history of zeno's arguments on motion

7/27/2019 History of Zeno's Arguments on Motion

1/9

History of Zeno's Arguments on Motion: Phases in the Development of the Theory of LimitsAuthor(s): Florian CajoriReviewed work(s):Source: The American Mathematical Monthly, Vol. 22, No. 6 (Jun., 1915), pp. 179-186Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2973956 .Accessed: 14/05/2012 08:30

Your 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.jsp

JSTOR 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 [email protected].

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to

The American Mathematical Monthly.

http://www.jstor.org
http://www.jstor.org/action/showPublisher?publisherCode=maahttp://www.jstor.org/stable/2973956?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/2973956?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=maa


2/9

THEAMERICAN

MATHEMATICAL MONTHLYVOLUME XXII JUNE, 1915 NUMBER 6

HISTORY OF ZENO'S ARGUMENTS ON MOTION:PHASES N THE DEVELOPMENT F THE THEORYOF LIMITS.By FLORIAN CAJORI, olorado ollege.

VII.7. KANT AND OTHER PRE-CANTORIANISCUSSION.We now come to a commandingfigure n philosophicthought-EmmanuelKant. He took Zeno's dialecticsmore eriously han had been the custombefore.Kant says that critics harged Zeno with a complete denial of both of two self-

contradictory ropositions. "But," says Kant, "I do not think that he canbe rightly harged with this."'1 Zeno was not as much ofa skepticas has beenpretended. Kant did not writeon Zeno's arguments n motion,but he touchedon other arguments f Zeno. Kant's first ntinomy, r "the first onflict f thetranscendental deas," contains parts whichremind ne of the followingnnihila-tion ofthe notion of space, as given by Zeno: If there s space, it is in something,forevery hing hat is, s in something; but that which s in something,s also inspace. Space, then,must also be in space, and so on infinitely: herefore here sno space. While Kant did not contribute irectly o a clearerunderstandingfZeno's arguments n motion, the effect f his writingswas a morepainstakingand searching xamination f that subject.In 1794there ppeared inHalle a monograph n Zeno's arguments n motionby C. H. E. Lohse,which s permeatedby the atmosphere f Kantian philosophy.It is theearliestpublicationon our topic whichappeared in theform fa mono-graph.2 Of tsfourparts,the first eals withZeno's systemngeneral, he secondgives his arguments gainst motion, he thirdelucidatesAristotle'srefutation fZeno, hefourthealswith the onlyway" ofrefutingeno. The astargument,the"stade," is not discussed t all. Aristotle'sistinctionetween potential

1Kant's Werke, d. III, "Kritikder reinenVemunft," . Aufl. 1787), Berlin, 904,p. 345.2 Car. Henr. Erdm.Lohse, Diss. (praesideHoffbauer)e argumentis,uibusZeno Eleatesnullum sse motum emonstravitt de unica horum efutandorumatione.Halle,1794. Allourinformationn Lohse'spaper s takenfrom . Wellmann,p. cit.,pp. 12-14.179


3/9

180 ZENO S ARGUMENTS ON MOTIONand an actual division to infinity s pronouncedarbitrary. Whatever can bedivided to infinity, ays Lohse, actually consists of an infinite umberof partswhich exist even before division. He decides on this point against Aristotleand in favor of Zeno, as Bayle had done, thoughhe does not mentionBayle.Lohse claims that Zeno's fundamental rror ay in a wrong conceptionof timeand space. These are not qualities subject to our senses,but are formswhichdetermine he manner n which our senses are affected;theyare a priori deas.Time and space can both be divided to infinity, ut one cannot considertimeas made up ofindivisiblepointsin the manner ofZeno, else what happens in amoment ftimewould happen in no time. Rest is not,as Zeno and his followersclaim,the absence of motion; it is the least velocityofsuccession. A body canbe perceivedonlyas it moves. "Without doubt," says Lohse, "all mistakesoftheir ystem prangfrom hat error. Thence it came that reason and the sensesseemed to contradict ach other."Presiding t the time when Lohse presentedhis dissertation oran academicdegree at Halle was Joh. ChristophHoffbauer 1766-1827) who, many yearslater, prepared a cyclopaedia article, "Achilles (Der Trugschluss)"1 Afterexpressinghis disapproval of Facciolati's argument previouslyreferred o) hestates that Zeno's argument s true only on a condition which has not beenstated explicitly:Zeno's contention that the faster runnerwill always onlyarriveat the places where the slowerhas been, and will be behind the slowerrunner, s true only on conditionthat the fasterhas not overtaken the slower.The only thingproved by Zeno is therefore hat the fasterrunner annot haveovertaken he slower s long as the slower s still n advance!A replyto Hoffbauer's rgumentwas made by ChristianLudwig Gerling,professor f mathematics, stronomy, nd physics at theUniversity fMarburg,in a prorectorat ddress.2 The claim that Zeno's argument s valid only forcertainpoints, not for all, is no objectionat all, unless it is first hownto be amistake o assert as true for ll points whatis in fact true of an infinite umber fpoints; a defender f Zeno may always demand that the points be shown,forwhichthe proofdoes not hold. Gerling nsists that Hoffbauerhimself easonsin a circlewhen he accuses Zeno ofreasoning n a circle,forwhoeverhas still toprovethepossibility f an overtaking s not yet permitted o speak of the timebefore rafterwhichthe overtaking akes place.Against Waldin's argument,advanced at this same university Marburg)forty-threeearsprevious, o theeffect hatZeno assumes theexistence fmotion,the very thingthat is in dispute, Gerlingargues that Zeno's argument s anindirect ne, a reductio d absurdum,he form f which s quite valid.

1 Allg.Encycl. . Wissensch. . Kuinste, on J. S. Erschu. J. G. Gruber, eipzig,1818.2 De Zenonis leaticiparalogismis otum pectantibus,issertatiouctore hr.Lud. Gerling.Marburg, 825. We know hisdissertationnly rom hedescriptionf t given y E. Wellmann,op. cit., p. 14, 15, and by Dr. Johann einrich oewe, "Ueber die Zenonischen inwturfeegendie Bewegung,"n Bohm.Gesellsch.. Wissensch.,I Folge, 1 Bd., 1867,pp. 30, 34. In Poggen-dorff's andworterbuch,he date of Gerling's issertations given s 1830. We have seen refer-ences o an editionnGerman f the year 1846. Fromthiswe infer hat several ditions f thave appeared, nd that t enjoyed considerableirculation.


4/9

ZENO S ARGUMENTS ON MOTION 181Lohse's metaphysical pparatus Gerlingdeclaresneedlessand useless. In theconstructivepart of his dissertation,Gerlingdwells on the distinctionbetweencontinuousand discretequantity,admits the infinite ivisibility f space andtime,and constructs he infinite eometric rogressionswhose sums giverespec-

tively hedistanceand the timeofrunning, eforeAchillesovertakes hetortoise.GerlinghererepeatswhatGregory t. Vincenthad donelongbefore, nlyGerlinguses letters,while Gregory ssumed a special numericalcase. Gregory s no-wherementionedby Gerling. The sums of the two geometricprogressions revalues which n no way conflictwith the estimateobtained from ensuousper-ception; Zeno's paradox, as interpretedby aid of the mathematicalformulas,conflicts n no way with experience. Hence the puzzle is solved. Though amathematician,Gerlingdoes not feel the need of explaining he possibility f avariable reaching ts limit.As to the "Arrow" a sharpdistinction etween the continuous nd the dis-crete s sufficient. n continuousquantitythe number ofpossible subdivisionsis arbitrary,nd each subdivision is itselfcontinuous. Hence Zeno's allegeddenial of the infinitedivisibility oes not follow. Gerlingtreats the "Stade"withmore than customaryrespect,and admitsthat, if one assumes withZenothatspace and timebe not infinitelyivisible, hen t follows, s Zeno says,thathalfthetime s equal to the wholetime.An entirelydifferentype of discussion,morealong the lines of Kant, pro-foundand obscure, s given by Georg WilhelmFriedrichHegel. He holds the

view that "Zeno's dialectic of matterhas not been refuted o the presentday;even now we have not got beyond it, and the matter s left n uncertainty."1He protectsAristotle gainstBaylewhoobjectedto Aristotle's istinction etweena potentialand an actual subdivisionofa lineto infinity. Hegel keenlyrealizesthe speculative mportanceof Zeno's paradoxes and pointsout that the dialec-ticianof Elea had analyzed ourconceptsof timeand space and had pointedoutthe contradictionsnvolvedtherein; "Kant's antinomiesdo no morethan Zenodidhere."2 Movementappears "in its distinction fpureself-identitynd purenegativity,he point as distinguished rom ontinuity."3 This continuitys anabsolute hangingtogether, n annihilationof all differences,f being by itself;thepointon the otherhand is pure existenceby itself, he absolutedistinctnessfrom thers, he suspensionof all self-identitynd all hanging ogether. In timeand space the oppositesare united in one, hence the contradiction s exhibitedinmotion. Hegel's position s a longway, still,fromGeorgCantor's continuum,with ts skilfulunion of continuity nd discreteness. In the "dichotomy" theassumptionofhalf a space is incorrect, ays Hegel, "there is no half of space,for pace is continuous;a piece of wood maybe broken nto twohalves,but notspace, and space onlyexists n movement."4 Motion is connectivity, isintegra-tionintoan indefinite umberofaggregates s its opposite.1G. W.F. Hegel,HistoryfPhilosophy,ransl. yE. S. Haldane,Vol. , London, 892,p.265.See also Hegel'sSamtl.Werke,d. 13, 1833,pp. 314-327.2 Hegel ed. Haldane],Vol. I, p. 277.3 Hegel,op.cit.,Vol. I, p. 268.4Hegel, op. cit.,Vol. I, p. 271.


5/9

182 ZENO S ARGUMENTS ON MOTIONSomewhatmore specific nd comprehensible re the ideas set forthby hisphilosophical pponent,JohannFriedrichHerbart. Zeno's paradoxesare takenup byhim n twoworks,his popularEinleitung n die Philosophie 1813) and hismore technical and scientificAllgemeineMetaphysik 1828-9). Only in thelatterwork s the solutionof the contradictions ttempted. Fromit we quote:'"The argumentnevitablyonfuseshose,who admit the infinite ivisibilityf thepathand then onsole hemselves ith correspondingnfiniteivisibilityfthetime, o such degreethatthought firstwillingo considerhe process f dividing, hichmustcontinueo infinity,they oon n one eapconsiderhe nfiniteumber ftime ntervalss passed over, ince hey eethat heymust ombinehe nfiniteumberfparts fthetime s well s ofthepathto theplaceofovertaking,hich hey annotdo. The leapand the doublynfiniteivisionre bothfaultyand amount o naught."Thus, this infinite ubdivisionof the time and space is rejectedby Herbart,becausethemind s not able to imagine ll thesteps n theprocess. Imaginability

is madethe criterion f truth r error. This criterion hrows ut infinityt once;it throwsout non-euclidean geometryand other parts of mathematics. Wecannotreallymagine hingswhichwe have never een. Our sensesare naccurate,our ntuitions re crude; hence it would seemto us impossible o buildup soundmathematical heory,feverythingnimaginablewere to be cast aside. Herbarttriesto explainmotionby the conceptofvelocity, hich seems itself o involvea contradiction,hatHerbartendeavorsto resolvebyhistheory fa "rigid ine,"a sortofcontinuum,whichmighthave givenrise to greatpossibilities pon morecarefuldevelopment. As it is, it offers reaterobstacles by farthan does theoriginal Dichotomy" or "Achilles" which t is intendedto explain.A stilldifferentttitudetoward Zeno's paradoxesis takenby FriedrichAdolfTrendelenburgof the Universityof Berlin, in his Logische Untersuchungen,1840,wherehe constructshis philosophic systemupon the concept of motion.Constructivemotion s common o the externalworldofbeingand to the nternalworldofthought, o that thought, s the counterpart f externalmotion,pro-duces from tselfspace, time,and the categories. Motion is undefinable. Inaccordancewith this view it is only throughmotion that Zeno's argumentsagainstmotionhave cometo be. For theydepend uponthe divisionof time andspace, and the synthesisof those divisions. But division and synthesis renothing ut specialforms f motion. What theproofs ombat,they themselvesuse as themeansofcombat,and thereby estify o the controlling ature ofmo-tion. Trendelenburg nd Kant evidentlybegin at opposite ends; Kant takestimeand space as a priori deas, and motionas secondaryand dependentuponthem; Trendelenburgmakes motionthe a prioriidea, and pretendsto derivetimeand space fromt.FriederichUeberwegof the Universityof Kdnigsberg2 efers o our subjectin differentartsof hisLogik. He says in one place that the "Achilles" provestoo little; itprovesmerely hat thetortoise annot be overtakenwithin definiteseries, nd then claimsthat the tortoise cannot be overtakenanywhere nd at

1 J.T. Herbart, amtl.Werke, erausgeg.onKarlKehrbach, angensalza, ol.VIII, 1893,p. 177.2 F. Ueberweg,ystemerLogik, . Aufl., . 387 if.


6/9

ZENO S ARGUMENTS ON MOTION 183any time. True as this criticismmay be, it does not illuminate the mattersufficientlyo satisfy hereader.Of the same type, but fuller n statement, s a criticismby Carl Prantl,'professor t the University fMunich. He claimsthat Zeno discardedthe con-cept of continuity y considering nly some particularpointson a line and onlysome particular moments n time. By drawinghis inferences romthese dis-integrated ragments f timeand space, Zeno was able to advance contradictionsin a picturesque manner. This conversionof the general and continuous ntothe particular and momentarywill be encounteredoften, ays Prantl,in thosewho care more forrhetorical orm han fortrue philosophy.Much confidence n his ability to clear up the mystery fZeno's paradoxes isdisplayed by Eugen Carl Diihring in his Kritische Geschichte er Philosophie,first dition,1869. Threeconcepts renecessaryhere: rest,motion nd position.Usually only the first wo are considered. At each moment point) of time amovingbody has a definite ositionbut no motion. This factmakes it difficultto explain motion. He says further2 "The compelling orce nd real conclusive-ness of the Eleatic contentions s to be found chiefly nd almost exclusively nthe logical necessitywhich does not permit he infinite o be thoughtof as com-pleted, as enumerated so to speak, and concluded. . . It is the conceptofinfinity hichproves tself verywhere nd also where t is not readilyrecognized,as the true cause of the contradictions." Diihringdiscussesinfinityn severalplaces ofhis works. He believesin the infinity suallyset forthn the studyofthe calculus,-a variable which ncreases without imit,but at any momenthasreallya finite alue. He makes war against the conceptof an actual infinity-"jene wiiste, sich widersprechendeUnendlichkeit." "The infinitedivisibilityindicates . . . only this, that I can conceive the divisionof a quantityas far asI choose,without imit. If, on the otherhand, I consider he divisionto infinityas reallyexisting utsideofmypresentation f t,then there oon result he mostmanifold ontradictions. . . As regards motion, t must be recognizedthat itbelongs o theempirical oncepts, . e., in ourthinkinghereremainsherealwaysan unrecognizable esidue,forwe must give up the attemptto penetrate to thereasons of the phenomena." Georg Cantor criticizesDiihringin thesewords:"The proofs fDuhring gainst heproperly-infiniteould be given n muchfewerwordsandappear o meto amount o this, ither hata definiteinite umber, oweverarge tmaybethoughtobe,cannever e an infiniteumber,s followsmmediatelyrom heconcept f t,orelse thatthe variable, n unlimitedlyargefinite umber, annot e thoughtfwith hequalityofdefinitenessnd thereforeot with he qualityofexistence,s follows gainfrom henatureof thevariability. That not the east is hereby stablishedgainst heconceivabilityftrans-finiteumber, feel ertain; nd yet, hoseproofs re taken s proofs gainst hereality ftrans-finite umbers. To me thismode ofargumentationppears he same as if, rom heexistence finnumerablehadesof green,wewereto conclude hat there anbe nored."3Diihring's explanation of infinitynd of Zeno is accepted by Eduard Well-mann,in his historicalmonograph4 f 1870. Anotherresearch,partlyhistorical'Carl Prantl,GeschichteerLogik m Abendlande,. Bd., Leipzig, 855,pp. 10, 11.2 Kritische esch. . Philosophie, r. E. Duihring,eipzig, 894,p. 49.3GeorgCantor,Grundlageniner llg.Mannichfaltigkeitslehre,eipzig, 883,p. 44.4E. Wellmann,p. cit., . 23.


7/9

184 ZENO S ARGUMENTS ON MOTIONand partly expository,was published in 1867, by JohannHeinrichLoewe, apupilofthephilosopher,AntonGunther fVienna. It is referred o by Knauer'as the most acute and satisfactory xplanationthat has yetbeen offered. "Thesolutionoftheriddle,"says Loewe,2"appears to us to lie in theknowledge hatcontradictionsmust arise inevitably, s soon as space,time, nd motion are con-sidered at the same time from he stand-pointof sensuouspresentation nd ofnon-sensuous onceptualreasoning." One point ofview appeals to the imagina-tion; the other o abstractthought. Sensuousperception an follow he processof infinite ivision only a littleway, everythingbeyond is a matter of purereason. Gerling'spresentation f "Achilles" is an appeal to reason. As longas one considers he infinitemultiplicity f small distances and oftime-intervals,one approaches the riddles from he standpointof abstractthought; when oneappeals to the imagination, henthe finite ime and the finiteengthoftheracestand out. Loewe seems still to hold to the old view that thought an recognizeno end to a motionwhichextendsover an infinite rocess. Hence the contradic-tionmuststand,the antinomy s evident.Thus we see that Germanphilosophydown to the last quarter of the nine-teenthcenturycontinuallyaccentuates the existence of contradictions n theproblem f motion.Some English thinkersof the nineteenthcentury,who were interested nZeno's arguments,came under the influence of Kantian philosophy. TheKantian attitude toward Zeno is described n the article "Zeno" in the eightheditionof theEncyclopaediaBritannica 1860) thus:"He brought most owerful ind o his task, nd, curious o say, ubsequenthinkers aveverygenerallygreed nmisunderstandingoth hisreasoningnd hismethod,nd it is only flate years that Kant, in hisAntinomies fthePure Reason (see Kritik derReinen Vernunft) eizeduponthemuchmaligned octrines f theEleatic,and heldthem p to theadmiration fall truethinkerss rareexamples facuteand just thought. Bayle, n a cleverpaper on Zeno, nhisDictionnaire, akeshim,according o custom, sceptic. Brucker inds hat Zeno surpasseshis ntelligence,ndhe s content o makehim pantheist. Others gain,have charged imwithnihilism. Zeno,fortunately,an affordo sit quiteeasy to all those ffrontsfferedo hisreason. . .they [argumentsgainstmotion] ll take theirrise, s Kant and Hamilton LecturesnMetaphysics)aveshown, romhe nability f themind o conceive ither he ultimate ndivisi-bility, r theendlessdivisibility,fspace and time, s extensive nd as protensive uantities.Thepossibilityfmotion, oweverertains an observed act, s thus hown obe inconceivable.To havediscoveredhispeculiarityfourmental onstitution,nd to havestated t with minentclearness, elongs oZeno theEleatic, nd to him lone."SirWilliamHamiltonputs thismatterthus '

"Time is a protensiveuantity,nd, consequently,ny part of it,however mall, annot,without contradiction,e imagineds not divisiblento parts, nd theseparts nto others dinfinitum. ut the opposite lternative s equallyimpossible;we cannotthinkthis infinitedivision. Oneis necessarilyrue;butneither an be conceived ossible. It is on this nabilityofthemind o conceive ither heultimatendivisibility,r theendless ivisibilityfspaceandtime, hat theargumentsftheEleatic Zeno against hepossibility fmotion re founded,-argumentshich t leastshow, hatmotion, owever ertain s a fact, annotbe conceived os-1VincenzKnauer,Die HauptproblemeerPhilosophie, ienu. Leipzig,1892,p. 54.2 J.H. Loewe,op.cit., . 32.3Lectures n Metaphysicsnd Logic, ySirWilliamHamilton, ol. I, Boston, 863,Lecture38,p. 530.


8/9

ZENO S ARGUMENTS ON MOTION 185sible, s it involves contradiction. . . Now the aw of mind, hat theconceivables in everyrelation ounded y the nconceivable, call the Law of the Conditioned."

JohnStuartMill, in his Logic,' refers o Thomas Brownwho consideredthe"Achilles" insoluble, nd then offers solutionto the invention f whichhe laysno claim. It presentsno new pointsof view. Herbert Spencer discusses ques-tions of time and space in his First Principles2 nd concludes in generalthat"ultimate scientificdeas, then, are all representative f realitiesthat cannotbecomprehended." In particular,"halve and again halve the rate of movementforever,yet movement till exists; and the smallestmovement s separated byan impassablegap fromno movement."It is readilyseen that the nineteenth enturyphilosophershad penetrateddeeperthan mostoftheirpredecessors nd had encountered ifficultiesreviouslyneglectedby Hobbes and otherswho seemedto thinkthat theyhad solved the"Achilles" paradoxby the mere statement hat time,as well as space, was infi-nitelydivisible. What came to be thoroughly ealized since the time of Kantwas the impossibilityf magining he "Achilles" from he standpointof nfinitedivisibilityfa distance, hat all appeals to intuitionwere futile. WhenSpencer,says that nfinite ivisibilityannotbe "comprehended," nd Thomas BrownandSir William Hamilton say that motion is "insoluble" and "inconceivable," Itake it that theymean simplythat these processes are unimaginable, hat theyare beyond thereach ofour sensual intuitions. I do not interprethemto meanthat these processes are beyond the reach of logic, beyond the reach of thereasoningfaculty o as to be, and forever emain,whollymysterious. Mathe--matics ncludes amongits results numerousteachingswhichone cannot "imag-ine." Probablyno one claims to be truly able to visualize to himself he non--euclideangeometries; analystsdo not claimto be able to imagine or see a con-tinuouscurvewhich has no tangentline at any of its points. Yet no modernmathematician ejectsnon-euclidean eometriesndnon-differentiableontinuouscurves.These unimaginablemathematicalcreationsare admitted into the scienceoas a matter fnecessity. Felix Klein states the ssueas follows: "As thesubjectsof abstract geometry annot be sharplycomprehended hrough pace intuition,,one cannot resta rigorousproof n abstractgeometry pon mere ntuition, utmustgo back to a logicaldeductionfrom xioms assumedto be exact."3It so happensthat England's twofamousopiumeaters,Thomas De Quinceyand Samuel Taylor Coleridge,were interested n the "Achilles." Coleridge'scriticalpowerswere set forth y De Quincey n the following erms:'

"I had remarkedo himthat thesophism, s it is usuallycalled,but thedifficulty,s itshould e called, fAchilles nd theTortoise,whichhadpuzzled ll thesagesofGreece,was, nfact,merelynother orm fthe perplexity hich esetsdecimal ractions; hat,for xample,fI A SystemfLogic,Vol. II, London, 851,p. 381.2H. Spencer, irstPrinciples fa NewSystemfPhilosophy,ew York,1882,pp. 47-67.-3 F. Klein,AnwendungerDifferential-nd ntegralrechnungufGeometrie. eipzig,1907rP. 19.4 Tait'sMagazine, ept. 1834,p. 514.


9/9

186 ZENO 'S ARGUMENTS ON MOTIONyouthrow into decimal orm,t willnever erminate,utbe .666666, tc., d infinitum.Yes,'Coleridgeeplied,theapparentbsurdityn theGrecian roblemrises hus,-because t assumesthe nfiniteivisibilityf pace,butdrops ut ofviewthecorrespondingnfinityftime.' Therewas a flash f ightning,hich lluminated darkness hathadexisted or wenty-threeenturies."

As a matterof fact,Aristotlehad seen that far. But Coleridgeproceededsomewhatfarthern an essay on Greek sophists n TheFriend,'where he says:"The few remains of Zeno the Eleatic, his paradoxes against the reality ofmotion, are mereidentical ropositionspunout nto sort fwhimsicalonundrums,s inthecelebrated aradoxentitled chilles nd theTortoise,hewholeplausibilityfwhich ests n the trick fassumingminimumf imewhile o minimums allowed ospace, oinedwith hatofexactingromntelligi-bilia,vo' eva, the conditionseculiar o objectsof the sensespoaLv6eva r aOvlxVa06jEVa."What belongsto Coleridgehimself n thispassage is the contention hat thesophismconsists n applyingto an idea conditionsonly properly pplicable tosensuous phaenomena. Coleridge's argumentwas elaborated many years laterin dialogue form, y ShadworthH. Hodgson. We give the critical part of thediscussion2

". . . being infinitely ivisible is not the same thing as being infinitely ivided. Actually todivide o infinityhat hundredthartof a minute,nwhich phenomenallys yousay) Achillesovertakes he tortoise,s an infinitelyongoperation. . . And thisdivisionyou call uponAchillesto perform, efore he tortoise can be overtaken,and to perform henomenally. . . Yourequire hatAchilles hallexhibit o the senses he nfiniteivisibilityf time nd space,whichappertainsto themtruly ndeed,but onlyas objects ofimaginationand thought. . . The worldofthoughtndrealitys not a world part,butis identicalwith hephenomenal orld, nlydif-ferentlyreated. . . Neither s there nycontradictionetween hem. Phenomenalmotion sas infinitelyivisiblenthoughts time ndspaceare."This explanationdoes not explain. Even as "objects of the imagination"the nfinite ivisibility ftime and space is a sourceofperplexity. Our imagina-tion is unable to followAchillesto the end, throughthe infinities f time andspace intervals. Moreover, "thought nd reality" are indeed worlds "apart"whenever he time ntervals, orrespondingo the space-intervals assed over byAchilles, re so taken that theyform ogether n infinite eries that is divergent,so that, in thought,Achilles never overtakesthe tortoise; in Zeno's traditoinalargument, thought and reality" were"apart."1Complete orks fS. T. Coleridge,ol. II, New York,1856,p. 399.2 Mind,London,Vol.V, 1880,pp. 386-388.[The remainingparts of this series are: D. VIEWED IN THE LIGHT OF AN IDEALISTIC CONTIN-UUM G. Cantor); E. POST-CANTORIAN DISSENSION.]

history of zeno's arguments on motion

Documents