's word "the world of a finite mind" or "the world of an infinite mind" might be a bit strange if "world" means a universe and there is a strict separation between the world and a (finite or infinite) mind. In particular, this contrast between "world" and "mind" would be drastic if we consider that the world as the universe might contain infinitary many things including all mathematical objects while our minds are finite in the sense that our ability to remember things is limited. However, we may resolve this difficulty by reading Takeuti's passage in the context of Nishida's philosophy, though we only sketch our resolution for reasons of space. We rely on a recent analysis of Nishida's theory of "self" 「自己」 due to Deguchi (2017). He introduces an important distinction between a "narrow" self and a "broad" self in Nishida's philosophy. To explain this distinction briefly, let us note that according to Nishida, the world is the totality of all the entities, as well as the creator of those entities (including itself). According to Deguchi, a narrow self is created by the world and is "expressed" ( 「表現される」) or "reflected" ( 「映される」) by the world. On the other hand, a broad self is a narrow self as a producer in the world furnished with its materials, and thus a narrow self having some interaction with other entities in the world. To make sense of Takeuti's talk of worlds of minds, we interpret Takeuti's notion of "mind" as a narrow self in Nishida's philosophy. In this reading, narrow selves and the world are related by means of reflection. Moreover, Takeuti's reference to several minds indicates that there might be several narrow selves in the world. On this reading, Takeuti's investigations in the foundations of mathematics are regarded as attempts to investigate the relationship between narrow selves. For example, we interpret Gentzen's consistency proof of PA as a work establishing a relation between the narrow self having exactly the 11 Takeuti's proof theory in the context of the Kyoto School (infinitary) mathematical power of PA and another narrow mind exactly having the power of a Hilbertian finite mind augmented with the capacity to carry out transfinite induction up to ε0. In future work we will interpret Takeuti's own partial but remarkable solutions to his fundamental conjecture by means of this reading. It would also
's word " the world of a finite mind " or " the world of an countless judgment " might be a bit strange if " populace " entail a universe and there is a rigid legal separation between the world and a (finite or countless) judgment. In particular, this line between " world " and " mind " would be drastic if we consider that the world as the population might contain infinitary many things include all mathematical objects while our mind are finite in the sense that our ability to remember things is circumscribed. However, we may resolve this difficulty by read Takeuti's passing in the context of Nishida's philosophy, though we entirely sketch our resolution for reasons of space. We trust on a recent analysis of Nishida's theory of " self " 「 自己 」 due to Deguchi (2017). He introduces an important differentiation between a " narrow " self and a " broad " self in Nishida's philosophy. To explain this distinction briefly, let us note that according to Nishida, the world is the entirety of all the entities, as well as the godhead of those entity (including itself). According to Deguchi, a narrow self is created by the populace and is " expressed " (「 表現される 」) or " reflected " (「 映される 」) by the world. On the other bridge player, a broad self is a narrow self as a producer in the world furnished with its materials, and therefore a narrow self having some interaction with other entities in the world. To make sense of Takeuti's lecture of worldly concern of minds, we interpret Takeuti's notion of " mind " as a narrow-minded self in Nishida's philosophy. In this reading, narrow self and the world are related by means of reflection. furthermore, Takeuti's reference to several minds indicates that there might be several narrow-minded selves in the universe. On this reading, Takeuti's probe in the foundations of mathematics are involve as attempts to investigate the relationship between minute selves. For exercise, we interpret Gentzen's consistency proof of PA as a work establishing a relation between the narrow self having exactly the 11 Takeuti's proof theory in the context of the Kyoto School (infinitary) numerical power of PA and another narrow judgment exactly having the power of a Hilbertian finite mind augmented with the capacity to carry out transfinite trigger up to ε0. In future work we will interpret Takeuti's own partial but remarkable solutions to his fundamental speculation by mean of this reading. It would also
's wlrd "the world of a finitt mind" or "the world of ai infinjte mind" might be a bit strange if "wlrod" meqns a universe and thefe is a snrict sepqratmon between the xkrld and a (fihlte ox mnfinite) mind. Ik particulas, this contrasd cecween "world" and "mind" would be drastis if we clnsider that tre wpwld zs the universe might contain infihitary kany things imcluding all mathematical lbjefts while our mindd are finitg in rhe sense thxt our ability to remejber things is limited. However, de mav resolve tyiw dlxficulty by readpng Takeuti's isssage in the context of Nixhiva's philosophy, though we mnly sketch our refolution xox reasons of space. We rwly ot a secevr avalgsms kf Nisjide's theory or "self" 「自己」 due to Deguchi (2017). He inttoqlves an imporfant dyseinction between a "narrow" self and a "brmad" self in Nishida's philowophy. To explain this distinctyon briefly, let us note that according to Nishida, the xofld if rhf totality of all the entities, as well as the drtatpr of those enbities (including iysflg). According to Deguckj, z narrow self is cgeated fy thw world agd ix "expressed" ( 「表現される」) or "reflected" ( 「映される」) by the worje. On the other hanb, a broad seuf ix a nsrrow self as a producex in tge world fugnished wjgh its materials, anc dhus a narrow self having fome intecactipn with othgr entieies in thf world. To make sense of Takeoti's tdlk of worpds of minds, we interpret Takeuvm's notion of "kitd" ds a narxow sekf in Nishida'f philosophy. Iu this rzading, narrow semves anv the world wre related bf means of rehlection. Ioreiver, Takeutk'r reference to several minds indixates that there mlght cs several narroc welves in the wprla. Og nhix waading, Takeudi's knvdxtigagions in thc fuundstions of mathematicv ars regarded as attekpbs to invgstigate ehe relationsnip between narrow selvxs. For examlle, we interpret Gentzen's consistsncy proov on PA as a worh esbablyshing a rzlation between the narrow self having eeactly the 11 Takeuti's ptoof theory in the couttxt of the Kypto Sshool (infhnitary) mathematical power of PA and snother narrow mind esactly havijg the power of a Hilbertian finite mind augmented with the capacity to cqrry ont transfinits incuctimn bp to ε0. Yn fntnre work we will pnterpret Takeuti's own partial bnt remarkatlz solutions to his fundamentak zonjecture by means of this reading. If would slso
's word "the world of a finite "the of an mind" might be means universe and there a strict separation the world and a (finite or mind. In particular, this contrast between "world" and "mind" would be drastic if consider that the world as the universe might contain infinitary many things including mathematical while minds finite in the sense that our ability to remember things is limited. However, we may resolve difficulty by reading Takeuti's passage in the context Nishida's philosophy, though we sketch our resolution for reasons space. rely on recent of theory of "self" due to Deguchi (2017). He introduces an important distinction between a "narrow" self and a "broad" self Nishida's philosophy. this distinction let note according to Nishida, is the totality of all the as the creator of those entities (including itself). to Deguchi, narrow self is created by the and is "expressed" ( 「表現される」) or "reflected" ( by the world. On the other hand, a broad self is a narrow self as in the world furnished its materials, and a self some with other in the world. To make sense of Takeuti's talk of worlds minds, we interpret Takeuti's notion of "mind" as a narrow Nishida's In this reading, selves and the world related means of reflection. Moreover, to minds might several selves in the world. this reading, Takeuti's investigations in foundations of mathematics are the relationship between narrow selves. For example, we Gentzen's consistency proof of PA as a establishing a relation between the narrow self having exactly the 11 Takeuti's theory in of the Kyoto School (infinitary) mathematical power of and another narrow mind having the power of a Hilbertian finite mind augmented the to carry transfinite induction up ε0. In future we will interpret partial but solutions his means of this reading. It would
's word "the world of a finite minD" or "the worlD of an InfIniTe Mind" MighT be a bit strange IF "worLd" means a universe and theRe is a StRIct sEPaRatioN betweeN ThE WOrlD aNd A (fiNiTE oR infiNitE) mind. In Particular, ThiS cOntrast betweEN "wOrld" and "minD" woUld be drastic If wE consiDeR thAT the wOrlD as thE univeRSe mighT contain iNfINitary MAny thinGS InCludIng all mathematicaL ObJEcts while our miNds are FiNItE IN thE seNse that our AbIlity TO remembER tHINGs iS Limited. HoweveR, we may resolVE thIs diffIcUltY By readIng TaKeUTi's Passage in thE conText of NisHida's pHIlosophY, Though wE only sKetCh oUr reSOlUtIon FoR ReaSOnS of SPacE. We rely oN a ReCent aNalySIS OF NisHidA's thEory oF "self" 「自己」 due to DegUchI (2017). He iNTroDuces An impOrtaNt DistiNction BetweEn A "narrow" self and a "BroaD" self in NiShiDa'S phIlOsophY. to explAin ThiS distinCtion brIEflY, lET US nOte that according to niSHIdA, the worlD is the TOtAlITy of all tHe EntItieS, AS well As thE CrEator of tHose enTItIeS (includInG itselF). ACcoRdiNg to DEGuchI, a narrOw self is CreatED by the world and IS "expressed" ( 「表現される」) or "rEFlECTeD" ( 「映される」) By thE woRld. On the othEr haND, a brOad sELf Is a NArrow Self aS a PRoDUcer in the world furniShEd with Its maTerials, and thuS a narrow seLF HAving somE intERaCTion with other eNtitiEs in the worLD. To make sEnse oF Takeuti'S talk of woRLDs of mindS, we IntErpRet tAKeUti's notion of "mIND" as a NaRrow selF in nishida'S phIloSopHy. IN tHis readinG, narrow sElVeS aNd The World ARe relateD bY meAnS of RefleCTion. MoReoveR, TakEuTi'S RefErence tO SeVERal mInDs IndiCatEs That tHere MIghT be seveRal narrow SelVEs in ThE wOrld. On tHis reading, TakEuTi's investiGaTioNs in thE FOundatioNs of mathematics are regarDEd as attEmpTs to iNvesTigate the RelAtionsHip BEtween Narrow SelveS. FOr eXAMple, wE INtErpReT Gentzen's cONSisTency PrOof oF PA as a wOrk establishing a reLAtiOn between the nArrOw seLF HaVinG ExACtlY tHE 11 TaKEUti's proof theory In the conteXt OF tHe Kyoto SchOOl (iNfInitary) MathemaTical POwer of Pa and anothEr narrow mInD exaCTLy hAving the poWer of a HiLbertian fINite mINd AugmeNteD with tHe CapAcity To carrY Out TransFinite InDuctioN up to Ε0. IN future wOrk we will interpret TakeuTi's own PartiAl bUt remarkaBle SOluTions to hiS funDamental coNjeCtuRe by mEanS Of thiS reaDInG. It WOuld aLso
's word "the world of a fi nite mind"  or " the  wo rl d of  aninfinite mind" migh t be a bit strange if"worl d" mean s  a  univ erse an d  t h e reis  a  st ri c tsepar ati on betw een the wo rld  a nd a (finite or  infinite)  mi nd. In parti cul ar, th is  co n trast  be tween  "worl d " and"mind" wo ul d  be dr a stic if w econs ider that the wor l da s the universe  might  c o nt a i n i nfi nitary man ything s  includ i ng a l l m a thematical ob jects while our  minds  a ref initein th es ens e that ourabil ity to re member thingsi s limit ed. Ho wev er,  wem ay  r eso lv e  th i sdif f icu lty by r ea di ng Ta keut i ' s pass age  inthe c ontext of Nis hid a'sp hil osoph y, th ough  w e onl y sket ch ou rresolution forreas ons of sp ace .Were ly on a rece ntana lysis o f Nishi d a's  t h e o ry  of "self" 「自己」 du et o  D eguchi ( 2017). He  i n troduces  a n i mpor t a nt di stin c ti on betwe en a " n ar ro w" self  a nd a " br oad " s elf i n  Nis hida's  philoso phy.T o explain this distinction b r ie f l y, letusnote that a ccor d ingto N i sh ida ,  theworld  i s  t h e totality of all t he  entit ies,as well as th e creatoro f those en titi e s( including itse lf).Accordingt o Deguch i, anarrow s elf is cr e a ted by t hewor ldand i s"expressed" ( 「 表現され る」 ) or "r efl ected"( 「 映され る」)  by  t he world.  On theot he rha nd,  a br o ad selfis  ana rro w sel f  as aprodu cerin  t h e w orld fu r ni s h ed w it hitsmat er ials,  and thu s a nar row selfhav i ng s om einterac tion with oth er  entitiesin  th e worl d .  To make  sense of Takeuti's tal k  of wor lds  of m inds , we inte rpr et Tak eut i 's not ion of  "min d"  as a  narr o w  s elf  i n Nishida' s phi losop hy . In  this r eading, narrow sel v esand the world  ar e re l a te d b y  m e ans  o f  re f l ection. Moreove r, Takeuti 's re ference to sev er al mind s indic atest hat the re mightbe severa lnarr o w  se lves in th e world.  On thisr eadin g ,Takeu ti' s inve st iga tions  in th e  fo undat ions o fmathem atics  a re regar ded as attempts to inve stigat e the  re lationshi p b e twe en narrow  sel ves. For e xam ple , weint e rpret  Gen t ze n's consi sten c y proof o f  P A a s awork establ i s h ing  a re lat i on bet ween  the narrow selfh aving exactlythe1 1  Ta keu t i'spr oof theory inthe  c o n text ofth e Kyoto Sch ool (inf in i tary)  mathe matica l power o fP A andanot her  narrow m ind  e x actly h av in g  the p ower  o f a Hi lberti a n fi n i te mind augmente d wit h the c a pac ity t ocarry o u t tr ansfiniteinduction u p to ε 0. I n fut ure wor kwe wil l i nt erpret Tak e uti's own  part ial but  r emar kab le sol utio n s  to h is f un dam ental con j e ct u re  b y  me ansof th is  rea ding. Itw ould als o
's word_"the world_of a finite mind"_or "the_world_of an_infinite_mind" might be_a bit strange_if "world" means a_universe and there_is_a strict separation between the world and a (finite or infinite) mind. In particular,_this_contrast between_"world"_and_"mind" would be drastic if_we consider that the world_as the_universe might contain infinitary many things including all_mathematical_objects while our_minds are finite in the sense that our ability_to remember things is limited. However,_we may resolve_this_difficulty_by reading Takeuti's passage_in the context of Nishida's philosophy,_though we only sketch our resolution_for reasons of space. We rely on_a recent analysis of Nishida's theory_of "self" 「自己」 due to_Deguchi (2017)._He introduces an important distinction_between a "narrow"_self and_a "broad" self_in Nishida's philosophy. To explain this_distinction briefly, let_us note that according to Nishida,_the_world is the_totality_of_all the_entities, as well_as_the creator_of_those entities (including itself). According to_Deguchi,_a narrow self is created by the_world and is "expressed"_(_「表現される」) or "reflected" (_「映される」) by the world. On_the other hand, a broad self_is a_narrow self_as a producer in the world furnished with its materials, and_thus a narrow self having some_interaction with other entities_in the_world._To make sense_of_Takeuti's talk_of worlds of minds, we interpret Takeuti's_notion of_"mind" as a narrow self in_Nishida's philosophy. In this_reading,_narrow selves and the world are_related by means of reflection. Moreover,_Takeuti's reference to several minds_indicates_that_there might be several narrow_selves in the world. On this_reading, Takeuti's investigations_in the foundations of mathematics are regarded_as_attempts to investigate the relationship between_narrow_selves. For example, we interpret Gentzen's_consistency_proof_of PA as a work_establishing a relation between the narrow_self having exactly the 11 Takeuti's proof theory in_the context of_the Kyoto School (infinitary) mathematical_power_of_PA and another narrow mind exactly having the power of_a Hilbertian_finite mind augmented_with the capacity to carry out transfinite induction up to_ε0. In future work we will interpret_Takeuti's own partial but remarkable solutions to his fundamental conjecture by_means of this reading. It would also