The Growth of Mathematical Knowledge

455 Pages·2000·15.47 MB·English

by  Jaakko Hintikka (auth.)

THEGROWTH OFMATHEMATICAL KNOWLEDGE
SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
JAAKKOHINTIKKA, Boston University
Editors:
DIRKVANDALEN, University ofUtrecht,The Netherlands
DONALD DAVIDSON, University ofCalifornia, Berkeley
THEaA.F.KUIPERS, University ofGroningen, The Netherlands
PATRICKSUPPES, Stanford University, California
JAN waLEN-sKI, Jagiellonian University, Krakow, Poland
VOLUME 289
THE GROWTH
OF MATHEMATICAL
KNOWLEDGE
Editedby
EMILY GROSHOLZ
ThePennsylvania State University, U.S.A.
and
HERBERT BREGER
LeibnizArchives,Hannover and Universityof Hannover.Germany
,
....
Springer-Science+Business Media, B.Y:
AC.I.P.Cataloguerecord for this book isavailablefrom the LibraryofCongress.
ISBN978-90-481-5391-6 ISBN978-94-015-9558-2(eBook)
DOI10.1007/978-94-015-9558-2
Printed onacid-free paper
All Rights Reserved
©2000SpringerScience+BusinessMediaDordrecht
OriginallypublishedbyKluwerAcademicPublishersin2000.
Softcoverreprintofthehardcover1stedition2000
No part ofthe material protectedby this copyrightnotice may bereproduced or
utilizedinany form or byany means, electronicor mechanical,
including photocopying,recording orbyany information storageand
retrieval system,without written permissionfrom the copyrightowner.
TABLEOFCONTENTS
ACKNOWLEDGMENTS ix
INTRODUCTION xi
NOTES ON CONTRIBUTORS xxxix
PARTI:THEQUESTIONOFEMPIRICISM
THEROLEOF SCIENTIFICTHEORYAND EMPIRICALFACTINTHE GROWTH
OF MATHEMATICALKNOWLEDGE
JAAKKO HINTIKKA/KnowledgeofFunctions inthe Growthof
MathematicalKnowledge
MICHAELS.MAHONEY/Huygensand the Pendulum:From Device
to MathematicalRelation 17
2
DONALD GILLIES/An EmpiricistPhilosophyofMathematics
and Its Implications forthe HistoryofMathematics 41
IVO SCHNEIDER/ The MathematizationofChance inthe Middle
ofthe 17th Century 59
MICHAEL LISTON /Mathematical Empiricism and
the MathematizationofChance:CommentonGillies and Schneider 77
3
EMILYGROSHOLZ/The Partial Unification ofDomains,Hybrids,
and the Growth ofMathematicalKnowledge 81
CRAIG FRASER/Hamilton-JacobiMethods and Weierstrassian
Field Theory inthe Calculus ofVariations 93
4
PAOLOMANCOSU/ On MathematicalExplanation 103
FRAN<;OIS DEGANDT/Mathematicsandthe ReelaborationofTruths 121
v
vi TABLEOF CONTENTS
5
MARK STEINER/Penrose and Platonism 133
MARK WILSON /Onthe MathematicsofSpilt Milk 143
PARTII:THEQUESTIONOFFORMALISM
THE ROLE OFABSTRACTION, ANALYSIS, AND AXIOMATIZATION IN THE
GROWTHOFMATHEMATICAL KNOWLEDGE.
CARLOCELLUCCI/TheGrowth ofMathematicalKnowledge:
An Open World View 153
DETLEFLAUGWITZ/Controversiesabout Numbersand Functions 177
CARLPOSY/ Epistemology, Ontology, and the Continuum 199
2
HERBERTBREGER/Tacit Knowledgeand MathematicalProgress 221
MADELINEM.MUNTERSBJORN/The Quadratureof
Parabolic Segments 1635- 1658:AResponse to Herbert Breger 231
MICHAELLISTON/MathematicalProgress:Ariadne'sThread 257
COLIN MCLARTY/Voir-Dire inthe Case ofMathematical Progress 269
3
HOURYA BENIS-SINACEUR/The Nature ofProgress inMathematics:
The SignificanceofAnalogy 281
EBERHARDKNOBLOCH/Analogyand the Growth of
Mathematical Knowledge 295
TABLE OFCONTENTS vii
4
ALEXEI BARABASHEV/ Evolution ofthe Modes ofSystematization
ofMathematicalKnowledge 315
ISABELLA BASHMAKOVAAND G.S.SMIRNOVA/Geometry,
the First UniversalLanguageofMathematics 331
III.THEQUESTIONOFPROGRESS
CRITERIAFORAND CHARACTERIZATIONSOF PROGRESS IN
MATHEMATICALKNOWLEDGE
PENELOPEMADDY / MathematicalProgress 341
MICHAELD. RESNIK / SomeRemarks on MathematicalProgress
from aStructuralist'sPerspective 353
2
VOLKERPECKHAUS/ Scientific Progress and Changes
inHierarchiesofScientific Disciplines 363
SERGEI DEMIDOV/Onthe Progress ofMathematics 377
KLAUS MAINZER/Attractors ofMathematicalProgress:
The Complex Dynamics ofMathematicalResearch 387
CHRISTIANTHIEL /On Some DeterminantsofMathematical Progress 407
ACKNOWLEDGMENTS
Emily Grosholz and Herbert Breger would like to thank the Alexander von Humboldt
Foundation for a Transatlantic Cooperation Grant (1994-1997) in the context ofwhich
the two conferences that gave rise to this book were planned. The Humboldt grant
represents quite precisely the intention of this project, to bring together in a more
effective waytwo communities ofscholars ontheEuropean andNorth American shores
oftheAtlantic Ocean.
Wewould also like to thank the Office ofContinuing and Distance Education
ofthe PennsylvaniaStateUniversity for itsgenerous support oftheconferences andthis
volume, and in particular Patricia Book, Associate Vice-President for Outreach and
Executive Director ofthe Division ofContinuing Education, forher faith inthe project.
We would also like to thank a variety ofunits at the Pennsylvania State University for
their support: the Institute for the Arts and Humanistic Studies, the Department of
Philosophy,the College ofthe LiberalArts, the College ofEarth and Mineral Sciences,
the College ofScience, and the College of Engineering, as well as the Department of
Mathematics and the Department ofGermanic and Slavic Languages and Literatures.
Finally,wewould liketothank Abe Schenitzer andAnik and Pierre Kerszberg fortheir
generous help with the translation of some of these essays; and Professors Boris
Kushner, Fritz Nagel, Abe Schenitzer, and Robert Thomas, and Drs. Lisa Shabel and
Aaron Lercher fortheir interestinandattendance oftheconferences.
Emily Grosholz would like to thank the American Council of Learned
Societies fora Fellowship (1997) which allowed herto work onthe Introduction to this
book, andto Clare Hallaswellasthe Department ofHistory and Philosophy of Science
atthe University ofCambridge, which welcomed her asa Visiting Fellow during 1997
98. The 1997-98 lecture series at King's College, London (in which Grosholz, Breger,
Mancosu and Gillies participated) wasespecially helpfuland illuminating.
She would also like to thank the graduate and undergraduate students at Penn
State in Philosophy 570 (Spring 1995) and Philosophy 512 (Spring 1996) for their
support ofthe conference inmanyrespects(Englishing, copy-editing, translation aswell
as hosting participants): Nathan Anderson, Derrick Calandrella, Evgenia Cherkasova,
Peter Costello, Russell Ford, Albert Frantz, Michael Jarrett, Ravinder Koul, Franklin
Perkins,Gregory Recco, DianaRhodes,CoryStyranko, andErnst-Jan Wit.
Finally, she would like to thank Evgenia Cherkasova, Gregory Recco, and
Rebecca Wayland for their role as research assistants in the preparation of this
manuscript. The many hours they have spent laboring with her over the linguistic,
computer-related, and mathematical complexities of the text are effectively
uncountable. Rebecca Wayland's ability to organize a large,multifarious plan ofwork
wasespecially valuable.
This volume is dedicated to Isabella Bashmakova in honor of her great
achievement as an historian of mathematics and to Evgenia Cherkasova, Professor
Bashmakova'sstudent andmine,inhonorofhergreat promise asaphilosopher.
IX
EMILYGROSHOLZ
INTRODUCTION
Mathematics has stood as a bridge between the Humanities and the Sciences since the
days ofclassical antiquity. For Plato, mathematics was evidence ofBeing in the midst
of Becoming, garden variety evidence apparent even to small children and the
unphilosophical, and therefore of the highest educational significance. In the great
central similesofTheRepublicitisthetouchstone ofintelligibilityfordiscourse, and in
the Timaeus itprovides in an oddly literal sense the framework ofnature, insuring the
intelligibility ofthe materialworld.
For Descartes, mathematical ideas had a clarity and distinctness akin to the
idea of God, as the fifth of the Meditations makes especially clear. Cartesian
mathematicals are constructions as well as objects envisioned by the soul; in the
Principles, thework ofthe physicistwhoprovides aquantified account ofthe machines
ofnature hovers between description and constitution. For Kant, mathematics reveals
the possibility of universal and necessary knowledge that is neither the logical
unpacking ofconcepts nor the record ofperceptual experience. Inthe Critique ofPure
Reason, mathematics is one of the transcendental instruments the human mind uses to
apprehend nature, and by apprehending to construct it under the universal and
necessary lawsofNewtonian mechanics.
The doctrines of these three great philosophers have been typical of the
Western tradition as a whole, up to the present century. The mathematician is both a
visionarywho testifies to an intelligibledomain, andademiurge whomakes worlds out
ofnumbers, figures,and their offspring.Mathematics isboth aknowing and amaking;
it leads the philosopher to both metaphysics and science. But in our era, philosophers
have lost sight ofthe full significance ofour own tradition. Contemporary philosophy
ofmathematics, whichtendstousethe instrumentoflogicalanalysistotheexclusion of
the methods ofhistory initsstudy,focusesonthe constructive, scientific dimensions of
mathematics. This approach is surely sound, but ought to be augmented by a more
historical and reflective approach, that will do justice to the other, metaphysical and
visionary, dimensions ofmathematicalthought.
Thisgeneral insightgainsurgency andconcreteness when brought intorelation
with the current state ofaffairs in philosophy of mathematics, where a crisis has been
emerging in the last decade. Since the days ofFrege and Russell, philosophers have
been examining mathematical rationality,methods ofdiscovery and methods ofproof,
anddeep questions aboutthe nature ofmathematical objects, by means offormal logic.
That is,they haveusedfirstandsecond orderpredicate logicandthetheory ofrecursive
functions as tools for analyzing mathematical knowledge. While this approach has
contributed immensely to reflection on mathematics in the twentieth century, modem
philosophy of mathematics has left certain questions unanswered and has only
incompletely addressed others.
xii INTRODUCTION
The philosophy of mathematics, tied to logic so closely in this way, has
exhausted many of its original possibilities and failed to attract renewed interest and
support from working mathematicians. This situation characterized philosophy of
science thirty-five years ago, when Thomas Kuhn wrote his ground-breaking The
Structure ofScientific Revolutions (1962), abookwhich insistedthatthe analytic tool of
logic must be supplemented by a philosophical study ofthe history ofscience. Kuhn
argued that philosophers must weigh their pronouncements about scientific rationality
against the evidence ofhistory, that is,against the actual practice ofscientists past and
present. He pointed out that a philosophy ofscience thus renovated should also prove
more interesting and more helpful to scientists.CriticismandtheGrowth ofKnowledge
(1970) edited by Imre Lakatos andAlan Musgrave, TheStructure ofScientific Theories
(1977) edited by Patrick Suppe, and Scientific Revolutions (1981) edited by Ian
Hacking, are three collections ofessays that chart the response to Kuhn's book during
the past decades.
Philosophy of mathematics has been slow to draw the analogy from the
Kuhnian sea-change in philosophy of science, but during the last decade, a growing
number of younger philosophers of mathematics have turned their attention to the
history ofmathematics and tried to make use of it in their investigations. The most
exciting ofthese concern howmathematical discovery takes place,how new discoveries
are structured and integrated into existing knowledge, and what light these processes
shed onthe existence andapplicability ofmathematical objects.
The collections of essays that best document this recent philosophical
development are History and Philosophy ofModern Mathematics (1988) edited by
Philip Kitcher and William Aspray; The Probabilistic Revolution (1987) edited by
Krueger et al.; Revolutions in Mathematics (1992) edited by Donald Gillies; and The
Space ofMathematics: Philosophical. Epistemological. and Historical Explorations
(1992) edited by Echeverria et al. These books show clearly that the younger North
American and British philosophers ofmathematics at itsforefront take their inspiration
from the primary archival and editorial work of European scholars who are
rediscovering, translating, and editing forgotten or overlooked mathematical
manuscripts, journals, and books. Impressive as these collections are, each one has its
limitations. The collections edited by Aspray and Kitcher and byEcheverriaetal. bring
together essays by philosophers and historians, but these essays sit, as it were, side by
side. The sheer fact ofjuxtapositiondoesnot amount to substantive interaction between
representatives ofthe two disciplines.The collection edited by Krueger et al., while an
inspiring example ofinterchange among Europeans and Americans, is focussed on the
single domainofprobability theory. Perhaps themost satisfactory synthesis isproduced
in the Gillies volume, though it tends to subordinate historical to philosophical
considerations.
The present collection of essays arose from the editors' conviction that
important issues in the philosophy of mathematics could be illuminated by a more
highly structured dialogue between philosophers and historians, perhaps inthe form of
case studies drawn from the history ofmathematics answering to philosophical essays.
Wewanted to seephilosophers ofmathematics makedeeper andmore systematic useof