Incompleteness

by Rebecca Goldstein

  7 Flexner’s resignation was the result of his having made two appointments, both of economists, without first consulting the touchy faculty. The mathematicians were particularly sensitive on this matter, foreshadowing events that were to cast a pall over Gödel’s last years at the Institute.

  8 Unfortunately, Morgenstern’s account, and so all the others that derive from it, omits mention of the precise constitutional flaw. I asked John Dawson whether he knew what it was supposed to be, and he e-mailed back: “No, I don’t, though many have asked that question. There is a set of shorthand notes in Gödel’s Nachlass concerned with American government (presumably made while he was studying for the citizenship examination) that might contain the answer, but transcribing that particular item has never had as much priority as the mathematical material” (3 January 2004).

  9 A few months after this talk, Robinson succumbed to pancreatic cancer, a death which reportedly struck Gödel hard.

  10 Oppenheimer had managed to merge the floundering School of Economics and Politics with the flourishing School of Humanities to create the new School of History. But by the time he stepped down—with less than six months to live—he found that half his faculty weren’t speaking with him. All the mathematicians—with the exception of Gödel—were in the enemy camp. In speaking to mathematical survivors of that period even today, Oppenheimer’s character is described in uncharitable terms. Memories seem never to fade at the Institute.

  11 His scholarly work primarily concerned American anti-trust policy; he’d done a study of United States v. United Shoe Machinery Corporation. Said the mathematician André Weil, “I think he wrote his thesis about a shoe factory.”

  12 Weil had come to the Institute from France and had been one of the original participants in the pseudonymous existence of “Nicolas Bourbaki.” It was under this name (Bourbaki was identified as “formerly of the Royal Poldavian Academy”) that a group of young mathematicians published at least two dozen mathematical treatises of the highest order, bringing the level of proof up to a new standard of rigor.

  13 Menger is referring here to Gödel’s outrage that his rights as a university Dozent had been tampered with by the Third Reich.

  14 Gödel wrote his mother on 4 and 19 July 1962, of his “recent discovery” of the “modern poet,” Franz Kafka. Gödel also enjoyed abstract and surrealist painting. In other respects, his cultural tastes are notable for their childlike quality. Not only did he prefer fairy tales to Goethe and Shakespeare, writing to his mother that only in such tales is the world represented as it ought to be, but he was also a great fan of Disney movies and saw Snow White at least three times.

  15 An Op-ed piece in The New York Times, which ran, appropriately enough, on New Year’s Day, beautifully delineated the chasm between subjective and objective time: “A hundred years ago today, the discovery of special relativity was still 18 months away, and science still embraced the Newtonian description of time. Now, however, modern physics’ notion of time is clearly at odds with the one most of us have internalized. Einstein greeted the failure of science to confirm the familiar experience of time with ‘painful but inevitable resignation.’ The developments since his era have only widened the disparity between common experience and scientific knowledge. Most physicists cope with this disparity by compartmentalizing: there’s time as understood scientifically, and then there’s time as experienced intuitively. For decades, I’ve struggled to bring my experience closer to my understanding.”

  16 Schilpp had first approached Gödel in 1946 to contribute an essay for this Festschrift in honor of Einstein. Gödel immediately agreed, but there were to be many postponements of delivery of the final paper. Schilpp had hoped to have his volume ready for Einstein’s seventieth birthday (14 March 1949). Gödel did not finish his article until a month before and even then held onto it. Schilpp got Gödel to agree to present Einstein with the paper at the gala birthday celebration thrown for him at the Princeton Inn on 19 March. Soon after, Schilpp received a copy of the paper.

  Notes

  Introduction

  p. 13 “Hitler shakes the tree”: Quote unattributed, in “Bad Days on Mount Olympus: The Big Shoot-Out in Princeton,” Atlantic Monthly, February 1974.

  p. 19 “after its driver”: Helen Dukas, Einstein’s secretary, in a letter to C. Seelig as quoted in Abraham Pais, “Subtle is the Lord . . .”: The Science and the Life of Albert Einstein (Oxford: Oxford University Press, 1982), p. 473.

  p. 20 All of his thinking: Harry Woolf, ed., Some Strangeness of Proportion: A Centennial Symposium to Celebrate the Achievements of Albert Einstein (Reading, MA.: Addison-Wesley, 1980), p. 485.

  p. 23 The Encyclopedia of Philosophy’s article “Gödel’s Theorem”: J. van Heijenoort, The Encyclopedia of Philosophy, Vol 3, ed., Paul Edwards (New York: Macmillan Publishing Co., Inc., and the Free Press, 1967), pp. 348–9.

  p. 24 “He is the devil”: David Foster Wallace, “Approaching Infinity,” Boston Globe, 12 December 2003.

  p. 29 “Science without epistemology”: Paul A. Schilpp, Albert Einstein, Philosopher-Scientist (New York: Tudor, 1949), p. 684.

  p. 29 “No story of Einstein”: Woolf, op. cit.

  p. 31 “the logic of Aristotle’s successor”: Armand Borel, “The School of Mathematics at the Institute for Advanced Study,” in A Century of Mathematics in America, ed. Peter Duren (Providence, RI: American Mathematical Society, 1989), p. 130.

  p. 31 “I don’t believe in natural science”: Quoted in Ed Regis, Who Got Einstein’s Office? Eccentricity and Genius at the Institute for Advanced Study (Cambridge, MA: Perseus, 1987), p. 58. I also corroborated the story with John Bahcall.

  p. 32 “After that . . . I just gave up”: Private conversation, May 2002.

  p. 32 “the laws of nature are a priori”: Reported to me by Paul Benacerraf, who was told of the conversation by Chomsky.

  p. 33 “um das Privileg zu haben”: In a letter to Bruno Kreisky (Bundesmeister für Auswartige Angelegenheiten of Austria) dated 25 October 1965.

  p. 34 “two-membered ‘natural kind’ ”: Hao Wang, Reflections on Kurt Gödel (Cambridge, MA: MIT Press, 1987), p. 2.

  p. 37 “It works, yes”: Michael Frayn, Copenhagen (New York: Anchor Books, 1998), pp. 71–2.

  p. 39 “Gödel’s findings seem to have even more far-reaching consequences”: William Barrett, Irrational Man: A Study in Existentialist Philosphy (New York: Anchor Books, 1962, 1990), p. 39.

  p. 42 “the religious paradise of youth”: Schilpp, op. cit., p. 5.

  Chapter 1: A Platonist among the Platonists

  p. 54 a brief “History of the Gödel Family”: Reprinted in Gödel Remembered: Salzburg 10–12 July 1983, ed. P. Weingartner and L. Schmetterer (Napoli: Bibiopolis, 1987).

  p. 58 John Dawson, Logical Dilemmas: The Life and Work of Kurt Gödel (Wellesley, MA: A K Peters LTD, 1997), p. 14.

  p. 58 “an exile in Czechoslovakia”: Dawson 1997, op. cit., p. 15.

  p. 58 “his interest in precision”: Wang 1987, op. cit., p. 41.

  p. 60 until John Dawson undertook the formidable task: See also Dawson’s (1997) excellent biography of Gödel that has already been cited.

  p. 60 manuscripts for articles that he had promised to deliver: For example, the article for Rudolf Carnap’s Festschrift, which he had promised to P. A. Schilpp in 1953. Gödel produced no less than six drafts of the “brief note” which grew into a lengthy manuscript. He kept revising until 1959, when he finaly wrote to Schilpp, calling it quits. Two of the six versons were posthumously published under the title “Is Mathematics the Syntax of Language?” in volume III of Gödel’s Collected Works.

  p. 60 a hysteria of discretion: Solomon Feferman expressed this double aspect of Gödel’s personality this way: “What I find striking here is the contrast on the one hand between the depth of Gödel’s convictions which underlay his work, combined with his sureness of insight leading him to the core of each problem, and on the other hand the tight rein he placed on the
expression of his true thoughts.” See Solomon Feferman, “Kurt Gödel: Conviction and Caution,” in Gödel’s Theorem in Focus, ed. S. G. Shanker (London: Croom Helm, 1988), p. 111.

  p. 63 I remember my own: See Rebecca Goldstein, “Writers on Writing: On the Wings of Enchantment,” The New York Times, 19 December 2002.

  p. 63 “Here is the life, Socrates”: Symposium, 211d-211e. Translated by William S. Cobb (State University of New York, 1993).

  p. 64 Gödel had been drawn toward number theory: Wang, 1987, op. cit., p. 22.

  p. 66 “It is a measure of Gödel’s status”: Jaakko Hintikka, On Gödel (Belmont, CA: Wadsworth Thomson Learning, 2000), p. 1.

  p. 72 “Theory of Café Central”: Quoted in Alan S. Janik and Hans Veigl, Wittgenstein in Vienna: A Biographical Excursion Through the City and its History (New York: Springer, 1998), p. 88.

  p. 74 The philosopher Karl Popper . . . waited with impatience: See David Edmonds and John Edinow, Wittgenstein’s Poker: The Story of a Ten-Minute Argument Between Two Great Philosophers (New York: HarperCollins, 2001), for a lively discussion of Popper’s exclusion from the Vienna Circle and the antagonism to Wittgenstein that lay behind it.

  p. 74 “The Circle’s voice,” Edmonds and Edinow, op. cit., p. 163.

  p. 75 He was dismayed: He himself acknowledged in a letter to the mathematician Menger that he had contributed (presumably because of his disinclination to address it) to the misapprehension: “In conseq[uence] of frequent tiredness I hardly ever answer letters before a week’s time. But in this case there was moreover a sp[ecial] reason, namely that I always have inhibitions in writing about my relationship to the Vienna Circle because I never was a logical pos[itivist] in which the term is commonly understood and explained in the [manifesto of] 1929. On the other hand by some publ[ications] (probably in part through my own fault) the impr[ession] is created that I was.” Quoted in Wang 1987, op. cit., p. 49.

  p. 80 “The pleasant atmosphere”: Rudolf Carnap, “Intellectual Autobiography,” in The Philosophy of Rudolf Carnap, ed. Paul A. Schilpp (La Salle, IL: Open Court, 1963).

  p. 80 footnote 7: Boltzmann had succeeded: For a fascinating discussion of Boltzmann’s life and works see David Lindley, Boltzmann’s Atom: The Great Debate That Launched A Revolution in Physics (New York: Free Press, 2001).

  p. 81 “especially interested in the formal-logical”: Herbert Feigl, “The Wiener Kreis in America,” in The Intellectual Migration: Europe and America, 1930–1960, ed. Donald Fleming and Bernard Bailyn (Cambridge, MA: Harvard Univeristy Press, 1969), p. 635.

  p. 81 “that almost seemed to exude sincerity”: Karl Menger, Reminiscences of the Vienna Circle and the Mathematical Colloquia, ed. Louise Golland, Brian McGuinness, and Abe Sklar (Dordrecht: Kluwer, 1994), p. 63.

  p. 81 When anything came up in conversation: Menger, op. cit., p. 64.

  p. 82 “Schlick especially seemed to resent this”: Menger, op. cit., p. 65.

  p. 84 “Uncritical, run-down aristocrats”: Menger, op. cit., pp. 61–2.

  p. 85 “our philosophical movement with its international trade name”: Feigl, op. cit., p. 630.

  p. 85 “The truths of pure mathematics”: Feigl, op. cit., p. 652.

  p. 88 “a rather dingy room,” Menger, op. cit., p. 55.

  p. 90 “the Austrian equivalent”: Ray Monk, Ludwig Wittgenstein: The Duty of Genius (New York: Penguin, 1990), p. 72.

  p. 94 “We were both cross from the heat”: Bertrand Russell to Ottoline Morrel, 27 May 1913.

  p. 95 footnote 13: “were taken from authors”: Allan Janik and Stephan Toulmin, Wittgenstein’s Vienna (New York: Simon and Schuster, 1973), p. 27.

  p. 95 Partly it was the Viennese aspect in his thinking: This is a dominant theme in Janik and Toulmin, op. cit. “Those of us who attended his lectures [at Cambridge University] . . . still found ourselves looking upon his ideas, his methods of argument and his very topics of discussion as something totally original and his own. . . . If there was an intellectual gulf between him and us, it was not because his philosophical methods, style of exposition and subject matter were (as we supposed) unique and unparalleled. It was a sign, rather, of a culture clash: the clash between a Viennese thinker whose intellectual problems and personal attitudes alike had been formed in the neo-Kantian environment of pre-1914, in which logic and ethics were essentially bound up with each other and with the critique of language (Sprachkritik), and an audience of students whose philosophical questions had been shaped by the neo-Humean . . . empiricism of Moore, Russell and their colleagues” (pp. 21–2).

  p. 95 “a quintessentially Viennese figure”: Monk, op. cit., p. 20.

  p. 96 pronounced a genius by Russell: “Wittgenstein’s recurring thoughts of suicide between 1903 and 1912, and the fact that these thoughts abated only after Russell’s recognition of his genius, suggest that he accepted this imperative (Weininger’s) in all its terrifying severity.” Monk, op. cit., p. 25.

  p. 99 “stranger than others by orders of magnitude”: Jaakko Hintikka, op. cit, p. 3. Hintikka is here speaking even more widely than of logic; he is speaking of all of mathematics.

  p. 101 “We shall see contradiction”: Remarks on the Foundations of Mathematics (Cambridge, MA: MIT Press, 1967), p. 110.

  p. 101 the later Wittgenstein came to regard the entire field as a “curse”: For example, he writes: “The curse of the invasion of mathematics by mathematical logic is that now any proposition can be represented in a mathematical symbolism, and this makes us feel obliged to understand it. Although of course this method of writing is nothing but the translation of vague ordinary prose.” Wittgenstein, op. cit., 155.

  p. 103 footnote 16: “an Indian poet much in vogue”: Monk, op. cit., p. 243.

  p. 104 footnote 17: “Although Gödel had not persuaded Carnap on this fundamental issue”: Eckehart Köhler, “Gödel and the Vienna Circle: Platonism versus Formalism,” in History of Logic, Methodology, and Philosophy of Science, section 13 (Vienna Institute for Advanced Studies). Later cited in S. G. Shanker, ed., Gödel’s Theorem in Focus (London: Croom Helm, 1988).

  p. 105 “I shall relate to you”: Quoted in Janik and Veigl, op. cit. p. 63.

  p. 105 “a mythological character”: Monk, op. cit., p. 284.

  p. 105 “Schlick adored him”: Feigl, op. cit., p. 638.

  p. 105 “Feigl had always had an unusual ability to get along with everyone”: Menger, op. cit., p. 66.

  p. 105 “limitless admiration for Carnap”: Ibid.

  p. 106 “I once wanted to give a few words in the foreword”: Letter to Ludwig von Ficker, quoted in Monk, op. cit., p. 178. The letter is undated, but Monk says it was almost certainly written 19 November 1919.

  p. 110 “a slim, unusually quiet young man”: Menger, op. cit., p. 201.

  p. 110 “a very unassuming, diligent worker”: Feigl, op. cit., p. 640.

  p. 111 “Some reductionism is correct”: Hao Wang, A Logical Journey: From Gödel to Philosophy (Cambridge, MA: MIT Press, 1996).

  p. 113 “[A] friend described going to Beethoven’s door”: Bertrand Russell to Ottoline Morrell, 23 April 1912.

  p. 114 “I had my tonsils out”: Fania Pascal, “Wittgenstein: A Personal Memoir,” in Recollections of Wittgenstein, ed. R. Rhees (Oxford: Oxford Press, 1984), pp. 28–9.

  p. 117 “In the early 1970’s”: Menger, op. cit., p. 230.

  p. 117 “ ‘logische Kunststücke’ ”: The relevant passage is in Remarks on the Foundations of Mathematics, Appendix I, p. 19. “You say: . . . , so P is true and unprovable. That presumably means: Therefore P. That is all right with me—but for what purpose do you write down this ‘assertion’? (It is as if someone had extracted from certain principles about natural forms and architectural style the idea that on Mount Everest, where no one can live, there belonged a chalet in the Baroque style. And how could you make the truth of the assertion plausible to me, since you can make no use of it except to do these little conjuring tricks?”

  p. 118 “As far a
s my theorems about undecidable propositions are concerned”: Menger, op. cit., p. 231.

  p. 118 “Wittgenstein’s views on mathematical logic”: Georg Kreisel, “Wittgenstein’s ‘Remarks on the Foundations of Mathematics,’” British Journal for the Philosophy of Science, IV (1958), pp. 143–4.

  Chapter 2: Hilbert and the Formalists

  p. 124 “So the geometrical figures are signs or mnemonic symbols”: David Hilbert, “Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematischer-Kongress zu Paris 1900.” Nachrichten von der Könglichen Gesellschaft der Wissenschaften zu Göttingen, 253–97. English translation in Felix Browder, ed., “Mathematical Developments Arising from the Hilbert Problems,” Proceedings of Symposia in Pure Mathematics XXVIII, parts 1 and 2. (Providence, RI: American Mathematical Society, 1976).

  p. 128 “In mathematics we must always strive after a system”: Gottlob Frege, Begriffsschrift, eine der arithmetischen nachgebildete Normal-sprache des reinen Denkens (Halle: Nebert, 1879). English translation in Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Cambridge, MA: Harvard Univeristy Press, 1967), p. 279.

  p. 128 “no science can be so enveloped in obscurity as mathematics”: Jean Heijenoort, ed., op. cit., p. 242.

  p. 136 “meaningless marks on paper”: Quoted in John de Pillis and Nick Rose, Mathematical Maxims and Minims (Raleigh, NC, 1988).

  p. 142 “Admittedly, the present state of affairs”: David Hilbert, “On the Infinite,” in Philosophy of Mathematics, ed. Paul Benacerraf and Hilary Putnam (Englewood Cliffs, NJ: Prentice-Hall, 1964), p. 141. A translation of a talk delivered 4 June 1925 before a congress of the Westphalian Mathematical Society in Münster, in honor of Karl Weierstrass. Translated by Erna Putnam and Gerald J. Massey from Mathematische Annalen (Berlin) no. 95 (1925), pp. 161–90.