First of all, he’d said, the Harvard department was too “empiricist,” and he thought they’d have been critical of what he had to say. Second of all—and this part of the answer, White told me, had really interested him—Gödel felt he would have been doing an injustice to the ideas themselves, because he hadn’t yet completed them; to expose them prematurely to an unsympathetic audience would be acting unjustly toward them.
So it seems, at least from this story, that his reluctance to voice his unfashionable intuitions in any form that fell short of a proof was not only a matter of his own distaste for intellectual wrangling but also connected with a perceived ethical obligation toward the ideas themselves, which is appropriate for an impassioned Platonist.
In 1964, Paul Benacerraf, of Princeton University’s philosophy department, and Hilary Putnam, of Harvard’s, edited a book entitled Philosophy of Mathematics and they wanted permission to include two of Gödel’s articles, “Russell’s Mathematical Logic” and “What is Cantor’s Continuum Problem?” In the latter essay, revised and expanded for this volume, Gödel allows himself to state in clear and distinct terms the metaphysical Platonism to which he had subscribed, even as he’d sat, a backbencher among the positivists, quietly listening to the members of the Vienna Circle proclaiming the everlasting end of metaphysics, that is, of any assertions of existence that go beyond the empirically verifiable:
[T]he objects of transfinite set theory . . . clearly do not belong to the physical world and even their indirect connection with physical experience is very loose. . . .
But despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perceptions will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in the future. The set-theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics.
Gödel explains in the article how Cantor’s continuum hypothesis has been shown to be independent of the axioms of set theory, and his reasons for believing that the hypothesis is actually false. (His was the part of the proof [of the undecidability of the continuum hypothesis] that showed that the continuum hypothesis can’t be proved false on the basis of present axioms of set theory, in other words that it’s consistent with the axioms of set theory, so his believing that it’s nonetheless false is particularly interesting. Paul Cohen proved that the continuum hypothesis, on the basis of the axioms of set theory, can’t be proved true either. So together they proved the undecidability of the continuum hypothesis.) He connects his Platonist belief in the objective truth or falsity of such undecidable propositions as the continuum hypothesis with his own incompleteness result:
What, however, perhaps more than anything else, justifies the acceptance of this criterion of truth in set theory is the fact that continued appeals to mathematical intuition are necessary not only for obtaining unambiguous answers to the questions of transfinite set theory, but also for the solution of the problems of finitary number theory (of the type of Goldbach’s conjecture [which, you may remember, asserts that every even number larger than two is the sum of two primes]) where the meaningfulness and unambiguity of the concepts entering into them can hardly be doubted. This follows from the fact that for every axiomatic system there are infinitely many undecidable propositions of this type.
Benacerraf recounted for me how Gödel would call either him or Putnam every day to voice his ambivalence and misgivings about having his articles included in their volume, extending his permission one day only to withdraw it the next, and then rethinking the withdrawal the day after that. He was afraid that the two “positivist” editors would use their introduction to attack his ideas. Only when the two, singly and repeatedly, had promised him that they intended only to place each of the articles in its proper context and had no intention whatsoever of evaluating any of the chosen contributions did he finally agree to have his articles included.
Was there any basis, I asked Benacerraf, for Gödel’s thinking that either he or Putnam was a positivist?
“Well, Putnam, at some stage at least, sure. After all, his dissertation advisor had been Reichenbach.”
The Benacerraf /Putnam volume was divided into four sections: Part One: The Foundations of Mathematics; Part Two: The Existence of Mathematical Objects; Part Three: Mathematical Truth; and Part Four: Wittgenstein on Mathematics. Though the volume includes the writings of Frege, Hilbert, Gödel—all the leaders in foundations—only Wittgenstein, who had never accepted Gödel’s theorems as important, is judged sufficiently significant to merit an entire section. How, one wonders, did Gödel react to his author’s copy?
Gödel had clearly expected that the full implications of his theorems would be as transparent to others as they had seemed to him, and he was not beyond the perfectly normal human reactions of disappointment and even resentment (though this was usually hidden behind the opacity of his thick reserve). He’d complained to Olga Taussky-Todd, as she reports in her memoir of him, that Hilbert still, even after Gödel’s proof, had continued to espouse formalism. “He spoke to me about this, I think in Zurich, and lashed out against Hilbert’s paper ‘Tertium non datur’ [Goettinger Nachr. 1932], saying something like ‘how can he write such a paper after what I have done?’ ”
He seemed to have felt increasingly alone and embattled in the highest turret of Reine Vernunft and took refuge in the sort of profound isolation that few spots on earth can afford with such abundant completeness—if that’s what one is after—as the Institute for Advanced Study.
The Coffee Is Wretched
Gödel first came to the Institute for the academic year 1933–34, which was the first year of its operation. The mathematician Veblen, one of Flexner’s first appointments to the nascent Institute, had met the young logician in Vienna and had been sufficiently impressed to bring him to New Jersey for a temporary visit. Of course, von Neumann, who was also spending at least some of his time at Princeton, was quite interested in the logician who had spoken of his revolutionary theorems in one crisp compressed sentence delivered in Königsberg. Gödel had not wanted to lecture his first semester in America, being still uncertain of his English, but by the second semester he gave a series of lectures on incompleteness. This is how it came to be that the talk among von Neumann and his circle was filled with references to Gödel when Alan Turing came to spend the academic year 1936–37 in Princeton, so that he returned to Cambridge determined to pursue Gödelian lines of reasoning, with such successful results.
Sometime in the course of this first stay, Gödel met Einstein. Einstein had already moved permanently to Princeton, since Nazified Germany was no longer a possibility for him. It was Veblen who introduced them, but the famous friendship didn’t begin until several years later when Gödel himself moved permanently to Princeton.
Gödel, as a classified Aryan, went back to Vienna at the end of the academic year. Menger noted that when Gödel returned he seemed even more fragile:
Gödel was more withdrawn after his return from America than before; but he still conversed with visitors to the Colloquium. . . . To all the 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.
Gödel in fact spent a few weeks in a sanatorium after returning to Vienna, where the psychiatrist Julius Wagner-Jauregg, who had won a Nobel prize in 1927
, diagnosed a nervous breakdown brought on by overwork. Of course, it might not have been just overwork. Gödel’s crisis came soon after Moritz Schlick had been shot dead on the stairs of the university. This event was profoundly unsettling for even the most stable of people. The murder signified the effective end of the Vienna Circle, although its influence would continue to spread, especially with most of its former members soon required to find refuge outside Europe. In any case, Gödel recovered sufficiently, after a few weeks in the sanatorium, and returned to teaching his course on topics in mathematical logic.
Gödel’s position at the University of Vienna was rather a lowly one. He became a Privatdozent in Vienna in 1933. A Privatdozent is granted the right to lecture, though he receives no salary. For the honor, a candidate has to write a second dissertation. Gödel’s proof of the completeness of the predicate calculus (limpid logic) had constituted his Ph.D. dissertation. His proof of the incompleteness of arithmetic was submitted for his second dissertation, the Habilitationsschrift. The commission to consider Gödel’s application met on 25 November 1932. A candidate for the Dozentur requires a sponsor, and Hans Hahn, his dissertation advisor, served as Gödel’s sponsor. Hahn testified to the committee that Gödel’s dissertation was of great scientific worth and that the Habilitationsschrift was “an achievement of the first rank” that had “attracted the greatest attention in scientific circles” and was already destined to go down in the history of mathematics (which seems, on the whole, rather faint praise).4 Hahn made the not terribly audacious judgment that Gödel’s work far exceeded the requirements for the Habilitation and the committee unanimously agreed.
However, this wasn’t the last hoop a candidate had to jump through before being declared a Privatdozent. The entire faculty had to take a vote, not only on the candidate’s scientific worth but also on his personal worthiness as well. “The results, as recorded by the dean in his report of 17 February 1933 to the ministry of instruction were, on the question of his character, fifty-one in the affirmative and one ‘no.’ On the question of his scientific merit, there were forty nine ‘yes’s and one quite astounding ‘no.’ ” (Two professors must have left between the two vote-takings.) John Dawson writes that in a private communication, “Dr. Werner Schimanovich has reported that the naysayer was Professor Wirtinger, who thought that the incompleteness paper overlapped too much with the dissertation,” an almost inconceivably wrongheaded appraisal, since the dissertation had proved the completeness, not incompleteness, of a formal system, viz. that of the predicate calculus (or limpid logic).5 But that there should have been any dissent at all on the question of whether Gödel’s incompleteness theorems were worthy of earning him a place on the lowest rung of the University of Vienna is bizarre. What was Gödel’s inner response, one wonders, to the less-than-unanimous vote? Disdain, resentment, an increased sense of insecurity? Gödel, of course, knew the significance of his theorems in all their mathematical and metamathematical splendor. But his intellectual audacity was so strangely coupled with diffidence that, again, it would be foolhardy to try to guess the affect behind the opacity.
Gödel was never to learn the most elementary of lessons with regard to maneuvering for position and status. One can well imagine that any Viennese professor trying to judge the importance of the mathematical work from the self-importance of the mathematician would have been misled. It was to be no different at the Institute. It was only in 1953—after he had received an honorary degree from Harvard, which cited the incompleteness theorems as the most important mathematical discovery of the past century, and had also been elected to the National Academy of Sciences—that he was, at long last, made a permanent member of the Institute. Once again, the impetus came from von Neumann, who is reported to have said, “How can any of us be called professor when Gödel is not?” Given his lack of worldly status, it is perhaps no wonder that his wife always considered his older brother the more successful of the two, since he was a medical doctor.
Gödel had married Adele Nimbursky (nee Porkert) in 1938. Hao Wang, who in the spring of 1976 (the year of Gödel’s death) had decided to write a précis of Gödel’s intellectual development, giving the logician the opportunity to comment on it, writes (in an endnote): “G married Adele Porkert on 20 September 1938. He asked me to delete this information from an early draft on the ground that his wife has no direct influence on his work.” The Gödel marriage was, according to just about everyone, weird. Gödel’s mother, in particular, found her son’s matrimonial choice inexplicable. His father had already died; but then he had never been close to his father. His relationship with his mother, however, was entirely different, so his unexpected choice in a bride was some cause for vexation.
The catalogue of maternal complaints against Adele were primarily these: she was a divorced woman of the wrong religion (Catholic), wrong class (lower), wrong age (six years older than Gödel), wrong appearance (she had a port-colored stain on the side of her face), and, perhaps most seriously, wrong occupation (she’d been a cabaret dancer; she would tell people it had been ballet, but it hadn’t).
Princeton’s “society” for the most part concurred with Gödel’s mother. When Adele arrived in Princeton, Oskar Morgenstern described her as a typical Viennese washerwoman and correctly predicted that she would be a dismal social failure. “Of course Gödel himself is half crazy,” he recorded in his journal. The verification of Morgenstern’s prediction caused Adele Gödel much grief, though of course her husband could not have cared less. Gödel was as indifferent to the snubs as he was indifferent to the causes for them. He had married a Hausfrau-caretaker and Adele proved as capable of caring for his frail body and soul as could be expected of a mere mortal. Adele told a neighbor friend that even back in Vienna, when she first became involved with Gödel, she used to taste his food for him, foreshadowing the dark sieges of paranoia that would increasingly seize hold of him.
Soon after his marriage in Vienna, Gödel set sail, without his bride, once more for Princeton, where von Neumann had drummed up great interest in Gödel’s theorems. He lectured that semester at the Institute on his discoveries in set theory (concerning both the axioms of choice6 and the continuum hypothesis). After spending the autumn term at the Institute, he went on to South Bend, Indiana, to spend the spring term at Notre Dame. His visit to Notre Dame was due to Menger, who had decided to emigrate and had ended up in Indiana.
While Gödel was in South Bend, Czechoslovakia was handed over to Hitler. This was the fateful year of 1939, the entire world holding its breath. To the dismay of Menger, Gödel, instead of sending for Adele and making plans to resettle in America, insisted on returning to Vienna. He was extremely incensed that his Dozentur was in danger of being revoked by the New Order, and felt that he must hurry back and see that his rights weren’t violated. Menger, devoted to Gödel, found his fondness severely tried.
He had complained about the revocation of his dozentship and had spoken about violated rights. “How can one speak of rights in the present situation?” I asked. “And what practical value can even rights at the University of Vienna have for you under such circumstances?” But despite pleas and warnings by all his acquaintances at Notre Dame and Princeton, he was determined to go to Vienna; and he went.
A man who could be terrorized by a refrigerator, convinced that it was emitting poisonous gases, returned to a Vienna overrun with Nazis to secure “his rights.”
Gödel’s world back in Vienna was now thoroughly Nazified. Menger had written to Veblen, while still in Vienna, that “whereas I . . . don’t believe that Austria has more than 45% Nazis, the percentage at the universities is certainly 75% and among the mathematicians I have to do with . . . [apart from] some pupils of mine, not far from 100%.”
Gödel was decidedly not anti-Semitic. He never took the slightest offense when others assumed he was Jewish; he simply corrected the error “for the sake of the truth,” as he had written in the unmailed correction to Bertrand Russell’s autobiography. The g
roup of thinkers with whom he associated in Vienna were for the most part Jewish. Gödel neatly instantiates the tongue-in-cheek advice that the satirist Leon Hirshfeld gave to travelers: “Be careful during your stay in Vienna not to be too interesting or original, otherwise you might behind your back suddenly be called Jewish.”
Though he didn’t in the least partake of the crudely simplifying racial theories that were having such a popular run in Vienna and elsewhere, Gödel was dismayingly indifferent to the plight of the victims of those racial theories. The German (Jewish) positivist philosopher, Gustav Bergmann, recalled to John Dawson that shortly after arriving in America in October 1938, he was invited to lunch with Gödel, who asked, with clueless charm, “And what brings you to America, Herr Bergmann?”
For Menger, there was a limit to how much one could forgive clueless genius:
During the summer I heard nothing from Gödel. But on August 30, 1939, one of the few days between the Hitler-Stalin pact and the entry of German troops into Poland which unleashed the second world war, he wrote me a letter that may well represent a record for unconcern on the threshold of world-shaking events: “Since the end of June I have been here in Vienna again and had a great many tasks to perform so it was unfortunately not possible to write up anything for the Colloquium. How did the examinations turn out for my logic lectures? . . . In the fall I hope to be back in Princeton.”
Menger’s affection for Gödel considerably cooled, not to return until decades later, at the end of Gödel’s life, when he came to understand more completely the deep and abiding strangeness of the logician.
So there was Gödel in Vienna in 1939. But it was a Vienna sadly changed, as even Gödel must have apprehended, however dimly. The old Circle was no more, of course. Schlick had been murdered by a psychotic student who was then transformed into a hero in the Nazi press; Feigl, Carnap, and Menger had fled from the increasingly poisonous atmosphere. Hans Hahn had died of cancer on 24 February 1934 at the age of 55—one day before a group of Nazis stormed the chancellery in Vienna and assassinated Dollfuss in a failed putsch.
Incompleteness Page 19
