THE CONCEPT OF THE INFINITE.

Transcription

1 THE CONCEPT OF THE INFINITE. EvERY student of the deeper problems of theology fmilir withwht oftenknown s theproblemoftheinfinite. Under the meningofthone phrse mybebrought number of dtinguhble, but closely connected questions. Someofthesequestions pper,inmoreorlessveiledform, even in the bckground of the dcussions of dily life. We llredposedtoregrdtimesendless,nd spcesbound less. Problems bout wht lsts forever, or bout wht hd nobeginning, resuggestedtousbyfmilirconsider tions. Even children sk questions tht imply the instence nd the interest of th conception of infinite time. The dult mind,inourmodern dys, remindedconstntly freshof thconceptionbythefctsofgeology,ndbythetheoryof evolution. On the other hnd, stronomy just s constntly suggests the problem of the boundlessness of the world in spce. AndtheologyknowstheproblemoftheInfiniteinthe formofwell-known questions concerning theinfinityofgod, nd concerning wht th infinity, if it dmitted, implies. Even if one regrds ll such problems s insoluble, there remins, fornystudentofhumnnturein generl, ndof the religious consciousness in prticulr, the question: VVht rethedeepermotivesthtmkemnsodposed toconceive both the universe nd God s infinite? YettheproblemoftheInfinite,innyofitsforms,so ncient, nd hs been so often dcussed, tht nyone who resitnewhstomeettoncetheobjectionththecnonly thresh gin theoldstrw. Imyswell syttheoutset,

2 22 THE HIBBERT JOURNAL therefore, tht the following pper seems to me to be justified bythefctthtcertinofthe recent dcussions ofthe conceptoftheinfinite, towhich mytitle refers, hve set these ncient problems in decidedly new light. Th pper inthemin,therefore, reportuponwht,infrnce,hs oflte been clled, in philosophicl dcussion, the New Infinite. I myself cre little for th modern fshion of recommending ides merely by prefixing the djective new. Truth never essentilly new, being lwys eternl. But if thedjective new servestomkereder ptientenough tottendtoonemoreessyontopic which Artotleso skilfully outlined, which the Scholstics so ptiently elborted, nd which the modern dcussions of Knt s Antimonies my seemtosometohvelongsinceexhusted,iwillnothesitte toemploythesomuchbused word. Asfct,recentd cussion hsputtheconceptoftheinfinitein wht,tome, seemsdecidedlynovellight. VVeseemtobetthebegin ningofthettinmentofquiteunexpectedinsight stothe logic of ll dcussions bout infinite collections, complexities, nd mgnitudes. VVhile the dcussions to which I refer hve beenbegun,ndhvebeen,inthemin,crried onbycertin mthemticins of somewht philosophicl turn of mind, theyhvenowreched point where, sithink,thegenerl students of philosophy nd of theology should no longer ignore them. Inrecentpublictionofmyown, Ihveendevoured in severl pssges to pply the results of these mthemticl studentsofthelogicoftheinfinitetotheconsidertionof centrlmetphysiclproblems. Inthe presentpper, however, Ishllttemptlittletht originl. Ishllbecontentif whtisyservestoindictetonyfellow-student thtthe problemoftheinfiniteslivingproblemto-dysitws when Artotle first ttcked it, nd tht new results, of unlooked for exctitude nd clerness, hve ltely been obtined inthncientfieldofwork. 1 The World ndthe Individul, 2vols.,London, 1899nd See especilly the Supplementry Essy ppended to the first volume.

3 THE CONCEPT OF THE INFINITE 23 I. The scope ofthe presentessymustfirstbe brieflyindicted. I hve mentioned the fct tht some rther mysterious motives, lying very. deep in humn nture, hve led mny mentobelieve thttheworld infinite, ndtosserttht God infinite. Such beliefs nd ssertions, in their origin, ntedte ny cler consciousness, on the prt of those who first mintin them, both regrding wht these motives for such doctrines my be, nd regrding wht the very concept of infinity itself mens. Tht th unconsciousness bout themeningndthegroundsofourbeliefintheinfinitedoes go long with our erly ssurnces bout the infinity of things cnbeshownbothbythecseofanximnder, ndbytht of ny thoughtful modern child who sks questions tht presuppose n ide of the infinity of the imiverse. Accord ingly, when wetryto come to clerer insight bout the problem of the Infinite, we nturlly hve to dtinguh two questions. The one purely logicl question: Wht do wemenby the concept oftheinfinite? Theother metphysicl question: VVht grounds hve we, if we hve ny grounds, for sserting tht the rel universe, whether divine or mteril, whether sptil or temporl, infinite? The rtionl nswer to the ltter question presupposes tht the first question hs been nswered. On the other hnd, n nswer to the first question might leve the second question wholly open. Nowthepresentessywillbeminly devotedtothe first ofthesetwoquestions. I shll dcuss, for themost prt, the concept of the Infinite. The question whether the rel world, or whether God, ctully infinite, will merely be touchedupon siclose. Itthelogicndnotthemet physic oftheproblemoftheinfinitethtwillhereformmy min topic. YetIdmit,ndinfctinst, thtthewholeinterestof thelogiclsuethusdefinedliesinitsreltiontothemet

4 24 THE HIBBERT JOURNAL physicl sue. I m well wre how brren considertion ofthemereconceptoftheinfinite would be,ifitdidnot helpustowrds decion oftheproblem whethertherel world infiniteornot, nd neverthelessifeeltht,inthe present stte of philosophicl study, we must tke the trouble to dwell somewht crefully upon the merely preliminry prob lem,eventtherkofbeingccusedofelbortingmere concept, nd of neglecting n ppel to the concrete fcts of therelworld. ForIfind,sIlookoverthehtoryofthe problem of the Infinite, tht much of the ordinry tretment ofthemtterhsbeen confinedtocertinftlcircle,in whichthestudents ofourproblemhvebeenledroundnd round. First, the foresid motives, vguely felt, hve forced men tomkethehypothesthttheworld infinite. As soon s one hstried to nlyse these motives, one hs observed thtcertin spects ofour experience do indeed furnh pprent grounds for believing in the infinity of the universe. But hereupon, becoming criticl, one hs sid: Yet the concept of wht the Infinite nd mens seems to trnscend the limits of humn intelligence. Andsoonehsrefused toconsiderfrthertheevidences for therelityoftheinfinite, simply becuseofth supposed incomprehensibility of the conception. On the other hnd, nyefforttocleruptheconceptionoftheinfinitehsoften metwiththeobjectionthtmerenlys ofidestedious, ndthtonewntslightstothefcts. Thus,however,the problem of the Infinite hs oflen filed to receive fir tretment from either side. The fcts bering upon the mtter re ignored,becusethe concept toodifficult; ndtheconcept neglected onthe ple thtthefctslonecnbedecive. I desirenewtobrekintothftlcircle. Letus mke t lestour concept oftheinfinitecler,ndthenweshllbe prepredtobejusttothefctswhichindictetheinfinityof the universe. In expounding thenewer conceptions oftheinfinite, I shll follow, s I hve lredy indicted, the led of certin

5 THE CONCEPT OF THE INFINITE 25 mthemticins, in prticulr of Richrd Dedekind nd George Cntor. Ishlluse,however,in prt, my ownillustrtions, ndshlltryto emphse inmyownwythephilosophicl,s opposed to the mthemticl, significnce of the ides in question. I shll then briefly indicte how the new ides ought,inmyopinion,tomodifyllfuturedcussion ofthe evidences regrding the ctul extence of infinite beings. Imylsosy,tonce,thtmydcussion oftheconcept oftheinfinitewillhvereltionnotsomuchtothe concept of infinitemgnitudes (such s ordinryeucliden spce when itviewedspossessingvolume),butrthertotheconceptof collections, whose units exceed in number the number of ny finite collection of units. The conception of n infinite mgni tude, such sninfinite volumeorninfinite mss,would require for its sttement certin conventions regrding the mesurement of mgnitude, which do not here need our ttention. I shll confine myself to defining infinite collec tions, nd infinitely complex systems of objects. We shll see thtthe metphysicl, ndinprticulrthetheologicl,pplic tionsofourconceptoftheinfinitereespecillyreltedtoth spect of our topic, while the conception of n infinite mgni tude, in the nrrower sense of tht term, hs less philosophicl interest. II. In ordertohelpustowrdsthnewconceptionofthe Infinite, let us begin by reminding ourselves of very simple 1AfullerccountoftheliterturethnherepossibleIhve given in the course of the Supplementry Essy just cited. The definition of Dedekind contined inhnow clssic essy: WsSind und WsSollen diezhlen? Th pper hs recently been trnslted into Englh, nd publhed in volume entitledessy; onnumber,bytheopencourtcompny of Chicgo. George Cntor's numerous ppers re widely scttered. Their substnce hs been in prt summred in the dmirble book by Lou Couturt: L I1;fini Mlhé mliqu(pr, 1897). A fuller sttement of the technicl results hs ltely been given, from the mthemticl point of view, by Schonfliess, in h Berichi iiber die Mengenlehre, in the eighth volume of the Proceedings of the Deutsche Mthemtilrer-rereinigu-ng.

6 26 THE HIBBERT JOURNAL observtion, whichmnyofusmyhve mdeinthesedys whendvertements resoconstntlybeforeour eyes. Iths occsionlly occurred to some ingenious mnufcturer, when in serchoftrde-mrk,touse,ssuchtrde-mrk,pictureof one of the pckges wherein h own mnufctured product putupforsle. Crrying outthpln,themnufcturerin question ccordingly puts upon every pckge of h goods. lbel whereon engrved th trde-mrk. We cn ll recll, I fncy, pckges of proprietry rticles lbelled in th wy. Someofusmyhvenoticed,however, in pssing, certin logicl consequence which th pln involves, if only we suppose the pln rigidly crried out. Ech lbelled pckge to ber upon itself, in curiously egotticl fshion, pictureofitself. Butthe pckge, thuslbelledwithitsown picture, inevitbly requires the picture to contin, for ccurcy s ske,sprecerepresenttionspossible ofthe pper nce,notonlyofthewhole pckge, butofeveryvibledetil thereof. The lbel, however, itself detil belonging to the ppernce thtthe pckge presents. Accordingly,the picture tht constitutes the lbel must contin, s prt of its own detil,pictureofitself. Wht we see,then,onthe ctul pckge, pictureofthpckge;whilethrepre sented pckge hs upon itself, in the picture, second trde mrk lbel, which gin contins picture of the first pckge, ndsooncemoreofthelbelitself;ndthseriesofpictures within pictures continues before our eyes s fr s the ptience orthewgesoftheengrverofthetrde-mrk hveledhim to proceed intheworkofdrwingtherequireddetils. Now itmyhveoccurredtosomeofustht,iftheplnofsuch trde-mrk s th were to be exhustively crried out, without nyfilureinthe engrver orinthemteriltohinderits expression, the pictures within pictures, which the pln de mnds, would soon become invibly smll. In fct, it not hrdtoseehow,bymensofsingledefinble pln,viz.,by mens of the one requirement tht the pckge shll ber upon itself, s lbel, perfectly ccurte pictoril representtion of

7 THE CONCEPT OF THE INFINITE 27 itself, including in th representtion the lbel which the pckge bers, one logiclly prescribes n undertking tht could not be exhustively crried out if the lbel itself con tined only finite series of pictures within pictures, however long tht series might be, or however minute the detil. Just sthelbelwouldfiltopicturethewhole pckge ofwhich itselfvible prt, unlessthelbelcontinedpictureof itself,sonypictureofthelbelthuscontinedwithinlrger pictureofthelbel,ndofthe pckge, would beimperfect unless, howeversmllitmight be,itcontinedpictureof itself; ndthustherecouldbenolstmemberoftheseriesof pictures within pictures, which the one pln of mking the lbel perfectpictureofthe pckge wouldprescribe. Nowthsystemofthe pckge, thepictureofthepckge, thepictureofthpicture,ndsoon,system defined by single,ndinonesense,verysimplepln. Wemytonce givethplnnme. Weshllcllitplnofprticulr sortofself-representtion, plnwherebywholetobe picturedor representedbyoneofitsownprts. Itsimple pln,becuseinordertodefineityouhveonlytodefine:- first, the forml conception of perfect pictoril representtion ofnobject ( conceptionwhich,ofcourse,remins forusn idel, just s ny geometricl definition n idel, but which perfectly comprehensible idel); nd secondly, the eqully forml conception tht the picture shll be con tined in, or lid upon, the object tht pictured, nd shllformprtthereof. Putthesetwopurelyforml nd perfectly definite ides together, nd the proposed pln exctly defined. Letusconsider thetwo ides for moment seprtely. \Veknowwhtittoconceive tht vibleobject,o,shll hve picture, R, which shll precely represent its every vible detil. In order to form th conception prt from the otheronethtijustcombinedwithit,werenotobligedto conceivethtthepicturertobeslrge stheobjecto. Tht smller picture should still be perfect representtion

8 28 THE HIBBERT JOURNAL of lrger object perfectly definble idel. Wht we menbythidelmerelyth,thttoeveryvrietyofdetil in the object there shll correspond some precely similr vriety of detil in the picture. Thus, if the object consted oftwolines,rrnged incross,thepicturewould simplybe nother cross. If the object consted of seven dtinct points, rrnged inrow,thepicturewouldberowofsevenpoints. Sofrthere indeed norequirementthteitherobjector picture should be infinite, or even modertely complex. Andnextwemyviewtheotheroneofourtwoidesby itself. Tht vibleobject,r,shouldbeprtoflrger object,thlsoprecelydefinbleide,ndverysimple one. Thide, moreover,,uponitsfce,nottllincon stent with the former ide. Buthereupon,inordertodefine whtwehve clled the plnofself-representtion,wehveonlyto suppose thesetwo seprtely definble ides, tht of the perfect picture, nd tht oftheprtcontinedwithinnduponthewhole,tobecom bined, so tht vible object should be produced tht contined, s prt of itself, perfect representtion of itself. But t once, so soon s, by th combintion of two perfectly compre hensible nd constent ides, we define the pln of self representtion, we observe tht no finite degree of compliction ofobject ndpicturewould enble usto conceive thepln perfectly crried out. An object tht contined, s prt of itself, perfect picture of itself, in other words, self-repre senttiveobjector system ofthetypehereinquestion, would of necessity prove to be n object whose complexity of structure no finite series of detils could exhust; for it would continpictureofitself,withinwhich there wstobefound pictureofthpicture,nd pictureofthsecondpicture, ndsoonwithoutend. III. The trivil illustrtion of the nture of Self-Representtive Systemwhichwehvejustused, hsthusdeepermening

9 THE CONCEPT OF THE INFINITE 29 thn we should t first suppose. We define comprtively simple pln;buthereuponwe come to see tht thepln demnds, for its complete expression, n infinite series of detils. And we see t once tht the self-representtive chrcteroftheplnthelogiclgroundforthinfinityof the required series. The self-representtion of whole by one ofitsownprtswould,ifcrriedout,implythtthewholein question hd n infinitely complex constitution. But here uponletusturnformomentfromthstudyoftheexplicitly self-representtive systems to the considertion of n object tht wellofus re ccustomed to regrd s t lest possibleobjectofthought,ndthtwerell dposedtocon ceive s, t lest potentilly, n infinitely complex object. I refer to the mthemticl object known s the series of whole numbers,1,2,3,4,5,ndtherest. Wellgreetht,inour conceptionstlest,nowholenumberthtyoucnnmecn beregrded sthelstofthe possible wholenumbers. Any seriesofnumbersthtwecntpresent writedown,ortht wecncountinfinitetime,willbefiniteseries. Butno such finite seriescnexhustthe possible whole numbers. Ontheotherhnd,whtwemenbytheobjectsclled whole numberssomethingperfectlyprece. The possible whole numbersformno finite collection; but theydoformperfectly definite collectionof objects,- definite inthesensethtth collection excludes from its own domin ll other objects. VVehvenodifiicultyintelling,whennyobjectbrought beforeournotice,whetheritwholenumberornot. Thirty wholenumber; but or Tl ; not whole number. A treeornngel not whole number. Thus the collection of possible objects clled wholenumbers,lthough,in one 31; perfectly definite sense, boundless collection, hving no lst term, still fr from being n ll-inclusive collection. It it infinite in one sense; but, in nother sense, strictly limited nd exclusive of whtever lies outside of it. Cntor wouldcllsuch ninfinitecollection well-defined collection (wo/ddefim'rte Mnge) of possible objects, -endless, but in no it

10 30 THE HIBBERT JOURNAL sense vguely endless, since of ll possible objects you cn exctly sy whether they belong to the collection in question or not. Let us, then, ccept for moment the whole-number-series s collective objectofourthought. Let usregrdits infiniteinthemerelynegtive senseofhvingnolstterm. I now wh to cll ttention to n interesting consequence of viewing the number series thus. If you choose, you cn, nmely, view the whole number series s contining within itselfperfectlydefiniteprtofitself,which,inprece sense, complete representtion or picture of the whole series. For the series of whole numbers essentilly chrctered by thefctthtiths firstmember,second member, third member,ndsoonwithoutend. Grnting th,stheessen til chrcter of the series, let us consider certin perfectly definite portion of the whole number series, nmely, the series ofevennumbers. Thtserieshsfirstmember,2; second member,4;thirdmember,6;fourthmember,8;ndsoon without end. Now, suppose thtunderseriesofthewhole numbers,iwritetheseriesofevennumbersinorder,thus: 3? :*~} 95 9? >-101 P. It plin tht, just s conceive thtnonumberintheupper series thelstofthe whole numbers, so m forced to I I conceivethtnoevennumberinthelowerseries the lst oftheevennumbers. It lso plin tht, however fr I mightextend theupperseries,bywriting inorderthewhole numbers up to ny whole number, n, however lrge, I might stillextendthe seriesofevennumbers bywriting them in orderupto2n. Thelowerseriesmightthuslwys remin justscomplexndjustswell-orderedseriessthewhole numbers oftheseriesbove it. Andthusthelower series would form, s possible fct, prece picture of the upper series. Speking in generl terms, cnsythttonywhole I number n, however lrge, there lwys corresponds, in th

11 THE CONCEPT OF THE INFINITE 31 wyofrrngingmtters,nevennumber,viz.2n,sothtthe lower series ble tofiunh, from its stores of possible members, the resources for the picture or representtion of every whole number, however gret, nd of every series of whole numbers, however long. The world of the possible even numbers,sofrsthepossessionoffirst,second,third, ndnolstmemberconcerned,precelysrichsthewhole numberseries. Thus,then,therenexctsenseinwhichl cnsy,thecomplexobject clled thetotlity ofthe even numbersprecelymirrors, depicts, corresponds incomplexity to, the complex object clled the totlity of the whole numbers. But,ontheotherhnd,theevennumbersformmerely prt, nd perfectly definite prt, of the whole numbers. For fi'om the totlity clled the collection of the even numbers, ll theoddnumbersreexcluded. Yetthmereprtsrich initsstructuresthewhole. Th illustrtion of the even numbers, viewed s constitut ingprtofthewholenumbers, but prtwhichneverthe lesscnbemdeto represent precely thewhole, hs been muchusedintherecentdcussionsofthe newinfinite. A more striking illustrtion still furnhed, I think, by nother series of whole numbers, selected, ccording to definite prin ciple, from mongst the totlity of the whole numbers. Let us consider, nmely, the series of the integrl powers of 2, rrnged in their nturl order, thus: 2,2,2,2,2.... Nowitplin,tglnce,thtthseriesofthe powers of 2infiniteinprecelythesense inwhichtheseriesofthe wholenumbersinfinite. Fortherepowerof2tocorre spond to every whole number without exception, since every wholenumbercnbeusedsnexponent,indictingpower towhich 2cnbered,nornywholenumber possible whichcnnotbeusedssuch nexponent. Hencetheseries ofthepowersof2,shererrnged inorder,precelycorre sponds, member for member, to the series of thewhole

12 32 THE HIBBERT JOURNAL numbers. But,ontheotherhnd,everyintegrlpowerof2 itself whole number. Thus2*=4;23=8; nd so on without end. Andthewholenumbersthtrepowersof2, tkenlltogether,constitutenotonlymereprt,butin very exct sense n extremely smll prt, of the entire collec tion of the whole numbers. For there re infinitely numerous groups of whole numbers which re not powers of 2. Thus, llthewholenumbersthtrepowersof3,ndllthepowers of 5, swell sllthepowersof7,orofnyotherprime number, nd, in ddition, ll the products of different prime numbers (17.0. llnumberssuch s 3x7,or5x11),ndfinlly, llthosenumberswhich reproductsof powers ofdifferent primenumbers (i.e. llnumberssuchs2 x7,or 5 xll )re excluded from mongst those whole numbers which re powers of 2. And, nevertheless, tht prt of the whole numbers which constsofthepowersof2hsseprtemembertocorrespond to every single whole number without exception. In other words,thprt,smllsit precely s rich s the whole., IV. Butletushereuponlookbck. Asweswincseofthe trde-mrk,thesystemofpicturesdefined bytheoneplnof requiring given object to contin, s prt of itself, com plete representtion of itself, would prove to be n infinitely complex system in cse we supposed the pln crried out. Or, in brief: ny Self-Representtive system of the sort tht we before defined, in pln or idel, infinitely complex. But, s thewholenumbersystemhsjustillustrtedforus,thecon verse of th proposition lso holds true. Any system of possible objects tht we lredy recogne s infinite in the negtive sense of hving no lst member, it inevitbly such tht we cn t plesure dcover within prtwhich,in complexity, fully dequte to represent the whole. Thus Infinity nd Self-Representtion (using the ltter term in the specil sense bove defined) prove to be inseprbly connected

13 THE CONCEPT OF THE INFINITE 38 propertiesofnysystemof objects thtwecnprecelydefine. If system to be self-representtive in the foregoing sense, it must beinfinite;ontheotherhnd,ifsomehowwelredy knowittobeinfinite,wecnproveittobesuchthtinsome (yes, in infinitely numerous) definite wys it self-represent tiveintheforegoingsenseofthtterm. Inviewofthesefcts,ithsoccurredtoDedekindtooffer, sthedefinitionofwhtwemenbytheinfinityofsystem orofnobject,formulthtwemyexpressinourownwy thus:--an objector system infinite itcn berightly regrdedscpbleofbeingprecelyrepresented,in complexity of structure,orinnumber of constituents, byone of itsown prts. Ihvetogivethdefinitionfirstinformthtnotyet idelly exct. Dedekind pproches inhessyuponthe number concept, in more bstrct nd exct fshion. But hve sid enough to show, I tthentureoftheinfinite,there it, I hope,thtinthwyoflooking something worth following up. And swehvehere little spce forgetting closer cquintncewiththesenew spects ofourtopic,letustonce remind ourselves of wht interest philosophicl student my hveinsuch view of the infinite. V. Any self-representtive system, if -complete, would be infinite. We pproched ourrecognitionofthttruthby trivil instnce. But the philosophicl student knows of one of h own most centrl nd beutiful problems which the formul now reched sets in somewht new light. Tht problem the problem of the Self. Whtever our view of the psychology of self-consciousness, or of the mentl limit tionsunderwhichwenowreforcedtoliveinthworld,we must ll of us recogne tht one chrctertic function of the Self the eflort refiectivelytoknowitself. Self-consciousness weneverfullyget,butweimtit; our ethicl s well Von.I.--No. 1. it 3

14 34 THE HIBBERT JOURNAL s our metphysicl gol. Now wht would be the conscious stte of being who hd ttined complete self-consciousness, who reflectively knew precely wht he ment, nd did, nd ws? To such being we esily scribe godlike chrcters. GodHimselfweoftenconceive ssuch completedself. If other selves thn God re cpble of such complete self consciousness, they re in so fr formlly similr in nture to the divine. But wht our observtion of the self-representtive systems hs shown us tht in their form, however trivil their content, these systems possess structure correspondent to the onethtwemustscribe tonyidellycompleteself,insojr it, s conceiveds self-conscious. Acompletelyself-conscious being would contin within himself, s prtofhwhole consciousness, not, of course, mere picture, but complete rtionl representtion of h own nture, nd of the whole of thnture. In consequence, swe hvenowseen,hewould be, lpsofcto, n infinite being. To define the idellyor formlly complete Self, to define the infinite, thusto define the infinite. Conversely, to define n object tht inevitbly hs the forml structure which we must ttribute to n idel Self. The two conceptions re convertible. To question whether the infinite rel, or whether ny rel being infinite,, there fore, simply to sk whether the Self, in its idel completion, concept tht stnds for ny ctul entity, or whether, in turn, RelityhstheformoftheSelf. Thustheproblemofthe infinite becomes centrl in philosophy in new sense. Menwhile,whenoncewelern toviewthemtterthus, theconceptoftheinfinite loses its vgueness, its negtive spect,its ppernce ofmeningsimplywhtlcksboundry, or hs no outlines. The conception of n idelly completed Selfmybehrdoreven remote one, but certinly not merely negtive, or vgue one. VVereyou lltht s Selfyouidellymight be,youwould notlosedefiniteness of outline,or prece chrcter,ordtinction fromotherselves. Yet,swenowsee,youwould become,informlcomplexity, infinite. Hence,tobethusinfinite would notmen tobe it

15 THE CONCEPT OF THE INFINITE 35 nothinginprticulr. Norwoulditmentobeeverythingt once. Northexct conceptoftheinfiniteonewhichwe cmiot grsp. Onthecontrry,noconceptmore prece; nd not mny importnt concepts re simpler. To conceive thetruentureoftheinfinite, we hvenottothinkofits vstness, or even negtively of its endlessness. VVe hve merely to think of its self-representtive chrcter. VI. Butifthnew concept simplendexct,it ppers toour common-sense unquestionbly prdoxicl. For we ll erly lerned certin so-clled xiom, used by Euclid, nd very generlly regrded s typicl cse of fundmentl verity. Ththeprincipletht theprtcnnotbeequltothe whole, ortht thewholemustbegreterthnthe prt. Nowitmyppertosomeredertht,intheforegoingstte ments bout the even numbers, nd their reltion to the whole numbers, ndinourillustrtionoftheseriesofthe powers of 2,we seemto hve comedngerously ner todenying the truthofthxiominitspplictiontoinfiniteor self-repre senttive systems. Th seeming well founded. As fct, our definition of infinite systems s self-representtive depends uponctullydenyingthtthxiom pplies tothem. It quitetruethtthexiomboutprtndwhole pplies toll finite systems nd collections. But common-sense, in tlking bout the vguely pprecited ides of the infinite which we llforminconnectionwiththenotionofinfinite spce nd endlesstime,hsoften expressed, inmoreorlesshltingwy, its sense tht to infinite systems the xiom in question would somehow fil to pply. Subtrct finite from n infinite mgnitude, nd the reminder, s we sometimes feel, must be sgretsever. Butthenewerconceptionoftheinfinite depends,notuponsuchvgue sense offiluretopplythe oldxiom,butupondefining,in prece wy,thtproperty of infinite systems (nmely, their property of being self-repre

16 36 THE HIBBERT JOURNAL senttive) which, s property, ensures tht the xiom of whole nd prt does not pply to these infinite systems. As fct, it perfectly possible to investigte mny mthemti clly defined infinite objects nd collections in very prece fshiontoseewhetherornotheyreequl. Itpossible to define two infinite collections tht re unequl to ech other. Itpossibletodefinethesortofequlityorofinequlitytht,insuchinstnces,inquestion,withsmuchprecionsyou cnuseindefiningtheequlityoftwofinitenumbers. And nevertheless it possible, while retining ll the definiteness of one s conceptions, to mke the whole investigtion of infinite mgnitudes nd collections depend upon sserting tht, in their cse,theprtmyequlthewhole. Escpe from bondge to rbitrry xioms, infct, necessry condition of exct thinking upon fundmentl topics. Whenyoussume nrbitrryxiom, s,ofcourse,youhve right to do, in ny prticulr investigtion, still necessry, you wnt to think in thoroughgoing fshion, to know tht th xiom rbitrrysolong sits opposite not if self-contrdictory. Consequently, in considering the rnge of possibilities, you cn lwys suppose the contrdictory of your originlly ssumed xiom to hold true for some conceived rnge of t lest possible being. Now, the so-clled xiom boutwholendprtcomestousinthefirst plcenotsn bsolutely necessry presupposition of thought, but s n empiriclgenerltion,foundedonour experience offinite collections nd mgnitudes. VI/by th xiom holds true for finite collections we do not ordinrily see. It something tolerntht th xiom pplies tothem precely becuse theyrefinite;ndtht relm of eqully exct nd definite objects of thought it possible, to which th xiom does not pply Let metrytoshowthewyinwhichdedekind, inh essy on number, nd Cntor in h theory of the reltionships of infinite ssemblges of objects, gree, both s to the exct definition of the concept of the equlity of two collections of

17 'I HE CONCEPT OF THE INFINITE 37 objects,ndstotheprece senseinwhich,incseofinfinite collections,prtmybeequltowhole. \Vhtdowemenbycllingnytwocollectionsofobjects numericlly equl to ech other? The nswer esily sug gestedbynillustrtion. Supposeustoknowthtthere compny of soldiers mrching long street, nd tht every soldierinthcompny hsgunupon h shoulder. We neednotinthcsecounthowmnysoldierstherereinthe compny in order to know, with precion, tht there re precely smnygunsintht compny s equipment sthere resoldiersinthecompny. Heretheequlityofthetwo collections defined in terms of wht the mthemticins cll reltion of one to one correspondence. By hypothes, the lwholdsthtto every soldierthere corresponds one, nd only one, musket, while to every musket in question there corresponds one, nd only one, soldier, nmely, the mn who crriesit. Toknowthlwtoknowthenumericlequlity of the two collections. Counting in th cse unnecessry. It mkes no difference whether the compny contins fifty or twohundredsoldiers. Innycse,ifthe supposed lwholds true,thereresmnygunsssoldiers. \Vith the conception of equlity thus illustrted, we re fi-ee from the necessity of lwys counting definble collections of objects before we know whether they re equl. We my thendefine equlity ingenerlthus:--ifandbretwo collections of objects, nd if generl lw known whereby werebletobesurethtto every individul objectina therecorresponds, ormybemdeto correspond, oneobject, nd oneobjectonly,inthecollectionb,ndiftheinversere ltionholdstrue,thenthetwocollectionsandb, byvirtue ofthonetoonecorrespondence, reequltoechother. NowDedekind, inhmentioned essy, firstdefines the conceptionofequlityinthese terms,ndthen gives toh definitionofninfinitecollection moreexctformthnwe hve yet used, by combining th conception with one other eqully simple nd exct notion. Th second notion tht of

18 38 THE HIBBERT JOURNAL VVholendPrt. The prece definitionofthereltionof wholendprt,sppliedtothecseofcollectionsofobjects, sfollows:letthere betwocollections,andb. Letit be known, either directly through definition, or otherwe, thteveryobjectwhich belongs tothecollectionb, belongs to thecollectiona,whileitlsoknownthtthere reobjects ofthecollectiona,whichdonotbelongtothecollectionb. ThenthecollectionBtobe clled prtofthewhole collection A. Preming these two dtinct conceptions, tht of equlity ndthtofthereltionofwholend prt,then Dedekind proceeds to h definition of n infinite collection s follows: Acollectioninfiniteifitcnbeputinonetoonecorrespond ence,orcnthusbe foundequlto,one of itsown prts. Th definition Dedekind introduces, in h essy upon the number concept, in dvnce of ny definition of the whole numbers themselves. He thus defines the infinity of collection while usingonlythetwoconceptsof theonetoone co'rre.spondence, nd of thewholend prtreltion. Hethuslogicllyexpresses h conception of the infinite quite in dvnce of stting ny definite conception of wht finite collection ; nd, in the order of h definitions, tells us wht the infinite, before he showsushowtocountthree,orten,ornyotherfinitenumber. But,snobjectormyheresy,meredefinitionsdonot of themselves ensure the possibility of their objects. Cn Dedekind show us, prt from mere empiricl illustrtions of the plusibility of h ide,- cn he show us, sy,tht collection defined s infinite in h sense possible col lection? Is not the very notion contrdictory one? How I cnthewholebeequltotheprt? nswer, Dedekind esily shows tht h conception of I the infinite cn be pplied without ny self-contrdiction. Or,shesys,hecnshowthtthererepossible systemsof objects,infiniteinhsenseoftheterm. Henmestonce such system. The relm, he sys, of the totlity of mypossiblethoughts,inhexct sense,ninfiniterelm.

19 THE CONCEPT OF THE INFINITE 89 For, s Dedekind continues, to ny thought of mine,-let ussytothethoughtss,for exmple, tomythoughtofmy c0zmtry, theremybemdeto correspond, intherelmofmy possible thoughts, nother thought which we shll cll nd which we my suppose to be the thought whose expression wouldbe: The thought s s, (viz., the thought of my cozm/try) oneofmythoughts. Iftheworldofmypossiblethoughts contins the possible thought certinly lso contins the s, it possiblethoughts. Nowletuscllllthoughtsoftheform.s~, reflective thoughts. Thoughts of th reflective type re thoughts tht const in thinking, concerning some other possiblethought,tht th oneofmythoughts. Now,to every possible thought of mine, without exception, there cn bemdeto correspond, intherelmofmypossible thoughts, onendonlyonedtinctthoughtoftheform nd vice vers. Hence, the whole collection of my possible thoughts, nd the collectionofthe possible thoughts ofthetype s, s, i.e.ofthe reflective thoughts, re precely equl, just s the two collec tionsofthemuskets ndofthesoldiers reequl. Forthe two collections of thoughts correspond to ech other, in one to one fshion, precely s the guns correspond to the soldiers. Yet the collection perfectly definite prt, nd not the s wholeoftherelmofmypossible thoughts. For there re thoughts, such s the simple thought of my country, which re notreflective. Inthrelmofmypossiblethoughts,prt my,therefore, beequltothewhole,notvguely,butin perfectly definble fshion. Hence, by the definition, th relm or collection of the totlity of my possible thoughts infinite. Yetsurelythe conceptionoftherelmofllmy possible thoughts not contrdictory conception. VII. Thus,then,thelogicl bsforthenewconceptofthe Infinite,initsoutlines, complete. Onecndefineinfinite collections without mking use, in the definition, of their

20 40 THE HIBBERT JOURNAL merely negtive chrcter of being without end. One cn definethembytellingwhttheyre,rtherthn whtthey renot. Onecnformbsfordtinguhingsuchcollec tions,indefinitefshion,bothfromonenother,ndfromll finite collections. One cn, consequently, nme criterion upon which to bse rguments regrding the question whether infinite collections ext in the rel world. For the question s to the rel extence of infinite collections becomes identicl with theproblem whether therelworldcontinsfcts,or.s-ystevrw Q] fcts,which possess certinsort of self-representtive structure. Or,inotherwords,theproblemoftherelityoftheInfinite becomes identicl with the problem whether the universe, or nyportionoftheuniverse,hsthe smeformortypewhich we re obliged to ttribute to n idelly completed Self. Whtever considertions mke for n ideltic interpret tion of relity, thus become considertions which lso tend to prove tht the universe n infinitely complex relity, or tht certin infinite system of fcts rel. For Ide.lm, in defining the Being of things s necessrily involving their ezteiwefor some formofknowledge, committedtothe thesthtwhtever,ipsofctoknown (e.g. tothe Absolute). Buttheknowledgeofnyfct,ifthknowledgeextstll, itselffct. Hencetheessence ofidelmliesinitsthes tht to every fct corresponds the krwwkdge of thtfet, while every ctofknowledgeitselfbelongs totheworldoffcts. Since, however, the fct-world, even for Idelm, contins mnyspects(such sthe spects clled feeling,will,worth, nd the like), which re not identicl with knowledge, lthough, fornidelt,theyllextsknown spects oftheworld, it follows tht, for n idelt, the fcts which constitute the extence of knowledge re themselves but prt, nd re not thewholeoftheworldoffcts. Yet, byhypothes, th prt, since it contins cts of knowledge corresponding to every relfct, dequte tothewhole,or,indedekind ssense, equl to the whole. Hence the idelt s system of fcts must, by Dedekind s definition, be infinite. Or in brief for the

21 THE CONCEPT OF THE INFINITE 4-1 idelt, the rel world self-representtive system, nd therefore infinite. But I hve myself lso endevoured to show, in my Supplementry Essy h-edy cited, tht similr consequence holds for ny metphysicl system, even if such system notideltic. For, sihve there ttemptedto explin t length, every metphysicl interprettion of the universe, whtever its chrcter, must imply tht the rel world self-representtive, ndconsequentlyninfinite system. In consequence Iconceive thtdedekind s definitionofthe Infiniteleds ustotheimportnt resultofbeingbleforthe first time to show explicitly tht the rel world, whtever else it,ninfinitelycomplexsystemoffcts. The ncient objections to supposing nything rel to be infinite in its complexity of structure, the time-honoured rgu ments ginst sserting tht the infinite rel, hve ll rested, in the end, upon the supposed indeterlnzinteness of the concept ofninfinitecollection,oroftheinfinitein generl. Butthe exct definition of Dedekind enbles us to conceive the Infinite, in ny one of its specil instnces, s something perfectly pre ce nd determinte. For instnce, let us suppose the collection ofllthewholenumberstoextsfctinthe world. Th collection hs positive properties, which, s Dedekindhsshown,follow necessrily fromhdefinitionof the infinity of the collection. Now th collection contins prt, precely equl in complexity to the whole, nmely, the collection before mentioned, of ll the powers of 2. Now,lthough thprtofthewhole collectionofthe wholenumbersninfinite prt, whoseinfinitycnlsobe defined in Dedekind s positive terms, yet it nevertheless perfectlydeterminte prt. For,ifweskwhtwholenumbers re Id? over,when,fromtheinfinitecollectionofllofthem, tken together, we remove or subtrct the entire infinite collec tion,orprt,clled thepowersof2,thenswer perfectly definite. For the whole numbers tht re not powers of two, themselves form precely definble collection. VVe cn even go much further. From the infinite collection of the whole

22 42 THE HIBBERT JOURNAL numbers we cn suppose subtrcted or removed, in succession, n infinite series of collections of whole numbers, ec/1 of which collectionsinfinite;ndyet,ifthe process exctlydefined, wecntellprecelywhtwillbeleftover fter llthinfinite series of subtrctions crried out. For, to exemplify th fct:- theprimenumbers,2,3,5,7,etc.,formofthemselves demonstrbly endless series of whole numbers. For there nolstprimenumber. Nowlet us suppose tht from the collection consting of ll the whole numbers, we first remove or subtrct the infinite collection of ll the prime numbers. Suppose tht we next remove the infinite collection of ll the squresoflltheprimenumbers. Thenletusremovethe infinite collection of ll the cubes of the prime numbers; then llthe fourth powersofllthe prime numbers. Let us continue th process without end, ech timeremoving n infinity of whole numbers, but continuing to infinity the process of removing higher nd higher powers of ech prime number. Will the finl result of th entire infinite series of successive subtrctions of infinite prts from the originlly infinite whole be in the lest indeterminte? On the con trry,weknow toncewhtwholenumberswillsurvivethe process. For the numbers tht will remin over fter the completion of the infinite series of removls will be those whole numberswhichreeithertheproductsofdifferent primes, or elsetheproductsof powers ofdifferent primes. Thus prece my be the results of reckoning with infinite collections, if only we use the right, which Dedekind s view of the positive infinite gives us, toregrdevery suchcollection, s soon sit precely defined, s n ctully possible nd given totlity, with prece reltions to other totlities, finite nd infinite. Norresuchelementryinstncesofthe possible exctness of our conceptions of infinite processes by ny mens the principl exmples of the essentil determinteness of the infinite. Cntor, whose reserches hve wrought such revolution in our knowledge of infinite collections, hs been bletoshowtht, despite thewonderful plsticitywhichthe

23 THE CONCEPT OF THE INFINITE 43 foregoing concept of the equlity of two infinite collections obviouslypossesses,the concept, sdefinedbove,nevertheless hsnexctly limited rnge ofppliction. Fortherere definble collections, infinite in the foregoing sense, which re demonstrbly not equl to one nother. Tht, there re cses where we cn show tht, of two given infinite collections, onesoexceedsincomplexitytheotherthtonetoonecorre spondence cnnot be estblhed between them. In such cse,oneofthetwocollectionsmyindeed beprtofthe other,butwillthenbe,inth cse, prtwhichlthough infinite, not equl to the whole. Our previous definition of theinfinite,infct,whileit dependeduponpointing outtht, in infinite collections, the prt my equl the whole, did not ssertthtninfinitecollectionmustbeequltoeveryoneof itsown prts, butsserted onlytht ninfinite collection equltosomeofits p.rts. Incse,however, ninfinitecollec tioneontinscertininfinite prts towhichitnot equl, but whichitexceedsinsuchfshionthtonetoonecorrespond encebetweenthewholendsuchprtimpossible,thenthe greter infinite collection sidbycntor tobehigherin Mdcbtigkeit orindignitythnsuchlesserprt. Thecon cept of the Dignities of the infinite, which Cntor thus intro duced, dependsuponproving thtprecelysuchgrdtionsof infinity re to be found in cse of certin definble collections ofpossibleobjects. Asfct,itcnbeshownthtthecollec tion consting ofll the possible frctions, rtionl nd irrtionl,between 0nd 1,ofhigherdignitythnthe collectionofllthewhole numbers. Ontheotherhnd, collection consting merely of ll the rtionl frctions, of thesmedignitysthecollectionofllthewholenumbers. Theproofofboththese results cnbegivenin perfectly elementry form, which indeed too lengthy to be stted here, but which cn be mde comprehensible to lmost ny creful student who retins the slightest knowledge of elementry rith metic nd lgebr. Yet the first dcovery of these Dignities or grdtions of the infinite, s mde by Cntor, constitutes one

24 44 THE HIBBERT JOURNAL of the most ingenious dvnces of recent exct thinking. Cntor himselfhsshown (nd independentlymrchrless. Peircehsdonethesme),thttherenendless seriesof these possible Dignities of the infinite. Theresultofsuchreserches,however,toshowinnew wy how determinte n object n infinite collection, once exctlydefined,proves tobe. Forninfinitecollectionof lower Dignity, lthough unquestionbly boundless in its own grde, remins in perfectly definite sense incomprbly smll when considered with reference to n infinite collection of higher Dignity. Infinity, nd prece limittion, re thus shown to be perfectlycomptible chrcters. For no process of numericl multipliction, even pursued d injinitum, cn directlycrryonefromninfinityofnylowerdignitytoone of higherdignity. The trnsition from one grde to highergrdecnbemde onlybymensofcertinprecely definble opertions which re not expressible in merely quntittive terms. The lower nd the higher Dignities re thus seprted by logiclly shrp boundries of which erlier specultions upon the infinite gve notthe slightest hint. Buttheseboundries, exting intherelmofwhtwsonce the voidndformlessinfinite, showusththenceforthno one who identifies the infinite with the indeterminte wre whereofhespeks;ndthtnoonewhoconceivestheinfinite merelyintermsofthe negtive endless process cnbe regrded shvinggrsped thetruentureoftheproblemof the infinite. Menwhile, to look in yet nother direction, the concept tht,inninfinitesystem,theprtcn,ininfinitiesofthesme Dignity, beequltothewhole,throws whollynewlight upon the possible reltions of equlity which, in perfected stte, might ext between wht we now cll n Individul, or CretedSelf,nd God, stheabsoluteself. Perhps being,whoinone sense ppered infinitely lessthngod,or whotlleventswsbut oneofninfinitenumberofprts within the divine whole, might nevertheless justly count it not

25 THE CONCEPT OF THE INFINITE 45 robberytobeequltogod,ifonlythprtilbeing,byvirtue of n immortl life, or of perfected process of self-ttinment, received, in the universe, somewhere n infinite expression. The possible vlue of such conception for theology seems to mke it deserving of somewht creful ttention. I conclude, then, by urging the conceptofthe New Infinite uponthettentionofstudentsof deepertheologicl problems. I believe it to be demonstrble tht the infinite, in generl, neithersomethingindeterminte,norsomethingde finble only in negtive terms, nor something incomprehensible. Ibelieveittobedemonstrblethtthe rel universe n exctly determinte but ctully infinite system, whose struc ture tht reveled to us in Self-Consciousness. And I believe tht the newer reserches regrding the infinite hve setthtruthinnewndwelcomelight. JOSIAH ROYCE. HARVARD Umvt-znsrrv.