 centered at Kyoto University called the "Kyoto School" 「京都学派」. As we explain below, he uses some keywords due to Kitaro Nishida(西田幾 多郎) for explaining his philosophical ideas about the foundations of mathematics. To understand this point, we focus on another passage in the paper in 1956. In the above example, I have frequently used the word "illusion", but indeed I hope the readers to replace it by "intuition". In this sense, I redcall the intuition for grasping a function or a set in general, the active volitional intuition. Indeed, when we try to grasp a function or a set, this active volition works. And, what would be thought by this active volition, the object of this active volition, or the volition itself would be called a function or a set. (Takeuti, 1956b, p. 299) This passage occurs just after the previous one and it looks very abrupt. Indeed, this kind of abruptness is a feature of Takeuti's writings about philosophical ideas. Here, we would like to understand his philosophical background in the context of Japanese philosophy. We can then better understand his peculiar conception of formalism. 9 Takeuti's proof theory in the context of the Kyoto School One point to be noted here is that he uses the word "intuition"「直観」 or "volition"「意 志」for explaining the power of symbols in mathematics. At first sight, this word suggests Hilbert's use of the word "intuition" in characterizing his finitist standpoint. However, if we read the last sentence carefully, it turns out that the intuition or volition itself is a mathematical object, e.g., a function. This is one feature of Japanese philosophy which is articulated by Nishida. In particular, the word "active intuition"「行為的直観」is a keyword in Nishida's philosophy (3). In Nishida's view, active intuition comprises the fundamental mode of experience of the world that does not suppose a separation between object and subject (4). In the context of mathematics, this can be understood as making no separation between the act by a subject of constructing an object such as a set, and the object so constructed itself. Thus both this act and the object of this act can be identified as an active intuition, as Takeuti says in the passage in question
centered at Kyoto University called the " Kyoto School " 「 京都学派 」. As we explain below, he use some keywords ascribable to Kitaro Nishida(西田幾 多郎) for explaining his philosophical idea about the foundation of mathematics. To understand this point, we concenter on another passing in the paper in 1956. In the above model, I have frequently used the word " magic trick ", but indeed I hope the readers to replace it by " intuition ". In this common sense, I redcall the intuition for grasping a function or a set in general, the active volitional intuition. Indeed, when we try to grasp a function or a stage set, this active volition works. And, what would be thought by this active volition, the object of this active volition, or the volition itself would be call a function or a set. (Takeuti, 1956b, p. 299) This passage occurs merely after the previous one and it looks very abrupt. Indeed, this kind of abruptness is a feature of Takeuti's writings about philosophical mind. Here, we would like to understand his philosophical background in the context of Japanese philosophy. We can then well understand his peculiar conception of formalism. 9 Takeuti's proof theory in the context of the Kyoto School One point to be noted here is that he uses the word " intuition"「直観 」 or " volition"「意 志」for explaining the power of symbol in mathematics. At first sight, this parole suggest Hilbert's use of the word " intuition " in characterizing his finitist standpoint. However, if we read the last sentence cautiously, it turns out that the intuition or volition itself is a mathematical object, e.g., a function. This is one feature of Japanese philosophy which is articulated by Nishida. In particular, the word " active intuition"「行為的直観」is a keyword in Nishida's philosophy (3). In Nishida's view, active intuition constitute the fundamental manner of experience of the world that does not suppose a separation between object and subject (4). In the context of mathematics, this can be sympathize as making no separation between the act by a subject of constructing an object such as a set, and the object so construct itself. Thus both this act and the aim of this act can be identified as an active intuition, as Takeuti says in the passage in question
 cejtered at Kyoto Universiuy called the "Kyojo Schoon" 「京都学派」. As se explakn below, he uses some keyworvs dye to Kitaro Nishida(西田幾 多郎) for ebplaining his phioosokhical ideas abouv the fomudatikks of nathematics. To understang this point, wa woeus on another passage in the paper yn 1956. In yhf above examplg, I hsde fdvqmently used the word "illusion", buf indeev I hope the resders to replace it by "intkitiln". In this sense, I redcall thg ineyition for gfasping a function or z set in general, the active volktioncl intuitiob. Undfgd, when we tcy to drasp a funcbpon or d set, tnis active volltion woeks. And, what would be thought by this astive volhtnon, the object of thiw qctivg volhtiov, or tht vpljtion ltsxlf would bs called a dunction or a set. (Tskqlyi, 1956b, p. 299) This passade occurs just after the previous one and it looks very abrupt. Indewd, this kind of abrupjness is a feature of Takeuti's writings about philosophical ideaa. Hexc, dw aould like to understand his philosophical basigtolnd in the contexb of Japanese philpslpni. We can then cetter unserstand his pecullar conseptiin of forialixm. 9 Takeuti's proof theory ib the contexn of the Kyoto School Lne point tu be notec here is that he uses che wodd "intuitioj"「直観」 or "volifkon"「意 志」for explainivg nhe [ower of symbols in mathemwtics. At hirst sight, ghis word fuggests Hllberb'v use of the word "lntuijion" it charactegizing his finitist standpoint. Ipwever, if we raad the lasc sentcnce carefully, yt turns out tkat the nntuitkon or volption itsxlf is a matrematical objaft, e.g., a funcvion. This is ine deature uf Japanese pholosophy ckich is aeticulated by Nishlda. Iv particular, the wote "active intuiyiov"「行為的直観」is a kxyworq in Nishida's phiuosulhy (3). Kn Nishida's xiew, active intuition cokpriaes the fundamentak iode of gxperiencq of the worlc that does not sukpose e sepacation bejween object and subject (4). In ths context of mathematics, thiw can be undzrstood as making no separation between vhe act by a subject of constructing an oblegt such as a set, wnd the otject so constructed itself. Thus both this act and the objsct of this act can be identified as an active intuition, as Takeuti says in the passqge in question
centered at Kyoto University called the "Kyoto As explain below, uses some keywords for his philosophical ideas the foundations of To understand this point, we focus another passage in the paper in 1956. In the above example, I have used the word "illusion", but indeed I hope the readers to replace it "intuition". this I the intuition for grasping a function or a set in general, the active volitional intuition. Indeed, we try to grasp a function or a this active volition works. what would be thought by active the object this volition, the volition itself be called a function or a set. (Takeuti, 1956b, p. 299) This passage occurs just after the one and very abrupt. this of is a feature writings about philosophical ideas. Here, we understand his philosophical background in the context of philosophy. We then better understand his peculiar conception formalism. 9 Takeuti's proof theory in the context the Kyoto School One point to be noted here is that he uses the word "volition"「意 志」for explaining the of symbols in At sight, word Hilbert's use the word "intuition" in characterizing his finitist standpoint. However, if we the last sentence carefully, it turns out that the intuition itself a mathematical object, a function. This is feature Japanese philosophy which is Nishida. particular, intuition"「行為的直観」is keyword Nishida's philosophy (3). In view, active intuition comprises the mode of experience of suppose a separation between object and subject (4). the context of mathematics, this can be as making no separation between the act by a subject of constructing object such set, and the object so constructed itself. Thus this act and the of this act can be identified as an active as says in passage in question
 centered at Kyoto University Called the "KYoto SChoOl" 「京都学派」. AS wE expLain Below, he uses somE KeywOrds due to Kitaro Nishida(西田幾 多郎) For exPlAIninG HiS philOsophicAL iDEAs aBoUt The FoUNdAtionS of MathemaTics. To undeRstAnD this point, we FOcUs on anotheR paSsage in the paPer In 1956. In thE aBovE ExampLe, I Have fRequenTLy used The word "ilLuSIon", but INdeed I hOPE tHe reAders to replace it bY "InTUition". In this seNse, I reDcALl THE inTuiTion for graSpIng a fUNction oR A sET IN geNEral, the active Volitional iNTuiTion. InDeEd, wHEn we trY to grAsP A fuNction or a seT, thiS active voLition WOrks. And, WHat woulD be thoUghT by This ACtIvE voLiTIon, THe ObjECt oF this actIvE vOlitiOn, or THE VOlitIon ItseLf wouLd be called a fuNctIon oR A seT. (TakeUti, 1956b, p. 299) this PaSsage Occurs Just aFtEr the previous onE and It looks veRy aBrUpt. inDeed, tHIs kind Of aBruPtness iS a featuRE of taKEUTi'S writings about philOsOPHiCal ideas. here, we WOuLd LIke to undErStaNd hiS PHilosOphiCAl BackgrouNd in thE CoNtExt of JaPaNese phIlOsoPhy. we can THen bEtter uNderstanD his pECuliar conceptiON of formalism. 9 TAKeUTI's PRoof TheOry in the conText OF the kyotO scHooL one poInt to Be NOtED here is that he uses thE wOrd "intUitioN"「直観」 or "volition"「意 志」foR explaininG THE power of SymbOLs IN mathematics. At First Sight, this wORd suggesTs HilBert's use Of the word "INTuition" iN chAraCteRizING hIs finitist staNDPoinT. HOwever, iF we Read the LasT seNteNce CaRefully, it Turns out ThAt ThE iNtuItion OR volitioN iTseLf Is a MatheMAtical ObjecT, e.g., a FuNcTIon. this is oNE fEATure Of jaPaneSe pHiLosopHy whICh iS articuLated by NiShiDA. In pArTiCular, thE word "active inTuItion"「行為的直観」is a keYwOrd In NishIDA's philosOphy (3). In Nishida's view, activE IntuitiOn cOmpriSes tHe fundameNtaL mode oF exPEriencE of the World ThAt dOES not sUPPoSe a SeParation beTWEen ObjecT aNd suBject (4). In The context of mathemATicS, this can be undErsTood AS MaKinG No SEpaRaTIon BETween the act by a sUbject of coNsTRuCting an objECt sUcH as a set, And the oBject SO constrUcted itseLf. Thus botH tHis aCT And The object oF this act Can be idenTIfied AS aN actiVe iNtuitiOn, As TAkeutI says iN The PassaGe in quEsTion
 centered at Kyoto Univers ity called  the"Ky oto  S choo l" 「 京都学派」. As we e x plai n below, he uses somekeywo rd s  due to  Kita ro Nish i da ( 西 田幾多郎 )for  e x pl ainin g h is phil osophicalide as  about the f o un dations of  ma thematics. T o u nderst an d t h is po int , wefocuso n anot her passa ge in the paper i n 19 56.In the above exam p le ,  I have freque ntly u se d  t h e  wo rd"illusion" ,but i n deed Ih op e t her eaders to rep lace it by" int uition ".  In this s ense,  I red call the in tuit ion for g raspin g  a func t ion ora setingen eral ,  t he  ac ti v e v o li tio n alintuitio n.  I ndeed , wh e n w e tr y t o gr asp a  function ora s et,t his  acti ve vo liti on  work s. And , wha twould be though t by  this act ive  v oli ti on, t h e obje ctofthis ac tive vo l iti on , o rthe volition itsel fw o ul d be cal led af un ct i on or ase t.(Tak e u ti, 1 956b ,  p . 299) T his pa s sa ge  occurs  j ust af te r t heprevi o us o ne and  it look s ver y  abrupt. Indee d , this kind o f  a b r up t ness  is  a featureof T a keut i'sw ri tin g s abo ut ph il o so p hical ideas. Here,we  would  like  to understan d his phil o s o phical b ackg r ou n d in the conte xt of  Japanesep hilosoph y. We  can the n betteru n derstand  hi s p ecu lia r co nception of f o r mali sm . 9 Tak eut i's pro ofthe ory  in  t he contex t of the  K yo to  S cho ol On e  point t obeno ted  here is tha t heuses  t he wor d "intu i ti o n "「直観 」or  "vo lit io n"「意志」fo r  ex plainin g the pow ero f sy mb ol s in ma thematics. At  f irst sight ,thi s word s uggestsHilbert's use of the wo r d "intu iti on" i n ch aracteriz ing  his f ini t ist st andpoi nt. H ow eve r ,  if w e re adth e last sen t e nce  care fu lly,  it tur ns out that the in t uit ion or voliti onitse l f  i s a ma t hem at i cal o bject, e.g., afunction.Th i sis one fea t ure  o f Japan ese phi losop h y which  is artic ulated by  N ishi d a . I n particul ar, theword "act i ve in t ui tion" 「行為 的直観」is  a  ke yword  in Ni s hid a's p hiloso ph y (3).  In N is hida's v iew, active intuition c ompris es th e f undamenta l m o deof experi ence  of the wo rld  th at do esn ot su ppos e  a  se p arati on b e tween obj e ct  an d su bject (4).I n the  cont ext of mat hema tics, this can be understood asmaki n g  no  se p arat io n between theact  b y a subjec tof construc ting anob j ect s uch as  a set , and t h e  o b ject s o co nst ructed it sel f. Thus bo th  t h is act  and  t he obj ect of this a ct can be identi fieda s  an a c tiv e int ui tion, a s  Tak euti saysin the pass age in  que stion 
 centered_at Kyoto_University called the "Kyoto_School" 「京都学派」._As_we explain_below,_he uses some_keywords due to_Kitaro Nishida(西田幾 多郎) for_explaining his philosophical_ideas_about the foundations of mathematics. To understand this point, we focus on another passage_in_the paper_in_1956._In the above example, I_have frequently used the word_"illusion", but_indeed I hope the readers to replace it_by_"intuition". In this_sense, I redcall the intuition for grasping a function_or a set in general, the_active volitional intuition._Indeed,_when_we try to grasp_a function or a set, this_active volition works. And, what would_be thought by this active volition, the_object of this active volition, or_the volition itself would be_called a_function or a set. (Takeuti,_1956b, p. 299)_This passage_occurs just after_the previous one and it looks_very abrupt. Indeed,_this kind of abruptness is a_feature_of Takeuti's writings_about_philosophical_ideas. Here,_we would like_to_understand his_philosophical_background in the context of Japanese_philosophy._We can then better understand his peculiar_conception of formalism. 9_Takeuti's_proof theory in the_context of the Kyoto School_One point to be noted here_is that_he uses_the word "intuition"「直観」 or "volition"「意 志」for explaining the power of symbols_in mathematics. At first sight, this_word suggests Hilbert's use_of the_word_"intuition" in characterizing_his_finitist standpoint._However, if we read the last sentence_carefully, it_turns out that the intuition or_volition itself is a_mathematical_object, e.g., a function. This is_one feature of Japanese philosophy which_is articulated by Nishida. In_particular,_the_word "active intuition"「行為的直観」is a keyword_in Nishida's philosophy (3). In Nishida's_view, active intuition_comprises the fundamental mode of experience of_the_world that does not suppose a_separation_between object and subject (4). In_the_context_of mathematics, this can be_understood as making no separation between_the act by a subject of constructing an object_such as a_set, and the object so_constructed_itself._Thus both this act and the object of this act_can be_identified as an_active intuition, as Takeuti says in the passage in question