,,on ,,! ,,!ory ,,( ,,! ,,inf9ite ,,9 ,,mod]n ,,?\ght ,,on ,,! ,,!ory ,,( ,,! ,,inf9ite ,,9 ,,mod]n ,,?\ght ,,two ,,introductory ,,/udies ,,by ,e4 ,f4 ,,j\rda9 ,,doctor ,,( ,,! ,,univ]sity ,,( ,,p>is ,,vice-,,pr9cipal1 ,,/4 ,,hugh',s ,,hall1 ,,ox=d ,,longmans1 ,,gre5 ,,& ,,co4 #ci ,,pat]no/] ,,r[1 ,,london ,,new ,,york1 ,,bombay1 ,,& ,,calcutta #aiaa ,all rights res]v$ ,( ! two pap]s "h reproduc$1 ! fir/ 0 giv5 9 #aije to a meet+ ( ! wom5 sci5ce /ud5ts 9 ,ox=d2 ! second1 9 #aijh1 to ! ,philosophical ,society ( ?is ,college4 ,!y are pr9t$ by reque/1 ) ! au?or's apologies = _! incomplet5ess4 ,! lecture =m has be5 reta9$4 ,i am indebt$ to my bro!r1 ,mr4 ,p4 ,j\rda91 = help 9 prep>+ ! fir/ lecture1 & = 8 revision ( ! text4 ,e4 ,f4 ,,j\rda94 ,,/4 ,,hugh',s ,,hall1 ,,ox=d1 ,janu>y1 #aiaa4 ,,cont5ts ,i ,,page ,,! ,,problem ,,( ,,! ,,f9ite ,,& ,,! ,,inf9ite #a ,,ii ,,pragmatism ,,& ,a ,,!ory ,,( ,,k #ca ,i ,,! ,,problem ,,( ,,! ,,f9ite ,,& ,,! ,,inf9ite ,! influ5ce ( ma!matics on philosophy & vice v]s c be inf]r$ f ! hi/orical progress ( bo? /udies1 ?\gh x has n be5 possible till ab\t )9 ! la/ fifte5 ye>s to give a logical explanation = ! relations betwe5 !m4 ,z long z x 0 believ$1 accord+ to ! ,kantian view1 t ! sci5ce ( ma!matics 0 bas$ on intuitions ( "t & space1 ! alliance betwe5 philosophy & ma!matics c\ld n be prov$ to be clos] ?an t betwe5 philosophy & exp]im5tal sci5ce1 al?\gh ! hi/orical fact rema9$ t philosophy & ma!matics ex]cis$ a mutual /imulus1 & develop$ at ! same p]iods ( hi/ory4 ,b ma!matics1 z n[ def9$1 is indep5d5t ( intuitions ( space & "t1 & also ( axioms & hypo!ses4#a ,ma!matics1 z n[ und]/ood1 is bas$1 l =mal logic1 on ! pr]equisites ( ?\ght1 n on ! notions ( space & "t4 ,"h !re is no def9ition ( numb] or space1 b ! conception ( numb] & space1#b :i* is m complicat$1 c be d]iv$ f !m4 ,all o!r complicat$ m9d processes c1 9 ! same way1 be r$uc$ to ! simple elem5ts ( ! pr]equisites ( ?\ght4 ,su* a sci5ce might exi/ out ( conditions ( "t & space z we "k !m4 ,x is a sci5ce ( relations r ?an ( m]e numb]4 ,f\nd$1 !n1 on ! laws ( symbolic logic1 x is a valuable aid & illu/ration to philosophy2 philosophy1 on ! o!r h&1 c imag9e l9es = ! ex]cise ( ! con/ructive p[] involv$ 9 ma!matics4 ,x is ! object ( ?is pap] to %[ t ! close ?\gh app>5tly accid5tal union ( philosophy & ma!matics ?r\gh\t ! hi/ory ( ?\ght c n[ be expla9$1 & t ! problems ) :i* pure ma!matics is n[ conc]n$ are ^? :i* lie at ! core ( philosophic ?\ght & speculation4 7,symbolic logic has develop$ to meet ! new dem&s made ^u x4 ,x does n n[ r$uce itself to ! syllogism1 z ,ari/otle ?\ght x did2 ! pr]equisites ( ?\ght are %[n to be manifold in/ead ( s+le4#c7 ,! use ( ! ^w philosophic 9 ?is connection sugge/s a necessity = fur!r def9ition4 ,philosophy is held to include at lea/ two great bran*es--,metaphysics & ,e?ics4 ,! influ5ce ( ma!matics is mo/ evid5t on ! metaphysical side ( philosophy2 9 fact1 ! gr\p+ ( ma!matics & metaphysics z alli$ sci5ces t5ds to br+ out ! ess5tial di/9ction betwe5 metaphysics & e?ics1 &--?\gh n by any means to imply a break 9 _! real relation--to %[ :]e ?is has be5 misund]/ood4 ,no philosophy has be5 equally /rong on bo? sides2 !y repres5t diff]5t =ms ( activity ( ! human m9d2 b x is /ill true1 & f ! conditions always mu/ be1 t an e?ical sy/em gr[s out ( metaphysics z practice foll[s precept & conduct implies belief4 ,! new def9ition ( ma!matics does n t\* ^! consequ5ces2 x m]ely m>ks ! limits )9 :i* philosophy on ! metaphysical side c submit to1 or re/ ^u1 ! conclusions ( ma!matics4 ,z to ! hi/orical relation betwe5 metaphysics & ma!matics1 ! subject is s va/ t we %all only attempt a v rapid g5]alisation ( its results on ! gr[? ( ! conception ( ! ,f9ite & ! ,inf9ite4 7,( c\rse1 !re are _m o!r sides ( ! relation :i* might be /udi$47 ,! g5]al result ( ! inquiry has be51 z f> z we c judge1 t metaphysics has ex]cis$ an inspir+ =ce on ma!matics1 & ma!matics has def9$ & /r5g!n$ ! conceptions ( metaphysics at e critical /age 9 ! hi/ory ( philosophy4 ,b :]e metaphysics has be5 treat$ z ! pro( ( sci5ce1 :]e x has be5 laid d[n z ! f\ndation = exact k1 ! results h n correspond$ ) ! tru? ( exp]i5ce1 & ! quality ( ?\ght has become deg5]ate4 ,progress dep5ds on ! "r p]ception ( ! relations betwe5 ! sci5ces & p>ts ( philosophy4 ,su* progress is especially evid5t 9 ! e>ly ,greek & 9 ! mod]n p]iods1 :ile ! l>ge p]iod f ! ,*ri/ian era to ! ,r5aissance gives examples ( ! un=tunate rev]sal ( ! p>ts ( metaphysics & sci5ce & consequ5t confusion ( ?\ght4#d ,! problems ( ! metaphysician are no d\bt 9 a s5se always ! same2 b ?is is equally true ( ! problems ( any o!r sci5ce4 ,! me?ods by :i* ! problems are attack$ & ! adequacy ( ! solutions !y receive v>y1 f age to age1 9 close correspond5ce ) ! g5]al developm5t ( sci5ce4 ,e great metaphysical conception has ex]cis$ its influ5ce on ! g5]al hi/ory ( sci5ce1 & 9 return e important movem5t 9 sci5ce has affect$ ! developm5t ( metaphysics4 ,! metaphysician c\ld n if he w\ld1 & w\ld n if he c\ld1 escape ! duty ( e/imat+ ! be>+ ( ! great sci5tific !ories ( 8 "t ^u our ultimate conceptions ( ! nature ( ! _w z a :ole4 ,e fundam5tal advance 9 sci5ce ?us calls = a re/atem5t & reconsid]ation ( ! old metaphysical problems 9 ! light ( ! new discov]y4 99 99 99 99 99 ,dur+ ! ,greek p]iod ma!matics 0 ! only bran* ( sci5ce :i* 0 at all develop$1 & its developm5t co9cid$ ) ! age ( ! philosoph]s4 ,?us :5 ,plato spoke ( sci5ce he always meant ma!matics4 ,& ev5 lat]1 :5 ! physical sci5ces _h begun to develop1 ,ari/otle put ma!matical ideas into close connection ) metaphysical ones :5 he /at$ t !y occupi$ ! middle t]m betwe5 ! ideal & ! s5sible4 ,bo? ,plato & ,ari/otle ref]r$ to & dep5d$ ^u ma!matical pro(s & illu/rations ( philosophical que/ions4 ,dur+ ?is ,greek p]iod ! conception ( ,inf9ity took %ape4 ,! pre-,platonic notion1 reproduc$ aga9 lat] 9 ! decl9e ( ,socratic !ory by ! ,/oics1 0 t ! ,inf9ite 0 ! aggregate ( ! ,f9ite2 ! ,platonic & ,ari/otelian !ory1 t1 namely1 ( ! mo/ vigor\s mom5t ( ,greek ?\ght1 0 t ! ,inf9ite 0 m ?an ! aggregate ( ! ,f9ite2 t x _h a self-det]m9$ exi/5ce f :i* ! ,f9ite _h be5 d]iv$4 ,exi/5ce1 z kn[n to man1 0 treat$ z a compromise betwe5 ! ,f9ite & ! ,inf9ite4 ,neo-,platonism altoge!r sep>at$ ! ,inf9ite f ! ,f9ite4 ,9 ! ,alex&r9e metaphysics1 :i* repres5t$ a decad5t /age ( philosophy & its deviation f ! sci5ces1 ! conception ( ! ,inf9ite became less cle> & logical2 x div]g$ f ! view :i* _h be5 affect$ by ma!matical ?\ght1 & t5d$ to assimilate to itself ! ideas ( p]fection & univ]sality1 :i*1 philosophically speak+1 are conceptions di/9ct f t ( ,inf9ity--univ]sality ref]r+ to a common pr9ciple ( unity1 & p]fection involv+ ! moral ideal4 ,real progress 0 def]r$ by ! too rapid coh]5ce ( ideas only p>tially analys$ & und]/ood4 ,?9k]s pass$ quickly f ! exclusive contemplation ( subject to t ( object & back aga91#e ea* new p]iod negativ+ all previ\s exp]i5ce1 till ! result 0 ! exclusion ( an imp]fectly analys$ ,relative & ,f9ite f an insuffici5tly appreh5d$ ,absolute & ,inf9ite4 ,aft] ! ,*ri/ian era ,greek philosophy drift$ (f into s*ola/icism & lo/ t\* ( reality1 ! gramm> ( ,ari/otelian logic replac+ ! vital connection ( ideas4 ,/4 ,anselm1 x is true1 attempt$ to f9d a rational pro( ( ! exi/5ce ( ,god1 & id5tifi$ ,him ) ! ,inf9ite ( ,greek ?\ght2 b ,/4 ,?omas ,aqu9as l$ away ! argum5t to a discussion z to h[ f> =m & matt]1 sep>ately consid]$1 %>$ 9 ! quality ( ,inf9ity4 7,he ?\ght =m did1 b n matt]47 ,an ov]p[]+ s5se ( my/]y1 jo9$ to a premature desire = def9ition )\t sci5tific analysis1 sapp$ ! vig\r ( m$ival ?\ght4 ,?r\gh\t ! middle ages1 !n1 we see ! conditions ( ! ,greek p]iod rev]s$3 philosophy dur+ ! second p]iod is n1 z 9 ! fir/1 engag$ 9 giv+ a /imulus to ! ef=ts ( pure reason2 r ! intuitions ( philosophy are treat$ z axiomatic1 & a false sup]/ructure ( k1 ali5 to exp]i5ce & reality1 is erect$ ^u ^! f\ndations4 ,philosophy1 9 fact1 is us$ z a g5]al basis = sci5ce4 ,! p>ts ( philosophy & ma!matics1 correctly ?\gh imp]fectly se5 by ! ,greeks1 are 9 ! second p]iod ex*ang$1 & ! result is confusion ( idea4 ,! notion ( ! ,inf9ite1 z 9 ! ,alex&r9e metaphysics1 is held to include p]fection & univ]sality1 & does n exi/ z a conception ap>t f ^!4 ,aft] ! ,r5aissance1 ! s*ola/ic philosophy fall+ into disuse1 ! attempt to f9d an explanation ( ! ,cosmos1 a syn!sis ( ! univ]se1 0 ab&on$1 & replac$ by ! ,c>tesian idea--! inf]5ce ( exi/5ce f ?\ght1 & ! limitation ( ! sph]e ( inquiry to t :i* c\ld be kn[n by ! ego4 ,new sci5tific & ma!matical discov]ies kept pace ) ?is new analysis & developm5t ( ?\ght1#f & ! sur] gr\nd 9 philosophy 0 def9itely alli$ ) ! "w ( ! ma!matical m9d4 ,! philosophical !sis develop$ f 8,! ,inf9ite is ! negation ( ! ,f9ite10 to 8,! ,inf9ite presupposes ! ,f9ite & does n exclude x40 ,! problem ( ! ,f9ite & ! ,inf9ite became ! great idea ( ! age1 & !re 0 a rev]sion to ! ,greek notion ( exi/5ce z a compromise betwe5 ! two1 & almo/ ! h9t ( a com+ explanation ( !m4 ,9 ! decl9e ( ,c>tesian philosophy1 :5 x drift$ (f into ,pan!ism1 !re 0 only a vague conception ( ! ,inf9ite1 & we trace a t5d5cy to id5tify ! notion ( ,inf9ity ) t ( ! ,cosmos4 ,9 m$ival ?\ght ! idea ( ! ,inf9ite _h become confus$ ) t ( ! ,p]fect & ,univ]sal2 9 ! mod]n p]iod ! ef=t to give a concrete expression to ! notions ( ,inf9ity1 ,p]fection1 & ,univ]sality div]t$ ! ideas f _! relation to ! ,creator & appli$ !m to ! ,creation4 ,kant1 :o gave a new impulse to "s p>ts ( ! ,c>tesian idea1 neglect$ bo? ma!matical pro(s & ! se>* = a metaphysical ,absolute4 ,9 avoid+ ! subject he help$ to p]petuate ! vague descriptions ( ! ,f9ite & ! ,inf9ite1 uncorrect$ by ma!matical ?\ght1 :i* _h be5 ! curr5cy ( ! philosophy ( 8 age1 & :i* corrupt$ ! philosophy ( ! succe$+ c5tury4 ,! n9ete5? c5tury produc$ no?+ m ?an guesses at tru?1 :i* 71 p]haps1 n v f> wrong1 & :i* ! pres5t c5tury is engag$ 9 correct+ & sub/antiat+4 ,! same vagu5ess afflict$ bo? ma!matics 7!ory ( functions7 & philosophy4 ,fi*te1 ,s*ell+1 & ,hegel1 p>ticul>ly ,hegel1 id5tify ! metaphysical ,absolute ) reality1 inf9ity1 & ! univ]sal4 ,! ideas ( cont9uity & inf9ity are n sep>at$ by !m f ^? ( p]fection & univ]sality1 nor f "o ano!r1 & _! nature is n und]/ood4 ,leav+ aside ! ,fr5* neo-critical ,s*ool 7,r5\vi]7 & ! ,engli% ,s*ool 7,sp5c]7--! fir/ ( :om d5y ! ,inf9ite1 ?us act+ 9 opposition to ma!matical reason+1 :ile ! second p]petuate ,kant's error ( consid]+ ! ,inf9ite1 ?\gh ?9kable1 z unkn[able 7,dr4 ,caird has po9t$ out t ?is position is illogical7--we arrive at a mom5t 9 hi/ory :i* is m fruitful 9 result on ! ma!matical side1 & w1 no d\bt1 h an effect on metaphysics4 ,=1 ow+ to rec5t discov]ies 9 ,g]many & ,engl&1 ma!matics is n[ 9 a position to give great] support ?an be=e to ! intuitions ( philosophy4 ,hi!rto1 philosoph]s h be5 reluctant to all[ full value to ! ma!matical conceptions ( ,inf9ity1 & ) "s ju/ice1 z ! notion _h n be5 suffici5tly analys$4 ,philosoph]s1 :o nev] attempt$ ! analysis1 h be5 incl9$ to accept c]ta9 contradictions 9 _! conception z inh]5t 9 ! nature ( ,inf9ity4 ,)9 ! la/ tw5ty-five ye>s ,cantor & ,d$ek9d h cle>$ up ! notion ( cont9uity1 & ,russell has giv5 great] precision to ! idea1 & has appli$ ?is reason+ to philosophy4 ,pres5t-day metaphysicians seem to be divid$ into two gr\ps2 on ! "o side1 ^? :o consid] 9 philosophy ! value ( a !ory ( be+1 &1 on ! o!r1 ^? :o *iefly consid] ! value ( a !ory ( k1 i4e4 ! ,epi/emologi/s4 ,! fir/ gr\p1 devot+ !mselves to psy*ology1 evolution1 & hi/ory1 h no necess>y belief 9 ! ,inf9ite4 ,! ,epi/emologi/s1 ^: "w is f\nd$ on ,kant1 discuss ! !ory ( k & enum]ate ! conditions ( k4 ,_! argum5t may n t\*1 b does n exclude1 ! notion ( ! ,inf9ite4 ,! position ( ! ,epi/emologi/ has be5 made inf9itely m secure by rec5t ma!matical "w4 ,t ( ! psy*ologi/ rema9s almo/ unt\*$4 ,x is necess>y n[ to exam9e m closely ! ma!matical results to :i* ref]5ce has be5 made4 ,9 g5]al t]ms x may be said t ma!matics has1 z a /udy1 l$ imm$iately f ! nature ( ! subject to ! p]ception ( ! ,inf9ite1 & to a k ( ! connection betwe5 ! ,inf9ite & ! ,f9ite4 ,! simple/ =m 9 :i* ! idea c be put is /at$ by ,/4 ,augu/9e1 :o said t numb]s consid]$ individually 7 f9ite1 b consid]$ z an aggregate 7 inf9ite4#g ,be=e ,/4 ,augu/9e1 & aft] him d[n ! long /ream ( philosophic ?\ght1 ! !ologian & ! philosoph] h turn$ to ma!matics = illu/rations ( ! inf9itely great & inf9itely little1 z develop$ f ! concrete processes ( ari?metic & geometry4 ,! recurr+ decimal 9 ari?metic1 ! prop]ties ( ! circle & ellipse 9 geometry1 ( ! cone 9 conic sections1 & ( ! surd 9 algebra1 all t\* ! problem ( numb] & space on ! side ( ,inf9ity4 ,9 high] ma!matics x is possible to />t f ! idea ( ! ,f9ite & rea* ! conception ( ! ,inf9ite2 or to rev]se ! process1 & f ! ,inf9ite to d$uce ! ,f9ite4 ,?us 9 ! famili> puzzle ( ! subdivision ( ! p>ts ( a /raight l9e by halv+ ! rema9d]1 !re w be a cr[d+ & a coalesc+ ( ! po9ts ( division t[>ds "o end ( ! l9e1 ! po9ts ( division gett+ inf9itely ne>]1 b ! /eps w nev] meet4 ,"h 9 ! c5tre ( a /raight l9e--a limit$ /raight l9e--we are confront$ ) ! problem ( ,inf9ity4 ,aga91 f a s]ies ( f9ite numb]s we c ga9 ! notion ( an inf9ite s]ies4 ,take two s]ies :i* h a correspond5ce ) "o ano!r4 ,if = e elem5t ( ! "o we c *oose an elem5t ( ! o!r1 & ( ! o!r !re is an elem5t = ! "o1 :5 at any po9t we cut (f its progress to inf9ity1 ?is happ5s3-- ,"o s]ies1 if summ$ up1 w give a l>g] num]ical result ?an ! o!r1 & !re=e c be said to be great] ?an ! second4 ,let u call ! fir/ s]ies ,a1 & ! second ,b4 ,let u n[ imag9e ! two s]ies1 ?\gh />t+ at a def9ite po9t1 are nev] cut at ! fur!r end4 ,!n to all inf9ity s]ies ,b is )\t c]ta9 numb]s :i* s]ies ,a possesses1 & z an inf9ite s]ies is small] ?an s]ies ,a4 ,b1 on ! o!r h&1 :5 nei!r s]ies is cut1 s]ies ,b reta9s its correspond5ce ) s]ies ,a4 ,?us we atta9 a def9ition ( an inf9ite s]ies4 ,x is su* t ! "p1 :ile be+ less ?an ! :ole1 has yet a complete correspond5ce ) ! :ole4 ,! :ole is great] ?an ! "p1 b take away ! "p f ! :ole & t :i* rema9s corresponds to x 9 inf9ity1 because ! te/ ( summ+ ! s]ies 7:i* w\ld give a contr>y result7 involves limitation1 & ?us _c be appli$4 ,subtraction c take place 9 ,inf9ity )\t loss4 ,by rev]s+ ?is process1 & by />t+ f ! !ory ( ! ,inf9ite1 we may ga9 "s idea ( ! discov]y ( ! ,f9ite4 ,s ,d$ek9d & ,russell def9e f9ite numb]s n only 9 ! usual way z ^? :i* c be rea*$ by ma!matical induction1 />t+ f #j & increas+ by #a at ea* /ep1 b also z ^? ( classes :i* are n simil> to ! p>ts ( !mselves obta9$ by tak+ away s+le t]ms4 ,t is ! rev]sal ( ! process appli$ j n[4 ,d$ek9d also has d$uc$ ! ,f9ite f ! ,inf9ite by a novel process4 ,he pr$icates a _w ( ?\ght :i* we ea* & all possess1 fill$ ) ?\ghts & ?+s1 to ea* ?+ correspond+ a ?\ght4 ,!re are ?us two 8trans-f9ite0 s]ies 9 ! m9ds ( ea* & all ( u2 we _c say :5 ! s]ies ( ?\ghts & ?+s w end2 b !y h numb]1 ?\gh x is inf9ite numb]4 7,numb] exi/s :]ev] !re is a correspond5ce1 "o to "o1 betwe5 two aggregates47 ,b 9 ?is ,g$ank5welt1 says ,d$ek9d1 !re is "o ?+ to :i* !re is no correspond+ ?\ght3 t is ! ego4 ,ea* man is "p ( 8 own _w ( ?\ght1 b !re is no ?\ght ( himself 9 8 m9d correspond+ exactly to himself1 z a ?\ght 9 8 m9d corresponds to ano!r object4#h ,two important results foll[ f ,d$ek9d's !ory3 fir/1 ! exi/5ce ( a f9ite numb] "o1 ! numb] ( ! ego1 z d$uc$ f ! ,g$ank5welt ( two inf9ite sy/ems2 second1 by putt+ toge!r all ! ,g$ank5welts !re are or may be1 we get ! notion ( s]ies ( s]ies1 :i* seems to transc5d ,inf9ity1 & x gives u ! conditions :i* are possibly ga!r$ up 9 ! ,absolute4 ,n[ ! argum5t f ! ,f9ite to ! ,inf9ite & ! conv]se process may bo? be employ$ 9 ma!matics 7or bo? may be neglect$1 z 9 ! elem5t>y me?ods ( calculation us$ 9 ari?metic74 ,a discussion has tak5 place 9 ! ,hibb]t ,j\rnal on ! relative value ( ! two me?ods4 ,keys]1 9 an article call$ ! ,axiom ( ,inf9ity1 argu$ t "o me?od1 t ( ,d$ek9d1 %\ld be exclusively develop$4 ,russell answ]$ him1 /at+ t x 0 n necess>y to hold exclusively to ei!r4 ,if ! ,f9ite & ! ,inf9ite c 9 turn be d$uc$ f "o ano!r1 nei!r conception c be truly call$ an axiom4 ,! real axiom is exi/5ce1 :i* includes bo?1 & :i* is def9$ by ma!maticians z t :i* is n self-contradictory4 ,n[ ! problem ( ,inf9ity includes also t ( cont9uity2 9 o!r words1 ! problem ( numb] includes t ( c>d9al & ord9al numb]4 ,x is "t to get to ! ma!matical def9ition ( numb]1 :i* we h f\nd z a conception c be atta*$ bo? to ! ,f9ite & to ! ,inf9ite4 ,:at is numb] 9 ma!matics8 ,take any collection ( ?+s--we call t an aggregate4 ,if an aggregate corresponds "o to "o ) ano!r aggregate1 !y are bo? said to h a numb]1 & ! same numb]4 ,subtract f ! idea ( an aggregate ! idea ( quality or k9d1 & ord] or arrangem5t1 :at is left is its c>d9al numb]4 ,if y subtract quality & n ord]1 ! result is an ord9al numb]4 ,?is reason+ applies bo? to f9ite & inf9ite aggregates2 9 fact1 ! ,inf9ite may be said to possess mo/ ( ! prop]ties :i* we atta* to ! ,f9ite4 ,two inf9ite aggregates1 = example1 c h an ord9al correspond5ce1 & inf9ite aggregates submit1 l f9ite ones1 to ari?metical processes4 ,! ma!matician analyses /ill m closely ! relation betwe5 ! ,f9ite & ! ,inf9ite1 z foll[s3-- ,he />ts f ! aggregate1 :i* he analyses into ! ,f9ite & ! ,inf9ite1 & ! latt] he analyses into ! ,transf9ite & ! ,absolute4 ,( ^! two elem5ts1 "o only has till j lately be5 ! subject ( ma!matical treatm5t--! "o call$ ! ,transf9ite4 ,x is ! transf9ite subdivision ( ! ,inf9ite to :i* ! idea ( numb] is applicable1 & :i* is1 !re=e1 9 a s5se insep>able f ! ,f9ite4 ,inf9ite numb]s or s]ies ought !n to be m correctly describ$ z ,transf9ite4 ,b ! processes ( ma!matics d n end "h2 !y rea* up to ! idea ( ! ,absolute ,inf9ite1 ! conception ( :i* has be5 atta9$ 9 rec5t ye>s by ma!matical "w4 ,! results ( ?is "w may n[ be briefly summ>is$4 ,i4 ,! ,absolute appe>s to h ! same relation to ! ,transf9ite z ! ,transf9ite to ! ,f9ite4 ,if ! ,f9ite deals ) numb]s1 & ! ,transf9ite ) s]ies ( numb]s1 ! ,absolute deals ) s]ies ( s]ies4 ,?us !re are at lea/ two examples ( ! ,inf9ite )9 our grasp :i* lead up to ! idea ( ! ,absolute4 ,"o is ! class ( all classes ( propositions2 ! o!r is ! s]ies ( all worlds ( ?\ght1 9 ,d$ek9d's s5se4 ,,ii4 ,! ,f9ite1 ,transf9ite1 & ,absolute c be fur!r def9$ 9 ?is way4 ,!re is no greate/ f9ite numb]1 b !re is a lea/ transf9ite numb]1 :i* has be5 call$ ,aleph #j1 & :i* c be prov$ to be great] ?an any possible f9ite numb]1 h[ev] l>ge1 because if !re 7 a la/ numb] x mu/ be small] ?an ! sum ( ! :ole s]ies4 ,!re are un5d+ s]ies ( ,alephs or inf9ite numb]s1 :i* are z di/9ct f "o ano!r 9 idea z #a is f #j1 & :i* c no m be d]iv$ f "o ano!r by a ma!matical process ?an #a c be d]iv$ f #j1 b c be rea*$ 9 ! same way by induction4 ,beyond ! ,transf9ite we _c discov] 9 ! ,absolute ! idea ( lea/ or ( greate/4 ,,iii4 ,! relation ( c>d9al & ord9al numb] also ?r[s "s light on ! ,f9ite1 ,transf9ite1 & ,absolute4 ,9 ! ,f9ite1 c>d9als & ord9als are p>allel to "o ano!r2 9 ! ,transf9ite !y /rik+ly div]ge2 9 ! ,absolute we _c trace any connection betwe5 c>d9als & ord9als1 i4e41 x is possible to h an ord9al s]ies to :i* !re c be no correspond+ c>d9al numb] or type4#i ,,iv4 ,if ari?metical processes are appli$ to ! ,f9ite1 ,transf9ite1 or ,absolute1 we get int]e/+ results4 ,we "k ! effect ( addition1 multiplication1 & rais+ to a p[]1 on ! ,f9ite4 ,! fir/ two processes h be5 appli$ to ! ,alephs2 ! la/ has be5 =mulat$1 b ! ma!matical results h n yet be5 br\ght to a satisfactory conclusion4 ,broadly speak+1 we may say t ! rais+ ( an ,aleph to a p[] may make x transc5d ! ,f9ite & ! ,transf9ite & melt into ! ,absolute4 ,?us all ma!matical processes :i* f9d _! goal 9 ! ,absolute w\ld f9d _! annihilation !re4 ,no f9ite ma!matical conception w\ld be applicable to x4 ,n[ ! conception ( ?is ,absolute ,inf9ite1 ( :i* ! aggregate ( all ord9al numb]s is p]haps a symbol1#aj has be5 subject$ to criticism4 ,"s ma!maticians#aa ?9k t x exi/s1 b has no numb]4 ,x is discov]$ by a logical process1 b defies analysis & ! application to x ( ! notion ( numb]4 ,all ma!matical conceptions f9d 9 x _! aim & conclusion4 ,! importance ( ?is !ory1 its practical importance1 lies 9 ! v mu* simpl] ma!matical =mul t c be produc$ n[ t ! logical process is %[n to ext5d f ! ,f9ite to ! ,absolute ,inf9ite 79 ! same way t ! lab\r ( summ+ a s]ies ari?metically by /atem5t & addition is %ort5$ by ! application ( algebraical pr9ciples :i* dep5d on l>g] k74 ,its philosophical importance is great3 ! ,absolute is "h1 z else:]e1 ! goal ( human ?\ght1 & is ! ma!matician's "n = ! highe/ p[] discov]able by human reason4 ,x w\ld be v int]e/+ to discuss ! probable attitude ( a ,pascal or a ,hegel to ^! ma!matical conceptions1 if !y _h be5 aw>e ( !m4 ,take ,pascal's puzzle ( ! ,f9ite & ! ,inf9ite4 ,he ?\ght t if ! ,f9ite c\ld be subtract$ f ! ,inf9ite1 ! ,inf9ite w\ld !reby lose "s ( its quality ( inf9ity4 ,h[ diff]5tly w\ld x h appe>$ to him _h he realis$ t an aggregate inf9ite c>d9al c h subtract$ f x ei!r f9ite or transf9ite t]ms3 if transf9ite t]ms1 _m diff]5t answ]s result1 giv+ diff]5t degrees ( transf9ity3 if only f9ite t]ms are tak5 away1 ! ,inf9ite rema9s 9 its entirety4 ,h[1 aga91 w\ld ,hegel h rejoic$ 9 a def9ition ( ?\ght & exi/5ce :i* w\ld bridge ov] ! logical gulf 9 8 sy/em6 ,hegel ass]t$ t ?\ght & exi/5ce 7 "o4 ,he is object$ to by _m philosoph]s1 :o ask :]e is ! t]tium quid :i* makes x possible to rea* f "o to ! o!r1 or pr$icate _! ess5tial unity8 ,b ! ma!matician def9es exi/5ce z some?+ :i* is n self-contradictory4 ,?\ght1 !n1 to him is a =m ( exi/5ce1 = ?\ght is n self-contradictory2 b ! two1 ?\ght & exi/5ce1 are n necess>ily cont]m9\s4#ab ,h5ce1 to say t non-contradiction is a fundam5tal condition ( true ?9k+ is z mu* z to say t x is a fundam5tal *>act]i/ic ( real exi/5ce1 & he id5tifies ?\ght ) reality4 ,dr4 ,caird rem>ks t ! secul> consci5ce conceives ( ! ,inf9ite z oppos$ to ! ,f9ite2 ! religi\s consci5ce treats ! ,inf9ite z real1 presuppos$ by ! illusory ,f9ite4 ,:]e does ! tru? lie8 ,ma!matics does n admit ! necessity ( adopt+ ei!r view at ! exp5se ( ! o!r4 ,metaphysics /&+ alone produces results t may be disprov$1 b _c be prov$4 ,ma!matics /&+ alone produces results t are susceptible ( pro(4 ,bo? are bas$ on logic1 & re/ on ! pr]equisites ( ?\ght4 ,toge!r !y are a field = ! be/ p[]s ( human reason3 metaphysics supplies insight1 intuition1 imag9ation2 ma!matics (f]s ! indubitable pro( & translates ! ideal into ! actual4 ,b ! elem5t 9 philosophical ?\ght :i*1 employ+ ! psy*ological me?od1 t5ds to ! discussion ( a !ory ( be+ r ?an t ( k1 & ?us to ! realisation ( an e?ical sy/em r ?an to metaphysical discov]y1 is av]se f accept+ ^! conclusions4 ,x rema9s1 !re=e1 = u to exam9e ! criticism (f]$ by ! psy*ological s*ool on :at !y call ! ma!matis+ ( philosophy2 & x w be f\nd t ! attack deals bo? ) ! gr\nd ( ! alliance & its results4 ,a typical expon5t ( ?is s*ool is ,moisant1 :o1 9 ! ,revue ,philosophique = ,janu>y #aije1 attack$ :at he consid]$ to be ! *>act]i/ic ( mod]n philosophy & also its vice4 ,x w be obs]v$ t at ! outset he rev]ses ! rles ( philosophy & ma!matics z we h appreh5d$ !m4 ,philosophy1 he says1 %\ld expect to be inspir$ by ma!matics1 b %\ld avoid its me?od4 ,next1 he connects ! mod]n movem5t ) ! !ories ( ,leibniz1 :o aim$ at sub/itut+ g5]al =mul = elem5t>y =ms ( reason & calculation4 ,^! %ort cuts1 :i* seem to ! ma!matician to lib]ate ! m9d f a burd5 :i* prev5ts x f employ+ its full activity1 seem to ! psy*ologi/ to t5d to a me*anical me?od1 9 :i* ! ?9k] is only aw>e ( premises & results1 & 9 :i* ! ma!matical concept t5ds to replace ! real idea4 ,!n he attacks ! new def9ition ( ma!matics z ! sci5ce ( relations1 ass]t+ t x /ill conta9s notions ( space4#ac ,f9ally1 he comes to ! real "q at issue1 & ent]s into ! comp>ison ( a metaphysical & a ma!matical problem4 ,he takes z 8 subject ! argum5t f ! kn[n to ! unkn[n4 ,desc>tes _h said t argum5t %\ld lead f ! kn[n to ! unkn[n1 simple to complex1 & _h def9$ ! fir/ z t :i* c\ld be kn[n )\t ! help ( ! second4 ,?is logical ord] ( reason+ has be5 attribut$ to ma!matics1 b has be5 consid]$ to be inapplicable to philosophy4 ,ma!matics1 9 its rec5t developm5t1 by ! argum5t f ! ,f9ite to ! ,inf9ite & back aga91 />ts f two propositions1 nei!r ( :i* c be said to be axiomatic1 because ea* 9 turn c be prov$ f ! o!r1 b 9 ! c\rse ( argum5t f ei!r ma!matics makes use ( ! logical process4 ,! real axiom1 z has be5 %[n1 is t ( exi/5ce or be+4 ,a metaphysical argum5t has ! same root--t ( exi/5ce--b a metaphysical problem deals ) p>adoxes1 ) que/ions :i* are sometimes def9$ z hav+ two answ]s1 ea* equally correct1 & sometimes z yield+ no answ] at all4 ,! me?od ( !sis1 anti!sis1 & syn!sis is 9 ! ,hegelian logic appli$ to _! solution4 ,a ma!matical & a metaphysical problem are n1 !n1 problems ( ! same k9d to be solv$ by ! same me?od2 nor is ! conception ( ! ma!matical ,absolute rea*$ 9 ! same way z t ( ! metaphysical ,absolute4 ,we are ev5 unable to say h[ f> !y correspond except 9 respect ( _! absolut5ess4#ad ,b ! cont5tion ( ! ma!matician to-day & ( ! epi/emologi/ s*ool ( philosophy is n ! id5tity ( me?ods & results 9 ! two sci5ces4 ,x is ! axiom ( exi/5ce on :i* !y bo? dep5d3 ! law ( ?\ght by :i* all me?ods are develop$1 &1 above all1 ! correlative value ( ea* sci5ce to ! o!r1 :i* all[s u1 9 develop+ our k f ! /&po9t ( ! two sci5ces1 to recognise some?+ ( ! greatness ( ! ,absolute pr9ciple to :i* !y bo? rea* up1 & 9 :i* _! be+ consi/s4 ,,footnotes3 #a ,( c\rse1 if we compreh5d 9 our view only elem5t>y geometrical & algebraical sci5ce1 x is easy to %[ t !y d dem& bo? axioms & intuitions4 ,take1 e4g4 ,euclid ,i4 ,i41 :]e 9 ! con/ruction x is necess>y to employ intuition = ! ass]tion t ! arcs really cut "o ano!r4 ,!re is no logical c]ta9ty t !y d2 9 fact1 9 "s o!r conditions1 e4g4 9 ^? ( o!r space dim5sions1 !y might n4 #b ,?is is1 ( c\rse1 n ! space ( exp]i5ce4 ,logic & ma!matics deal ) implications ( ?\ght4 ,see ,b4 ,russell 7,hibb]t ,j\rnal1 #aijd1 pp4 #hji-ab71 :o has %[n t 9 all pure ma!matics x is only ! implications t are ass]t$1 n ! premiss or ! consequ5ce1 z ma!maticians us$ =m]ly to assume4 #c ,de ,morgan1 ,peirce1 ,s*rd]1 & ,b4 ,russell h work$ out ! logic ( relations z well z ! syllogism4 #d ,see ,taylor1 8,elem5ts ( ,metaphysics10 p4 #ac4 #e ,see ,dr4 ,caird1 8,evolution ( ,!ology 9 ! ,greek ,philosoph]s40 #f ,s ,galileo1 ,newton1 ,huyg5s 7 philosoph]s 9 sci5ce4 ,desc>tes1 ,pascal1 ,leibniz 7 ma!maticians z well z philosoph]s4 #g ,see ,s4 ,augu/9e1 ,de ,civitate ,dei1 ,book ,,xii4 *4 xix43 8,ita v]o suis quisque num]us proprietatibus t]m9atur1 ut nullus eorum p> esse cuicumque alt]i possit4 ,ergo et disp>es int] se atque div]si sunt1 et s+uli quique f9iti sunt1 et omnes inf9iti sunt40 #h ,see ,r4 ,d$ek9d1 ,0 s9d und 0 soll5 die ,zahl58 #ahic4 #i ,two transf9ite aggregates c h an ord9al correspond5ce ) "o ano!r4 #aj ,see ,g4 ,cantor1 ,zur ,lehre vom ,transf9it54 #ahij4 #aa e4g4 ,mr4 ,p4 ,j\rda91 ,philosophical ,magaz9e4 #aijd4 #ab ,! same result is h9t$ at by ,mr4 ,taylor4 ,taylor1 8,elem5ts ( ,metaphysics10 p4 #bb4 #ac ,l9e> ord]1 #a1 #b1 #c1 c4 ,circul>4 ,a ,,cd ,b1 ,a ,,cd ,b4''' ,! latt]1 x is true1 involves ! idea ( sep>ation4 ,b ?is idea c be develop$ f ^? ( inclusion & exclusion1 :i* belong to ! fundam5tal laws ( ?\ght4 #ad ,! ,absolute1 accord+ to a rec5t metaphysical ?9k]1 is 8a consci\s life :i* embraces ! totality ( exi/5ce1 all at once1 & 9 a p]fect sy/ematic unity1 z ! cont5t ( its exp]i5ce40--,taylor1 8,elem5ts ( ,metaphysics10 p4 #fj4 ,,ii ,,pragmatism ,,& ,a ,,!ory ,,( ,,k ,! "q be=e u is ! relation ( ,pragmatism to a body ( k4 7a7 ,"o "q at issue betwe5 ! ,ideali/#ae & ! ,pragmati/ has to d ) ! way 9 :i* ea* def9es k & gauges its ultimate aim4 ,bo? say t k is relative1 b "o s*ool ass]ts t ! human m9d sl[ly & labori\sly uncov]s or discov]s :at ,goe! calls ! 8,liv+ g>m5t ( ,deity10 i4e4 ! _w ( nature1 & comes into a h]itage ( sci5tific tru? :i* increas+ly corresponds to ! subject ( 8 fai?2 ! o!r claims t we live 9 a self-evolv+ univ]se 9 :i* 9 ! c\rse ( long ages a new heav5 & a new e>? may be creat$ :i* are n =ese5 or impli$ 9 pres5t conditions4 ,9 o!r words1 ! ,ideali/ f9ds ! ,div9e 9 human life2 he f9ds 9 8 own small corn] ( ! univ]se ! microcosm & symbol ( ,inf9ity3 ! ,pragmati/ consid]s t no?+ is :i* is n a result ( human action1 & l[]s ! ,div9e elem5t to ! result ( individual human activity4 ,a compromise betwe5 ! two ideas on new & int]e/+ l9es has rec5tly be5 made by ,b]gson4 ,! ,*ri/ian doctr9e ( ,imman5ce & ,transc5d5ce also comb9es !m4 ,n[ ! increase ( a body ( k w\ld seem to dep5d on ! comp>ison ( ! successful work+ out ( hypo!ses ) ! discrepancies f !ory t f "t to "t appe>4 ,tak5 toge!r1 pro(s & discrepancies po9t to ! evid5ce ( a l>g] law4 ,?is is ,hegel's logic1 & ! pr9ciple1 s f> z x is "h impli$1 is n d5i$ 9 mod]n times1 = no "o wi%es to f\nd a logic on a /udy ( discrepancies z su*4 ,ev5 ,w4 ,james says1 8,:5ev] y once place y\rself at ! po9t ( view ( any high] syn!sis y see exactly h[ x does1 9 a fa%ion1 take up opposites into itself40#af ,9 fact1 )\t ! notion ( unity1 t ( discrepancy c\ld n exi/3 !re mu/ be a backgr\nd on :i* ! diff]5ces appe>4 ,! ultimate unity is symbolis$ 9 ! ,ideali/ doctr9e ( an ,absolute4 ,! ,absolute ( ,ideali/ic ?\ght is n1 h[ev]1 n[ conceiv$ ( 7z ! ,pragmati/ w\ld h u believe7 z an ab/ract unity1 b z "o involv+ a social bond1 & h5ce relations :i* c be describ$ z p]sonal1 if we rememb] t ! ,p]sonality ( ! ,absolute transc5ds our notion ( human p]sonality4 ,su* a conception ( ! t]m ,absolute1 a new read+ ( ! !ory ( ! ,"o & ! ,_m1 has be5 l$ up to by ,bradley & ,royce by me?ods ( logic1 & )\t any ref]5ce to dogma4 ,x has be5 conv5i5tly express$ by ,taylor4 ,! argum5t is briefly t ultimate ,reality mu/ be ,"o1 ,_m1 & ,p]sonal4 8,= our conclusion t m]e tru? _c be ! same ?+ z ultimate reality 0 itself bas$ ^u ! pr9ciple t only h>moni\s individuality is f9ally real1 & ?is is ! v pr9ciple employ$ by ! intellect itself :5ev] x judges "o ?\ght-con/ruction relatively high] or tru] ?an ano!r40#ag ,& aga93-- 8,if we speak ( exi/5ce z a society1 !n we mu/ be c>eful to rememb] t ! individual unity ( a society is j z real a fact ( exp]i5ce z ! individual unity ( ! memb]s :i* compose x1 & t :5 we call ! ,absolute a society r ?an a self1 we d n d s ) any int5tion ( ca/+ d\bt ^u its complete spiritual unity z an individual exp]i5ce40#ah ,! ,absolute has be5 /at$ 9 mod]n ?\ght to be ,"o1 ,_m1 ,real1 & ,p]sonal or ,social1 & ^! t]ms ( its qualification h be5 successively arriv$ at4 ,w4 ,james's words r+ holl[ :5 he attempts to dissociate su* a conception f ! reality ( :i* x is ! cr[n & inclusive symbol1 & type & ess5ce4 8,i p]sonally10 he says1 8give up ! ,absolute4 ,i f9d x entangles me 9 metaphysical p>adoxes t are inacceptable40 ,he all[s t !re may be a ,god1 ?\gh limit$ 9 p[] & goodness1 8"o help] among/ o!rs1 primus int] p>es 9 ! mid/ ( all ! %ap]s ( ! great _w's fate40 ,9 su* a sy/em1 z ,h4 ,jones has po9t$ out1 8!re is nei!r 9 ! univ]se nor 9 ,god any pr9ciple to inspire or guide1 or 9 any way to br+ ab\t ! amelioration desir$4 ,! process is guid$ by no end4 ,! univ]se beg9s by be+ an aggregate ( accid5ts1 plurali/ic1 discont9u\s1 irrational1 &1 ( itself1 _c become o!rwise4 ,!re is no?+ actual )9 to *ange its "*4''' ,god is himself f9ite1 helpless to br+ ab\t ?is great *ange1 a "p1 & no m1 ( a univ]se brok5 9 fragm5ts40 ,ano!r =m1 & a v s*ol>ly "o1 ( ! argum5t aga9/ ! exi/5ce ( an ,absolute has be5 /at$ by ,bax 9 ! 8,roots ( ,reality40 ,he appe>s to h rea*$ ! conclusion t ! telos1 ! goal ( human ?\ght1 is n an ,absolute involv+ any notion ( fixity1 b t x may be conceiv$ ( z a 8mov+ syn!sis40 ,he argues t ev]y?+ ( :i* we are consci\s 9 ! univ]se is se5 aga9/ a backgr\nd :i* itself moves1 & is only realisable or di/+ui%able if x %ifts ^u some?+ relatively motionless beh9d x4 ,he concludes1 !re=e1 t by analogy !re is no ,absolute1 s9ce :at we p]ceive always implies some?+ aga9/ :i* we p]ceive x2 ?us t !re is no goal by :i* & at :i* ! _s ( man c f9d re/4 ,on 8 !ory we c\ld nev] claim to rea* ! conception ( an ,absolute1 ?\gh he admits ! progressive "* ( human ?\ght1 & ! increas+ rea*1 lucidity1 & dep? ( ! human m9d4 ,! true answ] to ?is argum5t is t x proves exactly :at x sets out to disprove4 ,z x is ackn[l$g$ t only ! p]man5t or ! relatively p]man5t c produce ! ph5om5a ( *ange1 s ! appe>ance ( ! goal ( ?\ght z a mov+ syn!sis w\ld presuppose an ,absolute z a gr\nd reality4#ai ,if 9 tru? we 7 able to appreh5d entirely ! s\rce ( all life & ! backgr\nd ( all exp]i5ce1 we might say t x did n exi/ = u z an ,absolute1 b ! fact t :atev] we p]ceive po/ulates an un5d+ s]ies beh9d x1 c>ries ) x ! pro( ( an ,absolute ,inf9ite4 7,?is conclusion is l$ up to by ! ma!matician's idea ( ! s]ies ( all f9ite & transf9ite ord9al numb]s47 ,"s "p ( ?is argum5t has be5 already sugge/$ 9 ,ormond's 8,f\ndations ( ,k10 & s f> 0 us$ by ,mr4 ,ill+wor? 9 ! 8,doctr9e ( ! ,tr9ity40#bj 8,f a deep] metaphysical po9t ( view x is ! concept ( evolution itself t mu/ submit to ! det]m9ation ( k1 = x w be f\nd t 9 s f> z x becomes epi/emologically necess>y to gr\nd relative processes 9 an ,absolute exp]i5ce1 j s f> w x become necess>y also to connect ! evolution>y aspect ( ! _w itself ) a gr\nd reality t is /able1 & involves ! flux ( *ange only z transc5d+ & includ+ x40#ba ,! fur!r answ] t any judgm5t1 ev5 ! ,pragmati/'s 8judgm5t ( value10 implies an ,absolute has be5 /at$ 9 8 ,ox=d ,lectures#bb by ,pr(essor ,h4 ,jones4 7b7 ,! next po9t we %\ld l to "w out is ! relation ( fact to law4 ,! ,pragmati/ d5ies sci5tific law & also logic1 & makes 8 appeal to facts4 ,no conclusion c be drawn f t d5ial except by ! use ( logic itself4 ,if he consi/5tly d5i$ logic1 8 position w\ld be unassailable by logic1 b he uses ! me?od he d5ies1 & is ?us op5 to attack4 ,on ! subject ( ! ,laws ( ,sci5ce ! ,pragmati/ po9ts out truly t !re is no actual cont9uity betwe5 a fact & a law4 ,b laws are concepts1 ! result ( m5tal activities2 !y are !mselves subject to ! laws ( logic4 8,!y 7 means1 & y make !m ends10 compla9s ! ,pragmati/4 ,t is j :at nature h]self does4 ,%e p]fects means1 su* z ! means ( support+ life1 & !n ^! become ends4 ,language1 aga91 is at fir/ a means1 & !n becomes an end4 ,s does any sci5ce *ange its "* to ! onlook]4 ,a law1 too1 ?\gh x g5]alises facts1 is a limit on absolute g5]alisation4 ,x ?us /&s midway betwe5 ! ab/raction & ! fact4 ,! ,pragmati/1 h[ev]1 opposes to law :at he calls a new fact--:at %\ld r be call$ a hypo!sis4 ,he ass]ts t 9 e ev5t1 action1 exp]im5t1 !re is a m>g9 unse5 & unrecognis$ by u2 t at e mom5t1 !re=e1 ! unkn[n1 ! unexpect$1 may take %ape & voice & d5\nce all our c>eful & reason$ conclusions4 8,:y %\ld ! sun rise to-morr[ because he has ris5 to-day & ye/]day80 asks ! ,pragmati/4 8,we are mak+ an enorm\s assumption10 he says1 89 claim+ ! uni=mity ( ,nature & ! pr9ciple ( causality40 ,! ,ideali/ answ]s t ! ,pragmati/ makes a l>g] assumption 9 d\bt+ ! tru? ( ! pr9ciples1 :i* ?\gh relative & n absolute1 /ill d "w out 9 practice1 ?an ! ,ideali/ does 9 8 act ( fai?4 ,9 fact1 ! act ( fai? is rational z well z natural2 x is ! act ( d\bt+ t is 9 ?is case due to a m]e s*ola/ic quibble4 ,x is ! ,ideali/ & n ! ,pragmati/ :o makes 8 appeal to ! tru? ( facts4 ,ea* day t ! sun goes on ris+ f9ds ! ,ideali/ 9 a bett] philosophical position & ! ,pragmati/ 9 a worse1 except on ! assumption t ! l9k betwe5 man & ! ext]nal _w is a false imag9ation4 ,let u emphasise3--,x is ! ,pragmati/ :o quibbles ) logic1 & ! ,ideali/ :o appeals to facts4 7c7 ,n[ !re are c]ta9 facts & c]ta9 d$uctions f facts1 well kn[n to ma!maticians1 :i* we %\ld l to quote "h z hav+ a be>+ on ! !ory ( ! ,absolute1 because !y deal ) aspects ( ,inf9ity1 & m>k a connection betwe5 ! _w z we "k x & ! concepts ( ! philosoph]4 ,all h ! support ( sci5ce1 & furni% ! ,ideali/ philosoph] ) examples :i* support 8 !ories1 & /r5g!n 8 position 9 ! face ( ! ,pragmati/ attack4 ,!y h to d ) ! !ory ( ,inf9ity z %[n 93-- ,i4 ,! ,indef9ite ,regress4 ,,ii4 ,inf9ite s]ies4 ,,iii4 ,dim5sions 9 space & "t4 ,be=e ent]+ ^u !m we mu/ repeat t ! "q ( numb] & s]ies 9 ma!matics is indep5d5t ( ! assumptions ( space & "t4 ,z a sci5ce1 ma!matics c\ld exi/ outside !m3 ord] is n necess>ily spatial or temporal4 ,our conclusions1 !re=e1 _c be attack$ on ! gr\nd t !y are bas$ on ,euclidean conceptions ( space3 !y are bas$ on ! laws ( logic4 ,i4 ,,! ,,indef9ite ,,regress ,hume &1 lat]1 ,kant argu$ t by ! pr9ciple ( association :5 we ?9k ( "o quality ( a ?+ ! o!rs are naturally br\ght be=e our m9ds1 & ?us t we get into ! habit ( attribut+ to ! notion ( ! ?+ a c]ta9 gr\p ( qualities4 ,& x is true t we d att5d to a ?+ all at once1 includ+ 9 ! notion ( x all ! qualities :i* we "k belong to x4 ,n[ exp]i5ce1 accord+ to ,leibniz1 gives u an example ( a unity :i* embraces a multiplicity ( detail4 ,?us a ?+ is "o sub/ance z embody+ an individual exp]i5ce1 & its qualities belong to x 9 ! same s5se z ! con/itu5ts ( exp]i5ce belong to ! s+le exp]i5ce4 ,^! qualities are 9 relation4 7,! ,pragmati/ d5ies ! exi/5ce ( relations z "p ( a high] unity4#bc7 ,b !y are n only relation1 s9ce relation always implies some?+ m ?an itself4 ,let u take ! example ( numb]4 ,numb]s c\ld nev] h be5 c\nt$ if !re _h n be5 ?+s to c\nt4 ,n[ suppose ea* quality c\ld be analys$ into a new relation1 we %\ld /ill n get rid ( ! quality4 ,at ea* /age !re rema9s a quality 9 relation1 & ?is goes on to ,inf9ity4 ,su* a con/ant subdivision p]haps results f our f9ite exp]i5ce seiz+ facts 9 a disjo9t$ way4 ,:5 we analyse a law 9 its work+1 we always d seem to come to ?is ,indef9ite ,regress4 ,n[ x has be5 ! reproa* aga9/ metaphysics1 z utt]$ by ! ,pragmati/1 t !re is no correspond5ce 9 sci5tific fact to ?is road into ,inf9ity4 ,w4 ,james ass]ts3 8,b 9 po9t ( fact1 nature doesn't make eggs by mak+ fir/ half an egg1 !n a qu>t]1 !n an eigh?1 c41 & add+ !m toge!r4 ,%e ei!r makes a :ole egg at once or none at all1 & s ( all h] o!r units4 ,x is only 9 ! sph]e ( *ange1 !n1 :]e "o phase ( a ?+ mu/ ne$s come into be+ be=e ano!r phase c come1 t ,z5o's p>adox gives tr\ble4 ,& x gives tr\ble !n only if ! successive /eps ( *ange be inf9itely divisible40#bd ,! sph]e ( *ange1 h[ev]1 "o w\ld answ]1 includes all nature1 & sci5ce 9 its discov]ies acts on ! hypo!sis t ^! /eps ( *ange may be inf9itely divisible4 ,royce held to x firmly t any consi/5t attempt to make an ord]ly arrangem5t ( ! t]ms ( an inf9ite :ole mu/ lead to ! ,indef9ite ,regress4 ,& he fur!r %[s ! connection ) ! fact t an inf9ite s]ies c be adequately repres5t$ by a "p ( itself4 ,9 ! ,boyle ,lecture1 deliv]$ 9 ,ox=d 9 #aijh1 on ! prop]ties ( radium1 two facts em]g$ :i* %[ t ! ,indef9ite ,regress is n[ recognis$ 9 sci5ce4 ,fir/1 t 9 ! region ( exp]im5t we become aw>e ( gr\ps ( elem5ts alli$ to radium1 :i* seem1 9 ! numb] ( individuals 9 _! gr\ps1 to foll[ a simple ari?metical progression4 ,secondly1 t radio-active elem5ts lose 9 activity at a c]ta9 rate1 :i* always repres5ts an exact proportion ( ! mass :i* rema9s4 ,! trem5d\s dis9tegrat+ =ce slack5s 9 exact relation to ! "t :i* passes1 s t ! small] ! morsel ! less ! relative loss ( mass4 ,"h1 !n1 is ! ,indef9ite ,regress4 ,9 ! _w ( fact z well z ( ideas we are deal+ ) aspects ( ,inf9ity4#be ,,ii4 ,,inf9ite ,,s]ies ,!re are o!r aspects ( ,inf9ity :i* we c get at by /udy+ s]ies1 & :i* 9 ! conception ( s]ies ( s]ies give /r5g? & po9t to ! philosophic conception ( an ,absolute4 ,pr(4 ,c4 ,keys] develops ?is ?\ght1 & %[s 79 two rec5t articles1 ,janu>y & ,april #aiji1 9 ! ,hibb]t ,j\rnal7 t c]ta9 !ological dogmas1 su* z ! doctr9e ( ! ,tr9ity1 & c]ta9 attributes ( ! ,div9e ,be+1 su* z ,omnisci5ce & ,omnipres5ce1 are entirely conceivable by ! human m9d if reg>d$ )\t ! p>alys+ limitations ( ! ,f9ite4 ,he %[s t 9 our ma!matical =mul :i* h to d ) inf9ite s]ies we h ! exact replica ( :at to ! lay1 non-ma!matical m9d seem to be ! p>adoxes ( ! ,a?anasian ,cre$4 ,he fir/ %[s t 9 a ma!matical analogy po9ts ( view ab\t an ,inf9ite ,be+1 ev5 if p>tially discordant1 may all be true if reg>d is _h to ,8 ,inf9ity4#bf ,fur!r1 he %[s t c]ta9 assumptions1 su* z ! :ole is great] ?an its "p1 are inapplicable to ,inf9ite ,be+4 ,! conception ( a ,tr9ity 9 ,unity 9 :i* 8none is a=e or aft] o!r1 none is great] or less ?an ano!r1 b ! :ole ?ree p]sons are co-et]nal toge!r & co-equal0 is rationally conceivable by ! ma!matician :o is famili> ) ! !ory ( manifolds4#bg ,we h1 he %[s1 ?ree inf9ite manifolds3-- ,e ( ! ev5 integ]s4 ,o ( ! odd ones4 ,f ( ! fractions hav+ integ]s = _! t]ms4 ,no two ( ^! h a s+le elem5t 9 common1 yet ! ?ree toge!r con/itute "o manifold ,m1 t is exactly equal 9 weal? ( elem5ts to ea* ( its inf9ite compon5ts4 ,aga91 !re is ! app>5t opposition betwe5 ! ,omnisci5ce ( ,god & ! fre$om ( man4 ,! anti!sis disappe>s if we realise t f ! po9t ( view ( ,inf9ites ! dignity & p[] ( ,omnisci5ce rema9 ! same1 ev5 if "s "p ( exp]i5ce is n yet drawn into ! sph]e ( ,omnisci5ce4#bh ,"h we h ! pres5t conceiv$ ( z a mov+ plane sep>at+ ! unkn[n f ! kn[n4 ,! 8pa/0 c be said to be kn[n1 ?\gh its cont5t *anges e in/ant4 ,?is is ! real answ] to ,w4 ,james's cry t he c\ld accept an ,absolute if x _h ev5 ! fragm5t ( an 8o!r40 ,!re c be ?is 8o!r10 & yet ! ,absolute /ill rema9s an ,absolute4 ,! doctr9e ( ,omnipres5ce foll[s f ! argum5t ( ! ,cont9uum 7:i* is ! aggregate ( all real numb]s74 ,?us ! numb] ( po9ts 9 space ( inf9ite dim5sion is no great] ?an ! numb] ( po9ts 9 any "p ( space z kn[n to u4 ,! :ole is inc>nate 9 e "p1 because to ea* "p1 9 h[ev] small an atom1 corresponds a po9t 9 ! univ]sal :ole1 & ! numb] ( po9ts 9 a space ( inf9ite dim5sions is equal to ! numb] ( po9ts 9 a /raight l9e h[ev] small4 ,& ?is is true n m]ely ( po9ts b also ( =ces4 8,! ,univ]se is dynamic1 *>g$ ?r\gh\t ) innum]able modes ( motion4 ,ea* po9t1 h[ev]1 ( any mov+ ?+--an ion ( gas1 a vibrat+ fibre ( bra9--is repres5t$ by a correspond+ po9t 9 ,s 7a small typical atom71 & s )9 ! t9y sph]e--inde$1 9 e room1 h[ev] small--! :ole dynamics ( ! univ]se is depict$ completely & co-enact$ by motion ( po9ts & trans=mation ( po9t configurations4 ,!re 9 m9iature proce$ at once ! c\ntless play & int]play ( e k9d ( motion1 small & l>ge1 simple & complex1 ! quiv]+ dance ( ! molecule1 ! wave & sw+ ( univ]sal !r40#bi ,,iii4 ,,dim5sions ,,9 ,,space ,!re is ano!r argum5t1 "o relat+ to ! !ories ( "t & space1 :i* greatly affects ! conception ( omnipres5ce4 ,?is is ! argum5t ( ! _m dim5sions1 call$ by ,keys] ! 8radiant concept ( hyp]-space1 :i* only a g5]ation ago 0 reg>d$1 ev5 by ma!maticians--mo/ adv5tur\s ( m5--z be+ purposeless & va91 b :i* mean:ile has advanc$ s rapidly to comm&+ position t ev5 ! foll[+ /atem5t by ,po9c>1 9 8 rec5t address be=e ! ,int]national ,ma!matical ,congress at ,rome on #8,l',av5ir des ,ma?matiques1' is well )9 ! limits ( cons]vatism3 #8,n\s sommes auj\rd'hui tellem5t famili>iss avec cette notion que n\s p\vons en p>l]1 mme dans un c\rs d'univ]sit1 sans provoqu] trop d'tonnem5t4' ,! fact is t ! doctr9e already exi/s 9 a va/ & rapidly gr[+ lit]ature1 fl\ri%es 9 all ! sci5tific languages ( ! _w1 & 9 its ess5tial pr9ciples has become = ma!matics z or?odox z ! multiplication table40 ,! pres5t position ( ! !ory is briefly ?is3 ,if !re did n exi/ a f\r? dim5sion1 we c\ld n be aw>e ( a ?ird z su*1 & s on4 ,are we !n look+ out ^u a ?ird dim5sional _w1 & realis+ x z su* because we are m5tally capable ( conceiv+ dim5sions beyond x8 ,our _w s5sibly conta9s "o dim5sional & two dim5sional facts--! fir/ su* z a "t s]ies1 = :i* "o numb] is suffici5t to fix a po9t1 & ! second su* z a plane :]e position c be fix$ by two numb]s4 ,does our _w conta9 facts ( o!r dim5sions8 8,all p>ticles ( air are f\r-dim5sional 9 magnitude :51 9 addition to _! position 9 space1 we also consid] ! v>iable d5sities :i* !y assume1 z !y are express$ by ! diff]5t heights ( ! b>omet] 9 ! diff]5t p>ts ( ! atmosph]e4 ,simil>ly all conceivable sph]es 9 space are f\r-dim5sional magnitudes1 = _! c5tres =m a ?ree-dim5sional po9t-aggregate1 & ar\nd ea* c5tre a "o-dim5sional totality ( sph]es1 ! radii ( :i* c be express$ by e num]ical magnitude f z]o to inf9ity4 ,fur!r1 if we imag9e a measur+-/ick ( inv>iable l5g? to assume e conceivable position 9 space1 ! positions s obta9$ w con/itute a five-dim5sional aggregate4 ,= 9 ! fir/ place "o ( ! extremities ( ! measur+-/ick may be conceiv$ to assume a position at e po9t ( space1 & ?is det]m9es = "o extremity alone ( ! /ick a ?ree-dim5sional totality ( position1 &1 secondly1 z we h se5 above1 !re proce$s f e su* position ( ?is extremity a two-dim5sional totality ( directions1 & by conceiv+ ! measur+-/ick to be plac$ l5g?wise 9 e "o ( ^! directions1 we %all obta9 all ! conceivable positions :i* ! second extremity c assume1 & consequ5tly ! dim5sions mu/ be #c #b or #e '''0 c41 c4#cj ,ma!maticians h = long done problems 9 ! sev5? & eigh? dim5sions4 ,!y h told u t y _c tie a knot 9 ! second dim5sion1 because !re is no up or d[n1 & ! ?reads w\ld n cross--nor 9 ! f\r?1 because ! knot w\ld pull out 9 a new direction & w\ld n hold4 ,b x has only lately be5 realis$ t f\r? & o!r dim5sions may be actual fact 9 ! _w r\nd u4 ,( c\rse1 f ! po9t ( view ( a po9t !re are only ?ree dim5sions to be kn[n1 b to a l9e 9 ! same space !re are five1 to ! surface probably six4 ,our intellig5ce at pres5t does n g beyond ! po9t2 b if we c\ld ?9k ( space f ! po9t ( view ( a solid1 worlds ^u worlds w\ld rise be=e our view4 ,( ! f\r? dim5sion we c discov] "s facts by analogy4 ,we c c\nt ! edges ( its typical figure1 & apply ?\ght to det]m9+ "s ( its conditions4 ,b a m int]e/+ subject ( rese>* is ! inquiry into ! light ?r[n by ! !ory ( f\r dim5sions on ! det]m9ation ( c]ta9 atoms 9 *emi/ry1 t are kn[n to be di/9ct elem5ts1 b c\ld only be det]m9$ actually 9 ano!r dim5sion4#ca 8,9 *emi/ry1 ! molecules ( a comp\nd body are said to consi/ ( ! atoms ( ! elem5ts :i* are conta9$ 9 ! body1 & ^! are suppos$ to be situat$ at c]ta9 di/ances f "o ano!r & to be held 9 _! relative positions by c]ta9 =ces4 ,at fir/ ! c5tres ( ! atoms 7 conceiv$ to lie 9 "o & ! same plane4 ,b ,wislic5us 0 l$ by rese>*es 9 p>alactic acid to expla9 ! diff]5ces ( isom]ic molecules ( ! same /ructural =mul by ! diff]5t positions ( ! atoms 9 space4 ,9 fact1 f\r po9ts c always be s arrang$ 9 space t e two ( !m may h any di/ance f ea* o!r2 & ! *ange ( "o ( ! six di/ances does n necess>ily involve ! alt]ation ( any o!r4 8,b suppose our molecule consi/s ( five atoms8 ,f\r ( ^! may be s plac$ t ! di/ance betwe5 any two ( !m c be made :at we please4 ,b x is no long] possible to give ! fif? atom a position su* t ea* ( ! f\r di/ances by :i* x is sep>at$ f ! o!r atoms may be :at we please4 ,on ! contr>y1 ! f\r? di/ance is dep5d5t on ! ?ree rema9+ di/ances1 = ! space ( exp]i5ce has only ?ree dim5sions4 ,if1 !re=e1 ,i h a molecule :i* consi/s ( five atoms1 ,i _c alt] ! di/ance betwe5 two ( !m )\t at lea/ alt]+ "s second di/ance4 ,b if we imag9e ! c5tres ( ! atoms plac$ 9 a f\r-dim5sion$ space1 ?is c be done2 all ! t5 di/ances :i* may be conceiv$ to exi/ betwe5 ! five po9ts w !n be indep5d5t ( "o ano!r4 ,to rea* ! same result 9 ! case ( six atoms we mu/ assume a five-dim5sional space1 & s on40#cb ,"h we see t if *emi/ry z a sci5ce is b\nd to take acc\nt ( all its facts1 ! sci5ti/ is confront$ ) a problem ( dim5sions t is really a problem ( ,inf9ity appli$ n1 z 9 ! o!r cases quot$1 to numb]1 b to space4 ,& !re is a reason :i* expla9s :y ! same problem t5ds to appe> 9 ^! diff]5t ways4 ,bo? "t & space c be mo/ correctly ?\ght ( z s]ies3 ! =m] kn[n to u z possess+ "o direction1 ?\gh possibly involv+ m1 & ! latt] ?ree1 ?\gh possibly involv+ m4 ,"t is n a ?+ nor a condition1 b x is ! way 9 :i* we are enabl$ to appreh5d ! relations ( actions to "o ano!r4 ,! assumption ( ! ,pragmati/1 t a diff]5t date 9 hi/ory is a new condition :i* might affect a *emical exp]im5t1 is mean+less1 unless by t he int5ds to say t at ! diff]5t date new conditions prevail$4 ,! g5]al conclusion ( rec5t ?\ght is !n to e/abli% ! ,ideali/ position m /rongly by an appeal to ma!matical argum5t4 ,?is argum5t is /r5g!n$ by f9d+ at ! pres5t "t "s support 9 sci5tific fact & exp]im5t4 ,! ,ideali/ !re=e appeals to fact1 & 8 position re/s ultimately on a tru? :i* has its aspects ( con=mity ) sci5tific exp]im5t & ) logical or ma!matical pro(4 ,,footnotes3 #ae ,?is ^w is us$ "h 9 ! mo/ g5]al & inclusive s5se z apply+ to all ?9k]s :o accept ! reality ( relations z "p ( a high] ,unity4 #af 8,a ,plurali/ic ,univ]se10 p4 #ii4 #ag ,taylor1 8,elem5ts ( ,metaphysics10 p4 #cab4 #ah ,ibid41 p4 #cej4 #ai ,a succession ( :at is disconnect$ is n *ange4 ,*ange is a succession )9 an id5tity3 if n )9 ! id5tity1 !re is no *ange1 only analysis & re-gr\p+4 ,! clos] our k is ( ourselves or any?+ else1 ! m we see t *ange is ! expression 9 "t ( an id5tity4 #bj ,ill+wor?1 8,! ,doctr9e ( ! ,tr9ity10 p4 #f4 #ba ,ormond1 8,f\ndations ( ,k10 p4 #ai4 #bb #aijh-aj4 #bc ,james1 8,a ,plurali/ic ,univ]se10 p4 #hj4 #bd 8,a ,plurali/ic ,univ]se10 p4 #bcj4 #be ,see ,a4 ,t4 ,cam]on1 8,radio-,*emi/ry10 p4 #ag3 8,! curves illu/rate two fur!r po9ts4 ,!y approa* con/ant value t[>ds ! end ( a mon?1 b x is se5 t !y rea* a f9al value only at inf9ite "t4 ,?is prop]ty is common to all su* curves2 x illu/rates ! fact t ! life ( a radio-active elem5t is inf9ite40 ,x is expla9$ 9 ! same book 7p4 #ca7 t 8inf9ity is only a relative t]m2 9 ?is connection x only means a long] "t ?an we c measure40 #bf ,8 !ology is n s good z 8 ma!matics2 he seems to ?9k t 9 ! ,cre$ we ass]t our belief 9 ! ,incompreh5sible1 9 ! s5se ( t :i* is 8n capable ( be+ seiz$ by ! m9d10 in/ead ( 9 t :i* is 8untrammell$ by limitations40 ,! ^w is imm5sus1 be/ translat$ inf9ite4 #bg ,hibb]t ,j\rnal1 #aiji1 pp4 #fbf-bh4 #bh ,hibb]t ,j\rnal1 #aiji1 p4 #fbi4 #bi ,hibb]t ,j\rnal1 #aiji1 p4 #fcb4 #cj ,s*ub]t1 8,ma!matical ,essays & ,recreations10 pp4 #gj1 #ga4 #ca ,see ,van t' ,h(f1 ,la ,*imie dans l'espace1 & ,s*ub]t1 8,ma!matical ,essays & ,recreations10 pp4 #hh-hi4 #cb ,s*ub]t1 8,ma!matical ,essays & ,recreations10 pp4 #hh-hi4 ,see also ,ma*1 8,cons]vation ( ,en]gy0 7trans4 ,op5 ,c\rt ,publi%+ ,co474 ,pr9t$ by ,,ballantyne1 ,,hanson ,,co4 ,ed9burgh ,london ,transcrib]'s ,note3 ,words & phrases 9 italics are surr\nd$ by und]scores1 l ?is4 ,footnotes 7 r5umb]$ sequ5tially & 7 mov$ to ! end ( ! *apt]4 ,words may h inconsi/5t hyph5ation 9 ! text4 ,"o misspell$ ^w 0 correct$4