The Indispensability of Mathematics

by Mark Colyvan

Department of Philosophy
University of Sydney
Sydney, NSW, Australia

Publication Details

Publisher: Oxford University Press, New York
Publication Date: 2001
ISBN: 019513754X (hardcover); 0195166612 (paperback)
Pages: x + 172

You can purchase a copy of this book from Oxford University Press or Amazon.


This book looks at the Quine-Putnam indispensability argument in the philosophy of mathematics. This argument urges us to place mathematical entities on the same ontological footing as other theoretical entities indispensable to our best scientific theories. The argument has come under serious scrutiny in recent times with many influential philosophers unconvinced of its cogency. This book outlines the indispensability argument in considerable detail before defending it against various challenges.

Although the focus of attention is squarely on the indispensability argument, in order to appreciate the argument's full force it is necessary to consider many other interesting and related topics. These include questions about ontological commitments and the applications of mathematics to physical theories. Of particular interest here is the Quinean backdrop from which the indispensability argument emerges. This backdrop consists of the doctrines of holism and naturalism. The latter is crucial to the whole indispensability debate, so a considerable portion of this work is spent discussing naturalism and its relevance to the debate.


1. Mathematics and Its Applications
2. The Quinean Backdrop
3. The Eleatic Principle
4. Field's Fictionalism
5. Maddy's Objections
6. The Empirical Nature of Mathematical Knowledge
7. Conclusion

What the Reviewers Say

"The book is clearly written and contains many interesting and novel ideas that push forward the horizons of the debate. Colyvan's defence of the indispensability argument is impressive and powerful yet always fair-minded: opponents are given a sympathetic hearing. Moreover, Colyvan is also an excellent expositor, difficult ideas and positions are explained simply and perspicuously, and the result is a book that will be of use and interest to undergraduates and researchers alike ... Colyvan has written an excellent book, and one that everyone interested in the philosophy of mathematics should read."
– Joseph Melia (in Metascience)

"This book will be of particular interest to specialists and advanced students in the philosophy of mathematics and science, but also to epistemologists and metaphysicians in general. I do not say that this book is indispensable – indispensability is a rare commodity, not easily established. Nor do I say that if it were indispensable it would be true – I am not convinced of that entailment. I do say that this book is both useful and pleasurable – like mathematics."
– Colin Cheyne (in The Australasian Journal of Philosophy)

"[T]his book is provocative and interesting ... One important, lasting value of the book is that it presents a clear picture of what the Quinean world view is like. since Quine's own pronouncements on indispensability are scattered throughout his writings."
– Julian Cole and Stewart Shapiro (in Mind)

Reviews and Critical Notices

Arko, M. (2007) Croatian Journal of Philosophy, 7(9): 118–21.

Castro, E. (2004) Intelectu, 10 (in Portuguese).

Cheyne, C. (2002) Australasian Journal of Philosophy, 80(3): 378–9.

Cole, J. and Shapiro, S. (2003) Mind, 112(446): 331–6.

Decock, L. (2001) Tijdschrift, 63(4): 805–6 (in Dutch).

de Lorenzo, J. (2003) History and Philosophy of Logic, 24(2): 167–9 (in Spanish).

Melia, J. (2003) Metascience, 12(1): 55–8.

Peressini, A. (2003) Philosophia Mathematica (3), 11(2): 208–23.

Sayward, C. (2003) Philosophical Books, 44(3): 293–4.

Shapiro, S. (2004) Mathematical Reviews,


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