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Naturalism in Mathematics (Hardcover, New)
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Naturalism in Mathematics (Hardcover, New)
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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.
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