The New Yorker:

In “Journey to the Edge of Reason,” Stephen Budiansky’s wonderfully engrossing new biography of the Austrian-born logician Kurt Gödel, some light is shed on a famous story. Gödel is legendary for having published, in 1931, at the age of twenty-four, the most important logical proof of the past century, the incompleteness theorems. Gödel showed—though any non-mathematician who pretends to understand this in more than a rudimentary way is not being honest—that you can’t “axiomatize” arithmetic or geometry, and that any systems containing arithmetic will have arguments that can be neither proved nor disproved. The foundations of even secure-seeming systems, such as mathematics, are subject to “recursive” paradoxes, and can always end up referring to themselves, like an Escher illustration or the lyrics to “Monster Mash.

Gödel was a remarkable character, who sadly descended into mental illness as he aged, though his ailment always took in the context of his fiercely logical mind. One piece of evidence, for instance, of what was termed his “paranoia” was that he insisted to his psychiatrists that the Institute for Advanced Study, at Princeton, where he had taken a position in 1940, after the Nazi Anschluss, would have to fire him, since he had done nothing there of value equal to the work he’d done when he was twenty-five. This “delusion” was, in fact, perfectly true and acutely seen—he hadn’t achieved anything equivalent, but then no one could. An institute benefits from the presence of the most famous logician of his age, even as a genial presence. (The same was true of Albert Einstein, who was at the I.A.S. as much to be Einstein as he was to do physics.)

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