Hilbert’s Programs and Beyond

452 Pages·2013·2.102 MB·English

by  Wilfried Sieg

H ’ P
ILBERT S ROGRAMS
AND
B
EYOND
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Wilfried Sieg
Hilbert’s Programs
and Beyond
3
3
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LibraryofCongressCataloging-in-PublicationData
Sieg,Wilfried,1945-
Hilbert’sprogramsandbeyond/WilfriedSieg.
p. cm.
ISBN978-0-19-537222-9(hardback:alk.paper)—ISBN978-0-19-970715-7(e-book)
1.Mathematics—Philosophy.2.Hilbert,David,1862-1943.I.Title.
QA8.4.S5452013
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2012022356
9780195372229
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Preface
As the plural in Hilbert’s Programs and Beyond suggests, the essays
I selected for this volume go beyond the analysis of the finitist consistency
program Hilbert pursued during the 1920s. They show how Hilbert, since
theearly1890s,attemptedindifferentwaystocometogripswithachanged
andchangingsubject,i.e.,withmathematicsinitsmodernabstractformas
it began to evolve during the 19th century. They also argue that Hilbert’s
questcanbecontinuedbybroadeninghismethodologicalperspective,refin-
ing the mathematical investigations, and expanding proof theoretic tools.
ThesearchforfinitistconsistencyproofsoftheHilbertProgramisdiscussed
in detail, but it is seen in an intellectual and scientific context with deep
roots in the past and with remarkable future impact transcending logic
andthefoundationsofmathematics.
The essays have been published over a period of 25 years and are pre-
sented here in a uniform format and with a common bibliography.1 I cor-
rected typos silently and addressed, in explicit notes, the real mistakes I
amawareof.Someoftheessaysoverlap,astheywerewrittenfordifferent
audiencesandweretobeself-contained.Myphilosophicalviews,aswellas
my understanding of mathematical and historical facts, have evolved: the
Brief Guides that precede each of the four groups of essays indicate con-
nections and changes; the Introduction to the volume articulates a broad
perspectiveforHilbert’sPrograms.Iamawareoftheshortcomingsimplicit
inthefactthattheessayswerewrittenoveraquartercentury;butIthink
thattheyilluminate,coherently,centralthemesinthephilosophyofmath-
ematics, significant aspects of the history of mathematical logic, and the
foundationalgoalsofprooftheory.
I feel fortunate to have been drawn to the mathematical develop-
ments, proof theoretic investigations, and methodological reflections that
are richly represented not only in Hilbert’s papers and lectures, but
also in the essays of Paul Bernays, the two volumes of Grundlagen der
Mathematik,andtheworkinthetraditionoftheHilbertSchool.Threeuni-
versityteachersdeeplyinfluencedmymathematicalandlogicaleducation,
namely, Karl Peter Grotemeyer in Berlin, Dieter Rödding in Münster, and
SolomonFefermanatStanford.Grotemeyerintroducedmetostructuralist
1Theyarereprintedherewiththecontractualorexplicitpermissionofthecopyrighthold-
ers;thetwopapersthatwereco-authoredwithCharlesParsons,respectivelyDirkSchlimm,
appearalsowiththeirpermissions.
vii
viii Hilbert’sProgramsandBeyond
mathematics, and Rödding to the craft of logical work. Feferman, most
importantly,guidedmeinthecrucialsteptoindependentresearchinproof
theory; he has been mentor, collaborator, and friend. At the institutions
whereIhaveworked,anumberofcolleaguesinfluencedmedeeply:Dagfinn
Føllesdal,GeorgKreisel,andPatrickSuppesatStanfordUniversity;Isaac
Levi, Sidney Morgenbesser, and Charles Parsons at Columbia University;
Clark Glymour, Dana Scott, and Teddy Seidenfeld at Carnegie Mellon
University. Three colleagues, who do not fall fully into these institutional
categories, have stimulated my work for a long time and continue to do
so. They are Martin Davis, Howard Stein, and especially William Tait.
Iamalsoindebtedtomanyothercolleagues,friends,andstudentsforkeen
insightsandobservations;theyarementionedintheindividualessays.
Early in our academic careers, Wilfried Buchholz, Wolfram Pohlers,
and I did complementary proof theoretic work on theories of general-
ized inductive definitions. That work, together with papers by and with
Feferman, was published as volume 897 of the Springer Lecture Notes
in Mathematics; it was also the starting-point for my philosophical and
historical explorations. The latter have profited, for much longer than a
decadenow,fromworkingwithWilliamEwald,MichaelHallett,andUlrich
Majer on the edition of Hilbert’s unpublished lecture notes on logic and
arithmetic. Those notes have opened a new view on the development of
proof theory and mathematical logic that is reflected in my essays pub-
lished after 1999. I am grateful for the permission to use source material
fromtheNiedersächsischeStaats-undUniversitätsbibliothekinGöttingen
pertaining not only to Hilbert, but also to Dedekind; Dr. Helmut Rohlfing
hasbeenextremelyhelpful.
Academicworkisdoneininstitutionalsettings,andIhavehadthegood
fortune to work at three great universities. The departments of philoso-
phy at Stanford and Columbia provided rich, stimulating environments.
SpecialappreciationgoestoCarnegieMellon,whichhasbecome“myplace”.
Whileinterdisciplinaryworkisimportantatmanyuniversities,atCarnegie
Mellonitissoimportantastobeordinary,i.e.,itispartoftheregularorder
ofthings.Itreasurethataspectoftheuniversityandthriveparticipatingin
itsvigorousintellectuallife.Morespecificthanksareowedtoanotherinsti-
tution,theSwedishCollegiumforAdvancedStudy(SCAS)inUppsala;my
stayasaFellowatthatscholars’utopiaallowedmetofocusonthisvolume
in spring and early summer of 2009. Since then I have written two addi-
tional essays, (II.3) and (III.3), as well as the long Introduction. Last, but
certainly not least, my thanks go to Dawn McLaughlin. My essays would
hardly have seen the light of day in this collected form, had it not been
for her excellent editorial and TEX- nical support, patiently sustained over
almostfiveyears.
WilfriedSieg
July2012
Pittsburgh
Published with Permission
I.1 RichardDedekind.Wikimedia.“RichardDedekind,”
http://commons.wikimedia.org/wiki/File:Dedekind.jpeg.
I.2 LeopoldKronecker.Wikimedia.“LeopoldKronecker,”
http://commons.wikimedia.org/wiki/File:Leopold_Kronecker.jpg.
I.3 David Hilbert. Courtesy of Niedersächsische Staats- und Universi-
tätsbibliothek Goettingen, Abteilung Spezialsammlungen und
Bestandserhaltung.
II.1 David Hilbert. Attributed to Elisabeth Reidemeister. Courtesy of the
Emilio Segrè Visual Archives, Nina Courant Collection, American
InstituteofPhysics.
II.2 Johan von Neumann. Courtesy of Los Alamos National Laboratory
Archives. Unless otherwise indicated, this information has been
authored by an employee or employees of the Los Alamos National
Security, LLC (LANS), operator of the Los Alamos National Labo-
ratory under Contract No. DE-AC52-06NA25396 with the U.S.
Department of Energy. The U.S. Government has rights to use,
reproduce, and distribute this information. The public may copy and
use this information without charge, provided that this Notice and
anystatementofauthorshiparereproducedonallcopies.Neitherthe
Government nor LANS makes any warranty, express or implied, or
assumesanyliabilityorresponsibilityfortheuseofthisinformation.
II.3 Kurt Gödel. Photographer unknown. From The Shelby White and
Leon Levy Archives Center, Institute for Advanced Study, Princeton,
NJ,USA.
II.4 GerhardGentzen.CourtesyofEckartMenzler.
III.1 PaulBernays.CourtesyofWilfriedSieg.
ix