Wednesday, April 28, 2010

Happy Birthday, Kurt Godel

Today would be the 104th birthday of Kurt Godel, one of the great geniuses of the 20th century. Born in Austria (a part that is now in the Czech Republic), he attended the university in Vienna where began his studies in mathematics. While a mathematical mind of the highest caliber, he was anything but a one-dimensional thinker. He took courses in the history of philosophy with Heinrich Gomperz and was struck by the thought of Plato, something that would shape his approach to the deepest mathematical and meta-mathematical problems of the day.

The University of Vienna in the late 20s and early 30s was the place to be if one cared about mathematics. Math had long been placed on an intellectual pedestal, elevated to the unique status of template for all other fields of study, as the one place where absolutely certain knowledge could be found. But then non-Euclidean geometry, Cantor's work on transinfinite sets, and the paradoxes that were appearing in set theory cast deep doubts upon the underlying foundations of all of mathematics. There was a full-blown intellectual crisis at hand and the Vienna Circle, a group of philosophers, mathematicians, physicists, and a sociologist, were meeting to try to save human knowledge.

The wunderkind of the group, Rudolf Carnap, was teaching a class in the foundations of mathematics and Godel took the course. Carnap quickly realized Godel's genius and procured him an invitation to join the Circle and then the mathematical subcircle, a series of colloquia run by Karl Menger, an assistant professor.

The Vienna Circle had tried to bring together aspects of Einstein's theory of relativity, Ludwig Wittgenstein's thought in his Tractatus Logico-Philosophicus, and Alfred North Whitehead and Bertrand Russell's Principia Mathematica, in order to create a new firm foundation for all knowledge. They began by trying to formulate a criterion of cognitive significance, a rule that would tell meaningful propositions from nonsense. Of the meaningful claims, they divided them into those that were necessary truths and those that just happened to be true. They then looked for a philosophy of mathematics to justify those that were necessary and a philosophy of science to justify those that were contingent. Their approach to mathematics followed Russell in trying to ground all of mathematics in logic. To be a true mathematical claim, they argued, was to be provable.

As a quiet member, sitting in the back of the room at these conversations, Godel became intrigued by the question of provability. What and how much of the mathematics we accept was in fact provable? It was a technical question at the heart of the Logical Positivist mathematical project. But while the question fascinated him, the underlying philosophy behind the Positivist approach did not. He thought they were dead wrong. He was a Platonist. He thought that mathematical truths were absolute truths, not just truths within a logical system. He thought there was a higher realm of being, mathematical being and we were accessing its higher nature by doing math. This sort of metaphysical mumbo-jumbo was despised by the Positivists who wanted a completely metaphysics-free, scientific basis for understanding everything.

But Godel was fascinated and that meant something big was about to happen. In 1930, Godel did something amazing. He proved that the Positivist approach and any other like it, had to fail. He developed what is called his Incompleteness Theorem (extended in 1931). What he did in the most ingenious way was to figure out how to create a dictionary that lets you translate statements ABOUT mathematics, about what is true, about what is provable,... into actual mathematical equations such that there is a perfect mapping between true sentences IN mathematics and true sentences ABOUT mathematics. He then created a sentence that said "This sentence is unprovable" and showed that either it was true and therefore we had a mathematical sentence that was both true and unprovable or it was provable which made it false. This showed that we could not, as the Positivists wanted to do, contend that mathematical truth and provability are the same thing thereby undermining the most promising line of reasoning we had for saving the certainty of mathematics.

Personally, Godel was as strange and conflicted as his work. He was the nerd's nerd. Skinny with big round glasses, he was nervous and perpetually cold, wearing a heavy coat even in the summer. Yet, he was a driven ladies-man, or at least tried very hard to be. He eventually married a dancer. It is not entirely clear what sort of dancer, but from the fact that she was brash and lower class, there is good reason to suppose that her dancing was of a particular sort. The two of them bonded in a way that allowed them to fill the other's needs. When Nazi thugs would attack him on the street, figuring that since he was small and nerdy he must be Jewish, she would defend him, beating them off with an umbrella. She would taste his food -- Godel was perpetually concerned that someone was trying to poison him. He gave her position and status. Being a professor's wife was a big deal, and she wielded this in ways that put off those who were used to significant class status. They were both fascinated by the supernatural and attended seances and believed in much of what we would now call "new age."

After the Anschluss, once the Nazis took over Austria and the rest of his colleagues fled, Godel stayed, oblivious to politics. But once the attacks became frequent enough, he and his wife accepted an invitation to come to the U.S. He stayed at the Center for Advanced Study in Princeton, where he became very close with another refugee of the war Albert Einstein.

Those at the Institute convinced Godel that he ought to apply for American citizenship. This meant that one had to swear to uphold and defend the American Constitution, something most took as nominal. Not Godel. If he was going to adhere and defend something, he would know exactly what it is and so he undertook an exhaustive study of the document. And being Godel, he found a contradiction. He discovered that through the amendment process one could create a system that was both a democracy and a non-democracy. It is a basic theorem in logic that if a single contradiction is true, then all sentences are true, in other words, truth disappears, becomes meaningless. For a Platonist, this is disaster. Godel was frantic. He could not in good conscience say that he was willing to accept and defend it. He could not become an American. On the drive from Princeton to Philadelphia, Einstein and Oskar Morgenstern, mathematician and economist, tried to calm him down and convince him to just take the oath. He could help fix it once he was an American. They succeeded in getting him to agree to go through with it and as they walked into the immigration office, the judge who was to administer the oath saw Einstein. It happened to be the same judge who granted citizenship to Einstein, and the chance to have some face time with one of the world's most famous people was something he was not going to pass up. So, he invited Einstein back to his chambers. After some pleasantries, Einstein explained that they were here to get Godel his citizenship and the judge agreed to do it there personally, so they wouldn't have to wait in line. Before he administered the oath, he looked over at the mathematician and casually remarked that he must be relieved to be here in the United States, a democracy that could never become a fascist dictatorial state like his home in Europe. Godel jumped out of his skin and began citing chapter and verse explaining how the U.S. Constitution was rendered meaningless by its own structure, lecturing the judge on the intricacies of the document and formal logic. It took some time to calm him down, but eventually, he agreed to the oath and became a citizen.

But he became a lonely one when his wife passed away. She was his connection to the world and now he was alone and in a strange place. He shut himself off and, without his taster, stopped eating, sure that he was being poisoned. The poison didn't kill him, but the lack of food did as he all but starved himself to death. The irony of the man who showed that the provable had to be unprovable dying from that which was an attempt to stave off death is the sort of twist that one could only get with someone like Godel. And there was only one like Godel.