Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be
proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an
objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view--realism--is assessed and finally rejected in favour of another--naturalism--which attends less to metaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in
the assessment of candidates for axiomatic status in set theory. Maddy's 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.
Industry Reviews
`presentation is clear and concise ... historically informed and comprehensive.'
A.M.Coyne, MATH, Vol.931
`An excellent book ... The philosopher's task is not to provide external criticism, but rather to clarify debates in this so-called mathematical community. And Ms Maddy makes some fascinating (and very technical) steps in this direction.'
The Economist Review
`Maddy's important new idea - mathematical naturalism - is a view about the proper relationship between mathematics and philosophy ... This is a powerful book: wonderfully erudite and a pleasure to read. There is no better introduction to the methodology of modern set theory.'
Gideon Rosen, British Journal for the Philosophy of Science 50 (1999)
`The book is beautifully written, tightly argued and makes compelling reading. I believe the position Maddy introduces and defends - set theoretic naturalism - is a significant and original addition to the philosophy of mathematics landscape, and one that will certainly attract a great deal of attention ... In summary this a very important book covering some fascinating terrain on the border between philosophy and mathematics. The book will thus be of
considerable interest to philosophers of mathematics and mathematicians alike. Indeed, Maddy's engaging style and clear explanations of the more technical material makes this book, like her last, accessible and
of interest to the non-specialist as well. Naturalism in Mathematics should enjoy a wide readership and it will no doubt promote fruitful debate on the many important issues it raises.'
Mark Colyvan, University of Tasmania, Mind, Vol 108, no 431, July 1999
An excellent book. The Economist (UK), February 2000
The philosopher's task is not to provide external critisism, but rather to clarify debates in this so-called mathematical community. And Ms Maddy makes some fascinating (and very technical) steps in this direction. The Economist (UK), February 2000
`The central problem discussed in the book concerns the foundations of contemporary set theory. Drawing on detailed historical studies on the origins and development of set theory... she shows that the ensuing methodological debates have been settled on mathematical grounds and not on the basis of philosophical considerations.'
Eduard Glas, Mathematical Reviews.
