Who the hell is Kurt Gödel, and why should I care?

The Austrian mathematician and logician Kurt Gödel is mostly known nowadays for his incompleteness theorem. Simply put, his theorem states that any sufficiently strong axiomatic system is either inconsistent or incomplete. The implications of this discovery for the fields of mathematics and logic were massive: Logicians had long been trying to explain more or less the entire universe from a few simple logical axioms. Notably, Bertrand Russell's mammoth work Principia Mathematica was an attempt to explain all of mathematics from such a logical, axiomatic system. Gödel's theorem, it can be said, forced logic to learn humility. His discovery was a landmark, showing that mathematics is not a finished (and possibly not finishable) object. The way he arrived at the theorem (using a modified version of Epimenides' paradox) seems almost provokingly simple in hindsight. Most people who know of him nowadays do so either because they themselves are mathematicians, or because they've read Douglas Hofstadter's famous work Gödel, Escher, Bach: An Eternal Golden Braid (which would typically imply that they're geeks anyway).

Gödel's Childhood and Education

Kurt Gödel was born on the 28th April 1906 in Brünn, in what was then Austria-Hungary (the town is now called Brno, and is part of the Czech Republic). He went to school in his town of birth, and completed the Gymnasium (European senior secondary schools; roughly equivalent to US high schools) in 1923. At that time, he had already mastered university-level mathematics. According to his brother Rudolf Gödel, he had gained himself quite a reputation not only because of his mathematical talent, but also for the fact that he was a quite adept linguist, reputedly never having made a single grammatical error in Latin during his entire time at school. He entered the University of Vienna the same year as he finished the Gymnasium, doing undergraduate work in the field of mathematical philosophy. From there, he quickly developed an intense interest in formal logic, and according to his fellow students, showed incredible talent in this field right from the start. He completed his doctorate dissertation under the supervision of professor Hans Hahn, and until 1938, he belonged to the philosophical school of logical positivism.

The Incompleteness Theorem

One of the things that caught Gödel's interest during his time at university was Bertrand Russell's work. As an undergraduate he had studied Russell's book Introduction to Mathematical Philosophy, and he had later done an intensive study of Russell's main work, the Principia Mathematica. By 1931, Gödel published his work Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems"), in which he presented his famous incompleteness theorem. His work was like a kick in the face of the hundreds of years of desperate attempts of finding axioms in order to put the entire structure of mathematics on an axiomatic foundation. One of the "related systems" that were demolished by Gödel's theorem was David Hilbert's formalism, which attempted to describe mathematics as a formal system. One of the more practical implications of Gödel's incompleteness theorem is that it is impossible to program a computer to answer all mathematical questions. He arrived at his theorem using a modified form of Epimenides' so-called "liar paradox", which in its normal form says "this sentence is false". The paradox should be obvious, but for the benefit of those who forgot to load logic.so while getting out of bed:

  • If the sentence "this sentence is false" is indeed false, it is in fact speaking the truth. It says it's false, after all.
  • If, on the other hand, it is true, the sentence is a dirty liar. It claims to be false!

Gödel's theorem was expressed in a more complicated mathematical lingo (and I'll admit I've never personally read his aforementioned book; but that's because I'm a lazy-assed non-mathematician. So sue me.), but it boils down to the idea that in any sufficiently strong formal system, it is possible to express a theorem which says "this theorem is not provable". If the theorem is indeed provable, the system is inconsistent, because it houses a self-contradiction. If the theorem is not provable, it is saying that the system is incomplete. Ha-ha! A smack in the face, a logic bomb planted straight into the foundation of those systems that tried to plant mathematics on an axiomatic foundation. A related implication is that provability is a weaker notion than truth.

Gödel's Life during World War II

When Adolf Hitler seized political power in Germany in 1933, Gödel didn't particularly care. He didn't live in Germany, and in the time-honoured tradition of geeks everywhere, he didn't particularly care about politics. That changed when a Nazi student of Gödel's former teacher Moritz Schlick (who had taught Gödel's undergraduate class in mathematical philosophy) murdered his old teacher. Schlick had been the man to spur Gödel's interest in logic, and the event caused a full-scale emotional breakdown in Gödel. He had a serious nervous breakdown, which he recovered from in late 1934. At that time, he was offered a guest professorship at Princeton, and moved to the United States to teach. His 1934 lectures at Princeton have been published by Stephen Cole Kleene, under the title "On undecidable propositions of formal mathematical systems". In 1938, Gödel returned to Vienna to marry Adele Porkert. War broke out shortly afterwards, leaving the two trapped in Nazi-"reunited" Austria. Determined to return to teach at Princeton and just as determined to stay alive, Gödel decided to flee Europe. Going to the States across the Atlantic was impossible, so he took the long way: Travelling through Russia and then Japan, Gödel and his wife eventually found themselves in the US in 1940, at which time they formally emigrated there.

Death of a Logician: Paranoid delusions

From 1953 to his death in 1978, Gödel held a chair at Princeton's Institute for Advanced Study, and received the US National Medal of Science in 1974. He had written and published several acclaimed scientific works, most famously including the aforementioned "On Formally Undecidable....." as well as "Consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory" (released 1940). His work is considered classics of modern mathematical logic, and he had much to be proud of. He had many quite intelligent friends with whom he loved to debate philosophy, famously including Albert Einstein and John von Neumann.

Unfortunately, he was also gloriously insane. He was strongly opinionated about everything in his life, not only mathematics but also things he really didn't have much knowledge about, such as medicine. Added to that, he had a quite profound case of paranoia, and believed that unseen enemies were stalking him and trying to kill him. He suffered a duodenal ulcer and had severe bleeding, and put together an extremely strict diet for himself, which defied the advice his doctors had given him and caused him to slowly lose weight. Near the end of Gödel's life, his wife Adele was hospitalized for cardiac problems, and Gödel refused to eat. He himself could not cook, and he trusted nobody other than Adele to cook for him, believing that they would put poison in his food. He was found dead in his bed in 1978, curled up in a fetal position and starved to death. He was survived by Adele, and they never had any children.

"Either mathematics is too big for the human mind or the human mind is more than a machine."
--Kurt Gödel

References:

  • http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html
  • http://www.andrews.edu/~calkins/math/biograph/biogodel.htm
  • Douglas R. Hofstadter: "Gödel, Escher, Bach: An Eternal Golden Braid", ISBN 0465026850
Note: His name is pronounced roughly like "Koort GURdle".

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