Patterns of Change: Linguistic Innovations in the Development of Classical Mathematics

277 Pages·2008·3.171 MB·English

by  Ladislav Kvasz

Science Networks . Historical Studies
Founded by Erwin Hiebert and Hans Wußing
Volume 36
Edited by Eberhard Knobloch, Erhard Scholz and Helge Kragh
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I. Grattan-Guinness, London V.P. Vizgin, Moskva
J. Gray, Milton Keynes
Ladislav Kvasz
Patterns of Change
Linguistic Innovations in the Development
of Classical Mathematics
Birkhäuser
Basel · Boston · Berlin
Ladislav Kvasz
Comenius University
Faculty of Mathematics, Physics and Informatics
Mlynska dolina
842 48 Bratislava
Slovakia
and
Charles University
Faculty of Education
M. D. Rettigove 4
116 39 Praha
Czech Republic
e-mail: [email protected]
2000 Mathematical Subject Classification: 01-02, 00A30, 00A35, 97C30, 97D20
Library of Congress Control Number: 2008926677
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ISBN 978-3-7643-8839-3 Birkhäuser Verlag AG, Basel-Boston - Berlin
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ForMilena
Acknowledgements
Thepresentbookistheresultofmyresearchinthefieldofhistoryand
philosophy of mathematics that spanned more than twenty-five years.
During this time I was helped by contacts with many experts in the
field in my home country, Slovakia, as well as abroad. I would like to
expressmyappreciationandgratitudetoallofthem–myteachers,col-
leagues,studentsorsimplyfriends–whoduringthisratherlongperiod
of time encouraged and supported my work. In Slovakia these were
myteachers,friendsandlatercolleagues: PeterCvik,Vladim´ırBurjan,
Milan Hejny´, Egon Ga´l, Pavol Zlatosˇ, and Peter Buga´r. With many
of them I met at different private seminars during the communist era.
Somebecameuniversityprofessors,othersneverenteredacademia,but
they all were part of an unforgettable atmosphere of parallel culture in
the1980s.
I feel a special debt to Professor Dimitri D. Sokoloff, whom I met
duringmystaysattheMoscowStateUniversityin1989–1991. Thanks
to this encounter I enjoyed a glimpse into the scientific culture of the
circleroundJakovB.Zeldovich. ThankstoaHerderscholarshipfrom
the Stiftung F.V.S. zu HamburgI had the privilegeof spendingthe aca-
demic year 1993/1994 at the University of Vienna, discussing with
Professor Friedrich Stadler, Michael Sto¨ltzner, and Martin Dangl the
philosophy of Ludwig Wittgenstein and the Vienna circle. Thanks to
a Masaryk scholarship from the Foreign and Commonwealth Office I
spent the academic year 1994/1995 at King’s College London study-
ing the work of Karl Popper and Imre Lakatos with Professor Donald
Gillies. In 1998 I was awarded a Fulbright scholarship and spent a
semesterattheUniversityofCaliforniaatBerkeleystudyingwithPro-
fessor Paolo Moncosu the role of visual representations in mathemat-
ics. And finally, thanks to a Humboldt scholarship, I spent two fruit-
VI Acknowledgements
ful years at the Technische Universita¨t Berlin working with Professor
EberhardKnobloch. Theyallinfluencedthepictureofthedevelopment
ofmathematicspresentedinthisbook.
IwouldespeciallyliketothankProfessorsDonaldGilliesandGra-
ham D. Littler for their help with the final version of the manuscript,
to Donald Gillies for writing the preface and to Professor Eberhard
Knoblochforhissupportofthepublicationofthebook.
Some chapters of the present book are based on material published
earlier: History of Geometry and the Development of the Form of its
Language. Synthese, 116 (1998), 141–186; On classification of scien-
tificrevolutions. JournalforGeneralPhilosophyofScience,30(1999),
201–232; Changes of Language in the Development of Mathematics.
Philosophiamathematica,8(2000),47–83;Lakatos’MethodologyBe-
tween Logic and Dialectic. In: Appraising Lakatos. Eds. G. Kampis,
L. Kvasz and M. Sto¨ltzner, Kluwer (2002), pp. 211–241. Similari-
tiesanddifferencesbetweenthedevelopmentofgeometryandofalge-
bra. In: Mathematical Reasoning and Heuristics, (C. Cellucci and D.
Gillies, eds.), King’s College Publications, London 2005, pp. 25–47;
and History of Algebra and the Development of the Form of its Lan-
guage. Philosophia Mathematica, 14 (2006), pp. 287–317. I extend
thanks to Kluwer Publishers, Philosophia Mathematica, and Oxford
UniversityPressforpermissiontousethesepassages.
The financial support of a grant from the Slovak Scientific Grant
Agency VEGA 1/3621/06 Historical and philosophical aspects of ex-
actdisciplinesisherewithgratefullyacknowledged.
Preface
Kvasz’sbookisacontributiontothehistoryandphilosophyofmathe-
matics, or, as one might say, the historical approach to the philosophy
of mathematics. This approach is for mathematics what the history
and philosophy of science is for science. Yet the historical approach
to the philosophy of science appeared much earlier than the historical
approach to the philosophy of mathematics. The first significant work
inthehistoryandphilosophyofscienceisperhapsWilliamWhewell’s
PhilosophyoftheInductiveSciences,foundedupontheirHistory. This
was originally published in 1840, a second, enlarged edition appeared
in 1847, and the third edition appeared as three separate works pub-
lished between 1858 and 1860. Ernst Mach’s The Science of Mechan-
ics: ACriticalandHistoricalAccountofItsDevelopmentiscertainlya
work of history and philosophy of science. It first appeared in 1883,
and had six further editions in Mach’s lifetime (1888, 1897, 1901,
1904,1908,and1912). Duhem’sAimandStructureofPhysicalTheory
appearedin1906andhadasecondenlargededitionin1914. Sowecan
say that history and philosophy of science was a well-established field
th th
bytheendofthe19 andthebeginningofthe20 century.
By contrast the first significant work in the history and philosophy
of mathematics is Lakatos’s Proofs and Refutations, which was pub-
lishedasaseriesofpapersintheyears1963and1964. Giventhislate
appearance of history and philosophy of mathematics relative to his-
toryandphilosophyofscience,wewouldexpecttheearlydevelopment
of history and philosophy of mathematics to be strongly influenced by
ideaswhichhadbeenformulatedanddiscussedbythoseworkinginthe
VIII Preface
historyandphilosophyofscience. Thisprovestobethecase. Lakatos’s
own pioneering work was, as the title indicated, developed from Pop-
per’s model of conjectures and refutations which Popper had devised
toexplainthegrowthofscience.
In1992,Ieditedacollectionofpaperswiththegeneraltitle: Revo-
lutions in Mathematics. Once again the title showed clearly that ideas
drawn from the history and philosophy of science were being applied
to mathematics. The publication of Kuhn’s 1962 The Structure of Sci-
entificRevolutionsledtodebatesaboutwhetherrevolutionsoccurredin
science and, if so, what was their nature. The 1992 collection carried
overthisdebatetomathematics.
Kvasz’s book on history and philosophy of mathematics breaks to
someextentwiththistraditionofimportingideasfromthehistoryand
philosophy of science into mathematics. This is because he adopts a
linguisticapproach. Kvasz’sidea(p. 6)1istointerpretthedevelopment
of mathematics as a sequence of linguistic innovations. Kvasz points
out (p. 7) several advantages of this approach. These include the fact
that languages have many objective aspects that can easily be studied,
andsoaremoreaccessibletoanalysisthan,forexample, heuristics,or
psychologicalactsofdiscovery.
As a result of his linguistic approach, Kvasz draws more on ideas
from general analytic philosophy than from philosophy of science.
More specifically he makes use of the classic works of Frege and the
early Wittgenstein. However, Kvasz develops the ideas of Frege and
the early Wittgenstein in a number of novel ways. Perhaps most im-
portantlyheintroducesahistoricaldimensiontothestudyoflanguage.
Both Frege and Wittgenstein treat language as timeless, but Kvasz, by
contrast,focussesonthehistoricalchangesbywhichanolderlanguage
candevelopintoastronger,richernewlanguagewhichhasgreaterex-
pressivepower.
Perhaps,however,Kvaszhasnotbrokenawaycompletelyfromphi-
losophyof science, because it is worth notingthat Kuhn too adopteda
linguistic approach in his later period. This is shown in the 2000 book
TheRoadSinceStructurewhichcontainsessaysbyKuhnfromthepe-
riod1970–1993. Onp. 57ofthisbook,Kuhngoesasfarastosay:
1 PagereferencesontheirownaretoKvasz’sbookofwhichthisisthepreface.