The evolution of Principia Mathematica: Bertrand Russell's manuscripts and notes for the second edition

419 Pages·2011·1.962 MB·English

by  Linsky, Bernard

THE EVOLUTION OF PRINCIPIA MATHEMATICA
BertrandRussell’sManuscriptsandNotesfortheSecondEdition
Originally published in 1910, Principia Mathematica led to the development of
mathematical logic and computers and thus to information sciences. It became a
modelformodernanalyticphilosophyandremainsanimportantwork.
In the late 1960s, The Bertrand Russell Archives at McMaster University in
CanadaobtainedRussell’spapers,lettersandlibrary.Thesearchivescontainedthe
manuscripts for the new Introduction and three appendices that Russell added to
thesecondeditionin1925.Alsoincludedwasanothermanuscript,“TheHierarchy
of Propositions and Functions”, which was divided up and reused to create the
finalchangesforthesecondedition.Thesedocumentsprovidefascinatinginsight,
including Russell’s attempts to work out the theorems in the flawed Appendix B,
“On Induction”. An extensive introduction describes the stages of the manuscript
material on the way to print, and analyzes the proposed changes in the context of
thedevelopmentofsymboliclogicafter1910.
bernard linsky isaProfessorintheDepartmentofPhilosophyattheUniver-
sityofAlberta.

THE EVOLUTION OF
PRINCIPIA MATHEMATICA
Bertrand Russell’s Manuscripts and Notes
for the Second Edition
BERNARD LINSKY
UniversityofAlberta
cambridge university press
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Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Linsky,Bernard.
TheevolutionofPrincipiaMathematica:BertrandRussell’smanuscriptsandnotesforthesecondedition/
BernardLinsky.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-107-00327-9
1.Russell,Bertrand,1872–1970.Principlesofmathematics. 2.Mathematics–Philosophy. 3.Logic,
Symbolicandmathematical. I.Title.
QA9.R883L56 2011
510.1–dc22 2011006848
ISBN978-1-107-00327-9Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Contents
1 Introduction page1
1.1 Themanuscripts 9
1.2 Acknowledgements 10
2 Writingthesecondedition 13
2.1 ThelistofdefinitionsforCarnap 14
2.2 A.N.Whiteheadandthesecondedition 15
2.3 Hierarchyofpropositionsandfunctions 19
2.4 WorkingpapersforAppendixC 21
2.5 Amendedlistofpropositions 22
2.6 Correctionstothefirstedition 23
2.7 FrankRamseyandthesecondedition 30
2.8 Printingthesecondedition 36
3 Logicsincethefirstedition 39
3.1 CambridgeandDrWilliams’Library 41
3.2 Contributionssincethefirstedition 47
4 Notationandlogic 73
4.1 Primitivesymbolsandpunctuation 73
4.2 Propositionalfunctionsandlogicaltypes 83
4.3 Realandapparentvariables 88
4.4 Theaxiomofreducibility:∗12–Identity:∗13 91
4.5 Definitedescriptions:∗14 94
4.6 Classes:∗20 97
4.7 Generaltheoryofrelations:∗21 100
4.8 Logicofrelations:∗30–∗40andbeyond 102
5 Improvementsinthenewedition 108
5.1 Occurthroughitsvalues 112
v
vi Contents
5.2 Nologicalmatrixoftheformf!(φ!zˆ) 115
5.3 Variablesandquantifiers 122
5.4 TheShefferstroke 124
5.5 Reducibilityandextensionality 128
5.6 AppendixC 134
6 InductionandtypesinAppendixB 138
6.1 Inductionandinductiveclasses 138
6.2 TheprojectofAppendixB 146
6.3 Myhill’schallengeto∗89·24 148
6.4 TheerrorinAppendixB 151
6.5 Theproofof∗89·16 154
6.6 TypesinAppendixB 158
6.7 IdentityinAppendixB 163
6.8 The“rectification”ofinduction 166
6.9 SummaryofresultsaboutAppendixB 168
7 Thereceptionofthesecondedition 170
7.1 “Thefoundationsofmathematics” 170
7.2 Carnap’sAbrissderLogistik 178
7.3 HilbertandAckermann 180
7.4 Other“Germanauthorities” 181
7.5 Reviewsofthesecondedition 183
8 ThelistofdefinitionsforCarnap 189
8.1 Editorialnote 189
8.2 Thelist 189
Introductiontothesecondedition 214
Editorialnote 214
Themanuscript 215
AppendixA 260
Editoralnote 260
AppendixA:manuscript 260
AppendixB 279
Editorialnote 279
AppendixB:manuscript 279
AppendixC 291
Editorialnote 291
AppendixC:manuscript 291
Contents vii
Hierarchyofpropositionsandfunctions 301
Reconstructingtheoriginalmanuscript 301
TheremainingHPFleaves 303
Amendedlistofpropositions:notes 339
Editorialnote 339
Russell’snotes 339
Bibliography 396
BertrandRussell’spublishedworks 396
Unpublishedletters 397
References 398
Index 406

1
Introduction
ThesecondeditionofAlfredN.WhiteheadandBertrandRussell’sPrincipiaMath-
ematica was published by Cambridge University Press in three volumes between
1925and1927.Itconsistsofareprintofthefirstedition,whichappearedbetween
1910and1913,withtheadditionofanewIntroductionandthreeAppendices(A,
B, and C) written by Russell alone, and a List of Definitions.1 The new material
takesuponly66pages,yetitproposedradicalchangestothesystemofPrincipia
Mathematica,someofwhichrequirefundamentalrethinkingofthenatureoflogic.
What Russell oddly introduces as the “...most definite improvement resulting
from work in mathematical logic during the past fourteen years ...” is the pro-
posal to replace the familiar fundamental logical connectives “or” and “not” with
the single “Sheffer stroke”, “not-both.” This technically trivial change is in fact
carriedoutinarigorousfashionanddoesnotrequireanyrewritingofthebodyof
Principia Mathematica in order to be properly implemented. A second and gen-
uinelyfundamentalchangeistheadoptionof“extensionality”inthesecondedition.
Thisrequiresthatallpropositionalconnectivesaretobetruth-functional,andthat
co-extensive propositional functions, true of the same arguments, are identified.
Russellcharacterizesthisdoctrineastheresultoftwotheses,that:
...functionsofpropositionsarealwaystruth-functions,andthatafunctioncanonlyoccur
inapropositionthroughitsvalues.(PM,p.xiv)
What this change amounts to, and how it fits with the details of the various
traces of the non-extensional system of the first edition that are unaltered, such
as the definition of identity, will be discussed below. The third major change in
the second edition is the proposal to abandon the axiom of reducibility in the
development of mathematics that is the project of Principia Mathematica. The
1 Thenewmaterialwasalladdedtothefirstvolumeaspagesnumberedxiiitoxlvi,atthebeginning,following
thePrefacefromthefirstedition,andtheappendicesattheend,pages635to666attheendofthevolume.For
completebibliographicalinformationontheeditionsofPM,seeBlackwell&Ruja(1994,pp.19–25).
1