Stewart Shapiro

Thinking about mathematics

Even the least mathematical of us will deal quite a bit with numbers, but exactly what are they - do they have an independent existence? In Thinking about mathematics : the philosophy of mathematics , Stewart Shapiro discusses what philosophers throughout the ages have thought about this question.

Plato thought that numbers existed 'out there' -they had a reality independent of everything else, but Aristotle's view was that they were much more tied to the ways we use them in our lives. The book goes on to consider the views of Kant and Mill, and then looks at the three points of view of the early 20th century, Logicism, Formalism and Intuitionism. Shapiro then discusses more recent arguments about whether numbers actually exist or not, finishing with a chapter on structualism, where he puts forward his own point of view.

The book does not require too much prior experience in either mathematics or philosophy, but it's scope seemed considerably narrower than the title suggests - it is essentially just the ontology of numbers. Those wanting a wider view of the philosophy of mathematics - for instance the struggles with infinity, the incompleteness of arithmetic, and more recently the status of computers in mathematics - would be advised to look elsewhere. So who might want to read this book? Well those wanting to study the philosophy of mathematics will find it useful as a starting point for the ontology of numbers. If you're interested Shapiro's views then the final chapter gives a good indication. The book is also useful as an introduction to the works of other authors - I have added Hartry Field's Science without Numbers to my 'Books to Read' list.

Amazon.com info
Paperback 328 pages  
ISBN: 0192893068
Salesrank: 522057
Weight:1.09 lbs
Published: 2000 Oxford University Press
Amazon price $36.70
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Amazon.co.uk info
Paperback 328 pages  
ISBN: 0192893068
Salesrank: 271068
Weight:1.09 lbs
Published: 2001 Oxford University Press, U.S.A.
Amazon price £24.29
Marketplace:New from £20.62:Used from £18.46
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Amazon.ca info
Paperback 328 pages  
ISBN: 0192893068
Salesrank: 62065
Weight:1.09 lbs
Published: 2000 Oxford University Press
Amazon price CDN$ 38.37
Marketplace:New from CDN$ 38.37:Used from CDN$ 36.67
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Product Description
This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (logicism), the view that the essence of mathematics is the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline.

This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy.