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Maddy, Penelope, , . Naturalism in Mathematics
1997, Oxford: Oxford University Press.
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Publisher’s Note: 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.

Comment: Good further reading in advanced undergraduate or postgraduate courses on metaphysics, naturalism or philosophy of mathematics. Sections from the book – for instance, the chapters in Part II on indispensability considerations in scientific and mathematical practice – could be profitably read on their own. These sections may also be of interest in philosophy of science courses, as they provide a careful analysis of scientific practice (as it relates to what scientists take themselves to be ontologically committed to).

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Maddy, Penelope, , . Three Forms of Naturalism
2005, in The Oxford Hanbook of Philosophy of Mathematics and Logic, ed. S. Shapiro. New York: Oxford University Press.
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Summary: A clear introduction to mathematical naturalism and its Quinean roots; developing and defending Maddy’s own naturalist philosophy of mathematics. Maddy claims that the Quinian ignores some nuances of scientific practice that have a bearing on what the naturalist should take to be the real scientific standards of evidence. Historical studies show that scientists sometimes do not take themselves to be committed to entities that are indispensably quantified over in their best scientific theories, hence the Quinian position that naturalism dictates that we are committed to entities that are indispensably quantified over in our best scientific theories is incorrect.

Comment: Good primary reading in advanced undergraduate or postgraduate courses on metaphysics, naturalism or philosophy of mathematics. This would serve well both as a clear and fairly concise introduction to Quinean naturalism and to the indispensability argument in the philosophy of mathematics.

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