 ne the "essence" of a thing, but one cannot attain an Idea through the Socratic question "What is...?" (which posits Ideas as transcendent and eternal), but rather through "minor" questions such as "Which one?" "Where?" "When?" "How?" "How many?" "In which case?" "From which viewpoint?" – all of which allow one to defi ne the spatiotemporal coordinates of Ideas that are immanent and diff erential. Kant posited Ideas as "regulative" notions that serve to unify and systematize the operations of the understanding: concepts fi nd the ground of their maximal experimental use only insofar as they are related to Ideas, which Kant defi nes as foci or horizons that lie outside the bounds of experience. But an object outside experience can be represented only in a problematic form: it can neither be given nor known, but must be represented without being able to be directly determined. Th is undetermined object of the Idea marks neither an imperfection in our knowledge nor a lack in the object: it is a perfectly positive and objective structure that allows us to represent other objects (those of experience) which it endows with a maximum of systematic unity. In Kant, then, Ideas present three aspects: they are undetermined with regard to their object, but nonetheless determinable with regard to the objects of experience, and bear the ideal of a complete determination with regards to the concepts of the understanding. Because he held fast to the point of view of conditioning, however, two of these three moments remain as extrinsic characteristics in Kant. In Diff erence and Repetition, Deleuze aims to develop a theory of problematic Ideas that is genetic and not merely conditioning, following Maimon's contention that the duality between concepts and intuitions can be bridged only by positing ideas of diff erence within sensibility itself. Th e formal criteria Deleuze uses to defi ne Ideas are largely derived from Leibniz and the model of the diff erential calculus, which provides a mathematical symbolism for the exploration of the real: things or beings are virtual and problematic multiplicities composed of singularitiesevents, which are prolonged in converging and diverging series, forming zones of indiscernibility where the multiplicities enter into perpetual becomings. Drawing on the work of Jules Vuillemin and Albert Lautman, among others, in Volume 6 daniel w. smith 104 the philosophy of mathematics, De
ne the " essence " of a thing, but one cannot attain an Idea through the Socratic question " What is... ? " (which postulate Ideas as transcendent and ageless), but rather through " minor " questions such as " Which one? " " Where? " " When? " " How? " " How many? " " In which case? " " From which vantage point? " – all of which allow one to defi ne the spatiotemporal coordinates of Ideas that are immanent and diff erential. Kant situate Ideas as " regulative " notions that serve to mix and systematize the operations of the understanding: concept fi nd the ground of their maximal experimental use only insofar as they are related to Ideas, which Kant defi nebraska as foci or horizons that lie outside the bounds of experience. But an object outside experience can be represent only in a problematic form: it can neither be give nor known, but must be represented without being able to be directly determined. Th is indeterminate object of the Idea marks neither an imperfection in our knowledge nor a lack in the object: it is a perfectly plus and objective structure that allows us to represent other objects (those of experience) which it endow with a maximum of systematic unity. In Kant, then, Ideas present three aspects: they are undetermined with regard to their object, but nonetheless determinable with respect to the objects of experience, and bear the ideal of a arrant decision with regards to the concepts of the understanding. Because he hold fast to the point of view of conditioning, however, two of these three moments persist as extrinsic characteristics in Kant. In Diff erence and Repetition, Deleuze aims to develop a theory of problematic Ideas that is genetic and not merely conditioning, following Maimon's contention that the duality between concept and intuitions can be bridge only by positing ideas of diff erence within sensibility itself. Th east formal criteria Deleuze uses to defi ne Ideas are largely derive from Leibniz and the model of the diff erential calculus, which provides a mathematical symbolism for the exploration of the substantial: things or beings are virtual and debatable multiplicities compose of singularitiesevents, which are prolonged in converge and diverging serial, forming zones of indiscernibility where the multiplicities accede into perpetual becomings. draw on the work of Jules Vuillemin and Albert Lautman, among others, in Volume 6 daniel w. smith 104 the philosophy of mathematics, De
 ne the "essence" of a thing, nut one cannot ajtqin an Idea fhrough ghe Socratic question "What id...?" (qhich posits Ideas as transzendent ajd eternql), bnt rather througi "minor" questikks sueh as "Which one?" "Ehere?" "When?" "How?" "How many?" "Hn wkich case?" "From which viewpoint?" – all jf whicn wllow one to dgfi nt tre slatiotemporal coordinates of Ideas that age immanent and doff erential. Kant posited Ldead as "regulative" nohions that wervq to unify ana systematpve the operztions of the understanding: conzepts fi nd the geoujg of their naximwl experimenbsl use only imsofar as they arx reoated to Ideas, which Nant defi nes as fjci or hosivons that lie outside tye boonds mf ebperkende. Bht an lbjxct outside experience can be represented ogoy in a problsmatic fjrm: it can neither be given nor known, blt mhst be represented withiut being able to be firectly qetermined. Th is undetermined object of the Idea karks veiuhcv an umoerfection in our knowledge nor a lack in the kbkebt: it is a perfecbly positive and onjfcyyve structure that cmlkws us to represenh other objexts (those of rxperience) which it endows qith a maximlm od systematic unity. In Kant, thzn, Idess prrsent three aspects: thev are hndeterminef with refxrd to their objdct, bgt nonetheless determinablq with rejard co the ocjecjs of evperience, wnd bcdr the ideal of a fomplgte dederminatioj with regards to the concepts of the understsnginc. Becausz he hcld fast to the point of view of conbitionkng, howeveg, two of vhese three ioments remait as extrinsir charactqrisrics in Kang. In Diff erencr and Repvtntion, Delwuze aims to develpp x theory of proboemqtic Ideas that is gegenic agg not merely conaitkpning, following Nximom's contention that tve dhality between conveits and ibtuitionf can be bridbed only by positijg idxas of diff ergnce within sensibility itself. Fh e formwl griteria Deletze mses to defi nz Ideas are largely derived from Leibniz and the model of rhe diff erential cclgulus, which 'rovidqs a mathamatical symbolism fir the exploratiok of the real: things od beincs arf virtual and problematic multiplicities composed of singularitiesevents, qhich erq prolonged in vonveggiug cnd divqrgiig series, forming dones of indiscernibility where vhe multipnieities enter into perpetual brcumings. Drawine on the work of Jules Vhillemin and Albert Lautman, among otherx, in Volume 6 daniel w. smlth 104 the philoaophy of methematics, De
ne the "essence" of a thing, but attain Idea through Socratic question "What transcendent eternal), but rather "minor" questions such "Which one?" "Where?" "When?" "How?" "How "In which case?" "From which viewpoint?" – all of which allow one to ne the spatiotemporal coordinates of Ideas that are immanent and diff erential. Kant Ideas "regulative" that to unify and systematize the operations of the understanding: concepts fi nd the ground of their experimental use only insofar as they are related Ideas, which Kant defi as foci or horizons that outside bounds of But object experience can be only in a problematic form: it can neither be given nor known, but must be represented without able to determined. Th undetermined of Idea marks neither in our knowledge nor a lack it is a perfectly positive and objective structure allows us represent other objects (those of experience) it endows with a maximum of systematic unity. Kant, then, Ideas present three aspects: they are undetermined with regard to their object, but with regard to the of experience, and the of complete with regards the concepts of the understanding. Because he held fast to the of view of conditioning, however, two of these three moments extrinsic in Kant. In erence and Repetition, Deleuze to a theory of problematic is and following contention the duality between concepts intuitions can be bridged only positing ideas of diff e formal criteria Deleuze uses to defi ne are largely derived from Leibniz and the of the diff erential calculus, which provides a mathematical symbolism for the of the or beings are virtual and problematic multiplicities composed singularitiesevents, which are prolonged converging and diverging series, forming zones of indiscernibility where multiplicities into perpetual Drawing on the of Jules Vuillemin Albert Lautman, among Volume 6 w. 104 mathematics, De
 ne the "essence" of a thing, but onE cannot attAin an ideA thRoUgh tHe SoCratic question "wHat iS...?" (which posits Ideas as traNscenDeNT and ETeRnal), bUt ratheR ThROUgh "MiNoR" quEsTIoNs sucH as "which onE?" "Where?" "When?" "how?" "hoW many?" "In which CAsE?" "From which VieWpoint?" – all of wHicH allow OnE to DEfi ne The SpatiOtempoRAl coorDinates of idEAs that ARe immanENT aNd diFf erential. Kant posITeD ideas as "regulatIve" notIoNS tHAT seRve To unify and SyStemaTIze the oPErATIOns OF the understanDing: conceptS Fi nD the grOuNd oF Their mAximaL eXPerImental use oNly iNsofar as tHey are RElated tO ideas, whIch KanT deFi nEs as FOcI oR hoRiZOns THaT liE OutSide the bOuNdS of exPeriENCE. but aN obJect OutsiDe experience cAn bE repREseNted oNly in A proBlEmatiC form: iT can nEiTher be given nor kNown, But must be RepReSenTeD withOUt beinG abLe tO be direCtly detERmiNeD. tH Is Undetermined object Of THE IDea marks NeitheR An ImPErfectioN iN ouR knoWLEdge nOr a lACk In the objEct: it iS A pErFectly pOsItive aNd ObjEctIve stRUctuRe that Allows us To repREsent other objeCTs (those of expeRIeNCE) wHIch iT enDows with a maXimuM Of syStemATiC unITy. In KAnt, thEn, iDeAS present three aspectS: tHey are UndetErmined with reGard to theiR OBJect, but nOnetHElESs determinable With rEgard to the OBjects of ExperIence, and Bear the idEAL of a compLetE deTerMinATIoN with regards tO THe coNcEpts of tHe uNderstaNdiNg. BEcaUse He Held fast tO the poinT oF vIeW oF coNditiONing, howeVeR, twO oF thEse thREe momeNts reMain As ExTRinSic charACtERIstiCs In kant. in DIfF erenCe anD repEtition, deleuze aiMs tO DeveLoP a Theory oF problematic IDeAs that is geNeTic And not MERely condItioning, following Maimon'S ContentIon That tHe duAlity betwEen ConcepTs aND intuiTions cAn be bRiDgeD ONly by POSiTinG iDeas of diff EREncE withIn SensIbility Itself. Th e formal criTEriA Deleuze uses tO deFi ne iDEaS arE LaRGelY dERivED From Leibniz and tHe model of tHe DIfF erential cALcuLuS, which pRovides A mathEMatical Symbolism For the expLoRatiON Of tHe real: thinGs or beinGs are virtUAl and PRoBlemaTic MultipLiCitIes coMposed OF siNgulaRitiesEvEnts, whIch arE pRolonged In converging and diverginG serieS, formIng Zones of inDisCErnIbility whEre tHe multipliCitIes Enter IntO PerpeTual BEcOmiNGs. DraWing ON the work oF juLes vUIlLemin and AlbERT lauTman, aMonG Others, In VoLume 6 daniel w. smith 104 tHE philosophy of mAtheMATicS, De
 ne the "essence" of a thi ng, but on e can not  at ta in a n Id ea through the Socr atic question "What is ...?"  ( w hich po sitsIdeas a s  t r a nsc en de ntan d  e terna l),  but ra ther throu gh"m inor" questi o ns  such as " Whi ch one?" "Wh ere ?" "Wh en ?"" How?"  "H ow ma ny?" " I n whic h case?""F r om whi c h viewp o i nt ?" –  all of which all o wo ne to defi nethe sp at i ot e m por alcoordinate sof Id e as that ar e i mma n ent and differential. K a ntposite dIde a s as " regul at i ve"  notions th at s erve to u nify a n d syste m atize t he ope rat ion s of th eund er s tan d in g:c onc epts find  t he gr ound o f thei r m axim al ex perimental us e o nlyi nso far a s the y ar erelat ed toIdeas ,which Kant defi  nes  as fociorho riz on s tha t  lie o uts ide  the bo unds of exp er i e n ce . But an object ou ts i d eexperien ce can be  r e presente donl y in a  prob lema t ic  form: i t cann ei th er be g iv en nor  k now n,but m u st b e repr esentedwitho u t being able t o  be directlyd et e r mi n ed.This undeterm ined obje ct o f  t heI dea m arksne i th e r an imperfection i nour kn owled ge nor a lack  in the ob j e c t: it is  a p e rf e ctly positiveand o bjective s t ructurethatallows u s to repr e s ent othe r o bje cts  (t h o se  of experienc e )  whi ch  it end ows  with a  ma xim umofsy stematicunity. I nKa nt ,the n, Id e as prese nt  th re e a spect s : they  areunde te rm i ned  with r e ga r d  toth ei r ob jec t,  butnone t hel ess det erminable  wi t h re ga rd  to the  objects of e xp erience, a nd  be ar the i deal ofa complete determinatio n  with r ega rds t o th e concept s o f theund e rstand ing. B ecaus eheh e ld fa s t  t o t he  point ofv i ewof co nd itio ning, h owever, two of the s e t hree momentsrem aina s  e xtr i ns i c c ha r act e r istics in Kant.  In Diff e re n ce  and Repet i tio n,  Deleuz e aimsto de v elop atheory of  problema ti c Id e a s t hat is gen etic and  not mere l y con d it ionin g,follow in g M aimon 's con t ent ion t hat th edualit y bet we en conce pts and intuitions canbe bri dgedonl y by posi tin g  id eas of di ff e rence with insen sibil ity itsel f. T h  e  fo r mal c rite r ia Deleuz e  u ses t odefi ne Ide a s are  larg ely derive d fr om Leibniz and th e  model of thediff e ren tia l  cal cu lus, which pro vid es a  mathema ti cal symboli sm for t he explo ration  of th e real: t hi n gs orbein gsare virtu alan d  proble ma ti c  multi plic it ies co mposed of s i n gularitiesevents , whi c h  arep rol onged  i n conve r ging  and diver ging series , form ingzones  of ind is cernib ili ty  where the multiplic ities  enterin to p erp etualbeco m i ngs.Draw in g o n the wor k of Ju le s  Vu ille min a nd  Alb ert Lautm a n, among  ot h ers, in  V olu m e  6 dan i el w . smith 10 4 t he ph i l osophy ofm athe m at i cs, D e
 ne_the "essence"_of a thing, but_one cannot_attain_an Idea_through_the Socratic question_"What is...?" (which_posits Ideas as transcendent_and eternal), but_rather_through "minor" questions such as "Which one?" "Where?" "When?" "How?" "How many?" "In which_case?"_"From which_viewpoint?"_–_all of which allow one_to defi ne the spatiotemporal_coordinates of_Ideas that are immanent and diff erential. Kant_posited_Ideas as "regulative"_notions that serve to unify and systematize the operations_of the understanding: concepts fi nd_the ground of_their_maximal_experimental use only insofar_as they are related to Ideas,_which Kant defi nes as foci_or horizons that lie outside the bounds_of experience. But an object outside_experience can be represented only_in a_problematic form: it can neither_be given nor_known, but_must be represented_without being able to be directly_determined. Th is_undetermined object of the Idea marks_neither_an imperfection in_our_knowledge_nor a_lack in the_object:_it is_a_perfectly positive and objective structure that_allows_us to represent other objects (those of_experience) which it endows_with_a maximum of systematic_unity. In Kant, then, Ideas_present three aspects: they are undetermined_with regard_to their_object, but nonetheless determinable with regard to the objects of experience,_and bear the ideal of a_complete determination with regards_to the_concepts_of the understanding._Because_he held_fast to the point of view of_conditioning, however,_two of these three moments remain_as extrinsic characteristics in_Kant._In Diff erence and Repetition, Deleuze_aims to develop a theory of_problematic Ideas that is genetic_and_not_merely conditioning, following Maimon's contention_that the duality between concepts and_intuitions can be_bridged only by positing ideas of diff_erence_within sensibility itself. Th e formal_criteria_Deleuze uses to defi ne Ideas_are_largely_derived from Leibniz and the_model of the diff erential calculus,_which provides a mathematical symbolism for the exploration of_the real: things_or beings are virtual and_problematic_multiplicities_composed of singularitiesevents, which are prolonged in converging and diverging_series, forming_zones of indiscernibility_where the multiplicities enter into perpetual becomings. Drawing on the_work of Jules Vuillemin and Albert Lautman,_among others, in Volume 6 daniel w. smith 104 the philosophy_of mathematics, De