What Went On in Austria That Week: Gödel, Mozart, BachmannDecember 22, 2006
Courtesy of the Gödels Century Exhibition.
Philip J. Davis
Kurt Gödel: The Album. By Karl Sigmund, John Dawson, and Kurt Mühlberger, Vieweg, Wiesbaden, 2006, 225 pages, 29.90. Text both in German and in English.
After ten days in Austria last spring, as a participant in a symposium of the Austrian Society for the History of Mathematics, I walked into an exhibit in the Palais Palffy (in the heart of Vienna ) and my eyes opened wide. There, in front of me, on large poster boards, were blown up pictures of men and a woman who had been important influences in my early mathematical education!
W.V.O. Quine (logician and philosopher), with whom I had a course in mathematical logic, stared at me sternly and magisterially. A young-looking Philipp Frank (mathematical physicist and philosopher) reminded me of the courses I had taken with him in mechanics, relativity, and philosophy of science. Richard von Mises (World War I pilot, aerodynamicist, philosopher of randomness and of science, expert and collector of material on the poet Rilke), looking quite the elegant Herr Professor Doktor, was the man who, after interviewing me, said to Stefan Bergman: "OK. I guess you can put this young man on your Washington contract." Olga Taussky Todd (algebraist and matrician), in the few years we were together at the National Bureau of Standards, got me to pay some attention to the numerics of matrices. Alfred (Freddy) Ayer (British positivist philosopher, radio and TV personality, and bon vivant) revealed to me in our correspondence the most burning philosophic question of the day. More on that later.
All these figures, and others of whom I knew but had never met---Karl Menger, Karl Popper, Alfred Tarski, Ludwig Wittgenstein, and Oskar Morgenstern---loomed large in the exhibit. What, you may well ask, were all these spirits of the past doing in the Palais Palffy (other than making me feel uneasy about what I've done over the years with what I'd learned from them)?
The question is easily answered. When I got off the plane at the Vienna airport on May 17, 2006, I realized immediately that Vienna was in the throes of celebrating the 250th anniversary of the birth of Wolfgang Amadeus Mozart. Walking around the city, one couldn't avoid Mozart: He was on billboards, in leaflets, in concerts and talks; you name it. When I got to my hotel, slept off my jet lag, and took a look at the What's Doing in Vienna This Week flyers available in the hotel lobby, I found that Vienna was also celebrating (much more modestly) the 100th anniversary of Kurt Gödel and the 80th anniversary of Ingeborg Bachmann (19261973), prize-winning Austrian novelist, playwright, and poet. Exhibits marking these anniversaries were on display in one and the same place: the Palais Palffy. I had to go, of course; I knew a bit about Ingeborg Bachmann, which was an additional reason for going.
Early on a very rainy and windy Monday afternoon, I found my way to the Palais Palffy, just opposite the great Österreichische Nationalbibliothek and a few blocks from the renowned Café Mozart. Outside the Palais, a huge rectangular pennant bearing Gödel's name flapped in the wind. Inside, the exhibits devoted to Gödel and Bachmann were one floor apart. The Gödel exhibition rooms were open and free of charge, but no docent was there to offer brochures or to note the fact that I was the only visitor. (Later, I was joined by a young man in the garb and backpack of the stereotypical student.)
The Ingeborg Bachmann exhibit was better monitored and attended. I paid a few Euros for admission and encountered perhaps a half dozen visitors looking at the posterboards and listening to tapes of the author reading from her works.
The Gödel exhibit displayed, with artifacts and posters, not only Gödeliana---his family, his citations, descriptions of his great accomplishments in mathematics and logic, biographic material, encomia galore---but also material on the numerous personalities of the famed Wienerkreis (the Vienna Circle). These men and women, who had known Gödel, were part of the heady intellectual atmosphere that existed in Vienna prior to the tragic Anschluss of March 9, 1938, the annexation of Austria by Nazi Germany. The logical positivist position of the Vienna Circle was that statements that cannot be tested empirically are meaningless. After the Anschluss, members of the Circle fled to England and to the United States, and their doctrine became part of my early training and my early rebellion. Gödel claimed that he killed logical positivism---and this may be so---and in my own rejection of the position I have at the very least one thing in common with him.
And now to the book under review. It's a book that every Gödel Groupie, Aficionado, or Gödelolater will want for his library: an apotheosis of the man. The materials for the exhibition were assembled by Karl Sigmund of the University of Vienna, John Dawson of Pennsylvania State University, and Kurt Mühlberger, director of the University of Vienna Archives, and the book is a spin-off from the exhibition. It contains reproductions of all the displays of the exhibition and much, much more.
Put together with care and beautifully produced, the book includes letters, telegrams, official documents for the infant Kurt, Papa and Mamma, brother Rudy, Kurt's wife, Adele, the receipt for their frugal wedding reception at the Rathauskeller, which he as a compulsive paper collector preserved. There are many pictures of Gödel in the book: Gödel with Adele, with Einstein, with Tarski, with Abraham Wald, with Hao Wang (who wrote Reflections on Kurt Gödel), and others. All this is embedded in a text that contains a skeletonic biography and leads up to the renowned Incompleteness Theorems. Against the unlikely event that any reader is unfamiliar with them, here is a brief statement of the first: Given a consistent formalization of arithmetic, there are arithmetic truths that are not provable in that system.
As a bonus that buffs will appreciate, an appendix contains two documents of historic interest: The first is a synopsis of the seminal paper on the Incompleteness Theorems that Gödel submitted to the editor of the philosophic journal Erkentniss prior to publication. The second is the text of Karl Menger's semi-popular talk at the University of Vienna (1932) in which he explains the whole business.
Incidentally, John Dawson is the author of a splendid biography of Gödel; readers interested in the details of Gödel's career are encouraged to consult the book (or my review in the October 1997 issue of SIAM News).
Vienna is not the whole of Austria, any more than Hollywood is the whole of the United States, and the scene shifts now to the symposium I went to Austria to attend. It was held in Miesenbach, in the foothills of the Schneeberg, a drive of about an hour and a half south from Vienna. The symposium had as its theme the influence of the media on the development of mathematics. Relevant to the Gödel celebration and to this article was the paper presented by the British historian of mathematics Ivor Grattan-Guinness, titled "The Receptions of Gödel's Incompletability Theorems by Mathematicians, 19311960."
Grattan-Guinness, who had surveyed sixty-one important books from this period, pointed out that twenty-three of them, though they might appropriately have mentioned the Gödel theorems, failed to do so. Included in this category were such distinguished mathematicians and logicians as H. Hahn, B. Russell, H. Cartan, A.A. Fraenkel, P. Bernays, and P. Suppes. Why, Grattan-Guinness wondered, was this so? He suggested the following reasons:
"The authors had not heard of the theorems.
The authors had heard of the theorems but did not grasp their fundamental consequences.
The authors had heard and did comprehend, but they knew that Gödel's conception of a proof was and had to be much tighter' than they were likely ever to use themselves--far beyond even normal rigorous' maths. Thus, his results would not apply to their work; so there was no need to worry about them."
Grattan-Guinness pointed out that these omissions are "evidence of the ambiguous attitude that mathematicians have usually shown towards logic." His third reason comes close to what I have called the Paradox of the Irrelevance of the Gödel Theorems. Harvard University, in awarding an honorary doctorate to Gödel in 1953, called the Incompleteness Theorems "the most significant mathematical truth of the century," yet they are ignored by the overwhelming majority of research mathematicians. Well, in the words of Dr. Samuel Johnson, "in [composing] lapidary inscriptions, a man is not under oath," and I suppose that the same goes for honoris causa citations.
For me, Gödel's Incompleteness Theorems give rise to two parallel narratives. In one of them, Gödel destroys logicism, i.e., the view that mathematics is totally characterized as a hypthetico-deductive enterprise. In the other, reflecting the irrelevance of the theorems, is the view that the role of deductive proof in mathematics has been seriously misappraised. I referred earlier to a letter I received from Alfred Ayer. In his view, the most significant problem for philosophy was: "What is the case and how do we know it?" Ask this question of the two narratives.
On that rainy, windy Monday afternoon, the public was not beating a path to the Gödel Ausstellung. Yet it is not impossible that the young student who, other than me, was the sole visitor, will ponder what he saw and go away to create a whole new vision of what mathematics is all about.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at firstname.lastname@example.org.