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An Introduction to the Philosophy of Mathematics (Cambridge Introductions to Philosophy) Paperback – June 14, 2012
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"...Colyvan's introduction eschews a historically based, comprehensive survey. His final chapter contains a brief description of 20 selected mathematical theorems of philosophical interest. This skillfully written work, including liberal use of analogy and extensive exercises and recommended readings, is a stimulating introduction to some of the most discussed topics in contemporary philosophy of mathematics. Accessible to undergraduates with a background in mathematical logic... Highly recommended..."
--L.C. Archie, Lander University, CHOICE
"The present book is like a warm breeze after a cold winter in the rarefied atmosphere of the philosophy of mathematics.... The book is very well written and a pleasure to read. The chapters are short, clear and well structured.... include a list of discussion questions and recommended readings at the end of each chapter. ...it is a wonderful initiative.... I would not hesitate to use this book in an advanced undergraduate class in the philosophy of mathematics.... the philosophical discussions are always clear, provocative and stimulating. One of the challenges an instructor will face by adopting this book will undoubtedly be to contain the desire of students to discuss in depth some of the issues presented and to curb their enthusiasm and desire to know more or find answers to the questions."
--Jean-Pierre Marquis, Mathematical Reviews
Topics covered include the realism/anti-realism debate in mathematics, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Intended as a primary text for an introductory undergraduate course in the philosophy of mathematics.
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Colyvan isn't always careful with his language, which makes some of the discussion misleading, but overall, the book is suitable enough for a first look at the subject. As is typical, the book does not presuppose acquaintance beyond high-school mathematics and elementary logic, but Colyvan admits this and encourages the reader to venture into advanced mathematics, as he himself has done.
Colyvan says that those two articles by Benacerraf established the agenda for three problems in contemporary philosophy of mathematics.
(1) Set theory, the current standard foundation for mathematics, underdetermines mathematics. For example, in the von Neumann ordinals each ordinal comprises all of the ordinals preceding it (0 is an element of 1, and both 0 and 1 are elements of 2), but in the Zermelo ordinals each ordinal contains as its only element the ordinal immediately preceding it and no other. The question of whether 1 is an element of 3 is dependent on the definition of ordinals, and so (says Colyvan) set theory cannot answer the question with any definitude. [my language]
(2) The semantics of truth must be uniform across languages, and so the semantics of mathematical truth should not differ from the semantics of empirical truth. From this arises the problem of the ontology of mathematical entities. [my language]
(3) Knowledge claims about mathematical entities must be justified and the justification must show a connection from the mathematical entities to the person making the claim. [my language]
Along the way, Colyvan discusses how philosophers have responded to these three points. What he doesn't do is discuss how earlier writing on the philosophy of mathematics dealt, either explicitly or implicitly, with them (although prior to mentioning Benacerraf, he allows six pages to the classic positions). During his discussion of naturalism, indispensability, fictionalism, and nominalism, he never questions or even mentions the presupposition inherent in these views that mathematics needs science for its justification, and that if science can't justify it we need to construe mathematics in a manner that removes our ontological worries. The book, in fact, mainly deals with philosophical puzzlement about mathematics in relation to science. Mathematical structure and the philosophical views of structuralism are given only a cursory mention.
Not until page 78 does Colyvan say what he should have said at the beginning: "The job of philosophers of science and mathematics is to help make sense of, and contribute to, science and mathematics as practiced. The role of philosophers is not to overrule the pronouncements of mathematics and science on philosophical grounds - at least not pronouncements on matters of mathematics and science." How philosophers actually contribute to mathematics isn't clear from this book; and whether mathematicians need philosophers to "make sense of" mathematics for them isn't clear either. The philosophers' puzzlement is more likely their own and they're probably only making sense of mathematics for themselves.