Naturalism in Mathematics [Paperback]

Penelope Maddy (Author)

Book Description

ISBN-10: 0198250754 | ISBN-13: 978-0198250753 | Publication Date: September 14, 2000

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Editorial Reviews

Review

"Stylistically the book is an enjoyably readable and beautifully clear. In particular Maddy has a near-miraculous gift for explaining mathematical technicalities in a way that is comprehensible to the non-expert. The book is therefore well worth reading not just for the new and bold philosophical position it promotes, but also as a philosopher's guide to what is going on in contemporary set theory." The Philosophical Review

"Maddy's knowledge of early and contemporary set theory, and of philosophically significant parts of the history of mathematics generally, is impressively wide and deep, and her discussions of these matters are illuminating and rewarding. She writes in a clear, forthright and challenging style. Her book is eminently readable, instructive, and thought-provoking."--The Journal of Symbolic Logic

About the Author

Penelope Maddy is Professor of Philosophy at the University of California, Irvine, having previously held positions at the University of Illinois, Chicago, and the University of Notre Dame.

Product Details

• Paperback: 264 pages
• Publisher: Oxford University Press, USA (September 14, 2000)
• Language: English
• ISBN-10: 0198250754
• ISBN-13: 978-0198250753
• Product Dimensions: 8.3 x 5.3 x 0.7 inches
• Shipping Weight: 14.4 ounces (View shipping rates and policies)
• Average Customer Review: 3.5 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #1,196,920 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

16 of 18 people found the following review helpful:

4.0 out of 5 stars To what does our inquiry commit us?, December 6, 2005

By 

John Paul Jaubert (Houston, TX) - See all my reviews
(REAL NAME)   

This review is from: Naturalism in Mathematics (Paperback)

Well, I'm a little reluctant to write a review for this book for a few reasons. Firstly, it will only have a very narrow readership. I take it that it's meant to be read as part of the curricula of an upper undergraduate or graduate course on philosophy of mathematics. Only philosophers and mathematicians might read it, with the former possibly not understanding significant portions of it, and the latter possibly scoffing at the very project which the book represents. Secondly, I'm in the first camp (the philosophers) so I'm not the best reviewer but I'll give it a try anyway.

To begin, it's not a text. Don't come to the book thinking to take away something more in the way of *knowledge* than you came to it with. It's purpose is to contribute to a broader sort of understanding than that. You'll get the most out of it if you already come armed with a good acquaintance with the developments in set and model theory that have gone on since the '60's right up to the time of the book's publication.

Maddy's earlier book, Realism in Mathematics, was an elaboration and defense of the on-again-off-again unfashionable view (or family of views) according to which mathematical objects (numbers, sets, transfinites and other infinites, functions, their values, etc.) all exist in some more-or-less robust sense. This book can be variously interpreted as something of a minor retreat from or else a clarification of that realism (or both).

The book pays special attention to Godelian and Quinean arguments for realistic views according to which, to be distortingly brief, we should robustly believe in at least some mathematical objects because we must necessarily *act as if* we do in the very doing of mathematics.

Then too, and this isn't really a criticism but just a note, it has a rather narrow focus on just ZF and its extensions. For example, if I recall correctly, there's next to nothing in the book about Quine's New Foundations... I think there's one comment only, though in it she explains why the neglect. Similarly, there's nothing directly about very different efforts at foundations such as category theory.

I'm familiar with similar work by Hartry Field, Stewart Shapiro, and a few others, and I think Maddy's prose compares favorably. But I have to confess that I started off more sympathetic to her position than that of, say, Mr. Field.

The closing chapters have some good insights on the relationship of mathematics to philosophy, science, and pseudoscience and presents a nice "apology" for the pure mathematician. On the whole I recommend this slim volume.

5 of 9 people found the following review helpful:

3.0 out of 5 stars Mathematical lapdogism, June 26, 2009

By 

Viktor Blasjo - See all my reviews
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This review is from: Naturalism in Mathematics (Hardcover)

The "naturalism" that Maddy espouses reduces philosophers to lapdogs. Current mathematical practice is glorified as the highest and only court in which every question is to be settled: "look not to traditionally philosophical matter about the nature of mathematical entities, but to the needs and goals of mathematics itself" (p. 191). She pretends that this does not preclude a critical outlook (p. 181), but obviously the only type of criticism that can be offered by such a philosopher is a critique within the doctrinal framework itself (i.e., what amounts to debates between commissars at the Kremlin) rather than critiques of the framework itself. Consider for example Poincaré's critique of logicism. Since Poincaré's philosophical critique was that of an outsider who did not participate in this research, Maddy presumably thinks it appropriate to dismiss it without even listening to the arguments. Such arrogance is hardly very healthy.

The most important trait that Maddy inherits from her masters the mathematicians is their intolerance. Consider for example how Gödel's "realism" (i.e., Platonism) is rejected.

"A Gödelian realism along these lines has been subject to considerable philosophical criticism, most prominent being the simple objection that we have not been given a convincing account of this mathematical intuition." (p. 92-93).

Thus the "considerable philosophical criticism" amounts to this: we like to think that we know a lot of stuff. The idea that mathematical intuition plays a role is rejected not for any actual arguments against it (of which there are none) but because it violates this axiom. Ancient Greek atomism could have been rejected on similar grounds, but I hope no one would take such a rejection as a rejection of atomism as such.

The extent of Maddy's dogmatism is made clear when she tries to put her own theory in the mouth of Gödel:

"The irrelevance of philosophical realism to Gödel's real concern is most explicit in [the following] passage ... 'the question of the objective existence of the objects of mathematical intuition ... is not decisive for the problem under discussion here [i.e. the meaningfulness of the continuum problem]. The mere psychological fact of the existence of an intuition which is sufficiently clear to produce the axioms of set theory and an open series of extensions of them suffices to give meaning to the question of the truth or falsity of propositions like Cantor's continuum hypothesis' He goes on to cite 'the relevance of such new axioms to finitary number theory and the previously mentioned possibility of verifying axioms by their consequences. Here, quite explicitly, it is the mathematical considerations, not the philosophical ones, that are decisive. In these passages, Gödel is not producing and argument for mathematical realism based on a strong analogy between mathematics and physics. Rather, his concern are particular issues in the actual practice of mathematics ... and his arguments ultimately rest on considerations also drawn from within the practice..." (p. 175).

So Maddy can support her claim that Gödel cared only about practice and not philosophy only with a quotation that contains no reference to practice but on the contrary a crystal clear philosophical thesis, namely that "the mere psychological fact of the existence of an intuition which is sufficiently clear" is the appropriate basis for mathematics. It is a tribute to Maddy's narrow-mindedness that she can see her own thesis confirmed in a blatant assertion of the opposite.