Incompleteness
Page 26
Platonism, 44–51, 59, 61–64, 73, 75, 87–88, 104n, 110–13, 116–17, 135, 143–44, 148, 154, 163, 185, 192, 194, 213–19, 244, 255, 256, 260
Polgar, Alfred, 72–73
Popper, Karl, 74, 265n
postmodernism, 24–26, 37, 49, 69, 101, 135–36, 189
Princeton University, 13, 20n, 30–32, 36, 51, 59, 68, 124–26, 224, 226, 227, 229–30, 231–32, 238–39, 246, 258–59
Principia Mathematica (Russell and Whitehead), 83, 92–93, 96, 101, 144, 160
Prior Analytics (Aristotle), 55n–56n
proof theory, 144, 251
Protagoras, 38, 44, 62, 86–87
“pseudo-propositions,” 78
psychology, 70, 71, 80, 204–5
Putnam, Hilary, 111–12, 216–18
Pythagoreans, 39
quantum mechanics, 37–38, 40–41, 43
Quine, Willard van Orman, 88, 214
Ravel, Maurice, 90n
reality:
abstract, 43–45, 50–51, 62–64, 214
contingency and, 20–21, 30, 32n, 40–41
delusion vs., 202–5
empirical, 32, 50n, 74–80, 84–90, 107, 111, 137, 215, 258–59, 267n
mathematical, 44–48, 64, 121–22, 134–35, 140–41
meaning and, 44, 78–80, 81, 85–86, 97–98, 100, 101, 111, 131, 140–41, 191–92
objective, 23, 24–29, 33, 35–48, 50–51, 61–64, 85, 111–12, 131, 140–41, 188, 202–5, 214, 253–54
perception of, 46, 85, 111, 122, 134, 216
subjective, 24–25, 28, 36–40, 62, 84–90, 202–5
transcendental, 89, 191–92, 255–57, 259–61
as unknowable, 106–7
recursion theory, 132–33, 194, 252
Redlich, Friedrich, 54
Reichenbach, Hans, 82, 147, 158, 214, 218
Reidemeister, Kurt, 97
Reine Vernunft (pure reason), 15–16, 18, 67, 123, 218–19, 238, 258
relativity theory:
as conceptual revolution, 21–22, 36–43, 85
coordinate system in, 41n
development of, 35, 65
field equations for, 34n, 256, 257–58
general theory of, 19, 41
Gödel’s work on, 34n, 213n, 253–61
objective reality and, 36–38, 41–43, 85, 253–54
popular misinterpretation of, 37–40, 253–54
space-time continuum in, 41n, 42, 45, 253–61
special theory of, 19, 35, 41, 65
religion, 27n, 42–43, 77–79, 97, 192, 209–10, 243
Remarks on the Foundations of Mathematics (Wittgenstein), 117, 190, 268n–69n
Richard, Jules, 165n
Richard’s paradox, 165n–66n, 179n
Robinson, Abraham, 118n, 240–41
Rorty, Richard, 208–9
Rubin, H., 224n
Rubin, J., 224n
Russell, Bertrand, 78, 83, 111, 160, 198
Gödel and, 116–17, 216–18, 226, 248
paradox formulated by, 90–93, 100, 102, 142, 144
Wittgenstein and, 93–96, 97, 101–2, 108, 113, 119, 267n
“Russell’s Mathematical Logic” (Gödel), 216–18
Schilpp, P. A., 42, 257n, 265n
Schimanovich, Werner, 222
Schlick, Moritz, 73–74, 78–81, 82, 83, 84, 88, 96, 97, 104, 105, 110, 117, 148–49, 213, 220, 228
Schönberg, Arnold, 70
science:
methodology of, 29, 31, 76
pure, 16–17, 26–27, 85–86, 99–100
revolutions in, 21–22, 161, 231–32
subjectivity in, 36–40, 84
Secession, 70–71
self-referential statements, 66–67, 90–93, 165n–66n, 195
Shadows of the Mind (Penrose), 25, 201
Snow, C. P., 47n
Snow White, 252n
sociology, 242–45
Socrates, 63, 255
Solovay, Robert, 251
“Some Remarks on Axiomatized Set Theory” (Skolem), 154n
Sophists, 40, 62, 84, 112–13, 191, 193, 244
Spencer, Herbert, 85
Spinoza, Benedictus, 27n, 97, 126, 199
Sprache, Die (Kraus), 71
Sprachkritik (Mauthner), 103
Stalin, Joseph, 32, 227, 241
Straus, Ernst Gabor, 20–21, 29–30, 34
Structure of Scientific Revolutions, The (Kuhn), 161
Survey of Symbolic Logic (Lewis), 82
Tagore, Rabindranath, 103n–4n
Tarski, Alfred, 89
Taussky-Todd, Olga, 108, 204n, 218
“Tertium non datur” (Hilbert), 218
thaulamazein (ontological wonder), 54–55
Theory of Types, 92–93, 100, 144–45
thermodynamics, 80n
Thorne, Kip, 258, 259
Time, 221n
time, nature of, 34n, 41n, 42, 45, 253–61
Tractatus Logico-Philosophicus (Wittgenstein), 95n, 96–109, 119, 190–94
Tractatus Theologico-Politicus (Spinoza), 97
transfinite set theory, 216–17
truth:
absolute, 114, 124–26
beauty and, 28, 63–64, 107
inheritance laws of, 126n–27n
linguistic, 98–99, 104, 189–90
in logic, 44–45, 49–51, 98–102, 116–17, 152–54, 196n, 268n–69n
mathematical, 86–87, 98–102, 107–11, 137–38, 140–41, 148–49, 155–59, 188, 202–5
morality and, 62, 71–72, 86–87, 107, 124–26, 267n
objective, 23–25, 77–78, 85–86, 116–17, 216–17, 244
Turing, Alan, 166n, 190, 195–97, 219, 221n
“Two Dogmas of Empiricism” (Quine), 214
“Über formal unentscheidbare Sätze der Principia Mathematica und verwanter Systeme I” (Gödel), 24
see also incompleteness theorems
uncertainty principle, 21–22, 36–37, 39
undecidable propositions, 117–18
universal instantiation, 126n–27n
universal set, 93n
Veblen, Oswald, 19, 219, 226, 229
Vienna Circle, 43–44, 73–84, 89–113, 115, 116, 117, 119, 135–36, 147, 158, 160–61, 193, 213–14, 216, 228, 236, 266n, 267n
Voltaire, 247
von Neumann, John, 19–20, 33, 88, 148, 161–63, 187, 195, 212, 219, 223, 225n, 229, 238
Wagner-Jauregg, Julius, 220
Waismann, Frederich, 84, 104–5, 110, 148–49, 160
Wang, Hao, 34, 54, 58, 64, 83, 111, 154n, 159, 202–3, 213, 223, 249–50, 251, 259–60
Weil, André, 242n, 243, 251
Weil, Simone, 243
Weininger, Otto, 95–96
well-formed formulas (wffs), 86, 131–32, 141, 144n, 168–75, 178, 181
Werfel, Franz, 72
“What is Cantor’s Continuum Problem?” (Gödel), 111–12, 216–18
Wheeler, John Archibald, 258–59
White, Morton, 210, 215
Whitehead, Alfred North, 83, 92–93, 96, 101, 111, 144, 160, 198
Whitney, Hassler, 240, 250, 251
“Wiener Kreis in America, The” (Feigl), 213–14
Wiles, Andrew, 155n
Wirtinger, Professor, 222
Wissenschaftliche Weltauffassung: Der Wiener Kreis, 84
Wittgenstein, Karl, 70n
Wittgenstein, Ludwig:
background of, 90, 102
at Cambridge University, 94–95, 104, 105, 113–14, 195–97, 267n
early vs. late philosophy of, 100–102, 188–89
Gödel compared with, 109, 114–15, 118–19, 188–94, 218
Gödel’s theorems as viewed by, 89–90, 99, 100, 102, 111–20, 158, 188–95, 218, 268n–69n
influence of, 44, 104–9, 111, 113–14, 119
linguistic analysis by, 71, 96–102, 106–9, 118, 267n
logical positivism and, 43–44, 103–9, 119, 188–89, 191
personality of, 94–95, 104–7, 113–15, 118–19, 267n
/>
reputation of, 148–49, 214, 218, 221n
Russell and, 93–96, 97, 101–2, 108, 113, 119, 267n
Vienna Circle and, 43–44, 89–109, 117, 119, 267n
Wittgenstein, Paul, 90n
Wittgenstein’s Poker (Edmonds and Eidinow), 74–75
World War I, 68, 69–70, 96
Zen Buddhism, 103
Praise for Incompleteness:
The Proof and Paradox of Kurt Gödel
“Goldstein does a formidable job.”
—Anthony Doerr, Boston Globe
“Beguilingly empathetic. . . . Incompleteness sticks in the mind like Longitude, another book about an enigmatic scientific genius who registers on the reader’s mind as a mesmerizing void in human form. . . . Goldstein has a real knack for grounding even the loftiest theoretical disquisitions in reassuringly earthbound particulars. . . . To Goldstein’s enduring credit . . . we come away from Incompleteness with a sense of Gödel both at his most brilliant and, later, at his most neurotic.”
—David Kipen, San Francisco Chronicle
“Magnificent. . . . Goldstein is an excellent choice for this installment of Norton’s Great Discoveries series: Her philosophical background makes her a sure guide to the under-lying ideas, and she brings a novelistic depth of character and atmosphere . . . to her sympathetic depiction of the logician’s tortured psyche, as his relentless search for logical patterns . . . gradually darkened into paranoia.”
—Publishers Weekly
“Rebecca Goldstein has managed to get inside the head of Kurt Gödel and see what makes him tick. She presents Gödel’s mathematics as a consequence of his broader philosophical concerns. In her view Gödel is primarily a philosopher, but one who expressed his views through his mathematics. The strong connection between Gödel and Leibniz, which is little-known but was very important to Gödel, is particularly well documented. Highly recommended!”
—Gregory J. Chaitin,
author of Meta Math!: The Quest for Omega
“This is difficult material, at the borders of what we understand about human knowledge. The author has skillfully humanized it by showing us Gödel, Wittgenstein, and Einstein in their work, their friendships, and their disagreements. Perhaps only a novelist could have done this. Rebecca Goldstein has, in any case, done it superbly well.”
—John Derbyshire, New York Sun
“[Goldstein] writes with a light touch that readers are sure to enjoy.”
—Martin Davis, Nature
“Incompleteness is an artfully written and thoroughly engaging account of one of the greatest mathematical minds of the last century. By interweaving well-chosen episodes in Kurt Gödel’s life with a detailed yet remarkably accessible account of his most stunning breakthrough—a proof that there are true but unprovable statements—Goldstein reveals both Gödel’s torment and his genius. By the book’s end, we understand well why Einstein would look forward to ‘the privilege of walking home with Gödel,’ and we can’t help but wish that we’d been able to join them.”
—Brian Greene, author of
The Elegant Universe and The Fabric of the Cosmos
“In this penetrating, accessible, and beautifully written book, Rebecca Goldstein explores not only the work of one of the greatest mathematicians but also the relation of the human mind to the world around it.”
—Alan Lightman, author of Einstein’s Dreams
Copyright
Photograph Credits
p. 22 The American Institute of Physics/Emilio Segré Visual Archives
p. 253 Leonard McCombe/Timepix
Copyright © 2005 by Rebecca Goldstein
All rights reserved
Printed in the United States of America
First published as a Norton paperback 2006
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Library of Congress Cataloging-in-Publication Data
Goldstein, Rebecca, date.
Incompleteness : the proof and paradox of Kurt Gödel / Rebecca Goldstein.— 1st ed.
p. cm. — (Great discoveries)
Summary: “An introduction to the life and thought of Kurt Gödel, who transformed
our conception of math forever”—Provided by publisher.
Includes bibliographical references and index.
ISBN 0-393-05169-2
1. Gödel, Kurt. 2. Logicians—United States—Biography. 3. Logicians—Austria—
Biography. 4. Proof theory. I. Title. II. Series.
QA29.G58G65 2005
510’.92—dc22
2004023052
ISBN 0-393-32760-4 pbk.
ISBN 978-0-393-24245-4 (e-book)
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