 for sets............................... 22 5.11 Whence the Born Rule?.................................. 22 5.12 Measurement in QM/sets................................. 23 5.13 Summary of QM/sets................................... 23 5.14 Whence von Neumann's Type 1 and Type 2 processes?................. 24 6 Final remarks 26 1 Two types of reality 1.1 Objective definiteness and objective indefiniteness Our thesis in this paper is that mathematics (including logic) can be used to attack the problem of finding a realistic interpretation of (standard Dirac-von-Neumann) quantum mechanics (QM). Mathematics itself contains a very basic duality that can be associated with two meta-physical types of reality: 1. the common-sense notion of objectively definite reality assumed in classical physics, and 2. the notion of objectively indefinite reality suggested by quantum physics. The problem of interpreting quantum mechanics is essentially the problem of making sense out of the notion of objective indefiniteness. The approach taken here is to follow the lead of the mathematics of partitions, first for sets (where things are relatively "clear and distinct") and then for complex vector spaces where the mathematics of full QM resides. There has long been the notion of subjective or epistemic indefiniteness ("cloud of ignorance") that is slowly cleared up with more discrimination and distinctions (as in the game of Twenty Questions). But the vision of reality that seems appropriate for quantum mechanics is objective or ontological indefiniteness. The notion of objective indefiniteness in QM has been most emphasized by Abner Shimony ([37], [38], [39]). From these two basic ideas alone - indefiniteness and the superposition principle - it should be clear already that quantum mechanics conflicts sharply with common sense. If the quantum state of a system is a complete description of the system, then a quantity that has an
for sets............................... 22 5.11 Whence the Born Rule? .................................. 22 5.12 Measurement in QM / sets................................. 23 5.13 Summary of QM / sets................................... 23 5.14 Whence von Neumann's Type 1 and Type 2 processes? ................. 24 6 Final remark 26 1 Two type of reality 1.1 Objective definiteness and objective indefiniteness Our thesis in this newspaper is that mathematics (including logic) can be used to assail the problem of finding a realistic rendition of (standard Dirac - von - Neumann) quantum automobile mechanic (QM). Mathematics itself contains a very basic dichotomy that can be associated with two meta - physical types of reality: 1. the coarse - sense notion of objectively definite reality assumed in classical purgative, and 2. the notion of objectively indefinite reality suggested by quantum purgative. The problem of interpreting quantum mechanics is essentially the problem of construct sense out of the notion of objective indefiniteness. The approach taken here is to follow the tip of the mathematics of partitions, first for sets (where things are relatively " clear and discrete ") and then for complex vector spaces where the mathematics of full QM resides. There has long been the notion of subjective or epistemic indefiniteness (" cloud of ignorance ") that is slowly cleared up with more discrimination and distinctions (as in the game of Twenty Questions). But the sight of reality that seems appropriate for quantum automobile mechanic is objective or ontological indefiniteness. The notion of objective indefiniteness in QM has been most emphasized by Abner Shimony ([ 37 ], [ 38 ], [ 39 ]). From these two basic ideas entirely - indefiniteness and the superposition rationale - it should be clear already that quantum mechanics conflicts aggressively with common sense. If the quantum state of a system is a complete description of the system, then a quantity that has an
 fog sets............................... 22 5.11 Whence the Born Rule?.................................. 22 5.12 Measurement in QM/seta................................. 23 5.13 Summxry of QM/sets................................... 23 5.14 Whence von Nxumabn's Ttpe 1 and Type 2 processds?................. 24 6 Finap remarkw 26 1 Uwo types of realmfy 1.1 Objcetive fefiuiveness and objegtive indefhniteness Our dhdsns in this paper is that mathematics (includonh logic) can be usec to zntcck the problem of finding a rezlistic interpretatipn of (standard Dirac-von-Neulann) quantum mechanics (QM). Mathemajjcs utself contakns a very basic dualijy that can be associated with tdo meca-physical jvpes mf reality: 1. the bommon-sense notion of objectovely definite reelitt assumed in classican physics, and 2. the notion ox kbjectively indefunute rgalitf suetesged bb qhantum phbsics. The pdoblem of ibterpreting quantum mqbnanics is esaentiajlr the problem of making sense out of tht notjon of objective indefibiteness. The approach taken hewe is to follow the lead of the mathematics of pastitikvs, ylvst wir sets (where things are relatively "clear and dyatonbt") and then for cjmplex vectpr slwces where thg mathematjcs of full QM resldes. Thgre haw long betn thr notion of subjective or epistemic indvfinuteness ("cloud of iynorance") thac is skowly cleared up with more dnscrimjnation and distinctjuns (as in the gaoe pf Twenty Questions). But the dision of reakity thxt sgems ap[ropriate vor qmdntum mechanics is objeetive or ontolohical indefiniteness. The notion of objective ongefpniteness in QK has been moft emphasized yy Abner Shimuny ([37], [38], [39]). Froj these two basic ydeas alone - hjdefinitenesv and thq superpisition orinciple - it xhould be clear alrwady that quantum kecfznics conflicts shqrply with commpn renfe. Ih the xuantum stata of a rustem is a complcte desvription of the systam, tgen a quantity thay mas an
for sets............................... 22 5.11 Whence the Born 5.12 in QM/sets................................. 5.13 Summary of Neumann's 1 and Type processes?................. 24 6 remarks 26 1 Two types of 1.1 Objective definiteness and objective indefiniteness Our thesis in this paper is that (including logic) can be used to attack the problem of finding a realistic of Dirac-von-Neumann) mechanics Mathematics itself contains a very basic duality that can be associated with two meta-physical types of 1. the common-sense notion of objectively definite reality in classical physics, and the notion of objectively indefinite suggested quantum physics. problem interpreting mechanics is essentially problem of making sense out of the notion of objective indefiniteness. The approach taken here is to the lead mathematics of first sets things are relatively distinct") and then for complex vector mathematics of full QM resides. There has long the notion subjective or epistemic indefiniteness ("cloud of that is slowly cleared up with more discrimination distinctions (as in the game of Twenty Questions). But the vision of reality that seems quantum mechanics is objective ontological indefiniteness. The of indefiniteness QM been most by Abner Shimony ([37], [38], [39]). From these two basic ideas - indefiniteness and the superposition principle - it should be that mechanics conflicts sharply common sense. If the state a system is a of system, that an
 for sets............................... 22 5.11 Whence the Born Rule?.................................. 22 5.12 MEasurement In QM/sEts................................. 23 5.13 sumMaRy of qM/seTs................................... 23 5.14 Whence von NeuMAnn's type 1 and Type 2 processes?................. 24 6 FiNal reMaRKs 26 1 TwO TyPes of Reality 1.1 oBjECTivE dEfIniTeNEsS and oBjeCtive inDefiniteneSs OUr Thesis in this PApEr is that maTheMatics (includIng Logic) cAn Be uSEd to aTtaCk the ProbleM Of findIng a realiStIC interPRetatioN OF (sTandArd Dirac-von-NeumanN) QuANtum mechanics (Qm). MatheMaTIcS ITseLf cOntains a veRy Basic DUality tHAt CAN Be aSSociated with tWo meta-physiCAl tYpes of ReAliTY: 1. the coMmon-sEnSE noTion of objecTiveLy definitE realiTY assumeD In classIcal phYsiCs, aNd 2. thE NoTiOn oF oBJecTIvEly INdeFinite reAlItY suggEsteD BY QUantUm pHysiCs. The Problem of inteRprEtinG QuaNtum mEchanIcs iS eSsentIally tHe proBlEm of making sense Out oF the notioN of ObJecTiVe indEFiniteNesS. ThE approaCh taken HEre Is TO FOlLow the lead of the matHeMATiCs of partItions, FIrSt FOr sets (whErE thIngs ARE relaTiveLY "cLear and dIstincT") AnD tHen for cOmPlex veCtOr sPacEs wheRE the MathemAtics of fUll QM REsides. There has LOng been the notIOn OF SuBJectIve Or epistemic IndeFInitEnesS ("ClOud OF ignoRance") ThAT iS Slowly cleared up with MoRe discRiminAtion and distiNctions (as iN THE game of TWentY quEStions). But the viSion oF reality thAT seems apProprIate for qUantum mecHANics is obJecTivE or OntOLOgIcal indefinitENEss. THe Notion oF obJective IndEfiNitEneSs In QM has beEn most emPhAsIzEd By ABner SHImony ([37], [38], [39]). FroM tHesE tWo bAsic iDEas aloNe - indEfinItEnESs aNd the suPErPOSitiOn PrInciPle - It ShoulD be cLEar Already That quantUm mEChanIcS cOnflictS sharply with cOmMon sense. If ThE quAntum sTATe of a sysTem is a complete descriptiON of the sYstEm, theN a quAntity thaT haS an
 for sets................. .......... ....225.1 1When ce t he Born Rule?. . .... ...................... ..... .2 2 5. 1 2Measu rementi nQ M /se ts .. ... .. . .. ..... ... ....... ........ 2 3 5 .1 3 Summary of QM /sets..... ... ............ ... ...... .. ... .  23 5 .14  When ce von Neuman n's Type1a nd Typ e  2 proc e s se s?.. ............... 2 4  6 Final remarks26 1 T wo ty p e s o f r eality 1.1  O bject i ve defi n it e n e ssa nd objectiveindefiniten e ssOur th es isi n this  pape ri s t hat mathema tics  (includi ng log i c) canb e usedto att ack  th e pr o bl em  of  f i ndi n ga r e ali stic int er pr etati on o f ( s tand ard  Dir ac-vo n-Neumann) qu ant um m e cha nics(QM).  Mat he matic s itse lf co nt ains a very bas ic d uality th atca n b eassoc i ated w ith  tw o meta- physica l  ty pe s o freality: 1. the co mm o n -s ense not ion of ob je c tively d ef ini te r e a lityassu m ed  in clas sicalp hy si cs, and  2 . theno tio n o f obj e ctiv ely in definite  real i ty suggested b y  quantum phys i cs . Th e  pro ble m of interp reti n g qu antu m  m ech a nicsis es se n ti a lly the problem ofma king s enseout of the no tion of ob j e c tive ind efin i te n ess. The appro ach t aken herei s to fol low t he leadof the ma t h ematicsofpar tit ion s ,  f irst for sets ( wher ethingsare  relati vel y " cle aran d distinc t") andth en  f or  co mplex vector s pa ces  w her e the mathem atics  offu ll QMresides .  T h e re h as  l ongbee nthe n otio n  of  subjec tive or e pis t emic  i nd efinite ness ("cloudof  ignorance ")  th at iss l owly cle ared up with more discr i minatio n a nd di stin ctions (a s i n thegam e  of Tw enty Q uesti on s). B ut th e vi sio nof reality t hat  seem sappr opriate  for quantum mecha n ics  is objective  or  ont o l og ica l  i n def in i ten e s s. The notion o f objectiv ei nd efinitenes s  in  Q M has b een mos t emp h asizedby AbnerShimony ( [3 7],[ 3 8],  [39]). Fr om these  two basi c  idea s  a lone- i ndefin it ene ss an d thes upe rposi tion p ri nciple  - it  s hould be  clear already that qua ntum m echan ics  conflict s s h arp ly with c ommo n sense. I f t hequant ums tateof a sy ste m  is a  com p lete desc r ip tio n of  the system , t hen  a qu ant i ty tha t ha s an
 for_sets............................... 22_5.11 Whence the Born_Rule?.................................. 22_5.12_Measurement in_QM/sets................................._23 5.13 Summary_of QM/sets................................... 23_5.14 Whence von Neumann's_Type 1 and_Type_2 processes?................. 24 6 Final remarks 26 1 Two types of reality 1.1 Objective_definiteness_and objective_indefiniteness_Our_thesis in this paper is_that mathematics (including logic) can_be used_to attack the problem of finding a realistic_interpretation_of (standard Dirac-von-Neumann)_quantum mechanics (QM). Mathematics itself contains a very basic_duality that can be associated with_two meta-physical types_of_reality:_1. the common-sense notion_of objectively definite reality assumed in_classical physics, and 2. the notion_of objectively indefinite reality suggested by quantum_physics. The problem of interpreting quantum_mechanics is essentially the problem_of making_sense out of the notion_of objective indefiniteness._The approach_taken here is_to follow the lead of the_mathematics of partitions,_first for sets (where things are_relatively_"clear and distinct")_and_then_for complex_vector spaces where_the_mathematics of_full_QM resides. There has long been_the_notion of subjective or epistemic indefiniteness ("cloud_of ignorance") that is_slowly_cleared up with more_discrimination and distinctions (as in_the game of Twenty Questions). But_the vision_of reality_that seems appropriate for quantum mechanics is objective or ontological indefiniteness._The notion of objective indefiniteness in_QM has been most_emphasized by_Abner_Shimony ([37], [38],_[39])._From these_two basic ideas alone - indefiniteness and_the superposition_principle - it should be clear_already that quantum mechanics_conflicts_sharply with common sense. If the_quantum state of a system is_a complete description of the_system,_then_a quantity that has an