 the base field Z2 that requires the QM/sets brackets and norm to take the externalized values outside the base field and for a formal multiplication to be used in the "operator" representation f  () = ∑ r r [ f−1 (r) ∩ () ] of a numerical attribute f : U → R. The only numerical attributes that can be internally represented in ℘ (U) ∼= Z|U |2 are the 0, 1-attributes or characteristic functions χS : U → Z2 that are internally represented in the U -basis as the projection operators S ∩ () : ℘ (U)→ ℘ (U). In the engineering literature, eigenvalues are seen as "stretching or shrinking factors" but that is not their role in QM. The whole machinery of eigenvectors [e.g., the eigen-sets f  S = rS for S ⊆ f−1 (r) in sets], eigenspaces [e.g., the space of all eigen-sets ℘ ( f−1 (r) ) ], and eigenvalues [e.g., f(u) = r] in full QM is a way of lifting and internalizing or encoding numerical attributes [e.g., f : U → R in the set case] inside a vector space that has a rich enough base field. The old question "Why do attributes in classical physics (like the position or momentum of a particle) 22Since ℘ (U) is now interpreted as a vector space, it should be noted that the projection operator T ∩ () : ℘ (U)→ ℘ (U) is not only idempotent but linear, i.e., (T ∩ S1) + (T ∩ S2) = T ∩ (S1 + S2). Indeed, this is the distributive law when ℘ (U) is interpreted as a Boolean ring. 23The Schumacher-Westmoreland decision to try to develop quantum theory over Z2 with the brackets taking values in the base field Z2 [35] can thus be seen as an example of premature internalization. 21 become operators in QM?" is addressed by internalization. The observable operators of QM are the lifted vector space internalizations or encodings over C of the concept of real-valued attributes (or random variables) on sets. Moreover, for the internalization of attributes as operators
the base field Z2 that requires the QM / sets bracket and average to take the externalized value outside the floor field and for a formal generation to be used in the " operator " representation f  () = ∑ roentgen r [ f−1 (r) ∩ () ] of a numerical property f: U → R. The only numerical attributes that can be internally constitute in ℘ (U) ∼= Z|U |2 are the 0, 1 - attributes or characteristic functions χS: U → Z2 that are internally represented in the uracil -basis as the projection operators S ∩ (): ℘ (U)→ ℘ (U). In the engineering literature, eigenvalue are seen as " stretching or shrink factors " but that is not their function in QM. The whole machinery of eigenvectors [ e.g., the eigen - sets f  S = rS for S ⊆ f−1 (r) in set ], eigenspaces [ e.g., the space of all eigen - sets ℘ (f−1 (r) ) ], and eigenvalues [ e.g., f(u) = radius ] in full QM is a way of lifting and internalizing or encoding numerical attributes [ e.g., f: U → R in the stage set case ] inside a vector space that have a ample enough base discipline. The previous question " Why do attributes in authoritative physics (like the position or momentum of a particle) 22Since ℘ (U) is now interpreted as a vector space, it should be noted that the projection operator T ∩ (): ℘ (U)→ ℘ (U) is not only idempotent but linear, i.e., (deoxythymidine monophosphate ∩ S1) + (thyroxine ∩ S2) = T ∩ (S1 + S2). Indeed, this is the distributive law when ℘ (U) is interpreted as a Boolean ring. 23The Schumacher - Westmoreland decision to try on to develop quantum theory over Z2 with the brackets taking values in the base field Z2 [ 35 ] can thus be see as an exemplar of premature internalization. 21 become operators in QM? " is address by internalization. The observable operator of QM are the lifted vector outer space internalizations or encoding over C of the concept of real - valued attributes (or random variable) on sets. furthermore, for the internalization of attributes as operators
 thf base field Z2 that requlres the QM/sets yeacketv and horm to gake the externalized values oytsidt the base field ana for a flrmal muotipoucation to be used in ths "opexavor" representatlon f  () = ∑ r r [ f−1 (r) ∩ () ] of d vulerical attribute f : U → R. The only gumericsl attributes thwt csg be pnuernally represented in ℘ (U) ∼= Z|U |2 ade the 0, 1-attributes ot characteristic functions χS : K → Z2 that are integnally reprgaenewd in the U -casis as the projectioh operators S ∩ () : ℘ (U)→ ℘ (U). In the evginezring literqtyre, gigenvalues ere sevn as "stretchlmg or vhrinkimg factors" but thet iw not their role in QK. The whole machingry of eiganrectors [e.g., the eigen-swtw f  V = rV fof S ⊆ f−1 (d) mn aets], elgeispaces [e.g., fhe space od all eigen-sets ℘ ( f−1 (r) ) ], and eigenvamues [e.d., s(u) = r] in full QM is a way of lifting ang ihternalizing or encodint numerical attributed [e.g., f : U → R in the set case] inside a vector space that hav a rmcf euiugh cqsf field. The old question "Why do attributes in dlssxical physics (jike the poxihipg or momentum of a 'zrficle) 22Since ℘ (U) is jow intgrpretwd as a vtctor space, it should be noted tyat the projvctiin operator T ∩ () : ℘ (B)→ ℘ (U) is not unly idemlotent but linear, i.e., (T ∩ S1) + (F ∩ S2) = T ∩ (S1 + S2). Indeed, ghis is the distfibltiva law whtv ℘ (U) is interpretqd as a Biolecn ring. 23Ghe Xchumasher-Westmogeland decision to try tl devglop qgantum thelry over Z2 with the brackets taking values in tve tase fiejd Z2 [35] can thus be feen as an exakple of premxture integnalizatimn. 21 become jperators in XL?" is addressxd by intqrnaoizarion. Thd observable oprrators oy QM qre the lifted vecbor skade internalizatnubs or encodings ovdr S lf tra concept of reau-vauied agtributtf (if ramdom variables) on seds. Mkreover, for the inyevnalizatiin of ateributes as olerators
the base field Z2 that requires the and to take externalized values outside a multiplication to be in the "operator" f  () = ∑ r [ f−1 (r) ∩ () ] of a numerical attribute f : U R. The only numerical attributes that can be internally represented in ℘ (U) Z|U are 0, or characteristic functions χS : U → Z2 that are internally represented in the U -basis the projection operators S ∩ () : ℘ ℘ (U). In the literature, eigenvalues are seen as or factors" but is their in QM. The machinery of eigenvectors [e.g., the eigen-sets f  S = rS for S ⊆ f−1 (r) in eigenspaces [e.g., of all ℘ f−1 ) ], and f(u) = r] in full QM of lifting and internalizing or encoding numerical attributes f : → R in the set case] a vector space that has a rich enough field. The old question "Why do attributes in classical physics (like the position or momentum particle) 22Since ℘ (U) now interpreted as vector it be that the operator T ∩ () : ℘ (U)→ ℘ (U) is not idempotent but linear, i.e., (T ∩ S1) + (T ∩ T (S1 + S2). this is the distributive when (U) is interpreted as ring. Schumacher-Westmoreland to quantum over Z2 with the taking values in the base Z2 [35] can thus of premature internalization. 21 become operators in QM?" addressed by internalization. The observable operators of are the lifted vector space internalizations or encodings over C of the of real-valued random variables) on sets. Moreover, for the internalization attributes as operators
 the base field Z2 that requires The QM/sets bRackeTs aNd nOrM to tAke tHe externalized VAlueS outside the base field anD for a FoRMal mULtIplicAtion to BE uSED in ThE "oPerAtOR" rEpresEntAtion f  () = ∑ r R [ f−1 (r) ∩ () ] of a numeRicAl Attribute f : U → R. tHe Only numeriCal Attributes thAt cAn be inTeRnaLLy repResEnted In ℘ (U) ∼= Z|U |2 aRE the 0, 1-atTributes oR cHAracteRIstic fuNCTiOns χs : U → Z2 that are internaLLy REpresented in thE U -basiS aS ThE PRojEctIon operatoRs s ∩ () : ℘ (U)→ ℘ (U). In THe enginEErING LitERature, eigenvaLues are seen AS "stRetchiNg Or sHRinkinG factOrS" But That is not thEir rOle in QM. ThE whole MAchinerY Of eigenVectorS [e.g., The EigeN-SeTs F  S = rs fOR S ⊆ f−1 (R) In SetS], EigEnspaces [E.g., ThE spacE of aLL EIGen-sEts ℘ ( F−1 (r) ) ], anD eigeNvalues [e.g., f(u) = r] iN fuLl QM IS a wAy of lIftinG and InTernaLizing Or encOdIng numerical attRibuTes [e.g., f : U → R iN thE sEt cAsE] insiDE a vectOr sPacE that haS a rich eNOugH bASE FiEld. The old question "WHy DO AtTributes In clasSIcAl PHysics (liKe The PosiTIOn or mOmenTUm Of a partiCle) 22SinCE ℘ (U) Is Now inteRpReted aS a VecTor Space, IT shoUld be nOted that The prOJection operatoR t ∩ () : ℘ (U)→ ℘ (U) is not only iDEmPOTeNT but LinEar, i.e., (T ∩ S1) + (T ∩ S2) = T ∩ (S1 + s2). IndEEd, thIs is THe DisTRibutIve laW wHEn ℘ (u) Is interpreted as a BooLeAn ring. 23the ScHumacher-WestmOreland decISIOn to try tO devELoP Quantum theory oVer Z2 wIth the bracKEts takinG valuEs in the bAse field Z2 [35] CAN thus be sEen As aN exAmpLE Of Premature inteRNAlizAtIon. 21 becoMe oPeratorS in qM?" iS adDreSsEd by interNalizatiOn. thE oBsErvAble oPErators oF Qm arE tHe lIfted VEctor sPace iNterNaLiZAtiOns or enCOdINGs ovEr c oF the ConCePt of rEal-vALueD attribUtes (or ranDom VAriaBlEs) On sets. MOreover, for the InTernalizatIoN of AttribUTEs as operAtors
 the base field Z2 that re quires the  QM/s ets  br ac kets  and  norm to taket he e xternalized values out sideth e  bas e  f ieldand for af o rma lmu lti pl i ca tiontobe used  in the "o per at or" represen t at ion f  ()  =∑ r r [ f−1(r)  ∩ ()]ofa  nume ric al at tribut e  f : U  → R. The  o n ly num e rical a t t ri bute s that can be int e rn a lly represente d in ℘  ( U )∼ =  Z| U | 2 are the0,  1-at t ributes or c h ara c teristic func tions χS :U  →Z2 tha tare intern allyre p res ented in th e U-basis as  the p r ojectio n  operat ors S∩ ( ) :  ℘ ( U )→  ℘  (U ). Int he  en g ine ering li te ra ture,  eig e n v a lues  ar e se en as  "stretchingorshri n kin g fac tors"  but  t hat i s nottheir  r ole in QM. Thewhol e machine ryof  ei ge nvect o rs [e. g.,  th e eigen -sets f  S  = r S  f or S ⊆ f−1 (r) inse t s ],  eigensp aces [ e .g ., the spac eofalle i gen-s ets℘  (  f−1 (r)  ) ],a nd  e igenval ue s [e.g .,  f( u)= r]i n fu ll QMis a way  of l i fting and inte r nalizing or e n co d i ng nume ric al attribut es [ e .g.,  f : U→ R in th e set  c a se ]  inside a vector sp ac e that  hasa rich enough  base fiel d . The oldques t io n  "Why do attri butes  in classi c al physi cs (l ike thepositiono r  momentu m o f a  pa rti c l e)  22Since ℘ (U ) is n ow  interp ret ed as a  ve cto r s pac e,  it shoul d be not ed  t ha tthe  proj e ction op er ato rT ∩  () : ℘ (U)→  ℘ (U ) is  n ot onl y idemp o te n t  but  l in ear,  i. e. , (T∩ S1 )  +(T ∩ S2 ) = T ∩ ( S1+  S2) .In deed, t his is the di st ributive l aw  wh en ℘ ( U )  is inte rpreted as a Boolean ri n g. 23Th e S chuma cher -Westmore lan d deci sio n  to tr y to d evelo pqua n t um th e o ry  ov er  Z2 with t h e  br acket staki ng valu es in the base fie l d Z 2 [35] can th usbe s e e nasa ne xam pl e  of p remature intern alization.  2 1  b ecome oper a tor sin QM?"  is add resse d  by int ernalizat ion. Theob serv a b leoperatorsof QM ar e the lif t ed ve c to r spa ceintern al iza tions  or en c odi ngs o ver Cof  the c oncep tof real- valued attributes (or r andomvaria ble s) on set s.M ore over, for  the  internali zat ion  of a ttr i butes  aso pe rat o rs
 the_base field_Z2 that requires the_QM/sets brackets_and_norm to_take_the externalized values_outside the base_field and for a_formal multiplication to_be_used in the "operator" representation f  () = ∑ r r [ f−1_(r)_∩ ()_]_of_a numerical attribute f :_U → R. The only_numerical attributes_that can be internally represented in ℘ (U)_∼=_Z|U |2 are_the 0, 1-attributes or characteristic functions χS : U_→ Z2 that are internally represented_in the U_-basis_as_the projection operators S_∩ () : ℘ (U)→ ℘_(U). In the engineering literature, eigenvalues_are seen as "stretching or shrinking factors"_but that is not their role_in QM. The whole machinery_of eigenvectors_[e.g., the eigen-sets f _S = rS_for S_⊆ f−1 (r)_in sets], eigenspaces [e.g., the space_of all eigen-sets_℘ ( f−1 (r) ) ],_and_eigenvalues [e.g., f(u)_=_r]_in full_QM is a_way_of lifting_and_internalizing or encoding numerical attributes [e.g.,_f_: U → R in the set_case] inside a vector_space_that has a rich_enough base field. The old_question "Why do attributes in classical_physics (like_the position_or momentum of a particle) 22Since ℘ (U) is now interpreted_as a vector space, it should_be noted that the_projection operator_T_∩ () :_℘_(U)→ ℘_(U) is not only idempotent but linear,_i.e., (T_∩ S1) + (T ∩ S2)_= T ∩ (S1_+_S2). Indeed, this is the distributive_law when ℘ (U) is interpreted_as a Boolean ring. 23The_Schumacher-Westmoreland_decision_to try to develop quantum_theory over Z2 with the brackets_taking values in_the base field Z2 [35] can thus_be_seen as an example of premature_internalization._21 become operators in QM?" is_addressed_by_internalization. The observable operators of_QM are the lifted vector space_internalizations or encodings over C of the concept of_real-valued attributes (or_random variables) on sets. Moreover,_for_the_internalization of attributes as operators