,International Library of Philosophy and Scientific MethodThe Philosophy of 'As if'1) 11 i 1() s() 1) 11 Y () f, As i f'A Systemof the Theoretical, Practical anLl Religious Fictionsof MankindByII. VAIII I Nt;{l',HC. K. O(;{)ENIJONDONKEGAN PAUL, TRIi:NCH, TRUBNER &: CO., LTD.~ E W YORK: lIARCOUl1T. >.'lRACE & COMPANY.1935'"Made and Primed in Great Britai" b}'PEnCY L,mn>. HtJMPllmEB Il1J co. I:ro.u Bedford Sqt,are, T,om{on, W.C.tand at BradfordPREFACETO THE ENGLISH EDITIONTHE publication of this work in an English translation gives mevery great pleasure. From early youth I have studied Englishliterature, and later English philosophy. During the periodwhen my philosophical views were taking sha.pe, at1d especialIy,in the years 1874-1876, it was David Hume and still more J. S.Mill whose influence on my thought was paramount. Thus I wasearly attracted by English philosophy, and I formed the pro-ject of writing a History of English Thought.. But, like" manysimilar hopes, this plan was destined to remain unrealized. ! soonfound that importance of Fictions had already been partlyrecognized by English philosophers. English Notninalism ofthe Ages showed traces of such a recognition. W;,thJoh.n Dum Scotus, who died in 1308 1n Cologne, when' only inhIS thirty-fourth year, there began a Hceptical movement whichtended in the same direction. nut it was in William of Occam,who took refuge with Ludwig of Bavaria, and died in Munich in1347 at the age of 77, that we find for the first time a clear anddefinite treatment of the fictional nature of general ideas, de"veloped in a manner which is still a model for to-day. He fullyunderstood that flcta, as they were called in the writings of theMiddle Ages, although their theoretical non-existence might beadmitted, art;,.practically necessary and must be recognized inthis sense. Un the other hand this was not realized by Baconor even by Hume, though in Berkeley there are at least in-dications of an understanding of Fictions. But in Hobbes wefind a considerable knowledge both of Fictions themselves andof the theory of their use. Empty space, the idea of a 'bellumomnium contra omnes, and of an H contract II are forHobbes conscious Fictions. A special study of Hobbes' theoryof Fictions had been contemplated by O1y late colleagueProfessor Frischeisell-Kohler who was well versed in Englishviiviii THE PHILOSOPHY OF 'AS IF'philosophy and hoped to write a history of 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, both in practice and theory; more so than in othercountries. There is room for a special monograph on thissubject, covering the use of Fictions both in Adam Smith'spolitical economy and in Jeremy Bentham's political philosophy.In the present work the methods of Adam Smith and Benthamhave been treated 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 The Philosophy of I As if} " Prag-matism," too, so widespread throughout the English-speakingworld, has done something to prepare the ground for Fiction- in spite of their fundamental difference. Fictionalismdoes not admit the principle of Pragmatism which runs: "Anidea which is found to be useful in practice proves thereby thatit is also true in theory, and the fruitful is thus always true." principle of Fictionalism, on the other hand, or ratherthe outcome 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 an idea, in spite of its theoretical nullity mayhave great practical importance:' But though Fictionalism andPragmatism are diametrically opposed in principle, in practicethey may find much in common. Thus both acknowledge thevalue of metaphysical ideas, though for very different reasonsand with very different consequences.It can be shown, and has been demonstrated at length inthe present volume, that the theory of Fictions was more orless clearly stated by Kant, who was proud of his Scottishdescent. Nearly 100 pages of the work are devoted to thisquestion and it is there proved in detail that for Kant a largenumber of ideas, not only in metaphysics but also in matics, physics and jurisprudence, were Fictions. The metaMphysical ideas were somewhat confused by Kant himself inhis Critique of Pure Reason'" (Theory of Method), but weredefinitely called II heuristic Fictions." This was overlooked TO TIU"; IJ:NGLISH EDrfION ixand not understood in Kant's own day and for a long timeafter j and Kant was quite right when he said of himself" I ama century too early with my works; 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 ripe for thishis profoundest contribution, which I !Uay mention has nowbeen given due value by Professor Norman Kemp Smith ofEdinburgh in his admirable commentary on the Critique(recently published in a second Edition).HANS VAIHINGER.ANALYTICAL CON'l'EN'fSPAGIISAUTOBIOGRAPHICAL xxiii-xlviiiGENERAL INTRODUCTION 1-13CHAPTER I.-Thought. considered from the point 01 view 01a plltPosive organio Funotion . 1-8Empirical utility or adaptation to a purpose manifested by organicand logical functions-The organic formative force of the psyche--Stein.thal-The teleological treatment of the logical functions heur-istically permissible-Psychical mechanism and pm'posiveness notmutually exclusive-The test whether the logical functions haveattained their pUl"pose can only be pYacticrLl and not theoretical-Thought primarily an instrument of self-prcservation--Hcrbartand Schoponbauet-Logical purposiveness expressed ill the inven-tion of logical aids-The ways of thought not the ways of reality;they are only subjective, but are expedient-The devious ways ofthollgbt--Fundamental O1'1"or of dogmatism; its cQnfusion ai'thought and reality. I l.-Thougbt as an Art, Logic ll4I Technology . . 8-9The purposive organic fl1nctiOll attains the level of an AJ:t-'l'huliIarise Rules, wWch arc collected by Logic lW the" '!'()chnology ofThought." UI.-The Diflel'(luce between the Artifices andRules of Thought 10"! 2Distinction between the artificial Rules. as Qtdirtary, regularmethods, and the Artifices, as irregular methods 0: thought-Origin of these artifices: Leibnh: aud Newton-They show thepurposiveness of the logical function.CHAP'ER IV.-The Transition to Fiotions 12-13Fictions are devices of this sort and auxiliary operations of thought-Preliminary sketch of the fictional thought-construct-Epis-temological SIgnificance of fictional accessory concepts.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 to orientate ourselves inthe real world, but is not 4 copy of that world-The logical func-tions are an integral part of the cosmic process, but not a copy of it-Within the world of ideas logic disj;inguishes again between rela-tively objective ideational constructsand-thosewhich are subiectiveor fictional-Pure 'fictions and half.fictions.xixii THE PHILOSOPHY OF 'AS IF'VAGUSA. THE ENUMERATION AND DIVISION OF SCIENTIFICFICTIONS 17-77CHAPTER I.-Artiflcia.l Classitlcation 17-19Natural and artificial systems-Their relation to question species-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.CHAPTER I1.-Abstractive (Neglective) Fictions 19-24The deliberate omission of certain elements of reality in compli-cated phenomena-Standard example: Adam Smith's fiction in hispolitical economy-This is not an hypothesis, but a subjective,fictive method, by which an abstract system is created-Applica-tion of this method in the complicated phenomena of sociology andalso in mechanics and psychology (Herbnrt and Steinthal)-Theuselessnells of lI11ch ideational COlllltrllCts not to be deduced fromtheir unreality-Disputed examples: e.g, a period in which lan-guage consisted only of "roots.' Fictions of isolation-Relationto the method of approximation and tentative fictions-Averagefictions, e.g. l'hontnte moyen in statistics.CHAPTER Ur.-Schematic, Paradigmatic, Utopian and TypeFictions . . 24-27Schemata, models, and schematic drawings-The fiction of simplecases-Example: ThUnen's idea in economics-Paradigmata:method of imaginary cases to facilitate scientific pl"oof-Rhctodcalfictions-Utopias-The original state and the like-Value of suchideas and theu misuse-Imaginary archetypes-Goethe's fiction ofa, plant archetype" and Schiller's opinion of it.CHAPTER IV.-Symbolic (Analogical) Fictions 27-32The psychical mechanism in analogical fictions-Schleiermacller'stheological method rest\; on the conversion of dogmas into ana-logical fictions-This epistemological device derives from Kant-The Kantian " as if "-Categories as analogical fictions-Throughthe categories there can be no understanding of the world-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 according to Maimon-If such fictions are convertedinto hypotheses contradictions result-Errors andillusory problemsdue to the misuse of analogical fictions-Critical Positivism dis.tin-guishes these additions of the psyche from pure experience-Resignation, wise and unwise: negative understan.ding of theworld-Other analogical fictions.CHAPTER V.-luriBtic Fictions . 33-35A special form of the analogical fiction is found in'the legal fiction,a juristic device of frequent application-Importance of judsticmethods for logical theory-Thelogical function applies the samedevices in very various fields-The [ictio juris and its essentialdifference from the pra8sumptio-Examples from the GermanCo.mmercial. of these juristic methods to theeplstemologlcal nction-Both are valuable in practice, but them:eti-cally worthless, since they are based on a deviation from reality.CHAI;'TER VI.-Personiftoatory Fictions 36-38The form in which perSOll'!-lity is apperceived-Hypostatization ofphenomena-Abbreviations, Nommal fictions, Auxiliary words';['autological fictions. 'CHAPTER VII.-Summationai Fictions (General Ideas) 38-39The general idea as a mere fiction.ANALYTICAL CONTENTSCHAPTER VI 1I.--Heuristio FiotiQI1lI . 39-42The assumption of unl" freedom and ini;ellectualintuition-The retention of a fictton as 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-tion of objectively false hypotheses into subjectively expedientfictions-Significance of such fictions for thought in general.C. CONTRIBUTIONS TO THE HISTORY AND TBE()ll.Y OFFICTIONS. . . . . . . . . 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-tion from immed1ate pe:rception and from the belief that thoughtis identical with reality-Scepticlsm-elumsy of theancients, particulaxly in mathematics, though signa of scientificfictions are not altogether wanting-Express avoidance of nctionalconstructs-HyPotheses and fiCtionS-Platonic myths and similarfictions-Parmenidean fiction of the elements of the 'wQrld ofappearance-----The world of appeal anee as a fictional C011Struct-The fiction of the !lpheriGal form of the Absolute in Parm.enides-Symoolic thought.CHAPTER XIX.-Begjnnings of a Theory of Fictions &lXt.ong theGreeks . . . . . . . . . 139-1420Illsuflicient familiarity with the use of nctions naturally t:esults inan absence of adequate theory-No critical distinction between anecessity of thouglit and reality-Aristotle's methodology deficientin this respect-llll'6t1E/Ttr and V7('01'tfUPl1oL in Plato attd Aris-totle-Aristotle recognil:ed the fictional nature of math.e1UllticalANALYl'lCAL abstl"ilctions-Latl'l' Greek philosophy-The gradual differentia-tion of hypothct.icl and fictional aSHumptions-Thc Sccptics-Their negativism cOl1trD,stcll with modern sceptical Positivism,CHAPTER XXX.-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'ER XXXI I.-Medieval Terminology . 145The significance of Nominalism which recognized general ideas asfictiones-One-sid.... negative sense of the term in Nominalism-Impor.tance of schoiiistic terminology for modern times.CHAPTER XXXI II.-The Use of Fictions in ModemTimes 146-154More extensive use of legal fictions in modern times-Discussion oflegal fictions in relation to logic by Lcibniz-Utopia.n fictions-Fictions chiefly applied in mathematics-The development ofmodern mathematics by means of fictional auxiliary concepts-Mcrtschinsky's fiction of minima of constant size-His relation toBruno-" Inlillitcly ,Hstant points "-Lack of a. methodology o1ictional concepts, ()spedally in mathematics.Mail11on'g vIew of t.he Mona: my philosophy of ' As if', which also leuds to a more thoroughstUdy of Kant's' As If' theory.ORIGIN OF PHILOSOPHY OF 'AS IF' xxxviiof the material given to the senses has remained with me eversince.1 derived great profit from Avenarius in so far as he was apungent critic of Kant's theories. This preventecl me fromregarding Kant's philosophy as dogma, but anyway I was notinclined to do this. I could not follow Avenarius, however, inhis radical empiricism, or rather positivism. He realized quiterightly that the ideas of substance, causality etc. are imposedsubjectively by the psyche on the given, yet for this very reason,according to "the principle of the least energy", he wanted to(eliminate them completely from human thought. But I heldthat they are fictions, which must be retained because oftheir utility.In the autumn of 1875 Wundt came to Leipzig. His firstlecture was on logic and I listened to it with great interest andprofit. He appealed_ to me in every way. For his sake Ishould have liked to remain all in Leipzig, and I hau alreauyplanned a Joumat of Pure aud AppNed Lagle, in which I hopedto interest him. But family matters called me back to SouthGermany. I was only able to have one more term in the North,and that was to be in Berlin, where the Swahial1, Eduard Zellet,waS actively at work. The help which I got (rom him and fromhis friend Helmholtz, and also from Steinthal, Lazarus, Lassonand Paulsen was more or less valuable to me, but what wasreally important was that I came across the writings of Gruppe,who had died shortly before this, and they were useful formy theory of fiction. My private studies were devoted mostlyto David Hume and John Stuart Mill, whose exact knowledgewas decisive for my philosophic attitude.At the same time, during my Berlin days in the summer of1876, my first book on philosophy was published, Hartmmzn,Diihrng and Lange-a critical Essay on the History of Philo-sophy in the Nineteeth Century. It consisted of lectures whichI had given in the Academic Philosophical Society at Leipzig.The author of the History of M aterialt'sm, with his Kantiantendencies, seemed to me to strike the happy medium betweenthe spiritualistic metaphysics of E. von Hartmann on the onehand and the materialistic positivism of E. Dtihring on theother hand. In Berlin I had got to know these two menpersonally. 111 my book I also announced the early publicationof my investigation of Fictions.For family reasons I had to choose a University near myxxxviii THE PHILOSOPHY OF 'AS IF'South German home in which to take up ll1y residence as alecturer; so in the autumn of 1876 I moved to Strassburg,where I received a welcome from Laas. In his 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 attitude which he took up some yearslater in his three-volume treatise on Idealism and Positivism.He was the unprejudiced man of whom I stood in need, Hewas able to do justice to my own attitude. He was busy justthen with the study of John Stuart Mill's Examinatiolt of SirWiltz'am Hamilton's PMlosophy, in which I joined him, all themore readily because this was really a continuation of myBerlin studies of Hume and Mill. The resolution of so-calledreality, from an epistemological or psychological point ofview, into" Sensations and possibilities of sensation" seemedboth to him and to me to be the correct analytical way. Onthe other hand Laas resembled Avenarius, who was relatedto him, in his positivist tendency to eliminate all further s u b ~jective additions as unjustified and useless, whereas I was alwaysanxious to emphasize and keep hold of tbe practical value anduse of these theoretically unjustifiable conceptions of the oldm"idealism.During the latter part of the year J876, for my inauguraldissertation, I wrote down my thoughts in a large manuscript,to which I gave the title "Logical Studies. Part I: TheTheory of Scientific Fictions," As I had been carefullycollecting the material for several years and had gone into itmost thoroughly many times, the writing of it did not take melong, I handed in my MS. in the New Year and at the endof February 1877 I received my vem41egendi. The work whichreceived this recognition from the Faculty is exactly the sameas what was published in 19/1 as the" Part I: Basic Principles IJof The Philosophy of 'As if'. In it I developed the wholesystem of scientific fictions, that is to say the' As if' treatment,applied practically to the most varied aspects of science, andI tried to give an exhaustive theory of this manifold' As if'process.But like Laas I regarded this dissertation only as a roughoutline) in need of much supplementing and correction, so Imade use of the next two years, so far as my lectures allowedme, to work at my MS. My father's death compelled me toOllItiIN OP l>IIILOSOPHY OF '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 on Kant for the centena.ry in 1881 of his Critiqueof Pure Reasolt, I had then just started a far more thoroughstudy of Kant, particularly his' As if' theory, and in the courseof this I had found in his Prolegomena that" misplacement ofpages" which had passed unnoticed by many thousands ofKant readers for nearly a hundred years, but which is generallyrecognized by science nowadays. 50 I hoped, by applicationof the philological method, and by penetrating logical analysis,to further the study of Kant. But, as I have said, this newwork was only a means to an end, and I hoped in a few yearsto be able to return to my researches on Fiction.The above-mentioned" Law of the Preponderance of theMeans over the End ", which unfortunately I neglected toformulate theoretically and publish at the right moment. hasproved in a practical sense very momentous in my own life,Wheoj in l8841 the first volulne of my Commentary on Kant 2brought me un appointment as special Professor at Halle, Ihoped SOOIl to be able to finish the other volumes there. ButI At thut time r w a ~ also considering the plan of writing a History of EnglishPhilosophy, mentioned in my Preface to this transl(\tion, But there WM then so littleinteresland understanding in Germany for the development of English philosophythat on the advice of the experts of the time the publishers did not regard thesuggestion with favour." On my journey from Strassburg to Halle, I pllid another visit to Friedrich T.Vischer, whom r had often seen in the interval. Our talk was concerned chieflywith his philosophical novel Auch Et'tzt1- (1897) in which he e x p r e s ~ e s his lavouriteidea of the decline of the German people since 1871. He had shown in this bookhow the Germans by their arrogance would become involved in a world-war, inwhich after hard struggles and a moral revival they would eventually be victorious.Even then I did not agree with this optimism, and my political pessimism grewstronger in the following years, particularly after 1888. After 1908, and particUlarly19u, I contemplated following Leibniz' example and entering the arena of worldhistory with an anonymous pamphlet, Ft'nis Germam'cu, with the motto" QuosDeus vult perdere, prius dementat", and with the clevie-e, of Schiller's Cassandra"The Thunderer's clouds loom heavy over Ilion ", I thought of having thispamphlet printed in Switzerland, but my eyesight became rapidly worse and pre-vented me from doing this, I also said to myself that I would be a voice cryingin the wilderness, for it seemed impossible to penetrate the blindness of my seventymillion fellow-countrymen. I felt afraid too that the publication of my views mightonly increase the number of our enemies and the weight of their opposition and thatmy action might thus hasten the impending catastrophe, Even then I would havementioned most of the factors that are recognized to-day-or at least ought to berecognized-as the causes of the disaster, An unjustified optimism (if I do not goso far as Schopenhal1er in calling it a .. criminal optimism ") had [or a long time beenleading German policy astray in the dlfectlOn of improvidence, rashness andarrogance, A ratiohal pessimism might have saved us from the horrors of a world-war, World"philosophyand practical politics have a closer connection than is gen-e"ll1y realized,xl THE PHILOSOPHY Oli' 'AS IF'my lectures on the one hand and bad health on the other heldup the publication of the second volume until 1892. In 1894I was appointed reg'ular Professor in Halle, and in 1896 Ifounded the Kantstudim as a means of helping on my 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 my duty, inorder to promote the Kantstudien, to start a Fund to defraythe costs. This Fund was a success, but its organizationnecessitated the foundation of a Kant Society and this gradu-ally became more and more an end in itself and took up toomuch of my time and strength, although I was fortunate inhaving most 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, in the midst of all these clIrious complications andcrossings of my original intentions, a misfortune unexpectedlybrought a happy solution, and enabled me after twenty-sevenyears to return to my original plan, which I had given up in1879. The misfortune was the weakening of my eyesight, sothat it became impossible for me to continue fTlY lectures, or thespecial classes which I particularly enjoyed. So I had to giveup my official duties. The eyesight still remaining to me wasjust sufficient to allow me to publish my MS. I got myDissertation of 1876 copied, and introduced a number of smalleditorial alterations. This comprehensive MS. now forms" Part I: Basic Principles JI of The Philosophy of {As if'. Ialso completed the revision which I had made between 1877and the beginning of 1879 on the basis of the reviews of thattime, and this forms the Part II (Special) of the complete work.This part took me two and a half years because of my badeyesight, and Part II I (Historical) took me another two anda half years. Between 1877 and 1879 I had made a note 01the most important' As if' passages in Kant's works, and Inow completed this in an exhaustive manner, so that I wasable to produce a monograph on Kant's' As if' theory of nearlyone ~ u n d r e d pages. The exposition of Forberg's religlon of, As if' also took me a long time, and so did the developmentof F. A. Lange's" Standpoint of the Ideal," with which I hadmuch in common. But what took longer still was the finalORIGIN OF PHILOSOPHY 01" 'AS IF' xlisection on Nietzsche's theory of Fictions, which he had con-densed into ~ few pages. It was the Spring of 19I1 beforethe work appeared.I called this work Tilt) Plu'losoplty of' As if' because itseemed to me to express more convincingly than any other Ipossible title what I wanted to say, namely that I As if', i.e.appearance, the consciously-false, plays an enormous part in\science, in world-philosophies and in life. I wanted to give acomplete enumeration of all the methods in which we operateintentionally with consciously false ideas, or rather judgments.I wanted to reveal the secret life of these extraordinary methods.I wanted to give a complete theory, an anatomy and physiologyso to speak, or rather a biology of' As if'. For the method offiction which is found in a greater or lesser degl'ee in all thesciences can best be expressed by this complex conjunctionI As if'. Thus I had to give a survey of all the branches ofscience from this point of view. But it was not only a methodological investigation that I wasattempting. The study of fictional thought in all branches ofscience had led me gradually to extend these investigations tophilosophy itself, particularly to epistemology, ethics and thephilosophy of \cligion. Just as my investigations into thefunction of' As if' had arisen out of a definite view of the worldso again this developed independently into a universal systemof philosophy-I gave it the name of "Positivist Idealism"or" Idealistic Positivism ". As I have already mentioned, ErnstLaas had published between 1884 and 1886 a three-volumework on Idealism and Positivism, in which he attacked Idealismand championed Positivism. The positivist attitude was alsorepresented in Germany by Mach, Avenarius and to a certainextent by Schuppe, and it found particular favour with thescientifically inclined (but the name Positivism was neverplaced in the forefront of any programme). The chief currentsof Gentian philosophy, however, wei'e certainly idealistic, thoughin different ways. Between these one-sided views 1 it seemedto me that a compromise was necessary, all the more so becauseJ. The growing tendency of the" idealistic" philosophers and the Neo-Kantiansto return to Fichte and Hegel seemed to me to be becomlng more and morc dangerous.I was always convinced that this one-sided idealistic tendency, which was partlyforeign and partly hostile to reality, was the more dangerQus to the whole of Germancivilization 10 that it led our youth to undereslimate foreign philosophy, and there-w i t ~ the whole civilization of neighbouring peoples, their capacity and. in gene(al,then mental and moral power.xlii 'fHE PHILOSOPHY 01' 'AS Ill"attempts of this kind had met with success in other I considered that the time had come to announce the union of PaSltl"Vls"m. The result has proved thns in Modern TimesWE now proceed to an account of the use of the scientific fictionin modern times, Here its employment is incomparably moreextensive,So far we have found in the legal 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 science of arbitrary human regulations.Moreover the fiction was applied rather in the practice of law.On the other hand it was not yet as extensively employed as inmodern law, where it has been used specifically in the foundationof Public law and where, moreover, the fiction of juristic personsis very widely adopted, even to the extent of including theState itself in so far as the State can be regarded as a juristicperson. Both in the special jJractz'ce and in the theory of law,the fiction has been far more extensively employed in recenttimes than in the classical period. In England especially ithas been much used and abused. The fiction serves to subsumea given case under some general rule, when the case in questioncan thereby be treated juristically. For instance, it is assumedthat a husband is the father of a child if he was in the countryat the time of the child's conception, i.e. since every single casecannot be investigated, the general assumption is made thatevery husband is to be regarded as the father of a child ifhe was in the country when the child was conceived. This ex-ample is given by Leibniz in his Nouveaux Essaz's [E.T. p. 260],but it is rather a jJraesumjJtz'o than a true fiction. A fiction inthe juristic sense can only be spoken of if a husband, whosewife has committed adultery, is nevertheless regarded, if hewas in the country at the time, as the father of the child. Hewould then be regarde"d as if he were the father of the child,although he is not and although we know he is not. This lastaddition is what differentiates the jictz'o from the fraesumjJtio, forin the jJraesumptz'o, a presupposition is made until the contrary isproven, whereas the fictlo is the acceptance of a statement ora fact although we are certain of the contrary. An example ofa real fiction is the fact, for instance, that in England (in theeighteenth century) every crime could be treated as ifpersonallydirected against the king, and every plaintiff had the right tobring his action under this fiction. The practical value of thisfiction lay in the fact that.trials under'this fiction were far morestringent than those under the ordinary laws, for charges thusmade were brought before a special court. Here we have the"as if" in all its force. The Code Napoleon also allows anumber of legal fictions; for example, the household goods ofa woman are regarded as immobilt"a. Similarly we find fictitiousproperty, etc. and under certain conditions an {( enfant c o n ~ u "148 PART I: BASIC PRINCIPLEScan be regarded as "nc" if important legal consequences areinvolved.In legal practice the employment of :fiction may lead bothto benefits and also to the grossest forms of injustice, as when allwomen were treated as if they were minors.In legal theory the fiction was particularly used in thetheory of contract, in so far as the State was regarded as theresult of a contract and was treated as a juristic person.This fiction, which was already known to the ancients, hasbeen very extensively used in recent times.Another favourite method was the ideal or Utopian fiction.In the nineteenth century the French Socialists, Fourier, forexample, were still employing this metpod of spreading theirideas by the description of towns and states as if the ideas theypromulgated had been there realized. Such a method passesvery easily over into the realm of phantasy and forms thetransition from a scientific treatment to pure poetry. But thiswhole group of scientific methods must not be overlooked,though they are neither very important nor do they presentany theoretical difficulties.With the growth of science the fiction began to be moreextensively employed.The first of the main fields where really great results wereachieved was mathematics. Modern mathematics is charac-terized specifically by the freedom with which it forms thesefictional constructs. A careful study of the development ofmathematics brings to light a number of such fictions. We donot so much mean thereby such substitutions as the ~ m p l o y ~ment of. letters instead of figures as a notation, though eventhis simple method is strictly speaking a fiction. By the fictionthat a, h, c, z, yare numbers, and by ,treating them as if theyactually were, enormous progress is made; results can begeneralized and calculations simplified. This is usually calledan application of symbols, but taken logically, we are dealinghere with a substitutive fiction. Thought itself, in general, whenoperating with words instead of perceptions, makes use of suchsymbols.But quite apart from this, fictions have been more andmore used in recent mathematics. Their most famous andmost fertile application was in the measurement of curves byDescartes, Leibniz and Newton. This is really the classicalexample. By means of the fiction of coordinates, of artificialFICTIONS IN MODERN TIMES 149lines (all artificial lines are fictional methods), and by means ofdifferentials or fluxions, a treatment of curves became possible.]The methods of unjustified transference, of zero-cases, ofabstract generalization, etc. are modern mathematical devices.They were generally known by these names; great mathe-maticians have always been distinguished by the invention ofdevices, and these devices are always essentially based uponfictions. Even the drawing of artificial lines is such a device.Schopenhauer called attention to the fact that no real know-ledge can be obtained by their means. But such devices arenot meant for this but for practical purposes.It is upon such devices and fictions that the concepts of theinfinitely large, and of negative, fractional, imaginary andirrational numbers, are based, all of them serving the purposeof simplifying calculation and all in a strict sense logicallycontradictory.The utilization of these devices, to which the progress ofmodern mathematics is due, has continued right into our owntime, and every really new discovery in mathematics rests uponsuch a device. The device of abstract generalization has nowbeen applied to space, and spaces of more than three dimensionshave been imagined.The method of determinants depends on such an artifice.Of special interest are the fictions of line, surface and volumeelements as a foundation for the use of measure-numbers.Mertschinsky, in particular, has utilized the fiction of minimao['constant size for purpose. This fiction had already beenemployed by Giordano Bruno in his De trz'plci mnt"mo etmensura, and De 1lZonade, numero et figura. But Bruno stillhesitates whether to treat his minima as fictions or hypotheses.1 In this connection let me refer to a remarkable and instructive book byA. Mouchot, La rt!jorme etenaue malh!maliques pures (Paris, 187]). On the analogy of the theory of the two coordinates,invented by Descartes for dealing with curved lines, Mouchot regards every realpoint as consisting of two imaginary points. He also treats imaginary numbersfrom this point of view. He then formnlates a " Principe des relations contingentes,"that bears some relationship to Herbart's "Method of chance aspects," and to speak of cordes ideales, of rayons et centres imaginaires, of imaginary variables,imaginary triangles, imaginary dimensions, angles-aU of whicb are dedllced from thetheory cif imaginary points. The object here is to approach reality by contingentand arbitrary methods and thus to see it in various lights and render it amenable totreatment. The author relates his theory to that of Charles (Apper;u etc.) in order to explain the connection of the imaginary and tbe contingent. The'elations COnl'ingentes are the key to lhe imaginary. In this sense the comme siplays an important part in Mouchot's work. What is imaginary is regarded as ifIt were real and is substituted for the real. Mouchot speaks of various conceptionswhich serve as sltiles secours en geometrie superieure.150 PART I: BASIC PRINCIPLESThe same uncertainty is found in Leibniz who, on the one hand,declared that minima infinite parva were only a modus diceltdi,but in the interests of his monodology was inclined to assumethat they were hypotheses. Whether Leibniz hit upon hisidea through the influence of Bruno has not yet been determined. It is not improbable. But Bruno's principle of applica-tion was different, for he used his minima in order to lay thefoundations of mensuration, while Leibniz was concerned withthe measurement of curves.Other mathematical fictions refer particularly to the infinite;as, for instance, infinitely distant points, infinite stretches,limits of infinite surfaces, convergence at infinity, etc.In modern mathematics the employment of such fictionalconcepts is quite general, but mathematicians and philosophershave so far not developed any methodology for these devices,though such a methodology would certainly be very illuminat-ing as regards the use both of the infinite and the absolute froma 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 of themsciences which have been fully developed only in modern times.Numerous other examples of the modern use of fictionshave already been given in our classificatory chapters. Wethere saw how a number of sciences have successfully utilizedthe scientific fiction in all its different forms. The true natureof these devices was frequently realized, but they were oftenemployed quite instinctively without any methodological understanding. Hence a number of famous controversies, turning onthe question whether certain concepts were legitimate or not.This question has already been partly discussed in detail above.The fiction may find some employment in philosophytoo, but here if anywhere caution is necessary. It can neverserve as an explanation of anything but only as a means ofsimplifying thought and for the purposes of practical ethics.1Maimon put forward the view that Leibniz' monodologyand pre-established har-mony were only fictions; but with this.1 Descatles .created. methodological fictions: DUhring in his Kritische Ges.chlchte d4r Phttosophte. p. ::61, well calls the 1dea of II deceiving god a Valuablefiction, and also other "tlopes." Absolute doubt is fOT Descartes also melely amethodological fiction.FICTIONS IN MODERN TIMES 151we cannot agree, for Leibniz interpreted his doctrines otherwise.But had they been fictions they certainly would have been veryuseless constructs. It is one question whether Leibniz desiredhis doctrines to be understood in this way and quite anotherwhat value we are to attach to such constructs. Leibniz un-doubtedly regarded his doctrines as hypotheses and not asfictions. Whether after they have ceased to function as 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 of parallelism. For us this is only afiction but one of tremendous scientific and heuristic value.On the other hand, metaphysically the relation between thephysical and psychical can scarcely be such as Spinoza and themodern Spinozists, such as Bain (following Hartley), Lange,Wundt and others, assume.Whether Kant's fiction of a Ding an sz"ch is still really ofvalue to us requires a special investigation. But a sharpdistinction must be drawn between Kant's realization that theDng an sch is a fiction and his actual employment andutilization of this fiction. He himself employs it for scientificpurposes and in his own hands it was transformed into anhypothesis.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 also a fiction.This error prevented him from 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 remain as a scaffolding.Had he destroyed that scaffolding and rejected them both hewould have found that sensation was the sole reality left.When, therefore, Jacobi says that "without the presup-position of objects as Things-in-themselves, and of ideationalfaculties upon which they work, it is not possible to enterthe Kantian system, though with them it is quite impossibleto remain in it "-in other words that the beginning and thecontinuation of the Kritik are mutually "destructive "-he152 PART I: BASIC PRINCIPLESwas quite correct. Kant, after having discovered and assertedin the Kritik that Thlllgs-in-themselves are merely fictions,had only to recognize frankly that these presuppositions ofhis were nothing more than provisional devices for the purposeof 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 not look at these methods of visualization and calculation-for as such Faraday, 5chonbein, Magnus, Du Bois-Reymond,1 cr. Wundt, UbeY die Au/gabe dey Plii/osophie in dey Gegmwart, p. 6.~ [Joey die A1ifgalJe fkr Naturwissutsduift, Jena 1878, p. 7.222 PART II: SPECIAL STUDIESFick, etc., regarded the atomic theory-as an objective processof nature. Many scientists speak of atoms without reallymeaning to assume them: some even reject the reality ofempty space and yet continue to speak of atoms, although theassumption of empty space is a necessary' consequence of theatomic theory. Unquestionably this conceptual method isthe most convenient one, but this constitutes, of course, noproof of its objective-metaphysical validity.According to the more recent views of physicists, Kirchhoff,for instance, all phenomena are reduced to forces and relativeeffects of forces. For the physical specialist, matter is in noway dependent upon the assumption of extended minimalparticles. Matter forms an entirely empty and meaninglesssubject for the forces and is but an inaccurate after-effect of aview which has gl'own accustomed to the idea of extendedand separated bodies, and which also assumes substances asbearers of the elementary forces. But this conceptual methodprovides a simplification of the theory, not only because particlesof matter are looked upon as the supporters of these forces,but because they are regarded as infinitely small. The formerattitude is of greater value in making the abstract concept offorce concrete, the latter in simplifying the calculation. Thatis why the atoms are allowed to remain, though everythingthat exists has found adequate expression in the forces, Weinterpolate this conceptual aid because it is so convenient.It is literally an hypostasized Nothing with which we aredealing, in the case of the atom; for if everything has beendissolved and absorbed into the forces, what becomes of matter?And if the atoms are to be represented as infinitely small, howare they to be distinguished from the mathematical point whichis also merely an hypostasized nothing? 18Fictions in Mathematica.l Physics.!IN physics, and particularly in mathematical physics, as wellas in m'echanics, we make 'use of a number of fictional constructswhich are in part merely convenient, in part absolutely indis-1 Sl4pplem4ntary to Pat'I), Ckaptet XVI.MATHEMATICAL PHYSICS 223pensable. Faraday's" lines of force" possessing no mass 01"inertia, for instance, are to be regarded as auxiliary ideas forthe purpose of visualization. Maxwell tried to see in theselines of force !iomething more than mere mathematiCal symbols.But that Maxwell in this interpretation was contradict-ing the intention of -Faraday, the actual originator of theconcept, that, in other words, he committed the frequent errorof transforming a fiction into an hypothesis, a mathematicalauxiliary idea into a physical theory, is best proved by Faraday'sown words. The lines of the magnetic force of gravitation,the lines of eclectro-static force and the bent lines of force, areall, according to him, imaginary.! No special meaning was tobe ascribed to these terms: he is convinced that he is notgiving expression by means of them to any real fact of nature,although this method of conceiving things apparently fits thesituation and is very neat.2He desires to limit the meaningof the words" line of force" in such a way that they designatenothing except the state of the force with respect to its 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 through space we do not know.DAccording to these state 'ments of Faraday, Zollner is unquestionably right in rejectingMaxwell's interpretation of these lines' of force as physicalentities as a gross misunderstanding. It is also quite clear thatMaxwell made this confusion through a lack of methodologicalinsight into what constituted the difference between a fictionand an hypothesis. We know this definitely because Faradayexpresses himself quite clearly in a letter to Tyndall (14thMarch 1855), who, he says, is aware that he (Faraday) treatsthe lines of force only as n}resmtatiotzs of magnetic power,and that he does not profess to say to what physical idea theymay thereafter point, or into what they will resolve themselves.Faraday did not allow himself to be led astray by the greatmathematical utility of his new conception, which was ofextraordinary value in the analytical deduction of physicalphenomena, into seeing in it more than a "representative"idea. He protested against the misunderstandings of hiscoptemporaries, men like the Dutch mathematician van Rees,t .. Experimmtale Untersuchungen," 13041 in Zollner'. Wiumscba/I. Abhand-IUH,,'H, 82. ,.I Ibia, 84, 3 IMrI, 84.224 PART n: SPECIAL S1'UDIESwho also seemed to find a physical hypothesis in this idea, indirect opposition to Faraday's clear declarations.This differentiation between hypothesis and fiction alsocoincides with the distinction drawn by Wilhelm Weber 1between real and ideal hypotheses.An ingenious artifice of thought is that of the "fictitiousmean" of Jevons (we follow his Principles if Scil21lce), which,h'as been used in various connections.2It is exceedinglypopular in mathematical physics, in those cases where a chainor group of force relationships belonging together, are thoughtof as 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 too complicateda calculation if every relationship were taken into consideration,a single unit is substituted for the many, which are regardedas combined within it.We owe the first a.pplication of this method to Archimedes.He hit upon the very ingenious idea of constructing in a givenbody, a point in which the weight of all the parts was thoughtof as being concentrated, so that the weight of the whole bodycould be accurately represented by the weight of this point.The centre of gravity thus takes the place of the weight ofinnumerable, infinitely small particles. each of which is activein its particular position. In order to obviate the tremendouscomple:x:ity in calculation necessitated by this circumstance-for the simplest mechanical problem would otherwise break upinto innumerable particular ones-a centre of gravity is imaginedwhich is thought of as a point and treated as if all the forcesof the individual parts were united there. Archimedes ex-plained the method for determining this centre. Thus inplace of a sphere as centre of gravity, 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 Messbestimmungen insbesondere itber Dio.magnetismus ",Abkmldl. d. Sacks. Ges. d. w., I, 560. Cf. Zollner, p,.inzipiett diller eld,trodylla-misckm Theo,i, d,r JlJat,rie. 1876, I, 91 ; and the same author's r-Vissenschaf/licheAb/"xndlu"r''', 1878, I, 45.2 The fictitious mean is employed in other sciences too, whenever the needarises of taking the average of ll. number of gradually varying phenomena andma.king this the basis of further calculation or consideration; for instance, instatistics, meteorology, etc., where it is important to substitute for a large numberof qllantities that oscillate aronnd an ideal paint, a common quantity valid for themall. An average is therefore constructed, by means of which we make our computa-tion as if everyone of the phenomena in question corresponded to it. A famousfiction 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 of having the points of applica-tion of the forces in the form of a circle, we find the centrefalling in the vacant interspace. The same holds for two ormore bodies, whether these are separated or united. Here,too, a point can be found that can be treated as if the resultantforce of both bundles of forces were concentrated in it. Wecan, for instance, imagine a common centre of gravity of theearth and the sun, that is, a point that can be regarded andintroduced into calculations as though it took the place ofboth these celestial bodies as an indivisible centre exercisingexactly the same influence upon a third point as the two bodiesdo in fact.We must also mention here as a peculiar and valuableauxiliary idea, the fiction of an absolutely fixed point.The empirical perception of all change and motion is alwaysconnected with empirical points of reference, and it is onlywhen related to these that we can recognize it as motion. Inother words all observed motion is relative, relative to us, to animaginary origin, relative to a fixed background or relative tothe apparently stationary earth or sun. These are all merepoints of reference which we must assume in succession. Manbegins by assuming himself as a point of reference and scienceconstantly postulates other points of reference because thosetaken first prove to be illusory, since they turn out to be inmotion themselves. In order to prove definitely and absolutelythe existence of motion, we must have an absolutely fixedpoint by means of which the speed and the direction of themotion can be measured. Since, however, according to modernviews, no such absolutely fixed body can be discovered in theuniverse, science is faced with a peculiar difficulty.Neumann 1 deserves the credit for having first demonstratedthat Galileo and Newton formulated their laws in such afashion that an absolute motion was assumed. Galileo's law ofinertia cannot, according to Neumann, possibly remain as astarting point for mathematical deductions. We do not indeedknow what we are to understand by motion in a direct line jindeed we know that these words can be interpreted in variousways and are capable of innumerable meanings. Motion, {or1Neumann, Ober die Pn'nzjpien der GalileiNewton'sclun Theone, Leipzig, 1870.p226 PART II: SPECIAL STUDlIsSinstance, that takes place in a straight line with regard to onecelestial body, will appear as curved with regard to every other.We must therefore begin with a special body in the universeand employ it as a basis for our judgment; use it, in otherwords, as the particular object with respect to which all motionis to be calculated. Only then shall we be in a position toconnect a definite meaning with the above words.To what body shall we assign this place of pre-eminence?Galileo and Newton give us no answer. They simply assumeabsolute motion without being clear in their own minds orbeing conscious of the fact that this presupposition involves theexistence of such an absolute and fixed point of reference.That this condition is necessarily involved was first clearlybrought out by Neumann, though there is a definite reference tothis matter in Descartes. It is for that reason that Neumannsets 'up as the first principle of the GalileoNewton theory theproposition that all conceivable motions existing in the worldare to be referred to one and the same absolutely fixed body,whose configuration, position and dimensions are unalterablefor all time. He calls this body "the body Alpha." We areto understand then by the motion of a point, not a change ofposition with regard to the earth or the sun, but one with regardto this body.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 in regard to this body alpha. This may also beexplained as meaning that every motion from now on isthought of as absolute. The nature and the really essentialcharacter of this 50-called absolute motion, consists in the factthat all change of position is brought into relation with oneand the same object, indeed with an object which, as Neumannexpresses it, is spatially extended and unalterable hut whichcannot be further described. If, however, we do not assumeabsolute motion, then the whole Galileo-Newton theory falls tothe ground i for, in that case, since every body in the universeis actually in motion, motion could only be defined as a relativechange of position of twp points with regard to one another.We should then arrive at a theory which is fundamentallydifferent from the Galileo-Newtonian, and whose agreementwith the actually observable phenomena might be very doubtful.We must insist again, therefore, that an absolute motion inABSOLUTE SPACE 221absolute space is a necessary presupposition of the Galileanlaw of inertia. In order to simplify the conception of absolutespace we have the body alpha.We shall never be able to find an empirical point that willsatisfy the above conditions. For that reason we assume anideal point that serves the same purposes. This is howNeumann understands his body alpha.We can thus perceive what a very peculiar construct thisfiction is. It represents an accretion to reality, an intercalationthat is to make conceptual mobility and the determination ofconcepts easier. In the final examination of reality, this inter-polated element must therefore drop out and be eliminated.Indeed, as soon as the connections and mediation, 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 as soonas the mathematical formulc.e have been discovered and applied.The same service is rendered by other auxiliary aids ofmechanics and physics; for example, the ether of light andthe electric fluid which, according to Neumann, serve only forpurposes of visualization and of connecting the calculation;and the intermediate element drops out as soon as this con-nection has been achieved. These scientific interlopers are notincluded in the council that definitely determines the relationsexisting in actual reality and, like all temporary makeshifts,are excluded from the principles in the real and narrower sense. 19The FictioJJ of Pure Absc>lute Space 1IT is the false assumption that mathematics can proceed in asense different iii a priori from that of any other science, and that,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 space from the logical standpoint? What logicall SZlppltmmtary to Part I, Cna}ttrs X and X PI, p. 52 If. andp. 73.a The a priori and deductive procedure of mathematics does not differ from thedeductive procedure possible 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" may mean something that is empiri-cally given and upon which mathematics is essentially based, orit can mean that space is an hypothesis without which mathe-matics could not exist. Mathematical space is unquestionablyan essential presupposition, but in neither of these two senses.It is quite easy to prove that space, in the mathematical sense,namely as a pure extension in three dimensions, is not thing actually given, a real fact. Empirically we find onlyindividual bodies possessing the fundamental character of ex-tension but never a general or pure space. It is tt,ue that thecircumstance that all objects are perceived as from auniform background (generally a light one), and the trans and colourlessness of the air, make it appear as ifindividual objects lay in a uniformly perceptible vacant space.This peculiar circumstance has unquestionably facilitated theemergence of an independent and absolute idea of space, butwe have no right to interpret this as implying that mathematicalspace is something empirically given. Indeed, for that veryreason, nobody has seriously contended that it is. The mathe-matical idea of space has not, therefore, the logical value of anexperience. Perhaps it possesses that of an hypothesis? Butthen we encounter even greater difficulties. How can an ideaso absurd and so contradictory make any claim to be anhypothesis? Mathematical space is a something that is anothing and a nothing that is a something. The contradictionsinherent in the concept of an unoccupied mathematical space arewell-known. A vacuum would be something contiguous andseparated where we find nothing contiguous and nothingseparated. If space is the relation of co-existence of real.objects, then, in the absence of these, it must be nothing andwould disappear with them. Since, however, the primarycharacteristic of a useful hypothesis, is its freedom fromcontradiction, such a contradictory concept as an absolute,unoccupied, mathematical space cannot be an hypothesis. Andit is this very contradictoriness that prevents us from contentingourselves, without further ado, with the favourite expressionof the mathematicians that these and similar concepts are1 We are not concerned here primarily witl) the psychological question nor withthe epistemological, but more particularly with the logical problem.ABSOLUTE SPACE 229"postulates": for this last concept is vague and indefinite. Weare consequently forced to ask the very pointed and embarrassingquestion: what logical position then can the idea of space occupy?In view of the fundamental importance of this point forour subsequent argument and the remarkable clarity with whichLeibniz in the main treated it, let us pause to examine hiscontroversy with Clarke. This controversy, in so far as it boreon the question of space, turned on the problem whether theconcept of an absolute, geometrical or vacant space, wasjustified or not, i.e. whether there was any actual vacuum inreality. Clarke, together with Newton, defended the existenceof an absolute space (and consequently of an absolute motion).Within this 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 in space, is alsoto be found absolute and vacant interspace. This theoryLeibniz attacks. "II n'y a point de vuide du tout" (Erdmann'sEdition, 748); such is the thesis that Leibniz tries to establishon theological, physical, mathematical and logical grounds."L'espace reel absolu" (Ibid. 751), is nn iI idole de quelquesAnglois modernes. Je dis 'idole J non pas dans un sensTheologique mais Philosophique, comme Ie chancelier Bacondisoit autrefois qu'il y a Idola Tribus, Idola Speeus." Herepeatedly enumerates the grandes dijJicultl!s and contmdictionsto which this concept leads. It is particul\l.rly by means ofhis "Principe de la raison suffisante" that he attempts todisprove the U imaginations," the "suppositions chimeriques ",and the" fictions impossibles" of his opponents. As a matterof fact, of course, the idea of absolute space and absolute timedoes lead to peculiar absurdities, and Leibniz' refutation is quitejustified. He constantly calls these concepts of absolute timeand space "chimeres toutes pures" and "imaginations super-ficielles". They are "fictions impossibles" (771). We might,for instance, at will think of any spatial position in the worlddisplaced any distance in absolute space; since, however, thetwo points cannot be distinguished, they will remain merelyideal and imaginary and the presupposition that this dis-placement is possible, i.e. the presupposition of absolute space,is a mere fiction.1The fact that there is no sufficient reason1 We see here clearly how, in Leibniz, "imagination" and "fiction" aresometimes used in a derogatory sense, as when he rejects the above idea. as unrealin the metaphysical sense, and sometimes in a good sense, as when he yet recognizesthat an idea is methodologically justified and eltpedient.PART II: SPECIAL why God should have created the world at an earlier momentthan he actually did, proves that this whole method of lookingat the matter and the presupposition of absolute time uponwhich it is based, is false. What holds of time holds also forspace. This idea that absolute space is a chimerical suppositionand an impossible fiction runs in all possible variations throughthe whole of Leibniz' correspondence, which is so important forhis philosophy.Let us now attempt to show how this dispute can be adjustedby means of a very simple methodological distinction, for herewe are concerned with the logical and metaphysical value ofthe concept of absolute space. As it certainly is not an experi.ence, the only question which can be involved is whether weare dealing with a justifiable hypothesis or a justifiable fiction,a fiction in the sense we originally fixed. We saw how Leibnizproved that this concept was contradictory and impossible andhow, for that reason, he rejected it. On the other hand we shallsee that Clarke stressed its practical necessity and utility, basedupon Newton's mathematical natural philosophy. Leibniz callsthe idea a fiction in a derogatory sense. He uses this concept,indeed, very frequently and, as we pointed out above, in thetwo different meanings of a good and a bad fiction. Had nothis enmity against the Newtonians, and the bad feeling thatarose on both sides in consequence, disturbed Leibniz' clearvision, and had his correspondence with Clarke not taken placein the later period of his life when he was isolated and em-bittered by misfortune, he would probably have applied here,too, the fundamental discovery that he had arrived at in con-nection with other questions; namely that there are necessaryand justified fictions. He might thus, perhaps, have found thecorrect solution, that the idea of absolute space is an able auxiliary idea, i.e. that, although it is in itself contradictoryand therefore imaginary and ideal, it must of necessity beformed for the building up of mathematics and mathematicalphysics.This simple solution clears up at one stroke the wholepassionate strife between Leibniz and the Newtonians. Allthe reasons advanced by Leibniz go to show that the conceptis imaginary, all the counter-reasons advanced by Clarke thatit is necessary. As is so often the 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 see it pass over into the genel"al consciousness, becomean everyday idea, until it is again attacked and, thoughdeprived in the end of its reality, yet admitted and allowedto persist because of its indispensability.In the ambiguity and double-edged nature of the concept"supposition ", we again recognize the duality in logical m e a n ~ing that gives to these concepts of absolute space, of the atom,etc., so varying and uncertain an appearance. Clarke proceedsfrom the necessity of this supposition, from the fact thatLeibniz himself makes it j Leibniz, on the other hand, callsit chimerical, sophistical and imaginary. In the meaning ofII 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 thetrue idea of 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 Replique aux Reflexions de Bayle" (Erdmann, 189; writtenseventeen years before the discussion with Clarke) he remarksthat the mathematical ideas of time, extension, motion andcontinuity are merely "des chases ideales." He agrees withHobbes who calls space a jJhantasnza i!xz'stentis. Extension is"the arrangement of possible coexistences." Of particularimportance is a passage (190) where he says that although themeditations of the mathematicians are ideal, this does notdeprive them of any of their utility. He consequently knewhow to value the usefulness of such concepts (191), althoughquite conscious of 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-without282 PART n: SPECIAL Sl'UDIESdeparting too much from Leibniz' meaning-in one direction,at least, the Theory of 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,useful and fruitful, because the mathematicians only investigatethe characteristics and Jaws of extended objects, qua extended,and not their materiality or other physical properties. Theconcept of pure space arises from retaining the relation ofobjects after the things themselves have already been thoughtaway. While permitting matter and its intensity to begradually reduced to zero, we still preserve the relation ofmaterial objects. Although, strictly speaking, space shoulddisappear at the very instant in which mattel' has been reducedto zero and thus disappears, we still retain the relation evenafter the related things have valiished. If we observe anobject in continuous extension and if we mentally allow thematter 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 hold of. At the very next moment n o t h i n g ~ness begins, 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 something which we experience only in Some-thing else (whether as property or as relation) from this otherentity-from something to which it is so firmly and in-extricably bound that when what has been detached isaccurately analysed we are forced to admit to ourselves thatnothing remains in our hands. Abstraction takes from thesubstratum and the elements their attributes and relations.ABSOLUTE SPACENow, strictly speaking, these detached pieces, apart from theiroriginal context, have no meaning: they evaporate intonothingness and lead to absurdities. Imagination, by reasonof its specific and peculiar gifts, comes to the aid and rescuesabstraction which, as described above, has dissolved the givenworld 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 thatthe space of the mathematicians is nothing but a scientific andartificial preparation, which differs from the schematic auxiliaryconstructs, etc., of other sciences, only in the nature of theobjects that are to be investigated and not in the method ofinvestigation. This unity of method must be strongly empha-sized. Only a methodological approach can purge us of ourold prejudices about the objects of "mathematics. Only themethodologist, by following the devious routes 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, fictional abstractions, abstract fictions, as weshall prove in the following pages in connection with particularmathematical constructs. Here we are concerned with theconcept of space in general, with pure absolute space, - aperfect example of a normal and scientific fiction. There istherefore no object in trying to argue away the blatant con-tradictions inherent in' this concept. To be 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 to say,of the honour of serving as an ideal example of a true andjustified fiction.234 PAUT II: SPECIAL STUDIES 20Surface, Line, Point, etc.. as lllictions 1WHAT holds for pure absolute space holds also - mutatismutandis - for the single mathematical spaces and parts ofspace, and for the idea of the s o ~ c a l l e d mathematical bodies,such as sphere, cylinder, cube, prism, etc. The psychologicaland logical foundations of these constructs are the correspondingempirical corporeal objects. And here again we find abstractionand imagination participating in the manner described above.The corporeal is reduced to a minimum, finalty to zero; andtherewith, from Cl; strictly logical standpoint, the boundaries ofthese corporeal objects must fade away and, so to speak, mergeinto themselves. But since we are abstracting only from theoccupying content, the form is still retained, and before allthese boundaries, completely deprived of their content, collapse,they are supported by the imagination which, as the contentdisappears and becomes infmitely thin, holds them in place asinfmitely thin shells, empty husks, as a skin, a covering, indeedeven as a mere frame. Such forms, without a content, are, assuch, nothing, indeed worse than nothing, for they are contra-dictory constructs, a nothing that is nevertheless conceived as asomething, a something that is already passing over into anothing. And yet just these contradictory constructs, thesefictional entities, are the indispensable bases of mathematicalthought. The boundaries of the empirical bodies are taken assuch, and are abstracted and hypostasized i and with theseimaginary constructs mathematics, and particularly geometry,operates.The same-mutatis mutandis-holds for the surface, the lineand the point. That the surface is the boundary of a bodyis a very old definition. Historically and psychogenetically,of course, we are first concerned with real II planes," i.e. flatboundaries. The concept of curved surfaces arises later. Thereare flat surfaces, i.e., really plane constructs, in nature, as wellas the great number due to the primitive participation of man;here we abstract from the material that forms the surface, andthe formal element is taken alone in itself and made indepen-dent by imagination. In this case also it is a contradiction to1 Supplementary tQ Part 1, Chapter.Y, p. 5I if.SURFACE, LINE, POINT 285speak of a surface as such; and yet scientific thought proceedsunconcerned along its path, in the face of these and even morepronounced contradictions. If thought were to allow itself tobe held up by such contradictions it would never be able tomove at all.The same is true of the line-as the "boundary of thesurface." Of lines, too, there is no lack, either in nature or inprimitive art, but they are, so to speak, immersed in thecorporeal. It is abstraction that first picks out these lines assomething special, with an existence of their own, and then callsin the aid of imagination to hypostasize these structures. Butthat they are merely fictional concepts is self-evident. Whatthe mathematician, the geometrician, draws on the blackboardor on paper, and calls a line, is not a line in the mathematicalsense, for it always possesses a second (and even a third)dimension even if that has been reduced to a minimum. Aline, in the mathematical sense, can never be sensuously repre-sented, for it is a matter of abstraction and imagination and, inall cases, remains a contradictory construct.The same naturally holds for the point which we are accus-tomed to call the limit of a line. Here, likewise, mathematics,on the basis of certain sense-experiences of which there aremany both in nature and among mankind, has constructed thenon-sensuous, we might say the super-sensuous idea, of a pointwithout extension in any dimension-an idea in itself bothuntenable and contradictory, a monstrous concept despite itsinfinitesimal size, of a something that is already a nothing, of anothing that is neyertheless supposed to be a something. Themathematical point IS, in all respects, a true