文档介绍：PHILOSOPHY of
MATHEMATICS
Structure and Ontology
STEWART SHAPIRO
1
3
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Copyright © 1997 by Stewart Shapiro
First published in1997 by Oxford University Press, Inc.
198 Madison Avenue, New York, New York 10016
First issued as an by Oxford University Press paperback, 2000
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All rights reserved. No part of this publication may be reproduced,
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Library of Congress Cataloging-in-Publication Data
Shapiro, Stewart, 1951–
Philosophy of mathematics : structure and ontology / Stewart
Shapiro.
p. cm.
Includes bibliographical references and index.
ISBN 0-19-509452-2; 0-19-513930-5 (pbk)
1. Mathematics—Philosophy. I. Title.
1997
510'.1—dc20 96-31722
135798642
Printed in the United States of America
on acid-free paper
For Beverly
Numbers . . . are known only by their laws, the laws
of arithmetic, so that any constructs obeying those
laws—certain sets, for instance—are eligible . . .
explications of number. Sets in turn are known
only by their laws, the laws of set theory . . . arith-
metic is all there is to number . . . there is no say-
ing absolutely what the numbers are; there is only
arithmetic.
Quine [1969, 44–45]
If in the consideration of a simply infinite system
. . . set in order by a transformation . . . we entirely
neglect the special character of the elements; sim-
ply retaining their distinguishability and taking
into account only the relations to one anothe