International Phenomenological Society is collaborating with JSTOR to digitize, preserve and extend access to Philosophy and Phenomenological Research.http://www.jstor.orgInternational Phenomenological SocietyFrege on the Statement of Number Author(s): David Sullivan Source:Philosophy and Phenomenological Research, Vol. 50, No. 3 (Mar., 1990), pp. 595-603Published by:International Phenomenological SocietyStable URL:http://www.jstor.org/stable/2108167Accessed: 01-08-2015 19:23 UTCYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jspJSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.This content downloaded from 164.73.224.2 on Sat, 01 Aug 2015 19:23:46 UTCAll use subject to JSTOR Terms and ConditionsPhilosophyand PhenomenologicalResearch Vol.L,No.3,Marchi990 Frege on the Statement of Number DAVIDSULLIVAN Universityof Illinois at Chicago Recent interest in the history of analytic philosophyhas focused attention quitenaturally onthe"foundingfathers" ofanalysis:Moore,Russell, Wittgenstein, and Carnap. This, in turn, has led to a re-examination of the nineteenth century and to intense speculation about the work of the Ger- man mathematician Gottlob Frege (i848-i9z5).Yet Frege's proper posi- tion within the historical canon of the discipline has been difficult to deter- mine.IfFregeisaphilosopher,whyarephilosophicaldoctrines, particularly traditional ones, so hard to find in Frege's writings - even the Grundlagen? Criticism toohas surrounded the very prospect of a historical account ofFrege's work.Some have questioned whether a credible investigation canbemadeintoFrege's intellectualbackgroundorwhethersuchan investigation, once made, can be adequately evaluated. These sceptics fear that the only connectionshistorical research will be able to bring to light are "parallels," that is, merely analogical features. What such an investi- gation willbe unable to determine are definitive "sources" for any of the key ideas in Frege's thought. Instead, it will be limited to making compari- sons, possibly superficial, between the ideas of Frege and those of his pre- decessors(or successors). The search for distinct sources quickly fixes upon the few direct refer- ences toworksand figures that Frege actually proffers. This brief paper will do likewise in its examinationof a possible source for Frege's central doctrine that "the content of a statement of number is an assertion about a concept."'ThisdoctrineprovidesakeyelementinFrege'slogicist project: it is, as Frege himself later described it, the "most fundamental" of his results on the concept of number and that upon which his account in theGrundgesetze "rests."'But couldthis fundamental idea conceivably haveappeared before Frege? I GottlobFrege, TheFoundationsof Arithmetic,trans. J.L. Austin (Oxford: Basil Black- well,1974),p.59. Gottlob Frege, The Basic Laws of Arithmetic, ed. and trans. MontgomeryFurth (Berke- leyand Los Angeles: University ofCalifornia Press,i967),p.5. FREGEONTHESTATEMENTOFNUMBER595 This content downloaded from 164.73.224.2 on Sat, 01 Aug 2015 19:23:46 UTCAll use subject to JSTOR Terms and ConditionsIt did. In one of his major works, the Psychologie als Wissenschaft, the nowobscurenineteenth-century philosopherJohannFriedrich Herbart (I776-i841)set forth a theory of number surprisingly similar to Frege's, in direct oppositiontothe onesuggested by Kant. Kant, remember, had characterized number as the result of our consciousawareness of the suc- cessiveadditionofhomogeneousunits.ForKant,number,generally speaking,isa"plurality consideredasunity."3 Itisaunitywhich,in counting,isdependent uponthe generation oftime.4 Herbart is unsympathetic to this viewpointor toany account of num- ber whichisbased upontheideaofcombiningunits intimetoform a totality. Herbart starts off his discussion in the Psychologie by noting that "since Kant, it has oftenenoughbeen repeated that the successive addi- tionofonetooneyieldsthisidea[of number], whichherewithiscon- nectedwithtime."5 But heisdriven toconcludehisowninvestigation with the thesis that number "has no more in common with time than do a hundred othertypesofideas,whichalsocanbe producedonlygradu- ally."6 Howthenisheabletoremovetheconceptofnumber from the domainoftime? Herbarteffectsthisseparationbydivorcingtheconceptofnumber properfromthatofhomogeneousunits.AccordingtoHerbart,time enters the picture in the "common operation of counting," for in counting it appears that "the number n+iis produced from the immediately pre- ceedingnumbernthroughtheadditionoftheunit."7 Butthispicture assumes that numbers actually consist of units. Abandon this assumption, Herbart believes, and oneremoves the need for temporal succession.To this end, Herbart brings to light an often overlookedfeature of our state- mentsofnumber; thisfeature isthat: . duringcounting,somethingis alwayspresentwhichis beingcounted;andtheideaof thissomethingmustalwaysremainsimilar,sincenotedlydissimilarthings(forexample, quills,sheetsof paper,sticksof sealing-wax)cannotbecountedtogetherunlessoneregards themas similar(bymeansof thegeneralconceptof 'writingmaterials').8 Immanuel Kant, Critique ofPure Reason,trans. NormanKemp Smith (NewYork: St. Martin'sPress, i965),p.ii6[B iii]. 4 Kant,CPR, p.i84[Bi82]. JohannFriedrich Herbart,SamtlicheWerkein chronologischerReihenfolge,ed.Karl Kehrbach and OttoFlugel (reprinted Aalen: Scientia Verlag,i964),vol.6,p.iio.All translationsfrom German-only sourcesare theresponsibility ofthis writer. 6 Herbart, SW, vol.6,p.iii. 7 Herbart, SW, vol.6,p.iio. Herbart, SW, vol.6,p. ii i. 596DAVIDSULLIVAN This content downloaded from 164.73.224.2 on Sat, 01 Aug 2015 19:23:46 UTCAll use subject to JSTOR Terms and ConditionsThat is to say, whenever we count, or make a statement of number, there must be something thatwe count: this something will always be indicated for usby thegeneral conceptwhichappears inthosestatements. But then what is counted or collected together are not the elusive units, but rather objects picked outby that general concept. Number,Herbart alleges, needs torely on the"homogeneous,"but not on"homogeneous units." Rather, number is now revealed to have an intimate connection to the general concept. Hence, our numerical predications may be character- izedby thefact that in them: Eachnumber...relatesto a generalconceptof the counted.This conceptcan, however, remainentirelyundetermined,sinceit is completelyirrelevantto the determinationof num- berwhatone is counting.However,one mustalwayskeepin mindthe relationto this unde- terminedconcept,sinceone would otherwisebe seducedby the falseconception of unitsas componentsof numbers.To the numberi z one mayaddin thoughtthe generalconceptof 'a chair'or 'a thaler'andso one wouldbecomeawarethatthe determinationof numberis undi- vided and at once attachesto the concept ... one can by all means successivelycount throughthe dozen ... but the numberi2z does not consistof twelveunits, since the unit would assumein this way the place of the generalconceptof the countable.9 As should be apparent, this view bears a strong resemblance to the one which Frege adopts as his own:in making a statement ofnumber, Frege claims that weare in fact making an assertion about(a predication of)a concept.'0 So whenHerbart concludestheabove-cited passage withthe remark that "a number is ... one of the possible answers to the question: Howmany?" it is no surprise that here toowefind a prominent motif of the Grundlagen:"What answers the question Howmany? is number."" Of course, if Frege never read Herbart or was otherwise unaware of his doctrines, then the effect of this parallelism of theme wouldbe nugatory. But Frege did read Herbart - and for this claim there are several pieces of evidence.First,amongtherecordeditemsoftheNachlass(now destroyed) we find that Frege made excerpts from Herbart specifically on the topicofthe"sense ofa statement ofnumber."'Second, Frege's yet wider knowledge of Herbart may be indicated by his occasional reference toHerbartianconcepts,suchasthe"travellinglimit."'3Third,Frege 9 Ibid. We are in fact ascribing to our general concepta certain property: as Frege explains it, our general concept is subsumed by a"higher or, so tosay, second order concept." See Frege, FA, p. 65. Frege, FA,p.57.See alsopp. 5n, 26,and98. AlbertVeraart,"GeschichtedeswissenschaftlichenNachlassesGottlobFregesund seinerEdition.MiteinemKatalogdesurspruglichen Bestandsdernachgelassenen Schriften Freges," Studien zu Frege I,ed. MatthiasSchirn (Stuttgart: Frommann-Holz- boog,1976),pp.96and 103. '3Gottlob Frege, CollectedPapers on Mathematics,Logic, and Philosophy,ed. Brian FREGEONTHESTATEMENTOFNUMBER597 This content downloaded from 164.73.224.2 on Sat, 01 Aug 2015 19:23:46 UTCAll use subject to JSTOR Terms and Conditionsconspicuouslyannounceshis allegianceto Herbart'sdoctrineon the con- cept of numberat the very beginningof the Grundlagen. In the Introduction,Fregeworriesthat few people even consider"the conceptof positive whole number"to be problematic,assuminginstead that "this concept is adequatelydealt with in the elementarytextbooks, wherethe subjectis settledonce and for all."'4Fregeinsiststhat we turn away from this "schoolboy"mentalityand examine "what others have thought."The result of our not doing so is .that we still rest content with the crudest of views, even though since Herbart's day a better doctrine has been available. It is sad and discouraging to observe how discoveries once made are always threatening to be lost again in this way,and howmuch work promises to have been donein vain,because wefancy ourselves so welloffthat weneed notbother to assimilate itsresults."5 Now the footnoteto Herbart'sname,markedin the text, does not referto the passages cited above from the Psychologie but toan educational work,theUmrisspadagogischerVorlesungen.i6Dotherelevantpas- sages from the Umrissinclude statementssimilaror equivalentto those alreadydisclosed? They do indeed.The applicablequote - which Fregeindicateswith the extract, "Two does not meantwo things,but doubling" - againsharply disavows any view of numbersas "sumsof units" and promulgatesthe doctrinethat numericalpredicationsare composedinsteadof "theundi- vided number"and "the generalconcept": Still more might be said against that false view of numbers, as if they were sums of units. This is so as little as that sums are products. Two does not mean two things, but doubling, no mat- ter if the doubled is one or many. The concept of'a dozen chairs' does not comprise twelve ideas of single chairs; rather there are contained only twoideas: the general concept'chair' and the undivided multiplication by twelve. The concept of 'one hundred men' likewise con- tains only two concepts: the general concept 'man' and the undivided number 'one hundred'. Likewise 'sixfoot','seven pound' -expressionsin whichlanguage comestoour aid with theuse ofthesingular."7 McGuinnessand trans. MaxBlack et al.(Oxford:Basil Blackwell,i984),p.178.The term derives ultimatelyfrom Herbart, SW, vol.8,p.68. '4Frege, FA, p.iii. '5Ibid. 6 Frege's possible educational interests - and their root in Herbart's work -would not be unusual for the time. Since i 865, there had been a strong revival of enthusiasm for Her- bart's writingsonscientific pedagogy.Karl VolkmarStoy(i815-i885),a Herbartian, founded a pedagogical seminar at the University of Jena in I 874.After Stoy's retirement, Wilhelm Rein (i847-i929),also a Herbartian and one of Frege's colleagues, turned Jena into a Mecca for educators all over the world.On these and other points of interest, see HaroldDunkel,HerbartandHerbartianism(Chicago:UniversityofChicagoPress, 1970). 17 Herbart, SW, vol.IO,p.10. 598DAVIDSULLIVAN This content downloaded from 164.73.224.2 on Sat, 01 Aug 2015 19:23:46 UTCAll use subject to JSTOR Terms and ConditionsNotice that Herbartsuggeststhat the truth of his doctrinemay be illus- tratedby the appearanceof the singularin certainlinguisticconstructions. This thoughtFregealso stronglyechoeswhen he remarksthat corrobora- tion for his centraldoctrinemay be foundin Germanidiom, that the "use ofthesingular...mayindicatethattheconceptisintended,notthe thing."8 Furthermore,we should also recallthat in the GrundlagenFregedoes not presentthe centraldoctrineexclusiveof a long section devotedto a discussionof number'srelationto the unit. In fact, Fregeput forthhis doc- trineas a "solution"to a veryparticular"difficulty."This difficultyFrege specifiesas that of reconcilingthe purportedidentityof units with their distinguishability: If we try to produce the number by putting together different distinct objects, the result is an agglomerationin whichthe objects containedremain still in possessionofprecisely those properties which serve to distinguish them from one another; and that is not number. But if wetry todoit in the other way,by putting together identicals, the result runs perpetually together intooneand wenever reach a plurality.'9 We are bound, Fregethinks, to uncovera solution to this difficultythat will preservethe "distinctionbetween one and many."" Our dilemmais that if we give the name 'unit'to the thingsto be num- bered, then, strictlyspeaking,we cannot call them identical;and yet if they are identical,then we can neverget beyondthe numberone. That is to say, if the things to be numberedhave distinguishingproperties,then they cannotbe, by definition,the samethingandyet if theyhaveno distin- guishing properties,then there isnoway toreach aplurality. In the attemptto fix upon homogeneousunitseitherwe "obliterate"the distinc- tion betweenthe one and the manywe were boundto preserve,or we are "forced... to ascribeto unitstwo contradictoryqualities,namelyiden- tity and distinguishability.""' The solutionFregeprescribes,which reliesupon the centraldoctrine,is designedto resolvejust this dilemma.Once we realizethe role of the gen- eralconceptin statementsof number,we may appropriatelyidentifywhat we have called the unit with the concept appearingin such statements: In the proposition"Jupiterhas four moons",the unit is "moon of Jupiter". Under this con- cept falls moon I, and likewise also moon II, and moon III too,and finally moon IV. Thus we can say: the unit to which I relates is identical-with the unit to which II relates, and so on. This x8 Frege, FA, p.64. '1Frege, FA, p. 50. ?Frege, FA, p. 58. 2'Ibid. FREGEONTHESTATEMENTOFNUMBER599 This content downloaded from 164.73.224.2 on Sat, 01 Aug 2015 19:23:46 UTCAll use subject to JSTOR Terms and Conditionsgives us our identity.But when we assertthe distinguishabilityof units, we meanthat the things numberedare distinguishable." The identity(or unity)lies in the abilityof differentthingsto fall underthe same concept; the difference(or plurality)lies in the fact that this sub- sumptiondoes not make them into the same object.We can numberdis- tinct objects by bringingthem togetherundera propertythey all share: but this does not meanthat they have no other distinguishingproperties, for then they would be the same object. Now the verysort of dilemmaoutlinedby Fregehas its roots, according to Herbart,in a singleperplexity - one which lies at the baseof the major metaphysicalquandaries."3From the fundamentalEleaticaporia of 'the one and the many'stem such difficultiesas reconcilingthe one thingwith many propertiesor the single ego with many presentations.24All such difficultieshave somethingin common:they ascribeboth unity and plu- ralityto a singlething.In the case of number,the sole corrective,Herbart would counsel,is to keepin mindwhat he callsthe "pointof the relation," which hereis "theconceptof the counted."25If we keep our eyes fixedon this concept,Fregeconcurs,we may "easilysolve the problemof reconcil- ing the identity of units with their distinguishability."6 This close affinity between the views ofFrege and Herbart, while hardly manifest tomodern English readers, did notgounnoticed in Frege's ownday. Reinhold Hoppe (i8i6-i900)inhis review ofthe Grundlagenadvertedtoitseveral times, and sodid Edmund Husserl (i859-1938)in his early work Philosophieder Arithmetik.If we briefly examine some quotations from these sources, then the cognizance of Frege'scontemporarieswill surely become clear. Hoppe, as a proponentof psychologism,takespains to point out some of Frege'schief errors. One of these, not surprisingly,involves Frege's approachto the concept: Frege, accordingto Hoppe, fails to take into proper account the "developmentalhistory" of concepts. Hoppe chas- tizes Fregefor his allegianceto the enemy camp, for being a logicianfor whom a conceptis "not an achievementbut ratheran item lightedupon, which he takes into his work as a chemist would a fossil or a meteor- " Frege, FA, pp.66-67. 3 Onthis pointsee George H.Langley,"The MetaphysicalMethodofHerbart," Mind, n.s. zz