The Austrian-American mathematician and philosopher-scientist Kurt Gödel (1906-1978) developed the celebrated "Gödel's proof" which provided extraordinary insight into the basis of mathematical thought and revolutionized modern logic.
Kurt Gödel was born on April 28, 1906, in Brno, now in the Czech Republic but then part of Austria-Hungary. His father was a well-off textile manufacturer and his life with his parents and brother has been described as "happy." His inquisitive nature by age 6 earned him the family name "Mr. Why." By age 14 he had become interested in mathematics, and a year later, in philosophy. At 17, he mastered university-level mathematics and excelled at other subjects as well his brother Rudolph said, "it was rumored that in the whole of his time at high school not only was his work in Latin always given the top marks but that he had made not a single grammatical error."
Gödel entered the University of Vienna to study theoretical physics; two years later, he shifted to mathematics, and then to mathematical logic. He joined the university faculty in 1930 after receiving his doctorate. In 1931 Gödel published "On Formally Undecipherable Propositions of Principia Mathematica and Related Systems". It was an extremely specialized paper but it attracted early attention and became famously known as Gödel's proof. Gödel was 25.
Gödel's proof denies the possibility that a mathematical system supported on axioms can be verified within that system and ends 100 years of attempts by previous mathematical inquiry to establish a system of axioms which might embody the whole of mathematical reasoning that is, to put all of mathematics on an axiomatic base. This work had been brought to a high level of attainment in the sections on the elementary logic of propositions in Bertrand Russell's Principia Mathematica, and it had been apparently completed in the brilliant achievements of David Hilbert in his "axiomatic period" from 1922 to 1930.
Gödel devised a method of converting the symbols of mathematical logic into numbers (Gödel numbers) so as to achieve the arithmetization of metamathematical statements, that is, statements about mathematical arrangements and formulas. He was able to illustrate how a metamathematical statement could be shown to be demonstrable even when postulating its own indemonstrability. From this it would follow that any arithmetical formula is undecidable on the basis of any metamathematical reasoning which could be represented arithmetically. At the same time it could be shown that an undemonstrable formula can nevertheless be established as an arithmetic truth.
Gödel showed in this highly complex chain of reasoning that it is not possible to prove the self-consistency of a system on the basis of metamathematical statements except by going outside that system for the methods of proof. Further, he showed that statements can be constructed within such a system which can be neither proved nor disproved within that system but which can be shown to be arithmetical truths. These conclusions revolutionized mathematical thinking and stimulated the branch of mathematics known as proof theory.
Gödel's life was devoted to the activity of doing fundamental theoretical work. His work in mathematical logic lasted until 1942, when he became primarily occupied with philosophy, intensely studying Leibniz (with whom he closely identified), Kant, and Husserl, until his death in 1978. Gödel arrived at the Institute for Advanced Study in Princeton, NJ, in the fall of 1933, where he met Einstein for the first time, and lectured there for several months in 1934. He married in Adele Porkert in Vienna in 1938. After several commutes between Princeton and Vienna, the Gödel's moved to Princeton permanently in 1940. He became a permanent member of the Institute in 1946 and was appointed to a professorship in 1953.
Gödel distanced himself from the affairs of the world and took part in almost no practical activities: such were the demands of his concentration on fundamental theoretical work. He restricted himself to few contacts with the outside world and most of its inhabitants. He was inclined to caution and privacy; he avoided controversies and appeared to be "exceptionally sensitive" to criticism. He published little (but left a large body of notes and unpublished works), lectured infrequently, accepted few invitations, and disliked travel to the point of declining several honorary degrees because accepting them meant traveling. He was not interested in operating motor vehicles. His few interests were in surrealist and abstract art, his favorite writers included Goethe and Franz Kafka, he enjoyed light classics and some 'pop' music and Disney films, especially Snow White.
Gödel and Einstein found each other to be intellectual equals, and as it happened they shared the same cultural background. Beginning in 1942 in Princeton, they saw and talked with each other almost daily until Einstein's death in 1955. Einstein told a colleague that in the later years of his life, his own work no longer meant much and "that he came to the Institute merely to have the privilege to be able to walk home with Gödel"
Gödel's physique was frail and he was in relatively poor health for much of his life, suffering at times from depression enough to be hospitalized. Gödel's brother, a physician, observed that Kurt's diet was excessively stringent and was harmful. Gödel did not obey doctor's orders, "even at the point where most people would," and he himself admitted that he was a difficult patient. It was widely believed that he was paranoid and constantly worried about food poisoning. In 1978, he died of malnutrition and "inanition" (starvation) caused by "a personality disorder" (according to his death certificate).
Since his death, Gödel's fame has spread more widely, beginning almost immediately with the 1979 publication of Douglas R. Hofstader's Godel, Escher and Bach. The mathematician John von Neumann has called Gödel's achievement in modern logic "a landmark which will remain visible far in space in time." George Zebrowski had said that "No other example of human thought is as far-reaching as Gödel's proof." Gödel's friend and biographer Hao Wang observes that to find work of comparable character in both science and philosophy, "one has to go back to Descartes (1596-1650) and Leibniz (1646-1716), and he adds that it may take "hundreds of years" for the more definite confirmation or refutation of some of [Gödel's] larger conjectures."
In layman's terms, what Gödel did was show conclusively that humans do not live in a universe in which they can solve all problems and learn everything. It can never be done because the universe is infinite and human minds are not. In a way, Gödel's proof is a truth about systems of thought, not about the universe; it is about maps, and not about the territory they represent. What Gödel set out to prove is that the actual territory will always transcend the map.
As one writer has put it, "Unpredictable things happen to finite beings." Gödel's proof suggests a universe that is an open-ended, infinite, eternal existence, requiring no beginning, and in this universe our knowledge may become extensive and significant but will never be complete. An unfalsifiable idea is complete within itself; little green men may live in all refrigerators, but we can't know that since they disappear when the door is opened. Religious dogma is another example of an unfalsifiable idea, for part of its appeal is that it has its own internal resistance to answering questions about its truth. Dogmas are outside Gödel's universe because they try to end all discussions and tests of truth, whereas Gödel's universe asks that we appreciate the practical value of imperfection, serendipity, and wildness. Open-endedness: legal systems can never be more than "good enough;" political systems which are closed impoverish cultural and economic lives, and ultimately fail.
In even simpler terms, as Zebrowski puts it, Gödel's proof can be explained this way: an elderly woman attends a meeting of philosophers concerned with the nature of the universe and tells them that the world rests on the back of a turtle. The chairman asks her to explain what this turtle stands on; she snaps back that it stands on the back of yet another turtle. "And what does that turtle stand on?" demands the chairman. The elderly woman shakes her finger and replies, "You can't fool me, sonny it's turtles all the way down!"
Further Reading on Kurt Gödel
For a model of expository biography, see Hao Wang, Reflections on Kurt Gödel (1987); also, Pelle Yourgrau, The Disappearance of Time (1991), and John W. Dawson, Logical Dilemmas: The Life and Work of Kurt Gödel. George Zebrowski's "Life in Gödel's Universe: Maps All The Way" Omni (April 1992) is very helpful for non-mathematicians.