Platonism and Anti-Platonism in Mathematics

by Mark Balaguer

Overview

In this deft and vigorous book, Mark Balaguer demonstrates that there are no good arguments for or against mathematical platonism (ie., the view that abstract, or non-spatio-temporal, mathematical objects exist, and that mathematical theories are descriptions of such objects). Balaguer does this by establishing that both platonism and anti-platonism are defensible positions. In Part I, he shows that the former is defensible by introducing a novel version of platonism, which he calls full-blooded platonism, or FBP. He argues that if platonists endorse FBP, they can then solve all of the problems traditionally associated with their view, most notably the two Benacerrafian problems (that is, the epistemological problem and the non-uniqueness problem). In Part II, Balaguer defends anti-platonism (in particular, mathematical fictionalism) against various attacks, chief among them the Quine-Putnam indispensability argument. Balaguer's version of fictionalism bears similarities to Hartry Field's, but the arguments Balaguer uses to defend this view are very different. Parts I and II of this book taken together clearly establish that we do not have any good argument for or against platonism. In Part III, Balaguer extends his conclusions, arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument, and indeed, that there is no fact of the matter as to whether platonism is correct (ie., whether there exist any abstract objects). This lucid and accessibly written book breaks new ground in its area of engagement and makes vital reading for both specialists and anyone else interested in the philosophy of mathematics or metaphysics in general.

Product Details

ISBN-13:	9780190284053
Publisher:	Oxford University Press
Publication date:	08/20/1998
Sold by:	Barnes & Noble
Format:	eBook
File size:	434 KB

About the Author

California State University, Los Angeles

Table of Contents

1	Introduction	3
1.	The Project of This Book	3
2.	Mathematical Platonism and Anti-Platonism	5
3.	Synopsis of the Book	14
1	Platonism	19
2	The Epistemological Argument Against Platonism	21
1.	Introduction	21
2.	Formulating the Epistemological Argument	22
3.	A Taxonomy of Platonist Responses	24
4.	Contact with Other Worlds: Godel	25
5.	Contact in This World: Maddy	28
6.	Knowledge Without Contact	35
3	A New Platonist Epistemology	48
1.	Introduction	48
2.	Skeleton of the Refutation of the Epistemological Argument	48
3.	Internalist vs. Externalist Explanations	53
4.	Defending and Motivating FBP	58
5.	Consistency	69
4	Non-Uniqueness Embraced	76
1.	Introduction	76
2.	Trying to Salvage the Numbers	77
3.	Structuralism	80
4.	The Solution	84
5.	Two Loose Ends	90
2	Anti-Platonism	93
5	The Fregean Argument Against Anti-Platonism	95
1.	Introduction	95
2.	The Argument	95
3.	In Defense of Fictionalism	98
4.	Nonfictionalistic Versions of Anti-Realistic Anti-Platonism	100
5.	The Refutation of Realistic Anti-Platonism	104
6.	Platonism and the Issue of Applicability and Indispensability	109
6	Denying the Existence of Indispensable Applications: Toward a Nominalization of Quantum Mechanics	113
1.	Introduction	113
2.	How Field Nominalizes	114
3.	Malament's Objection	117
4.	The Strategy for Nominalizing QM	120
5.	The Nominalistic Status of Propensities	126
7	Accounting for Indispensable Applications from a Fictionalist Point of View	128
1.	Introduction	128
2.	What, Exactly, Needs to Be Accounted For?	128
3.	A Fictionalist Account of the Applicability of Mathematics	130
4.	Problems with Platonism Revisited	142
3	Conclusions	149
8	The Unsolvability of the Problem and a Kinder, Gentler Positivism	151
1.	Introduction	151
2.	The Strong Epistemic Conclusion	152
3.	The Metaphysical Conclusion	158
4.	My Official View	178
	Notes	181
	Bibliography	207
	Index	213

Editorial Reviews

"excellent book...exceptionally clear, insightful, and useful critical survey"—The Review of Modern Logic "Platonism and anti-platonism in mathematics is an impressive work. Balaguer presents forceful arguments for the viability of both FBP and fictionalism, and against the feasibility of any substantially different Platonist or anti-Platonist position. ... an admirable achievement."—The Bulletin of Symbolic Logic

From the Publisher