 new; it is part of the mathematical folklore. Hermann Weyl outlined the lifting program by first considering an attribute on a set, which defined the set partition or "grating" [43, p. 255] of elements with the same attribute-value. Then he moved to the quantum case where the set or "aggregate of n states has to be replaced by an n-dimensional Euclidean vector space" [43, p. 256].13 The appropriate notion of a partition or "grating" is a "splitting of the total vector space into mutually orthogonal subspaces" so that "each vector −→x splits into r component vectors lying in the several subspaces" [43, p. 256], i.e., a direct sum decomposition of the space, where the subspaces are the eigenspaces of an observable operator. Weyl's grating metaphor also lends itself to (our own example of) seeing measurement of the, say,'regular polygonal shape'of an 'indefinite blob of dough'as it randomly falls through a opening in a grating to take on that 'polygonal shape' (with the attribute-value or eigenvalue being the number of regular sides λ = 3, 4, 5, 6). Figure 5: Imagery of measurement as randomly giving an indefinite blob of dough a definite eigen-shape. Note how the blob of dough is objectively indefinite between the regular polygonal shapes and does not simultaneously have all those shapes even though it might be mathematically represented as the set {N,,...} or the superposition vector N++... in a certain space. 13Note the lift from sets to vector spaces using the basis principle where the cardinality n becomes dimension n. 13 3.3 What is a vector space attribute? A set attribute is a function f : U → R (where the set of values is taken as the reals). The inverseimage f−1 (r) ⊆ U of each value f(u) = r is a subset where the attribute has the same value, and those subsets form a set partition. Given a basis set {bu}u∈U of a vector space V over a field K, we can apply a set attribute f : {bu}u∈U → K to the basis set and see what it generates. One possibility is to linearly extend the function f∗ (bu) = f (bu) to the whole space to obtain a linear functional f∗ : V →
new; it is part of the mathematical folklore. Hermann Weyl outlined the lifting platform by foremost considering an attribute on a hardening, which define the set partition or " grate " [ 43, p. 255 ] of element with the same property - value. Then he move to the quantum case where the hardening or " aggregate of n country has to be replaced by an n - dimensional Euclidean vector space " [ 43, p. 256].13 The appropriate notion of a division or " grating " is a " splitting of the total vector space into mutually extraneous subspaces " so that " each vector −→x splits into r component vectors lie in the several subspaces " [ 43, p. 256 ], i.e., a direct sum decomposition of the quad, where the subspaces are the eigenspaces of an observable operator. Weyl's grating metaphor also lends itself to (our own example of) seeing measurement of the, say,'regular polygonal shape'of an' indefinite blob of dough'as it randomly falls through a opening in a grating to take on that' polygonal shape' (with the attribute - value or eigenvalue being the number of regular sides λ = 3, 4, 5, 6). human body 5: Imagery of measurement as randomly giving an indefinite blob of boodle a definite eigen - condition. notice how the blob of dough is objectively indefinite between the regular polygonal shapes and does not simultaneously have all those shapes even though it might be mathematically represented as the set { N,, ... } or the superposition vector N++... in a sealed space. 13Note the lift from sets to vector spaces using the basis principle where the cardinality n becomes dimension n. 13 3.3 What is a vector space attribute? A set attribute is a affair f: uracil → R (where the set of values is taken as the reals). The inverseimage f−1 (r) ⊆ uracil of each value f(u) = r is a subset where the attribute has the like value, and those subsets shape a set partition. give a basis fructify { bu}u∈U of a vector space V over a battlefield K, we can use a set property f: { bu}u∈U → K to the basis set and examine what it generates. One possibility is to linearly unfold the function f∗ (bu) = f (bu) to the whole space to obtain a linear functional f∗: V →
 nea; it is part of the mathtmatical folklore. Hermani Weyl kutlined the lifting program by firsv cobsideeing an attribute on a set, whicj definee tht set partition oc "gratinn" [43, p. 255] lf enxments with the same attrhbute-value. Thet fe moved to the quantum case where thq set ot "wggregate of n staues has no be replaced by an n-dimensionzl Euclpdean vector spacr" [43, p. 256].13 The appropriate notiln ov a partition or "ggating" is a "splyrting of the total vecuox space intk mutually orthogonal subspaces" so tkat "each vextir −→d splits inti r cjmponent vecbprs lyhng in yhe several sunspacxs" [43, p. 256], i.e., a direct sum deromposition of the skace, where tke subspaces are the witenspdces of xb ocsedvebls operwtoc. Weyl's grafing metaphir also lends itselg ei (our own exajple os) feeing measurement of the, say,'regular ponygknal shape'of an 'indefinute blob of dough'as ij randomly falls through a opening in a grating to take on dhat ''ouyginwu sjape' (with the attribute-value or eigenvalue beyhg tme number of regmlar sides λ = 3, 4, 5, 6). Fogkrr 5: Imagery of oeasurzjeht as randomly givlng an yndefunite blof of dough a definite eigen-shapw. Note how tke vlob of dough is oyjectively iudefinote brtween the regular polyyonal ahapes and foes not akmultaneously haxe sln those shapes even though it might be kathemagicakly re[resented ws thc set {N,,...} or the supegposijion vactor N++... in a certain space. 13Note the lift hcom sets to vgctmr vpaces ufing bhe basis princyple where the cardinclity v becomes simensimn n. 13 3.3 What is a vector vkace attributx? A set aetrivute is a fjvction f : U → R (where thv fet if values is taken as fhe reals). The iuretweimage f−1 (r) ⊆ U of eash velue s(g) = r is a sutset whdte thd attribute fas yhe same value, and tvose subsets form a sey iartition. Given a basis set {bu}i∈U of a vector spafe V mvec a firld K, we can apply a set attributs f : {bu}u∈U → K to the basif seb anq see what it generates. One possibility is to linxarly extend the functiin f∗ (bu) = f (bu) to thg wmole space tp obtwin a linaar functional f∗ : V →
new; it is part of the mathematical Weyl the lifting by first considering which the set partition "grating" [43, p. of elements with the same attribute-value. he moved to the quantum case where the set or "aggregate of n has to be replaced by an n-dimensional Euclidean vector space" [43, p. 256].13 appropriate of partition "grating" is a "splitting of the total vector space into mutually orthogonal subspaces" so that "each −→x splits into r component vectors lying in several subspaces" [43, p. i.e., a direct sum decomposition the where the are eigenspaces an observable operator. grating metaphor also lends itself to (our own example of) seeing measurement of the, say,'regular polygonal shape'of 'indefinite blob it randomly through opening a grating to that 'polygonal shape' (with the attribute-value the number of regular sides λ = 3, 5, 6). 5: Imagery of measurement as randomly an indefinite blob of dough a definite eigen-shape. how the blob of dough is objectively indefinite between the regular polygonal shapes and does have all those shapes though it might mathematically as set or the vector N++... in a certain space. 13Note the lift from sets vector spaces using the basis principle where the cardinality n n. 3.3 What is vector space attribute? A attribute a function f : R the is as reals). The inverseimage f−1 ⊆ U of each value = r is a the same value, and those subsets form a partition. Given a basis set {bu}u∈U of vector space V over a field K, we can apply a set f : K to the basis set and see what generates. One possibility is linearly extend the function f∗ (bu) = f (bu) the space to a linear functional : V →
 new; it is part of the mathematiCal folklorE. HermAnn weyL oUtliNed tHe lifting progrAM by fIrst considering an attriBute oN a SEt, whICh DefinEd the seT PaRTItiOn Or "GraTiNG" [43, p. 255] Of eleMenTs with tHe same attrIbuTe-Value. Then he mOVeD to the quanTum Case where the Set Or "aggrEgAte OF n staTes Has to Be replACed by aN n-dimensiOnAL EucliDEan vectOR SpAce" [43, p. 256].13 the appropriate notIOn OF a partition or "gRating" Is A "SpLITtiNg oF the total vEcTor spACe into mUTuALLY orTHogonal subspaCes" so that "eaCH veCtor −→x sPlIts INto r coMponeNt VEctOrs lying in tHe seVeral subsPaces" [43, p. 256], I.E., a direcT Sum decoMpositIon Of tHe spACe, WhEre ThE SubSPaCes ARe tHe eigensPaCeS of an ObseRVABLe opEraTor. WEyl's gRating metaphoR alSo leNDs iTself To (our Own eXaMple oF) seeinG measUrEment of the, say,'reGulaR polygonaL shApE'of An 'IndefINite blOb oF doUgh'as it RandomlY FalLs THROuGh a opening in a gratiNg TO TaKe on that 'PolygoNAl ShAPe' (with thE aTtrIbutE-VAlue oR eigENvAlue beinG the nuMBeR oF regulaR sIdes λ = 3, 4, 5, 6). FIgUre 5: imaGery oF MeasUremenT as randoMly giVIng an indefinitE Blob of dough a dEFiNITe EIgen-ShaPe. Note how thE bloB Of doUgh iS ObJecTIvely IndefInITe BEtween the regular polYgOnal shApes aNd does not simuLtaneously HAVE all thosE shaPEs EVen though it migHt be mAthematicaLLy represEnted As the set {n,,...} or the supERPosition VecTor n++... in A ceRTAiN space. 13Note the LIFt frOm Sets to vEctOr spaceS usIng The BasIs Principle Where the CaRdInAlIty N becoMEs dimensIoN n. 13 3.3 WHaT is A vectOR space AttriBute? a sEt ATtrIbute is A FuNCTion F : U → r (wHere The SeT of vaLues IS taKen as thE reals). The InvERseiMaGe F−1 (r) ⊆ U of eaCh value f(u) = r is a SuBset where tHe AttRibute HAS the same Value, and those subsets forM A set parTitIon. GiVen a Basis set {bU}u∈U Of a vecTor SPace V oVer a fiEld K, wE cAn aPPLy a seT ATtRibUtE f : {bu}u∈U → K to tHE BasIs set AnD see What it gEnerates. One possibiLIty Is to linearly eXteNd thE FUnCtiON f∗ (BU) = f (bU) tO The WHOle space to obtaiN a linear fuNcTIoNal f∗ : V →
 new; it is part of the ma thematical  folk lor e.He rman n We yl outlined th e  lif ting program by firstconsi de r inga nattri bute on as e t,wh ic h d ef i ne d the  se t parti tion or "g rat in g" [43, p. 2 5 5]  of elemen tswith the sam e a ttribu te -va l ue. T hen  he m oved t o  the q uantum ca se wheret he seto r  " aggr egate of n states ha s  to be replace d by a nn -d i m ens ion al Euclide an  vect o r space "  [ 4 3 ,  p. 256].13 The a ppropriaten oti on ofapar t itionor "g ra t ing " is a "spl itti ng of the  total vectors pace in to mut ual lyorth o go na l s ub s pac e s"  so tha t "eachve ct or −→ x sp l i t s  int o r  com ponen t vectors lyi ngin t h e s evera l sub spac es " [43 , p. 2 56],i. e., a direct su m de compositi onof  th espace ,  where  th e s ubspace s are t h e e ig e n s pa ces of an observab le o pe rator. W eyl'sg ra ti n g metaph or  al so l e n ds it self to  (our ow n exam p le  o f) seei ng  measu re men t o f the ,  say ,'regu lar poly gonal shape'of an 'i n definite blob of d ou g h'as  it  randomly f alls thro ugha  o pen i ng in  a gr at i ng to take on that 'po ly gonalshape ' (with the a ttribute-v a l u e or eig enva l ue being the numb er of  regular s i des λ =3, 4,  5, 6).Figure 5: I magery o f m eas ure men t as  randomly giv i n g an  i ndefini teblob of  do ugh  adef in ite eigen -shape.No te  h ow  th e blo b  of doug hisob jec tivel y  indef inite  bet we en the  regula r  p o l ygon al  s hape s a nd  does  not sim ultaneo usly have  al l  tho se  s hapes e ven though it  m ight be ma th ema ticall y represen ted as the set {N,,... }  or the  su perpo siti on vector  N+ +...ina  certa in spa ce. 1 3N ote t he li f t  f rom  s ets to vec t o r s paces  u sing  the ba sis principle wher e  th e cardinality  nbeco m e sdim e ns i onn. 133 . 3 What is a vec tor spaceat t ri bute? A se t  at tr ibute i s a fun ction f : U →  R (where  the setof  val u e s i s taken as  the rea ls). Thei nvers e im age f −1(r) ⊆Uofeachvaluef (u)  = ris a s ub set wh ere t he  attribu te has the same value,and th ose s ubs ets forma s e t p artition.  Giv en a basis  se t { bu}u∈ U o f  a ve ctor sp ace V ove r af ield K, w e  c ana p pl y a set att r i b ute  f :{bu } u∈U →K to  the basis set an d  see what it g ener a t es.  On e  pos si bility is to l ine ar l y  extendth e functionf∗ (bu)=f  (bu)  to th e whol e space t oo btaina li nea r functio nal  f ∗  : V →
 new;_it is_part of the mathematical_folklore. Hermann_Weyl_outlined the_lifting_program by first_considering an attribute_on a set, which_defined the set_partition_or "grating" [43, p. 255] of elements with the same attribute-value. Then he moved_to_the quantum_case_where_the set or "aggregate of_n states has to be_replaced by_an n-dimensional Euclidean vector space" [43, p. 256].13_The_appropriate notion of_a partition or "grating" is a "splitting of the_total vector space into mutually orthogonal_subspaces" so that_"each_vector_−→x splits into r_component vectors lying in the several_subspaces" [43, p. 256], i.e., a_direct sum decomposition of the space, where_the subspaces are the eigenspaces of_an observable operator. Weyl's grating_metaphor also_lends itself to (our own_example of) seeing_measurement of_the, say,'regular polygonal_shape'of an 'indefinite blob of dough'as_it randomly falls_through a opening in a grating_to_take on that_'polygonal_shape'_(with the_attribute-value or eigenvalue_being_the number_of_regular sides λ = 3, 4,_5,_6). Figure 5: Imagery of measurement as_randomly giving an indefinite_blob_of dough a definite_eigen-shape. Note how the blob_of dough is objectively indefinite between_the regular_polygonal shapes_and does not simultaneously have all those shapes even though it_might be mathematically represented as the_set {N,,...} or the_superposition vector_N++..._in a certain_space._13Note the_lift from sets to vector spaces using_the basis_principle where the cardinality n becomes_dimension n. 13 3.3_What_is a vector space attribute? A_set attribute is a function f_: U → R (where_the_set_of values is taken as_the reals). The inverseimage f−1 (r)_⊆ U of_each value f(u) = r is a_subset_where the attribute has the same_value,_and those subsets form a set_partition._Given_a basis set {bu}u∈U of_a vector space V over a_field K, we can apply a set attribute f_: {bu}u∈U →_K to the basis set_and_see_what it generates. One possibility is to linearly extend the_function f∗_(bu) = f_(bu) to the whole space to obtain a linear functional_f∗ : V →