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to figure out whether it all comes from in here,” Alan said, reaching out to rap his knuckles on the side of Lawrence’s head, “or do you sometimes take in new ideas from other human beings?”
    “When I was a little boy, I saw angels in a church in Virginia,” Lawrence said, “but I think that they came from inside my head.”
    “Very well,” Alan said.
    But later Alan had another go at it. They had reached the fire lookout tower and it had been a thunderous disappointment: just an alienated staircase leading nowhere, and a small cleared area below that was glittery with shards of liquor bottles. They pitched their tent by the side of a pond that turned out to be full of rust-colored algae that stuck to the hairs on their bodies. Then there was nothing left to do but drink schnapps and talk about math.
    Alan said, “Look, it’s like this: Bertrand Russell and another chap named Whitehead wrote Principia Mathematica . . .”
    “Now I know you’re pulling my leg,” Waterhouse said. “Even I know that Sir Isaac Newton wrote that .”
    “Newton wrote a different book, also called Principia Mathematica, which isn’t really about mathematics at all; it’s about what we would today call physics.”
    “Then why did he call it Principia Mathematica? ”
    “Because the distinction between mathematics and physics wasn’t especially clear in Newton’s day—”
    “Or maybe even in zis day,” Rudy said.
    “—which is directly relevant to what I’m talking about,” Alan continued. “I am talking about Russell’s P.M., in which he and Whitehead started absolutely from scratch, I mean from nothing, and built it all up—all mathematics—from a small number of first principles. And why I am telling you this, Lawrence, is that—Lawrence! Pay attention!”
    “Hmmm?”
    “Rudy—take this stick, here—that’s right—and keep a close eye on Lawrence, and when he gets that foggy look on his face, poke him with it!”
    “Zis is not an English school, you can’t do zese kind of sing.”
    “I’m listening,” Lawrence said.
    “What came out of P.M., which was terrifically radical, was the ability to say that all of math, really, can be expressed as a certain ordering of symbols.”
    “Leibniz said it a long time before zen!” protested Rudy.
    “Er, Leibniz invented the notation we use for calculus, but—”
    “I’m not talking about zat!”
    “And he invented matrices, but—”
    “I’m not talking about zat eezer!”
    “And he did some work with binary arithmetic, but—”
    “Zat is completely different!”
    “Well, what the hell are you talking about, then, Rudy?”
    “Leibniz invented ze basic alphabet—wrote down a set of symbols, for expressing statements about logic.”
    “Well, I wasn’t aware that Herr Leibniz counted formal logic among his interests, but—”
    “Of course! He wanted to do what Russell and Whitehead did, except not just with mathematics, but with everything in ze whole world!”
    “Well, from the fact that you are the only man on the planet, Rudy, who seems to know about this undertaking of Leibniz’s, can we assume that he failed?”
    “ You can assume anything that pleases your fancy, Alan,” Rudy responded, “but I am a mathematician and I do not assume anything .”
    Alan sighed woundedly, and gave Rudy a SignificantLook which Waterhouse assumed meant that there would be trouble later. “If I may just make some headway, here,” he said, “all I’m really trying to get you to agree on, is that mathematics can be expressed as a series of symbols,” (he snatched the Lawrence-poking stick and began drawing things like + = 3) √-1π in the dirt) “and frankly I could not care less whether they happen to be Leibniz’s symbols, or Russell’s, or the hexagrams of the I Ching… .”
    “Leibniz was fascinated by the I Ching!” Rudy began.
    “Shut up about Leibniz for a moment, Rudy, because look here: You—Rudy—and I are on a train, as it were,