-two. Plus-' He set down two more. 'Another two. One-two. Equals four. One-two-three-four.'

'What's so bad about that?' Lawrence said.

'But Lawrence-when you really do math,in an abstract way, you're not counting bottlecaps, are you?'

'I'm not counting anything.'

Rudy broke the following news: 'Zat is a very modern position for you to take.'

'It is?'

Alan said, 'There was this implicit belief, for a long time, that math was a sort of physics of bottlecaps. That any mathematical operation you could do on paper, no matter how complicated, could be reduced-in theory, anyway-to messing about with actual physical counters, such as bottlecaps, in the real world.'

'But you can't have two point one bottlecaps.'

'All right, all right, say we use bottlecaps for integers, and for real numbers like two point one, we use physical measurements, like the length of this stick.' Alan tossed the stick down next to the bottlecaps.

'Well what about pi, then? You can't have a stick that's exactly pi inches long.'

'Pi is from geometry-ze same story,' Rudy put in.

'Yes, it was believed that Euclid's geometry was really a kind of physics, that his lines and so on represented properties of the physical world. But-you know Einstein?'

'I'm not very good with names.'

'That white-haired chap with the big mustache?'

'Oh, yeah,' Lawrence said dimly, 'I tried to ask him my sprocket question. He claimedhe was late for an appointment or something.'

'That fellow has come up with a general relativity theory, which is sort of a practical application, not of Euclid's, but of Riemann'sgeometry-'

'The same Riemann of your zeta function?'

'Same Riemann, different subject. Now let's not get sidetracked here Lawrence-'

'Riemann showed you could have many many different geometries that were not the geometry of Euclid but that still made sense internally,' Rudy explained.

'All right, so back to P.M.then,' Lawrence said.

'Yes! Russell and Whitehead. It's like this: when mathematicians began fooling around with things like the square root of negative one, and quaternions, then they were no longer dealing with things that you could translate into sticks and bottlecaps. And yet they were still getting sound results.'

'Or at least internally consistent results,' Rudy said.

'Okay. Meaning that math was more than a physics of bottlecaps.'

'It appeared that way, Lawrence, but this raised the question of was mathematics really trueor was it just a game played with symbols? In other words-are we discovering Truth, or just wanking?'

'It has to be true because if you do physics with it, it all works out! I've heard of that general relativity thing, and I know they did experiments and figured out it was true.'

'Ze great majority of mathematics does not lend itself to experimental testing,' Rudy said.

'The whole idea of this project is to sever the ties to physics,' Alan said.

'And yet not to be yanking ourselves.'

'That's what P.M.was trying to do?'

'Russell and Whitehead broke all mathematical concepts down into brutally simple things like sets. From there they got to integers, and so on.

'But how can you break something like pi down into a set?'

'You can't,' Alan said, 'but you can express it as a long string of digits. Three point one four one five nine, and so on.'

'And digits are integers,' Rudy said.

'But no fair! Pi itselfis not an integer!'

'But you can calculate the digitsof pi, one at a time, by using certain formulas. And you can write down the formulas like so!' Alan scratched this in the dirt:

'I have used the Leibniz series in order to placate our friend. See, Lawrence? It is a string of symbols.'

'Okay. I see the string of symbols,' Lawrence said reluctantly.

'Can we move on? Godel said, just a few years ago, 'Say! If you buy into this business about mathematics being just strings of symbols, guess what?' And he pointed out that any string of symbols-such as this very formula, here-can be translated into integers.'

'How?'

'Nothing fancy, Lawrence-it's just simple encryption. Arbitrary. The number '538' might be written down instead of this great ugly [sigma], and so on.

'Seems pretty close to wanking, now.'

'No, no. Because then Godel sprang the trap! Formulas can act on numbers, right?'

'Sure. Like 2x.'

'Yes. You can substitute any number for x and the formula 2x will double it. But if another mathematical formula, such as this one right here, for calculating pi, can be encoded as a number, then you can have another formula act on it. Formulas acting on formulas!'

'Is that all?'

'No. Then he showed, really through a very simple argument, that if formulas really can refer to themselves, it's possible to write one down saying 'this statement cannot be proved.' Which was tremendously startling to Hilbert and everyone else, who expected the opposite result.'

'Have you mentioned this Hilbert guy before?'

'No, he is new to this discussion, Lawrence.'

'Who is he?'

'A man who asks difficult questions. He asked a whole list of them once. Godel answered one of them.'

'And Turing answered another,' Rudy said.

'Who's that?'

'It's me,' Alan said. 'But Rudy's joking. 'Turing' doesn't really have an umlaut in it.'

'He's going to have an umlaut in him later tonight,' Rudy said, looking at Alan in a way that, in retrospect, years later, Lawrence would understand to have been smoldering.

'Well, don't keep me in suspense. Which one of his questions did you answer?'

'The Entscheidungsproblem,' Rudy said.

'Meaning?'