6/15/2020

Kurt Gödel’s Brilliant Madness



Arnold Newman’s iconic 1956 portraits of Kurt Gödel

Kurt Gödel’s Brilliant Madness



Jun 12 · 36 min read


“Modern math’s absolute Prince of Darkness” — David Foster Wallace
“Laying the foundation for today's flourishing study of mathematical logic”



Introduction


Left: Rudolf Gödel and his little brother Kurt around 1908. Right: Marianne and Rudolf Gödel with their children Kurt (left) and Rudolf (right)
“Gödel considered himself always Austrian and an exile in Czechoslovakia”

Left: The Gödel villa in Brno (Photo: Dawson, 1983). Right: The Evangelische Privat-Volks- und Bürgerschule (Photo: unknown)
“He was a rather silent man, but he offered his help whenever it was needed” — Olga Taussky-Todd
"At a mathematical conference preceding Hilbert's address, a quiet, obscure young man, Kurt Gödel, only a year beyond his PhD, announced a result which would forever change the foundations of mathematics. He formalized the liar paradox, "This statement is false" to prove roughly that for any effectively axiomatized consistent extension T of number theory (Peano arithmetic) there is a sentence σ which asserts its own unprovability in T.John von Neumann, who was in the audience immediately understood the importance of Gödel's incompleteness theorem. He was at the conference representing Hilbert's proof theory program and recognized that Hilbert's program was over.In the next few weeks von Neumann realized that by arithmetizing the proof of Gödel's first theorem, one could prove an even better one, that no such formal system T could prove its own consistency. A few weeks later he brought his proof to Gödel, who thanked him and informed him politely that he had already submitted the second incompleteness theorem for publication."- Excerpt, Computability. Turing, Gödel, Church and Beyond by Copeland et al. (2015)

Left: Kurt Gödel as a student, ca 1924–27 (Photo: Wikimedia Commons). Right: Gödel’s legendary 1931 publication “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I” in Monatshefte für Mathematik und Physik 38 (1931), pp. 173–198.



Around the same time, the President of Germany Paul von Hindenburg reluctantly appointed Nazi leader Adolf Hitler as the new Chancellor of Germany. After only two months in office, following the burning of the Reichstag building, the German parliament passed the "Enabling Act" giving the chancellor full legislative power for a period of four years. Following the death of von Hindenburg the next year, Hitler used the act to merge the offices of Chancellor and President, creating the new office of "Führer and Reichskanzler".

First visit to America (1933–34)

Dear Mr. Gödel,I just returned to Europe from Princeton, and I would be very glad if during my European stay there would be an opportunity to meet you. At the end of September I will go back to Princeton, where in the future I will spend two terms per year, as I received an offer from the new Bamberger-Flexner "Institute for Advanced Study", and I accepted.Perhaps you would like to know more than you have gathered from the brief correspondence up to now about the conditions in Princeton and about the structure of this new Institute,, of which you will fortunately be a member next year. Therefore, I write to you first of all, and also on Veblen's request, to tell you that I am happy to provide you with any information you may be interested in.Perhaps we can also meet sometime and somewhere--this summer I will probably still lecture in Berlin (for the last time), and I will repeatedly be in or travel through Vienna. What is your program?With best regards, which I ask you to extend also to Menger and Hahn, Yours sincerely, Johann von Neumann

Princeton University’s Fine Hall (now Jones Hall) in 1931, where the Institute for Advanced Study was housed from 1933–1939 (Photo: The Trustees of Princeton University)

First Mental Breakdown (1934–35)


The SS Rex Gödel traveled on from New York to Genoa

Following the Great Depression, a governmental crisis in 1932 had given way for a conservative politician named Engelbert Dollfuss (1892–1934) to ascend to power in Austria. In early 1933 responding to Hitler's own ascension to Chancellor of Germany, Dollfuss shut down the Austrian parliament and assumed dictatorial power. The next February a brief Austrian Civil War--sometimes known as the February Uprising--took place when austrofascists loyal to Dollfuss skirmished with socialist forces. By July, taking advantage of the unstable situation, Austrian Nazis loyal to Hitler had stormed the Chancellery in a failed coup and assassinated Dollfuss. In response, the head of the Italian fascist party Benito Mussolini mobilized part of the Italian army on the Austrian border, threatening Hitler with war in the event of a German invasion of Austria. With Mussolini's approval, Dollfuss was soon succeeded by Kurt Schuschnigg (1897–1977) who restored stability in Austria.
Gödel was more withdrawn after his return from America than before; but he still spoke with visitors to the Colloquium, [...] especially with Wald and Tarski [...]. To all members of the Colloquium Gödel was generous with opinions and advice in mathematical and logical questions. He consistently perceived problematic points quickly and thoroughly and made replies with greatest precision in a minimum of words, often opening up novel aspects for the inquirer. He expressed all this as if it were completely a matter of course, but often with a certain shyness whose charm awoke warm personal feelings for him in many a listener.- Excerpt, Recollections of Kurt Gödel by Karl Menger (1981)

Sanatorium Purkersdorf in Wien-Umgebung where Gödel spent a week in October 1934

Second visit to America (1935)

Second Mental Breakdown (1936)


Postcard illustrating Sanatorium Rekawinkel in the 1930s





Left: Adele Thusnelda Porkert (neé Nimbursky). Right: Kurt and Adele on their wedding day in 1938
At the time, pressure from pro-unification activists had forced the Austrian chancellor Schuschnigg to announce that there would be a referendum on the 13th of March on a possible union with Nazi Germany. Hitler portrayed the vote as an "obfuscation of the popular will in Austria and Germany" and threatened invasion. Under the pressure, Schuschnigg cancelled the referendum. Unopposed by the Austrian military, on March 12th the German Wehrmacht marched into Austria in what is now known as the "Anschluss", the annexation of Austria into Nazi Germany.

Third visit to America (1938–39)

War (1939)

The German invasion of Poland began on the 1st of September 1939, the day after the Supreme Soviet of the Soviet Union approved the Molotov-Ribbentrop Pact. The United Kingdom, France and their empires responded by declaring war on Nazi Germany.

Emigration to America (1939–40)

Dear Doctor Flexner,I am returning enclosed Gödel's wire [...] Under these conditions the situation is considerably better than we originally expected it to be. While it would have been difficult, if not impossible, to intercede with the German authorities, there should be a way to persuade the American consul in Vienna to grant a visa.As far as I can make out from Gödel's statements and from other cases of similar nature, his difficulty is this: professors' visas are granted only to applicants who go into teaching or research positions in this country and who have had such positions previously for two years in the country they come from. The objection made against Gödel - and against some other refugees - is that the two years teaching in the country of origin have to be immediately preceding their application; whereas Gödel was suspended from his position by the Nazis after the "Anschluss" in 1938. [...]
Couldn't you find it possible to intercede with the State Department in this sense?
2. The claim may be made with perfect justification that Gödel is unreplaceable for our educational program. Indeed Gödel is absolutely irreplaceable; he is the only mathematician alive about whom I would dare make this statement. He represents a very important branch of mathematics, formal logics, in which he outranks everybody else to a much higher degree than usually happens in any other branch of mathematics. Indeed, the entire modern development of formal logics concerning "undecidable questions", the solution of the famous "continuum hypothesis", and quite unexpected connections between this field and other parts of mathematics, are entirely his individual contribution.Besides, the ouvre of his scientific achievements is obviously still in steep ascent, and more is to be expected from him in the future.
“I am convinced that salvaging him from the wreck of Europe is one of the great single contributions anyone could make.”

Life in Princeton

“Gödel did some of his best work […] at a time when he was less normal than now.” —John von Neumann (1945)

Adele and Kurt Gödel in Princeton (Photo: Courtesy of IAS Archive)




Letter from John von Neumann to Oswald Veblen (November 30, 1945)
I talked to Aydelotte about Gödel. [...]1) Gödel's work on Leibniz was important in the history of mathematics. He might still do better work in mathematics proper, but this was solid work too, and a man of his caliber and record ought to be the sole judge of what he does.2) Gödel's whole intellectual behavior at present is such, that he may easily do more work in mathematics proper. In fact I judged, that his probability of doing some is no worse than that of most mathematicians past 35.3) Gödel's two past papers ("Widerspruchfreiheit" and Continuum Hypothesis), are in any case worth more than the total literary output, past, present and future, of most mathematicians, including many who would be good enough to be our colleagues.4) Gödel did some of his best work (Continuum Hypothesis) at the Institute - actually, at a time when he was less normal than now. The Institute is clearly committed to support him, and it is ungracious and undignified to continue a man of Gödel's merit in the present arrangement forever. Hence it is just as well to correct the situation as quickly as feasible.5) (Upon Aydolette's question.) Gödel's technical performance is vastly greater than B.Russel's.After some discussion Aydelotte seemed to agree with me on all counts. He stated:a) Gödel's situation is now very much on his mind and he proposes to correct it.
b) He recognizes that giving him Mayer's position is the necessary minimum.
c) He feels that he has to bring this matter, however, before the Faculty.I agree with c), and took the line, that Gödel's case will only gain if the existence of the problem is publicly recognized and his objective scientific importance is publicly discussed - that is before the Faculty.I think that it will be very good if you repeat these thoughts to Aydelotte. He definitely committed himself to taking the action indicated.With best regards, 
yours as ever,John

Becoming an American citizen (1947)


Left: Einstein and Gödel at the Institute for Advanced Study in 19xx, photograph taken by Oskar Morgenstern. Right: Morgenstern and Gödel, photograph likely taken by Albert Einstein (Photos: Courtesy of IAS Archive)
“How can any of us be called professor when Gödel is not?”

Relationship with Albert Einstein


Einstein and Gödel in the early 1950s (Photo: Richard Arens. Courtesy of the IAS Archives)


Einstein and Gödel in Princeton (Photo: Courtesy of the IAS Archive)

Third Mental Breakdown (1970)


Gödel and Morgenstern’s wife Dorothy in 1973 (Photo: A.G. Wightman. Courtesy of the IAS Archives)

Death (1978)


Adele and Kurt Gödel’s grave in Princeton, New Jersey (Photo: Wikimedia Commons)

Today [...] Kurt Gödel called me again [...] and spoke to me for about 15 minutes [...] He asserted that the doctors are not telling him the truth, that they do not want to deal with him, that he is in an emergency (exactly what he told me with the same words a few weeks ago, a few months ago, two years ago), and that I should help him get into the Princeton Hospital. [...][He] also assured me that [...] perhaps two years ago [...] two men appeared who pretended to be doctors, [...] They were swindlers [who] were trying to get him in the hospital [...] and he [...] had great difficulty unmasking them.[...]It is hard to describe what such a conversation [...] means for me: here is one fo the most brilliant men of our century, greatly attached to me [...] [who] is clearly mentally disturbed, suffering from some kind of paranoia, expecting help from me [...] and I [am] unable to extend it to him. Even while I was mobile and tried to help him [...] I was unable to accomplish anything [...] Now, by clinging to me --and he had nobody else, that is quite clear -- he adds to the burden I am carrying.
Gödel's demise was fraught with Pyrrhic irony: Unable to escape from the inner logic of his paranoia - to adopt, as it were, a "metatheoretical perspective" - he succumbed to starvation in the grip of his obsessive fear of being poisoned. Like the actions of an inhabitant in one of those timeless universes he had envisioned, his death was, and was not, of his own volition.- Excerpt, Logical Dilemmas by John W. Dawson (2006)





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Editor-in-Chief at Cantor’s Paradise. Research fellow at the Norwegian University of Science and Technology.

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