Kurt Friedrich Gödel
Kurt Friedrich Gödel
The names of most mathematical innovators, while they may loom large within their own community, are hardly ever known to the outside world. Kurt Gödel's name and achievements, however, are a part of the framework necessary to an understanding of the postmodern worldview. His incompleteness theorem, which holds that within any axiomatic system there are propositions that cannot be proved or disproved using the axioms of that system, would prove as earth shattering as the better-known relativity theory of his friend Albert Einstein (1879-1955).
The younger son of Rudolf (a textile worker) and Marianne Gödel was born on April 28, 1906. His hometown of Brünn—now Brno in the Czech Republic—was then part of Moravia, a province of the Austro-Hungarian Empire. Gödel's family were themselves German speakers and thus would remain more closely tied to the world of the Austrian capital than to that of their Moravian surroundings.
A devout Lutheran throughout his life, Gödel studied at a Lutheran school in Brünn and in 1916 entered a gymnasium, where he studied until 1924. Planning to study physics, he enrolled at the University of Vienna, but when a professor interested him in number theory he changed his major to mathematics. Gödel, however, retained a lifelong interest in physics.
In 1929 Gödel's father died, and his mother and older brother moved to Vienna. Meanwhile, Gödel was hard at work on his dissertation concerning the incompleteness theorem, for which he received his doctorate in 1930. Inspired by the self-contained system of Euclidean geometry, in which every precept could be derived from a few initial axioms, mathematicians had long been intrigued by the question of just how many axioms would be necessary to prove all true statements in geometry. In his dissertation, Gödel showed that the set of all true statements and the set of all provable statements were essentially the same—a reassuring answer to those who hoped to apply Euclidean methods to the whole of mathematics.
Among those hopefuls were two eminently qualified contemporaries, Bertrand Russell (1872-1970) and Alfred North Whitehead (1861-1947), who had earlier published their ponderous Principia Mathematica. The work, a three-volume tome whose publication costs Whitehead and Russell had to undertake personally, has been called "one of the most influential books never read." In it, the two distinguished mathematician-philosophers set out to derive all mathematical principles from a given set of axioms. Then, in September 1930, the 24-year-old Austrian mathematician who had already made a name for himself with his incompleteness theorem dropped a bomb—a paper called "On Formally Undecidable Propositions of Principia Mathematica and Related Systems."
In it, Gödel applied a new method, assigning numerical values to the symbols of logic, to prove that Russell and Whitehead's quest was doomed. At the heart of mathematics, Gödel showed, were statements that were clearly true but could not be proved by axioms, which, in turn, meant that the set of true statements and the set of provable statements were not identical. At some point, self-referential statements—logic loops on the order of "This statement does not say what it says"—would be inevitable.
The incompleteness theorem shook the foundations of mathematicians' assumptions about the absolute truths undergirding their discipline and forever changed the terms in which mathematics was discussed. Having turned the world upside-down, Gödel—an introverted young man now 27 years old—joined the Institute for Advanced Study at Princeton, where he was free to work without teaching responsibilities. He divided his time between Princeton and Austria and during this period devoted himself to the study of set theory. On September 20, 1938, he married a partner who was altogether his opposite yet with whom he proved compatible—Adele Nimbursky, a nightclub dancer. The two never had children.
In 1939 Gödel, alarmed by the changing situation in Austria, which had recently been absorbed by Nazi Germany, left his homeland for good. He remained at Princeton for the rest of his life and there he enjoyed the companionship of Einstein and mathematical economist Oskar Morgenstern (1902-1977), one of the founders of game theory. In his latter years, he was primarily concerned with questions of philosophy rather than mathematics.
Gödel received honorary degrees from Yale and Harvard as well as the Einstein Award in 1951 and the National Medal of Science in 1975. Increasingly eccentric in his latter years, he refused to eat for fear of poisoning and died of malnutrition on January 14, 1978.