The Information - James Gleick [86]
Enter Kurt Gödel.
He was born in 1906 in Brno, at the center of the Czech province of Moravia. He studied physics at the University of Vienna, seventy-five miles south, and as a twenty-year-old became part of the Vienna Circle, a group of philosophers and mathematicians who met regularly in smoky coffeehouses like the Café Josephinum and the Café Reichsrat to propound logic and realism as a bulwark against metaphysics—by which they meant spiritualism, phenomenology, irrationality. Gödel talked to them about the New Logic (this term was in the air) and before long about metamathematics—der Metamathematik. Metamathematics was not to mathematics what metaphysics was to physics. It was mathematics once removed—mathematics about mathematics—a formal system “looked at from the outside” (“äußerlich betrachtet”).♦ He was about to make the most important statement, prove the most important theorem about knowledge in the twentieth century. He was going to kill Russell’s dream of a perfect logical system. He was going to show that the paradoxes were not excrescences; they were fundamental.
Gödel praised the Russell and Whitehead project before he buried it: mathematical logic was, he wrote, “a science prior to all others, which contains the ideas and principles underlying all sciences.”♦ Principia Mathematica, the great opus, embodied a formal system that had become, in its brief lifetime, so comprehensive and so dominant that Gödel referred to it in shorthand: PM. By PM he meant the system, as opposed to the book. In PM, mathematics had been contained—a ship in a bottle, no longer buffeted and turned by the vast unruly seas. By 1930, when mathematicians proved something, they did it according to PM. In PM, as Gödel said, “one can prove any theorem using nothing but a few mechanical rules.”♦
Any theorem: for the system was, or claimed to be, complete. Mechanical rules: for the logic operated inexorably, with no room for varying human interpretation. Its symbols were drained of meaning. Anyone could verify a proof step by step, by following the rules, without understanding it. Calling this quality mechanical invoked the dreams of Charles Babbage and Ada Lovelace, machines grinding through numbers, and numbers standing for anything at all.
Amid the doomed culture of 1930 Vienna, listening to his new friends debate the New Logic, his manner reticent, his eyes magnified by black-framed round spectacles, the twenty-four-year-old Gödel believed in the perfection of the bottle that was PM but doubted whether mathematics could truly be contained. This slight young man turned his doubt into a great and horrifying discovery. He found that lurking within PM—and within any consistent system of logic—there must be monsters of a kind hitherto unconceived: statements that can never be proved, and yet can never be disproved. There must be truths, that is, that cannot be proved—and Gödel could prove it.
He accomplished this with iron rigor disguised as sleight of hand. He employed the formal rules of PM and, as he employed them, also approached them metamathematically—viewed them, that is, from the outside. As he explained, all the symbols of PM—numbers, operations of arithmetic, logical connectors, and punctuation—constituted a limited alphabet. Every statement or formula of PM was written in this alphabet. Likewise every proof comprised a finite sequence of formulas—just a longer passage written in the same alphabet. This is where metamathematics came in. Metamathematically, Gödel pointed out, one sign is as good as another; the choice of a particular alphabet is arbitrary. One could use the traditional assortment of numerals and glyphs (from arithmetic: +, −, =, ×; from logic: ¬, ∨, ⊃, ∃), or one could use letters, or one could use dots and dashes. It was a matter of encoding, slipping from one symbol set to another.
Gödel proposed