 be interesting to interpret Takeuti's results in set theory, and to make sense of Takeuti's view that proof theory and set theory are internally connected, by means of this Nishidean interpretation. One might wonder whether the notion of broad self is necessary in order to interpret Takeuti's thoughts in the context of Nishida's philosophy, since the notion of narrow self is seemingly sufficient for interpreting Takeuti's notion of mind. We reply as follows. Firstly, Takeuti says explicitly that minds can operate or create mathematical objects such as numbers, functions, or sets. In the context of Nishida's philosophy, such mathematical objects would not belong to a narrow self, but to the world. Hence, the notion of broad self is also necessary, or at least natural, for interpreting Takeuti's writings. Secondly, we discuss the notion of "self-reflection" 「自己反省」 in Takeuti's writings. We begin by noting its usage in the following passage: Now, for example, consider a finite mind. If the function of a finite mind is completely finite, then our mathematics remains in finite. However, our finite mind is the so-called potential infinity. In other words, we can indicate infinitary objects like 0, 1, 2, 3,... [...] Why? This is because our mind has the ability of self-reflection, that is, the ability to observe what we are doing and to know what we are doing. (Takeuti, 1972, p. 173; translated by the authors) To understand this passage, let us consider the number 1 as a stroke |. Since adding one stroke is clearly possible, we can consider 2 as ||. Iterating this process, we consider any finite number occurring in the sequence 0, 1, 2,.... Now, following Takeuti, we may reflect on what we are doing. It is not so easy to characterize the limit of the ability of selfreflection, but Takeuti clearly considers the possibility that we can observe the process of "adding a stroke beginning with the empty sequence". Therefore, we can consider the set of natural numbers as potentially infinite. Interestingly, Takeuti claims the same thing for an infinite mind as well. This holds true for an infinite mind as well. When an infinite mind creates sets 12 Takeuti's proof theory in the context of the Kyoto School from the empty set, though I explain the construction of ordinals very briefly in my lecture or explanation, if we think
be interesting to interpret Takeuti's results in set theory, and to seduce common sense of Takeuti's view that proof theory and specify hypothesis are internally connected, by means of this Nishidean interpretation. One might wonder whether the impression of across-the-board self is necessary in order to rede Takeuti's thoughts in the context of Nishida's philosophy, since the impression of narrow self is seemingly sufficient for interpreting Takeuti's notion of judgment. We reply as follows. Firstly, Takeuti says explicitly that minds can function or create mathematical objects such as act, functions, or sets. In the context of Nishida's philosophy, such mathematical object would not belong to a narrow self, but to the world. Hence, the notion of wide self is also necessary, or at least natural, for interpreting Takeuti's writings. Secondly, we discuss the notion of " self - reflection " 「 自己反省 」 in Takeuti's writings. We begin by noting its usage in the following passage: Now, for example, consider a finite mind. If the affair of a finite mind is completely finite, then our mathematics persist in finite. However, our finite judgment is the so - called potential eternity. In other word, we can indicate infinitary objects like 0, 1, 2, 3, ... [... ] Why? This is because our judgment has the ability of self - reflection, that is, the ability to observe what we are doing and to know what we are doing. (Takeuti, 1972, p. 173; understand by the authors) To understand this passage, let us consider the number 1 as a throw |. Since adding one stroke is clearly possible, we can consider 2 as ||. repeat this process, we consider any finite number occur in the sequence 0, 1, 2, .... Now, following Takeuti, we may reflect on what we are dress. It is not so comfortable to characterize the limit of the ability of selfreflection, but Takeuti clearly considers the possibility that we can detect the process of " total a stroke begin with the empty sequence ". Therefore, we can consider the set of natural number as potentially infinite. Interestingly, Takeuti claims the same thing for an infinite mind as well. This holds truthful for an infinite mind equally well. When an infinite mind creates sets 12 Takeuti's proof theory in the context of the Kyoto School from the empty set, though I explain the structure of ordinals very briefly in my lecture or explanation, if we think
 be interesting to interpreu Takeuti's results in sev theorg, and to make sense of Takeuti's view tyat peoof theory and set thdory are pnternallt coinected, by means of this Nishiscan iutxrpretation. One might wonger whether tha vocion of broad self is necessary in owder to ijterpret Takeuji's tnjughfs in the context of Nishida's philksophy, vince the notoon of narrow self is seemlnglj sufficient for ijterpreting Takqyti's notion uf mind. We reply as fomlows. Firstly, Takeuti says explkcitlv that mindw xan mperate or rreate mathematical objectv such ss numbers, fungtionv, oe sets. In the context of Nishida's philofophy, sucv jathematical objextw wound nmt bdoone tk e nzrrow delh, but to ths world. Henxe, the notion of brpaq self is also necesfawy, or at least natural, for interpreting Taieuti's writings. Secondlt, we discuss the notiln of "sels-reflection" 「自己反省」 in Takeuti's writings. We begin by nothng ivs uscnc in rhf following passage: Now, for example, consider w fonpte mind. If the fmnction of a finitr limq is completeuy finnfe, then our mathematlcs remwins un finite. Howrver, our finite mind is the so-called ponentual infinity. In otker words, we can indivate infinitary objects liks 0, 1, 2, 3,... [...] Why? Hhis is bszause our mind hxs nhe dbility of self-reflection, ehat is, tie abnlity to obsgrve whwt we are foing and to know what wf are dming. (Takeuhi, 1972, p. 173; translated by the authors) To understanc dhiv passagz, let ms consider the number 1 as a xtroke |. Sincd adding ohe strone is clearjy possible, wa can considec 2 as ||. Itqratung rhis pruzess, we considrr any fiunte numbee occurring in the sedhence 0, 1, 2,.... Now, fooloqing Takeuti, we mah rqfpert on fhat we are goine. Ig is nut so easy bo zharscterize the limit ox ths ability of selfrrfjection, vut Taketti clearly cpnsiders the possihilitb that we csn jbserve the process of "adding z stroke heglnning with tre enpty sequencz". Therefore, we can consider the set of netural numbers as potenrially infinite. Intgrextingly, Takxuti cjaims the same thing for an ibfinite mind as wtll. This holds true for an inxinitf mind as well. When an infinite mind creates sets 12 Takeuti's proof theory un the context of fhe Lyoto Sehopl froi thx xmpty set, though P explain the construction of orvinals verf yriefly in my lecture or explsnxtion, if we tfink
be interesting to interpret Takeuti's results in and make sense Takeuti's view that are connected, by means this Nishidean interpretation. might wonder whether the notion of self is necessary in order to interpret Takeuti's thoughts in the context of philosophy, since the notion of narrow self is seemingly sufficient for interpreting Takeuti's of We as Firstly, Takeuti says explicitly that minds can operate or create mathematical objects such as numbers, functions, sets. In the context of Nishida's philosophy, such objects would not belong a narrow self, but to world. the notion broad is necessary, or at natural, for interpreting Takeuti's writings. Secondly, we discuss the notion of "self-reflection" 「自己反省」 in Takeuti's writings. We by noting in the passage: for consider a finite the function of a finite mind then our mathematics remains in finite. However, our mind is so-called potential infinity. In other words, can indicate infinitary objects like 0, 1, 2, [...] Why? This is because our mind has the ability of self-reflection, that is, the observe what we are and to know we doing. 1972, 173; translated the authors) To understand this passage, let us consider the number as a stroke |. Since adding one stroke is clearly can 2 as ||. this process, we consider finite occurring in the sequence 2,.... following reflect what are doing. It is so easy to characterize the of the ability of the possibility that we can observe the process "adding a stroke beginning with the empty Therefore, we can consider the set of natural numbers as potentially infinite. Takeuti claims thing for an infinite mind as well. This true for an infinite as well. When an infinite mind creates sets 12 proof in the of the Kyoto from the empty though I explain of ordinals briefly my if we think
 be interesting to interpret TAkeuti's resUlts iN seT thEoRy, anD to mAke sense of TakeUTi's vIew that proof theory and sEt theOrY Are iNTeRnallY connecTEd, BY MeaNs Of ThiS NIShIdean IntErpretaTion. One migHt wOnDer whether thE NoTion of broaD seLf is necessarY in Order tO iNteRPret TAkeUti's tHoughtS In the cOntext of NIsHIda's phILosophy, SINcE the Notion of narrow selF Is SEemingly sufficIent foR iNTeRPRetIng takeuti's noTiOn of mINd. We repLY aS FOLloWS. Firstly, TakeuTi says expliCItlY that mInDs cAN operaTe or cReATe mAthematical ObjeCts such as NumberS, FunctioNS, or sets. in the cOntExt Of NiSHiDa'S phIlOSopHY, sUch MAthEmatical ObJeCts woUld nOT BELong To a NarrOw selF, but to the worlD. HeNce, tHE noTion oF broaD selF iS also NecessAry, or At Least natural, for InteRpreting TAkeUtI's wRiTings. sEcondlY, we DisCuss the Notion oF "SelF-rEFLEcTion" 「自己反省」 in Takeuti's writInGS. we Begin by nOting iTS uSaGE in the foLlOwiNg paSSAge: NoW, for EXaMple, consIder a fINiTe Mind. If tHe FunctiOn Of a FinIte miND is cOmpletEly finitE, then OUr mathematics rEMains in finite. hOwEVEr, OUr fiNitE mind is the sO-calLEd poTentIAl InfINity. IN otheR wORdS, We can indicate infiniTaRy objeCts liKe 0, 1, 2, 3,... [...] Why? This is beCause our miND HAs the abiLity OF sELf-reflection, thAt is, tHe ability tO Observe wHat we Are doing And to know WHAt we are dOinG. (TaKeuTi, 1972, p. 173; TRAnSlated by the auTHOrs) TO uNderstaNd tHis passAge, Let Us cOnsIdEr the numbEr 1 as a strOkE |. SInCe AddIng onE Stroke is ClEarLy PosSible, WE can coNsideR 2 as ||. ITeRaTIng This proCEsS, WE conSiDeR any FinItE numbEr ocCUrrIng in thE sequence 0, 1, 2,.... now, FOlloWiNg takeuti, We may reflect oN wHat we are doInG. It Is not sO EAsy to chaRacterize the limit of the aBIlity of SelFreflEctiOn, but TakeUti ClearlY coNSiders The posSibilItY thAT We can OBSeRve ThE process of "ADDinG a strOkE begInning wIth the empty sequencE". theRefore, we can coNsiDer tHE SeT of NAtURal NuMBerS AS potentially infInite. InterEsTInGly, Takeuti CLaiMs The same Thing foR an inFInite miNd as well. THis holds tRuE for AN InfInite mind aS well. WheN an infiniTE mind CReAtes sEts 12 takeutI's ProOf theOry in tHE coNtext Of the KYoTo SchoOl froM tHe empty sEt, though I explain the consTructiOn of oRdiNals very bRieFLy iN my lecturE or eXplanation, If wE thInk
 be interesting to interpr et Takeuti 's re sul tsin  set  the ory, and to ma k e se nse of Takeuti's viewthatpr o of t h eo ry an d set t h eo r y  ar ein ter na l ly  conn ect ed, bymeans of t his  N ishidean int e rp retation.One  might wonde r w hether  t hen otion  of  broa d self is nec essary in  o r der to interpr e t  T akeu ti's thoughts int he context of Nis hida's  p h il o s oph y,since theno tiono f narro w  s e l f  is seemingly suf ficient for int erpret in g T a keuti' s not io n  of  mind. We r eply  as follo ws. Fi r stly, T a keuti s ays ex pli cit ly t h at  m ind sc ano pe rat e  or  createma th emati calo b j e ctssuc h as  numb ers, function s,or s e ts.  In t he co ntex tof Ni shida' s phi lo sophy, such mat hema tical obj ect swou ld  notb elongtoa n arrow s elf, bu t  to  t h e wo rld. Hence, the no ti o n  o f broadself i s  a ls o  necessa ry , o r at l eastnatu r al , for in terpre t in gTakeuti 's  writi ng s.Sec ondly ,  wediscus s the no tiono f "self-reflec t ion" 「自己反省」 i n  T a k eu t i'swri tings. We b egin by n otin g  i tsu sagein th ef ol l owing passage: Now,  f or exa mple,  consider a f inite mind . I f the fu ncti o no f a finite min d iscompletely finite,thenour math ematics r e m ains infin ite . H owe v e r,  our finite m i n d is  t he so-c all ed pote nti alinf ini ty . In othe r words,  w eca nind icate infinita ry  ob je cts  like 0, 1,2, 3, ...[. .. ]  Wh y? This is b ecau se  o ur m ind  h as th e ab i lit y of se lf-reflec tio n , th at  i s, theability to ob se rve what w eare  doing a nd to kn ow what we are doing. ( T akeuti,  19 72, p . 17 3; transl ate d by t hea uthors ) To u nders ta ndt h is pa s s ag e,le t us consi d e r t he nu mb er 1  as a s troke |. Since add i ngone stroke is  cl earl y po ssi b le ,  we  c a n c o n sider 2 as ||.Iteratingth i sprocess, w e  co ns ider an y finit e num b er occu rring inthe seque nc e 0, 1 , 2 ,.... Now,  followi ng Takeut i , wem ay  refl ect  on wh at  we  aredoing. Itis no t so e as y to c harac te rize the  limit of the ability o f self refle cti on, but T ake u ticlearly c onsi ders the p oss ibi litytha t  we c an o b se rve the p roce s s of "add i ng  as t ro ke beginnin g w ith  theemp t y sequ ence ". Therefore, wec an consider th e se t ofnat u ralnu mbers as poten tia ll y infinite .Interesting ly, Take ut i  clai ms the  samething f o r  a n  infin itemin d as well . T hi s  holdstr ue for an  inf in ite mi nd asw ell. W hen an infinitemindc r eates set s 12Ta keuti's proo f theory i n the conte xt oftheKyoto  School  f rom th e e mp ty set, th o ugh I exp lainthe con st ruct ion  of or dina l s  very  bri ef lyin my lec t u re or  e x pla nati on, i fwe t hink
 be_interesting to_interpret Takeuti's results in_set theory,_and_to make_sense_of Takeuti's view_that proof theory_and set theory are_internally connected, by_means_of this Nishidean interpretation. One might wonder whether the notion of broad self is_necessary_in order_to_interpret_Takeuti's thoughts in the context_of Nishida's philosophy, since the_notion of_narrow self is seemingly sufficient for interpreting Takeuti's_notion_of mind. We_reply as follows. Firstly, Takeuti says explicitly that minds_can operate or create mathematical objects_such as numbers,_functions,_or_sets. In the context_of Nishida's philosophy, such mathematical objects_would not belong to a narrow_self, but to the world. Hence, the_notion of broad self is also_necessary, or at least natural,_for interpreting_Takeuti's writings. Secondly, we discuss_the notion of_"self-reflection" 「自己反省」_in Takeuti's writings._We begin by noting its usage_in the following_passage: Now, for example, consider a_finite_mind. If the_function_of_a finite_mind is completely_finite,_then our_mathematics_remains in finite. However, our finite_mind_is the so-called potential infinity. In other_words, we can indicate_infinitary_objects like 0, 1,_2, 3,... [...] Why? This_is because our mind has the_ability of_self-reflection, that_is, the ability to observe what we are doing and to_know what we are doing. (Takeuti,_1972, p. 173; translated_by the_authors)_To understand this_passage,_let us_consider the number 1 as a stroke_|. Since_adding one stroke is clearly possible,_we can consider 2_as_||. Iterating this process, we consider_any finite number occurring in the_sequence 0, 1, 2,.... Now,_following_Takeuti,_we may reflect on what_we are doing. It is not_so easy to_characterize the limit of the ability of_selfreflection,_but Takeuti clearly considers the possibility_that_we can observe the process of_"adding_a_stroke beginning with the empty_sequence". Therefore, we can consider the_set of natural numbers as potentially infinite. Interestingly, Takeuti_claims the same_thing for an infinite mind_as_well._This holds true for an infinite mind as well. When_an infinite_mind creates sets_12 Takeuti's proof theory in the context of the Kyoto_School from the empty set, though I_explain the construction of ordinals very briefly in my lecture or_explanation, if we think