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Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures (Philosophical Issues in Science) Hardcover – September 22, 1999

ISBN-13: 978-0415122740 ISBN-10: 0415122740
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Editorial Reviews

From Library Journal

This book offers an intriguing sample of both traditional and current ideas in the philosophy of mathematics. Its readable style is aimed at those with more background in philosophy than in mathematics, with detailed examples usually involving only secondary-level algebra and geometry. The author (philosophy, Univ. of Toronto) states that one of his goals is to argue for Platonism. This attempt is not entirely convincing, both because some of Browns positions need more substantiation and because alternative views are presented only insofar as they can be readily dismissed. As the subtitle suggests, a unique feature is the interesting argument in favor of the validity and usefulness of pictures (not limited to geometrical diagrams) as proofs in and of themselves. It is to be hoped that a few mathematically substantive typographical errors in the examples will be corrected in the finished edition. For academic mathematics and philosophy collections.Kristine Fowler, Mathematics Lib., Univ. of Minnesota, Minneapolis
Copyright 1999 Reed Business Information, Inc.


This is an excellent introductory text for philosophy of mathematics courses, something which is much needed.
–Mary Tiles, University of Hawa'ii

Philosophy of Mathematics is written in a lively and accessible style. It uses a wealth of stimulating examples. and covers a wide variety of topics in the philosophy of mathematics, including some new and little explored questions.
–Alexander Bird, University of Edinburgh

The book is filled with Brown's insightful views on many issues in the philosophy of mathematics; most importantly, his love of, and enthusiasm for, his subject is apparent on every page....Students, I think, will be provoked by Brown's up-front approach and inspired to think seriously about the philosophy of mathematics themselves.

All in all this is a wonderful introduction to the philosophy of mathematics. It's lively, accessible, and, above all, a terrific read. It would make an ideal text for an undergraduate course on the philosophy of mathematics; indeed, I recommend it to anyone interested in the philosophy of mathematics--even specialists in the area can learn from this book.
–Mrk Colyvan, University of Tasmania and University of California, Irvine, Philosophy in Review

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Product Details

  • Series: Philosophical Issues in Science
  • Hardcover: 232 pages
  • Publisher: Routledge (September 22, 1999)
  • Language: English
  • ISBN-10: 0415122740
  • ISBN-13: 978-0415122740
  • Product Dimensions: 6.1 x 0.6 x 9.2 inches
  • Shipping Weight: 1 pounds (View shipping rates and policies)
  • Average Customer Review: 3.8 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Best Sellers Rank: #14,485,450 in Books (See Top 100 in Books)

Customer Reviews

3.8 out of 5 stars
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Most Helpful Customer Reviews

15 of 19 people found the following review helpful By Pedro Rosario on May 9, 2002
Format: Paperback
James Robert Brown has exposed with excellence in his book a very good defense of Platonism in Philosophy of Mathematics. He shows the basic premises that are shared more or less by all Platonists, including Frege, Godel, and many others.
He shows why philosophers argue against Platonism, and which are the biases and confusions they make that apparently they show a rejection to it.
It discusses subjects such as numbers, sets, geometrical objects, graphs, and even fractals, and how Platonism can recognize all of them as abstract objects, and how pictures can help us psychologically to grasp these abstract objects.
However, with all of this I have only one problem. He proposes a kind of "mind's eye" by which we are able to "grasp" these abstract objects. Although he presents a very keen argument for refuting the argument that only through sensible experience we are able to know anything about the world, I still feel uneasy and not satisfied about his epistemological account. The hypothesis of this "mind's eye" is the reason why most philosophers find the Platonist proposal so objectionable. In order to account for our knowledge of abstract objects, we must posit a kind of myserious mystic faculty of the "mind's eye".
I think that Husserl's categorial intuition and categorial abstraction (which he proposes in his "Logical Investigations") or Katz proposal of intellectual intuition (in his "Realistic Rationalism") are much more acceptable proposals than Brown's "mind's eye".
Despite this difference, I highly recommend this book, along with Jerrold Katz's "Realistic Rationalism", as a great and serious exposition of the Platonist proposal in philosophy of mathematics that I have ever found. It also serves as a good introduction to philosophy of mathematics.
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18 of 24 people found the following review helpful By A Customer on November 16, 2002
Format: Paperback
The philosophy of mathematics contains many interesting issues. This makes it unavoidable that there should be some interesting material in JR Brown's _Philosophy of Mathematics_. On the other hand, while Brown's prose is certainly accessible, his treatment of the issues in the philosophy of mathematics does not aid but rather gets in the way of the reader's understanding and appreciation of them.
Brown is desperately in need of an editor, or at least a proofreader. There are countless grammatical and typographical errors which really ought not to be in a final edition; and while most of these do little more than make the text seem unpolished and amateurish, some obscure the subject matter: for example, because of Brown's extreme laziness in bracket-counting, his proof of Gödel's second incompleteness theorem (chapter 5) is quite incomprehensible unless the reader is adept enough at playing Sherlock Holmes to come up with the proof Brown *really* meant to give.
But Brown doesn't just need an editor to fix his mechanical errors: he needs someone to help him choose what to print. In chapter 8, on constructive mathematics, Brown fills up a full page with lengthy quotes from Brouwer, admitting that "This is pretty obscure stuff," (116) but declining to elucidate Brouwer's point except with more quotation, which he admits is "no better".
There is interesting material in this book, but the presentation is far from perfect, and often aggravating. Brown is to be credited with a sense for what is interesting, but he would put that sense to better use advising another writer than writing himself. If you can't find another introduction to the philosophy of mathematics, this book is worth reading; if you can, I strongly advise you to investigate your alternatives.
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1 of 1 people found the following review helpful By Kodoku on October 5, 2011
Format: Kindle Edition
I am currently taking a course taught by none other than the author himself, which is a third year undergraduate course, PHL346: Philosophy of Mathematics, offered at the University of Toronto in Canada, and uses this as the textbook.

The author is a Platonist. Most of this book centers around Platonism - other positions are largely discussed in the context of how they avoid problems in Platonism or introduce new problems. This may seem like a questionable way of doing things, yet I think there's a very good reason for it. The author thinks (and I agree with him, even if I'm not sold on the truth of Platonism) that this position offers the richest account of mathematics and how it's done. Just as physicists or biologists discover objectively real truths about the objectively real natural world, mathematicians discover objectively real truths about the objectively real mathematical world. A recurring theme in the book (the role of pictures and visualization in general in mathematics) also has an elegant place in Platonism - pictures offer a window into the realm of mathematical objects, and are therefore more than mere heuristic devices.

Whether one ultimately accepts the independent existence of mathematical objects, I think Platonism is a very engaging and insightful idea to mold the book around as an introduction to the subject. I also must disagree with a sentiment expressed that he casually dismisses arguments against his position. He openly admits that he has no resolution to the problem of access, for example, which is the problem of how we "observe" mathematical objects in any sense of the word. What he does instead is to show how this problem is not clearly worse than problems with the other major positions.
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