Kurt Gödel, Life of a Logician

Kurt Gödel, Life of a Logician

Allen Filson

Math 5010 - Spring Semester 2007

The Life

Kurt Friedrich Gödel was born in Brünn, Moravia (present day Brno, Czechoslovakia) on April 28th 1906 of the ethnic German family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh). Kurt was given the middle name of Rudolf Gödel's employer (Friedrich Redlich) who, at the time of Kurt's birth, also served as a god-parent to the newborn. The town of Brünn, during Kurt's early years, had a slight German-speaking majority and this was the language of his parents as well. Even after the collapse of the Austro-Hungarian empire following the end of World War I and the declaration of the independent Czechoslovakian nation, of which the city of Brünn now existed within, young Kurt would not assimilate into this newly created state. Automatically becoming a Czechoslovak citizen at age 12. He later told his biographer John D. Dawson that he felt like an "exiled Austrian in Czechoslovakia" ("ein österreichischer Verbannter in Tschechoslowakien") during this time[1]. He was never able to speak Czech and refused to learn it at school.

After he matriculated at the University of Vienna, and he was deeply involved with his studies, he would make the choice, at age 23 to became an Austrian citizen. When Nazi Germany annexed Austria, Gödel automatically became a German citizen at the age of 32. Then, as if 4 changes of citizenship were not enough, he would become a naturalized American citizen at the end of World War II at the age of 42. So before Kurt's 50th birthday, he could have claimed citizenship to no less then 5 different nations.

Kurt, the youngest of two children, had an older brother Rudolf (named after his father), who would become, over the course of both his and Kurt's life the unofficial family biographer. “According to his brother, Kurt had a happy childhood, though he was shy and easily upset. His family dubbed him Herr Warum (Mr. Why) because of his continual questions”[2]. Much of the information know about the Gödel family in the years preceding both brothers leaving for University would be recalled by Kurt's older brother Rudolf. Kurt himself would not respond to queries about his early years or be evasive in responses. This was probably more aligned with his overall reticence to engage anyone in what he would perceive as trivial banter, rather then some traumatic childhood experiences. Kurt's childhood was a secure and comfortable one, because of the elder Rudolf Gödel's stature within the textile company that he would eventually become part owner in.

In 1924, at the age of 18, Kurt joined his older brother Rudolf at the University of Vienna. By that time he had already mastered university level mathematics through his education at the Staats-Realgymnasium, a German language high school in Brünn. Although initially intending to study theoretical physics, Kurt also attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism. He read Kant's Metaphysische Anfangsgründe der Naturwissenschaft, and participated in the Vienna Circle with Moritz Schlick, Hans Hahn, and Rudolf Carnap among others. Around 1926, greatly influenced by the lectures he attended by the number theorist Philipp Furtwangler, Kurt would change his emphasis of study away from Physics and focus on Mathematics. Later in life he would recall that Furtwangler's lectures where some of the most interesting that he ever attended[3].

At this time, and by changing his area of study from physics to mathematics, young Kurt began to mingle within the circle of mathematicians, philosophers and physical scientist that would introduce him into the field of mathematics that he would forever define and shape. During 1927, when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book, “Introduction to Mathematical Philosophy”, Kurt became interested in mathematical logic. Moritz Schlick, more or less the leader of the Vienna Circle, defined the groups philosophical leanings, as well as it's followers, as “logical positivists”[4].

The Vienna Circle (in German: der Wiener Kreis) was a group of philosophers who gathered around Moritz Schlick when he was called to the Vienna University in 1922, organized in a philosophical association named Verein Ernst Mach (Ernst Mach Society). Among its members were Moritz Schlick, chairman of the Ernst Mach Society, Gustav Bergmann, Rudolf Carnap, Herbert Feigl, Philipp Frank, Kurt Gödel, Hans Hahn, Victor Kraft, Karl Menger, Marcel Natkin, Otto Neurath, Olga Hahn-Neurath, Theodor Radakovic, and Friedrich Waismann. With the exception of Gödel, members of the Vienna Circle had a common attitude towards philosophy. Gödel's Platonism[5] conflicted with the societies positivism and he drops away by 1933.

Although Kurt's own philosophical bent, Platonism, was orthogonal to the belief's of the members of the Vienna Circle, he would be active in the group for the latter years of the 1920's and would discover his research passion through interactions with this very group. Specifically, “Rudolf Carnap, whose 1928 lectures on mathematical logic and the philosophical foundations of arithmetic strongly influenced the future direction of his [Kurt's] research”[6].

The coming years of 1929 and 1930 would be some of Kurt's most productive in the field of mathematical logic. In the summer of 1929, shortly after the Carnap lectures Kurt completes his doctoral dissertation under Hans Hahn. Hans himself was an analyst concerned with general topology and logic, and had become Kurt's primary instructor while working for his doctorate degree at the University of Vienna. Kurt's doctoral dissertation establishes the completeness of the first-order predicate calculus (Gödel's completeness theorem) for which he received a doctorate in mathematics from the University of Vienna in February 1930. On the heels of that work, at the Koenigsberg conference on September 7th, 1930, Kurt announces his first incompleteness theorem. October 23rd 1930, the second incompleteness theorem is announced as well. The first and second incompleteness theorems would forever change the face of mathematics. Hans Hahn would serve as the referee for the work submitted to the University of Vienna as Kurt's “Habilitationsschrift”[7] and would write:

“a scientific achievement of the first rank . . . will find its place in the history of mathematics . . . The work submitted by Dr. Gödel surpasses by far the standard usually required for Habilitation. Today Dr. Gödel is already the principle authority in the field of symbolic logic and research on the foundations of mathematics.”[8]

During the academic years of 1931-1932 Kurt would become an assistant in Hahn's seminar in mathematical logic, selecting much of the material discussed. Then in 1933, he becomes Privatdozent at the University of Vienna and teaches his first class. His teaching style is defined as 'sedate' and his classes draw few students. His uneasiness with groups is evidenced by his lectures where in he “lectures to the board”[9] rarely engaging his students in free form discussion or conjecture. Teaching would never become one of Kurt's strengths as the interaction with students was difficult and awkward for the introverted and shy Gödel.

In the early years of the 1930's the newly established Institute for Advanced Studies (IAS) at Princeton, New Jersey facilitated several trips for the young Gödel to travel across the Atlantic and give lectures on his incompleteness theorems. Travels to the IAS in 33', 34' and 35' were facilitated by Oswald Veblean who was most interested in establishing a relationship with this young logician who had forever changed the nature of mathematical logic. Generally speaking, the trips to the IAS became a mental and physical drain on Kurt and he would return exhaust and depressed. On one such trip, in December of 1936, Gödel prematurely ended his alloted time at the IAS and returned to Vienna to be committed to a sanatorium for depression and overwork.

After the Anschluss in 1938, wherein Austria becomes a part of Nazi Germany, horizon's begin to darken in Vienna. Germany abolished the title of Privatdozent, so Gödel had to apply for a different position, 'dozent never ordnung' under the new order; and his former association with Jewish members of the Vienna Circle, especially with Hahn, weighed heavily against him. He was initially denied a post at the re-staffed University of Vienna. His predicament became worse when he was ordered to submit for an army physical exam and was found fit for military service. Gödel now was at risk of being conscripted into the German army.

A bright spot on the darkening horizon of European events for Kurt comes in 1938 when he marries Adele Porkert (six years his senior) whom worked at the Viennese night club called Der Nachtfalter (The Moth) as a dancer. Records show that the two had been involved with one another for many years and probably began seeing one another as early as 1927. The Gödel family, particularly the father had been adamantly against the relationship due to the age difference between the two. It is unknown whether the rejection to the relationship was indeed due to the age difference or more to Adele's occupation and lifestyle. Rudolf Gödel Sr. himself was 14 years older then Kurt's mother. Nevertheless, after Rudolf Gödel Sr.'s passing in 1929, it appears that the family began to warm to Adele which resulted in the marriage of 1938.

With Europe descending into the darkness of fascism and with the help of Oswald Veblen of the IAS, Kurt obtains an American non quota immigrant visa and leaves Europe in 1940, never to return. Fearing the journey across the Atlantic due to the increase U-Boat activity, he travels across Asia, the Pacific and all of North America to take up a post at the Institute for Advanced Study. Ironically, after fleeing Europe, from 1941 to 1945 he was listed at the University of Vienna as 'Dozent Fur Grundlagen Der Mathematik und Logik'. Even long after the end of World War II and up until his death, Kurt would refuse honors bestowed upon him by the University of Vienna. For that matter he felt so, betrayed by his treatment before the war that most European honor bestowed upon him were either ignored or refused. This did not stop the University of Vienna in bestowing upon him there highest order of achievement, posthumously.

In 1946, Gödel becomes a permanent member of the IAS and American citizenship following shortly thereafter in 1948. Albert Einstein, his coach for citizenship and a newly minted citizen himself accompanies him to the citizenship ceremony. By 1948, Kurt and Albert had become the best of friends and where observed each evening walking home from the IAS together. At his citizenship hearing, Gödel informs the judge that he has discovered a way in which a dictatorship could be legally installed in the United State by virtue of a flaw in the constitution of the United States. The judge, bemused, dismisses the comment.

The final years of Gödel's life see him become a full professor at the Institute in 1953 and an emeritus professor in 1976. On January 14th 1978, Kurt Gödel passes on with the cause of death listed as 'malnutrition and inanition caused by a personality disturbance'. He weighted a mere 65lbs.

The Mathematics and Philosophy

Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems set one of Gödel's life courses. In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik (Principles of Theoretical Logic), an introduction to first-order logic in which the problem of completeness was posed: Are the

axioms of a formal system sufficient to derive every statement that is true in all models of the system? This was the topic chosen by Gödel for his doctorate work.

In 1928, mathematical logic is not a well defined field. It was still a part of the “foundations of mathematics” - a subject that belonged more to philosophy then to mathematics. In short, mathematical logic had 3 competing foundation schools:

1) Formalism - Developed by David Hilbert treated mathematics as a purely formal and syntactical subject in which meaning was introduced only at the metamathematical level.

2) Intuitionism- Developed by L.E.J. Browner, stressed the role of intuition as opposed formal aspect and rejected much of both classical logic and classical mathematics.

3) Logicism - Developed by Bertraud Russell and embodied in his 'Principia Mathematic', asserted that mathematics was a part of logic and could be developed as a logical system.

All three influenced Gödel who followed none of them but established a new paradigm: mathematical logic as a part of classical mathematics, answering questions of genuine mathematical interest and only indirectly of philosophical importance.

Gödel's dissertation, grew out of a problem that Hilbert and Wilhelm Ackerman had posed in their book. Hilbert & Ackerman remarked that it was not know whether every valid axiom of first order logic is provable and, moreover, whether each axiom of first order is independent. His dissertation solved both problems. At the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, Gödel established the completeness of the first-order predicate calculus (this result is known as Gödel's completeness theorem). His thesis, along with some additional work, was published by the Vienna Academy of Science.

Even though Kurt announced his incompleteness theorems late in the 1930, he would not formally publish the results until two years later. One of Kurt's most interesting idiosyncrasies was his nature of delaying the publishing of research results until he was absolutely certain the arguments were precise and irrefutable. In 1932 there appeared in a German scientific periodical a relatively short paper entitled “Uber formal unetscheidbare Satze der Principia Mathematica und verwandter Systeme” (On Formally Undecidable Propositions of Principia Mathematica and Related Systems”).