 philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's prooftheoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project. 1. Introduction In this article we will introduce our project of showing that Takeuti's finitist program had its own character and goals, related to but distinct from the classical finitist program of Hilbert and Bernays. Takeuti's program, rooted in his readings of Hilbert and Bernays but also in his background in the philosophy of the Kyoto school of Japanese philosophy, envisions finitism as not only realizing our human limitations as provers but also our relation to what Takeuti called "infinite mind", modern mathematics with its full embrace of infinitary set theory. Finitism realizes the connection between our finite minds and this infinite mind. In embracing this connection between the finite and the infinite, Takeuti's Takeuti's proof theory in the context of the Kyoto School programme also permits the clarification of the mathematical means available to the finitist, going beyond primitive recursive arithmetic. Thus Takeuti's finitism permits his classical Gentzenian proof of the consistency of arithmetic, and indeed points forward toward finitist proofs of consistency of higher-order theories as well. 2. Proof-theoretic background Proof theory is the study of mathematical proofs as mathematical objects themselves, by means of ordinary mathematical methods. This branch of mathematical logic was created by David Hilbert with the goal of proving metamathematical results, like the independence of certain axioms from other axioms, and the consistency of axiom systems. Hilbert set his sights on proving the consistency of elementary arithmetic, because of its foundational importance for other parts of mathematics, especially analysis. For such a proof to have foundational force, the resources used by the proof should be more secure than those whose consistency is under investigation. Hilbert identified a part of mathematical reasoning that he judged to be ideally secure that he called finitary mathematics. He would describe his finitism as follows: Our treatment of the basics of number theory and algebra was meant to demonstrate how to apply and implement direct contentual inference that takes place in thought experiments on intuitively conceived objects and is free of axiomatic assumptions. Let us call this kind of inference "finitist" inference for short, and likewise the
philosophy. In this paper, we aim to describe a cosmopolitan outline of our undertaking to investigate Takeuti's doctrine of mathematics. In particular, after reviewing Takeuti's prooftheoretic results concisely, we describe some key elements in Takeuti's text. By explaining these texts, we indicate out the connection between Takeuti's proof theory and Nishida's doctrine and explain the future goals of our project. 1. Introduction In this article we will precede our project of testify that Takeuti's finitist program had its own fictional character and goals, related to but distinct from the authoritative finitist program of Hilbert and Bernays. Takeuti's program, settle in his readings of Hilbert and Bernays but besides in his background in the doctrine of the Kyoto school of Japanese philosophy, envisions finitism as not only realizing our human limitations as provers but besides our relation to what Takeuti called " infinite mind ", modern mathematics with its full embrace of infinitary set theory. Finitism realizes the connection between our finite minds and this infinite mind. In embracing this connection between the finite and the infinite, Takeuti's Takeuti's proof theory in the context of the Kyoto School program also permits the clearing of the numerical means available to the finitist, going beyond primitive recursive arithmetic. Thus Takeuti's finitism allow his classical Gentzenian proof of the consistency of arithmetic, and indeed points ahead toward finitist proofs of consistency of higher - order theories as well. 2. Proof - theoretic backdrop Proof theory is the study of mathematical proofs as numerical objects themselves, by mean of ordinary mathematical methods. This branch of mathematical logic was created by David Hilbert with the finish of proving metamathematical results, like the independence of certain axioms from other axioms, and the consistency of axiom systems. Hilbert set his sights on prove the consistency of elementary arithmetic, because of its foundational importance for other parts of mathematics, especially analysis. For such a proof to have foundational military unit, the resources practice by the proof should be more dependable than those whose consistency is under investigation. Hilbert identified a part of mathematical reasoning that he judged to be ideally dependable that he called finitary mathematics. He would describe his finitism as follows: Our treatment of the basics of number theory and algebra was meant to demonstrate how to use and implement direct contentual inference that subscribe place in thought experiment on intuitively conceived objects and is free of axiomatic assumptions. Let us call this kind of inference " finitist " inference for short, and likewise the
 phllosophy. In this paper, wt aim to describe a genecal outmine of uur project to investigate Tekeuri's pyilosophy of mathematizs. In parnicular, adter eeviewing Vzkeuti's prooffmeorecir results briefky, we descsibe some key aldmznts in Takeuti's texts. By explaining these yedts, we point oot tht cjnnedniin between Takeuti's proof thekry and Nishida's phikosophy and explain the fuhure goals of our projfct. 1. Intrododtijb In this argicle we wpnl introdude our project of showing that Gakeuci's finitisj lrlcram had itw own character akc goalv, relatrd to but distlnct hrom the classical finitivt program of Hilbgrt and Besncys. Takeuti's program, eoited hn hhs rdqdivgs oh Hjlbert anv Bernays bht also in yis background in tne inilosophy of the Kroeo school of Japanese philosophy, envisimns finitism as not only rwalizing our human lilitations as provers but also our relation to what Takeuti callxd "inylkite nijd", modern mathematics with its full embrace os imfpnitary set theorn. Finitism realizex hhr connection bgtween our finite minds and hhis insinitw mind. In embtacing this connection betwwen the finine abd the infinite, Taneuti's Takebti's ptoof tneory in the context of the Kyoto Scholl prograjoe also permits ghe cnarificaukon of the mathemwtical meens arailable to jhe finytist, goinh beyond primitive recurdive crithketic. Thus Takeuti's finitism permits his rkassical Genjzetiat proof jf thc consistency os arithmetic, aud indeeb poings forward toward finitist pwoofs of conshdtency of hijher-order theiriew as weuu. 2. Proof-theoreyic backggobnd Proof theory is the stuvy ow mathematical pxuifs as mathematocau ofjvctx evemselves, by meavs ug ordknary mathenxticsl methods. This branwh or mathematical logoc was creqted by Qavid Hilbert with the goal of krovinj metaoathekatycal results, like the independsnce of cfrtwin axioms fwom ither axioms, and the consistency of axiom systems. Hmlbert set his sights ob proving the consixuency of elemxntary arithmethc, because of its foyndational importsnce for other parts kf matvematlcs, especially analysis. For such a proof to have foundational force, the rwsourcxs used by ths prpof svobld be mowe sxcnre than those whpse consistency is under investmgation. Hinbzrt identified a part of mathrmxtical reasonkng that he judged to be ideally secure that he called finitary mathematics. He would dexcribe hus fihitism as hollows: Our treatmeny of tke basucs of bumbcr theory and xlgrbva was meajt tp bemonstrate how to apply and implwment eigect contentual mnference thwu takes place in thought expedimehhs on intuitively vonceived objevts and is free of aqioiatic assumptiihs. Let us cakl this kind of infxrence "finithst" inference for short, and likewise the
philosophy. In this paper, we aim to general of our to investigate Takeuti's after Takeuti's prooftheoretic results we describe some elements in Takeuti's texts. By explaining texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy explain the future goals of our project. 1. Introduction In this article we introduce project showing Takeuti's finitist program had its own character and goals, related to but distinct from the classical program of Hilbert and Bernays. Takeuti's program, rooted his readings of Hilbert Bernays but also in his in philosophy of Kyoto of philosophy, envisions finitism not only realizing our human limitations as provers but also our relation to what Takeuti called "infinite modern mathematics full embrace infinitary theory. realizes the connection finite minds and this infinite mind. connection between the finite and the infinite, Takeuti's proof theory the context of the Kyoto School also permits the clarification of the mathematical means to the finitist, going beyond primitive recursive arithmetic. Thus Takeuti's finitism permits his classical Gentzenian the consistency of arithmetic, indeed points forward finitist of of theories as 2. Proof-theoretic background Proof theory is the study of mathematical proofs mathematical objects themselves, by means of ordinary mathematical methods. This mathematical was created by Hilbert with the goal proving results, like the independence axioms other consistency axiom Hilbert set his sights proving the consistency of elementary because of its foundational mathematics, especially analysis. For such a proof to foundational force, the resources used by the should be more secure than those whose consistency is under investigation. Hilbert a part reasoning that he judged to be ideally secure he called finitary mathematics. would describe his finitism as follows: Our treatment of basics number theory algebra was meant demonstrate how to and implement direct that takes in experiments objects and is free of axiomatic Let call this kind of inference inference and likewise the
 philosophy. In this paper, we aiM to describE a genEraL ouTlIne oF our Project to invesTIgatE Takeuti's philosophy of mAthemAtICs. In PArTiculAr, after REvIEWinG TAkEutI's PRoOftheOreTic resuLts briefly, We dEsCribe some key ELeMents in TakEutI's texts. By expLaiNing thEsE teXTs, we pOinT out tHe connECtion bEtween TakEuTI's prooF Theory aND niShidA's philosophy and exPLaIN the future goalS of our PrOJeCT. 1. intRodUction In thIs ArticLE we will INtRODUce OUr project of shOwing that TaKEutI's finiTiSt pROgram hAd its OwN ChaRacter and goAls, rElated to bUt distINct from THe classIcal fiNitIst ProgRAm Of hilBeRT anD beRnaYS. TaKeuti's prOgRaM, rootEd in HIS REadiNgs Of HiLbert And Bernays but AlsO in hIS baCkgroUnd in The pHiLosopHy of thE KyotO sChool of Japanese PhilOsophy, envIsiOnS fiNiTism aS Not onlY reAliZing our Human liMItaTiONS As Provers but also our rElATIoN to what TAkeuti CAlLeD "Infinite MiNd", mOderN MAthemAticS WiTh its fulL embraCE oF iNfinitaRy Set theOrY. FiNitIsm reALizeS the coNnection BetweEN our finite mindS And this infiniTE mIND. IN EmbrAciNg this conneCtioN BetwEen tHE fIniTE and tHe infInITe, tAkeuti's Takeuti's prooF tHeory iN the cOntext of the KyOto School pROGRamme alsO perMItS The clarificatiOn of tHe mathematICal means AvailAble to thE finitist, GOIng beyonD prImiTivE reCURsIve arithmetic. tHUs TaKeUti's finItiSm permiTs hIs cLasSicAl gentzeniaN proof of ThE cOnSiSteNcy of ARithmetiC, aNd iNdEed PointS ForwarD towaRd fiNiTiST prOofs of cONsISTencY oF hIgheR-orDeR theoRies AS weLl. 2. Proof-Theoretic BacKGrouNd prOof theoRy is the study oF mAthematicaL pRooFs as maTHEmatical Objects themselves, by meanS Of ordinAry MatheMatiCal methodS. ThIs branCh oF MathemAtical Logic WaS crEATed by dAViD HiLbErt with the GOAl oF provInG metAmathemAtical results, like tHE inDependence of cErtAin aXIOmS frOM oTHer AxIOms, AND the consistency Of axiom sysTeMS. HIlbert set hIS siGhTs on proVing the ConsiSTency of ElementarY arithmetIc, BecaUSE of Its foundatIonal impOrtance foR Other PArTs of mAthEmaticS, eSpeCiallY analySIs. FOr sucH a prooF tO have fOundaTiOnal forcE, the resources used by the pRoof shOuld bE moRe secure tHan THosE whose conSistEncy is undeR inVesTigatIon. hIlberT ideNTiFieD A part Of maTHematical REaSonING tHat he judged TO BE idEally SecURe that He caLled finitary matheMAtics. He would deScriBE His FinITism As Follows: Our treaTmeNt OF The basicS oF number theoRy and algEbRA was mEant to DemonsTrate hoW TO aPPly and ImplEmeNt direct cOntEnTUal infeReNcE That taKes pLaCe in thOught eXPeriMENts on intuitively ConceIVEd objECts And is FrEe of axiOMatiC assumptioNs. Let us call This kiNd of InferEnce "finItIst" infEreNcE for short, aND likewise The
 philosophy. In this paper , we aim t o des cri beagene raloutline of our proj ect to investigate Tak euti' sp hilo s op hy of  mathem a ti c s . I npa rti cu l ar , aft erreviewi ng Takeuti 'spr ooftheoretic re sults brie fly , we describ e s ome ke yele m entsinTakeu ti's t e xts. B y explain in g  these texts,w e  p oint  out the connecti o nb etween Takeuti 's pro of th e o ryand  Nishida's  p hilos o phy and ex p l a int he future goa ls of our p r oje ct. 1.  I ntr o ductio n Inth i s a rticle we w illintroduce  our p r oject o f  showin g that  Ta keu ti's fi ni tis tp rog r am  ha d  it s own ch ar ac ter a nd g o a l s , re lat ed t o but  distinct fro m t he c l ass icalfinit istpr ogram  of Hi lbert  a nd Bernays. Tak euti 's progra m,ro ote din hi s  readi ngs  of  Hilber t and B e rna ys b u talso in his backgr ou n d  i n the ph ilosop h yof the Kyot osch oolo f  Japa nese ph ilosophy , envi s io ns  finiti sm  as no tonl y r ealiz i ng o ur hum an limit ation s  as provers bu t  also our rel a ti o n  t o  wha t T akeuti call ed " i nfin item in d", moder n mat he m at i cs with its full em br ace of  infi nitary set th eory. Fini t i s m realiz es t h ec onnection betw een o ur finitem inds and  this  infinit e mind. I n embracin g t his  co nne c t io n between the f init eand the  in finite,  Ta keu ti' s T ak euti's pr oof theo ry  i nth e c ontex t  of theKy oto  S cho ol pr o gramme  also  per mi ts the  clarif i ca t i on o fth e ma the ma tical  mea n s a vailabl e to thefin i tist ,go ing bey ond primitive  r ecursive a ri thm etic.T h us Takeu ti's finitism permits h i s class ica l Gen tzen ian proof  of  the c ons i stency  of ar ithme ti c,a n d ind e e dpoi nt s forwardt o war d fin it istproofsof consistency ofh igh er-order theo rie s as w el l.2 .P roo f- t heo r e tic backgroundProof theo ry is  the study ofma themati cal pro ofs a s  mathem atical ob jects the ms elve s ,  by  means ofordinary  mathemat i cal m e th ods.Thi s bran ch  of  math ematic a l l ogicwas cr ea ted by  Davi dHilbertwith the goal of provin g meta mathe mat ical resu lts ,  li ke the in depe ndence ofcer tai n axi oms fromothe r  a xio m s, an d th e  consiste n cy  of a xi om systems. H i lbe rt se t h i s sigh ts o n proving the con s istency of ele ment a r y a rit h meti c,  because of it s f ou n d ationalim portance fo r otherpa r ts of  mathe matics , espec i a ll y  analy sis.  Fo r such apro of to have  f ou n dation al f or ce, th e reso u rces u sed by the proof  shou l d  be m o resecur ethan th o se w hose consi stency is u nder i nves tigat ion. Hi lb ert id ent if ied a part of mathem atica l reaso ni ng t hat  he ju dged t o beidea ll y s ecure tha t he ca ll e d f init ary m at hema tics. Hew ould des cri b e his f in iti s m  as fo l lo w s : Our trea tme nt of t he basicso f nu m be r  theo ry and  algeb ra wasm ean tto demo nst r a te how to  apply an d  im pl emen t direct  c onte nt ual  i nference that t akes pla ce in th o u ght ex per i ment s  on in tu itively conc eived  o b jects a n d  is  free o f  ax iom ati c as sum ptio n s. Let us call this kind  ofinfer e nce  "fin itist "inferenc e  for  short,a nd likewise t he
 philosophy._In this_paper, we aim to_describe a_general_outline of_our_project to investigate_Takeuti's philosophy of_mathematics. In particular, after_reviewing Takeuti's prooftheoretic_results_briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we_point_out the_connection_between_Takeuti's proof theory and Nishida's_philosophy and explain the future_goals of_our project. 1. Introduction In this article we_will_introduce our project_of showing that Takeuti's finitist program had its own_character and goals, related to but_distinct from the_classical_finitist_program of Hilbert and_Bernays. Takeuti's program, rooted in his_readings of Hilbert and Bernays but_also in his background in the philosophy_of the Kyoto school of Japanese_philosophy, envisions finitism as not_only realizing_our human limitations as provers_but also our_relation to_what Takeuti called_"infinite mind", modern mathematics with its_full embrace of_infinitary set theory. Finitism realizes the_connection_between our finite_minds_and_this infinite_mind. In embracing_this_connection between_the_finite and the infinite, Takeuti's Takeuti's_proof_theory in the context of the Kyoto_School programme also permits_the_clarification of the mathematical_means available to the finitist,_going beyond primitive recursive arithmetic. Thus_Takeuti's finitism_permits his_classical Gentzenian proof of the consistency of arithmetic, and indeed points_forward toward finitist proofs of consistency_of higher-order theories as_well. 2._Proof-theoretic_background Proof theory_is_the study_of mathematical proofs as mathematical objects themselves,_by means_of ordinary mathematical methods. This branch_of mathematical logic was_created_by David Hilbert with the goal_of proving metamathematical results, like the_independence of certain axioms from_other_axioms,_and the consistency of axiom_systems. Hilbert set his sights on_proving the consistency_of elementary arithmetic, because of its foundational_importance_for other parts of mathematics, especially_analysis._For such a proof to have_foundational_force,_the resources used by the_proof should be more secure than_those whose consistency is under investigation. Hilbert identified a_part of mathematical_reasoning that he judged to_be_ideally_secure that he called finitary mathematics. He would describe his_finitism as_follows: Our treatment_of the basics of number theory and algebra was meant_to demonstrate how to apply and implement_direct contentual inference that takes place in thought experiments on intuitively_conceived objects and is free of axiomatic assumptions._Let us call this kind of inference_"finitist" inference for short,_and_likewise the