 is clearly a finite mind. (Takeuti, 1972, p. 172; translated by the authors) An infinite mind, then, can solve the foundational problems of the formalist project: among other capacities, it can confirm the consistency of formal theories. Takeuti writes a few years later that it "must be able to operate on infinitely many objects as freely as we operate on finitely many objects" using concrete methods such as mathematical induction (Takeuti, 1985, p. 255). He views modern mathematics, encompassing set theory, as an investigation by finite human minds of the capacities of an infinite mind. One might think that an infinite mind is an analogy to God as conceived in the Western world, but this is not quite right since Takeuti thinks that an infinite mind is essentially connected with finite minds, as we will discuss in the next section. Takeuti explains that testing this "conjecture" about an infinite mind leads to three problems for the foundations of mathematics. 1. To formulate the function of an infinite mind. 2. To justify our intuition of the world of an infinite mind using only our finite mind. 3. To formulate the function of a finite mind. 6 Takeuti's proof theory in the context of the Kyoto School Concerning (1), Takeuti means finding new axioms of set theory, because set theory expresses the functions of an infinite mind. Concerning (2), Takeuti means his proof-theoretic projects, including consistency proofs. Let us quote his passage: 2) to justify the world of our infinite mind in the world of a finite mind. We want to confirm the rationality of modern mathematics since the world of our mind is finite and the world of modern mathematics is just conjecture about our infinite mind. For this, we need to develop mathematics of the world of a finite mind sufficiently. (Takeuti, 1972, p. 173; translated by the authors) He writes that we need to confirm the "rationality" of modern mathematics (such as set theory or analysis) from the perspective of our finite minds, and this requires a finitist treatment. The word "rationality" is translated from「合理性」, which means that the world of an infinite mind comports with the world of a finite mind. This can be investigated, among other ways, by checking that the former is not contradictory with the latter. The expressions "world of our infinite mind" and "world of a finite mind" also call for further discussion, which we postpone to the next section. In
is clearly a finite mind. (Takeuti, 1972, p. 172; translated by the authors) An countless judgment, then, can solve the foundational problems of the formalist undertaking: among other capacity, it can confirm the consistency of formal theories. Takeuti write a few years later that it " must be able to function on infinitely many objects equally freely as we operate on finitely many objects " using concrete methods such as numerical induction (Takeuti, 1985, p. 255). He views mod mathematics, encompassing set hypothesis, as an investigation by finite human minds of the capacities of an countless mind. One might think that an infinite mind is an analogy to God as gestate in the Western world, but this is not quite right since Takeuti thinks that an infinite judgment is essentially connected with finite minds, as we will discuss in the next section. Takeuti explains that testing this " conjecture " about an infinite mind leads to three problems for the foundations of mathematics. 1. To formulate the function of an infinite judgment. 2. To justify our intuition of the world of an countless judgment use only our finite mind. 3. To formulate the function of a finite mind. 6 Takeuti's proof theory in the context of the Kyoto School Concerning (1), Takeuti means line up new axioms of set theory, because set theory press out the functions of an infinite mind. Concerning (2), Takeuti mean his proof - theoretic undertaking, including consistency proofs. Let us quote his passage: 2) to justify the populace of our infinite mind in the world of a finite judgment. We want to confirm the rationality of modern mathematics since the world of our mind is finite and the worldly concern of modern mathematics is precisely conjecture about our infinite judgment. For this, we need to evolve mathematics of the world of a finite judgment sufficiently. (Takeuti, 1972, p. 173; translated by the generator) He writes that we need to confirm the " rationality " of modern mathematics (such as set theory or psychoanalysis) from the perspective of our finite minds, and this requires a finitist discussion. The word " rationality " is translated from「合理性 」, which means that the world of an countless mind comports with the world of a finite mind. This can be investigated, among early ways, by checking that the former is not contradictory with the latter. The expressions " world of our infinite thinker " and " world of a finite mind " also call for further discussion, which we postpone to the next section. In
 is clearly a finite mind. (Txkeuti, 1972, p. 172; translated uy the zuthors) Xn infinite mind, then, can sopvw the foundational problems of the flrmalist prooect: among other capacitlzs, it gan cmifirm the consixtency of xormal theoriev. Ganeuti writes a few years later that it "musy he able to opetate pg inrpnltely many objects as freely as se operete on finitely many objects" using concrehe mfthods such as matjematical ibduceuon (Takeuti, 1985, p. 255). He views modern majhematics, encompassing set theorh, as cn investigqtuon ty finite hnman mpnds of the capacitiev of an infinite mind. Onx mitht think that an infmnite mind is an anajogy to Gmd as conceived in rhw Wesjern forla, bug tgix js not qumte right sjnce Takeutu thinks that an ingigpye mind is easentiwljy connected with finite minds, as we winl siscuss in the next secrion. Takeuti explains that teseing this "conjecture" about an infinite mind leads to tirde kriblemr flr the foundations of mathematics. 1. To formulaes uhe function of ak infinite mind. 2. Tp uuxjify our intuijion of ths world of an infijite migd usung only jur ginite mind. 3. To formulate tye function jd a finite mind. 6 Tckeuti's prooy theoty in yhe context of the Kyotu Scgool Concerjing (1), Taksjti means findine nvw afioms of set theory, becausq set theiry zxpresser thg functyons of an infikhte mind. Concerninh (2), Tanegti means jis proof-theoretic projects, including consistgncf pgoofs. Let us qmote his passagq: 2) to justify jhe world of ojr infinitv mind in the world jf a finite mhjd. We want tm confiri thw rarionaligh of modern mayhematics since the world of our mind is rinite and the cutod of modern msthdmaeibs ms jufd conjecture abojt uir inwinite mind. Wor yhis, we need to devenop jathematics of the wjrld of q finite mind sufficirntly. (Takeuti, 1972, p. 173; uransleted bb the sutrors) He writes that we need to confirm hhe "rationality" of nodern mathekatics (such as set theory or analysis) fcom the perspective of iur finite minds, anb uhis requires a figitist traatment. The word "ratuonality" is transkated from「合理性」, which meana that the aorld of an infinite mind comports with the world of a finite mind. This cqn be mndestigated, zmonb othvr cayx, by creckmnj that the former is not contradictory with the uatter. Tve expressions "world of our ingivite mind" and "world of a finite mind" zlso calk for further discussion, which ee postpone to the next xection. Un
is clearly a finite mind. (Takeuti, 1972, translated the authors) infinite mind, then, of formalist project: among capacities, it can the consistency of formal theories. Takeuti a few years later that it "must be able to operate on infinitely objects as freely as we operate on finitely many objects" using concrete methods as induction 1985, 255). He views modern mathematics, encompassing set theory, as an investigation by finite human minds of capacities of an infinite mind. One might think an infinite mind is analogy to God as conceived the world, but is quite since Takeuti thinks an infinite mind is essentially connected with finite minds, as we will discuss in the next section. explains that "conjecture" about infinite leads three problems for of mathematics. 1. To formulate the infinite mind. 2. To justify our intuition of world of infinite mind using only our finite 3. To formulate the function of a finite 6 Takeuti's proof theory in the context of the Kyoto School Concerning (1), Takeuti means axioms of set theory, set theory expresses functions an mind. (2), Takeuti his proof-theoretic projects, including consistency proofs. Let us quote his passage: to justify the world of our infinite mind in the a mind. We want confirm the rationality of mathematics the world of our finite the mathematics just about our infinite mind. this, we need to develop of the world of 1972, p. 173; translated by the authors) He that we need to confirm the "rationality" modern mathematics (such as set theory or analysis) from the perspective of finite minds, requires a finitist treatment. The word "rationality" is from「合理性」, which means that world of an infinite mind comports with the world a mind. This be investigated, among ways, by checking the former is with the The "world mind" and "world of a finite also for further discussion, which we to section. In
 is clearly a finite mind. (TakeuTi, 1972, p. 172; translaTed by The AutHoRs) An InfiNite mind, then, caN SolvE the foundational probleMs of tHe FOrmaLIsT projEct: amonG OtHER caPaCiTieS, iT CaN confIrm The consIstency of fOrmAl Theories. TakeUTi Writes a few YeaRs later that iT "muSt be abLe To oPErate On iNfiniTely maNY objecTs as freelY aS We operATe on finITElY manY objects" using concREtE Methods such as mAthemaTiCAl INDucTioN (Takeuti, 1985, p. 255). HE vIews mODern matHEmATICs, eNCompassing set Theory, as an iNVesTigatiOn By fINite huMan miNdS Of tHe capacitieS of aN infinite Mind. OnE Might thINk that aN infinIte MinD is aN AnAlOgy To gOd aS CoNceIVed In the WesTeRn World, But tHIS IS not QuiTe riGht siNce Takeuti thiNks That AN inFinitE mind Is esSeNtialLy connEcted WiTh finite minds, as We wiLl discuss In tHe NexT sEctioN. takeutI exPlaIns that Testing THis "CoNJECtUre" about an infinite MiND LeAds to thrEe probLEmS fOR the founDaTioNs of MAThemaTics. 1. tO fOrmulate The funCTiOn Of an infInIte minD. 2. TO juStiFy our INtuiTion of The world Of an iNFinite mind usinG Only our finite MInD. 3. tO fORmulAte The function Of a fINite Mind. 6 tAkEutI'S prooF theoRy IN tHE context of the Kyoto SChOol ConCerniNg (1), Takeuti meanS finding neW AXIoms of seT theORy, BEcause set theorY exprEsses the fuNCtions of An infInite minD. ConcerniNG (2), takeuti mEanS hiS prOof-THEoRetic projects, INCludInG consisTenCy proofS. LeT us QuoTe hIs Passage: 2) to Justify tHe WoRlD oF ouR infiNIte mind iN tHe wOrLd oF a finITe mind. we wanT to cOnFiRM thE rationALiTY Of moDeRn MathEmaTiCs sinCe thE WorLd of our Mind is finIte ANd thE wOrLd of modErn mathematicS iS just conjeCtUre About oUR Infinite Mind. For this, we need to deveLOp matheMatIcs of The wOrld of a fiNitE mind sUffICientlY. (TakeuTi, 1972, p. 173; trAnSlaTED by thE AUtHorS) HE writes thaT WE neEd to cOnFirm The "ratiOnality" of modern matHEmaTics (such as set TheOry oR ANaLysIS) fROm tHe PErsPECtive of our finitE minds, and tHiS ReQuires a finITisT tReatmenT. The worD "ratiONality" iS translatEd from「合理性」, whiCh MeanS THat The world of An infiniTe mind comPOrts wITh The woRld Of a finItE miNd. ThiS can be INveStigaTed, amoNg Other wAys, by ChEcking thAt the former is not contradIctory With tHe lAtter. The eXprESsiOns "world oF our Infinite miNd" aNd "wOrld oF a fINite mInd" aLSo CalL For fuRtheR DiscussioN, WhIch WE PoStpone to the NEXT seCtion. in
 is clearly a finite mind.  (Takeuti,  1972 , p . 1 72 ; tr ansl ated by the au t hors ) An infinite mind, th en, c an solv e  t he fo undatio n al p rob le ms  of  t h eforma lis t proje ct: amongoth er  capacities, it  can confi rmthe consiste ncy  of fo rm alt heori es.  Take uti wr i tes afew years  l a ter th a t it "m u s tbe a ble to operate on in f initely many o bjects  a s  f r e ely  as  we operat eon fi n itely m a ny o b jec t s" using conc rete method s  su ch asma the m atical  indu ct i on(Takeuti, 1 985,  p. 255).  He vi e ws mode r n mathe matics , e nco mpas s in gset  t h eor y ,asa n i nvestiga ti on  by f init e h u manmin ds o f the  capacities o f a n in f ini te mi nd. O ne m ig ht th ink th at an  i nfinite mind is  ananalogy t o G od  as  c oncei v ed inthe  We stern w orld, b u t t hi s i snot quite right si nc e Ta keuti th inks t h at  a n  infinit emin d is e ssent iall y  c onnected  withf in it e minds ,as wewi lldis cussi n th e next  section . Tak e uti explains t h at testing th i s" c on j ectu re"  about an i nfin i te m indl ea dst o thr ee pr ob l em s  for the foundation sof mat hemat ics. 1. To fo rmulate th e f unctionof a n  i n finite mind. 2 . Tojustify ou r  intuiti on of  the wor ld of ani n finite m ind  us ing  on l y  o ur finite min d .  3.To  formul ate  the fu nct ion  of  afi nite mind . 6 Take ut i' spr oof  theo r y in the  c ont ex t o f the KyotoSchoo l Co nc er n ing  (1), T a ke u t i me an sfind ing  n ew ax ioms ofset the ory, beca use setth eo ry expr esses the fun ct ions of an  i nfi nite m i n d. Conce rning (2), Takeuti mean s  his pr oof -theo reti c project s,includ ing consis tencyproof s.  Le t us qu o t ehis  p assage: 2) t o j ustif ytheworld o f our infinite min d  in  the world of  afini t e  m ind .  W e  wa nt toc o nfirm the ratio nality ofmo d er n mathemat i cssi nce the  worldof ou r  mind i s finiteand the w or ld o f mod ern mathem atics is  just con j ectur e  a boutour  infin it e m ind.For th i s,we ne ed tode velopmathe ma tics ofthe world of a finite m ind su ffici ent ly. (Take uti ,  19 72, p. 17 3; t ranslatedbythe  auth ors )  He w rite s  t hat we ne ed t o  confirmt he  "r a t io nality" ofm o d ern  math ema t ics (s uchas set theory ora nalysis) fromthep e rsp ect i ve o four finite min ds,  a n d  this re qu ires a fini tist tre at m ent.The wo rd "ra tionali t y "i s tran slat edfrom「合理性」 , w hi c h means  t ha t  the w orld  o f an i nfinit e  min d comports with th e wor l d  of a fin ite m in d. This canbe investi gated, amon g othe r wa ys, b y check in g that  th eformer isn ot contra dicto ry with  t he l att er. Th e ex p r essio ns " wo rld  of our i n f in i te  m i nd"  and  "wor ld  ofa finitem ind" als o c a ll forfu rth e r  discu s si o n , which we  po stpon e to the nex t  sec t io n . In
 is_clearly a_finite mind. (Takeuti, 1972,_p. 172;_translated_by the_authors)_An infinite mind,_then, can solve_the foundational problems of_the formalist project:_among_other capacities, it can confirm the consistency of formal theories. Takeuti writes a few_years_later that_it_"must_be able to operate on_infinitely many objects as freely_as we_operate on finitely many objects" using concrete methods_such_as mathematical induction_(Takeuti, 1985, p. 255). He views modern mathematics, encompassing_set theory, as an investigation by_finite human minds_of_the_capacities of an infinite_mind. One might think that an_infinite mind is an analogy to_God as conceived in the Western world,_but this is not quite right_since Takeuti thinks that an_infinite mind_is essentially connected with finite_minds, as we_will discuss_in the next_section. Takeuti explains that testing this_"conjecture" about an_infinite mind leads to three problems_for_the foundations of_mathematics._1._To formulate_the function of_an_infinite mind._2._To justify our intuition of the_world_of an infinite mind using only our_finite mind. 3. To_formulate_the function of a_finite mind. 6 Takeuti's proof_theory in the context of the_Kyoto School_Concerning (1),_Takeuti means finding new axioms of set theory, because set theory_expresses the functions of an infinite_mind. Concerning (2), Takeuti_means his_proof-theoretic_projects, including consistency_proofs._Let us_quote his passage: 2) to justify the_world of_our infinite mind in the world_of a finite mind._We_want to confirm the rationality of_modern mathematics since the world of_our mind is finite and_the_world_of modern mathematics is just_conjecture about our infinite mind. For_this, we need_to develop mathematics of the world of_a_finite mind sufficiently. (Takeuti, 1972, p._173;_translated by the authors) He writes_that_we_need to confirm the "rationality"_of modern mathematics (such as set_theory or analysis) from the perspective of our finite_minds, and this_requires a finitist treatment. The_word_"rationality"_is translated from「合理性」, which means that the world of an_infinite mind_comports with the_world of a finite mind. This can be investigated, among_other ways, by checking that the former_is not contradictory with the latter. The expressions "world of our_infinite mind" and "world of a finite mind"_also call for further discussion, which we_postpone to the next_section._In