Philosophy of Mathematics: Selected Readings

By Paul Benacerraf

Book Overview

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt G del. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and G del himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.

• Format: Paperback
• Language: English
• ISBN: 052129648X
• ISBN13: 9780521296489
• Release Date: January 1984
• Publisher: Cambridge University Press
• Length: 612 Pages
• Weight: 1.91 lbs.
• Dimensions: 1.5" x 6.0" x 9.1"

Customer Reviews

Predicated on "the customary manner of doing mathematics"

This is a useful anthology. I shall not argue its merits or demerits, but rather submit the following thesis: the shortcomings of virtually all of these theories are traceable to their unwarranted identification of mathematics with 20th century axiomatic mathematics. Bernays asserts that since "the customary manner of doing mathematics ... consists in establishing theories detached as much as possible from the thinking subject," any contrary view is "extreme" (p. 267). "The customary manner of doing mathematics" is thus glorified: it is tacitly assumed that "the customary manner" will reign supreme for all future. Otherwise it would not be "extreme" to suggest that "the customary manner" may not be infallible. Now of course one does not infer the absolute truth of an economic theory from its successful account of one particular society. But precisely this is being done in the case of philosophy of mathematics. Let us begin with the ever-foolish logical positivists. They define their position against Mill, who "maintained that [mathematical] propositions were inductive generalizations based on an extremely large number of instances" (Ayer, p. 317). Incidentally we later see Hempel making the exact same argument (p. 378), the positivist herd being predictable as always. But back to Ayer. The argument against Mill is that mathematical propositions are unfalsifiable: "Whatever instance we care to take, we shall always find that the situations in which a logical or mathematical principle might appear to be confuted are accounted for in such a way as to leave the principle unassailed" (p. 319). Therefore, the doctrine goes, mathematical propositions are analytic a priori, they are true "by virtue of definitions" (Hempel, p. 379), "none of them provide any information about any matter of fact" (p. 321), their truth do not depend on "facts about the world" (p. 316). Putting aside for the moment the trivial objection that some scientific laws (such as the law of inertia) are also analytic, this is still trivially false. Consider the discovery of the binomial series and other power series in the 17th century. These were obviously tested. There is a tacit appeal to "the customary manner of doing mathematics" here again. We now know that power series can be embedded in an axiomatic system that makes them true "by virtue of definitions." But this is a fact of experience. Only "facts about the world" tell us that mathematical propositions are susceptible to such treatment, i.e., that "the customary manner of doing mathematics" is all-pervasive. The claim that mathematics is analytic is the claim that another world would be impossible. Here is another passage in Ayer which plainly presupposes "the customary manner of doing mathematics": "Appeal to intuition [is] a source of danger to the geometer. It has, indeed, been shown that Euclid himself was guilty of ... make[ing] assumptions which are accidentally true of the particular figure he is using as an illustr

Indespensable compilation of the great papers.

I checked the first edition of this book out from the library when composing my senior thesis, and enjoyed it so much that I continued to read it long after I was finished, finally buying a copy of the second edition for myself. The first edition was an outstanding compilation of the most influential papers on the subject, hand-picked by two of the giants of modern philosophy. The second edition is, by and large, an improvement on the first, with more relevant selections added and less relevant selections removed (I would have liked to see "Logic Without Ontology" retained in the new edition, but it can be obtained elsewhere). A great benefit of this volume is that it is the only volume, as far as I know, still in print which contains the important essays on intuitionism by Brouwer and Heyting (the series edited by the two on the foundations of mathematics is, sadly, out of print). The inclusion of these essays is not only helpful for historical interest, but also keeps with the book's spirit of presenting the full spectrum of mathematical philosophy. One of the difficulties with presenting the philosophy of mathematics in a popular or summarized form is that, as with most analytic philosophy, it isn't possible to obtain any kind of understanding of the subject by partitioning it into schools of thought. Each author is unique. Because of this, it is much better to try to use a volume such as this one from the outset, rather than starting with an egregiously oversimplified account such as Korner's "The Philosophy of Mathematics: An Introductory Essay" (the book typically used to introduce the subject). So, not only would this make a superb text on the subject for a seminar, but it is also my first resort when I need a reference.