The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940

631 Pages·2011·2.94 MB·English

by  Paolo Mancosu

The Adventure of Reason
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The Adventure of
Reason
Interplay between Philosophy of
Mathematics and Mathematical Logic,
1900–1940
PAOLO MANCOSU
1
3
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PREFACE
In the last ten years I have extensively investigated a variety of topics in the history of
logic,thefoundationsofmathematics,andthephilosophyoflogic.Theperiodofinvesti-
gationgoesroughlyfromthebeginningofthetwentiethcenturytotheearly1940s.This
period corresponds to the ‘golden age’ of mathematical logic including the discovery
of Go¨del’s incompleteness theorems and the development of Tarskian semantics, to
mention perhaps the two most well-known achievements. These and other results in
mathematicallogicbothinfluencedandwereinfluencedbycontemporarydevelopments
inthephilosophyofmathematics,thephilosophyoflogic,andgeneralphilosophy.While
thegeneraloutlineofsuchinteractionsarebynowwellknown,amorenuancedpicture
ofsuchdevelopmentscanbegatheredbyexploiting,inadditiontothepublishedsources,
therichcollectionofmaterialsfoundinseveralarchives.Thepapersgatheredhereexploit
untapped archival sources and thus make available a wealth of new information that
deepensinsignificantwaysourunderstandingoftheabove-mentionedareas.Inpursuing
thisresearch,Iworkedatlengthin,amongothers,theHilbertarchive(Go¨ttingen),the
BernaysandWeylarchives(Zurich),theCarnaparchive(PittsburghandKonstanz),the
Neurath archive (Haarlem and Konstanz), the Tarski archive (Berkeley), the Woodger
archive (London), the Behmann archive (Erlangen, now Berlin), the Mahnke archive
(Marburg),theKaufmannarchive(Konstanz),andtheQuinearchive(Harvard).
Whiletheresultsofthisresearchwerepublishedseparately,Iampleasednowtobring
these papers together in a single volume. The essays are closely linked by the fact that
the subject matter is homogeneous and were written with a single major aim, namely
that of reaching a deeper understanding of the interaction between developments in
mathematicallogicandthefoundationsofmathematicsandlogicfrom1900to1940.
To the discerning reader, the developments described in these pages will justify the
title of this book TheAdventureofReason. I borrowed this beautiful image from a letter
from Becker to Mahnke, which also gives the title to Chapter 12. The adventure of
reason reflects itself in the ambitious mathematical and philosophical aims that drive
theconstitutionofmathematicallogic,thephilosophicalandmathematicalaimsthatit
is made to serve, and the unexpected surprises lying in wait. That’s the way I like to
think of the ambitious goals by Russell and Whitehead, of Hilbert’s program, and of
Husserl’sphenomenologicalambitions,tonameonlyafewmajordevelopmentstreated
in the book. But just like Columbus had set sail to reach the Indies and found other
unexpected—notlessrichandexciting—lands,inthesamewaytheadventureofreason
inlogicandthefoundationsofmathematicsledtosurprisingnewfindsthatdramatically
reshapedtheoriginalgoals.
Theessayscontainedinthisvolume,allpreviouslypublishedexceptChapters17and
18, fall naturally into the five parts that make up the book. The first part contains a
Preface vii
lengthyaccountofthehistoryofmathematicallogicfromtheearlypartofthetwentieth
century to the middle of the 1930s. On account of its length, I recommend using this
chapter as needed while reading the other chapters in the book. Part II contains seven
chaptersthatdealwithfoundationalissuesinmathematics,namelytheRussellianinflu-
enceinGo¨ttingen,constructivism,andGo¨del’sincompletenesstheorems.Thethirdpart
detailsadramaticmomentinthedevelopmentofphenomenologythroughthethreefig-
ureswho,besidesHusserl,hadthebestunderstandingofthenewdevelopmentsinlogic,
mathematics,andphysics.IamreferringtoHermannWeyl,OskarBecker,andDietrich
Mahnke.Inspiteoftheirdifferentsolutionstotheproblemoftherelationshipbetween
phenomenologyandtheexactsciences,theyallagreedontheunprecedentedstrainput
by the new scientific developments on the Husserlian phenomenological grounding of
the sciences described in Ideas. Part IV focuses on the discussion of nominalism as a
foundation for mathematics and science in the early stages of such discussions, that is
duringthemeetingsamongCarnap,Quine,andTarskiinHarvardin1940-41and,more
broadly in Quine’s and Tarski’s trajectory on such issues. Finally, part V both discusses
theimpactofTarski’stheoryoftruthandlogicalconsequenceinthephilosophicalmilieu
oftheViennaCircleandtreatsanumberofissuesrelatedtothepropercharacterization
oflogicalconsequenceinTarskiduringthatperiod.Inanarchivalappendix,Chapter18,
IpublishahithertounpublishedlecturebyTarski,whichcontainsimportantinformation
onthenotionoflogicalconsequenceandonthedevelopmentofcentralnotionsinthe
semantics of theories formulated in higher order logic. I would like to thank professor
JanTarskithesonandManagerofAlfredTarski’sliteraryestate,forhavinggenerously
grantedmepermissiontopublishthistext.
Each part is prefaced by a summary of the contents of the articles and in addition
contains a bibliographical update. These updates do not strive at completeness but
Ihopetheyarecomprehensiveenoughtoindicatethemajornewcontributionsthathave
appeared since the original publication of the articles. As a rule I have tried to repro-
duce the articles without major modifications. When I felt, as in the case of Chapter2
and Chapter 16, that a substantial comment needed to be added, I have done so by
appendinganAddendumtothosearticles.Other,veryminor,modificationsareindicated
by an insertion in the text within double brackets, as in [[modified from the original
introduction]].InafewcasesIhaveremovedinaccuraciesfromtheoriginallypublished
version and replaced reference to my published articles by the corresponding chapters
ofthisbookwithoutflaggingthisexplicitly.Thearticlesarereproducedintheirentirety
with the only exception being Chapter 11 from which the first four sections have been
removed.ThesesectionsareidenticalincontenttothefirstfoursectionsofChapter10,
thus no loss in content ensues. The details of the original publications with permission
to republish are contained in the Acknowledgments at the beginning of the book. The
original acknowledgements, if any, appear as an uncued footnote at the bottom of the
firstpageoftherelevantchapter.Thus,Iwillhereacknowledgeonlythosescholarswho
helpedmesinceconceivingthisbook.IwouldliketothankPeterMomtchiloff,theeditor
in Philosophy at OUP, Oxford, for having encouraged me to embark on this project
starting with a conversation in Vancouver in November 2006 and for having followed
viii Preface
the project through with the Press. I would like to thank Solomon Feferman, Marcus
Giaquinto, Donald Gillies, and Daniel Isaacson for having been so supportive of the
project from its very inception and for having provided many excellent suggestions on
avarietyofmatters.TomRyckmanandRichardZachhavehelpedmeimmenselywith
comments on the introductions and the bibliographical updates. Fabrizio Cariani has
takenon,onceagain,theungratefultaskofresettingtheoriginalarticlesintoabeautiful
LATEX format. His work was partially supported by a COR grant at U.C. Berkeley; but
only partially and I am very grateful for his unfailing dedication to the project. In this
connection,IwouldalsoliketothankMarcoDettweilerandVolkerPeckhausforhaving
provided us with the source files of Chapters 11 and 12 which saved Fabrizio from the
drudgeryofresettingthepapersinLATEX.Duringthelateststagesofthepreparationof
thebookIwasalsofortunatetoreceivethehelpofRebeccaMillsopwhosavedmefrom
thepainfultaskofinsertinganalmostuncountablenumberoftinycorrectionsintothe
source files. It is also a pleasure to acknowledge my co-authors for Chapter 1 (Richard
ZachandCalixtoBadesa),Chapter6(MathieuMarion),andChapters10and11(Thomas
Ryckman), for having allowed me to reprint our joint papers and, much more, for the
intellectualstimulusandpleasurewesharedwhilewritingthem.
Work on the final stages of the book has been made possible by a Guggenheim
FellowshipandbyamembershipattheInstituteforAdvancedStudyinPrincetoninthe
SpringTerm2009.Icannotthinkofbetterconditionsforbringingabooktocompletion
thantheonesIwasfortunatetohave.IamgratefultotheGuggenheimFoundationand
totheInstituteforAdvancedStudyfortheirgeneroussupportandtothelatterforhaving
providedsuchidealworkingconditions.
ThisbookisdedicatedtomywifeElenaRusso,forhavingbeenclosetome,through-
outthecompositionofthisbook,inspirit,ifnotalwaysinperson.
Princeton,
InstituteforAdvancedStudy,
March27,2009
ACKNOWLEDGMENTS
Chapter 1 was originally published in L. Haaparanta (ed.), The Development of Modern
Logic(Oxford:OxfordUniversityPress,2009),pp.318–470.
Chapter2wasoriginallypublishedinP.Mancosu(ed.),FromBrouwertoHilbert(Oxford:
OxfordUniversityPress,1998),pp.149–88.
Chapter 3 was originally published in The Bulletin of Symbolic Logic, 5(3) (1999),
pp.303–30.ReprintedwithkindpermissionoftheAssociationofSymbolicLogic.
Chapter 4 was originally published in Synthese, 137 (2003), pp. 59–101. Reprinted with
kind permission from Springer Science + Business Media, ©2003 Kluwer Academic
Publishers.
Chapter 5 was originally published in Wilfried Sieg, Richard Sommer, and Carolyn
Talcott (eds.), Reflections on the Foundations of Mathematics: Essays in Honor of Solomon
Feferman,AssociationforSymbolicLogic(2002),pp.349–71.Reprintedbykindpermis-
sionoftheAssociationofSymbolicLogic.
Chapter6wasoriginallypublishedinFriedrichStadler(ed.),TheViennaCircleandLogical
Empiricism:Re-EvaluationandFuturePerspectives(Dordrecht:Kluwer,2003),pp.171–88.
Chapter 7 was originally published in History and Philosophy of Logic, 20 (1999),
pp.33–45.ReprintedwithkindpermissionofTaylorandFrancis.
Chapter 8 was originally published in Notre Dame Journal of Formal Logic, 45(2) (2004),
pp. 109–25. ©2004, Yale University. Reprinted by permission of the publisher, Duke
UniversityPress.
Chapter9wasoriginallypublishedinP.Mancosu(ed.),FromBrouwertoHilbert(Oxford:
OxfordUniversityPress,1998),pp.65–85.
Chapter 10 was originally published in Philosophia Mathematica, 10(3)(2002),
pp.130–202.
Chapter11wasoriginallypublishedinOskarBeckerunddiePhilosophiederMathematik,ed.
Volker Peckhaus (Munich: Wilhelm Fink Verlag, 2005) (Neuzeit & Gegenwart Philoso-
phie in Wissenschaft und Gesellschaft), pp. 153–228. Reprinted by kind permission of
FinkVerlag.
Chapter 12 was originally published in Oskar Becker und die Philosophie der Mathematik,
ed. Volker Peckhaus (Munich: Wilhelm Fink Verlag, 2005) (Neuzeit & Gegenwart:
PhilosophieinWissenschaftundGesellschaft),pp.229–43.Reprintedbykindpermission
ofFinkVerlag.
Chapter 13 was originally published in History and Philosophy of Logic, 26 (2005),
pp.327–57.ReprintedwithkindpermissionofTaylorandFrancis.