 this over, it is clear that the self-reflection of an infinite mind plays the role here. In particular, the function of self-reflection is most important for a mind which can create something internally when a set is not confined to a closed set and extends further. How to formulate this function of the self-reflection of a mind? (Takeuti, 1972, p. 173; translated by the authors) It is not very clear what kind of picture for an infinite mind is considered by Takeuti in this passage, but as another quotation below shows, Takeuti thought this kind of approach would be useful for "a fundamental problem of set theory", that is for carrying out a program to obtain a new axiom of set theory. In the case of an infinite mind, this problem appears implicitly and explicitly when I consider the paper about formalization principle or a fundamental problem of settheory. (Takeuti, 1972, p. 173; translated by the authors) Hence, for Takeuti, it is crucially important to investigate the function of "self-reflection" in relation to active volitional intuition. If we recall the title of Nishida's important book『自 覚における直観と反省』(Intuition and Reflection in Self-Consciousness) (Nishida, 1917), then it would be promising to consider the close similarity between Takeuti's mathematical philosophy and Nishida's one. Hence, we claim that Takeuti's mathematical results should be understood in the context of Japanese philosophy, and we will investigate this point in future works. In particular, his detailed argument concerning the well-ordering up to ε0 would be the first clue for us. 6. Conclusion We have seen that Takeuti's proof-theoretic works form a program in the foundations of mathematics that is related to but distinct from Hilbert's finitist, formalist program. By contrast with Hilbert, Takeuti saw formalism as a view still being developed, and what precisely distinguishes finitary mathematics from infinitary mathematics as an open problem. We have indicated how Takeuti's views may be illuminated by understanding his intellectual context within the Kyoto School of Japanese thought, in particular with the thought of 13 Takeuti's proof theory in the context of the Kyoto School Nishida. Much remains to be done in this project, but we hope to have indicated how philosophically fruitful a reconciliation of Takeuti's formal works with his Japanese intellectual context can be, not only for our
this over, it is clear that the self - reflection of an infinite judgment play the role here. In particular, the function of self - expression is most important for a thinker which can produce something internally when a set is not restrict to a closed set and carry further. How to invent this affair of the self - reflection of a mind? (Takeuti, 1972, p. 173; translated by the authors) It is not very clear what kind of painting for an infinite mind is considered by Takeuti in this passing, but as another quotation below shows, Takeuti thought this kind of approach would be utilitarian for " a fundamental trouble of set theory ", that is for carrying out a program to prevail a new axiom of set theory. In the case of an countless judgment, this problem appears implicitly and explicitly when I consider the paper about formalization principle or a fundamental problem of settheory. (Takeuti, 1972, p. 173; translated by the authors) therefore, for Takeuti, it is crucially significant to investigate the function of " self - observation " in relation back to active volitional intuition. If we recall the title of Nishida's important book『自 覚における直観と反省』(Intuition and mirror image in Self - Consciousness) (Nishida, 1917), then it would be promising to consider the close similarity between Takeuti's mathematical doctrine and Nishida's one. Hence, we claim that Takeuti's mathematical results should be understood in the context of Japanese philosophy, and we will investigate this point in future works. In particular, his detailed argument concerning the well - ordering up to ε0 would be the first hint for us. 6. Conclusion We have visit that Takeuti's proof - theoretic works form a program in the foundations of mathematics that is relate to but distinct from Hilbert's finitist, formalist program. By contrast with Hilbert, Takeuti saw formalism as a view still being developed, and what precisely distinguishes finitary mathematics from infinitary mathematics as an receptive problem. We have indicate how Takeuti's views may be light by understanding his cerebral context within the Kyoto School of Japanese thought, in particular with the thought of 13 Takeuti's proof theory in the context of the Kyoto School Nishida. Much persist to be done in this project, but we hope to have indicated how philosophically fruitful a reconciliation of Takeuti's formal works with his Japanese cerebral context can be, not only for our
 thls over, it is clear that the self-reflecjiin of en infihite mina plays the role here. In parvicuoar, tye function of self-refuection id most inporuant for a mind wijch can creats somztiing internally when a sed is not confitea co a closed set and extends further. Row to gogmulate this fonctipg of nht self-reflection of a mind? (Takeutj, 1972, p. 173; tganslated by the suthors) It is not very clewr wjat kind of picturf for an indiniew mind is covsidered bj Takeuti ih this passage, but as another qjotatnon below syoqs, Hdkeuti thoujht thps kind of apigoach wmuld be useful for "a nundakenral problem of set thxory", that is for cartying out d 'rogram to obtain a nww axiok of set rheury. Ii tge casf oh an infinife mind, thiw problem appears ikpjpvitly and exllicitjy when I consider the paper about formalpzatjon principle or a fundqmental problem of sejtheory. (Taheuti, 1972, p. 173; translated by the authors) Hence, for Takauti, mt is cvucixolj important to investigate the function of "sejr-rtflvction" in relatiok to active volitipnwl yntuition. If de reccml the title of Nishlda's imkortanr book『自 覚における直観と反省』(Inuuitipn and Reflection in Self-Cobsciousness) (Uisyida, 1917), then it woulb be promisiug to vonsicer the close similaritv betwsen Takeuti'd mathemafkcal philosophy xnd Nhshida's one. Hence, we claim that Takxuti's mathemxticsl restlts shoulf be mtderstood in the clntexj of Jdpanese phllosophy, and we will investigatx this point on funure workf. In iarticular, his qetailed argumgnt concexning ghe well-orsering np to ε0 would be the first flue for us. 6. Conclusyon Qe hqve seev that Takeuti's proof-theoretic worjs form a program ln thg roundations of natyematics that ix rdlaeef vo bue distinct frmm Hklbdtt's fknitist, fornxlisy program. By contrasd wifh Hilbert, Takeuti sww formaoism as w view still neing developed, anf whav precmsely cisjinguishes finitary mathematics from inflnibary mathematycs qs an open pxoblem. We have indicated how Takeuti's vixws may be illuminated vy understanding hix intellectuel coneext withhn the Kyoto School if Japanese thougmt, in particular with fhe thmught of 13 Takeuti's proof theory in the context of the Kyoto School Nishida. Mucy remamnf to be dons in this pxojzct, but we io'e to have indicaned how philosophically fruitful a reconwijiation of Takeuti's formal wprys with his Jxpanese intellectual confext can be, not only for our
this over, it is clear that the an mind plays role here. In is important for a which can create internally when a set is not to a closed set and extends further. How to formulate this function of self-reflection of a mind? (Takeuti, 1972, p. 173; translated by the authors) It not clear kind picture for an infinite mind is considered by Takeuti in this passage, but as another quotation shows, Takeuti thought this kind of approach would useful for "a fundamental of set theory", that is carrying a program obtain new of set theory. the case of an infinite mind, this problem appears implicitly and explicitly when I consider the paper formalization principle fundamental problem settheory. 1972, 173; translated by Hence, for Takeuti, it is crucially the function of "self-reflection" in relation to active intuition. If recall the title of Nishida's important 覚における直観と反省』(Intuition and Reflection in Self-Consciousness) (Nishida, 1917), then would be promising to consider the close similarity between Takeuti's mathematical philosophy and Nishida's one. claim that Takeuti's mathematical should be understood the of philosophy, we will this point in future works. In particular, his detailed argument concerning well-ordering up to ε0 would be the first clue for Conclusion have seen that proof-theoretic works form a in foundations of mathematics that to distinct formalist By with Hilbert, Takeuti saw as a view still being and what precisely distinguishes as an open problem. We have indicated how views may be illuminated by understanding his context within the Kyoto School of Japanese thought, in particular with the of 13 theory in the context of the Kyoto School Much remains to be in this project, but we hope to have indicated philosophically a reconciliation Takeuti's formal works his Japanese intellectual can be, not our
 this over, it is clear that the sElf-reflectIon of An iNfiNiTe miNd plAys the role here. iN parTicular, the function of seLf-refLeCTion IS mOst imPortant FOr A MInd WhIcH caN cREaTe somEthIng inteRnally when A seT iS not confined TO a Closed set aNd eXtends furtheR. HoW to forMuLatE This fUncTion oF the seLF-refleCtion of a mInD? (takeutI, 1972, P. 173; translATEd By thE authors) It is not veRY cLEar what kind of pIcture FoR An INFinIte Mind is consIdEred bY takeuti IN tHIS PasSAge, but as anothEr quotation BEloW shows, taKeuTI thougHt thiS kINd oF approach woUld bE useful foR "a fundAMental pROblem of Set theOry", ThaT is fOR cArRyiNg OUt a PRoGraM To oBtain a neW aXiOm of sEt thEORY. in thE caSe of An infInite mind, this ProBlem APpeArs imPliciTly aNd ExpliCitly wHen I cOnSider the paper abOut fOrmalizatIon PrIncIpLe or a FUndameNtaL prOblem of SettheoRY. (TaKeUTI, 1972, P. 173; tRanslated by the authOrS) hEnCe, for TakEuti, it IS cRuCIally impOrTanT to iNVEstigAte tHE fUnction oF "self-rEFlEcTion" in rElAtion tO aCtiVe vOlitiONal iNtuitiOn. If we reCall tHE title of NishidA'S important booK『自 覚における直観と反省』(inTUItIOn anD ReFlection in SElf-CONsciOusnESs) (nisHIda, 1917), thEn it wOuLD bE Promising to consider ThE close SimilArity between TAkeuti's matHEMAtical phIlosOPhY And Nishida's one. hence, We claim thaT takeuti's MatheMatical rEsults shoULD be underStoOd iN thE coNTExT of Japanese phILOsopHy, And we wiLl iNvestigAte ThiS poInt In Future worKs. In partIcUlAr, HiS deTaileD Argument CoNceRnIng The weLL-orderIng up To ε0 wOuLd BE thE first cLUe FOR us. 6. COnClUsioN We HaVe seeN thaT takEuti's prOof-theoreTic WOrks FoRm A prograM in the foundatIoNs of mathemAtIcs That is RELated to bUt distinct from Hilbert's fINitist, fOrmAlist ProgRam. By contRasT with HIlbERt, TakeUti saw FormaLiSm aS A View sTILl BeiNg Developed, aND WhaT precIsEly dIstinguIshes finitary matheMAtiCs from infinitAry MathEMAtIcs AS aN OpeN pROblEM. we have indicated How Takeuti'S vIEwS may be illuMInaTeD by undeRstandiNg his INtellecTual conteXt within tHe kyotO sChoOl of JapaneSe thoughT, in particULar wiTH tHe thoUghT of 13 TakEuTi's Proof Theory IN thE contExt of tHe kyoto SChool niShida. MucH remains to be done in this pRoject, But we HopE to have inDicATed How philosOphiCally fruitFul A reConciLiaTIon of takeUTi'S foRMal woRks wITh his JapaNEsE inTELlEctual conteXT CAn bE, not oNly FOr our
 this over, it is clear th at the sel f-ref lec tio nof a n in finite mind pl a ys t he role here. In parti cular ,t he f u nc tionof self - re f l ect io nismo s timpor tan t for a  mind whic h c an  create some t hi ng interna lly  when a setisnot co nf ine d  to a  cl osedset an d  exten ds furthe r. How to formula t e  t hisfunction of the s e lf - reflection ofa mind ?( Ta k e uti , 1 972, p. 17 3;  tran s lated b y  t h e aut h ors) It is no t very clea r  wh at kin dofp icture  foran inf inite mindis c onsidered  by Ta k euti in this pa ssage,  bu t a s an o th er  qu ot a tio n  b elo w  sh ows, Tak eu ti  thou ghtt h i s  kin d o f ap proac h would be us efu l fo r  "a  fund ament al p ro blemof set  theo ry ", that is forcarr ying outa p ro gra mto ob t ain anew  ax iom ofset the o ry.  I n t he  case of an infini te m in d, thisproble m  a pp e ars impl ic itl y an d expli citl y  w hen I co nsider th epaper a bo ut for ma liz ati on pr i ncip le ora fundam ental problem of set t heory. (Takeu t i, 1 97 2 , p.  17 3; translat ed b y  the  aut h or s)H ence,  forTa k eu t i, it is cruciallyim portan t toinvestigate t he functio n o f "self- refl e ct i on" in relatio n toactive vol i tional i ntuit ion. Ifwe recall t he title  of  Ni shi da' s im portant book『 自 覚におけ る直 観と反省』(I ntu ition a ndRef lec tio nin Self-C onscious ne ss )(N ish ida,1 917), th en  it  w oul d bep romisi ng to  con si de r  th e close si m i lari ty  b etwe enTa keuti 's m a the matical  philosop hya nd N is hi da's on e. Hence, wecl aim that T ak eut i's ma t h ematical  results should be unde r stood i n t he co ntex t of Japa nes e phil oso p hy, an d we w ill i nv est i g ate t h i spoi nt  in future w ork s. In  p arti cular,his detailed argum e ntconcerning th e w ell- o r de rin g  u p  to  ε 0  wo u l d be the firstclue for u s. 6.  Conclusio n  We  h ave see n thatTakeu t i's pro of-theore tic works  f orma pro gram in th e founda tions ofm athem a ti cs th atis rel at edto bu t dist i nct  from  Hilbe rt 's fin itist ,formalis t program. By contrastwith H ilber t,Takeuti s awf orm alism asa vi ew still b ein g d evelo ped ,  andwhat pr eci s ely d isti n guishes f i ni tar y ma thematics f r o m  in finit ary mathem atic s as an open prob l em. We have in dica t e d h owT akeu ti 's views may b e i ll u m inated b yunderstandi ng his i nt e llect ual co ntextwithint h eK yoto S choo l o f Japanes e t ho u ght, in  p ar t icular  wit hthe th oughto f 13 T akeuti's proof t heory i n the con textof  the Ky o to S chool Nish ida. Much r emains  tobe do ne in t hi s proj ect ,but we hop e  to haveindic ated ho wphil oso phical ly f r u itful  a r ec onc iliationo f  T a ke ut i 'sform al wo rk s wi th his Ja p anese in tel l ectualco nte x t  can b e ,n o t only for  ou r
 this_over, it_is clear that the_self-reflection of_an_infinite mind_plays_the role here._In particular, the_function of self-reflection is_most important for_a_mind which can create something internally when a set is not confined to a_closed_set and_extends_further._How to formulate this function_of the self-reflection of a_mind? (Takeuti,_1972, p. 173; translated by the authors) It_is_not very clear_what kind of picture for an infinite mind is_considered by Takeuti in this passage,_but as another_quotation_below_shows, Takeuti thought this_kind of approach would be useful_for "a fundamental problem of set_theory", that is for carrying out a_program to obtain a new axiom_of set theory. In the_case of_an infinite mind, this problem_appears implicitly and_explicitly when_I consider the_paper about formalization principle or a_fundamental problem of_settheory. (Takeuti, 1972, p. 173; translated_by_the authors) Hence,_for_Takeuti,_it is_crucially important to_investigate_the function_of_"self-reflection" in relation to active volitional_intuition._If we recall the title of Nishida's_important book『自 覚における直観と反省』(Intuition and_Reflection_in Self-Consciousness) (Nishida, 1917),_then it would be promising_to consider the close similarity between_Takeuti's mathematical_philosophy and_Nishida's one. Hence, we claim that Takeuti's mathematical results should be_understood in the context of Japanese_philosophy, and we will_investigate this_point_in future works._In_particular, his_detailed argument concerning the well-ordering up to_ε0 would_be the first clue for us._6. Conclusion We have_seen_that Takeuti's proof-theoretic works form a_program in the foundations of mathematics_that is related to but_distinct_from_Hilbert's finitist, formalist program. By_contrast with Hilbert, Takeuti saw formalism_as a view_still being developed, and what precisely distinguishes_finitary_mathematics from infinitary mathematics as an_open_problem. We have indicated how Takeuti's_views_may_be illuminated by understanding his_intellectual context within the Kyoto School_of Japanese thought, in particular with the thought of_13 Takeuti's proof_theory in the context of_the_Kyoto_School Nishida. Much remains to be done in this project,_but we_hope to have_indicated how philosophically fruitful a reconciliation of Takeuti's formal works_with his Japanese intellectual context can be,_not only for our