Incompleteness: The Proof and Paradox of Kurt Godel
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.
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way. In brief, I think: All of that which many are babbling today, I have defined in my book by remaining silent about it. He took himself to have demonstrated how little one has actually said after one has finished saying all that can be said. The question is whether the requisite silence, imposed in proposition 7, hides nothing at all or rather all of the most important things. The positivists certainly interpreted Wittgenstein to be saying the former, which is almost certainly one of the
how seriously the young logician and confirmed Platonist ever took Wittgenstein is, in the end, unknowable. Beyond Wittgenstein towered the figure of the 120 R EBECCA G OLDSTEIN most influential mathematician of the day, David Hilbert, a figure Godel could not possibly dismiss as mathematically inadequate. Like Wittgenstein's, Hilbert's views on the nature of mathematics could not have been more incompatible with the mathematical result that the young Godel was soon to spring on an
into wffs, and special sequences of wffs (in other words, proofs). Everything is built out of the basic symbols of the alphabet: a wff is a sequence of these symbols, and a proof is a sequence of wffs (with the conclusion simply the last entry in the sequence). The Godel numbering thus begins by assigning each primitive symbol of the alphabet a number. Once each of the primitive symbols has a number, one continues with a rule for assigning numbers to the wffs themselves, based on the
of nature are a priori." Three magnificent minds, as at home in the world of pure ideas as anyone on this planet, yet they (and there are more) reported hitting an insurmountable impasse in discussing ideas with Godel. Einstein, too, was presented time and again, on their daily walks to and from the Institute, with examples of Godel's strange intuitions, his profound "anti-empiricism." Nevertheless Einstein consistently sought out the logician's company. In fact, 6 Godel's hostility to the theory