,InternationalLibrary ofPhilosophy and Scientific MethodThe Philosophy of'Asif'1) 11 i 1() s() 1) 11 Y () f, As i f'A Systemof the Theoretical, Practical anLl ReligiousFictionsof MankindByII. VAIII I Nt;{l',HC. K. O(;{)ENIJONDONKEGANPAUL,TRIi:NCH, TRUBNER&: CO., LTD.~ E W YORK: lIARCOUl1T. >.'lRACE & COMPANY.1935'"Madeand Primed in GreatBritai"b}'PEnCYL,mn>. HtJMPllmEB Il1J co. I:ro.u Bedford Sqt,are, T,om{on, W.C.tand at BradfordPREFACETOTHEENGLISHEDITIONTHE publication of this work in an Englishtranslation gives mevery great pleasure. Fromearly youthI have studiedEnglishliterature, and later English philosophy. Duringthe periodwhenmyphilosophical views weretakingsha.pe, at1despecialIy,in the years1874-1876, it wasDavidHume andstill more J.S.Mill whose influence on my thought was paramount. Thus I wasearlyattracted byEnglish philosophy, and I formedthepro-ject of writing aHistoryofEnglish Thought.. But, like" manysimilar hopes, this plan was destined to remain unrealized. ! soonfoundthat importanceofFictionshadalready beenpartlyrecognizedbyEnglishphilosophers. English Notninalismofthe Agesshowedtraces ofsucharecognition. W;,thJoh.n Dum Scotus, who died in13081nCologne, when' only inhISthirty-fourthyear, there beganaHceptical movement whichtended inthe same direction. nut itwasinWilliam of Occam,who tookrefuge with Ludwig of Bavaria, and diedinMunich in1347 at the age of 77, thatwe findforthe first time a clear anddefinitetreatment ofthefictional natureof general ideas, de"veloped in a manner whichis still a model for to-day. He fullyunderstoodthatflcta, asthey were calledinthe writings of theMiddleAges, althoughtheir theoreticalnon-existence might beadmitted, art;,.practically necessaryand must berecognizedinthis sense. Un theother handthiswasnot realizedby Baconor even by Hume, thoughin Berkeleythereareat least in-dications of anunderstandingofFictions. But in Hobbes wefind a considerable knowledgeboth of Fictions themselvesandofthe theory of their use. Empty space, theideaof a'bellumomnium contra omnes, andof an Hcontract II are forHobbes conscious Fictions. A special study of Hobbes' theoryof Fictions had been contemplated by O1y late colleagueProfessor Frischeisell-Kohler whowas well versed inEnglishviiviii THE PHILOSOPHYOF 'ASIF'philosophyand hoped to writeahistoryof its development;but owing to his early death neither project was t'ealized.Fictions, part of England's heritage from the Romans, haveplayed a large part in English jurisprudence and politicalphilosophy, bothinpracticeandtheory; more so thaninothercountries. There is roomfor a special monograph on thissubject, covering the use of Fictions both in Adam Smith'spolitical economy and in Jeremy Bentham's political philosophy.Inthe present work the methods of AdamSmith andBenthamhave beentreated in some detail, but they would appear inquite another light, if brought into relation with the wholehistory of English thought. Thus particularly in Englandconditions point to a favourable reception for the theory ofFictions as developed in ThePhilosophyof I Asif} " Prag-matism," too, so widespreadthroughout the English-speakingworld, has donesomethingtoprepare theground for Fiction- inspite of their fundamental difference. Fictionalismdoes not admit the principle ofPragmatism whichruns: "Anidea which is foundtobe useful inpractice provesthereby thatit is alsotrue intheory, and the fruitful isthus always true." principle of Fictionalism, on the other hand, or rathertheoutcome of Fictionalism, is as follows: II An idea whosetheoretical untruth or incorrectness, and therewith its falsity,is admitted, is not for that reason practically valueless anduseless; for such anidea, in spite of itstheoreticalnullitymayhave greatpractical importance:' But thoughFictionalism andPragmatismarediametrically opposed inprinciple, inpracticethey may findmuchincommon. Thus bothacknowledge thevalueof metaphysical ideas, thoughfor verydifferent reasonsand withvery different consequences.It canbe shown, andhas beendemonstratedat length inthe present volume, that the theoryofFictions was more orless clearlystated by Kant, who was proud of his Scottishdescent. Nearly 100 pages of the work are devoted tothisquestion and it is thereprovedindetail that for Kant a largenumber ofideas, not onlyin metaphysics but also inmatics, physics andjurisprudence, were Fictions. The metaMphysical ideas weresomewhat confused by Kant himself inhis Critique of Pure Reason'" (Theory of Method), but weredefinitely called II heuristic Fictions." This was overlooked TO TIU"; IJ:NGLISHEDrfION ixand not understood in Kant's own day and for a long timeafterj andKantwas quite right whenhe said of himself"Iamacentury too earlywith myworks; it will be a hundred yearsbefore they are properly understood." That was in 1797.The hundred years of incubation which Kant prophesied forhis theories have now gone by,and the times are ripefor thishis profoundest contribution, which I !Uay mention has nowbeen given due value by Professor Norman Kemp SmithofEdinburgh in his admirable commentary on the Critique(recentlypublished in a secondEdition).HANS VAIHINGER.ANALYTICAL CON'l'EN'fSPAGIISAUTOBIOGRAPHICAL xxiii-xlviiiGENERAL INTRODUCTION 1-13CHAPTERI.-Thought. considered fromthepoint 01view01aplltPosive organio Funotion . 1-8Empirical utility oradaptation toa purpose manifested by organicand logical functions-Theorganic formativeforce of the psyche--Stein.thal-The teleological treatment of the logical functionsheur-istically permissible-Psychical mechanismandpm'posivenessnotmutuallyexclusive-Thetest whetherthelogical functions haveattainedtheirpUl"posecanonly bepYacticrLl and not theoretical-Thought primarily an instrument of self-prcservation--Hcrbartand Schoponbauet-Logical purposivenessexpressed illthe inven-tion of logical aids-The ways of thought not the ways of reality;theyareonly subjective, but areexpedient-The devious ways ofthollgbt--Fundamental O1'1"or of dogmatism; its cQnfusion ai'thought andreality. I l.-Thougbt as an Art, Logic ll4I Technology. . 8-9The purposiveorganicfl1nctiOll attains thelevel of anAJ:t-'l'huliIariseRules, wWcharccollected byLogic lW the" '!'()chnology ofThought." UI.-The Diflel'(luce between the Artifices andRulesof Thought 10"! 2Distinction between the artificial Rules. as Qtdirtary, regularmethods, and the Artifices, as irregular methods 0: thought-Originof these artifices: Leibnh: audNewton-Theyshowthepurposiveness of the logical function.CHAP'ERIV.-The Transition to Fiotions 12-13Fictions are devices of this sort and auxiliary operations of thought-Preliminarysketch of the fictional thought-construct-Epis-temological SIgnificance of fictionalaccessoryconcepts.PART IBASIC PRINCIPLES 15-177General Introduotory Remarks on Fictional Construots I 5- I 6The regular, natural methods of thought-Their object is thecreation of objectively valid ideational cOllstructs--The wholeworld of ideas is an instrument to enable us toorientate ourselves inthe real world, butis not 4copy of that world-Thelogicalfunc-tions are an integral part of the cosmic process, but not acopy of it-Within the worldof ideas logic disj;inguishes again between rela-tively objective ideational constructsand-thosewhich are subiectiveor fictional-Pure 'fictions and half.fictions.xixiiTHE PHILOSOPHYOF 'AS IF'VAGUSA. THE ENUMERATION AND DIVISION OF SCIENTIFICFICTIONS 17-77CHAPTERI.-Artiflcia.l Classitlcation 17-19Natural andartificial systems-Theirrelation toquestionspecies-Difficulties of natural systems-The deVIce of a proVI-sIOnal artificial division-Practical and heul"istic advantages ofsnch artificial systems-Theoretical contradictions of artificialdivisions-Artificial definition.CHAPTERI1.-Abstractive (Neglective) Fictions 19-24The deliberateomissionofcertainelements of realityincompli-cated phenomena-Standard example: Adam Smith's fiction in hispolitical economy-Thisisnot anhypothesis, but a subjective,fictivemethod, bywhich an abstractsystemiscreated-Applica-tion of this method in the complicated phenomena of sociology andalsoinmechanics andpsychology (Herbnrt andSteinthal)-Theuselessnells oflI11ch ideationalCOlllltrllCts not tobededucedfromtheir unreality-Disputedexamples: e.g, a periodinwhichlan-guage consistedonlyof"roots.' Fictionsof isolation-Relationtothemethodofapproximationandtentativefictions-Averagefictions, e.g. l'hontntemoyen in statistics.CHAPTERUr.-Schematic, Paradigmatic, Utopian and TypeFictions . . 24-27Schemata, models, andschematic drawings-Thefiction of simplecases-Example: ThUnen's idea in economics-Paradigmata:method of imaginary cases to facilitate scientificpl"oof-Rhctodcalfictions-Utopias-Theoriginal stateandthelike-Valueof suchideas and theu misuse-Imaginary archetypes-Goethe's fiction ofa,plant archetype"andSchiller's opinion of it.CHAPTERIV.-Symbolic (Analogical) Fictions 27-32Thepsychicalmechanism inanalogicalfictions-Schleiermacller'stheological methodrest\;ontheconversionof dogmasintoana-logical fictions-Thisepistemological devicederivesfrom Kant-TheKantian"as if "-Categories as analogical fictions-Throughthecategories therecanbenounderstandingoftheworld-Thenecessity of a "Theory of Understan.ding" (comprehensiona.ltheory)-Analogical fictions and the illusion of knowledge towhich they give rise-Expediency of such fictions-Symbolicknowledge accordingtoMaimon-If suchfictions are convertedinto hypotheses contradictions result-Errors andillusory problemsdue to the misuse of analogical fictions-Critical Positivism dis.tin-guishes these additions of the psyche frompure experience-Resignation, wise andunwise: negative understan.dingof theworld-Otheranalogical fictions.CHAPTERV.-luriBtic Fictions . 33-35A special form of the analogical fiction is found in'the legal fiction,a juristic deviceoffrequent application-Importanceof judsticmethodsforlogical theory-Thelogical functionapplies the samedevices inveryvarious fields-The[ictiojuris andits essentialdifference from the pra8sumptio-Examples fromthe GermanCo.mmercial.of these juristic methods to theeplstemologlcal nction-Both are valuable in practice, but them:eti-callyworthless, since they are based on adeviation from reality.CHAI;'TERVI.-Personiftoatory Fictions 36-38Theform in which perSOll'!-lity is apperceived-Hypostatization ofphenomena-Abbreviations, Nommal fictions, Auxiliary words';['autological fictions. 'CHAPTERVII.-Summationai Fictions (General Ideas) 38-39The general idea asamerefiction.ANALYTICAL CONTENTSCHAPTERVI 1I.--Heuristio FiotiQI1lI . 39-42Theassumptionofunl"freedom and ini;ellectualintuition-The retention of aficttonas such requires energy-The creation of a logical consclenl'e--The development ofpost-Kantian philosophy from this stallc1p'oint-Harttiful andbeneficial effects of the Law of Ideational SbIfts-The transforma-tionof objectivelyfalse hypothesesintosubjectivelyexpedientfictions-Significanceof such fictions forthought ingeneral.C. CONTRIBUTIONS TO THEHISTORYANDTBE()ll.YOFFICTIONS. . . . . . . . . 135-156Preliminary Remarks. The 'l'heo17 and Practice of Fictilltll8 135CHAPTER XXVIII.-The Fiction in Greek Scientu\c Pro--c&dure . . . . . . . . . 135-139Relatively late a?pearance of the fiction-It implies an eJnancipa-tionfromimmed1ate pe:rception andfromthe belief thatthoughtis identical with reality-Scepticlsm-elumsyof theancients, particulaxlyinmathematics, thoughsigna of scientificfictions are not altogether wanting-Express avoidance of nctionalconstructs-HyPotheses andfiCtionS-Platonic myths andsimilarfictions-Parmenidean fiction of the elements of the 'wQrld ofappearance-----The worldofappeal aneeas afictional C011Struct-The fictionof the!lpheriGal form of theAbsolute in Parm.enides-Symoolic thought.CHAPTERXIX.-Begjnnings of aTheoryofFictions &lXt.ong theGreeks . . . . . . . . . 139-1420Illsuflicient familiarity with the use of nctions naturally t:esults inanabsenceofadequate theory-Nocritical distinctionbetween anecessity of thouglit and reality-Aristotle's methodology deficientin this respect-llll'6t1E/Ttr and V7('01'tfUPl1oL in Plato attd Aris-totle-Aristotlerecognil:edthefictional nature of math.e1UllticalANALYl'lCALabstl"ilctions-Latl'l' Greek philosophy-The gradual differentia-tion of hypothct.icl and fictional aSHumptions-Thc Sccptics-Their negativismcOl1trD,stcll with modern sceptical Positivism,CHAPTERXXX.-The Use of the Fiction among the Romans [43The forccd and arbitrary character of legalfietions-Their purpose. -Examples: actiones tltiles,xxxI.-Beginnings of a' Theory of Fictions amongthe Romans . 144The Romans realized the ambiguity of the term inr60EO't;-(i)Suppos'ilio and(ii) Fictio.CHAP1'ERXXXI I.-Medieval Terminology . 145The significance ofNominalism which recognizedgeneralideas asfictiones-One-sid.... negativesenseoftheterminNominalism-Impor.tanceof schoiiisticterminologyformodern times.CHAPTER XXXI II.-The Use of Fictions in ModemTimes 146-154More extensive use of legal fictions in modern times-Discussion oflegal fictions inrelationtologicbyLcibniz-Utopia.nfictions-Fictions chiefly applied in mathematics-The development ofmodernmathematics bymeans offictional auxiliaryconcepts-Mcrtschinsky's fictionofminimaof constant size-His relation toBruno-" Inlillitcly,Hstant points "-Lack of a. methodology o1ictional concepts, ()spedally in mathematics.Mail11on'gvIewof t.he Mona: my philosophy of ' As if', which also leuds to a more thoroughstUdy of Kant's' As If' theory.ORIGINOFPHILOSOPHYOF'ASIF' xxxviiof thematerialgiven to thesenses has remained withmeeversince.1 derived great profit fromAvenariusin so farashewasapungent critic of Kant's theories. This preventecl me fromregardingKant's philosophyas dogma, but anyway Iwas notinclined todo this. I could not followAvenarius, however, inhisradical empiricism, or rather positivism. He realized quiterightlythat the ideas of substance, causality etc. are imposedsubjectivelybythe psyche onthe given, yetforthis very reason,according to "the principle ofthe least energy", hewanted to(eliminate themcompletely fromhuman thought. But I heldthat they are fictions, which must beretainedbecause oftheir utility.In the autumn of 1875Wundtcame to Leipzig. His firstlecturewasonlogic andIlistened toitwith greatinterest andprofit. He appealed_to me in every way. For his sake Ishouldhaveliked to remain all in Leipzig, and I haualreauyplanned aJoumat ofPure audAppNed Lagle, inwhichI hopedtointeresthim. But family matterscalled me backto SouthGermany. I was onlyable to have one more terminthe North,andthatwasto be inBerlin, wherethe Swahial1, EduardZellet,waS actively at work. Thehelp whichIgot(romhimandfromhis friendHelmholtz, and also fromSteinthal, Lazarus,Lassonand Paulsen was more or less valuable to me, but what wasreally importantwas that Icame acrossthewritingsof Gruppe,whohaddiedshortlybefore this, andthey were useful formy theoryoffiction. Myprivate studieswere devotedmostlytoDavidHumeand John Stuart Mill, whoseexact knowledgewas decisiveformy philosophicattitude.At the sametime, duringmyBerlindays in thesummer of1876, myfirst bookon philosophy was published, Hartmmzn,Diihrng andLange-acritical Essayonthe HistoryofPhilo-sophy in theNineteethCentury. It consisted of lectures whichIhadgiveninthe Academic Philosophical Society at Leipzig.Theauthor of the History of M aterialt'sm, with his Kantiantendencies, seemedto meto strike thehappymediumbetweenthe spiritualistic metaphysics ofE. vonHartmann on the onehand and the materialistic positivismof E. Dtihringon theother hand. In Berlin I had got to know these two menpersonally. 111mybookIalso announced the earlypublicationof myinvestigationof Fictions.For family reasons I hadtochoose a UniversitynearmyxxxviiiTHEPHILOSOPHYOF'ASIF'SouthGerman home in which to take upll1yresidence as alecturer; so in the autumn of 1876 I moved to Strassburg,whereI receivedawelcome fromLaas. Inhis recent work onKant's Analogies of Experience he had ell'awn a sharp linebetween himself and the Kantian, or rather Neo - Kantian,A-priorism or "Transcendentalism," and he was graduallyapproaching that radical attitudewhich hetookupsome yearslater in his three-volumetreatiseon Idealismand Positivism.He was the unprejudicedmanof whomI stoodinneed, Hewasable to do justice tomy ownattitude. He was busy justthenwiththe study of John Stuart Mill's Examinatiolt ofSirWiltz'amHamilton's PMlosophy, in which I joined him, all themore readily because this was really a continuation of myBerlinstudies of Humeand Mill. The resolution of so-calledreality, from an epistemological or psychological point ofview, into" Sensations and possibilities ofsensation" seemedboth to himand tometobethecorrect analytical way. Onthe other hand Laas resembled Avenarius, who was relatedtohim, inhis positivist tendencytoeliminate all further s u b ~jective additions asunjustifiedanduseless,whereasIwas alwaysanxiousto emphasize and keep holdof tbepractical value anduse of these theoreticallyunjustifiable conceptions ofthe oldm"idealism.Duringthe latter part ofthe year J876, for myinauguraldissertation, I wrote downmy thoughts ina large manuscript,to which I gave the title "Logical Studies. Part I: TheTheory of Scientific Fictions," As I had been carefullycollectingthematerial for several years and had gone into itmost thoroughlymanytimes, thewriting ofit didnot take melong, I handedin myMS. inthe NewYear andat theendof February1877 I receivedmy vem41egendi. The workwhichreceivedthis recognition fromthe Faculty is exactly thesameas whatwaspublishedin19/1as the" Part I: Basic PrinciplesIJof ThePhilosophy of 'Asif'. In it I developed the wholesystemof scientific fictions, that is tosay the' As if' treatment,applied practicallytothe most varied aspects of science, andI tried to give an exhaustive theoryofthis manifold' Asif'process.But likeLaasI regarded this dissertation only as a roughoutline) in need ofmuch supplementing and correction, so Imadeuseofthenexttwoyears, sofar asmy lectures allowedme, toworkat my MS. Myfather's death compelledme toOllItiINOPl>IIILOSOPHYOF'AS IF"xxxixlook out fur some more remunerative occupation,! and so Imade a veryadvantageolls agreement with the generous andfar-sighted Stuttgart publisher, W. 5pemann, to produce aCommentary onKant for thecentena.ry in1881of hisCritiqueof Pure Reasolt, I had then just started a farmore thoroughstudy of Kant,particularly his' As if'theory, andin the courseof thisI had found in his Prolegomenathat" misplacement ofpages" which had passed unnoticed by many thousands ofKant readersfornearlya hundredyears, but whichis generallyrecognized by science nowadays. 50 I hoped, by applicationofthephilological method, andby penetratinglogical analysis,to further thestudy of Kant. But, as I have said, this newworkwas only ameans to anend, andIhopedin a fewyearstobe able to return tomyresearchesonFiction.The above-mentioned" Lawof the Preponderanceof theMeans over the End ", which unfortunately I neglected toformulate theoretically and publish at the right moment. hasproved in a practical sense verymomentous in my own life,Wheojin l8841 the first volulneof myCommentaryonKant2brought me un appointment as special Professor at Halle, IhopedSOOIl to beableto finishthe other volumes there. ButI At thut time rw a ~ alsoconsideringthe plan of writinga Historyof EnglishPhilosophy, mentioned in myPreface tothis transl(\tion, But thereWM then so littleintereslandunderstandinginGermany for the development ofEnglishphilosophythat on the adviceof theexperts of the time thepublishers did not regard thesuggestionwith favour." On my journeyfromStrassburgtoHalle, Ipllid another visit toFriedrich T.Vischer, whomrhadoften seeninthe interval. Our talk was concernedchieflywithhis philosophical novelAuch Et'tzt1- (1897) inwhichhe e x p r e s ~ e s his lavouriteidea ofthe declineofthe Germanpeople since 1871. He had shown in this bookhow the Germans by their arrogance would become involved in a world-war, inwhich afterhard struggles anda moral revival they wouldeventually bevictorious.Even then I did not agree with this optimism, and mypolitical pessimism grewstronger in the following years, particularly after 1888. After 1908, andparticUlarly19u,I contemplatedfollowingLeibniz' exampleand enteringthe arena of worldhistorywith an anonymous pamphlet, Ft'nis Germam'cu, with themotto" QuosDeus vult perdere, prius dementat", andwiththeclevie-e, of Schiller's Cassandra"The Thunderer's clouds loomheavy over Ilion ", I thought of having thispamphlet printedinSwitzerland, but myeyesight becamerapidlyworse and pre-ventedmefromdoing this, I also said tomyself that I would be a voice cryingin the wilderness, for it seemed impossible to penetrate the blindness ofmy seventymillionfellow-countrymen. I felt afraidtoothat the publication ofmyviews mightonly increase the number of our enemies andtheweight of their oppositionandthatmy action might thus hastenthe impendingcatastrophe, EventhenI would havementioned most of the factors that are recognized to-day-or at least ought to berecognized-as thecauses of thedisaster, Anunjustifiedoptimism(ifI do not goso faras Schopenhal1er in calling it a .. criminal optimism ") had [or a long time beenleading German policy astray in the dlfectlOn of improvidence, rashness andarrogance, Aratiohal pessimism might have saved usfromthe horrorsof a world-war, World"philosophyandpractical politics have a closer connection thanis gen-e"ll1y realized,xlTHEPHILOSOPHYOli' 'ASIF'my lectures onthe one handand bad healthonthe other heldupthe publicationofthesecondvolume until 1892. In 1894I was appointed reg'ular Professor in Halle, and in 1896 Ifounded theKantstudimas a means of helpingonmy work.But even this means preponderated over its own end. Mywork on the Commentary became secondary to the newperiodical. When in 1904 the centenary of Kant's deathwas celebrated, circumstances seemed to make it myduty, inorder to promote the Kantstudien, tostart a Fundto defraythe costs. This Fund was a success, but its organizationnecessitatedthe foundationofa Kant Society and thisgradu-ally became more andmore an endin itself and took up toomuch of mytime and strength, although I was fortunate inhavingmost efficient help in all these undertakings. Thus themeans always triumphed over the end for who::;e sake it hadbeen called into being, and robbed the original end of itslife.force.In" 1906, inthemidst of all these clIrious complications andcrossingsof my original intentions, a misfortune unexpectedlybrought a happysolution, and enabledme after twenty-sevenyears to return to my original plan, which I hadgivenupin1879. The misfortune was the weakening of my eyesight, sothat it became impossible for me tocontinuefTlY lectures, or thespecial classeswhich I particularly enjoyed. SoIhad to giveupmyofficial duties. The eyesight still remainingtome wasjust sufficient to allowme to publish my MS. I got myDissertationof 1876copied, and introducedanumber ofsmalleditorial alterations. This comprehensive MS. now forms" Part I: Basic PrinciplesJI of The Philosophy of {Asif'. Ialso completed the revision which I hadmade between1877and the beginning of 1879 onthe basis of the reviews ofthattime, and thisformsthePart II(Special) of the complete work.This part took me two and a halfyears becauseof mybadeyesight, and Part II I (Historical) took me another twoanda half years. Between 1877and1879 I had madea note 01the most important' As if' passages in Kant's works, and Inowcompleted this in an exhaustive manner, sothat I wasable to produce amonographon Kant's' Asif' theory of nearlyone ~ u n d r e d pages. The exposition of Forberg's religlon of, As if' also tookme a long time, and so did the developmentofF. A. Lange's" Standpoint of theIdeal," withwhich I hadmuch in common. But what took longer still was the finalORIGINOFPHILOSOPHY01" 'ASIF' xlisection on Nietzsche's theory of Fictions, which he had con-densed into ~ fewpages. It was the Spring of 19I1 beforetheworkappeared.Icalled this work Tilt) Plu'losoplty of' As if' because itseemedto me to express more convincingly than anyother Ipossible titlewhat I wanted tosay, namelythat IAs if', i.e.appearance, the consciously-false, plays an enormous part in\science, inworld-philosophies andinlife. Iwanted to give acomplete enumerationofall the methodsinwhichwe operateintentionallywithconsciously false ideas, or rather judgments.Iwantedto reveal the secret lifeof these extraordinary methods.Iwanted to give acomplete theory, ananatomy andphysiologyso to speak, or rather a biology of' As if'. For the methodoffictionwhich is found ina greater or lesserdegl'eeinall thesciences can best beexpressed by this complex conjunctionIAs if'. Thus I had to give a surveyofall the branches ofsciencefromthispoint of view. But it wasnotonly a methodological investigation thatI wasattempting. The study of fictional thought inall branchesofscience hadledme gradually toextendthese investigations tophilosophyitself, particularly to epistemology, ethics andthephilosophy of \cligion. Just as my investigations into thefunction of' As if' hadarisenout of adefinite view of theworldso againthis developed independently intoa universal systemof philosophy-I gaveit the name of "Positivist Idealism"or" Idealistic Positivism ". As Ihave alreadymentioned, ErnstLaas had published between 1884and 1886a three-volumework onIdealism andPositivism, in whichhe attacked Idealismandchampioned Positivism. Thepositivist attitudewas alsorepresentedinGermanybyMach, Avenarius and to a certainextent bySchuppe, and it found particular favour with thescientifically inclined (but the name Positivismwas neverplacedin theforefrontof anyprogramme). The chief currentsof Gentian philosophy, however, wei'e certainly idealistic, thoughin different ways. Betweenthese one-sided views1 it seemedtome that acompromise was necessary, allthemore so becauseJ. The growingtendency of the" idealistic" philosophersandtheNeo-Kantiansto returntoFichte and Hegel seemed to me to be becomlng more and morc dangerous.I wasalwaysconvincedthat this one-sided idealistictendency, which waspartlyforeign and partly hostile to reality, wasthe more dangerQus to the whole of Germancivilization 10 that it ledour youth to undereslimateforeign philosophy, andthere-w i t ~ the wholecivilizationofneighbouring peoples, their capacity and. ingene(al,then mental and moral power.xlii 'fHEPHILOSOPHY01' 'ASIll"attempts of this kind hadmet with success inother Iconsidered that the time had cometoannounce the unionofPaSltl"Vls"m. Theresult hasprovedthns inModernTimesWE nowproceedto an account of the use of the scientific fictionin modern times, Hereitsemployment is incomparablymoreextensive,Sofar wehave found inthelegal fiction the only reallyFICTIONS IN MODERN TIMES 147scientific fiction. We should, however, remember in thisconnection that jurisprudence is not really a science ofobjective reality but a scienceofarbitrary human regulations.Moreoverthefictionwasappliedrather in the practice of law.Onthe other handitwasnot yetasextensively employedasinmodern law, where it has been used specifically inthe foundationof Publiclaw andwhere, moreover, thefiction of juristicpersonsis very widely adopted, even to the extent of including theStateitselfin sofar as the State canberegardedas a juristicperson. Both in the special jJractz'ce and inthetheory of law,the fiction has been far more extensively employed in recenttimes than in the classical period. InEngland especiallyithasbeenmuch usedandabused. The fiction serves to subsumea givencase undersome general rule, when thecase in questioncantherebybe treatedjuristically. For instance, it is assumedthat a husbandis the fatherof achildif hewas in the countryat the timeof the child'sconception,i.e. sinceevery single casecannot beinvestigated, the general assumption is made thatevery husband is to be regarded as the father ofa child ifhe was inthe countrywhenthe child was conceived. This ex-ampleisgivenbyLeibnizinhisNouveauxEssaz's [E.T. p. 260],but it israther a jJraesumjJtz'o thanatruefiction. Afiction inthejuristicsensecan onlybespoken of ifa husband, whosewife has committed adultery, is nevertheless regarded, if hewas in the country at the time, as the father ofthe child. Hewouldthen be regarde"d as ifhe were the father ofthechild,althoughhe is notandalthoughwe know he is not. This lastaddition is what differentiatesthejictz'ofromthe fraesumjJtio, forinthe jJraesumptz'o, apresuppositionismade until the contrary isproven, whereas thefictlo is the acceptanceofa statement orafact although we arecertainof the contrary. Anexampleofa real fiction is the fact, for instance, that in England(intheeighteenth century) every crime could be treatedas ifpersonallydirected against the king, and every plaintiff had theright tobring hisactionunderthis fiction. The practical valueofthisfiction lay inthe fact that.trialsunder'this fictionwere far morestringent than those under theordinary laws, forcharges thusmade were brought beforeaspecial court. Herewehave the"asif"in all its force. The Code Napoleon also allows anumber oflegal fictions; for example, thehouseholdgoods ofa womanare regardedas immobilt"a. Similarly we find fictitiousproperty,etc. and under certain conditions an{( enfant c o n ~ u "148PART I: BASIC PRINCIPLEScan be regardedas "nc" ifimportant legal consequences areinvolved.Inlegal practice the employment of :fictionmay leadbothto benefits and also to the grossest forms of injustice, as whenallwomenweretreated as if theywere minors.In legal theory the fiction was particularly used in thetheory ofcontract, in so far as the Statewas regarded as theresult of a contract andwastreatedasa juristic person.This fiction, which was already knowntothe ancients, hasbeenvery extensivelyusedin recent times.Another favourite methodwas theideal orUtopianfiction.In thenineteenth centurythe French Socialists, Fourier, forexample, were still employingthis metpod of spreading theirideas by thedescription of townsandstates as if theideas theypromulgated had been thererealized. Such a method passesvery easily over into the realmof phantasy and forms thetransition froma scientific treatment topure poetry. But thiswhole group of scientific methods must not be overlooked,though they are neither veryimportant nor do theypresentany theoretical difficulties.Withthe growthof science thefiction begantobemoreextensively employed.The first ofthemain fields wherereally great results wereachieved was mathematics. Modern mathematics is charac-terized specificallybythe freedomwith which it forms thesefictional constructs. A careful studyof thedevelopment ofmathematics brings to light anumber ofsuch fictions. Wedonot so much mean thereby suchsubstitutions as the ~ m p l o y ~ment of. letters insteadoffigures as a notation, thougheventhis simple methodisstrictly speaking afiction. By the fictionthat a, h, c, z, yare numbers, andby,treatingthemas iftheyactually were, enormous progress is made; results can begeneralizedand calculations simplified. This is usually calledanapplication ofsymbols, but takenlogically, wearedealinghere witha substitutive fiction. Thought itself, in general, whenoperating with words instead of perceptions, makesuse ofsuchsymbols.But quite apart fromthis, fictions have been more andmore used in recent mathematics. Their most famous andmost fertile applicationwas in the measurement ofcurves byDescartes, Leibniz and Newton. This is reallythe classicalexample. Bymeans of thefiction of coordinates, ofartificialFICTIONSIN MODERN TIMES 149lines (allartificial linesarefictional methods), andbymeans ofdifferentialsor fluxions, atreatment of curvesbecamepossible.]The methods of unjustified transference, of zero-cases, ofabstract generalization, etc. are modernmathematical devices.They were generally known by these names; great mathe-maticians have always been distinguished bythe invention ofdevices, and these devices are always essentially baseduponfictions. Even the drawing ofartificial lines is such a device.Schopenhauercalled attention to the fact that noreal know-ledge can be obtained by their means. But such devices arenot meant forthis butfor practical purposes.It isuponsuch devices and fictionsthat theconcepts of theinfinitely large, and of negative, fractional, imaginary andirrational numbers, are based,all ofthemserving the purposeof simplifying calculation and all in a strict sense logicallycontradictory.The utilization of these devices, towhichthe progress ofmodernmathematics is due, has continued right intoour owntime, andeveryreallynew discoveryinmathematics rests uponsuch a device. The device ofabstract generalization has nowbeen applied tospace, andspaces of more than three dimensionshave been imagined.Themethod of determinants dependson such an artifice.Of special interest are the fictionsof line, surface and volumeelements as a foundation for the use of measure-numbers.Mertschinsky, in particular, has utilized thefiction ofminimao['constant size for purpose. This fictionhadalready beenemployed by Giordano Bruno in his De trz'plci mnt"mo etmensura, and De 1lZonade, numero et figura. But Brunostillhesitateswhether to treat his minima as fictions or hypotheses.1In this connection let me refer to a remarkable and instructive book byA. Mouchot, Lart!jormeetenaue malh!maliques pures(Paris, 187]). Onthe analogy of the theory of thetwocoordinates,invented by Descartes for dealing withcurved lines, Mouchot regards every realpoint as consistingof two imaginary points. He also treats imaginarynumbersfromthis point of view. He then formnlates a " Principe des relations contingentes,"that bears some relationship to Herbart's "Methodof chance aspects," andto speak of cordes ideales, ofrayonset centres imaginaires, of imaginary variables,imaginary triangles,imaginary dimensions, angles-aU of whicb are dedllced fromthetheorycif imaginary points. Theobject hereis to approachrealitybycontingentand arbitrary methods andthus to see it in variouslights andrender it amenable totreatment. Theauthor relateshis theory tothat ofCharles(Apper;u etc.) in order to explainthe connectionofthe imaginary andtbe contingent. The'elations COnl'ingentes are thekeyto lheimaginary. Inthis sense the comme siplaysanimportant part in Mouchot's work. What is imaginary isregardedas ifIt were real and is substitutedfor the real. Mouchot speaks of various conceptionswhich serve as sltiles secours en geometrie superieure.150 PART I: BASIC PRINCIPLESThe same uncertaintyisfoundinLeibniz who, onthe onehand,declaredthat minimainfinite parvawere onlya modusdiceltdi,but inthe interestsofhis monodologywas inclined toassumethat they were hypotheses. Whether Leibniz hit upon hisidea through the influence of Bruno has not yet beendetermined. Itis notimprobable. But Bruno's principle of applica-tionwas different, for he used his minima in order to lay thefoundations ofmensuration, whileLeibniz was concernedwiththemeasurement of curves.Other mathematical fictionsrefer particularly to the infinite;as, for instance, infinitely distant points, infinite stretches,limits of infinite surfaces,convergence atinfinity, etc.In modern mathematics the employment ofsuch fictionalconcepts isquite general, but mathematiciansandphilosophershave sofar not developed anymethodologyfor these devices,though such a methodologywouldcertainly be very illuminat-ing asregardstheuse both of the infiniteandthe absolutefroma philosophical standpoint. Generally speaking, these fictionsare methodological accessories for arriving at results whichcould otherwise not be obtained at all or only with greatdifficul ty.Extensive application of the fiction is also made in mechanics,in mathematical physics, and even in chemistry, all ofthemsciences whichhave beenfullydevelopedonly inmoderntimes.Numerous other examples of the modern use offictionshave already been given in our classificatory chapters. Wethere saw how a number ofsciences have successfully utilizedthe scientific fictioninall itsdifferent forms. Thetruenatureof these devices was frequently realized, but they wereoftenemployed quite instinctively withoutanymethodologicalunderstanding. Hence anumber of famouscontroversies, turning onthe question whether certain concepts were legitimate or not.This questionhas already beenpartly discussed indetail above.The fiction may find some employment in philosophytoo, but here if anywhere cautionis necessary. It canneverserve as an explanation ofanythingbut onlyas a means ofsimplifying thought andforthepurposes of practical ethics.1Maimon put forward the viewthat Leibniz' monodologyand pre-established har-monywere only fictions; but withthis.1Descatles.created. methodological fictions: DUhringinhis Kritische Ges.chlchte d4r Phttosophte. p. ::61, well calls the1deaofII deceivinggod aValuablefiction, andalso other "tlopes." Absolute doubt is fOT Descartes also melely amethodological fiction.FICTIONS IN MODERN TIMES 151we cannotagree, forLeibniz interpretedhis doctrines otherwise.Buthadtheybeen fictionsthey certainlywouldhavebeen veryuseless constructs. It is onequestionwhether Leibniz desiredhis doctrines to be understood inthis way and quite anotherwhat value weare to attachtosuchconstructs. Leibniz un-doubtedly regarded his doctrines as hypotheses and not asfictions. Whether after theyhave ceased tofunctionas hypo-theses they can stilI be used as fictions-as we saw waspossible in other cases-is doubtful. This is more likely tohold for Spinoza's theory ofparallelism. For usthisisonlyafiction but one of tremendous scientific and heuristic value.On theother hand, metaphysically the relation between thephysical andpsychicalcanscarcely be suchasSpinoza andthemodern Spinozists, such as Bain (following Hartley), Lange,Wundt andothers, assume.Whether Kant's fiction ofa Dingansz"ch is still reallyofvalue to us requires a special investigation. But a sharpdistinctionmust be drawnbetween Kant's realization that theDngan sch is a fiction and his actual employment andutilization ofthis fiction. He himselfemploys it for scientificpurposes and inhis own hands it was transformed intoanhypothesis.We have then to distinguish two fads, first that Kantrecognized the employment of the Dng an sich up to hisown time as based upon a fiction, and secondly, that hehimself created the same fiction. What he recognized inothers he did not recognize in himself, namely, that his Dingan sz'ch was alsoa fiction.This error prevented himfrom recognizing actual s e n s a ~tions as the sole reality and from discovering that all realknowledge comes only fron) observation of the sequence ofsensations.Kant allowed the tacit provisional assumption that thereare egos and Things-in-themselves, to remainas a scaffolding.Had he destroyed that scaffolding and rejected themboth hewouldhave foundthatsensationwasthesole reality left.When, therefore, Jacobi says that "without the presup-position of objects as Things-in-themselves, andofideationalfaculties upon which they work, it is not possible to enterthe Kantian system, thoughwiththemit is quiteimpossibleto remaininit "-inother words that the beginningandthecontinuation of the Kritik are mutually "destructive "-he152PART I: BASIC PRINCIPLESwas quitecorrect. Kant, after havingdiscovered andassertedin the Kritik that Thlllgs-in-themselves are merely fictions,had only to recognize frankly that these presuppositions ofhis werenothingmore thanprovisional devicesfor thepurposeof arriving at his conclusions; he had only to recognize, inother words, that there is only empirical knowledge, and hewould have been left, as was Maimon, with sensations asthe sole reality. B\lt he allowed his schematic frame tostand; and whenever fictions do not drop , cohesion, crystallization,etc. ;but this application does not transform subjective methodo-logical means into an objective-metaphysical reality. Wemust notlookat thesemethods of visualizationandcalculation-for as suchFaraday, 5chonbein,Magnus, DuBois-Reymond,1 cr. Wundt, UbeYdieAu/gabe deyPlii/osophie in deyGegmwart,p. 6.~ [Joey dieA1ifgalJe fkrNaturwissutsduift, Jena1878,p. 7.222 PART II: SPECIAL STUDIESFick, etc., regardedtheatomic theory-as anobjective processof nature. Many scientists speak of atoms without reallymeaning to assume them: some even reject the reality ofempty space andyet continue tospeak ofatoms, although theassumption ofemptyspaceis a necessary' consequenceof theatomic theory. Unquestionably this conceptual method isthe most convenient one, but this constitutes, of course, noproof of its objective-metaphysical validity.Accordingto themorerecent viewsof physicists, Kirchhoff,for instance, all phenomenaare reduced toforces andrelativeeffectsof forces. For the physical specialist, matter is in noway dependent upon the assumption of extended minimalparticles. Matter forms an entirely emptyand meaninglesssubjectforthe forces and is but aninaccurate after-effect ofaviewwhich has gl'own accustomed to the idea of extendedand separated bodies, and which also assumes substances asbearers ofthe elementary forces. But this conceptual methodprovides a simplification of the theory, not only because particlesof matter are lookedupon as the supporters oftheseforces,but becausethey are regardedas infinitely small. The formerattitude is ofgreater value in making theabstract concept offorce concrete, thelatter insimplifyingthecalculation. Thatis why the atoms are allowed to remain, though everythingthat exists hasfound adequate expression in theforces, Weinterpolate this conceptual aid because it is so convenient.It is literally an hypostasized Nothing with which we aredealing, in thecase of theatom; for ifeverything has beendissolved andabsorbedintothe forces, what becomes of matter?And if the atomsare to be representedas infinitely small, howare they to be distinguished fromthemathematical point whichis also merely anhypostasizednothing?18FictionsinMathematica.l Physics.!INphysics, and particularlyin mathematical physics, as wellasin m'echanics, we make 'use of a number of fictional constructswhichare in part merelyconvenient, in part absolutelyindis-1 Sl4pplem4ntary to Pat'I), CkaptetXVI.MATHEMATICAL PHYSICS 223pensable. Faraday's" lines of force" possessing no mass 01"inertia, for instance, are to be regarded as auxiliaryideas forthe purposeof visualization. Maxwell tried to see in theselines of force !iomething more thanmeremathematiCalsymbols.But that Maxwell in this interpretation was contradict-ing the intention of -Faraday, the actual originator of theconcept, that, in other words, hecommitted the frequent errorof transforming a fictionintoan hypothesis, a mathematicalauxiliaryidea into aphysical theory, is best provedby Faraday'sown words. The lines of the magnetic force of gravitation,the linesofeclectro-static forceand the bent linesofforce, areall, according to him, imaginary.! Nospecialmeaning was tobe ascribed to these terms: he is convinced that he is notgiving expression by means of them to anyreal fact ofnature,although thismethod ofconceivingthings apparentlyfits thesituationandis very neat.2He desires to limit the meaningof thewords" line of force"in sucha way that they designatenothingexcept thestate ofthe force with respect toits sizeand dirtction, and do not involve any idea concerning thenature of the physical cause of the phenomena. How, forinstance, magnetic power is carried through various bodiesor throughspace we donot know.DAccordingto these state 'ments of Faraday,Zollner is unquestionablyright in rejectingMaxwell's interpretation of these lines' of force as physicalentities as agrossmisunderstanding. It is also quiteclear thatMaxwell made this confusionthrough alack ofmethodologicalinsight intowhat constituted the difference between a fictionandan hypothesis. Weknow this definitelybecause Faradayexpresses himself quite clearlyin a letter to Tyndall (14thMarch 1855), who, he says, is awarethat he (Faraday) treatsthe lines of force onlyas n}resmtatiotzs ofmagnetic power,and that he does not profess tosay towhat physical idea theymay thereafter point, or into what theywill resolve themselves.Faradaydid not allowhimself to be led astray bythe greatmathematical utility of his new conception, which was ofextraordinary value in the analytical deduction of physicalphenomena, intoseeing in it more than a "representative"idea. He protested against the misunderstandings of hiscoptemporaries, men like the DutchmathematicianvanRees,t .. Experimmtale Untersuchungen," 13041 in Zollner'. Wiumscba/I. Abhand-IUH,,'H, 82. ,.I Ibia, 84, 3 IMrI, 84.224 PARTn: SPECIAL S1'UDIESwho also seemed tofind a physical hypothesisinthis idea, indirect oppositiontoFaraday's clear declarations.This differentiation between hypothesis and fiction alsocoincides with the distinction drawn by Wilhelm Weber 1betweenreal andideal hypotheses.An ingenious artifice ofthought is that of the "fictitiousmean" of Jevons (we followhis Principles ifScil21lce), which,h'as been used in various connections.2It is exceedinglypopular inmathematical physics,inthose caseswhere a chainor group offorce relationshipsbelongingtogether, are thoughtofas united in an ideal mean point so that, should circum-stances demand, this totality can be applied in a computa-tion immediately. Since it would necessitate toocomplicateda calculationif every relationship were taken into consideration,a single unit is substituted forthemany, whichareregardedascombinedwithinit.We owe thefirst a.pplication of thismethodto Archimedes.He hit upon the veryingenious idea of constructingina givenbody, apoint in whichtheweight ofall theparts was thoughtofas being concentrated, so that the weight ofthewhole bodycould beaccurately represented by the weight of this point.The centre ofgravity thus takes the place of the weight ofinnumerable, infinitely small particles. eachof which is activein its particular position. Inorder toobviate thetremendouscomple:x:ity in calculation necessitated bythis circumstance-for the simplestmechanical problemwouldotherwise breakupinto innumerable particular ones-a centre of gravity is imaginedwhich is thoughtof asa point and treated as ifall the forcesof the individual parts were united there. Archimedes ex-plained the method for determining this centre. Thus inplace ofa sphere as centreofgravity, we have its indivisiblecentre which, in this case, still lies within the body. Butin the case of a ring, this centre of gravity is entirelyI "Elektrodynamische Messbestimmungeninsbesondereitber Dio.magnetismus ",Abkmldl. d. Sacks. Ges. d. w., I, 560. Cf. Zollner, p,.inzipiett dillereld,trodylla-misckmTheo,i, d,r JlJat,rie. 1876, I, 91 ; andthesameauthor's r-Vissenschaf/licheAb/"xndlu"r''', 1878, I, 45.2 The fictitious meanis employed in other sciences too, whenever the needarises of takingtheaverage of ll. number of gradually varying phenomena andma.king this the basis of further calculation or consideration; for instance, instatistics, meteorology, etc., where itisimportant tosubstitute for a largenumberofqllantities that oscillatearonnd anideal paint, a common quantity validfor themall. An average is therefore constructed, by means of which we make our computa-tion asifeveryone ofthephenomenainquestioncorrespondedtoit. Afamousfiction of this type is that of Quetelet, nam.:lly, his" homme moyen ", i.e. thefiction of a normal average indhidual.MATHEMATICAL PHYSICS 225imaginary; for here, instead ofhavingthe points ofapplica-tion of the forces in the formofa circle, wefindthecentrefalling in the vacant interspace. Thesame holds for twoormore bodies, whether these are separated or united. Here,too, apoint canbe foundthat can be treated as if the resultantforce ofboth bundles of forces were concentratedinit. Wecan, for instance, imagine a common centre ofgravityoftheearthand the sun, that is, a point that can be regarded andintroduced into calculations as though it tookthe place ofboththese celestial bodies as anindivisible centreexercisingexactly the sameinfluence uponathird point as the twobodiesdo in fact.We must also mention here as a peculiar and valuableauxiliary idea, thefictionof anabsolutelyfixedpoint.The empiricalperception ofallchangeandmotionis alwaysconnected with empirical points of reference, and it is onlywhen related to these that we can recognize it asmotion. Inother words allobservedmotionisrelative, relativeto us, to animaginaryorigin, relative to a fixed backgroundor relative tothe apparently stationary earth or sun. Theseare all merepoints of referencewhich wemust assume insuccession. Manbegins by assuming himselfasapoint of reference andscienceconstantlypostulates other points of reference becausethosetaken first prove tobe illusory, since they turnout tobe inmotion themselves. In order to prove definitely andabsolutelythe existence of motion, we must have an absolutely fixedpoint by means of whichthe speedandthe direction of themotion can be measured. Since, however, accordingtomodernviews, nosuchabsolutely fixed body canbediscoveredintheuniverse, science isfacedwithapeculiar difficulty.Neumann 1 deservesthe creditforhaving first demonstratedthat Galileo and Newton formulated their laws in such afashion that an absolutemotion was assumed. Galileo's law ofinertia cannot, according to Neumann, possibly remain as astarting point formathematicaldeductions. Wedo notindeedknow what we areto understandbymotionin a direct line jindeedweknow thatthese wordscan beinterpretedin variousways andarecapable of innumerable meanings. Motion, {or1Neumann, Ober die Pn'nzjpien der GalileiNewton'sclun Theone, Leipzig, 1870.p226PART II: SPECIAL STUDlIsSinstance, that takes place inastraight linewithregard toonecelestial body,willappear ascurvedwithregardtoevery other.Wemust therefore begin witha special bodyin the universeand employ it as a basis for our judgment; use it, in otherwords, as the particular object withrespect towhichall motionis to be calculated. Only then shall we be in a positiontoconnect adefinitemeaning withtheabove words.Towhat bodyshall weassignthis place of pre-eminence?Galileo and Newton giveus noanswer. They simply assumeabsolute motion without being clear in their own minds orbeing conscious of thefact that thispresuppositioninvolvestheexistence of such an absolute and fixed point of reference.That this condition is necessarily involved was first clearlybroughtout by Neumann, thoughthere is adefinite reference tothis matter inDescartes. It is for that reasonthat Neumannsets 'upas the first principle ofthe GalileoNewton theory theproposition that all conceivable motionsexisting in theworldareto be referred toone and thesame absolutelyfixed body,whose configuration, position and dimensions are unalterablefor all time. He calls this body"thebodyAlpha." We areto understand then by the motionof apoint, not a change ofposition withregardtothe earth or the sun, but onewithregardtothisbody.What is attained by this conception? This, that a clearcontent is given for the first time to the determination ofrectilineality in Galileo's law: the rectilineal movement is tobe understood inregard tothisbody alpha. This may also beexplained as meaning that every motion from nowon isthought of as absolute. The nature and the really essentialcharacter ofthis 50-called absolutemotion, consists inthe factthat all change ofposition is brought into relation with oneandthesame object, indeed with anobject which, asNeumannexpresses it, is spatially extendedand unalterable hut whichcannot be further described. If, however, we do not assumeabsolutemotion, then the wholeGalileo-Newton theoryfalls tothe ground i for, in that case, sinceevery body in the universeisactuallyinmotion, motioncould onlybe definedas arelativechange of position of twppoints with regard toone another.We should then arrive at a theory which is fundamentallydifferent from the Galileo-Newtonian, and whose agreementwith the actually observable phenomena might be verydoubtful.We must insist again, therefore, that an absolute motion inABSOLUTE SPACE 221absolute space is a necessary presupposition of the Galileanlaw ofinertia. Inorder to simplify the conceptionofabsolutespace wehavethe body alpha.We shall never beable tofindanempirical point that willsatisfy the above conditions. For that reason weassumeanideal point that serves the same purposes. This is howNeumannunderstands his body alpha.We can thus perceive what avery peculiar construct thisfictionis. It representsanaccretionto reality, anintercalationthat is to make conceptual mobilityandthe determination ofconceptseasier. Inthe final examination ofreality, thisinter-polated element must therefore drop out andbe eliminated.Indeed, as soonasthe connections andmediation, for which thefiction has been created and introduced, have been accom-plished, the fiction loses its significance and drops out of thefinal calculation. We consequently find no mention of thisbody alpha in experimental physics, for it disappears assoonas the mathematical formulc.e have been discoveredandapplied.The same service is rendered by other auxiliary aids ofmechanics and physics; for example, the ether oflight andthe electric fluid which, according to Neumann, serveonly forpurposes ofvisualization and of connecting the calculation;and the intermediate element drops out as soon as this con-nection hasbeen achieved. Thesescientific interlopers arenotincluded inthe council that definitely determines therelationsexistingin actual realityand, like all temporary makeshifts,are excluded fromtheprinciples in thereal andnarrower sense.19TheFictioJJ of PureAbsc>luteSpace 1IT is thefalse assumptionthat mathematics can proceed inasensedifferent iiiapriori fromthat of any other science, andthat,in mathematics, everything is magically extracted from themind itself, which is immediately responsible for a distortedidea of the logical meaning of the space concept. The questionis: what is spacefromthe logical standpoint? What logicall SZlppltmmtary to Part I, Cna}ttrs Xand X PI, p. 52 If.andp. 73.a Theapriori and deductive procedureofmathematics does not differ fromthedeductive procedurepossible in ather sciences in essence or qU\1lity. but quantitativelyand in degree.228 PART II: SPECIAL STUDIESrank does mathematical space occupy? 1 It is the "pre-supposition" of mathematics. But"presupposition" is anambiguous word that does 110t express any definite logicalvalue. "Presupposition"maymeansomethingthat is empiri-cally givenanduponwhichmathematicsis essentially based, orit canmeanthat space is an hypothesis without which mathe-matics couldnot exist. Mathematicalspace is unquestionablyanessential presupposition, but in neither ofthese twosenses.It is quite easytoprove that space, inthemathematical sense,namelyas a pureextensionin three dimensions, is not thing actually given, a real fact. Empirically we find onlyindividual bodies possessingthe fundamental character ofex-tension but never ageneralor purespace. It is tt,ue that thecircumstancethat all objectsare perceivedas fromauniform background (generally a light one), and the trans and colourlessness of the air, make it appear as ifindividualobjects layina uniformlyperceptiblevacant space.This peculiar circumstance has unquestionably facilitated theemergenceofan independent andabsolute ideaofspace, butwe have no right to interpret thisas implying that mathematicalspace is something empirically given. Indeed, for that veryreason, nobodyhasseriously contended that it is. Themathe-matical ideaofspace has not,therefore, the logicalvalue of anexperience. Perhaps it possesses that ofan hypothesis? Butthenwe encounter evengreater difficulties. How can anideaso absurd and so contradictory make any claimto be anhypothesis? Mathematical space is a something that is anothing anda nothing thatisa something. The contradictionsinherent inthe concept of anunoccupiedmathematical space arewell-known. A vacuumwould besomethingcontiguous andseparated where we find nothing contiguous and nothingseparated. If space is the relation of co-existence of real.objects, then, in theabsence of these, it must benothingandwould disappear with them. Since, however, the primarycharacteristic of a useful hypothesis, is its freedom fromcontradiction, such a contradictory concept as an absolute,unoccupied, mathematicalspace cannot be anhypothesis. Andit is thisvery contradictoriness that prevents usfrom contentingourselves, without further ado, with the favourite expressionof the mathematicians that these and similar concepts are1 We are not concerned hereprimarily witl) the psychological questionnor withthe epistemological, but more particularly withthe logical problem.ABSOLUTE SPACE229"postulates": for thislast concept isvague and indefinite. Weare consequently forced to askthe very pointed and embarrassingquestion: what logical position then can the idea of space occupy?In viewof the fundamental importance of this point forour subsequent argument andtheremarkable clarity with whichLeibniz in the main treated it, let us pause toexamine hiscontroversy withClarke. This controversy, inso far asit boreon the question of space, turned on the problemwhether theconcept of an absolute, geometrical or vacant space, wasjustified or not, i.e. whether there was any actual vacuuminreality. Clarke, together with Newton, defendedthe existenceofanabsolute space (andconsequently ofanabsolute motion).Withinthis absolute space, in a general but definite position,the universe, i.e. the material world, is located; and between. these bodies that are, as it were, swimming inspace, is alsoto be found absolute and vacant interspace. This theoryLeibnizattacks. "IIn'y apointdevuide dutout" (Erdmann'sEdition, 748); such isthe thesis that Leibniz tries toestablishon theological, physical, mathematical and logical grounds."L'espace reel absolu" (Ibid. 751), is nn iI idole de quelquesAnglois modernes. Je dis 'idoleJ non pas dans un sensTheologique mais Philosophique, comme Ie chancelier Bacondisoit autrefois qu'il y a Idola Tribus, Idola Speeus." Herepeatedlyenumeratesthe grandesdijJicultl!s andcontmdictionsto which this concept leads. It is particul\l.rlyby means ofhis "Principe de la raison suffisante" that he attempts todisprove the U imaginations," the "suppositions chimeriques ",and the" fictions impossibles" ofhis opponents. As a matteroffact, ofcourse,the ideaofabsolute spaceand absolute timedoes lead to peculiarabsurdities, andLeibniz'refutationisquitejustified. Heconstantly calls these concepts ofabsolute timeand space"chimeres toutes pures" and"imaginations super-ficielles". Theyare "fictions impossibles" (771). We might,for instance, at will thinkof anyspatial position inthe worlddisplacedanydistance in absolutespace; since, however, thetwo points cannot be distinguished, they will remain merelyideal and imaginary and the presupposition that this dis-placement ispossible, i.e. thepresupposition of absolutespace,is a merefiction.1Thefact that there is nosufficient reason1 We see here clearly how, in Leibniz, "imagination" and "fiction" aresometimes usedinaderogatory sense, as whenhe rejects the above idea. as unrealin themetaphysical sense, and sometimes ina good sense,as when he yet recognizesthat an idea is methodologically justified andeltpedient.PART II: SPECIAL whyGod should have createdtheworldat anearlier momentthanhe actuallydid,proves that thiswhole method oflookingat the matter and the presupposition of absolute time uponwhich it isbased, is false. What holds oftime holds alsoforspace. This idea that absolute space is a chimerical suppositionand animpossiblefictionruns inallpossible variations throughthe whole of Leibniz' correspondence, whichissoimportant forhis philosophy.Letusnow attempt to show how this dispute can be adjustedby means ofa verysimplemethodological distinction, forhereweare concernedwiththelogical and metaphysical valueofthe concept of absolute space. As it certainlyis not an experi.ence, theonlyquestion whichcanbe involved is whether weare dealing witha justifiable hypothesis or a justifiablefiction,a fictionin the sense weoriginally fixed. We saw howLeibnizproved that thisconcept was contradictory and impossibleandhow, for that reason, he rejectedit. Onthe other handwe shallsee thatClarke stresseditspracticalnecessity andutility, baseduponNewton's mathematical natural philosophy. Leibniz callstheideaa fictionin aderogatory sense. Heuses this concept,indeed, veryfrequently and, as we pointedout above, in thetwo different meanings of a good anda badfiction. Had nothis enmityagainst the Newtonians, and the badfeeling thataroseon bothsides in consequence, disturbed Leibniz' clearvision, and hadhis correspondence withClarke not takenplaceinthe later periodof his life when he was isolatedandem-bittered by misfortune, he would probablyhave applied here,too, the fundamental discovery that he hadarrived at incon-nectionwithotherquestions; namely that there are necessaryandjustified fictions. He might thus, perhaps, have found thecorrect solution, that the idea of absolute space isanable auxiliary idea, i.e. that, althoughit isinitself contradictoryand therefore imaginary and ideal, it must of necessity beformed for thebuildingupofmathematics and mathematicalphysics.This simple solution clears up at one stroke the wholepassionate strife between Leibniz and the Newtonians. Allthe reasons advanced byLeibnizgo toshowthat the conceptis imaginary, all thecounter-reasons advancedbyClarkethatit is necessary. As is sooftenthe case we here find a con-,tradictory conception (whose exact definition we owe toNewton) at .first attacked because of its logical difficulties;ABSOLUTE SPACE 231we then seeit pass over intothegenel"al consciousness, becomean everyday idea, until it is again attacked and, thoughdeprived in the end of its reality, yet admittedand allowedto persist because of its indispensability.Intheambiguity anddouble-edged nature ofthe concept"supposition ", we againrecognize theduality inlogical m e a n ~ing that gives totheseconcepts ofabsolute space, of the atom,etc.,sovarying anduncertain anappearance. Clarke proceedsfrom the necessity of this supposition, from the fact thatLeibniz himself makes itj Leibniz, on the other hand, callsit chimerical, sophistical andimaginary. Inthe meaningofII fiction" developed by us, both views are united; the con-ception is nonsensical but fruitful.Leibniz, indeed, had this solution in his hand but he didnot express it clearly. At one place (769) he himself callsattention to the fact that such "choses purement ideales ",even if their unreality is recognized, are useful (" dont laconsideration ne laisse pas d'etre utile "). This gives us thetrueideaof the methodological fiction. That Leibniz merelyhinted at an idea with which he was quite familiar and didnot fully demonstrate it, can only be explained by the factthat he allowed himself to be carried away by passion.Otherwise in a calm \ discussion be would have recognizedthat the suppositions of Clarke were necessary and usefulfictions.Considering the fundamental nature of this point, it is ofinterest to cite other places in Liebniz' writings which showhis attitude without ambiguity. Thus, for instance, in hisII Repliqueaux Reflexions de Bayle" (Erdmann, 189; writtenseventeenyears beforethe discussion with Clarke)he remarksthat the mathematical ideas of time, extension, motion andcontinuity are merely "des chases ideales." He agrees withHobbes whocalls space a jJhantasnza i!xz'stentis. Extension is"the arrangement of possible coexistences." Of particularimportance is apassage (190) where hesays that although themeditations of the mathematicians are ideal, this does notdeprive themofany of their utility. He consequentlyknewhowto value the usefulness ofsuch concepts (191), althoughquite consciousof the fact that mathematics does not furnishthe most fundamental knowledge and that this is to besought in the more important calculus of Metaphysics, in the"Analysis of ideas ", for which we may substitute-without282PARTn: SPECIAL Sl'UDIESdepartingtoo much fromLeibniz' meaning-inonedirection,at least, the Theoryof Knowledge and a methodology connected with it.Pure mathematical space is a fiction. Its concept hasthe marks of a fiction; the idea of an extension withoutanything extended, of separation without things that are toto be separated, is something unthinkable, absurd and im-possible. For mathematics, however, the concept is necessary,usefulandfruitful, becausethemathematiciansonly investigatethe characteristics andJaws of extendedobjects, quaextended,and not their materiality or other physical properties. Theconcept of purespace arises from retaining the relation ofobjects after the things themselves have already beenthoughtaway. While permitting matter and its intensity to begradually reduced to zero, we still preserve the relation ofmaterial objects. Although, strictly speaking, space shoulddisappear at theveryinstant in whichmattel'has beenreducedto zero and thus disappears, we still retainthe relation evenafter the related things have valiished. If we observe anobject incontinuous extension and if we mentally allowthematter to become thinner and thinner until it reaches zero,then pure space is the limit when matter is conceived asdisappearing and the intensity of the occupying matter isconceived as being consumed and expiring. This is themoment we seize holdof. At the very next moment n o t h i n g ~nessbegins, the zero is substituted for matter, which is seizedand retained at the very last moment of its flight andexpiration.We have so far in the course of our investigation cometo the conclusion that the concept of space, i.e. the conceptof pure and mathematical space, is formed by a peculiarprocess of Ollr conceptual faculty in which abstraction andimagination work together in a remarkable manner. Abstrac-tion detaches somethingwhich we experience onlyin Some-thing else (whether as property or as relation) fromthis otherentity-from something to which it is so firmly and in-extricably bound that when what has been detached isaccuratelyanalysed we are forced toadmit to ourselves thatnothing remains in our hands. Abstraction takes fromthesubstratumand the elements their attributes and relations.ABSOLUTE SPACENow, strictlyspeaking, these detached pieces, apart fromtheiroriginal context, have no meaning: they evaporate intonothingness and lead to absurdities. Imagination, by reasonofitsspecific and peculiar gifts, comes to the aidand rescuesabstraction which, as described above, has dissolvedthegivenworld into nothing and stands looking round helplessly atthe result of its activity. Imagination reintroduces into theisolated relation the idea of the related elements, but in aform in which they are only shadows of what we find inreality. I t thus provides a support for the product of ab-straction and prevents it from falling into the abyss ofnothingness.What we must do, therefore, is to make clear to ourselves thatthespaceof themathematicians is nothing buta scientific andartificial preparation, which differsfromtheschematic auxiliaryconstructs, etc., of other sciences, only in the nature of theobjects thataretobeinvestigatedandnot inthemethodofinvestigation. This unity ofmethod mustbestronglyempha-sized. Onlya methodological approach can purge us of ourold prejudices about the objects of "mathematics. Onlythemethodologist, byfollowingthedeviousroutes of human under-standing, can demonstrate how, in mathematics also, exactlythe same methodological principles are valid as in the othersciences. The objects of mathematics a.re artificial preparations,artificial structures, fictionalabstractions, abstractfictions, asweshall prove in the followingpages inconnection withparticularmathematical constructs. Here we are concerned with theconcept of space in general, with pure absolute space, - aperfect example of a normal and scientific fiction. Thereistherefore no object in trying toargue away the blatant con-tradictionsinherent in' this concept. Tobe a true fiction, theconcept of space should be self-contradictory. Anyone whodesires to" free" the concept of space from these contradictions,would deprive it of its characteristic qualities, that is tosay,ofthe honour of serving as an ideal example of a trueandjustified fiction.234 PAUT II: SPECIAL STUDIES 20Surface, Line, Point, etc.. as lllictions1WHAT holds for pure absolute space holds also - mutatismutandis - for the single mathematical spaces and parts ofspace, andfor theideaof the s o ~ c a l l e d mathematical bodies,such as sphere, cylinder, cube, prism, etc. Thepsychologicaland logical foundationsof these constructs arethe correspondingempirical corporeal objects. Andhereagainwe findabstractionandimagination participating in the manner described above.The corporeal is reducedtoa minimum, finalty to zero; andtherewith, from Cl; strictlylogical standpoint, theboundaries ofthese corporeal objectsmust fadeawayand, sotospeak, mergeinto themselves. But sincewe areabstractingonly fromtheoccupying content, the formis still retained, and before alltheseboundaries, completely deprived of their content, collapse,they aresupported by the imagination which, as the contentdisappears and becomes infmitelythin,holds themin placeasinfmitely thinshells, emptyhusks, as askin, a covering, indeedevenasa mere frame. Suchforms, without a content, are, assuch, nothing, indeed worsethan nothing, for theyare contra-dictoryconstructs, anothingthatis neverthelessconceived as asomething, a somethingthat is already passing over into anothing. And yet just these contradictory constructs, thesefictional entities, are the indispensable bases of mathematicalthought. The boundariesofthe empirical bodiesaretakenassuch, and are abstracted and hypostasizedi and with theseimaginaryconstructs mathematics, and particularlygeometry,operates.The same-mutatismutandis-holds for the surface, thelineand the point. That the surface is the boundaryofa bodyis avery old definition. Historically and psychogenetically,of course, we are first concerned with real II planes," i.e. flatboundaries. The concept of curvedsurfaces arises later. Thereare flat surfaces, i.e., really plane constructs, in nature, as wellas the great number due tothe primitive participation of man;here we abstract fromthe material that forms thesurface, andtheformal element istaken alone initselfand made indepen-dent by imagination. In this casealsoitis a contradictionto1 SupplementarytQ Part 1, Chapter.Y, p. 5I if.SURFACE, LINE, POINT285speak of a surfaceas such; andyet scientific thought proceedsunconcerned alongits path, inthefaceof these and evenmorepronounced contradictions. Ifthought were toallow itselftobe heldupby such contradictions it would never be able tomove at all.The same is true of the line-as the "boundary of thesurface." Oflines, too, there is nolack, either in natureorinprimitive art, but they are, so to speak, immersed in thecorporeal. It is abstractionthat first picks out these linesassomethingspecial, with anexistence of theirown, andthencallsin the aid of imaginationtohypostasize thesestructures. Butthat theyare merelyfictional concepts is self-evident. Whatthemathematician, the geometrician, draws onthe blackboardor on paper, andcalls a line, is nota lineinthemathematicalsense, for it always possesses a second (and even a third)dimension even if that has been reduced to a minimum. Aline, inthe mathematical sense, cannever be sensuouslyrepre-sented,for it isamatter of abstractionandimaginationand, inall cases, remainsacontradictoryconstruct.The samenaturally holds forthe pointwhichweare accus-tomed to call the limit of aline. Here, likewise, mathematics,on the basis of certain sense-experiences of which there aremanybothin natureandamongmankind, has constructedthenon-sensuous, wemight saythesuper-sensuous idea, of a pointwithout extension in anydimension-an idea in itself bothuntenable and contradictory, a monstrous concept despite itsinfinitesimal size, ofa somethingthat isalready anothing, of anothing that is neyerthelesssupposedtobeasomething. Themathematical point IS, in all respects, a true andcompletemathematical fiction.A point as a zero-dimensional construct, is, in itself,entirely contradictory, though as necessary asit is absurd. Aconstruct without any dimensionis, in itself, a nothing. But theone-dimensional construct of the line andthe two-dimensionalconstruct ofthe surfacearecontradictory ideas. In reality weknowonly material, corporeal objects, out of whose peculiarcharacteristic of extension we abstract the threedimensions.The two-dimensional construct of the surface and the one-dimensional construct ofthe line that weoccasionallyappearto observe inthese bodies, are onlyabstractions individualizedby the imagination, in other words, fictions with which weoperate as if there were realities corresponding to them;286PART II: SPECIAL STUDIESnecessary conceptual aids and auxiliary concepts that helpus,indeed, in thinkingbut which cannot give usanyknowledge ofreality. We are hereoperatingwith unrealities and notwithrealities; but they are useful and indispensable unrealities.We regard these unrealities as real, however, because we areaccustomed toregard everythingas real towhich we give aname, without realizing that we can bestow anameon unrealas well as on real things. Anyonewhorealizesthis, andwhofurther realizes that certain unreal ideas are necessary anduseful,has graspedthetrue, scientific concept of afiction.In the examinationofthe surface, the line and the point,another point of view canalso be applied. Hithertowehavetakenthese constructs as limits, inthe sense that they are limitsmade self-subsistent byour imagination, although theyare, ofcourse, merelylimits ofa somethingthat has been detachedfromreal objects by our abstraction. But here, too, we canintroduce the concept of a flux, ofthe progressive diminutionfromsomething real tozero, andthus allowthe constructs inquestion toarise insuchawaythat we canstop theprocess ofdisappearance atthelast moment, as we did above in connectionwith the originof pure mathematicalspace. 21TheFictionof theInfinitelySmall1INorder tounderstandthe functionof the Infinitely Small it isnecessarytoexaminein detail the natureoftheobjects withwhich it is concerned. The mathematical constructs are theabstract forms of spatial and temporal contiguity and succes-sion. One of the fundamental characteristics of the latter,which, in the last analysis, is something definitely given, istheir division into genera and species. Thus we have forinstance the genus of the conic section which is subdividedinto the various species: circle, ellipse, parabola, hyperbola.Wehavehere clear-cut andaccurately definable modificationsofthe general formofaconcept, differentiating agenus into anumber of different species. We canonly proceedfromonespeciestothe other bymaking a conceptual leap.1. Supp/emmtary to Part I,ChaptersXIl(p. 56if.) and XlII.THE INFINITELY SMALLNowmathematical constructs possess one property, namely,the possibility ofa progressive andcontinuous diminutionandenlargement. To which must be added the property thatthrough this progressive diminution (or magnification) of anelement of such a concept, it constantly approximates moreclosely to a neighbouring element. The conceptual formulaofthe ellipse demands, for example,the presence of twofociwhichmust be afinite distance apart. This distance itselfisundetermined and can be made arbitrarilylargeor small; 50long as the necessary condition is adhered to, we have anellipse. It is, however, an objective and undeniable fact thatthe closer these foci approach toeach other, the nearer doesthe ellipse approximate to the circle. From this it followsthat when this distance disappears entirely, the ellipse haspassed into a circle. This transition from the one type tothe other is, however, in the last analysis, only possible by asudden jump which takes us in one instant into a differentfield. By definition, the ellipse must possess an eccentricityandtwo foci P and P, distance mapart. Anentirely differentformarises as soonas thisdistance disappears. But between,the presence of 1ft and thedisappearance ofm, absolutelynothird form exists. The concept of the ellipse has 'f1Z as avariable element.It is afact, then, that thegraduallyapproachesthecircle throughthesuccessive diminution of mi when mbecomes0, the circle takes the placeofthe ellipse..Fromthese factsour conceptual faculty forms somethingnew, which, however,remainsentirelywithin thedomain ofconception or imagina-tion. The more I divide 1ft the smaller it becomes. I cancontinue this sub - division ad nfinitum. What if I nowi.e. form the fiction, that this infinitely progressingdivisionhadbeencompleted? Iwould, of course, be indulgingina crazy logical contradiction, but I should also secure anadvantage. If-I am only assuming this in imagination,fictively-this infinite division had been completed. then thelast part would not be finite but infinitelysmall. Andifthedistancebecame infinitely sman, F and P' would coincideandyet ofcourse not coincide. There would still be a distance,but it wouh:l nolonger bea real distance, because it wouldnolonger befinite. Let us imagine this quite chimerical case.What is its purpose, what haveI gained thereby?Itwas our purpose in the previous paragraphs to show the238PART II: SPECIAL STUDIESexistenceofa constant transition between the ellipseand thecircle, inothet words, to thinkofthe circle as a special caseof the ellipse; we didnut wanttoleave thelimitsof our ,speciesin order to arrive at the circle. But what value can thispossibly possess? Is it not mere play? Not in the least.For if I cansaythecircleis tobe regarded as an ellipse, Ihave theright toapply the laws of the one to the other.Thematter can alsobeexpressed thus: insayingthat thecircle is an ellipse I make a mistake, for in the ellipsethereare twofoci,inthe circle only one centre. But I am making thiserror progressivelysmaller bycontinually decreasing m,' andthe error will become infinitely small when m isassumedto be" infinitelysmall ". It is true that indoing this, Iammakinguse of a concept full of contradictions and, in fact, make asecond mistake; nevertheless, I attain mygoal of beingableto treat the circle onthe analogy of the ellipse.Weare, then, here dealingwith a forced and compulsoryanalogy,with an unjustified transference. I proceedas if thecircle were an ellipse and I attainthis throughthe idea-as ifsuch an infinitely small distance existed-in otherwords, Iamoperatingwithpurely imaginaryconcepts, which are, however,fruitful fictions.Ifwenowconsider thefunction ofthe "infinitelysmall"from a more general point of view, we shall see that thisconcept serves the purpose of permitting us to regard asidentical, constructs which are closely related and of whichoneis constantly approachingtheother through a diminution(or anenlargement)in oneofitsconceptual elements, without,however, coincidingas longas the conceptual element remainsat alL But wherever onespecies ofconstruct can bereducedto another and tothelaws of theother, the task of thoughtis simplified. Thisconcept, then, by creatingaforced analogy,serves as a bond between different specieswith theobject ofsimplifying the thought process.lFor that reason the concept of the" infinitely small" mustof course be contradictory. There is, as we have shown, aconceptual jumpbetween the mathematical species, for there1 !,>s SQun asthis principle has beengrasped it Canbe veryextensively applied.For Instance, theline canbe regardedas a surfacewhoseseconddimensionis in- small and. theas a bodywhose third dimensionis infinitely ,small.SImIlarly, the straIght lme cail be regalded0.3so-called extended anglebythmkingofthelilledb as dividedat any selected point x andregardingthepartsaX andPGhas tbe sides of an anglewhose inclination is infinitely small.THE INFINI1'EJ..,Y SMALL289ois an eternal chasmbetween nothingand something; and sothe concept in question must itself be across between somethingandnothing. If it is tobe the intermediary betweentwo speciesthat differ by reasonoftheabsence or presence of an element,thenif we are tosucceed in regarding one as a special caseof the other it must be possible toimagine, conceive, or pictureeither thepresence of the elementas an absence, orits absenceasa presence. This contradictory taskthe conceptunder dis-cussiontakes uponitself by regarding the absence of anelementas the presence ofaninfinitelysmall part ofthis element. Inthe" infinitely-smallJ) we find, indeed, boththe nothingand thesomething at one and the same time. In order to act as amediating concept, the infinitely-small must combine within itselfthese contradictory conditions; anditistherefore atrue fiction.There are cases where this fictive method of approachrepresents more than mere dialectic playand an unnecessarymediation, namely, where a direct d e ~duction froma formula will not work;this indirect path, this circuitous mad,then becomes theonlymeans of a t t a j n ~ingour goal. Thisalreadyholdsfor themeasurement of the area of the circle.We should never be able to arrive atthe formula for the area of the circlefroma study of the circle as a drcle,simply because the desire to determinethe surface of the circle is based upon the fictional conceptthat a common measure can be found for rectilinear andcurved figures. Where such a formula cannot be directly de-duced, thecircuitous path through theinfinitely-small rendersexcellent service; byadmittedlyregarding the circle, in thiscase, as a special instance, a limiting case, of a polygon, weobtainthe formulaby this fictivetreatment. We act, we speak,we think, wecalculate-as ifthe circle were a regular polygonwith an infinite number ofinfinitely small sides. We createthe fictionofinfinitelymany, infinitelysmall sides. Thus thefiction of the infinitely-small is here not merely useless playbut possesses both a meaningand a justification and is, as wehaveshown, at least a convenient method of speech, a con-venient concept. Fromtheformula of the arc ofa polygonF=:!CM(AB+BC+CD+ ... +NA),(whereCM=theradiusof the inscribed circle), weobtaintheformulafor the areaof240PART II: SPECIAL STUDIESthe circle by considering that, in this case, if we regard thecircle asa regular polygonwithan infinitenumber of infinitelysmall sides, theradius of thecircumscribed circle (the sideofthetrianglesintowhichthe polygon is brokenup) andthat ofthe inscribed circle (the altitude of the triangles) differ infinitelylittle; so that we are justified in substituting the radius ofthe circle in question for the factor CMj by means ofwhichand other modifications, we obtain the well-known formulaF=r?r.These examples, which can be multiplied at will, provethat the" infinitely-small" and the "infinitely-large" are bothmediating concepts between dissimilar constructs and thatthere is nothing mysterious to be looked for behind them.lThe "passage through the infinite," as this artifice has beennamed, is a perfectly transparent methodological process, asour analysis has shown. The contradiction is absolutely in-dispensable for these two concepts since theyare toconnectfields that aredissimilar, exclusive by definition, and there isan element absent inthe concept of the one that is containedin that of the other. But weformthesecontradictory conceptsin the full consciousness of their contradictory nature: fullyrealizingthatwe are constructingfalseandimpossible conceptsfor a practical scientific purpose-in a word, that they arefictions.Ifwe nowput the question, howwe are to explain theriddle thatby means of such illogical, indeedsenseless concepts,correct results are obtained, the answer lies inwhat we foundabove tobe the generallaw of fictions, namelyin the correctionof the errors that havebeen committed. (Cf. p. 109 ff.)Thought obviously makes a mistake, as was sufficientlyobvious in the last example. This error consists simply inthefact that the circle is regardedas apolygonat all. Since,as isperfectly clear fromthe elementary definitions, these twoconstructs-circle and polygon-arespecificallydifferent, it isabsolutely impossiblelogicallytosubsume the one constructunder theother species. The error is thereforemanifest. It1 With regard to the infinite let me add here what Gauss says about it'(Bn'ifwec1ml, Vol. II,211). He says that theinfinite" is merely afafPtt de parler,for we art; really speaking of limits to which certain relations apprOXImateas closelyas we Wish th.emto, while others are allowed to increase without limitation.",li'afi>n tk parler-so tooLeibniz; had already designated the infinitein every respect&$IImPdus diclttai.THE INFINITELYSMALL241is self-evident, however, that an error, wherever introduced, isboundto be a disturbing element in the final result; but since inthecaseunder discussion this does not happen, the result andtheinferences drawnfromit being specifically true for c a l c u l a t ~ing the area, this canonly beexplained bythe fact that, insome way or other, this error is corrected. And such isactually the case. Letus consider the arcmn which we assumetobe equalto the angle mpn (mn=mp+pn). Here themistakeis obvious, and the equation positively false. This error is,however, corrected and made harmlessby assuming that rn ~ n both mcmbers of theequation are infin- ite1y small, the arc aswell as the angle, and that, as remarkedabove, we infer there- fromthat thiscircleislooked uponasa polygon with anin-finitenumber of infinitely smallsides. Bythe constant diminu-tion of mnand the correspondingmultiplicationofthe numberof angles of the inscribedpolygon, the error that has beenmadeis just as constantly decreased in size. Because this is thought ofas continued ad infinitum, the errorismade infinitely small, thatis, becomes zero, Thewhole secret consists, consequently, incompensatingfor anerror committed. This correction takeson a specificformin these and similar cases, sothat one erroris compensated for byanother; and for that reason we arejustifiedincalling thiswhole procedure" the method of doubleerror." This secondmistake consists in the illogicalassumptionofinfinitelysmaIl or, ifitispreferred, an infinitely-large entity(one referring to the size of the sides ofthe polygon, the otherto their number, both being dependent upon one another).Theabove equation asit stands is false: it loses infalsityasthe sidesbecome smaller andtheir number larger, but itsfalsityremains finite as longas the quantities connected withit arefinite. As we indicated above, the error becomes infinitelysmall,i.e. equal to zero, as soon as the quantities pass over intoinfinity, except that the positingofan infinitely.small entityconstitutes another error which compensates for the ::first onecommitted. After the two mistakes have thus nullifiedeachother, the calculation is freed fromboth; the result becomesquite correct after the'first error has been made good by asecond. Nowmany propositions that hold for polygons can,mutatis mutandis, be transferred to the circle; at any rate,those that admit of such a transformation. Adefinite andQ242PART II: SPECIAL STUDIESuseful object has consequentlybeen attained by this fictionalanalogy. The treatment ofthe circle as if it were apolygonhas thus proved itself a fruitful conception: I act, speak,think, and calculate as ifinfinitely small sides of polygonsexisted, as if an infinite number of such sides existed,and could be completely summed into a finite quantity;and with all these false concepts I arrive in the end at a correctresult.As is well-known, our example-thereduction ofthe circletoa.polygon-is not the onlyone of its kind. The same fictiveanalogical methodis veryfrequently applied. Onthe sameprinciples we treat the cylinder as a regulat' prismwith aninfinite number of sides and, by means of this fiction, theformula for the volume of the prismV::::: It. Fis, in the formV = Ie. r2 'lr,alsoapplicable to the cylinder. Similarly, we canlook upon the cone as a regular pyramid with an infinitenumberof sides andthe formulafor the volume of the pyramidV = th.. Fholds, inthe formV"= i h.,.27r, for the cone. It isthe same in calculating the area of the sphere. For thispurpose we first consider the surface described byapolygonrotating round an axis, and transfer the lawwhich we thusdiscover to the surface described by a semi-circle. Tode-termine thevolume we canpicture the sphere as broken upintoan infinite number (j)f trilateral pyramids withinfinitelysmallbases groupedaround acentre.In all these cases we find the same principle of fictiveanalogy, according to which the curved line is regarded asmade up of aninfinite number ofinfinitelysmall straight lines.Instrict logic, as we havealreadyremarked, we could neversubsumethe curved under the straight line. All the laws ofrectilinear figures hold only for such, and rectilinear figuresremain rectilinear even if we increase the number of theirangles to infinity. Weshould never beable to cometoanylimiting boundary and no point could be given where therectilinea.r would suddenly take a leap and become curved.Truly enough both would continually approximate t.o eachother, but approximation does not mean contact, nor is itcoincidence. Nopossible multiplication ofthesides can leadto a coincidence. Thus we see the error that is made byidentifyingthe two. Howthis is corrected wehave alreadyindicated.There ,area' number of other instructive examplesof thisTHE INFINITELY SMALL248method. It may, forinstance, beof value toadopt thereverseprocess andsubsume thestraight line under thecurve. Sinceacircular line with a very largediameter closelyapproximatestothe sttoaight line, the latter canberegardedas a segment ofa circle havingan infinitediameter; astraight line is treatedas ifit were a portionof the periphery of acircle withaninfinite diameter.Thesameholds for the point, i.e. the mathematical point,where it is