 with the definiteness given here by the markings p on the n elements distributed between the k boxes Ci, and then one erases the markings p so the blocks or boxes in the partition have only an occupation number ni with no distinctions between the ni elements in each box Ci. When this scheme for representing indefiniteness is applied in quantum mechanics, then it is an objective indefiniteness in that no further differentiation between the elements of a box Ci is possible. Since photons come into being and disappear, are emitted and absorbed, they are individuals without identity. No specification beyond what was previously called the effective state of the aggregate is therefore possible. Hence the state of a photon gas is known when for each possible state α of a photon the number nα of photons in that state is given (Bose-Einstein statistics of radiation). [43, p. 246] 4 Thus within QM, the treatment of the indefiniteness due to the indistinguishability of quantum particles of the same type is to artificially treat them as distinct and then collect together or superpose the permutations of the particles (in a totally symmetric or totally antisymmetric manner) that factors out their supposed distinctness (see any QM text such as [9]). Our point in this section is the general mathematical theme that indefiniteness is described by taking a partition or quotient of a set of definite entities. A partition is a mixture of indefiniteness and definiteness. Each block is indefinite between the elements within it, but the blocks of the partition are distinct from one another. 1.3 Mathematical description of definiteness = subsets The common-sense classical view of reality is that it is completely definite or determined and fully propertied all the way down. Every entity or thing definitely has a property P or definitely has the property ¬P. Peter Mittelstaedt quotes Immanuel Kant's treatment of the idea of complete determinateness: Every thing as regards its possibility is likewise subject to the principle of complete determination according to which if all possible predicates are taken together with the contradictory opposites, then one of each pair of contradictory opposites must belong to it. (Kant quoted in: [33, p. 170]) Given a universe set U, a predicate P is represented by the subset S ⊆ U of elements that have the property, and the complement subset Sc = U−S represents the elements that have the property ¬P. 1.4 Two logics for the two types of reality The two mathematical concepts of
with the definiteness given here by the markings p on the newton chemical element distributed between the k boxes Ci, and then one erases the marking p so the blocks or box in the partition have merely an occupation number ni with no distinctions between the ni element in each box Ci. When this scheme for present indefiniteness is practice in quantum mechanics, then it is an objective indefiniteness in that no further differentiation between the elements of a box Ci is possible. Since photon come into being and disappear, are emit and absorbed, they are individuals without identity. No stipulation beyond what was previously called the effective country of the aggregate is consequently possible. Hence the state of a photon gas is know when for each possible state α of a photon the number nα of photons in that state is move over (Bose - Einstein statistics of radiation). [ 43, p. 246 ] 4 Thus within QM, the treatment of the indefiniteness due to the indistinguishability of quantum particles of the same type is to artificially treat them as distinct and then roll up together or superpose the permutation of the particle (in a totally symmetric or totally antisymmetric manner) that factor out their supposed distinctness (see any QM text such as [ 9 ]). Our distributor point in this section is the general mathematical theme that indefiniteness is described by taking a division or quotient of a set of definite entities. A partition is a mixture of indefiniteness and definiteness. Each block is indefinite between the elements within it, but the block of the division are distinct from one another. 1.3 Mathematical description of definiteness = subsets The common - sense classical view of world is that it is completely definite or compulsive and fully propertied all the way down. Every entity or thing definitely has a place P or definitely has the property ¬P. Peter Mittelstaedt quotes Immanuel Kant's discussion of the idea of complete determinateness: Every thing as regards its hypothesis is likewise subject to the principle of arrant determination accord to which if all possible predicates are taken together with the contradictory opposites, then one of each pair of contradictory opposites must belong to to it. (Kant quoted in: [ 33, p. 170 ]) Given a population set U, a predicate P is represented by the subset S ⊆ U of element that have the property, and the complement subset Sc = U−S represents the chemical element that have the property ¬P. 1.4 Two logics for the two types of reality The two mathematical concepts of
 wihh the definiteness givek here by the matkungs p on ths n elemdnts distributed between the k boxew Ci, and then one erasds the magkings p wo tie blocks or boxxa in thc parfltion iave only an ocgupation nukber ni with nm aidtinctions between the ni elements yn each blx Ci. When thif scnqme ror representing indefiniteness is appliev in quantum mevhanics, then it is an objeftivf indefiniteness ij that no fodthqe differentixtion betwteu the elemehts of a box Ci is possible. Sinze phptons come ibto teing and dmsappewr, are emittcc and dbsorbec, they are indlviduels qithout identity. No s'ecification beyond rhat was [rzviously called the edfwctivg stade ow thd afgcegzte is thxrefore posaible. Hence the state of a phouon tas is known shen fjr each possible state α of a photon the nlmbed nα of photons in that wtate is given (Bose-Eijstein stwtistics of radiation). [43, p. 246] 4 Thus within QM, the tredtmenv uf uhc kbdffiniteness due to the indistinguishability os qiaktum particles on the same type is tl sttificially trgat them aa distinct and thej collest totether or suprrpose the permutations of rhe particlef (in a totally symmztric or totclly amtisykmetric manner) that faccors oht their suoposed diaginctness (see anh QK dext such as [9]). Our point in this secvion ns the gdnersl matrematical hheme that indefinitenesd is bescrhbed by taning a partition or quotient of a set of defimide vntities. C partltion is a mixttre of indefinnteness cnd dewiniteness. Each bnock is indqfinite betweaj the elemenvs within it, vut rhe blozys of the partotion are distinct drom one another. 1.3 Katfsmatical descri'uiob of definitenexs = sufsvts Tra common-sensa clxsskval vkew of reallty is yhat it is completelf derinite or determinrd and fuloy propewtied all the way down. Every enuity oc thinj defimitgly has a property P or definitsly has tje iroperty ¬P. Peeer Nittelstaedt quotes Immanuel Kant's treatment of the idea of complete eeterminateness: Evety uhing as regacds itf possibinity is likewise subhect to the princlple of complete deterjinatimn acfording to which if all possible predicates are taken together with the cintradmceory opposifes, yhen mnz oy each [air uf contradictory opposites must belong to it. (Kait quoted hn: [33, p. 170]) Given a universe set U, s oredicate P ir represented by the subaet S ⊆ U of elements that have the proprrty, and the complement xubset Sx = U−S representr the elements thay have the propertt ¬P. 1.4 Two logics fur yhc two typed of rzality The two mathematical concepts of
with the definiteness given here by the on n elements between the k erases markings p so blocks or boxes the partition have only an occupation ni with no distinctions between the ni elements in each box Ci. When scheme for representing indefiniteness is applied in quantum mechanics, then it is an indefiniteness that further between the elements of a box Ci is possible. Since photons come into being and disappear, emitted and absorbed, they are individuals without identity. specification beyond what was called the effective state of aggregate therefore possible. the of photon gas is when for each possible state α of a photon the number nα of photons in that state given (Bose-Einstein radiation). [43, 246] Thus QM, the treatment indefiniteness due to the indistinguishability of the same type is to artificially treat them distinct and collect together or superpose the permutations the particles (in a totally symmetric or totally manner) that factors out their supposed distinctness (see any QM text such as [9]). Our this section is the mathematical theme that is by a or quotient a set of definite entities. A partition is a mixture of and definiteness. Each block is indefinite between the elements within the of the partition distinct from one another. Mathematical of definiteness = subsets classical of it completely or determined and fully all the way down. Every or thing definitely has has the property ¬P. Peter Mittelstaedt quotes Immanuel treatment of the idea of complete determinateness: thing as regards its possibility is likewise subject to the principle of determination according if all possible predicates are taken together with contradictory opposites, then one each pair of contradictory opposites must belong to it. quoted [33, p. Given a universe U, a predicate is represented by S ⊆ of that and the complement subset Sc = represents elements that have the property 1.4 for the two types of reality The two mathematical concepts of
 with the definiteness given hEre by the maRkingS p oN thE n ElemEnts Distributed betWEen tHe k boxes Ci, and then one erAses tHe MArkiNGs P so thE blocks OR bOXEs iN tHe ParTiTIoN have OnlY an occuPation numbEr nI wIth no distincTIoNs between tHe nI elements in eAch Box Ci. WHeN thIS scheMe fOr repResentINg indeFinitenesS iS ApplieD In quantUM MeChanIcs, then it is an objeCTiVE indefinitenesS in thaT nO FuRTHer DifFerentiatiOn BetweEN the eleMEnTS OF a bOX Ci is possible. since photonS ComE into bEiNg aND disapPear, aRe EMitTed and absorBed, tHey are indIviduaLS withouT IdentitY. No speCifIcaTion BEyOnD whAt WAs pREvIouSLy cAlled the EfFeCtive StatE OF THe agGreGate Is theRefore possiblE. HeNce tHE stAte of A photOn gaS iS knowN when fOr eacH pOssible state α of A phoTon the numBer Nα Of pHoTons iN That stAte Is gIven (BosE-EinsteIN stAtISTIcS of radiation). [43, p. 246] 4 Thus wItHIN Qm, the treaTment oF ThE iNDefiniteNeSs dUe to THE indiStinGUiShabilitY of quaNTuM pArticleS oF the saMe TypE is To artIFiciAlly trEat them aS distINct and then collECt together or sUPeRPOsE The pErmUtations of tHe paRTiclEs (in A ToTalLY symmEtric Or TOtALly antisymmetric manNeR) that fActorS out their suppOsed distinCTNEss (see anY QM tEXt SUch as [9]). Our point iN this Section is tHE general MatheMatical tHeme that iNDEfinitenEss Is dEscRibED By Taking a partitION or qUoTient of A seT of defiNitE enTitIes. a pArtition iS a mixturE oF iNdEfIniTenesS And definItEneSs. eacH blocK Is indeFinitE betWeEn THe eLements WItHIN it, bUt ThE bloCks Of The paRtitIOn aRe distiNct from onE anOTher. 1.3 maThEmaticaL description oF dEfinitenesS = sUbsEts The COMmon-sensE classical view of reality IS that it Is cOmpleTely Definite oR deTerminEd aND fully ProperTied aLl The WAY down. eVErY enTiTy or thing dEFIniTely hAs A proPerty P oR definitely has the pROpeRty ¬P. Peter MittElsTaedT QUoTes iMmANueL KANt's TREatment of the ideA of completE dETeRminatenesS: eveRy Thing as Regards Its poSSibilitY is likewiSe subject To The pRINciPle of complEte deterMination aCCordiNG tO whicH if All posSiBle PrediCates aRE taKen toGether WiTh the cOntraDiCtory oppOsites, then one of each pair Of contRadicTorY oppositeS muST beLong to it. (KAnt qUoted in: [33, p. 170]) GiVen A unIversE seT u, a preDicaTE P Is rEPreseNted BY the subseT s ⊆ U Of eLEMeNts that have THE ProPerty, And THe compLemeNt subset Sc = U−S repreSEnts the elementS thaT HAve The PRopeRtY ¬P. 1.4 Two logics for The TwO TYpes of reAlIty The two maThematicAl COncepTs of
 with the definiteness giv en here by  themar kin gs  p o n th e n elements d i stri buted between the k bo xes C i, andt he n one  erases th e mar ki ng s p  s o  t he bl ock s or bo xes in the  pa rt ition have o n ly  an occupa tio n number niwit h no d is tin c tions  be tween  the n i  eleme nts in ea ch box Ci .  When t h i ssche me for representi n gi ndefinitenessis app li e di n  qu ant um mechani cs , the n  it isa no b j ect i ve indefinite ness in tha t  no  furth er  di f ferent iatio nb etw een the ele ment s of a bo x Ci i s  possib l e. Sinc e phot ons  co me i n to  b ein ga ndd is app e ar,  are emi tt ed  andabso r b e d , th eyareindiv iduals withou t i dent i ty.  No s pecif icat io n bey ond wh at wa spreviously call ed t he effect ive  s tat eof th e  aggre gat e i s there fore po s sib le . H en ce the state of aph o t on  gas isknownw he nf or eachpo ssi bles t ate α  ofa  p hoton th e numb e rnα  of pho to ns inth atsta te is give n (Bos e-Einste in st a tistics of rad i ation). [43,p .2 4 6] 4 Th uswithin QM,thet reat ment of  th e  inde finit en e ss due to the indistin gu ishabi lityof quantum pa rticles of t h e same t ypei st o artificially  trea t them asd istinctand t hen coll ect toget h e r or sup erp ose  th e p e r mu tations of th e part ic les (in  atotally  sy mme tri c o rtotally a ntisymme tr ic  m an ner ) tha t  factors  o utth eir  supp o sed di stinc tnes s(s e e a ny QM t e xt s uchas  [ 9]).  Ou rpoint  int his  sectio n is thegen e ralma th ematica l theme thatin definitene ss  is  descr i b ed by ta king a partition or quo t ient of  aset o f de finite en tit ies. A  pa r tition  is amixtu re  of i ndefi n i te nes sand defini t e nes s. Ea ch  blo ck is i ndefinite betweent heelements with init,b u tthe bl o cks  o f  th e partition are d istinct fr om on e another. 1.3  M athemat ical de scrip t ion ofdefiniten ess = sub se ts T h e  co mmon-sense  classic al view o f  real i ty  is t hat  it is  c omp letel y defi n ite  or d etermi ne d andfully  p ropertie d all the way down. Eve ry ent ity o r t hing defi nit e lyhas a pro pert y P or def ini tel y has  th e  prop erty ¬P . P e ter M itte l staedt qu o te s I m m an uel Kant'st r e atm ent o f t h e idea  ofcomplete determin a teness: Everythin g asreg a rdsit s possibilityisli k e wise sub je ct to the p rinciple  o f  comp lete d etermi nationa c co r ding t o wh ich  if all p oss ib l e predi ca te s  are t aken  t ogethe r with thec o ntradictory oppo sites , theno neof ea ch  pair o f  con tradictory  oppositesmust b elon g toit. (Ka nt  quote d i n:  [33, p. 1 7 0]) Given  a un iversese t U,  apredic ateP is re pres en ted  by the s u b se t  S  ⊆ U o f el ement sthat  have the property , a n d the c om ple m e nt sub s et S c = U−S re pre sents t he element s  tha t  h a ve th e prop erty ¬ P. 1.4T wolo gics fo r t h e  two type s of real i tyTh e tw o mathem at ical  c onc ep ts of
 with_the definiteness_given here by the_markings p_on_the n_elements_distributed between the_k boxes Ci,_and then one erases_the markings p_so_the blocks or boxes in the partition have only an occupation number ni with_no_distinctions between_the_ni_elements in each box Ci._When this scheme for representing_indefiniteness is_applied in quantum mechanics, then it is an_objective_indefiniteness in that_no further differentiation between the elements of a box_Ci is possible. Since photons come_into being and_disappear,_are_emitted and absorbed, they_are individuals without identity. No specification_beyond what was previously called the_effective state of the aggregate is therefore_possible. Hence the state of a_photon gas is known when_for each_possible state α of a_photon the number_nα of_photons in that_state is given (Bose-Einstein statistics of_radiation). [43, p._246] 4 Thus within QM, the_treatment_of the indefiniteness_due_to_the indistinguishability_of quantum particles_of_the same_type_is to artificially treat them as_distinct_and then collect together or superpose the_permutations of the particles_(in_a totally symmetric or_totally antisymmetric manner) that factors_out their supposed distinctness (see any_QM text_such as_[9]). Our point in this section is the general mathematical theme_that indefiniteness is described by taking_a partition or quotient_of a_set_of definite entities._A_partition is_a mixture of indefiniteness and definiteness. Each_block is_indefinite between the elements within it,_but the blocks of_the_partition are distinct from one another._1.3 Mathematical description of definiteness =_subsets The common-sense classical view_of_reality_is that it is completely_definite or determined and fully propertied_all the way_down. Every entity or thing definitely has_a_property P or definitely has the_property_¬P. Peter Mittelstaedt quotes Immanuel Kant's_treatment_of_the idea of complete determinateness:_Every thing as regards its possibility_is likewise subject to the principle of complete determination_according to which_if all possible predicates are_taken_together_with the contradictory opposites, then one of each pair of_contradictory opposites_must belong to_it. (Kant quoted in: [33, p. 170]) Given a universe_set U, a predicate P is represented_by the subset S ⊆ U of elements that have the_property, and the complement subset Sc = U−S_represents the elements that have the property_¬P. 1.4 Two logics_for_the two types of reality The two_mathematical_concepts_of