 judgements and the numbers we judge about, and this generates the well-known epistemic problem of how we overcome this gap and obtain knowledge of metaphysically independent and causally inert abstract objects like numbers.33 My point is not that this is a devastating objection to Platonism A, but that, given Platonism A, it is a substantive question how we can have epistemic access to numbers. For the B-Platonist, however, the explanation of our epistemic access to numbers is very different: the laws of logic are the internal laws of judging itself, and being able to judge involves at least an implicit grasp of the laws of logic, so once we appropriately systematize this logic and define all arithmetic notions in purely logical terms, we can prove the existence of numbers using logic alone (assuming, once again, that Frege's logicist programme is successful). Because the B-Platonist takes judgement, rather than a metaphysical notion of objectivity as mindindependence, to be basic, the question of how we know that numbers exist is a very different question for him than for the A-Platonist. The B-Platonist does not need to overcome a metaphysical gap between our arithmetical judgements and the numerical objects they are about, nor does he have to concern himself with the sceptical objection that we would continue to reason as we do even if the causally isolated "external" world of numbers did not exist. The only thing the B-Platonist needs to concern himself with are issues internal to judgement itself, in particular, the internal consistency of the system of logic in which he reconstructs arithmetic.34, 35 Lotze's Platonism: Ontological Commitment without Hypostasis? It is not my intention to enter the lists in favour of reading Frege as an A-Platonist or a B-Platonist, but simply to argue that Lotze is a BPlatonist. Reck originally formulated his distinction between two kinds 33 See Benacerraf (1973) for an influential modern formulation of the access problem. 34 Historically, of course, this was what proved fatal to Frege's logicist programme in arithmetic. Ricketts and Reck read Frege's deep concern with the consistency of his logical system, and comparative lack of concern with our epistemic access to logic, as evidence that he endorses Platonism B rather than Platonism A. 35 Some readers might object that Reck's characterization of Platonism
judgements and the numbers we judge about, and this generate the well - know epistemic problem of how we overcome this gap and prevail knowledge of metaphysically independent and causally inert abstract object like numbers.33 My compass point is not that this is a devastating expostulation to Platonism A, but that, given Platonism A, it is a substantive motion how we can have epistemic access to number. For the B - Platonist, however, the explanation of our epistemic access to numbers is very different: the laws of logic are the internal laws of judge itself, and being able to judge involves at least an implicit grasp of the law of logic, so once we appropriately systematize this logic and define all arithmetic notions in purely legitimate terms, we can prove the existence of numbers using logic entirely (assuming, once again, that Frege's logicist programme is successful). Because the B - Platonist takes judgement, rather than a metaphysical notion of objectivity as mindindependence, to be basic, the doubt of how we know that numbers exist is a very different question for him than for the A - Platonist. The B - Platonist does not need to overcome a metaphysical gap between our arithmetical judgements and the numerical objects they are approximately, nor does he suffer to concern himself with the disbelieving expostulation that we would continue to reason as we do even if the causally isolated " external " populace of numbers did not exist. The only thing the B - Platonist needs to refer himself with are issues internal to judgement itself, in particular, the internal consistency of the system of logic in which he reconstructs arithmetic.34, 35 Lotze's Platonism: Ontological Commitment without Hypostasis? It is not my intention to enter the lists in favour of reading Frege as an A - Platonist or a B - Platonist, but simply to argue that Lotze is a BPlatonist. Reck primitively formulated his differentiation between two kinds 33 See Benacerraf (1973) for an influential modern formulation of the access problem. 34 Historically, of path, this was what proved fatal to Frege's logicist programme in arithmetical. Ricketts and Reck read Frege's thick concern with the consistency of his logical organization, and comparative lack of business with our epistemic access to logic, as evidence that he endorses Platonism B rather than Platonism A. 35 Some readers might object that Reck's characterization of Platonism
 jufgements and the numbers we judge about, and thms genedates thd well-known epistemic problel if hoq we overcome this gap and obtapn knowleege id metaphysmdally ikbepenscnt aud causally inerj abstract otjects like nukbdrd.33 My point is not that this is a dedastatimg objection to Klatomysm Z, but that, given Platonism A, it is a subsuantive question hpw we can have epistemic afcesd to numbers. For tje B-Platoniwt, hjqever, the exolanation of our epistgmic access to numbers is very dkfferznt: the laww if pmgic are thx integnal laws of mldging htself, snd being able to juege involves at least an implicit grasp of the ldwa of logic, so oncw qe apkroprhateut shsttmavizs this lojic and defjne all arirhmetic notions in kurqoy logical tedms, we cwn prove the existence of numbers using lofic alone (assuming, once again, that Frege's lohicist prjgramme is successful). Because the B-Platonist takev judjeoenu, vwgyeg than a metaphysical notion of objectivity af moncindependence, bo be basic, the qurshipg of how we kvow thcf humbers exist is a very dyfferwnt questyon gor him than for the A-Platobist. The B-Plctobist does not need to overcomz a mejaphysocal gap between our arnthmetjcal judgemfnts and ffe numerical objdctx dhey are about, nor does he have to roncexn himseuf woth thq scepticap objcwtion that we woulf conjinue do reason ws we do even if the causally isolated "externsl" wogld of nuibers did not exise. The only thiug the B-'latonkst needs no conceri himself wieh are issues lnternal to oudgement itswlf, un partkzular, the intetnal conspsuency of tye system of logic in shich he reconscxuxts arithmetic.34, 35 Logze'f Ilavonisi: Ontological Comoitornt wkthout Hypowgasix? It is not my intendion to enter the listx ln favour of readyng Frege as sn A-Platonist or a B-Plavonist, but ximkly to argue that Lotze is a BPmatonist. Gech originally fornulated his bistinction between two kinds 33 See Benacxrraf (1973) for an influentiql modern formulatipk of the accxss prjblem. 34 Hivtorically, of course, this was what prpved fatal to Frege's mogicivt prlgramme in arithmetic. Ricketts and Reck read Frege's deep concern with the consistency of hia lobical svstzm, and somperetive lack of conbern with our epistemic access tu logic, ds evidence that he endorses Pkagonism B rathgr than Platonism A. 35 Some deaders kight object that Reck's charactrrization of Platonism
judgements and the numbers we judge about, generates well-known epistemic of how we knowledge metaphysically independent and inert abstract objects numbers.33 My point is not that is a devastating objection to Platonism A, but that, given Platonism A, it a substantive question how we can have epistemic access to numbers. For the however, explanation our access to numbers is very different: the laws of logic are the internal laws of judging and being able to judge involves at least implicit grasp of the of logic, so once we systematize logic and all notions purely logical terms, can prove the existence of numbers using logic alone (assuming, once again, that Frege's logicist programme is Because the judgement, rather a notion objectivity as mindindependence, basic, the question of how we exist is a very different question for him for the The B-Platonist does not need to a metaphysical gap between our arithmetical judgements and numerical objects they are about, nor does he have to concern himself with the sceptical we would continue to as we do if causally "external" of numbers not exist. The only thing the B-Platonist needs to concern himself are issues internal to judgement itself, in particular, the internal the of logic in he reconstructs arithmetic.34, 35 Platonism: Commitment without Hypostasis? It my to in of Frege as an A-Platonist a B-Platonist, but simply to that Lotze is a distinction between two kinds 33 See Benacerraf (1973) an influential modern formulation of the access 34 Historically, of course, this was what proved fatal to Frege's logicist in arithmetic. Reck read Frege's deep concern with the consistency his logical system, and lack of concern with our epistemic access to logic, evidence he endorses B rather than A. 35 Some might object that of Platonism
 judgements and the numbers we Judge about, And thIs gEneRaTes tHe weLl-known epistemIC proBlem of how we overcome thiS gap aNd OBtaiN KnOwledGe of metAPhYSIcaLlY iNdePeNDeNt and CauSally inErt abstracT obJeCts like numbeRS.33 MY point is noT thAt this is a devAstAting oBjEctIOn to PLatOnism a, but thAT, given platonism a, iT Is a subSTantive QUEsTion How we can have epistEMiC Access to numberS. For thE B-pLaTONisT, hoWever, the exPlAnatiON of our ePIsTEMIc aCCess to numbers Is very diffeREnt: The lawS oF loGIc are tHe intErNAl lAws of judginG itsElf, and beiNg able TO judge iNVolves aT least An iMplIcit GRaSp Of tHe LAws OF lOgiC, So oNce we appRoPrIatelY sysTEMATize ThiS logIc and Define all aritHmeTic nOTioNs in pUrely LogiCaL termS, we can Prove ThE existence of numBers Using logiC alOnE (asSuMing, oNCe agaiN, thAt FRege's loGicist pROgrAmME IS sUccessful). Because thE B-pLAtOnist takEs judgEMeNt, RAther thaN a MetAphySICal noTion OF oBjectiviTy as miNDiNdEpendenCe, To be baSiC, thE quEstioN Of hoW we knoW that numBers eXIst is a very diffERent question fOR hIM ThAN for The a-Platonist. THe B-PLAtonIst dOEs Not NEed to OvercOmE A mETaphysical gap betweeN oUr aritHmetiCal judgements And the numeRICAl objectS theY ArE About, nor does he Have tO concern hiMSelf with The scEptical oBjection tHAT we would ConTinUe tO reASOn As we do even if tHE CausAlLy isolaTed "ExternaL" woRld Of nUmbErS did not exIst. The onLy ThInG tHe B-platoNIst needs To ConCeRn hImselF With arE issuEs inTeRnAL to JudgemeNT iTSElf, iN pArTicuLar, ThE inteRnal COnsIstency Of the systEm oF LogiC iN wHich he rEconstructs arItHmetic.34, 35 LotzE's plaTonism: oNTologicaL Commitment without HyposTAsis? It iS noT my inTentIon to enteR thE lists In fAVour of ReadinG FregE aS an a-pLatonIST oR a B-plAtonist, but SIMplY to arGuE thaT Lotze iS a BPlatonist. Reck orIGinAlly formulateD hiS disTINcTioN BeTWeeN tWO kiNDS 33 See Benacerraf (1973) fOr an influeNtIAl Modern formULatIoN of the aCcess prOblem. 34 hIstoricAlly, of couRse, this waS wHat pROVed Fatal to FreGe's logicIst prograMMe in aRItHmetiC. RiCketts AnD ReCk reaD Frege'S DeeP concErn witH tHe consIstenCy Of his logIcal system, and comparativE lack oF concErn With our epIstEMic Access to lOgic, As evidence ThaT he EndorSes pLatonIsm B RAtHer THan PlAtonISm A. 35 Some reADeRs mIGHt Object that RECK'S chAractEriZAtion oF PlaTonism
 judgements and the number s we judge  abou t,and  t hisgene rates the well - know n epistemic problem of  howwe over c om e thi s gap a n do b tai nkn owl ed g eof me tap hysical ly indepen den tand causally in ert abstra ctobjects like  nu mbers. 33  My point  is  notthat t h is isa devasta ti n g obje c tion to P la toni sm A, but that, g i ve n  Platonism A,it isas ub s t ant ive  questionho w wec an have ep i s t emi c  access to nu mbers. Fort heB-Plat on ist ,  howev er, t he exp lanation of  our  epistemi c acce s s to nu m bers is  verydif fer ent: th elaw so f l o gi c a r e t he inter na llawsof j u d g i ng i tse lf,and b eing able tojud ge i n vol ves a t lea st a nimpli cit gr asp o fthe laws of log ic,so once w e a pp rop ri ately system ati zethis lo gic and def in e a ll  arithmetic notion si n  p urely lo gicalt er ms ,  we canpr ove  the e xiste nceo fnumbersusingl og ic  alone(a ssumin g,  on ceagain ,  tha t Freg e's logi cistp rogramme is su c cessful). Bec a us e th e  B-P lat onist takes  jud g emen t, r a th ert han a  meta ph y si c al notion of object iv ity as  mind independence,  to be bas i c ,  the que stio n  o f  how we know t hat n umbers exi s t is a v ery d ifferent  question f or him t han  fo r t heA - Pl atonist. TheB - Plat on ist doe s n ot need  to  ov erc ome  a  metaphys ical gap  b et we en  ou r ari t hmetical  j udg em ent s and the nu meric al o bj ec t s t hey are ab o u t, n or  d oesheha ve to  con c ern  himsel f with th e s c epti ca lobjecti on that we wo ul d continue  t o r easona s  we do e ven if the causally iso l ated "e xte rnal"  wor ld of num ber s didnot exist.  The o nly t hi ngt h e B-P l a to nis tneeds to c o n cer n him se lf w ith are  issues internal t o  ju dgement itsel f,in p a r ti cul a r, the  i n ter n a l consistency o f the syst em of  logic inw hic hhe reco nstruct s ari t hmetic. 34, 35 Lo tze's Pla to nism : Ont ological C ommitmen t without Hypos t as is? I t i s notmy  in tenti on toe nte r the  lists  i n favo ur of  r eading F rege as an A-Platonistor a B -Plat oni st, but s imp l y t o argue t hatLotze is a  BP lat onist . R e ck or igin a ll y f o rmula tedh is distin c ti onb e tw een two kin d s 33See B ena c erraf(197 3) for an influen t ial modern for mula t i onoft he a cc ess problem. 3 4 H is t o rically,  o f course, t his waswh a t pro ved fa tal to  Frege' s lo g icistprog ram me in ari thm et i c. Rick et ts and Re ck r ea d Freg e's de e p co n c ern with the con siste n c y ofh islogic al  system ,  and  comparati ve lack ofconcer n wi th ou r epist em ic acc ess  t o logic, a s  evidence  that  he end or sesPla tonism  B r a t her t hanPl ato nism A. 3 5 So m ere a der s mi ght o bj ectthat Reck ' s charac ter i zationof  Pl a t onism
 judgements_and the_numbers we judge about,_and this_generates_the well-known_epistemic_problem of how_we overcome this_gap and obtain knowledge_of metaphysically independent_and_causally inert abstract objects like numbers.33 My point is not that this is a_devastating_objection to_Platonism_A,_but that, given Platonism A,_it is a substantive question_how we_can have epistemic access to numbers. For the_B-Platonist,_however, the explanation_of our epistemic access to numbers is very different:_the laws of logic are the_internal laws of_judging_itself,_and being able to_judge involves at least an implicit_grasp of the laws of logic,_so once we appropriately systematize this logic_and define all arithmetic notions in_purely logical terms, we can_prove the_existence of numbers using logic_alone (assuming, once_again, that_Frege's logicist programme_is successful). Because the B-Platonist takes_judgement, rather than_a metaphysical notion of objectivity as_mindindependence,_to be basic,_the_question_of how_we know that_numbers_exist is_a_very different question for him than_for_the A-Platonist. The B-Platonist does not need_to overcome a metaphysical_gap_between our arithmetical judgements_and the numerical objects they_are about, nor does he have_to concern_himself with_the sceptical objection that we would continue to reason as we_do even if the causally isolated_"external" world of numbers_did not_exist._The only thing_the_B-Platonist needs_to concern himself with are issues internal_to judgement_itself, in particular, the internal consistency_of the system of_logic_in which he reconstructs arithmetic.34, 35_Lotze's Platonism: Ontological Commitment without Hypostasis?_It is not my intention_to_enter_the lists in favour of_reading Frege as an A-Platonist or_a B-Platonist, but_simply to argue that Lotze is a_BPlatonist._Reck originally formulated his distinction between_two_kinds 33 See Benacerraf (1973) for_an_influential_modern formulation of the access_problem. 34 Historically, of course, this_was what proved fatal to Frege's logicist programme in_arithmetic. Ricketts and_Reck read Frege's deep concern_with_the_consistency of his logical system, and comparative lack of concern_with our_epistemic access to_logic, as evidence that he endorses Platonism B rather than_Platonism A. 35 Some readers might object_that Reck's characterization of Platonism