Table Of ContentThe Oxford Handbook of
Philosophy of Mathematics
and Logic
STEWART SHAPIRO,
Editor
OXFORD UNIVERSITY PRESS
the oxford handbook of
.......................................................................................................................................
PHILOSOPHY OF
MATHEMATICS
AND LOGIC
.......................................................................................................................................
.......................................................................................................................................................
OXFORD HANDBOOKS IN PHILOSOPHY
PAUL K. MOSER, general editor
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Series AdvisoryBoard
ROBERT AUDI
University ofNebraska
MARTHA NUSSBAUM
University ofChicago
ALVIN PLANTINGA
University ofNotre Dame
ERNEST SOSA
BrownUniversity
the oxford handbook of
.......................................................................................................................................
PHILOSOPHY OF
MATHEMATICS
AND LOGIC
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Edited by
STEWART SHAPIRO
1
2005
1
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LibraryofCongressCataloging-in-PublicationData
TheOxfordhandbookofphilosophyofmathandlogic / editedbyStewartShapiro.
p. cm.—(Oxfordhandbooksinphilosophy)
Includesbibliographicalreferencesandindex.
ISBN-13 978-0-19-514877-0
ISBN0-19-514877-0
1.Mathematics—Philosophy. 2.Logic,Symbolicandmathematical—Philosophy.
I.Shapiro,Stewart,1951– II.Series.
QA8.4.O94 2004
510'.1—dc22 2004044847
1 3 5 7 9 8 6 4 2
PrintedintheUnitedStatesofAmerica
onacid-freepaper
P r e f a c e
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This volume provides comprehensive and accessible coverage of the disciplines of
philosophy of mathematics and philosophy of logic, including an overview of the
majorproblems,positions,andbattlelines.Inlinewiththeunderlyingthemeofthe
series,eachauthorwasgivenafreehandtodevelophisorherdistinctiveviewpoint.
Thus, the various chapters are not neutral.Readerssee exposition and criticism, as
well as substantial development of philosophical positions. I am pleased to report
thateachchapterbreaksnewground.Thevolumenotonlypresentsthedisciplines
of philosophy of mathematics and philosophy of logic, but advances them as well.
Formanyofthemajor positionsinthephilosophyofmathematics andlogic,
the book contains at least two chapters, at least one sympathetic to the view and
onecritical.Ofcourse,thisdoesnotguaranteethateverymajorviewpointisgiven
a sympathetic treatment. For example, one of my own pet positions, ante rem
structuralism, comes in for heavy criticism in two of the chapters, and is not
defendedanywhere(exceptbrieflyinchapter1).Inlightofthedepthandextentof
thedisciplinestoday,nosinglevolume,orseriesofvolumes,canprovideextensive
and sympathetic coverage of even the major positions on offer. And there would
hardly be a point to such an undertaking, since the disciplines are ever evolving.
Newpositions andnew criticisms ofold positionsemergewitheach issueofeach
majorphilosophyjournal.Mostofthechapterscontainanextensivebibliography.
In total, this volume provides a clear picture of the state of the art.
There is some overlap between the chapters. This is to be expected in a work
ofthisscope,anditwasexplicitlyencouraged.Authorsoftendrawinteresting,but
distinctive, conclusions from the same material. There is, of course, no sharp
separation between the philosophy of mathematics and the philosophy of logic.
The main issues and views of either discipline permeate those of the other. Just
about every chapter deals with matters mathematical and matters logical.
After the Introduction (chapter 1), the book begins with a historical section,
consisting of three chapters. Chapter 2 deals with the modern period—Kant and
his intellectual predecessors; chapter 3 concerns later empiricism, including John
StuartMillandlogicalpositivism;andchapter4focusesonLudwigWittgenstein.
The volume then turns to the ‘‘big three’’ views that dominated the philoso-
phyandfoundationsofmathematicsintheearlydecadesofthetwentiethcentury:
logicism, formalism, and intuitionism. There are three chapters on logicism, one
vi preface
dealing with the emergence of the program in the work of Frege, Russell, and
Dedekind (chapter 5); one on neologicism, the contemporary legacy of Fregean
logicism (chapter 6); and one called ‘‘Logicism Reconsidered,’’ which provides a
technical assessment of the program in its first century (chapter 7). This is fol-
lowedbyalengthychapteronformalism,coveringitshistoricalandphilosophical
aspects(chapter8).Twoofthethreechaptersonintuitionismoverlapconsiderably.
Thefirst(chapter9)providesthephilosophicalbackgroundtointuitionism,through
the work of L. E. J. Brouwer, Arend Heyting, and others. The second (chapter 10)
takes a more explicitly mathematical perspective. Chapter 11, ‘‘Intuitionism Re-
considered,’’ focuses largely on technical issues concerning the logic.
The next section of the volume deals with views that dominated in the later
twentieth century and beyond. Chapter 12 provides a sympathetic reconstruction
of Quinean holism and indispensability. This is followed by two chapters that
focusdirectlyonnaturalism.Chapter13laysouttheprinciplesofsomeprominent
naturalists, and chapter 14 is critical of the main themes of naturalism. Next up
are nominalism and structuralism, which get two chapters each. One of these is
sympathetic to at least one variation on the view in question, and the other
‘‘reconsiders.’’
Chapter19isadetailedandsympathetictreatmentofapredicativeapproachto
both the philosophy and the foundations of mathematics. This is followed by an
extensive treatment of the application of mathematics to the sciences; chapter 20
lays out different senses in which mathematics is to be applied, and draws some
surprising philosophical conclusions.
The last six chapters of the volume focus more directly on logical matters, in
three pairs. There are two chapters devoted to the central notion of logical
consequence. Chapter 21 presents and defends the role of semantic notions and
model theory, and chapter 22 takes a more ‘‘constructive’’ approach, leading to
prooftheory.Thenexttwochaptersdealwiththeso-calledparadoxesofrelevance,
chapter 23 arguing that the proper notion of logical consequence carries a notion
of relevance, and chapter 24 arguing against this. The final two chapters concern
higher-order logic. Chapter 25 presents higher-order logic and provides an over-
viewofitsvarioususesinfoundationalstudies.Ofcourse,chapter26reconsiders.
Throughout the process of assembling this book, I benefited considerably from
the sage advice of my editor, Peter Ohlin, of Oxford University Press, USA, and
from my colleagues and friends, at Ohio State, St. Andrews, and other institu-
tions. Thanks especially to Penelope Maddy and Michael Detlefsen.
C o n t e n t s
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Notes on the Contributors, xi
1. Philosophy of Mathematics and Its Logic: Introduction, 3
Stewart Shapiro
2. ApriorityandApplication:PhilosophyofMathematicsinthe
Modern Period, 28
Lisa Shabel
3. Later Empiricism and Logical Positivism, 51
John Skorupski
4. Wittgenstein on Philosophy of Logic and Mathematics, 75
Juliet Floyd
5. The Logicism of Frege, Dedekind, and Russell, 129
William Demopoulos and Peter Clark
6. Logicism in the Twenty-first Century, 166
Bob Hale and Crispin Wright
7. Logicism Reconsidered, 203
Agust(cid:1)ıın Rayo
8. Formalism, 236
Michael Detlefsen
9. Intuitionism and Philosophy, 318
Carl Posy
10. Intuitionism in Mathematics, 356
D. C. McCarty
viii contents
11. Intuitionism Reconsidered, 387
Roy Cook
12. Quine and the Web of Belief, 412
Michael D. Resnik
13. Three Forms of Naturalism, 437
Penelope Maddy
14. Naturalism Reconsidered, 460
Alan Weir
15. Nominalism, 483
Charles Chihara
16. Nominalism Reconsidered, 515
Gideon Rosen and John P. Burgess
17. Structuralism, 536
Geoffrey Hellman
18. Structuralism Reconsidered, 563
Fraser MacBride
19. Predicativity, 590
Solomon Feferman
20. Mathematics—Application and Applicability, 625
Mark Steiner
21. Logical Consequence, Proof Theory,
and Model Theory, 651
Stewart Shapiro
22. Logical Consequence From a Constructivist
Point of View, 671
Dag Prawitz
23. Relevance in Reasoning, 696
Neil Tennant
contents ix
24. No Requirement of Relevance, 727
John P. Burgess
25. Higher-order Logic, 751
Stewart Shapiro
26. Higher-order Logic Reconsidered, 781
Ignacio Jane´
Index, 811
