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The indispensability of mathematics. (English) Zbl 1078.00500

Oxford: Oxford University Press (ISBN 0-19-516661-2). x, 172 p. (2001).
In contemporary philosophy of mathematics, one of the most discussed considerations in favor of the existence of mathematical objects, such as numbers, is the so-called “indispensability argument”. It is usually attributed to the late W. V. O. Quine, but the standard articulation is in Hilary Putnam’s Philosophy of logic [{}Harper & Row, New York (1971)]. Roughly, the indispensability argument has the following premises: (1) Real analysis refers to, and has variables that range over, abstract objects called “real numbers”. (2) Real analysis is indispensable for physics. (3) If real analysis is indispensable for physics, then one who accepts physics as true of material reality is thereby committed to the truth of real analysis. (4) Physics is true. (5) If a theory is true, then the items in the range of its bound variables exist. The conclusion is that real numbers exist.
The purpose of this short book is to articulate and defend a Quinean version of the indispensability argument. After sketching the general form of indispensability arguments, and providing accounts of other non-Quinean versions, the author sketches the main Quinean theses that lead to the various premises of the indispensability argument. These include naturalism, an unrelenting empiricism, confirmational holism, and the thesis that the ontology of a theory is found in the range of its bound variables. The author goes on to deal with common views that stand opposed to the conclusion of the argument. Later chapters characterize the empirical and a posteriori nature of mathematics that flows from the theses that underlie the argument. Overall, the book presents a clear picture of the Quinean world view.

MSC:

00-02 Research exposition (monographs, survey articles) pertaining to mathematics in general
00A30 Philosophy of mathematics
03A05 Philosophical and critical aspects of logic and foundations
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