Mathematics Without Numbers: Towards a Modal-Structural Interpretation

167 Pages·1989·1.303 MB·English

by  Hellman G.

Mathematics Without Numbers
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Mathematics Without Numbers
Towards a Modal-Structural Interpretation
Geoffrey Hellman
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BritishLibraryCataloguinginPublicationData
Hellman,Geoffrey
Mathematicswithoutnumbers:towardsa
modal-structuralinterpretation
1.Mathematics
I.Title
510
ISBN0-19-824934-9
ISBN0-19-824034-1(pbk)
LibraryofCongressCataloginginPublicationData
Hellman,Geoffrey.
Mathematicswithoutnumbers:towardsamodal-structural
interpretation/ GeoffreyHellman.
Bibliography:p.Includesindex.
1.Mathematics—1961–I.Title.
QA37.2.H421989 510—dc1989-30517
ISBN0-19-824934-9
ISBN0-19-824034-1(pbk)
To The Memory
of My Parents
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Preface
. . . mathematics may be defined as thesubject in whichwe never know what we are talking about, nor whether
what we are saying is true.
Bertrand Russell
The idea that pure mathematics is concerned principally with the investigation of structures of various types in
completeabstractionfrom the nature of individual objects making up those structures is not a novel one, and can be
traced at least as far back as Dedekind's classic essay, ‘Was sind und was sollen die Zahlen?’ (originally published in
1888). It represents a striking contrast with the Fregean preoccupation with identifying “the objects” of particular
branches of mathematics, and it seems to have lain behind Hilbert's refusal to accept Frege's point of view on such
fundamental matters as the nature of mathematical definitions and axioms, mathematical existence, and truth.
With the rise of the comprehensive “logicist” systems of type theory and axiomatic set theory, however, the
structuralist idea was either neglected in favour of some arbitrarily chosen relative interpretation of ordinary
mathematics (number theory, analysis, etc.) within the comprehensive system, or else it was given metalinguistic lip-
service through the apparatus of Tarskian model theory, carried out within set theory itself.
Despite the attractive unifying power of modern set theory, embedding the structuralist intuition within it has its
disadvantages. Must we accept anything so powerful as, say, the Zermelo–Fraenkel axioms—categorically asserted as
truths about Platonic objects—in order to carry out a structuralist interpretation of number theory or classical
analysis? And what of set theory itself; can it not also be understood along structuralist lines, and would it not
constitute a philosophical advance so to understand it? “It”, after all, is actually a multitude of apparently conflicting
systems. On the standard Platonist picture, at most one of them can correctly describe “the real world of sets”. On a
structuralist interpretation, there is at least the
viii PREFACE
prospect of a healthy pluralism—that many different systems can be sustained as theories investigating different
structural possibilities.
A further main source of inspiration for the present study is Hilary Putnam's ‘Mathematics without Foundations’
[1967], which suggested that modal logic could be used as a framework for eliminating apparent reference to
mathematical objects entirely, including the objects of abstract set theory itself. Conundrums associated with a special
realmofmathematicalobjects,emphasized byanumberofcontemporaryphilosopherssuchas NelsonGoodmanand
Paul Benacerraf in terms strikingly reminiscent of Dedekind—how to reconcile talk of such objects with the
multiplicity of “ways of taking them”, how we ever manage to refer to such objects, and the like—such questions
wouldbeseennoteventoariseonthemodallogicaleliminativeinterpretation.Butthedetailsandimplicationsofsuchan
interpretation have remained to be worked out.
The present work is an exploratory effort at synthesizing these two strands of thought, “structuralism” and
“mathematics as modal logic”. The aim has been to provide for structuralism frameworks that are flexible, suitably
powerful, and as preciselydelineated as those that have grown out of the logicist tradition and which have dominated
non-constructive foundational thought.
When I began working on this project several years ago, Putnam was my principal source for the idea that set theory
itself ought to be understood along structuralist lines and that such an approach contains the germ of a resolution of
theproblemofproperclasses.Itwassomething ofarevelationwhenIstumbledupona classicopus ofZermelo, Über
Grenzzahlen und Mengenbereiche [1930], containing strikingly similar proposals. Although this work is known to
mathematicians for its leadingmathematical content(includingthe“discovery” of inaccessible cardinals), its intriguing
philosophical perspective has yet to receive the attention it deserves. (Unfortunately, an English translation has yet to
be published. Doctoral language requirements in this area still have their point!)
Concerning personalacknowledgements,Iowedeepandlasting gratitudetoNelsonGoodmanfor yearsofintellectual
guidanceand inspiration, althoughI am resigned tohis finding all toolittle evidence ofthisinthe presentwork.More
proximately, I am indebted to many people for encouragement, criticism, suggestions, and help. Among them are
students in graduate seminars in philosophy of mathematics that I have offered at Indiana University and the
PREFACE ix
University of Minnesota, and members of many audiences at conferences and colloquia who have endured earlier
versions of various parts of this work. I am especially grateful to the following individuals: John Burgess, Jeremy
Butterfield, Nino Cocchiarella, J. Alberto Coffa, William Craig, J. Michael Dunn, Hartry Field, Richard Grandy,
Michael Hallett, W. D. Hart, Daniel Isaacson, Ronald Jensen, Akihiro Kanamori, Saul Kripke, Penelope Maddy,
Charles Parsons, Hilary Putnam, Jack Silver, Howard Stein, W. W. Tait, and, in all probability, a number of others for
whose understanding I must now plead.
Much of the work for this monograph was carried out with the support of the National Science Foundation, Grants
No. SES-8420463 and No. SES-8605286, for which I am very grateful.
G.H.
Minneapolis, Minnesota
8 July 1988