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TitlePhilosophy of Mathematics
TagsMathematical Logic Immanuel Kant Mathematics Physics & Mathematics Euclidean Space
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Table of Contents
                            Contents
Preface
Introduction
Part I: Some Mathematicians as Philosophers
	1. The Kantian Legacy in Twentieth-Century Foundations of Mathematics
	2. Realism and the Debate on Impredicativity, 1917–1944
		Postscript to Essay 2
	3. Paul Bernays’ Later Philosophy of Mathematics
	4. Kurt Gödel
	5. Gödel’s “Russell’s Mathematical Logic”
		Postscript to Essay 5
	6 Quine and Gödel on Analyticity
		Postscript to Essay 6
	7. Platonism and Mathematical Intuition in Kurt Gödel’s Thought
		Postscript to Essay 7
Part II: Contemporaries
	8. Quine’s Nominalism
	9. Genetic Explanation in The Roots of Reference
	10. Hao Wang as Philosopher and Interpreter of Gödel
	11. Putnam on Existence and Ontology
	12. William Tait’s Philosophy of Mathematics
Bibliography
Copyright Acknowledgments
Index
                        
Document Text Contents
Page 2

P H I L O S O P H Y O F M A T H E M A T I C S

I N T H E T W E N T I E T H C E N T U R Y

Page 182

PLATONISM AND MATHEMATICAL INTUIT ION

167

to see itself as part of an argument that would “leave the platonistic
position as the only one tenable” (III 322– 323).27

The main dif� culty of the Gibbs lecture’s defense, however, is not
the omission he mentions at the end, of a case against Aristotelian real-
ism and psychologism, but that its central arguments are meant to be
in de pen dent of one’s standpoint in the traditional controversies about
foundations; the overall plan of the lecture is to draw implications
from the incompleteness theorems. Gödel’s main arguments aim to
strengthen an important part of his position, which he expresses by
saying that mathematics has a “real content.”28 But although this is
opposed to the conventionalism that he discerns in the views of the
Vienna Circle, and also to many forms of formalism, it is a point that
constructivists of the various kinds extant in Gödel’s and our own time
can concede, as Gödel is well aware. But it is probably a root convic-
tion that Gödel had from very early in his career; it very likely under-
lies the views that Gödel, in the letters to Wang, says contributed to his
early logical work. It would then also constitute part of his reaction to
attending sessions of the Vienna Circle before 1930.

4. Mathematical Intuition before 1964

I now turn to the conception of mathematical intuition, beginning with
some remarks about its development. I have outlined above the pre-
sen ta tion of Gödel’s realism in his early philosophical publications
RML, WCCP 1947, and the Gibbs lecture. For a reader who knows the
1964 version of WCCP, it is a striking fact about these writings that
the word “intuition” occurs in them very little, and no real attempt is
made to connect his general views with a conception of mathematical
intuition.

In RML the word “intuition” occurs in only three places, none of
which gives any evidence that intuition is at the time a fundamental

27 This remark appears to be an expression of a hope that Gödel maintained for
many years, that philosophical discussion might achieve “mathematical rigor” and
conclusiveness. As he was well aware, his actual philosophical writings, even at
their best, did not ful� ll this hope, and these remarks are part of an admission that
certain parts of the defense of mathematical realism had not been undertaken in
the lecture.
28 This conviction will come up in the discussion of intuition in §§4– 5; see also
Essays 5 and 6 in this volume.

Page 183

SOME MATHEMATIC IANS AS PHI LOS O PHERS

168

notion for Gödel himself. The fi rst (II 121) is in quotation marks and
refers to Hilbert’s ideas. The second is in one of the most often quoted
remarks in the paper, in which Russell is credited with “bringing to
light the amazing fact that our logical intuitions (i.e., intuitions con-
cerning such notions as truth, concept, being, class, etc.) are self-
contradictory” (II 124). Here “intuition” means something like a belief
arising from a strong natural inclination, even apparent obviousness.
In the following sentence these intuitions are described as “common-
sense assumptions of logic.” It’s not at all clear to what extent “intu-
ition” in this sense is a guide to the truth; it is clearly not an infallible
one. In the third place (II 138), Gödel again speaks of “our logical in-
tuitions,” evidently referring to the earlier remarks, and it seems clear
that he is using the term in the same sense.

One other remark in RML deserves comment. In his discussion of
the question whether the axioms of Principia are analytic in the sense
that they are true “owing to the meaning of the concepts” in them, he
sees the diffi culty that “we don’t perceive the concepts of ‘concept’
and ‘class’ with suffi cient distinctness, as is shown by the paradoxes”
(II 139– 140). Since “perception” of concepts is spoken of in unpub-
lished writings of Gödel, this seems to be an allusion to mathematical
intuition in a stronger sense. But the remark itself is negative; it’s not
clear what Gödel would say that is positive about perception of
concepts.

The word “intuition” does not occur at all in “Remarks Before the
Prince ton Bicentennial Conference” and only once in WCCP 1947.
Concerning constructivist views, he remarks:

This negative attitude towards Cantor’s set theory, however, is
by no means a necessary outcome of a closer examination of its
foundations, but only the result of certain philosophical concep-
tions of the nature of mathematics, which admit mathematical
objects only to the extent in which they are (or are believed to
be) interpretable as acts and constructions of our own mind, or
at least completely penetrable by our intuition. (II 180)

Since Gödel does not elaborate on his use of “intuition” at all, one
can’t on the basis of this text be at all sure what he has in mind. But it
appears that intuition as here understood, instead of being a basis for
possible knowledge of the strongest mathematical axioms, is restricted
in its application, so that the demand that mathematical objects be

Page 364

INDEX

349

theory, 222, 232– 242, 258; on structural-
ism, 234; on synonymy, 127, 206– 208,
274

Ramsey, F. P., 3, 41, 52– 53, 53n22, 59, 63,
108, 132, 159; predicative function, 53,
114

Rawls, John, 184– 185
realism, 40– 61, 190, 285, 286n30,

303– 309. See also concepts, realism
about; Dummett, Michael; Gödel, Kurt;
impredicativity; Platonism

reason, 75, 76, 101, 176– 178, 185n51, 291,
293. See also intrinsic plausibility;
intuition; Tait, William

reducibility, axiom of, 12, 42, 42n6, 45, 48,
52, 108, 116, 117, 120n37, 124, 125,
125n6

reference, 229
refl ection principles, 193, 193n14
refl ective equilibrium, 184, 184n50, 185,

185n51. See also extrinsic justifi cation
Reid, Constance, 67
replacement, axiom of, 58, 66, 111, 180,

239n27, 252, 253
Ricketts, Thomas, 149
Riehl, Alois, 69
Rosser, J. Barkley, 112
Rouilhan, Philippe de, 124– 126
Russell, Bertrand, 2, 103– 126, 194;

Principia Mathematica, 52, 138– 140;
theory of descriptions, 106, 106n9;
zig- zag theory, 113. See also reducibility,
axiom of

Russell- Myhill paradox, 125, 126
Russell’s paradox, 2, 107n11, 141n26, 213,

216, 219, 227, 236

Schilpp, Paul Arthur, 103, 143, 170, 259n33
Schütte, Kurt, 40, 119, 290
second- order logic, 64, 66, 208, 209, 211,

212, 213, 216, 234n16, 277
self- evidence, 71, 73, 77, 81, 252. See also

intrinsic plausibility; intuition
Sellars, Wilfrid, 285
separation, axiom schema of, 111, 237,

239n27, 250

set theory: Borel sets, 65, 66; concept of set,
43, 44, 47– 49, 100, 107, 144, 166, 178,
233– 240, 248– 253, 302, 318; defi nable
sets, 65, 117; defi nite totality, 65; naive
set theory, 235, 238, 316. See also axiom;
choice, axiom of; constructible universe;
descriptive set theory; foundation, axiom
of; inaccessible cardinals; iterative
conception of set; large cardinal axioms;
power set, axiom of; separation, axiom
schema of

Shapiro, Stewart, 90
Sieg, Wilfried, 3
Simpson, Stephen, 65
Skolem, Thoralf, 161, 238n27
Spector, Clifford, 117
Strawson, P. F., 224, 226
Stroud, Barry, 220n2
structuralism, 1, 59; Bernays on, 32, 34– 36,

47, 71, 82, 85– 91; Brouwer on, 22, 23,
26; Cassirer on, 14, 15; Hilbert on, 27;
Quine on, 234, 237, 238; Weyl on, 48

Sundholm, Göran, 131n12

Tait, William, 290– 319; axiomatic
conception, 33, 292– 302; on dialectic, 37,
293, 299, 300, 302; on fi nitism, 31n55,
309– 315; on Gödel, 181, 316– 319; on
intuition, 181, 309; on realism, 151,
303– 308

Takeuti, Gaisi, 194, 195, 290
Tarski, Alfred, 189, 201n3, 214, 215, 298
tautology, 6, 97, 132, 135, 256, 279; and

logical truths, 52– 54
theoretical entities, 86– 89, 86n44. See also

abstract objects; Bernays, Paul
tropes, 214
Turing, Alan, 3, 96, 144, 178, 254, 256
Type theory, 112n20, 114, 137, 216, 298.

See also Russell, Bertrand

undecidability. See continuum hypothesis;
incompleteness theorems

Väänänen, Jouko, 212
vagueness, 26, 43, 159, 160, 180, 192, 193,

289n39, 299

Page 365

INDEX

350

van Atten, Mark, 3, 19, 21n34, 23n39, 192,
195

van Dalen, Dirk, 18
vicious circle principle, 40, 41; Gödel on,

55– 59, 63n3, 107– 108, 111, 125,
155, 164, 191. See also Gödel, Kurt;
Poincaré, Henri; Russell, Bertrand; Weyl,
Hermann

Wang, Hao, 243– 266, 317; on analyticity,
256– 260; on intuition, 249, 250, 251,
251n15; reports on Gödel, 40, 58, 102,
117, 123, 129, 189, 191

Weyl, Hermann, 27, 40, 46– 49; arithmetic
analysis, 44; claim of a vicious circle,
42– 46; on Gödel, 104, 190; on intuition,
44, 49, 65; on sets, 43, 49

Wittgenstein, Ludwig, 8, 53, 137, 263, 265,
280, 291, 316; Tractatus, 2– 3, 52, 130,
150, 273n15, 291. See also Quine, W. V.;
Tait, William; Wang, Hao

Woodger, J. H., 219
Wright, Crispin, 38, 190, 209

Zach, Richard, 315
Zermelo, Ernst, 4, 238, 239; axiomatic

method, 240; on axiom of choice, 2, 4, 64;
on Cantor, 239; on defi nite properties,
158n9; well- ordering theorem, 241

Zermelo- Fraenkel Set Theory. See axiom;
choice, axiom of; foundation, axiom of;
power set, axiom of; separation, axiom
schema of

Zilsel, Edgar, 168, 318, 319

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