Kurt Gödel

Kurt Gödel (born April 28, 1906, Brünn, Austria-Hungary [now Brno, Czech Rep.]—died Jan. 14, 1978, Princeton, N.J., U.S.) was an Austrian-born mathematician, logician, and philosopher who obtained what may be the most important mathematical result of the 20th century: his famous incompleteness theorem, which states that within any axiomatic mathematical system there are propositions that cannot be proved or disproved on the basis of the axioms within that system; thus, such a system cannot be simultaneously complete and consistent. This proof established Gödel as one of the greatest logicians since Aristotle, and its repercussions continue to be felt and debated today.

Early life and career

Gödel suffered through several periods of poor health as a child, following a bout at age 6 with rheumatic fever, which left him fearful of having some residual heart problem. His lifelong concern with his health may have contributed to his eventual paranoia, which included obsessively cleaning his eating utensils and worrying over the purity of his food.

As a German-speaking Austrian, Gödel suddenly found himself living in the newly formed country of Czechoslovakia when the Austro-Hungarian Empire was broken up at the end of World War I in 1918. Six years later, though, he went to study in Austria, at the University of Vienna, where he earned his doctorate in mathematics in 1929. He joined the faculty at the University of Vienna the next year.

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During that period, Vienna was one of the intellectual hubs of the world. It was home to the famed Vienna Circle, a group of scientists, mathematicians, and philosophers who endorsed the naturalistic, strongly empiricist, and antimetaphysical view known as logical positivism. Gödel’s dissertation adviser, Hans Hahn, was one of the leaders of the Vienna Circle, and he introduced his star student to the group. However, Gödel’s own philosophical views could not have been more different from those of the positivists. He subscribed to Platonism, theism, and mind-body dualism. In addition, he was also somewhat mentally unstable and subject to paranoia—a problem that grew worse as he aged. Thus, his contact with the members of the Vienna Circle left him with the feeling that the 20th century was hostile to his ideas.

Gödel’s theorems

In his doctoral thesis, “Über die Vollständigkeit des Logikkalküls” (“On the Completeness of the Calculus of Logic”), published in a slightly shortened form in 1930, Gödel proved one of the most important logical results of the century—indeed, of all time—namely, the completeness theorem, which established that classical first-order logic, or predicate calculus, is complete in the sense that all of the first-order logical truths can be proved in standard first-order proof systems.

This, however, was nothing compared with what Gödel published in 1931—namely, the incompleteness theorem: “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (“On Formally Undecidable Propositions of Principia Mathematica and Related Systems”). Roughly speaking, this theorem established the result that it is impossible to use the axiomatic method to construct a mathematical theory, in any branch of mathematics, that entails all of the truths in that branch of mathematics. (In England, Alfred North Whitehead and Bertrand Russell had spent years on such a program, which they published as Principia Mathematica in three volumes in 1910, 1912, and 1913.) For instance, it is impossible to come up with an axiomatic mathematical theory that captures even all of the truths about the natural numbers (0, 1, 2, 3,…). This was an extremely important negative result, as before 1931 many mathematicians were trying to do precisely that—construct axiom systems that could be used to prove all mathematical truths. Indeed, several well-known logicians and mathematicians (e.g., Whitehead, Russell, Gottlob Frege, David Hilbert) spent significant portions of their careers on this project. Unfortunately for them, Gödel’s theorem destroyed this entire axiomatic research program.

International stardom and move to the United States

After the publication of the incompleteness theorem, Gödel became an internationally known intellectual figure. He traveled to the United States several times and lectured extensively at Princeton University in New Jersey, where he met Albert Einstein. This was the beginning of a close friendship that would last until Einstein’s death in 1955.

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However, it was also during this period that Gödel’s mental health began to deteriorate. He suffered from bouts of depression, and, after the murder of Moritz Schlick, one of the leaders of the Vienna Circle, by a deranged student, Gödel suffered a nervous breakdown. In the years to come, he suffered several more.

After Nazi Germany annexed Austria on March 12, 1938, Gödel found himself in a rather awkward situation, partly because he had a long history of close associations with various Jewish members of the Vienna Circle (indeed, he had been attacked on the streets of Vienna by youths who thought that he was Jewish) and partly because he was suddenly in danger of being conscripted into the German army. On Sept. 20, 1938, Gödel married Adele Nimbursky (née Porkert), and, when World War II broke out a year later, he fled Europe with his wife, taking the trans-Siberian railway across Asia, sailing across the Pacific Ocean, and then taking another train across the United States to Princeton, N.J., where, with the help of Einstein, he took up a position at the newly formed Institute for Advanced Studies (IAS). He spent the remainder of his life working and teaching at the IAS, from which he retired in 1976. Gödel became a U.S. citizen in 1948. (Einstein attended his hearing because Gödel’s behaviour was rather unpredictable, and Einstein was afraid that Gödel might sabotage his own case.)

In 1940, only months after he arrived in Princeton, Gödel published another classic mathematical paper, “Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory,” which proved that the axiom of choice and the continuum hypothesis are consistent with the standard axioms (such as the Zermelo-Fraenkel axioms) of set theory. This established half of a conjecture of Gödel’s—namely, that the continuum hypothesis could not be proven true or false in standard set theories. Gödel’s proof showed that it could not be proven false in those theories. In 1963 American mathematician Paul Cohen demonstrated that it could not be proven true in those theories either, vindicating Gödel’s conjecture.

In 1949 Gödel also made an important contribution to physics, showing that Einstein’s theory of general relativity allows for the possibility of time travel.

Turn to philosophy

In his later years, Gödel began writing about philosophical issues. Gödel had always been interested in this. Indeed, it is a little-known fact that Gödel set out to prove the incompleteness theorem in the first place because he thought he could use it to establish the philosophical view known as Platonism—or, more specifically, the subview known as mathematical Platonism. Mathematical Platonism is the view that mathematical sentences, such as “2 + 2 = 4,” provide true descriptions of a collection of objects—namely, numbers—that are nonphysical and nonmental and exist outside of space and time in a special mathematical realm—or, as it has also been called, “Platonic Heaven.” Gödel’s idea was that if he could prove the incompleteness theorem, then he could show that there were unprovable mathematical truths. This, he thought, would go a long way toward establishing Platonism, because it would show that mathematical truth is objective—i.e., that it goes beyond mere human provability or human axiom systems.

In 1964 Gödel published a philosophical paper, “What Is Cantor’s Continuum Problem?,” in which he proposed a solution to an ancient objection to Platonism. It is often argued that Platonism cannot be true, because it makes mathematical knowledge impossible: whereas humans seem to acquire all knowledge of the external world through sensory perception, Platonism asserts that mathematical objects, such as numbers, are nonphysical objects that cannot be perceived by the senses. Gödel responded to this argument by claiming that, in addition to the normal five senses, humans also possess a faculty of mathematical intuition, a faculty that enables people to grasp the nature of numbers or to see them in the mind’s eye. Gödel’s claim was that the faculty of mathematical intuition makes it possible to acquire knowledge of nonphysical mathematical objects that exist outside of space and time.

Unfortunately for Gödel, his philosophical views have not been very widely accepted. Everyone accepts his incompleteness theorem, but very few people believe that it establishes Platonism.

As Gödel aged, he grew more and more paranoid and eventually became convinced that he was being poisoned. He refused to eat unless his wife tasted his food first. When she became ill and had to be hospitalized for an extended period of time, Gödel essentially stopped eating and starved to death.