C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) L particular de solving with absolute truth.Hedidnotcontenthimself an of conception antics, he wanted to overcomephilosophicalidealismandwasagainstthe Moreover, theywereinasense motivated by those ideas.Buildingsem- withhisphilosophicalideasconcerninglogicandmathematics. nected the casewithsomePolishlogicians,e.g.,Stanisl The BertrandRussellResearchCentre,McMasterU. russell: ment oflogicandphilosophymathematics. or philosophers.Hisinvestigationswere his ownideaswithoutanycollaboration rational meta-mathematicswasnotdeve wassimilartothefateof conceptions way anominalisticinterpretationofthem asexperime theories other mathematical Hetreatedgeometry, of construction. matics, istheexpression being constructe claimed thattheobjectofdeductiv nominalism, whichfoundfullexpression concerning logicandmathematics.The The paperisdevotedtothepresentat —attempted toconstructa system containing the whole ofmathematics. Polskiej] isacknowledged. 1 The ON CHWISTEK’SPHILOSOPHYOF z z z theory of types. His logical investigations, however, were—as was were—as however, theory oftypes.Hislogicalinvestigations, in particularforhissimpli his logicalworks, for mainly known eon Chwistek(1894–1944)is W h ora fBrrn usl tde n.s.31(summer2011):121–30 the JournalofBertrand RussellStudies nancial support of Foundation forPoli Foundation of nancial support Mathematics andComputerScience/AdamMickiewiczU. W nite fragmentaryproblemsbut—similarlytoLes MATHEMATICS Roman Murawski 61–614 Poznan rmur @ amu.edu.pl W e sciences, hence in particular of mathe- e sciences,henceinparticular his logicalconceptions.The systemof cation ofWhiteheadandRussell’s ion ofChwistek’sphilosophicalideas arithmetic, mathematicalanalysisand loped by him in detail. He worked on Heworked indetail. loped byhim not inthemainstream of the develop- ´, Poland d inthemaccordingtoacceptedrules . The fateofChwistek’s philosophical ntal disciplines,andobtainedinthis in his philosophy of mathematics. He mathematics. in hisphilosophyof with otherlogicians,mathematicians main featureofhisphilosophywas sh Science[FundacjanarzeczNauki 1 issn 0036-01631; 5 aw Les ´niewski—con- online1913-8032 ´niewski C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) inscriptions thatcanbe obtained in a expressions or to only refer sequently ofclassicallogicandsettheory, assumption thatthetheoremsofsystembeingconstructed,andcon- the axiomandtheofchoice.Allthiswasbasedon duction construction duction tothesecondeditionof ChwistekandhiscontributionintheIntro- 1922b). Russellmentioned 1922aand (1921a, papers fromthe1920s,inparticularChwistek Cantor’s set theory. It should ful It Cantor’s settheory. should enablethereconstructionofaclassicallogicalcalculusand beasystemmorefundamentalthanlogic,andit ematics. Thiswould of expressionsand—onthebasis of it—hisso-called rationalmeta-math- of thesystem. that isthegreaterformalcomplexity for objects arerejected,buttheprice non-constructive theory types—a theoryofconstructivetypes.Inthis theory oflogicaltypesbyeliminatingnon-predicativede decided thentounifytheideasofRussellandPoincaré and to reform the ticular fromthemomentheattended a lecture byPoincaréin1909.He par- in Göttingen, in Chwistek’sinterestinlogicdatesfrom hisstudies murawski roman 122 compels ustosacri from substitute; any ity] withoutadopting took the heroiccourseof Chwistek Dr covered later by F. covered laterby it inanominalisticwaybyconstructingsimpletheoryoftypesredis- particular it should be free of any existential axioms, axioms, particular itshouldbefreeofanyexistential physical objects. as sions. Moreover,thoseexpressionsorinscriptionswereunderstood mainly inhis1935book, ofthedeductivesciences.Hedevelopedthem ing themethodology his ideasconcern- of part a as particular phy oflogicandmathematics,in 2 Those investigationsledChwistektotheconstructionof a full theory In Chwistek (1924 and 1925) he formulated a pure theory of logical He decided to rebuild the system ofWhiteheadandRusselldid system He decidedtorebuildthe Those ideas were developedbyChwist were Those ideas

Cf . also the correspondence between Russell and Chwistek (see Jadacki 1986). (seeJadacki Chwistek thecorrespondencebetweenRusselland . also W xed in advance, and not to the reference of those expres- those of in advance,andnottothereference xed W eagetda fodnr ahmtc.( ce agreatdealofordinarymathematics. y P. Ramsey. Itsfoundationswereformulatedinhis Ramsey. P. Granice nauki; Zaryslogikiimetodologiinauk nauki; Granice Principia Mathematica W l nominalisticassumptions,hencein dispensing with the axiom[ofreducibil- W hiswork,itisclearthatthiscourse nite number of steps by a rule of rule a by steps of number nite ek lateraspartofhisphiloso- W rst ofallthere- z : 2 W nitions. PM 1:xiv) C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) are the source of human su thesourceofhuman are because they are—in his opinion—incorrect ph ´cisl idealism in philosophy and mathematics. idealism inphilosophyand irrationalism, metaphysicsand (1948, p.3).Consequentlyhewasagainst simple andcleartruthsbaseduponexperienceexactreasoning” metaphysics,but confused constructing aworldviewshouldnotbe sciences andphilosophyaswell.Hewrote:“…thepointofdeparturein It concerns notonlymathematicsandtheexactsciences but experimental only twosources of knowledge,namelyexperience andstrictreasoning. was decidedlyagainstirrationalism.Rationalismconsistsinaccepting Chwistek refersheretorationalrelativism. ontheacceptedsystemofconcepts. depends and thesecondbecauseit corresponding tovariousrealities, experiments there arevarioustypesof is noabsolutereality.In ity ofobjectsleadtoinconsistencies.It cannot be absolutebecausethere It cannotbecompletebeca absolute. line ofLogicandtheMethodologyExact Sciences pirical and deductive knowledgearerelative.The and pirical logically reducedtosuchstatements.Bothem- simple facts,orcannotbe concerning periments, arenotbasedonreliableandcertainstatements withex- tions thatcannotbeexperimentallychecked,areinconsistent mon to allcorrectcognitiveprocesses.Itconsistsin rejecting all assump- schematization ofexperiencedobjectsandphenomena—afactorcom- sourceofknowledgeandthenecessity perience asafundamental ofex- And commonsenseis,accordingtohim—besidetheadmission con later questions. Theattempttosecuresuchknowledgewillsooneror i.e. knowle permit completeknowledge, It followsfromtheseconsiderationsthat highly appreciateddialectical materialism, seeing in fact almostnofun- theless appreciated itsepistemologicalconceptions.InadditionChwistek Hegel, HusserlandBergson.Seeing ihsudrao.(1948, p.42) with sound reason. s Chwistek accepted the principle of the rationalism of knowledge and Chwistek acceptedtheprincipleofrationalismknowledge According toChwistek,humanknowledgeisneithercompletenor 3 It is worth noting here that Chwistek wa Chwistek that here noting worth is It 5 ych . TheEnglishrevisionandtranslation, T ering, social injustice, cruel excesses and wars. cruelexcessesand injustice, social ering, The LimitsofScience On Chwistek’sPhilosophyofMathematics use statements concerning the total- the use statementsconcerning s againstirrationalismandidealismnotonly the defects of positivism,henever- the principle of contradictiondoesnot dge whichincludestheanswertoall ilosophical theoriesbutalsobecausethey 3 HesharplycriticizedPlato, hewrote: The Limits of Science: Out- Science: The Limitsof W rst isrelativebecause , appearedin1948. X 123 ict C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) matic rationalism. conceptions hedescribedascriticalrationalism,whichsetagainstdog- such asystem cannot beconstructedbecausephilosophical investigations isuseless—infact, philosophy that constructingdeductivesystemsin the helpoflaws(formal)logic. Later Chwistek cametotheconclusion terized inaxioms. On the basisofaxiomsonenowobtainstheoremswith It enablestheseparationofprimitivenotionswhosemeaningischarac- intuitive conceptsusedinagivendiscipline. the of analysis based onthe Itis ences, itcanbeusedalsoinempiricalsciencesandphilosophy. formmainlyindeductivesci- the constructivemethodinacomplete Though onecanapply Constructive MethodtoEpistemology](1923). tosowanie metodykonstrukcyjnej do teoriipoznania”[Applicationofa method.He tek—use aconstructive which mustbeemployedindealing wehave inmindnotsomeidealobjectbutthepatterns about reality the senses eventually assisted byinstruments.Hewrote:“…inspeaking onlywhatcanbeseenorexperienced by hence be giveninexperience, claimed thatanobjectofscienti damental con murawski roman 124 lible apparatuswhichsetso worked out,wewillbeabletosay, pletely iscom- When thismeta-mathematics] newsystem[i.e.,the systemofrational of Science Chwistek beginshis him. rational meta-mathematicsfoundedby struggle againstit,wasformallogic,inparticular a neously aweaponin never explained or explained in an insu an in orexplained never explained philosophical notions classical used often ing” (1948,p.23).HewritesalsointheIntroduction: reason- exact the destinyofsocietieswhoemployprinciples been thus: “Historyteachesthatultimatelyvictoryhasalways Introduction growth ofanti-rationalism”(1948,p.1).Andheendsthe unparalleled Both in science and philosophy one should—according toChwis- Both inscienceandphilosophyoneshould—according 4 Chwistek’s epistemologicalviewswereclosetoneo-positivism.He of thedi A wayout Notice a certain di certain Noticea bywritinginthe X icts betweenitandpositivism.Hisownepistemological 4 U culty ininterpretingChwistek’s U T culties causedbyirrationalism,andsimulta- exactthoughtfromotherformsofthought. W rst sentence:“Wearelivinginaperiodof W U c knowledge can be only what is or can is c knowledgecanbeonlywhat cient way. but gavethemaspecialmeaningwhichhe with agivencase”(1948,p.261). that we have atourdisposal aninfal- explained thisinhispaper“Zas- philosophicalview.In fact he (1948, p.22) Limits C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) of Realities](1921b),andhis existence andfullequalityoftheoreticalrightstoeachkindreality. the exactsciences).Heattributedindependent in reality (asconstructed everydaylife),andphysical (i.e. reality ofimages,thethings therealityofimpressions, have we sible typesofexperience.Hence of realitycorrespondingtofourpos- types four malization andaccepted edition containsnothingaboutthisthatisnotalreadyin the 1935 edition. this book publishedin1948—hencealreadyafterhisdeath—but the of edition English the Its foundationswereexplainedonceagainin (1935). odnosza realities. of theplurality theory losophical conceptionofChwistek,namelyhis types of experience. In this way we cometothebest-knownoriginalphi- experience canbeanobject of a cognition. There are,however,various are toocomplicated. formalize histheory.In tions bywordsanddonotexcludeinconsistentobjects”(1917,p.126). “exclude inconsistentobjects”;and descriptions bywords,butthey inconsistencies”; realistsdonotdemand cording to him, “nominalists demand descriptions by wordsing theproblemofexistence: excluding nominalism,realismandhyper-realism. Ac- ence” (1917,p.145). ac- of experi- elements with together cept asrealallmathematicalrelations should we then real, fact in is exists that everything that assumed we objects suchasofmathematics.Hewrote:“If abstract but reality him, has a more generalcharacterbecauseitconcernsnot only objects of Thelatter,accordingto “existence”. ing oftheconcepts“reality”and He developed hisconceptioninthebook realities wasalreadypresentinPascalandMach( many of superstition” (1917,p.145),andsuspectingthattheconcept a be to claiming that“theintuitivebeliefinonerealityseems Existence], of cept Having presentedChwistek’sgeneralmethodologicalandontological In the 5 an in saidabovethat,accordingtoChwistek,onlywhatisgiven We In the beginning he accepted only tworealitiesandattemptedto In thebeginningheacceptedonly three possiblepositionsconcern- distinguished In his(1917)Chwistek Thistheoryissometimescomparedwith 3 ce sie 5 Heexplaineditforthe W rst period (i.e., until 1925)Chwistekdistinguishedthemean- (i.e., rst period 3 dopoje 3 cia istnienia” [ThreeLecturesconcerningtheCon- cia Granice nauki On Chwistek’sPhilosophyofMathematics W nal versioncanbefound in W rst timeinthepaper“Trzyodczyty hyper-realists “do without descrip- without hyper-realists “do Popper’sconceptionofthreeworlds. z he abandonedtheattemptatfor- Wielos ´c ´ rzeczywistos cf Granice nauki . pp.149–50). ´ci z [Plurality 125 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) one cannotconstructsuchaspherebecause Chapter matize elementsofthelatter. particular domainsofreality,oneshouldsche- know to get generally to sciences—in hisopinionrefutedKantianidealism. tobethemostimportantachievementinexact considered tek Bolyai, GaussandLobatchevsky—whichChwis- of geometry Euclidean deductive theoriestodi apply to things. Tobeable statesof the expressionscanbeinterpretedasdescriptionsofparticular axioms arechoseninsuchawaythat and transformation of rems. Rules and formthebasisonwhichonededucestheo- play theroleofaxioms every given system oneacceptssuchrulesaswell some expressions that experience. Theycanbetransformedaccordingtoacceptedrules.In ematics are, in fact, expressions which are physicalobjectsgiventousin not idealobjectssuchaspoints,lines,numbersorsets.Objectsofmath- Consequentlyobjectsofmathematicsare cepted rulesofconstruction. mathematics, areexpressionsconstructedaccordingtoac- particular of of mathematics. his nominalism,whichfoundfullexpressioninphilosophy consider conceptions.Nowweshall logical his mathematics lyingatthebasisof mathematics. In factlet usturntohisviewsconnecteddirectlywiththephilosophyof ideas, we have already mentioned some murawski of roman his viewsof 126 from theGreeksasEuclid’s and thegeom geometry of experimental what isincluded i.e. garded asgeometry, this remained essentiallythesameupto segments, angles,andareas.TheEgyptian seems tobenotfullyjusti tulate the existence ofa tulate theexistence struc construction ofitpresupposesacertain consis inner objectthe ofan the existence guished postulatingtheexistence of an obje Geometry is, according to Chwistek, anexperimentaldiscipline.In Chwistek, Geometry is,accordingto 6 The developmentinthenineteenthcenturyofsystemsnon- Geometry isanexperimentalscience. Chwistek claimedthattheobjectsof the deductive sciences, hencein The claim that non-Euclidean geometries non-Euclidean that Theclaim ix of The LimitsofScience W ve-dimensional spherebecauseth W ed. In fact, one should take in oneshouldtake fact, ed. In Elements . (1948, p. 170) p. (1948, . hewrote: tency ofthegivenconceptisenough,buta ct anditsconstruction.Forapostulationof ture ofperceptualspace.Henceonecan pos- etrical metaphysicswhichwasinherited the perceptual space has onlythreedimen- space the perceptual It depends upon the measurement of Itdependsuponthemeasurement very day. To-day whatis generallyre- day. very intextbooks,isthepeculiarmixture s conceived it in this way and it has s conceiveditinthiswayand refutedKant’sphilosophyofgeometry to accountthatKantdistin- is conceptconsistent,but T erent disciplines,and 6 Those geometries Those C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) four-dimensional space-timesitisnecessarytoemploy in the hands of those who were reacting agai ofthosewhowerereacting thehands in the clothedin ofconven feathers idealism oftherulingcl leading tothereinforcement odczyty odnosza norwl.(1917,pp.144–5) on ourwill. one oranothertypedepending of illustration an as we draworthinkcanserve segment ofaline neither byexperimentalmeansnorintuitiveones,hencea geometry is conventionalism. Indeed, inhis Indeed, geometry isconventionalism. depends onadoptedaxioms. This suggests that a properphilosophyfor have shownthat,e.g.,theconceptofalinehas no objective character but sourceof reactionary social views and tems ofgeometryotherthanEuclideanone. wo that nothing sions. Infact,Kantclaimed science ofidealspatialco thereforeclearth is It using formulae. ofgeometrywithout It seemsthatitisimpossibletoattainageneralconcept the casealreadyinJ. hypotheticalentities—aswas sions. Infactconventionalismintroduces of expres- theory mental experimentalsciences—shouldbebasedona funda- that geometry—likeallother claiming conventionalism, rejected almostnofundamentaldi are to analyticgeometry.Hencethere Euclidean geometries]arefreeofinconsis Both systems[i.e.,thegeometry and systemofEuclidean systemsofnon- evsky’s theorems that onlyat tween themfromthetheoreticalpointof di geometriesar both that to theconclusion ua.(1948,pp.186–7) mulae. only has clearthatallthis Itis time. tendency. accept conventionalism,thoughheneverstateditexplicitly.Hewrote: of the systems ofnon-Euclideangeometriesrefutesthethesis tence ofconsistent the ConceptofExistence](1917),Chwistekstatesexplicitly that theexis- T 7 In erent lines;thedi It is worth addingherethatChwistek is It The Limits of Science The of Limits a priori 7 Chwistekwrote: character of geometry. It seems that he would be ready to be would characterofgeometry.Itseemsthathe 3 ce sie T y erences between those two types of lines can be caught can lines erences betweenthosetwotypesof 3 S. Mill or laterinPoincaré,thepropagatorofthis or Mill S. dopoje nstructions mustbe nulli On Chwistek’sPhilosophyofMathematics , however,Chwistekclearlyandcategorically W rst glance seem to be paradoxical…. So we come we So paradoxical…. be to seem glance rst 3 cia istnienia” [ThreeLecturesconcerning tendencies by reducingtruthtoe as much meaning as do mathematical for- mathematical do as muchmeaning tionalism becameaverydangerousinstrument claimed that conventionalism also became a became claimed thatconventionalismalso nst the old dogmatic idealism” (1948, p. 234). p. (1948, idealism” nst theolddogmatic uld excludethepossibili at theconceptionofgeometryas view.IntuitioneasilyacceptsLobatch- asses, writing: “It shouldbeobservedthat “It writing: asses, e equallytrue,eachofthemrefersto tencies—in fact they can be reduced be tencies—in facttheycan W W ed.… To speak of di of ed.… Tospeak rst papers,e.g.in“Trzy W ve-dimensional space- ty ofconsistentsys- U ciency andthus T erences be- T erent 127 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) Les stream ofthedevelopment of logic and philosophy ofmathematics.Like paths. Hisinvestigationswerenotinthemain- own his down went alone Chwistek War. cause allofthemwerekilledduringtheSecondWorld be- Kopelman, AbrahamMelamid,JózefPepisandKamilaWaltuch), Skarz ( been “earlierproclaimedthanchecked” had used, andhisconceptions Chwistek 1961,Preface,p.vii).Hedidnotexplainmany of conceptshe J. recognition.Inparticular, of mathematicsthatsomehisideasfound after 1945andthegrowing the interest in nominalismthephilosophy losophers (with the exception of his versionoftypetheory).Itwasonly amonglogiciansandphi- them. Hisworksdidnotstimulateinterest senseofresponsibility( full a they werenottreatedbyhimwith seems that it sophical investigationshadnosystematiccharacter,and an unparalleledresentmentbeingclose acted totheideaofpluralityrealitieseither with contempt or with of hisprofessionalreception:“theci Wolen had no close contacts with the Lvov–Warsaw philosophical school ( had noclosecontactswiththeLvov–Warsawphilosophicalschool of LvovUniversity,he ticians orphilosophers.Despitebeingaprofessor ideas withoutanycollaborationwithotherlogicians,mathema- tions and collaborators (among them Jan Herzberg, Wl collaborators (amongthemJanHerzberg, indetail.Itcouldnotbedonebyhis Chwistek by was notdeveloped fate ofhis logical conceptions. The system ofrationalmeta-mathematics nominalistic interpretationofallthem. wayasgeometry,obtaininginthisa same ematical theoriesinthe and othermath- analysis One shouldalsotreatarithmetic,mathematical murawski roman 128 Analysis ofMathematicalLanguage”(1962),wherehewrote: losophy ofmathematicsisfoundalsoin Henkin’s paper, “Nominalistic inthephi- A recognitionofChwistek’scontributiontonominalism in the versionproposedbyBourbakists( the consistencyofsettheory of proof a ematics canbeappliedtoobtain to whatextent,Chwistek’ssystemsofrationalmeta-math- whether, and ibid. y R. Myhill published a series of papers devoted to the problem of R. Myhillpublishedaseriesofpapersdevotedtotheproblem ´niewski ( What were the reasons for such reactions? In fact Chwistek’s philo- What werethereasonsforsuchreactions?InfactChwistek’s The fate of Chwistek’s philosophicalconceptionswassimilartothe of The fate 0 ). Hedidnotdevelophis systems in detail, being content to sketch yn ´ski 1989).Hisideashaveoftenbeensharplycriticized.Hewrote ´ski ´ski), orbyhisstudents(WolfAscherdorf, CelinaGildner,Kamila ´ski), cf . Murawski 2004), Chwistek worked onhisownconcep- rcles ofprofessionalphilosophersre- toawildfury”(1933,p.56). cf . Myhill1950,1951a, 1951b). 5 adysl 5 aw HetperandJan cf cf . . C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) tion of Tarski totheUnitedStatesin1938 tion of analysis of to de z z z z odnosza odczyty “Trzy 1917. L. Chwistek, Metaphysics referencetoChwistek’spluralism inhisbook, ourable philosopher, madefav- Australian the Note alsothatrecentlyR.Sylvan, speci theory ofReism(“Everythingthatis, associated withthisactivity.Amongthebe Lesniewski, T.Kotarbinski,Chwistekan the earl of Polish school oflogicians whereinshow allofitsassertionscould be to establishtheconsistencyofmathem not seem to be re seem tobe not mann, Lejewski,Sche byQuine,Goodman,Martin works ing From 1940ontherehasbeena steady seri itself evidentinthiscountry,and subs oiin (Henkin 1962,pp.187–8) position. the foundations of mathematics,andto made themselvescongenialbothtosome — — — — Chwistek 1961,pp.30–105. 51. ReprintedinChwistek1961,pp.3–29. Lectures concerning the Concept of Existence]. Lectures concerningtheConcept While thenominalistictraditioninphil 8 z z z z tische Zeitschrift Przegla la Aside from somewritingsofRussellthis Aside . 1922b. “Über die Antinomien der Prinzipien der . derPrinzipien Mathematik”. 1922b. “Über dieAntinomien . 1922a.“Zasadyczystejteoriitypów”[Pri . 1921b. . 1921a.“Antynomielogikiformalne [In fact Tarski left Poland in August1939. in leftPoland Tarski fact [In 3 d Filozo W W c concentrationofinterestinthisvi ne allcategoriesintermsofonelowestcategory,andChwistek’se 3 d Filozo mathematical Wielos W z (1997). czny ´c X ´ rzeczywistos W z ected outsideoftheirowncoun 24:164–71. Reprinted inChwistek 1963, pp.249–55. z 14: 236–43. czny V z 25: 359–91. Reprinted inChwistek 1963, pp. 256–85. er—and evenmyself.Nominalisticconsiderationshave z language, can be clearly discernedintheworkof language,canbeclearly On Chwistek’sPhilosophyofMathematics ´ci references [PluralityofRealities].Kraków.Reprintedin y decades of this century. The names of The names century. ofthis y decades equently in countries of western Europe. equently incountriesofwestern is apersonoranobject”)andhise j” [AntinomiesofFormal Logic]. others who haveadvancedaformalistic others d hisstudentHepter,and Tarski are all , Woodger, Church, Wang, G.Berg- holding a constructivistviewpointin holding atics byprovidinganinterpretation es ofpublicationsinthisarea,includ- 8 st known ofthis work is Kotarbinski’s n tobetruthsaboutphysical objects. z ewpoint, asappliedespeciallytothe 3 — theconcernwithnominalismmade ce sie osophy is of course very ancient, a interest of the Polish logicians did interestofthePolishlogicians nciples ofaPureTheory of Types]. z R. y M.] 3 dopoje try. Butwiththetransplanta- Przegla 3 3 d Filozo cia istnienia”[Three Transcendental W czny Mathema- z 20:122– Przeg- T T orts orts 129 z C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) Whitehead, A. Murawski, R.2004.“PhilosophicalRe z murawski roman 130 z z z z z z z z z Henkin, L. 1962. “Nominalistic Analysis Henkin, L.1962.“Nominalistic Jadacki, J. Myhill, J. Wolen Sylvan, R. 1997. 1997. R. Sylvan, — — — — — — — — — — printed inChwistek1963,pp.1–232. vols. Cambridge, ner &CoLtd,1948;reprintedLondon:Routledge,2000. and ArthurP.Coleman.LondonNewYork:KeganPaul,TrenchTrub- Exact Sciences Ksia bers”. Press. in Poland]. Warsaw. Reprinted inChwistek 1961, pp.149–277. Proceedings of the1960International Congress Nagel, P.Suppes,A.Tarski,eds., pp. 187–93. pp. Mathématique z z z ii z z z z z z ence”. 175–87. 130–6. cation of aConstructiveMethodtoEpistemology]. cation of Reducibility”. 1. Warsaw: Pan matics); Part Annales delaSociétéPolonaiseMathématique matics); Part1:General Principles of Interwar Period”. Boston, London: Kluwer. z . 1923. “Zastosowanie metodykonstrukcyjnejdoteoriipoznania”[Appli- . 1923.“Zastosowanie . 1951b. “TowardsaConsistentSetTheory”. . conc Investigations . 1951a.“Reporton . 1963. . 1961. . 1935. . 1933. . 1925. “The Theory ofConstructive 1924.“The Theory. ofConstructive . 1948. . Warsaw: Pan ´ski, J.1989. ´ski, 3 z0 nica-Atlas. Revised and translated as Chwistek 1948. Polish edn. re- edn. nica-Atlas. RevisedandtranslatedasChwistek1948.Polish Zagadnienia kultury duchowej w Polsce w Zagadnienia kulturyduchowej The JournalofSymbolicLogic Pisma Dialectics &Humanism y Pisma s Granice nauki.Zaryslogiki i metodologii nauk y R. 1950. “A Complete Theory of Natural, Rational and Real Num- R. 1950.“ACompleteTheoryofNatural,Rationaland J. 1986.“LeonChwistek—BertrandScienti Russell’s The LimitsofScience:OutlineLogicandtheMethodology y N., andB.Russell.1925–27. W W . RevisionofChwistek1935.Tr Transcendental Metaphysics ii ´stwowe WydawnictwoNaukowe. ´stwowe z lozo 3:92–141. lozo ´stwowe WydawnictwoNaukowe. ´stwowe Logic and Philosophy in the Lvov–Warsaw School Logic andPhilosophyintheLvov–Warsaw : CardinalArithmetic”. The JournalofSymbolicLogic uk Annals ofPureandAppliedLogic W W czne ilogiczne czne ilogiczne : Cambridge U.P.(1st edn., 1910–13.) z 13: 239–63. Logic, Methodology and Philosophy of Science: MethodologyandPhilosophy Logic, [Philosophical and Logical Writings]. Vol. Logical [Philosophicaland [PhilosophicalandLogicalWritings].Vol. X z 15: 185–96. 15: Logic: Theory of Classesand Relations”. ection onMathematicsinPoland the Types (PrinciplesofLogicandMathe- Types (Principles ofLogicandMathe- erning the Consistency of theAxiom the erning ofMathematicalLanguage”.InE. . Cambridge, Principia Mathematica Annales de la Société Polonaise de Annales delaSociétéPolonaise z [ProblemsofaSpiritual Culture z The JournalofSymbolicLogic (Stanford, ans. byHelenCharlotteBrodie z 16: 35–42. z 2:9–48. z 127:325–37. Przegla ´cisl uk ca 5 ych : TheWhiteHorse 3 : StanfordU.P.), d Filozo . Lvov-Warsaw: W c Correspond- . Dordrecht, . 2nd3 edn. W czny z 26: z 16: