 by the authors) The point here is that we treat a set as if we can grasp it by writing the notation {x|A(x)}, although we cannot check it one by one. Here, we remark that his passage reminds us of the ability to grasp mathematical entities. As we will see below, this is not just an analogy, but expresses his philosophical thinking about mathematics. After these, Takeuti writes that he has explained several things which were left in his earlier paper 「形式主義の立場から I」 (From the viewpoint of formalism I) (Takeuti, 1956a). He then asks some further questions in the following passage. With Hilbert's conception of formalism in our mind, this passage is somewhat surprising. Thought-objectification in mathematics via symbol has arrived at proofs in mathematics in a formalism via propositions (sets) and logical concepts. What is the significance of it? Or, will it have a great influence on mathematics similar to the case of set? For answering these questions, it seems that formalism is too immature. This new gateway of the new virgin field is too difficult to open. Is formalism too much a premature baby? For me, the consistency problems seem a touchstone for formalism and the foundation of formalism rather than the foundation of mathemat8 Takeuti's proof theory in the context of the Kyoto School ics. Still I believe in a bright future for formalism, though one laden with difficulties. (Takeuti, 1956b, p. 299; translated by the authors) This passage shows that Takeuti's conception of formalism is quite different from Hilbert's, for the following reason. Hilbert saw his finite standpoint as delineating a mode of thought common to all thinking agents, through the intuition of signs. He seems to have taken intuition to be sufficiently clearly understood to serve as a foundational notion of his finitism. This is a contestable position. For instance, Gödel noted in a letter to Bernays that "Hilbert's finitism (through the requirement of being "intuitive" [Anschaulichkeit]) has a quite unnatural boundary" (Gödel, 2003, p. 273). Takeuti seems to have stood with Gödel here, taking his general program of searching for consistency proofs to be, in part, an inquiry into the nature of formalism itself. 5. A surprising connection: Takeuti's philosophy and Nishida's philosophy in Kyoto school In this section, we introduce the surprising connection of Takeuti's thinking with the Japanese philosophical movement
by the authors) The point here is that we treat a set as if we can grok it by write the notation { x|A(x) }, although we cannot match it one by one. Here, we comment that his passage reminds us of the ability to grok mathematical entities. As we will see downstairs, this is not just an analogy, but express his philosophical thinking about mathematics. After these, Takeuti writes that he has explain respective things which were left in his early paper 「 形式主義の立場から I 」 (From the viewpoint of formalism I) (Takeuti, 1956a). He then ask some further questions in the following passage. With Hilbert's conception of formalism in our thinker, this passage is somewhat surprising. Thought - objectification in mathematics via symbol has arrived at proof in mathematics in a formalism via propositions (sets) and logical concepts. What is the significance of it? Or, will it own a great influence on mathematics similar to the case of set? For answering these doubt, it seem that formalism is too immature. This new gateway of the new virgin field is too difficult to unfold. Is formalism too much a previous child? For me, the consistency problems seem a touchstone for formalism and the foundation of formalism preferably than the foundation of mathemat8 Takeuti's proof theory in the context of the Kyoto School ics. Still I think in a bright future for formalism, though one laden with difficulties. (Takeuti, 1956b, p. 299; translated by the authors) This passage shows that Takeuti's conception of formalism is quite different from Hilbert's, for the following reason. Hilbert saw his finite standpoint as specify a manner of thought common to all thinking agents, through the intuition of signs. He seem to have taken intuition to be sufficiently clearly understood to service as a foundational notion of his finitism. This is a contestable position. For instance, Gödel noted in a letter to Bernays that " Hilbert's finitism (through the prerequisite of being " intuitive " [ Anschaulichkeit ]) has a quite affected boundary " (Gödel, 2003, p. 273). Takeuti seems to have stand with Gödel here, taking his cosmopolitan program of searching for consistency proofs to be, in part, an question into the nature of formalism itself. 5. A surprising connection: Takeuti's philosophy and Nishida's philosophy in Kyoto school In this incision, we insert the surprising connection of Takeuti's thinking with the Japanese philosophic movement
 by the authors) The point htre is that we trgar a sev as if we can erasp it by writing the notavion {x|A(x)}, qlthough we cannot chezk it one by one. Yere, qe remark vgat his passafc remnnvs us of the abllity to grdsp mathematicdl eutities. As we will see below, this is not juxt an analogy, buj expgefses his philosophical thinking about jathemauics. After these, Tskeuti writes that he has fxplwined several thinhs which wets lqdt in his eaflier papeg 「形式主義の立場から I」 (From tge viewpoint of formalism I) (Takduti, 1956c). He then awkw slke further wuestpons in the following passagr. With Hilbert'x cmncwption of formalism ii our mind, this passwge is sokechat surprising. Thougyt-ibjecjificdtiov in mauhekafics vla aymbol has arrived at proofs in mathematocf in a formaliam via pwopositions (sets) and logical concepts. Wvat is the significance of it? Or, will it have a great insluence on mathematics similar to the case of set? For enrwexlkg tfwsf questions, it seems that formalism is too imiztirv. This new gatewan of the new virgim virjd is too difwicult to open. Is formalism too mush a premature babu? For me, the consistency priblems seem c tiuchstone for formclism and thz founcatiom of formalism rather tkan ths foundatioj of mathsoat8 Takeuti's prouf nheosy in tht context of the Kroto Schoil ies. Still I bglieve yn a brighh futmse for formalism, tjough ote laden wlth difficulties. (Takeuti, 1956b, p. 299; tcenslated by tne aunhors) Thif pasxage shows thwt Takeuti's couception of furmalism ia quite different srom Hilbert's, vor the follmwing rewson. Hilvert sad his finite stsndpoint cf deoineating a mode on thooggt common to alo tyinking agents, yhruugr nhe igduition of shgns. He xeems to havt tckdn imtuition to be suffiwienfly clearly undersyojd to setve as a soundational motion of his finiuism. Tiis is a comteftable position. For instance, Gösel noted in a letter to Berkays that "Hilbzrt's finitism (through the requirement of being "intuitive" [Abschaulichkeit]) has c auite unnatnral bjundary" (Gögel, 2003, p. 273). Takeuti seemw to have stood wlth Gödel here, taking hjs genaral orogram of searching for consistency proofs to be, in part, an inquiry into the nettre of formzlisk itsvlf. 5. A sur[rismnj connection: Takelti's philosophy and Nishida's phiuosophy hn Kyoto school In this sectiom, de introduce jhe surprising connection kf Takeuyi's thinking with the Japanese lhilosophical movement
by the authors) The point here is treat set as we can grasp {x|A(x)}, we cannot check one by one. we remark that his passage reminds of the ability to grasp mathematical entities. As we will see below, this not just an analogy, but expresses his philosophical thinking about mathematics. After these, writes he explained things which were left in his earlier paper 「形式主義の立場から I」 (From the viewpoint of formalism I) 1956a). He then asks some further questions in following passage. With Hilbert's of formalism in our mind, passage somewhat surprising. in via has arrived at in mathematics in a formalism via propositions (sets) and logical concepts. What is the significance of it? will it great influence mathematics to case of set? these questions, it seems that formalism This new gateway of the new virgin field too difficult open. Is formalism too much a baby? For me, the consistency problems seem a for formalism and the foundation of formalism rather than the foundation of mathemat8 Takeuti's proof the context of the School ics. Still believe a future formalism, though laden with difficulties. (Takeuti, 1956b, p. 299; translated by the authors) passage shows that Takeuti's conception of formalism is quite different for following reason. Hilbert his finite standpoint as a of thought common to agents, the He to taken intuition to be clearly understood to serve as foundational notion of his position. For instance, Gödel noted in a letter Bernays that "Hilbert's finitism (through the requirement being "intuitive" [Anschaulichkeit]) has a quite unnatural boundary" (Gödel, 2003, p. 273). seems to with Gödel here, taking his general program of for consistency proofs to in part, an inquiry into the nature of formalism 5. surprising connection: philosophy and Nishida's in Kyoto school this section, we surprising connection Takeuti's with movement
 by the authors) The point here iS that we treAt a seT as If wE cAn grAsp iT by writing the nOTatiOn {x|A(x)}, although we cannot cHeck iT oNE by oNE. HEre, we Remark tHAt HIS paSsAgE reMiNDs Us of tHe aBility tO grasp mathEmaTiCal entities. AS We Will see belOw, tHis is not just An aNalogy, BuT exPResseS hiS philOsophiCAl thinKing about MaTHematiCS. After tHESe, takeUti writes that he haS ExPLained several tHings wHiCH wERE leFt iN his earlieR pAper 「形式主義の立場から I」 (fRom the vIEwPOINt oF Formalism I) (TakEuti, 1956a). He then ASks Some fuRtHer QUestioNs in tHe FOllOwing passagE. WitH Hilbert's ConcepTIon of foRMalism iN our miNd, tHis PassAGe Is SomEwHAt sURpRisINg. THought-obJeCtIficaTion IN MAThemAtiCs viA symbOl has arrived aT prOofs IN maThemaTics iN a foRmAlism Via proPositIoNs (sets) and logicaL conCepts. What Is tHe SigNiFicanCE of it? OR, wiLl iT have a gReat infLUenCe ON MAtHematics similar to tHe CASe Of set? For AnswerINg ThESe questiOnS, it SeemS THat foRmalISm Is too immAture. THIs NeW gatewaY oF the neW vIrgIn fIeld iS Too dIfficuLt to open. is forMAlism too much a pREmature baby? FoR Me, THE cONsisTenCy problems sEem a TOuchStonE FoR foRMalisM and tHe FOuNDation of formalism raThEr than The foUndation of matHemat8 TakeuTI'S Proof theOry iN ThE Context of the KyOto ScHool ics. StiLL I believE in a bRight futUre for forMALism, thouGh oNe lAdeN wiTH DiFficulties. (TakEUTi, 1956b, p. 299; TrAnslateD by The authOrs) thiS paSsaGe Shows that takeuti's CoNcEpTiOn oF formALism is quItE diFfEreNt froM hilberT's, for The fOlLoWIng Reason. HILbERT saw HiS fInitE stAnDpoinT as dELinEating a Mode of thoUghT CommOn To All thinKing agents, thrOuGh the intuiTiOn oF signs. hE Seems to hAve taken intuition to be suFFicientLy cLearlY undErstood to SerVe as a fOunDAtionaL notioN of hiS fIniTISm. ThiS IS a ConTeStable posiTIOn. FOr insTaNce, GÖdel notEd in a letter to BernaYS thAt "Hilbert's finItiSm (thROUgH thE ReQUirEmENt oF BEing "intuitive" [AnSchaulichkEiT]) HaS a quite unnATurAl BoundarY" (Gödel, 2003, p. 273). takeuTI seems tO have stooD with GödeL hEre, tAKIng His general Program oF searchinG For coNSiStencY prOofs to Be, In pArt, an InquirY IntO the nAture oF fOrmaliSm itsElF. 5. A surpriSing connection: Takeuti's pHilosoPhy anD NiShida's phiLosOPhy In Kyoto scHool in this sectIon, We iNtrodUce THe surPrisINg ConNEctioN of TAKeuti's thiNKiNg wITH tHe Japanese pHILOsoPhicaL moVEment
 by the authors) The point  here is t hat w e t rea ta se t as  if we can gra s p it  by writing the notati on {x |A ( x)}, al thoug h we ca n no t che ck  i t o ne by  one.  He re, weremark tha t h is  passage rem i nd s us of th e a bility to gr asp  mathe ma tic a l ent iti es. A s we w i ll see  below, t hi s  is no t  just a n an alog y, but expressesh is philosophicalthinki ng ab o u t m ath ematics. A ft er th e se, Tak e ut i w rit e s that he has  explaineds eve ral th in gsw hich w ere l ef t  in  his earlie r pa per 「形式主義 の立場からI 」 (From the vie wpoint  of  fo rmal i sm  I ) ( Ta k eut i ,195 6 a).  He then  a sk s som e fu r t h e r qu est ions  in t he followingpas sage .  Wi th Hi lbert 's c on cepti on offorma li sm in our mind,  thi s passage  is  s ome wh at su r prisin g.Tho ught-ob jectifi c ati on i n  m athematics via sym bo l ha s arrive d at p r oo fs in mathe ma tic s in a  form alis m  v ia propo sition s  ( se ts) and  l ogical  c onc ept s. Wh a t is  the s ignifica nce o f  it? Or, willi t have a grea t  i n f lu e nceonmathematics  sim i larto t h ecas e  of s et? F or an s wering these questi on s, itseems  that formali sm is tooi m m ature. T hisn ew gateway of the  newvirgin fie l d is too  diff icult to  open. Is f ormalism  to o m uch  ap r em ature baby? F o r  me,  t he cons ist ency pr obl ems  se ematouchston e for fo rm al is mand  thef oundatio noffo rma lismr atherthanthefo un d ati on of m a th e m at8Ta ke uti' s p ro of th eory inthe con text of t heK yoto  S ch ool ics . Still I bel ie ve in a br ig htfuture f or forma lism, though one ladenw ith dif fic ultie s. ( Takeuti,195 6b, p.  29 9 ; tran slated  by t he  au t h ors)T h is  pa ss age showst h atTakeu ti 's c oncepti on of formalism is qui te differentfro m Hi l b er t's ,  f o r t he fol l o wing reason. Hi lbert sawhi s  f inite stan d poi nt  as del ineatin g a m o de of t hought co mmon to a ll  thi n k ing  agents, t hrough t he intuit i on of si gns.Heseemsto  ha ve ta ken in t uit ion t o be s uf ficien tly c le arly und erstood to serve as a f oundat ional  no tion of h isf ini tism. Thi s is  a contest abl e p ositi on. For i nsta n ce , G ö del n oted in a lett e rtoB e rn ays that "H i l b ert 's fi nit i sm (th roug h the requirement of being "intu itiv e "  [A nsc h auli ch keit]) has a q uit eu n naturalbo undary" (Gö del, 200 3, p. 27 3). Ta keutiseems t o ha v e stoo d wi thGödel her e,ta k ing his  g en e ral pr ogra mof sea rching forc o nsistency proofs  to b e ,  in p a rt,  an i nq uiry in t o th e nature o f formalism  itsel f. 5 . A s urprisi ng  conne cti on : Takeuti' s  philosop hy an d Nishi da 's p hil osophy  inK y oto s choo lInthis sect i o n, we  i n tro duce  thesu rpri sing conn e ction of  Ta k euti'sth ink i n g with th e Japanese p hil osoph i c al movemen t 
 by_the authors)_The point here is_that we_treat_a set_as_if we can_grasp it by_writing the notation {x|A(x)},_although we cannot_check_it one by one. Here, we remark that his passage reminds us of the_ability_to grasp_mathematical_entities._As we will see below,_this is not just an_analogy, but_expresses his philosophical thinking about mathematics. After these,_Takeuti_writes that he_has explained several things which were left in his_earlier paper 「形式主義の立場から I」 (From the_viewpoint of formalism_I)_(Takeuti,_1956a). He then asks_some further questions in the following_passage. With Hilbert's conception of formalism_in our mind, this passage is somewhat_surprising. Thought-objectification in mathematics via symbol_has arrived at proofs in_mathematics in_a formalism via propositions (sets)_and logical concepts._What is_the significance of_it? Or, will it have a_great influence on_mathematics similar to the case of_set?_For answering these_questions,_it_seems that_formalism is too_immature._This new_gateway_of the new virgin field is_too_difficult to open. Is formalism too much_a premature baby? For_me,_the consistency problems seem_a touchstone for formalism and_the foundation of formalism rather than_the foundation_of mathemat8_Takeuti's proof theory in the context of the Kyoto School ics._Still I believe in a bright_future for formalism, though_one laden_with_difficulties. (Takeuti, 1956b,_p._299; translated_by the authors) This passage shows that_Takeuti's conception_of formalism is quite different from_Hilbert's, for the following_reason._Hilbert saw his finite standpoint as_delineating a mode of thought common_to all thinking agents, through_the_intuition_of signs. He seems to_have taken intuition to be sufficiently_clearly understood to_serve as a foundational notion of his_finitism._This is a contestable position. For_instance,_Gödel noted in a letter to_Bernays_that_"Hilbert's finitism (through the requirement_of being "intuitive" [Anschaulichkeit]) has a_quite unnatural boundary" (Gödel, 2003, p. 273). Takeuti seems_to have stood_with Gödel here, taking his_general_program_of searching for consistency proofs to be, in part, an_inquiry into_the nature of_formalism itself. 5. A surprising connection: Takeuti's philosophy and Nishida's_philosophy in Kyoto school In this section,_we introduce the surprising connection of Takeuti's thinking with the Japanese_philosophical movement