Chapter 3: The Proof of Incompleteness
p. 149 “the application of the Verification Principle to mathematics”: Monk, op. cit., p. 295.
p. 154 footnote 2: “The completeness theorem, mathematically, is indeed an almost trivial consequence”: Hao Wang, From Mathematics to Philosophy (New York: Humanities Press, 1974), pp. 8–9.
p. 158 No mention of Gödel . . . by Hans Reichenbach: Reichenbach’s account was published in Die Naturwissenschaften, Vol. 18 (1930), 1093–4.
p. 160 Gödel didn’t fully prove his second incompleteness theorem until after the conference: John Dawson, “The Reception of Gödel’s Incompleteness Theorems,” in Gödel’s Theorem in Focus, ed. S. G. Shanker (London: Croom Helm, 1988), p. 91, footnote 2.
p. 160 How, given Gödel’s Entdeckungen, could he not have questioned his former thinking?: See Dawson (1988) for a thorough discussion of the reaction, or initial lack thereof, to Gödel’s first incompleteness theorem.
p. 161 “In science . . . novelty emerges”: Kuhn, op cit., p. 64.
p. 179 What we use next is something called the diagonal lemma: See Hintikka, op. cit., p. 33.
p. 185 “Operating with the infinite can be made certain only by the finite”: Hilbert 1964, op. cit., p. 151.
p. 188 The last article that Gödel was to publish in his life: “Über eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes,” Dialectica 12 (1958), pp. 280–7.
p. 188 “doubtful . . . about the completeness of the formal systems”: In a letter to Constance Reid Bernays, 3 August 1966; quoted in Dawson 1997, p. 72.
p. 189 “Mathematics cannot be incomplete”: Wittgenstein 1967, op. cit., p. 158.
p. 190 “No calculus can decide a philosophical problem”: Philosophical Remarks, p. 296.
p. 190 “My task is not to talk about Gödel’s proof”: Remarks on the Foundations of Mathematics V, p. 16.
p. 191 “They are what is mystical”: Tractatus, 6.522.
p. 195 footnote 9: how Gödel “entered my intellectual life”: Stephen C. Kleene, “Gödel’s Impression on Students of Logic in the 1930’s,” in Gödel Remembered, ed. Paul Weingartner and Leopold Schmetterer (Naples: Bibliopolis, 1987) p. 52.
p. 196 footnote 10: “It is very queer”: “Wittgenstein’s Lectures on the Foundations of Mathematics: Cambridge 1939,” from the Notes of R. G. Bosanquet, Norma Malcolm, Rush Rhees, Yorick Smythies, ed. Cora Diamond (Ithaca, NY: Cornell University Press, 1976), Lecture XXI, pp. 206–7.
p. 200 “Gödel’s theorem seems to me to prove that Mechanism is false”: J. R. Lucas, “Minds, Machines, and Gödel,” Philosophy, XXXVI (1961), p. 112.
p. 201 “What did Gödel’s theorem achieve?”: Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness (Oxford: Oxford University Press, 1994), pp. 64–5.
p. 203 “Either the human mind surpasses all machines”: Wang 1974, op. cit., p. 324.
p. 205 “Delusions may be systematized”: Shervert H. Frazier and Arthur C. Carr, Introduction to Psychopathology (Jason Aronson, 1983), p. 106.
p. 205 “A paranoid person is irrationally rational”: James W. Anderson, Associate Professor of Clinical Psychology, Northwestern University, personal communication, 7 October 2003.
Chapter 4: Gödel’s Incompleteness
p. 210 He mentioned to . . . Morton White: Private conversation with Morton White, May 2002.
p. 213 footnote 3: according to Hao Wang . . . in 1970: Wang 1987, op. cit., p. 9.
p. 213 “Gödel would probably have published more”: Wang 1987, op. cit., p. 29.
p. 216 revised and expanded for this volume: The original appeared in the American Mathematical Monthly 54 (1947), pp. 515–25. The original had been published before Paul Cohen had proved that the continuum hypothesis could not be deduced from the axioms of set theory.
p. 220 “Gödel was more withdrawn after his return from America”: Menger, op. cit., p. 205.
p. 223 “How can any of us be called professor when Gödel is not?”: Stanislaw M. Ulam, Adventures of a Mathematician (New York: Charles Scribner’s Sons, 1976), p. 80. As Dawson remarks (Dawson 1997, p. 302, note 462), it “is worth noting that Gödel himself seems never to have complained about his status, either publicly or in private remarks or correspondence.”
p. 223 “G married Adele Porkert on 20 September 1938”: Wang 1988, op. cit., p. 47, footnote 7.
p. 226 “He had complained about the revocation of his dozentship”: Menger, op.cit., p. 123.
p. 227 “And what brings you to America, Herr Bergmann?”: Dawson 1997, op. cit., p. 90.
p. 227 “During the summer I heard nothing from Gödel”: Menger, op. cit., p. 124.
p. 230 “His case could hardly create a precedent”: Dawson 1997, op. cit., p. 148.
p. 241 Nonstandard analysis, he said, was not “a fad”: Dawson 1997, op. cit., p. 244.
p. 244 “He [Gödel] also pointed out that many scientists of great intelligence”: Morton Gabriel White, A Philosopher’s Story (University Park, PA: Pennsylvania State University Press, 1999), p. 303.
p. 245 “On every one of my admittedly infrequent trips to Princeton”: Menger, op. cit., p. 226.
p. 247 “Who could have an interest in destroying Leibniz’s writings?”: Menger, op. cit., p. 19.
p. 248 He also reported his suspicions that there were those who were trying to kill him: Dawson 1997, op. cit., pp. 249–50.
p. 248 “Today . . . Kurt Gödel called me”: Morgenstern papers, Perkins Memorial Library, Duke University, folder “Gödel, Kurt, 1974–1977.”
p. 250 “eyed him suspiciously”: Wang 1987, op. cit., p. 133.
p. 250 “I’ve lost the faculty for making positive decisions”: ibid.
p. 250 “It was said that G[ödel]’s weight was down to sixty-five pounds”: ibid.
p. 254 footnote 15: “The Time We Thought We Knew,” Brian Greene, Op/Ed, The New York Times, 1 January 2004.
p. 256 “temporal conditions in these universes show . . . surprising features”: Kurt Gödel, “A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy,” in Albert Einstein: Philosopher Scientist, ed. Paul Arthur Schilpp (New York: MJF Books, 1949), p. 560.
p. 257 “be excluded on physical grounds”: Einstein, op. cit., p. 687–8.
p. 258 “It turned out that he himself, as a preliminary step to get some evidence, had taken down the great Hubble atlas of the galaxies”: Jeremy Bernstein, Quantum Profiles (Princeton, NJ: Princeton University Press, 1991), pp. 140–1.
p. 260 “In philosophy Gödel has never arrived at what he looked for”: Wang 1987, op. cit., p. 46.
p. 260 “He also looked for (but failed to obtain) an epiphany”: Wang 1987, op. cit., p. 196.
Suggested Reading
Like many before and after me, my first substantive exposure to Gödel’s incompleteness theorems came not by way of studying the famous 1931 paper itself but rather by reading, as an undergraduate, the celebrated Gödel’s Proof by Ernest Nagel and James R. Newman (New York: New York University Press, 1968). This is a popular exposition that yet manages to go into some detail concerning the substance of the proof. My world was rocked. On rereading it after all these years, I was impressed all over again. It’s a wonderful little book, in its own way a classic.
Jaakko Hintikka’s very slim (70 pages) book, On Gödel (Belmont, CA: Wadsworth Thomson Learning, 2000), is also a clear and concise presentation of Gödel’s proof for the non-expert. Like the more expansive Gödel’s Proof, Hintikka’s proof is self-contained, requiring no previous knowledge of logic. He also has a good sense of humor.
So far as the life of the logician is concerned, Logical Dilemmas: The Life and Work of Kurt Gödel (Wellesley, MA: A K Peters, 1997) by John Dawson is definitive. As not only a logician but also Gödel’s archivist, whose wife learned to translate Gödel’s shorthand, Dawson was in an unrivaled position for presenting the life of Gödel. I was told by Institute mathematician Arm
and Borel that Gödel’s literary remains, which had been donated to the Institute for Advanced Study by Gödel’s widow, were in utter chaos, piled helter-skelter into decaying boxes; and then “a young man” (Dawson) had offered to put it all into order. “He did a good job, I’m told.” Indeed he did.
John Dawson also has two papers on Gödel that are accessible and interesting: “Kurt Gödel in Sharper Focus” and “The Reception of Gödel’s Incompleteness Theorems.” Both are reprinted in Gödel’s Theorem in Focus, edited by Stuart Shanker, as are other interesting essays, including Solomon Feferman’s “Kurt Gödel: Conviction and Caution.”
Hao Wang produced three rather eccentric but intriguing books out of the pickings of Gödel’s mind: From Mathematics to Philosophy (New York: Humanities Press, 1974), Reflections on Kurt Gödel (Cambridge, MA: MIT Press, 1987), and A Logical Journey (Cambridge, MA: MIT Press, 1996). The books recount conversations Wang had with Gödel, interlaced with history of the logician’s life and Wang’s own views on the topics he and Gödel discussed. What they lack in structure they compensate for in content.
There are several memoirs of Gödel, written by those who had first known him in Vienna, and they are fascinating and in their own way touching. There is first of all Georg Kreisl’s “Kurt Gödel: 1906-1978,” Biographical Memoirs of Fellows of the Royal Society, Vol. 26 (1980), pp. 148–224. Kreisl, an eminent mathematical logician, is in a unique position, having known Wittgenstein quite well when Kreisl was a student, and then, later, having gotten to know Gödel in Princeton. Karl Menger had been invited, together with Gödel, to join the Vienna Circle as favored students of Hans Hahn and his invaluable first-hand reminiscences of Gödel are recounted in “Memories of Kurt Gödel,” in Reminiscences of the Vienna Circle and the Mathematical Colloquium, ed. Louise Golland, Brian McGuinness, and Abe Sklar (Dordrecht: Kluwer, 1994). And then there is Olga Taussky-Todd, herself a number-theorist, who also had first come to know Gödel in their student days. Her “Remembrances of Kurt Gödel” is in Gödel Remembered (Naples: Bibliopolis, 1987).
If the reader is interested in seeing how a contemporary polymath applies Gödel’s theorems in his own creative scientific thinking, then he is advised to read Roger Penrose’s The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics (New York: Penguin, 1989) and his Shadows of the Mind: A Search for the Missing Science of Consciousness (Oxford: Oxford University Press, 1994). Like Gödel, Penrose is a confirmed mathematical Platonist; he interprets the incompleteness theorems exactly as Gödel did. There’s lots of further fascinating mathematics that he discusses—including Turing’s contributions to the work Gödel began, the Mandelbrot set, and Penrose’s own work on the tiling of the plane—all argued, by him, as pointing in the direction of Platonism. Penrose’s overall argument is that mathematical knowledge, the amazing fact that we have it, is evidence that the laws of physics are of a fundamentally different character than we have heretofore dreamt.
Douglas Hofstadter’s Pulitzer-prize-winning Gödel, Escher, Bach: The Eternal Golden Braid (New York: Basic Books, 1974) is a spirited romp through self-referentiality. Hofstadter does a wonderful job of braiding together ideas in logic, art, and music, just as the title promises. When, upon being asked what I’d been working on these past few years, I’d say “Gödel,” more often than not I got a blank stare in return. Then I’d mention the title of Hofstadter’s bestseller, and the blank stare would give way to a smile and an “oh yes.”
Raymond M. Smullyan is a mathematical logician who has written various exuberantly playful books that explore themes in logic, most especially self-referential paradoxes. His Gödel’s Incomplete-ness Theorems (Oxford Logic Guides, no. 19, 1995) is written with his characteristic lucidity and verve, approaching the proofs from various angles and making use of the reformulations of Gödel’s work that subsequent logicians, including Smullyan himself, have contributed. The book even comes with exercises that help the reader to assimilate the various subtleties of the proof, such as the strangeness of having arithmetical propositions that can talk to the condition of their own unprovability. No more logical background is required than a semester’s course in symbolic logic or, failing that, a good book in formal logic, such as Smullyan’s own First-Order Logic (Dover, 1992).
Finally, there is the writing of Gödel himself, his few published papers and his many unpublished works, in Collected Works, ed. Solomon Feferman et al. (Oxford: Oxford University Press, 1986–). There are five volumes to date.
Acknowledgments
If there is any aspect of being the perfect literary agent that Tina Bennett lacks, I have yet to discover it. The current project only served to reveal new aspects of Tina’s ways of unstintingly supporting her writers.
I am extremely grateful to the following people who shared their recollections of Kurt Gödel with me: John Bahcall, Paul Benacerraf, Armand Borel, Thomas Nagel, Morton White. Each one was stintless with his time. Simon Kochen not only spoke long hours with me but also generously read over my completed manuscript, catching some technical errors, for which I am profoundly grateful, and answering further queries by e-mail. Berel Lang also read the manuscript and his comments, too, were insightful, substantive, and helpful.
I thank John Dawson, not only for the Herculean work he did as Gödel’s archivist, which made the job of all scholars who follow possible, but also his prompt response to any question that arose.
As always, Sheldon Goldstein, physicist-philosopher, had insights that were invaluable. He, more than anyone, helped me to ease my way back into mathematical logic. There is not a man on Earth, I’d wager, quite equal to him for reminding one of the beauty and elegance of abstract thought. Steven Pinker generously read some early inchoate chapters, when I was feeling my way toward “popular technical writing,” and his comments and encouragement were sustaining. And when I threw up my hands, Yael Goldstein calmly placed them back on the keyboard, offering the sort of sage advice and substantive criticism and guidance without which this book, quite literally, would not have been written.
Accordingly, I dedicate the book to her, with gratitude, love, and stupefied admiration.
Index
Adler, Max, 73
algebra, 132n, 241
algorithms, 132, 171, 197n, 199–200
Alleluyeva, Svetlana, 241
Alt, Rudolf von, 70n
Altenberg, Peter, 72
analytic propositions, 44n
Anselm, Saint, 209
Aristotle, 31, 48, 55, 150, 205, 210, 259
arithmetic:
completeness vs. consistency of, 66–67, 134–45, 153–56, 158, 160, 162–63, 167–68, 169, 175–76, 181–88, 212, 221
Gödel’s consistency proof for (1958), 212
see also numbers
artificial intelligence, 201–2
Artin, Emil, 246
Atlantic Monthly, 241–42
atomic bomb, 22, 237
“Autobiographical Notes” (Einstein), 42–43
Autobiography (Russell), 116
“axiom of choice,” 224
Aydelotte, Frank, 229–30, 237
Ayer, Alfred Jules, 89, 108, 214
Bahcall, John, 31
Bamberger, Louis, 14–15, 16, 237
Barbara or Piety (Werfel), 72
Barrett, William, 38–39
Beethoven, Ludwig van, 96, 113
Bellah, Robert, 243–45
Benacerraf, Paul, 20n, 111–12, 216–18, 231
Berg, Alban, 70, 72
Bergmann, Gustav, 227
Bernays, Paul, 188, 212
Besso, Michele, 255
Bick, Auguste, 204n
Blackwell, Kenneth, 116
Bohr, Niels, 37–38, 39, 42, 43, 51, 214
Boltzmann, Ludwig, 80
Bolyai, Farkas, 130n
Bolyai, János, 130n
Borel, Armand, 31, 33–34, 236, 239–40
Born, Max, 43
Borsuk, Karol, 238–39
“Bourbaki, Nicolas,” 243n
Brouwer, Luitzen, 143
Bundy, McGeorge, 242
calculus, predicate, 35n, 150, 154n, 221, 222
Cantor, Georg, 93n, 139
Cantor’s continuum hypothesis, 111–12, 139, 216–18, 224–25
Cantor’s paradox, 93n, 142
Carnap, Rudolf, 80, 81, 82, 104, 105–6, 148, 158, 160–61, 213n, 228, 255, 265n
Chaitin, G. J., 166n
Chomsky, Noam, 32
Cocteau, Jean, 76n
cognition, 25–27, 31, 32, 36–40, 199–205, 211
Cohen, Paul, 139, 217, 225n
combinatorics, 118, 163–64, 202
computability, 196n–97n
computers, 19–20, 25, 133, 199–203
Comte, Auguste, 85
Concerto for the Left Hand (Ravel), 90n
Constitution, U.S., 232–34
constructive proofs, 143n, 148
Contributions to the Analysis of Sensation (Mach), 85
Copenhagen (Frayn), 37–38
Darwin, Charles, 199
Dawson, John, 54, 58, 59–60, 222, 227, 232n
decidability, 196n–97n
Descartes, René, 27n, 137, 260
descriptive propositions, 44
“diagonal argument,” 139
Dukas, Helen, 20n, 235
Edmonds, David, 74–75
Eidinow, John, 74–75
Einstein, Albert:
autobiographical notes by, 42–43
death of, 61, 234–36, 245–46, 255–56
Gödel compared with, 34–37, 40, 44, 47–48, 49, 113, 194, 248, 252
Gödel’s relationship with, 13–14, 20–23, 29–44, 48–49, 51, 213, 219, 232–36, 239, 245–46, 253–58
at Institute for Advanced Study, 13–14, 19–20, 48, 234–36, 237
intellectual isolation of, 35–38, 40–44, 48, 49
Jewish background of, 19
legends about, 30, 40–41
mathematics as interest of, 29–30, 35, 64
photographs of, 22, 253
as physicist, 29–30, 34; see also relativity theory
as political exile, 13–14, 34–35, 124–26
Incompleteness Page 24
