Turing's Cathedral: The Origins of the Digital Universe

by George Dyson

“Formal logic has to be taken over by mathematicians,” Veblen had announced on New Year’s Eve 1924, when the plans for what would become the Institute for Advanced Study were first taking form in his mind. “There does not exist an adequate logic at the present time, and unless the mathematicians create one, no one else is likely to do so.”10 It was Gödel, above anyone else—and now directly above von Neumann—who proved Veblen’s instincts correct. In 1924 both von Neumann and Gödel were working on the logical foundations of mathematics, before Gödel’s incompleteness theorems brought the Hilbert program to a close. Von Neumann “believed in Hilbert’s goal of a final and conclusive axiomatization of mathematics,” according to Stan Ulam, “and yet, in a 1925 paper, in a mysterious flash of intuition, he pointed out the limits of any axiomatic formulation of set theory. That was perhaps a sort of vague forecast of Gödel’s result.”11 The seeds of doubt were sown. In September of 1930, at the Königsberg conference on the epistemology of the exact sciences, Gödel made the first, tentative announcement of his incompleteness results. Von Neumann immediately saw the implications, and, as he wrote to Gödel on November 30, 1930, “using the methods you employed so successfully … I achieved a result that seems to me to be remarkable, namely, I was able to show that the consistency of mathematics is unprovable,” only to find out, by return mail, that Gödel had got there first.12 “He was disa...