Introduction to Mathematical Philosophy PDF
Preview Introduction to Mathematical Philosophy
Introduction to Mathematical Philosophy by BertrandRussell Originallypublishedby GeorgeAllen&Unwin,Ltd.,London.May. OnlineCorrectedEditionversion.(February,), basedonthe“secondedition”(secondprinting)ofApril ,incorporatingadditionalcorrections, markedingreen. [Russell’sblurbfromtheoriginaldustcover:] Thisbookisintendedforthosewhohavenoprevi- ousacquaintancewiththetopicsofwhichittreats, andnomoreknowledgeofmathematicsthancan beacquiredataprimaryschoolorevenatEton. It setsforthinelementaryformthelogicaldefinition ofnumber,theanalysisofthenotionoforder,the moderndoctrineoftheinfinite,andthetheoryof descriptionsandclassesassymbolicfictions. The morecontroversialanduncertainaspectsofthesub- jectaresubordinatedtothosewhichcanbynowbe regarded as acquired scientific knowledge. These are explained without the use of symbols, but in suchawayastogivereadersageneralunderstand- ingofthemethodsandpurposesofmathematical logic,which,itishoped,willbeofinterestnotonly tothosewhowishtoproceedtoamoreseriousstudy ofthesubject,butalsotothatwidercirclewhofeela desiretoknowthebearingsofthisimportantmod- ernscience. Contents Contents . . . . . . . . . . . . . . . . . . . . . iv Preface . . . . . . . . . . . . . . . . . . . . . . vi Editor’sNote . . . . . . . . . . . . . . . . . . . ix I. TheSeriesofNaturalNumbers . . . II. DefinitionofNumber . . . . . . . . . III. FinitudeandMathematicalInduction IV. TheDefinitionofOrder . . . . . . . V. KindsofRelations . . . . . . . . . . . VI. SimilarityofRelations . . . . . . . . VII. Rational,Real,andComplexNumbers VIII. InfiniteCardinalNumbers . . . . . . IX. InfiniteSeriesandOrdinals . . . . . X. LimitsandContinuity . . . . . . . . XI. LimitsandContinuityofFunctions . XII. SelectionsandtheMultiplicativeAx- iom . . . . . . . . . . . . . . . . . . . . iv XIII. The Axiom of Infinity and Logical Types . . . . . . . . . . . . . . . . . . XIV. IncompatibilityandtheTheoryofDe- duction . . . . . . . . . . . . . . . . . XV. PropositionalFunctions . . . . . . . XVI. Descriptions . . . . . . . . . . . . . . XVII. Classes . . . . . . . . . . . . . . . . . XVIII. MathematicsandLogic . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . Appendix:ChangestoOnlineEdition . . . . Preface Thisbookisintendedessentiallyasan“Introduc- v tion,” and does not aim at giving an exhaustive discussionoftheproblemswithwhichitdeals. It seemeddesirabletosetforthcertainresults,hith- ertoonlyavailabletothosewhohavemasteredlog- icalsymbolism,inaformofferingtheminimumof difficulty to the beginner. The utmost endeavour hasbeenmadetoavoiddogmatismonsuchques- tions as are still open to serious doubt, and this endeavourhastosomeextentdominatedthechoice oftopicsconsidered. Thebeginningsofmathemati- callogicarelessdefinitelyknownthanitslaterpor- tions,butareofatleastequalphilosophicalinterest. Muchofwhatissetforthinthefollowingchapters is not properly to be called “philosophy,” though themattersconcernedwereincludedinphilosophy solongasnosatisfactoryscienceofthemexisted. Thenatureofinfinityandcontinuity,forexample, belongedinformerdaystophilosophy,butbelongs nowtomathematics. Mathematicalphilosophy, in thestrictsense,cannot,perhaps,beheldtoinclude suchdefinitescientificresultsashavebeenobtained inthisregion;thephilosophyofmathematicswill naturallybeexpectedtodealwithquestionsonthe frontierofknowledge,astowhichcomparativecer- taintyisnotyetattained. Butspeculationonsuch questionsishardlylikelytobefruitfulunlessthe morescientificpartsoftheprinciplesofmathemat- icsareknown. Abookdealingwiththosepartsmay, therefore,claimtobeanintroductiontomathemati- calphilosophy,thoughitcanhardlyclaim,except where it steps outside its province, to be actually dealing with a part of philosophy. It does deal, | however,withabodyofknowledgewhich,tothose vi who accept it, appears to invalidate much tradi- tionalphilosophy,andevenagooddealofwhatis currentinthepresentday. Inthisway,aswellasby itsbearingonstillunsolvedproblems,mathemati- callogicisrelevanttophilosophy.Forthisreason,as wellasonaccountoftheintrinsicimportanceofthe subject,somepurposemaybeservedbyasuccinct accountofthemainresultsofmathematicallogicin aformrequiringneitheraknowledgeofmathemat- ics nor an aptitude for mathematical symbolism. Here, however, as elsewhere, the method is more importantthantheresults,fromthepointofview offurtherresearch;andthemethodcannotwellbe explainedwithintheframeworkofsuchabookas thefollowing. Itistobehopedthatsomereaders maybesufficientlyinterestedtoadvancetoastudy ofthemethodbywhichmathematicallogiccanbe madehelpfulininvestigatingthetraditionalprob- lemsofphilosophy. Butthatisatopicwithwhich thefollowingpageshavenotattemptedtodeal. BERTRANDRUSSELL. Editor’sNote [The note below was written by J. H. Muirhead, vii LL.D.,editoroftheLibraryofPhilosophyseriesin whichIntroductiontoMathematicalPhilosophywas originallypublished.] Thosewho,relyingonthedistinctionbetweenMath- ematicalPhilosophyandthePhilosophyofMath- ematics,thinkthatthisbookisoutofplaceinthe present Library, may be referred to what the au- thorhimselfsaysonthisheadinthePreface. Itis notnecessarytoagreewithwhathetheresuggests as to the readjustment of the field of philosophy bythetransferencefromittomathematicsofsuch problems as those of class, continuity, infinity, in ordertoperceivethebearingofthedefinitionsand discussionsthatfollowontheworkof“traditional philosophy.” Ifphilosopherscannotconsenttorel- egatethecriticismofthesecategoriestoanyofthe specialsciences,itisessential,atanyrate,thatthey shouldknowtheprecisemeaningthatthescienceof mathematics,inwhichtheseconceptsplaysolargea part,assignstothem. If,ontheotherhand,therebe mathematicianstowhomthesedefinitionsanddis- cussionsseemtobeanelaborationandcomplication ofthesimple,itmaybewelltoremindthemfrom thesideofphilosophythathere,aselsewhere,ap- parentsimplicitymayconcealacomplexitywhich itisthebusinessofsomebody,whetherphilosopher ormathematician,or,liketheauthorofthisvolume, bothinone,tounravel.
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