. Thus Takeuti's remark makes more sense in the context of Nishida's thought. Furthermore, we can make further sense of Takeuti's broad evaluative position of the relationship between infinitary mathematics and finitary mathematics, discussed in the previous section as "rationality"「合理性」, in which set theory and proof theory are interconnected to one another in an essential way. While set theory is often thought of as an investigation of the pure object that is a set, and proof theory as an investigation of the demonstrative powers of a subject, in neglecting the separation between object and subject Nishida's standpoint provides a way to interconnect these two investigations. Thus we can make better sense of Takeuti's understanding of formalism when seeing it within the framework of Nishida's work. We next make a historical remark about Takeuti's use of "active volitional intuition"「行 為的意志的直観」(5). As we have seen, this goes back to the quotation from "From the viewpoint of formalism II" in 1956 (Takeuti, 1956b). Takeuti attributed his use of the word "active volitional intuition" to Joichi Suetsuna(末綱恕一)(1898–1970), who was a mathematician with a strong interest in the foundations of mathematics and Buddhist philosophy (Kegon「華厳」) (6). Suetsuna had close contacts with Nishida and with the writer Daisetsu T. Suzuki(鈴木大拙)(1870–1966) about Zen in Japan (7). He already mentions Nishida's keywords like "active intuition" in the book 『論理と数理』 (Logic and Mathematics) 10 Takeuti's proof theory in the context of the Kyoto School published in 1947 (Suetsuna, 1947). It is much beyond the scope of this paper to describe Nishida's philosophy or the tradition of Japanese philosophy based on Zen, but we indicate the connection of Takeuti's thinking with Nishida's philosophy below. In particular, we will discuss two notions that we saw in Takeuti's writings in the last section, that of the "world" of finite and infinite minds, and that of "self-reflection", both of which play critical roles in Takeuti's views of the foundations of mathematics. Firstly, we discuss the notion of "world"「世界」. Takeuti
. Thus Takeuti's remark makes more sense in the context of Nishida's thought. Furthermore, we can seduce further common sense of Takeuti's broad evaluative placement of the kinship between infinitary mathematics and finitary mathematics, discussed in the previous section as " rationality"「合理性 」, in which arrange theory and proof theory are complect to one another in an essential way. While hardening theory is often intend of as an investigation of the pure object that is a set, and proof theory as an probe of the demonstrative powers of a subject, in neglecting the legal separation between object and subject Nishida's point of view provides a means to interconnect these two investigation. Thus we can make better common sense of Takeuti's understanding of formalism when seeing it within the framework of Nishida's work. We next make a historical remark about Takeuti's habit of " active volitional intuition"「行 為的意志的直観」(5). As we have seen, this goes back to the quotation from " From the viewpoint of formalism II " in 1956 (Takeuti, 1956b). Takeuti attributed his use of the word " active volitional intuition " to Joichi Suetsuna(末綱恕一)(1898–1970), who was a mathematician with a strong interest in the foundations of mathematics and Buddhist doctrine (Kegon「華厳 」) (6). Suetsuna have close contact with Nishida and with the writer Daisetsu T. Suzuki(鈴木大拙)(1870–1966) about Zen in Japan (7). He already mention Nishida's keywords like " active intuition " in the book 『 論理と数理 』 (Logic and Mathematics) 10 Takeuti's proof theory in the context of the Kyoto School published in 1947 (Suetsuna, 1947). It is much beyond the oscilloscope of this paper to describe Nishida's philosophy or the tradition of Japanese doctrine based on Zen, but we indicate the connection of Takeuti's thinking with Nishida's philosophy below. In particular, we will discuss two notions that we saw in Takeuti's writings in the last section, that of the " populace " of finite and infinite mind, and that of " self - reflection ", both of which play critical roles in Takeuti's views of the foundations of mathematics. first, we discuss the notion of " world"「世界 」. Takeuti
. Thks Takeuti's remark makes more sense in jhw contxxt of Hishida's thought. Furthermore, we can laje fuether sense of Takeuti'r broad enaluative posmtion of the relefionshii betscen iufmnitary mathemajics and finhtary mathemathcr, biscussed in the previous section as "rationsllty"「合理性」, in which sgt thtorr ans proof theory are interconnected fo one enother in an exsential way. While set thelry ls often thought ov as an invgatidqtion of the pure objebc that is a set, and proof theory as an invdstigction of thg selmnstrative 'owers of a subjecb, in neclectinb the separatipn uetwwen object and subjecv Nishida's standpoinj provides a way to interconnwcr theve tfo ivcesgigztmona. Thus we can make getter sensw of Takeuti's underxtwbding of formzlism rhqn seeing it within the framework of Nivhisa's work. We next make a historical remark ablut Takeuei's use of "active volitional intuition"「行 為的意志的直観」(5). As we hava seei, ghiw noes vafk to the quotation from "From the viewpoint os fprkalism II" in 1956 (Bakeuti, 1956b). Takeuti sthrofuted his use of thz wkrd "active volitiojal intoition" to Joichy Surtsuna(末綱恕一)(1898–1970), who was a mathematicuan with a snront interest in the youndations uf msthemstics and Buddhist philusopgy (Kegon「華厳」) (6). Sketsuna hza close contacts winh Nhshida and with the writer Daisetsu T. Sbzuki(鈴木大拙)(1870–1966) abuut Een in Tapan (7). He wlready mentions Nishida'd keycords like "actige intuition" in the book 『論理と数理』 (Logic and Mathemativs) 10 Tdkeuti's 'roof bheory in the cjntext of the Lyoto Sehool oublished pn 1947 (Suetsnna, 1947). It is mtch beyond tha scope of thms paper eo dwscrube Nisfkda's philosophu or the nrcdition od Japanese philosoihy bxaed on Zen, but ct ibdicate the conmecgiog lf Twneuti's thinkhng ditf Nishkda's philosiohy nelow. In particular, fe wjll discuss two noyijns that we saw yn Takeuti's wtitings in the lasu sectmon, thet of yhe "world" of finite and infinite jinds, and thwt of "self-reslecbion", both of wkich play critical roles in Takeuti's viexs of the foundations od mathematics. Firstkn, we discuss the gotion of "world"「世界」. Takeuti
. Thus Takeuti's remark makes more sense context Nishida's thought. we can make evaluative of the relationship infinitary mathematics and mathematics, discussed in the previous section "rationality"「合理性」, in which set theory and proof theory are interconnected to one another an essential way. While set theory is often thought of as an investigation the object is set, and proof theory as an investigation of the demonstrative powers of a subject, in neglecting separation between object and subject Nishida's standpoint provides way to interconnect these investigations. Thus we can make sense Takeuti's understanding formalism seeing within the framework Nishida's work. We next make a historical remark about Takeuti's use of "active volitional intuition"「行 為的意志的直観」(5). As have seen, back to quotation "From viewpoint of formalism 1956 (Takeuti, 1956b). Takeuti attributed his word "active volitional intuition" to Joichi Suetsuna(末綱恕一)(1898–1970), who a mathematician a strong interest in the foundations mathematics and Buddhist philosophy (Kegon「華厳」) (6). Suetsuna had contacts with Nishida and with the writer Daisetsu T. Suzuki(鈴木大拙)(1870–1966) about Zen in Japan (7). mentions Nishida's keywords like intuition" in the 『論理と数理』 and 10 proof theory the context of the Kyoto School published in 1947 (Suetsuna, 1947). is much beyond the scope of this paper to describe or tradition of Japanese based on Zen, but indicate connection of Takeuti's thinking philosophy In discuss notions we saw in Takeuti's in the last section, that the "world" of finite of "self-reflection", both of which play critical roles Takeuti's views of the foundations of mathematics. we discuss the notion of "world"「世界」. Takeuti
. Thus Takeuti's remark makes moRe sense in tHe conTexT of niShidA's thOught. FurthermoRE, we cAn make further sense of TaKeuti'S bROad eVAlUativE positiON oF THe rElAtIonShIP bEtweeN inFinitarY mathematiCs aNd Finitary mathEMaTics, discusSed In the previouS seCtion aS "rAtiONalitY"「合理性」, in Which Set theORy and pRoof theorY aRE interCOnnecteD TO oNe anOther in an essentiaL WaY. while set theory Is ofteN tHOuGHT of As aN investigaTiOn of tHE pure obJEcT THAt iS A set, and proof tHeory as an inVEstIgatioN oF thE DemonsTratiVe POweRs of a subjecT, in nEglecting The sepARation bETween obJect anD suBjeCt NiSHiDa'S stAnDPoiNT pRovIDes A way to inTeRcOnnecT theSE TWO invEstIgatIons. THus we can make bEttEr seNSe oF TakeUti's uNderStAndinG of forMalisM wHen seeing it withIn thE frameworK of niShiDa'S work. wE next mAke A hiStoricaL remark ABouT TAKEUtI's use of "active volitIoNAL iNtuition"「行 為的意志的直観」(5). as we haVE sEeN, This goes BaCk tO the QUOtatiOn frOM "FRom the viEwpoinT Of FoRmalism iI" In 1956 (TakeUtI, 1956b). TAkeUti atTRibuTed his Use of the Word "aCTive volitional INtuition" to JoiCHi sUEtSUna(末綱恕一)(1898–1970), wHo wAs a mathematIciaN With A strONg IntERest iN the fOuNDaTIons of mathematics anD BUddhisT philOsophy (Kegon「華厳」) (6). SuEtsuna had cLOSE contactS witH niSHida and with the WriteR Daisetsu T. sUzuki(鈴木大拙)(1870–1966) aboUt Zen In Japan (7). HE already mENTions NisHidA's kEywOrdS LIkE "active intuitION" in tHe Book 『論理と数理』 (LogIc aNd MatheMatIcs) 10 takEutI's Proof theoRy in the cOnTeXt Of The kyoto sChool pubLiSheD iN 1947 (SuEtsunA, 1947). it is muCh beyOnd tHe ScOPe oF this paPEr TO DescRiBe nishIda'S pHilosOphy OR thE traditIon of JapaNesE PhilOsOpHy based On Zen, but we indIcAte the connEcTioN of TakEUTi's thinkIng with Nishida's philosopHY below. IN paRticuLar, wE will discUss Two notIonS That we Saw in TAkeutI's WriTINgs in THE lAst SeCtion, that oF THe "wOrld" oF fInitE and infInite minds, and that oF "SelF-reflection", boTh oF whiCH PlAy cRItICal RoLEs iN tAkeuti's views of tHe foundatiOnS Of MathematicS. firStLy, we disCuss the NotioN Of "world"「世界」. takeuti
. Thus Takeuti's remark ma kes more s enseinthe  c onte xt o f Nishida's th o ught . Furthermore, we canmakefu r ther se nse o f Takeu t i' s bro ad  e val ua t iv e pos iti on of t he relatio nsh ip  between inf i ni tary mathe mat ics and fini tar y math em ati c s, di scu ssedin the previo us sectio na s "rat i onality " 「 合理 性」,in which set theo r ya nd proof theor y arein t er c o nne cte d to one a no theri n an es s en t i a l w a y. While settheory is o f ten  thoug ht  of as aninves ti g ati on of the p ureobject th at isa  set, a n d proof  theor y a s a n in v es ti gat io n  of th e d e mon strative  p ow ers o f as u b j ect,  in  neg lecti ng the separa tio n be t wee n obj ect a nd s ub jectNishid a's s ta ndpoint provide s away to in ter co nne ct  thes e  two i nve sti gations . Thusw e c an m a ke  better sense of T ak e u ti 's under standi n gof formalis mwhe n se e i ng it  wit h in  the fra mework of  N ishida' swork.We  ne xtmakea  his torica l remark  abou t  Takeuti's use of "active vo l it i o na l  int uit ion"「行 為的意志 的直観」 ( 5).As w e  h ave seen,  this  g o es back to the quotati on  from"From  the viewpoin t of forma l i s m II" in  195 6  ( T akeuti, 1956b) . Tak euti attri b uted his  useof the w ord "acti v e  volitio nal  in tui tio n "  t o Joichi Suet s u na(末 綱恕 一)(1898 –19 70), wh o w asa m ath em atician w ith a st ro ng  i nt ere st in the foun da tio ns  of  math e matics  andBudd hi st phi losophy (K e g on「華 厳」 )(6).  Su et sunahadc los e conta cts withNis h idaan dwith th e writer Dais et su T. Suzu ki (鈴木 大拙)(18 7 0 –1966) a bout Zen in Japan (7).H e alrea dymenti onsNishida's  ke ywords  li k e "act ive in tuiti on " i n the b o o k『論理 と数 理』 (Logica n d M athem at ics)  10 Tak euti's proof theor y  in  the contextoftheK y ot o S c ho o l p ub l ish e d  in 1947 (Suets una, 1947) .I tis much be y ond  t he scop e of th is pa p er to d escribe N ishida'sph ilos o p hyor the tra dition o f Japanes e  phil o so phy b ase d on Z en , b ut we  indic a tethe c onnect io n of T akeut i' s thinki ng with Nishida's philo sophybelow . I n particu lar ,  we  will dis cuss  two notio nstha t wesaw in Ta keut i 's  wr i tings  int he last s e ct ion , th at of the " w o r ld"  of f ini t e andinfi nite minds, and t h at of "self-re flec t i on" , b o th o fwhich play cri tic al r oles inTa keuti's vie ws of th ef ounda tionsof mat hematic s .  F i rstly,  wedis cuss thenot io n  of "wo rl d" 「 世界」. T akeu ti 
. Thus_Takeuti's remark_makes more sense in_the context_of_Nishida's thought._Furthermore,_we can make_further sense of_Takeuti's broad evaluative position_of the relationship_between_infinitary mathematics and finitary mathematics, discussed in the previous section as "rationality"「合理性」, in which_set_theory and_proof_theory_are interconnected to one another_in an essential way. While_set theory_is often thought of as an investigation of_the_pure object that_is a set, and proof theory as an investigation_of the demonstrative powers of a_subject, in neglecting_the_separation_between object and subject_Nishida's standpoint provides a way to_interconnect these two investigations. Thus we_can make better sense of Takeuti's understanding_of formalism when seeing it within_the framework of Nishida's work._We next_make a historical remark about_Takeuti's use of_"active volitional_intuition"「行 為的意志的直観」(5). As_we have seen, this goes back_to the quotation_from "From the viewpoint of formalism_II"_in 1956 (Takeuti,_1956b)._Takeuti_attributed his_use of the_word_"active volitional_intuition"_to Joichi Suetsuna(末綱恕一)(1898–1970), who was a_mathematician_with a strong interest in the foundations_of mathematics and Buddhist_philosophy_(Kegon「華厳」) (6). Suetsuna had_close contacts with Nishida and_with the writer Daisetsu T. Suzuki(鈴木大拙)(1870–1966)_about Zen_in Japan_(7). He already mentions Nishida's keywords like "active intuition" in the_book 『論理と数理』 (Logic and Mathematics) 10_Takeuti's proof theory in_the context_of_the Kyoto School_published_in 1947_(Suetsuna, 1947). It is much beyond the_scope of_this paper to describe Nishida's philosophy_or the tradition of_Japanese_philosophy based on Zen, but we_indicate the connection of Takeuti's thinking_with Nishida's philosophy below. In_particular,_we_will discuss two notions that_we saw in Takeuti's writings in_the last section,_that of the "world" of finite and_infinite_minds, and that of "self-reflection", both_of_which play critical roles in Takeuti's_views_of_the foundations of mathematics. Firstly,_we discuss the notion of "world"「世界」._Takeuti