Kurt Gödel

Kurt Gödel, Mathematician

• Born: 28 April 1906
• Birthplace: Brünn, Austria-Hungary (now Brno, Czech Republic)
• Died: 14 January 1978
• Best Known As: The Austrian mathematician known for his Incompleteness Theorem

Mathematician Kurt Gödel made his fame in 1931 with the publication of his Incompleteness Theorem, also known as Gödel's Theorem. Written while Gödel was a young faculty member at the University of Vienna, his paper demonstrated that any axiomatic system of arithmetic would have true but unprovable statements -- and that any formal system would therefore always be incomplete. This stomped all over the then-prevailing idea that the totality of mathematics could be neatly ordered with the correct set of axioms, or self-evident truths. Gödel's influence was also felt in science and in philosophy, which at the time was dominated by works such as Bertrand Russell and Alfred North Whitehead's Principia Mathematica (1913). Gödel left Austria and ended up joining the Institute for Advanced Study at Princeton University in 1940. There he spent time with his friend Albert Einstein and continued to work on number theory and on revisions of his classic work Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory. Gödel's work on recursive functions puts him in the company of Alan Turing as an influential figure in the history of computer science.

Gödel was a sensitive and fastidious man, and later in life he became so concerned with germs and food poisoning that he gave up eating and starved to death.

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Kurt Gödel

Kurt Gödel

Library of Congress

[b. Brünn, Austro-Hungary (Brno, Czechoslovakia), April 28, 1906, d. Princeton, New Jersey, January 14, 1978]

Kurt Gödel as a young man was part of the important Vienna Circle of logicians. Although Gödel made many contributions to logic (including showing that every statement in the most basic form of logic can either be proved or disproved), and some to physics, his 1931 proof that any system that contains the arithmetic of natural numbers is either not complete or not consistent, known as Gödel's incompleteness theorem, is his most famous. When World War II started, Gödel and his wife emigrated to Princeton, where he had lectured in 1934. From 1935 to 1937 he proved that some important unproved hypotheses in logic were consistent with set theory, work later extended by Paul Cohen, who showed that taking the opposite of these hypotheses was also consistent with set theory. In the 1940s and 1950s, Gödel was close to Einstein and formulated a mathematical framework for Einstein's theories.

Kurt Gödel

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

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.

Kurt Gödel

(born April 28, 1906, Brünn, Austria-Hungary — died Jan. 14, 1978, Princeton, N.J., U.S.) Austrian-born U.S. mathematician and logician. He began his career on the faculty of the University of Vienna, where he produced his groundbreaking proof (see Gödel's theorem) in the early 1930s. He emigrated to the U.S. in 1940 and taught at the Institute for Advanced Study at Princeton. There his close friendship with Albert Einstein led him into the field of general relativity theory and to solutions of some of Einstein's equations. A quiet and unassuming man, Gödel did not at first recognize the importance of his famous theorem, and later in life he declined many of the honours that it brought him.

For more information on Kurt Gödel, visit Britannica.com.

Kurt Gödel

Gödel, Kurt (1906-78) Mathematical logician. Born to German-speaking parents in Czechoslovakia, Gödel studied mathematics at the university of Vienna, where he also came into contact with the Vienna circle (see logical positivism). His ground-breaking results of 1931 became his Habilitationsschrift (postdoctoral thesis) in 1932 (see Gödel's theorems). In 1938 Gödel emigrated from Austria to America, and subsequently worked at the Institute for Advanced Studies at Princeton. Gödel's achievements are almost synonymous with those of mathematical logic in the middle years of the 20th century. They included the proof of the completeness of the first-order predicate calculus, and the ground-breaking results commonly referred to as Gödel's theorems. His proof that no system can show its own consistency effectively put an end to Hilbert's programme, although Gentzen's proof that if transfinite induction is allowed the consistency of arithmetic can be demonstrated, afforded a kind of consolation to Hilbert himself.

From 1943 onwards Gödel devoted himself largely to philosophy, including not only the philosophy of mathematics, but that of general relativity and cosmology. His philosophical views were diametrically opposed to those of the Vienna circle, and tended towards a Platonism that included abstract religious elements. In the following years Gödel was not free of eccentricities. Einstein tells the story of how Gödel called the game theorist Morgenstern on the evening of his citizenship interview in the United States, to explain how he had discovered a logical problem in the Constitution: a non-standard model that showed how a dictatorship could be created consistently with it. After a period of ill-health, he effectively starved himself to death for fear that he was being poisoned. Gödel's extensive writings on the foundations of logic and mathematics are published in Kurt Gödel: Collected Works, vol. i, Publications 1929-1936 (1986), and vol. ii, Publications 1938-1974 (1990).

Gödel, Kurt (gö'dəl) , 1906–78, American mathematician and logician, b. Brünn (now Brno, Czech Republic), grad. Univ. of Vienna (Ph.D., 1930). He came to the United States in 1940 and was naturalized in 1948. He was a member of the Institute for Advanced Study, Princeton, until 1953, when he became professor of mathematics at Princeton Univ. He is best known for his work in mathematical logic, particularly for his theorem (1931) stating that the various branches of mathematics are based in part on propositions that are not provable within the system itself, although they may be proved by means of logical (metamathematical) systems external to mathematics. Gödel shared the 1951 Albert Einstein Award for achievement in the natural sciences with Julian Schwinger, Harvard mathematical physicist. His writings include Foundations of Mathematics (1969).

Bibliography

See H. Wang, Reflections on Kurt Gödel (1987); E. Nagel et al., Gödel's Proof (rev. ed. 2001); R. Goldstein, The Proof and Paradox of Kurt Gödel (2005); P. Yourgrau, A World without Time: The Forgotten Legacy of Gödel and Einstein (2005).