Incompleteness: The Proof and Paradox of Kurt Godel
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KURT GODEL IS CONSIDERED the twentieth century's greatest mathematician. His monumental theorem of incompleteness overturned the prevailing conviction that the only true statements in math were those that could be proved. Inspired by Plato's philosophy of a higher reality, Godel demonstrated conclusively that there are in every formal system undeniably true statements that nevertheless cannot be proved. The result was an upheaval in mathematics. From the famous Vienna Circle and sparring with Wittgenstein to Princeton's Institute for Advanced Study, where he was Einstein's constant companion. Godel was both a towering intellect and a deeply mysterious figure, whose strange habits and ever-increasing paranoia led to his sad death by self-starvation. In this lucid and accessible study, Rebecca Goldstein, a philosopher of science and a gifted novelist whose work often focuses on science, explains the significance of Godel's theorems and the remarkable vision behind them, while bringing this eccentric, tortured genius and his world to life.
us—-mysteriously—with the means of accessing this ultimate "out yonder," of gaining at least partial glimpses of what might be called (in the current fashion in naming television shows: "Extreme Survival," "Extreme Makeover," "The Most Extreme") "extreme reality." Godel's mathematical Platonism was not in itself unusual. Many mathematicians have been mathematical realists; and 46 R EBECCA G OLDSTEIN even those who do not describe themselves as such, when they are cornered and asked pointblank
First World War, had singularly impressed Schlick's group. As stylistically arresting as its creator, this work achieved in its austere elegance a sort of poetry.14 The traditional tool of the philosopher—the argument—is dispensed with; each assertion is put forth, as Russell once remarked, "as if it were a Czar's ukase." The poet's obscurity of meaning is preserved despite (by means of?) the formal precision of its elaborate numbering system, which hierarchically arranges its assertions: so
syntax the meaning of a sign should never play a role. It must be possible to establish logical syntax without mentioning meaning of a sign: only the description of expressions may be presupposed. Interestingly, from this proposition alone, Wittgenstein claims to demonstrate the fundamental error of the Theory of Types: "3.331 From this observation we turn to Russell's 'theory of types'. It can be seen that Russell must be wrong, because he had to mention the meaning of signs when establishing
as a student, a reticent observer quietly taking in the opinions around him ... and drawing his own conclusions. Godel in the Vienna Circle: The Silent Dissenter Regardless of his profound, private disagreements with the positivists, Godel's association with the Circle led him into the most gregarious few years of his introverted life. He was meeting on a regular basis, not just on alternate Thursday evenings in the bare-bones room where the full Vienna Circle convened but also at late-night
when in the Introduction [to Mathematical Philosophy, first published in 1919, p. 169] he said " [Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features]." At that time evidently Russell had met the "not" even in this world, but later on under the influence of Wittgenstein he chose to overlook it. Coming from Godel, these are pointed words, which is of course why they still languish in a folder in Firestone Library. Some more