From Abstract Entities in Mathematics to Superposition States in Quantum Mechanics
David Ellerman




From Abstract Entities in Mathematics to 
Superposition States in Quantum Mechanics
    David Ellerman
University of California at Riverside
    
===================================================================================




Given an equivalence relation  on a set U, there are
two abstract notions of an element of the quotient set U/.
The #1 abstract notion is a set S=[  u] of equivalent
elements of U (an equivalence class); the #2 notion is an abstract entity
u_S that is definite on what is common to the elements of the equivalence
class S but is otherwise indefinite on the differences between those
elements. For instance, the #1 interpretation of a homotopy type is an
equivalence class of homotopic spaces, but the #2 interpretation, e.g., as
developed in homotopy type theory, is an abstract space (without points) that
has the properties that are in common to the spaces in the equivalence class
but is otherwise indefinite. In philosophy, the #2 abstract entities might
be called paradigm-universals, e.g., `the white thing' as
opposed to the #1 abstract notion of "the set of white things" (out of some
given collection U).

The paper shows how this #2 notion of a paradigm may be mathematically
modeled using incidence matrices in Boolean logic and density matrices in
probability theory. Then we cross the bridge to the density matrix treatment
of the indefinite superposition states in quantum mechanics (QM). This
connection between the #2 abstracts in mathematics and ontic indefinite
states in QM elucidates Abner Shimony's literal or objective indefiniteness
interpretation of QM.





§ INTRODUCTION


The purpose of this paper is to illuminate the late Abner Shimony's
objectively indefinite or `Literal' interpretation of quantum mechanics based
on seeing the superposition states as being objectively indefinite.


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 indefinite
value in that quantum state is objectively indefinite; its value is not merely
unknown by the scientist who seeks to describe the system. <cit.>


In addition to the Shimony's phrase "objective indefiniteness," other
philosophers of physics have used similar phrases for these indefinite states:



  * Peter Mittelstaedt's "incompletely determined" quantum states with
"objective indeterminateness" <cit.>;


  * Paul Feyerabend's "inherent indefiniteness" <cit.>;


  * Allen Stairs' "value indefiniteness" and "disjunctive facts"
<cit.>;


  * Steven French and Decio Krause's "ontic vagueness"
<cit.>; or


  * E. J. Lowe's "vague identity" and "indeterminacy" that is "ontic"
<cit.>.


But how can we understand the notion of an "ontic indefinite state"?



§ TWO VERSIONS OF ABSTRACTION


The claim is that we already have the notion of an indefinite state in the
mathematical notion of an entity that abstracts as definite what is common to
the distinct elements of a set S and rendering their differences as indefinite.

Given an equivalence relation on a set U such as "having the same color" and
if u u^' were white, then there are two notions of abstraction:



  * the #1 version of the abstraction operation takes equivalent entities
u u^' to the equivalence class [  u]  =[
u^'] of all white entities (in some universe U), and;


  * the #2 version of the abstraction operation takes all the equivalent
entities u u^' to the abstract entity "the white
entity" that is definite on what is common in the set of all particular white
things but is indefinite on how they differ (e.g., on all the other properties
that distinguish them).


For instance, there are two notions of an `element' of a quotient set or a
quotient group (or any other quotient object in algebra):



  * a quotient group element as an equivalence class or coset; or


  * a quotient group element as an abstract entity representing what is
common to the equivalence class.


Given any property S(  u) defined on the elements of
U, two abstract objects can be defined:












    < g r a p h i c s >








Figure 1: A property determines two types of abstract objects.



Intuitively the #2 abstract object u_S is `the paradigm
S-entity' (the blob-sum ⊞ is defined below) which is definite on
the S(  u) property and indefinite on (i.e., blobs out) the
differences between all the u∈ U such that S(  u).



§ AN EXAMPLE STARTING WITH ATTRIBUTES


Consider three predicates (binary attributes) P(  x), Q(
x), and R(  x) which could distinguish at most
2^3=8 definite-particular entities: u_1,...,u_8 called
eigen-elements and which can be presented in a table like a truth table:



c]|c|c|c|c|P(  x)     Q(  x)     R(  x)    
u
1     1     1     u_1
1     1     0     u_2
1     0     1     u_3
1     0     0     u_4
0     1     1     u_5
0     1     0     u_6
0     0     1     u_7
0     0     0     u_8



Table 1: Eight entities specified by 3 properties.



The general rule is if f,g,h:U→


ℝ are numerical attributes with the number of distinct values as n_f,
n_g, and n_h respectively, then those attributes could distinguish or
classify n_f× n_g× n_h distinct subsets of U. If the join
of the inverse-image partitions is the discrete partition, i.e.,{
f^-1}∨{  g^-1}∨{  h^-1}
=1_U <cit.>, then {  f,g,h} is a
complete set of attributes since they can distinguish or classify the
eigen-elements of U. Then we can distinguish the elements of U by their
triple of values, i.e., | f(  u_j)  ,g(
u_j)  ,h(  u_j)  ⟩ uniquely determines
u_j∈ U.

In the example, any subset S⊆ U={  u_1,...,u_8}
is characterized by a property S(  x), the disjunctive
normal form property, common to all and only the elements of S. If
S={  u_1,u_4,u_7}, then the DNF property is:



S(  x)  =[  P(  x)  ∧ Q(  x)
∧ R(  x)  ]  ∨[  P(  x)  ∧¬
Q(  x)  ∧¬ R(  x)  ]  ∨[  ¬
P(  x)  ∧¬ Q(  x)  ∧ R(  x)
].



But what are the #1 and #2 abstract entities?



  1. The #1 abstract entity is the set




S={  u_i∈ U|S(  u_i)  }  ={  u_1
,u_4,u_7}



of all the distinct S(  x)-entities; and



  2. The #2 abstract entity is the paradigm-universal
S(  x)-entity symbolized




u_S=u_1⊞ u_4⊞ u_7=⊞{  u_i∈ U|S(
u_i)  }

The `superposition' or `blob-sum' of u_1, u_4, and u_7.



that is definite on the DNF property S(  x) but indefinite on what distinguishes the different S(
x)-entities. Thus S(  u_S) holds but none of the
disjuncts hold since that would make u_S equal to u_1, u_4, or
u_7. Hence S(  u_S) is a `disjunctive fact' in the sense
of Allen Stairs <cit.>.



§ SOME PHILOSOPHICAL CONCERNS


It is best to think of S as the set of definite particular
S(  x)-entities in some universe U, while u_S is the
indefinite paradigm-universal S(  x)-entity is the
`superposition' u_S=⊞{u_i∈ U|S(  u_i)  } that
is, in general, "one over the many." Only when S={
u_j} is a singleton does the definite description `the
S-entity' refer to an element of U, i.e., u_{  u_j}=u_j.

Making the "one" u_S=⊞{u_i∈ U|S(  u_i)  }
over the many, i.e., more abstract than the u_i∈ U (for
| S| >1) avoids the paradoxes just as the iterative
notion of set does in ordinary set theory, i.e., for #1 type of
abstractions. Otherwise, if we ignore the given set U, then we can recreate
Russell's Paradox for R(  u_S)  ≡¬ S(
u_S) so:



u_R=⊞{  u_S|¬ S(  u_S)  } and
thus R(  u_R) implies ¬ R(  u_R), and
¬ R(  u_R) implies R(  u_R).



But if we define u_R=⊞{  u_S∈ U|¬ S(
u_S)  }, then assuming u_R∈ U leads to the
contradiction so u_R∉ U.

The paradigm-universal u_S is not universal `S-ness'. Where
S(  x) is being white, then u_white= `the white
thing`, not `whiteness'. This distinction goes back to Plato:


But Plato also used language which suggests not only that the Forms
exist separately (χωριστα) from all the
particulars, but also that each Form is a peculiarly accurate or good
particular of its own kind, i.e., the standard particular of the kind in
question or the model (παραδειγμα) to which other particulars approximate. <cit.>


Some have considered interpreting the Form as paradeigma as
an error.


For general characters are not characterized by themselves: humanity
is not human. The mistake is encouraged by the fact that in Greek the same
phrase may signify both the concrete and the abstract, e.g. λευκ oν (literally "the white") both "the white thing"
and "whiteness", so that it is doubtful whether αυτ o τ
o λευκ oν (literally "the white itself")
means "the superlatively white thing" or "whiteness in abstraction". <cit.>


Thus for the abstract property W(  u) "whiteness", we have:



  * the #1 abstraction is the set of white things W={
u∈ U:W(  u)  }, and;


  * the #2 abstraction `the white thing' u_W.




§ RELATIONS BETWEEN #1 AND #2 UNIVERSALS


For properties S() defined on U, there is a 1-1 correspondence between the
#1 and #2 universals:



∪{{  u}  |u∈ U&S(  u)  }  =S
⟷ u_S=⊞{  u_{  u}|u∈
U&S(  u)  }.



In each case, we may extend the definition of the property to the two
universals. For T() another property defined on U:



S(  T) iff (  ∀ u∈ U)  (  T(
u)  ⇒ S(  u)  ) iff S(
u_T).



In terms of the #1 universals, S(  S) holds by definition
and: S(  T) iff T⊆ S, and similarly S(
u_S) always holds. But what is the #2 universals equivalent of
T⊆ S? Intuitively u_S is `the S-thing' that is
definite on having the S-property but is otherwise indefinite on the
differences between the members of S. If we make more properties definite,
then in terms of subsets, that will in general cut down to a subset
T⊆ S, so u_T would inherit the paradigmatic property holding on
the superset S, i.e., S(  u_T).

This "process" to changing to a more definite universal u_S⇝
u_T for T⊆ S will be called projection and symbolized:



u_T u_S (or u_S u_T)

u_T is a "sharpening" or more definite version of u_S.

c]|c|c|c|S() defined on U     #1 abstraction     #2 abstraction
Universals for S()     S=∪{{  u}  |u∈ U&S(
u)  }     u_S=⊞{  u_{  u}|u∈
U&S(  u)  }
T() defined on U     S(  T) iff T⊆ S     iff
S(  u_T) iff u_T u_S



Table 2: Equivalents between #1 and #2 universals



In the language of Plato, the projection relation  is the relation of
"participation" (μεθεξις or
methexis). As Plato would say, u_T has the property S() iff it
participates in `the S-thing', i.e., S(  u_T) iff
u_T u_S.

Thus there are two theories of abstract objects:



  * Set theory is the theory of #1 abstract objects, the sets S, where
(taking ∈ as the participation relation), sets are never
self-participating, i.e., S∉ S;


  * There is a second theory about the #2 abstract entities, the
paradigms u_S, which are always self-participating, i.e.,
u_S u_S.


Like sets S, the #2 abstract entities u_S, the
paradigm-universals, are routinely used in mathematics.



§ EXAMPLES OF ABSTRACT PARADIGMS IN MATHEMATICS


There is an equivalence relation A≃ B between topological spaces which
is realized by a continuous map f:A→ B such that there is an
inverse g:B→ A so the fg:B→ B is homotopic to 1_B
(i.e., can be continuously deformed in 1_B) and gf is homotopic to
1_A. Classically "Homotopy types are the equivalence classes of spaces"
<cit.> under this equivalence relation. That is the #1 type of abstraction.

But the interpretation offered in homotopy type theory is expanding identity
to "coincide with the (unchanged) notion of equivalence" <cit.>
so it would refer to the #2 homotopy type, i.e., `the homotopy
type' that captures the mathematical properties shared by all spaces in an
equivalence class of homotopic spaces (wiping out the differences). Note that
`the homotopy type' is not one of the classical topological spaces
(with points etc.) in the #1 equivalence class of homotopic spaces.


While classical homotopy theory is analytic (spaces and paths are
made of points), homotopy type theory is synthetic: points, paths, and paths
between paths are basic, indivisible, primitive notions. <cit.>


Homotopy type theory systematically develops a theory of the #2
type of abstractions that grows out of homotopy theory and type theory in a
new foundational theory.


From the logical point of view, however, it is a radically new idea:
it says that isomorphic things can be identified! Mathematicians are of course
used to identifying isomorphic structures in practice, but they generally do
so by “abuse of notation”, or some other
informal device, knowing that the objects involved are not “really” identical. But in this new foundational scheme, such
structures can be formally identified, in the logical sense that every
property or construction involving one also applies to the other. <cit.>


Our purpose is rather more modest, to model the theory of paradigm-universals
u_S and their projections u_T–that is analogous to working with sets
and subsets, e.g., in a Boolean algebra of subsets. That is all we will need
to show that probability theory can be developed using paradigms u_S
instead of subset-events S, and to make the connection to quantum mechanics.

Another homotopy example is `the path going once (clockwise) around
the hole' in an annulus A (disk with one hole), an abstract entity 1∈π_0(  A)  ≅


ℤ:












    < g r a p h i c s >








Figure 2: `the path going once (clockwise) around the hole'



Note that `the path going once (clockwise) around the hole'
has the paradigmatic property of "going once (clockwise) around the hole" but
is not one of the particular (coordinatized) paths that constitute
the equivalence class of coordinatized once-around paths deformable into one another.

In a similar manner, we can view other common #2 abstractions such as:
`the cardinal number 5' that captures what is common to the
isomorphism class of all five-element sets; `the number 1
mod(  n)' that captures what is common within
the equivalence class {  ...,-2n+1,-n+1,1,n+1,2n+1,...} of
integers; `the circle' or `the equilateral triangle'–and so forth.

Category theory helped to motivate homotopy type theory for good reason.
Category theory has no notion of identity between objects, only isomorphism as
`equivalence' between objects. Therefore category theory can be seen as a
theory of abstract #2 objects ("up to isomorphism"), e.g.,
abstract sets, groups, spaces, etc.



§ THE CONNECTION TO INTERPRETING SYMMETRY OPERATIONS


The difference between the #1 abstract set and the #2 abstract entity
can also be visually illustrated in a simple example of the symmetry operation
(defining an equivalence relation) of reflection on the aA-axis for a fully
definite isosceles triangles:












    < g r a p h i c s >








Figure 3: Reflection on vertical axis symmetry operation.



Thus the equivalence class of reflective-symmetric figures in the
#1 or classical interpretation is the set:












    < g r a p h i c s >








Figure 4: The #1 abstraction of equivalence class.



But under the #2 or indefiniteness-abstraction(-quantum)
interpretation, the equivalence abstracts to the figure that is definite as to
what is the same and indefinite as to what is different between the definite
figures in the equivalence class:












    < g r a p h i c s >








Figure 5: The #2 abstraction of indefinite entity.



Note that the symmetry operation on the indefinite figure is the
identity. As noted in the discussion of homotopy type theory, the
movement from the #1 equivalence class S to the #2 abstract-indefinite
entity u_S replaces equivalence with identity. That is because the
symmetry operation goes from one element in an equivalence class S to
another element in S that differs in some definite aspects, but those are
precisely the aspects that are removed in the indefinite-abstract u_S–so
the symmetry just takes u_S to itself.

Since we are later going to relate the #2 entities to the indefinite states
of quantum mechanics, the example suggests that while classically a symmetry
operation is invariant on an equivalence class S (i.e., takes one definite
element in the equivalence class S to another definite element in S), in
the #2 quantum case, the symmetry operation on the indefinite entity
u_S is the identity.

This is illustrated in the transition from the classical Maxwell-Boltzmann
statistics to the quantum Bose-Einstein statistics. Suppose we have two
particles of the same type which are classically indistinguishable so,
following Weyl, we distinguish them as Mike and Ike. If each of the two
particles could be in states A, B, or C, then the set of possible states
is the set of nine ordered pairs {  A,B,C}×{
A,B,C}. Applying the symmetry operation of permuting Mike and Ike,
we have six equivalence classes.



c]|c|c|Equivalence classes under permutation     M-B
{(  A,B)  ,(  B,A)  }     2/9
{(  A,C)  ,(  C,A)  }     2/9
{(  B,C)  ,(  C,B)  }     2/9
{(  A,A)  }     1/9
{(  B,B)  }     1/9
{(  C,C)  }     1/9



Table 3: Maxwell-Boltzmann distribution.



Since the primitive data are the ordered pairs, we assign the equal
probabilities of 1/9 to each pair which results in the
Maxwell-Boltzmann distribution for the equivalence classes.

But in the quantum case, we don't have an equivalence class S of distinct
ordered pairs like {(  A,B)  ,(  B,A)
} under the symmetry; we have a single indefinite entity
u_{(  A,B)  ,(  B,A)  } where the
symmetry operation is the identity. Since there are now only six
primitive entities, we assign the equal probabilities of 1/6 to each
entity and obtain the Bose-Einstein distribution.



c]|c|c|Six indefinite states     B-E
1|c|u_{(  A,B)  ,(  B,A)
}     1/6
1|c|u_{(  A,C)  ,(  C,A)
}     1/6
1|c|u_{(  B,C)  ,(  C,B)
}     1/6
1|c|u_{(  A,A)  }    
1/6
1|c|u_{(  B,B)  }    
1/6
1|c|u_{(  C,C)  }    
1/6



Table 4: Bose-Einstein distribution.



Ruling out repeated states (i.e., the Pauli exclusion principle), there are
only three primitive entities and that gives the Fermi-Dirac
distribution.[For more of this pedagogical model of QM using sets
(where the sets may be given the #2 abstraction u_S interpretation),
see <cit.>.]



c]|c|c|Three possible indefinite states     F-D
1|c|u_{(  A,B)  ,(  B,A)
}     1/3
1|c|u_{(  A,C)  ,(  C,A)
}     1/3
1|c|u_{(  B,C)  ,(  C,B)
}     1/3



Table 5: Fermi-Dirac distribution.





§ HOW TO MODEL THE #1 AND #2 ABSTRACTS


There are simple but different models to distinguish the #1 and #2
interpretations for S⊆ U with a finite U={  u_1
,...,u_n} such as:












    < g r a p h i c s >








Figure 6: Universe U of figures



Ordinarily the set of solid figures S={  u_2,u_3
,u_4}⊆{  u_1,u_2,u_3,u_4}  =U would
be represented by a one-dimensional column vector | S⟩ =
[ 0; 1; 1; 1 ], but by using a two-dimensional matrix, we can represent
the two #1 and #2 versions of S as two types of incidence matrices.



  * The #1 (classical) representation of S (i.e., set of S-things or
set of solid figures) is the diagonal matrix In(  Δ
S) that lays the column vector | S⟩ along the
diagonal: In(  Δ S)  =
[ 0 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1 ]
= representation of set S of distinct S-entities. In
(  Δ S) is the incidence matrix of the diagonal
Δ S⊆ U× U whose entries are the values of the
characteristic function χ_Δ S(  u_j,u_k)
=δ_jkχ_S(  u_j).


  * The #2 (quantum) representation of S (i.e., the S-thing)
is the matrix In(  S× S) that uses a 1 in
the row j, column k cell to mean u_j and u_k are both in S:
In(  S× S)  =| S⟩(
| S⟩)  ^t=
[ 0 0 0 0; 0 1 1 1; 0 1 1 1; 0 1 1 1 ]
= representation of one indistinct S-thing, `the solid figure'
u_S=u_2⊞ u_3⊞ u_4. In(  S×
S) is the incidence matrix of the product S×
S⊆ U× U (instead of the diagonal Δ S) with the entries
χ_S× S(  u_j,u_k).


Note that for singletons S={  u_j},
In(  Δ S)  = In(
S× S) as expected, and for | S| >1,
In(  Δ S)  ≠ In(
S× S).

The two representations differ only in the off-diagonal entries. Think of the
off-diagonal In(  S× S)  _j,k=1's as
equating, cohering, or `blobbing' together u_j and u_k:



In(  S× S)  =
[ 0 0 0 0; 0 1 1 1; 0 1 1 1; 0 1 1 1 ] says

[          0          0          0          0;          0          1    u_2 u_3 u_ 2
 u_ 4;          0  u_ 3 u_ 2          1   u_3 u_ 4;          0  u_ 4 u_ 2   u
_4 u_3          1 ].



We now can represent the blob-sum #2 operation on entities: u_S
=⊞{  u_i∈ U|S(  u_i)  } as the
blob-sum ⊞ of the corresponding incidence matrices:



In(  S× S)  =⊞_u_i∈ S
In(  {  u_i}×{
u_i})



where the blob-sum ⊞ is defined for S_1
,S_2⊆ U with S=S_1∪ S_2:



In(  S_1× S_1)  ⊞In
(  S_2× S_2)  := In(  S×
S)  =In(  (  S_1∪ S_2)
×(  S_1∪ S_2)  )

=In(  S_1× S_1∪ S_2× S_2∪
S_1× S_2∪ S_2× S_1)

=In(  S_1× S_1)  ∨In
(  S_2× S_2)  ∨In(  S_1×
S_2)  ∨In(  S_2× S_1).

Disjunction: In(  S_1× S_1)
∨In(  S_2× S_2)  ∨ blobbing
cross-terms.[The disjunction of incidence matrices is the usual
entry-wise disjunction: 1∨1=1∨0=0∨1=1 and 0∨0=0, and similarly
for conjunction.]



For S={  u_2,u_4}, the blob-sum u_S=u_2⊞
u_4 is represented by:



In(  {  u_2}×{
u_2})  ⊞In(  {
u_4}×{  u_4})  =In
(  S× S)



where the blob-sum operation ⊞ means `blobbing-out' the
distinctions between entities in S (given by the cross-terms in {
u_2,u_4}×{  u_2,u_4}):



In(  S× S)  =In(  {
u_2}×{  u_2})  ⊞In(  {  u_4}×{  u_4})

=
[ 0 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 0 ]⊞
[ 0 0 0 0; 0 0 0 0; 0 0 0 0; 0 0 0 1 ]

=In(  {  u_2,u_4}×{
u_2,u_4})

=In(  {  u_2}×{
u_2})  ∨In(  {  u_4}×{  u_4})  ∨In(  {
u_2}×{  u_4})  ∨In
(  {  u_4}×{  u_2})

=
[ 0 0 0 0; 0 1 0 1; 0 0 0 0; 0 1 0 1 ].



Due to the development of Boolean subset logic and set theory, we are
perfectly comfortable with considering the #1 abstractions of sets S of
even concrete ur-elements like the set of entities on a table. The
representatives In(  Δ S) trivially form a
BA isomorphic to the BA of subsets ℘(  U)  .

To better understand abstraction in mathematics and indefinite states in QM,
we should become as comfortable with paradigms u_S as with sets S. The
paradigms u_S for S∈℘(  U) form a Boolean algebra
isomorphic to ℘(  U) under the mapping: for any Boolean
operation S#T for S,T∈℘(  U), u_S#u_T is the
paradigm represented by In(  (  S#T)
×(  S#T)  ).



  * The union of subsets S∪ T induces the operation on paradigms
represented by In(  (  S∪ T)  ×(
S∪ T)  )  =In(  S× S)
⊞In(  T× T), so the union or
join of paradigms is the blob-sum u_S∪ T =u_S⊞ u_T (note
as expected, for T⊆ S, u_S⊞ u_T=u_S);


  * The intersection or meet of paradigms u_S∧
u_T=u_S∩ T is represented by In(  S∩ T×
S∩ T)  =In(  S× S)  ∧In(  T× T) (note as expected, for
T⊆ S, u_S∧ u_T=u_T);


  * The negation of a paradigm ¬ u_S=u_S^c is
represented by In(  S^c× S^c)
=⊞{In(  {  u}×{
u})  |u∉ S} (note as expected, u_S⊞
u_S^c=u_U).




§ THE PROJECTION OPERATION: MAKING AN INDEFINITE ENTITY MORE DEFINITE


Now suppose we classify or partition all the elements of U
according to an attribute such as the parity of the number of sides, where a
partition is a set of disjoint subsets (blocks) of U whose union is
all of U. Let π be the partition of two blocks O={
Odd}  ={  u_1,u_3} and E={
Even}  ={  u_2,u_4}.

The equivalence relation defined by π is *indit(
π)  =(  O× O)  ∪(  E× E)
<cit.> and the disjunction is:



In(  O× O)  ∨In(
E× E)  =In(  *indit(
π)  )

[ 1 0 1 0; 0 0 0 0; 1 0 1 0; 0 0 0 0 ]∨
[ 0 0 0 0; 0 1 0 1; 0 0 0 0; 0 1 0 1 ]
=
[ 1 0 1 0; 0 1 0 1; 1 0 1 0; 0 1 0 1 ].



The #1 (classical) operation of intersecting the set of even-sided figures
with the set of solid figures to give the set of even-sided solid figures is
represented as the conjunction:



In(  Δ E)  ∧In(
Δ S)  =
[ 0 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 1 ]∧
[ 0 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1 ]
=
[ 0 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 1 ].



The #2 (quantum) operation of `sharpening' or `rendering more definite'
`the solid figure' u_S to `the even-sided solid figure'
u_{  u_2,u_4}, so u_{  u_2,u_4} u_S (suggested reading: u_{  u_2,u_4} is a
projection of u_S) is represented as:



In(  E× E)  ∧In(
S× S)  =
[ 0 0 0 0; 0 1 0 1; 0 0 0 0; 0 1 0 1 ]∧
[ 0 0 0 0; 0 1 1 1; 0 1 1 1; 0 1 1 1 ]
=
[ 0 0 0 0; 0 1 0 1; 0 0 0 0; 0 1 0 1 ].



But there is a better way to represent `sharpening' using matrix
multiplication instead of just the logical operation ∧ on matrices, and
it foreshadows the measurement operation in QM. The matrix In
(  Δ E)  =P_E is a projection matrix, i.e., the diagonal
matrix with diagonal entries χ_E(  u_i) so
P_E| S⟩ =| E∩ S⟩. Then the
result of the projection-sharpening can be represented as:



| E∩ S⟩(  | E∩ S⟩)  ^t=P_E| S⟩(  P_E|
S⟩)  ^t=P_E| S⟩(  |
S⟩)  ^tP_E

=P_EIn(  S× S)  P_E=In
(  E× E)  ∧In(  S× S).



Under the #2 interpretation, the parity-sharpening,
parity-differentiation, or parity-measurement of `the solid
figure' by both parities is represented as:



In(  *indit(  π)  )
∧ In(  S× S)  =P_OIn
(  S× S)  P_O+P_EIn(  S×
S)  P_E

=
[ 1 0 1 0; 0 1 0 1; 1 0 1 0; 0 1 0 1 ]∧
[ 0 0 0 0; 0 1 1 1; 0 1 1 1; 0 1 1 1 ]
=
[ 0 0 0 0; 0 1 0 1; 0 0 1 0; 0 1 0 1 ].



The results are `the even-sided solid figure' u_{
u_2,u_4} and `the odd-sided solid figure' u_{
u_3}=u_3. The important thing to notice is the action on the
off-diagonal elements where the action 1⇝0 in the j,k-entry
means that u_j and u_k have been deblobbed, decohered, distinguished,
or differentiated–in this case by parity:



In(  S× S) In
(  *indit(  π)  )  ∧
In(  S× S)

=P_OIn(  S× S)  P_O+P_EIn
(  S× S)  P_E

[          0          0          0          0;          0          1  1 deblob0          1;          0  1 deblob0          1 1 deblob
0;          0          1  1 deblob0          1 ].



We could also classify the figures as to having 4 or fewer sides ("few
sides") or not ("many sides") so that partition is σ={{
u_1,u_2}  ,{  u_3,u_4}} which is
represented by:



In(  *indit(  σ)  )
=
[ 1 1 0 0; 1 1 0 0; 0 0 1 1; 0 0 1 1 ] and

In(  *indit(  σ)  )
∧(  In(  *indit(
π)  )  ∧In(  S× S)  )
=
[ 0 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1 ]
= In(  Δ S).



Thus parity and few-or-many-sides properties suffice to classify the solid
figures uniquely and thus to yield the representation In
(  Δ S) of the distinct elements of S={  u_2
,u_3,u_4}. Thus making all the distinctions (i.e., decohering
the entities that cohered together in u_S) takes In(
S× S) In(  Δ S).

In QM jargon, the parity and few-or-many-sides attributes constitute a
"complete set of commuting operators" (CSCO) so that measurement of
`the solid figure' by those observables will take `the solid
figure,' to the separate eigen-solid-figures: `the few- and
even-sided solid figure' (the square u_2), `the many- and
odd-sided solid figure' (the pentagon u_3), and `the many- and
even-sided solid figure' (the hexagon u_4).



§ FROM INCIDENCE TO DENSITY MATRICES


The incidence matrices In(  Δ S) and
In(  S× S) can be turned into
density matrices by dividing through by their trace:



ρ(  Δ S)  =1/*tr[
In(  Δ S)  ] In
(  Δ S) and ρ(  S)  =1/*tr[  In(  S× S)  ] In(  S× S).



In terms of probabilities, this means treating the outcomes in S
as being equiprobable with probability 1/| S|.
But now we have the #1 and #2 interpretations of the sample space for
finite discrete probability theory.



  * The #1 (classical) interpretation, represented by ρ(  Δ
S), is the classical version with S as the sample space of
outcomes. For instance, the 6×6 diagonal matrix with diagonal entries
1/6 is "the statistical mixture describing the state of a classical
dice [die] before the outcome of the throw" <cit.>;


  * The #2 (quantum) interpretation replaces the "sample space" with the
one indefinite `the sample outcome' u_S represented by
ρ(  S) (like `the outcome of throwing a die') and,
in a trial, the indefinite outcome u_S `sharpens to' or becomes a definite
outcome u_i∈ S with probability 1/| S|.


Let f:U→


ℝ be a real-valued random variable with distinct values ϕ_i for
i=1,...,m and let π={  B_i}  _i=1,...,m where
B_i=f^-1(  ϕ_i), be the partition of U according to
the values. The classification of ρ(  S) according to the
different values is: In(  *indit(
π)  )  ∧ρ(  S) which distinguishes the
elements of S that have different f-values. If P_B_i is the diagonal
(projection) matrix with diagonal elements (  P_B_i)
_jj=χ_B_i(  u_j), then the probability of a trial
returning a u_j with f(  u_j)  =ϕ_i is:



(  ϕ_i|S)  =*tr[  P_B_iρ(
S)  ].



For instance, in the previous example, where f:U→


ℝ gives the parity partition π with the two values ϕ_odd and
ϕ_even, then:



P_evenρ(  S)  =
[ 0 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 1 ]
[   0   0   0   0;   0 1/3 1/3 1/3;   0 1/3 1/3 1/3;   0 1/3 1/3 1/3 ]=
[   0   0   0   0;   0 1/3 1/3 1/3;   0   0   0   0;   0 1/3 1/3 1/3 ]



so *tr[  P_evenρ(  S)  ]
=2/3 which is the conditional probability of getting `the
even-sided solid figure' starting with `the solid figure' in the
#2 (quantum) interpretation. And under the #1 (standard) interpretation,
(  ϕ_even|S)  =*tr[  P_even
ρ(  Δ S)  ]  =2/3 which is the probability
of getting an even-sided solid figure starting with the set of solid figures.

These two interpretations of finite discrete probability theory extend easily
to the case of point probabilities p_j for u_j∈ U, where:



  * (  ρ(  Δ S)  )  _jj=χ_S(
u_j)  p_j/(  S), so *tr[
P_evenρ(  Δ S)  ]  = probability of getting
an even-sided solid figure starting with the set of solid
figures, and


  * (  ρ(  S)  )  _j,k=χ_S(
u_j)  χ_S(  u_k)  √(p_jp_k)/(
S), so *tr[  P_evenρ(  S)
]  = probability of getting `the even-sided
solid figure' starting with `the solid figure.'


The whole of finite discrete probability theory can be developed in
this manner, mutatis mutandis, for the #2 interpretation paradigms.



§ DENSITY MATRICES IN QUANTUM MECHANICS


The jump to quantum mechanics (QM) is to replace the binary digits like 0,1
in incidence matrices or reals √(p_jp_k) in `classical' density
matrices by complex numbers. Instead of the set S represented by a column
| S⟩ of 0,1, we have a normalized column |ψ⟩ of complex numbers α_j whose absolute squares are
probabilities: |α_j| ^2=p_j, e.g.,



| S⟩ =
[ 0; 1; 1; 1 ]|ψ⟩ =
[ α_1; α_2; α_3; α_4 ]

where α_1=0 and |α_j| ^2=p_j for
j=2,3,4.





  * The density matrix ρ(  Δψ) has the
absolute squares |α_j| ^2=p_j laid out along
the diagonal.


  * The density matrix ρ(  ψ) has the
j,k-entry as the product of α_j and α_k^∗ (complex
conjugate of α_k), where p_j=α_j^∗α
_j=|α_j| ^2.


Thus:



ρ(  Δψ)  =
[   0   0   0   0;   0 p_2   0   0;   0   0 p_3   0;   0   0   0 p_4 ] and ρ(  ψ)  =
[        0        0        0        0;        0      p_2 α_2α_3^∗ α_2α_4^∗;        0 α_3α_2^∗      p_3 α_3α_4^∗;        0 α_4α_2^∗ α_4α_3^∗      p_4 ].




The] off-diagonal terms of a density matrix...are often
called quantum coherences because they are responsible for the
interference effects typical of quantum mechanics that are absent in classical
dynamics. <cit.>


The classifying or measuring operation In(
*indit(  π)  )  ∧ρ(
ψ) could still be defined taking the minimum of corresponding
entries in absolute value, but in QM it is defined as the Lüders
mixture operation <cit.>. If π={  B_1
,...,B_m} is a partition according to the eigenvalues ϕ
_1,...,ϕ_m on U={  u_1,...,u_n} (where U is an
orthonormal basis set for the observable being measured), let P_B_i be
the diagonal (projection) matrix with diagonal entries (  P_B_i
)  _jj=χ_B_i(  u_j). Then In
(  *indit(  π)  )  ∧ρ(
ψ) is obtained as:



∑_B_i∈πP_B_iρ(  ψ)  P_B_i

The Lüders mixture.



The probability of getting the result ϕ_i is:



(  ϕ_i|ψ)  =*tr[  P_B_i
ρ(  ψ)  ].





§ A POP SCIENCE INTERLUDE


The popular science version of the simplest case is Schrödinger's cat.












    < g r a p h i c s >








Figure 7: Usual "And" version of Schrödinger's cat.



This version of Schrödinger's cat as being "Dead & Alive" is
like the usual mis-interpretation of the unobserved particle as going through
"Slit 1 & Slit 2" in the double slit experiment. But the cat is not
definitely alive and definitely dead at the same time. The quantum
version is that the cat is indefinite between those two definite
possibilities; it's in cat-limbo.



Schrödinger's cat = dead-cat ⊞ live-cat.



It would be more accurate to say "Dead or Alive–but neither definitely," a
"disjunctive fact" <cit.>.











    < g r a p h i c s >








Figure 8: The disjunctive cat.



Technically the state vector is:












    < g r a p h i c s >








Figure 9: Schrödinger's cat state vector.



Using density matrices, we would represent Schrödinger's cat as being in
the state:



ρ(  cat)  =
[ 1/2live     1/2;     1/2 1/2dead ].





§ SIMPLEST QUANTUM EXAMPLE


Consider a system with two spin-observable σ eigenstates|↑⟩ and |↓⟩ (like
electron spin up or down along the z-axis) where the given normalized
superposition state is |ψ⟩ =1/√(2)
|↑⟩ +1/√(2)|↓⟩ =
[ α_↑; α_↓ ]
=
[ 1/√(2); 1/√(2) ] so the density matrix is ρ(  ψ)  =
[      p_↑ α_↑α_↓^∗; α_↓α_↑^∗      p_↓ ]
=
[ 1/2 1/2; 1/2 1/2 ] where p_↑=α_↑α_↑^∗ and
p_↓=α_↓α_↓^∗. The
measurement in that spin-observable σ goes from ρ(
ψ) to



In(  *indit(  σ)  )
∧ρ(  ψ)  =
[ 1 0; 0 1 ]∧
[      p_↑ α_↑α_↓^∗; α_↓α_↑^∗      p_↓ ]
=
[ p_↑   0;   0 p_↓ ]
= [ 1/2   0;   0 1/2 ]
=ρ(  Δψ).



Or using the Lüders mixture operation:



P_↑ρ(  ψ)  P_↑+P_↓ρ(
ψ)  P_↓

=
[ 1 0; 0 0 ]
[      p_↑ α_↑α_↓^∗; α_↓α_↑^∗      p_↓ ]
[ 1 0; 0 0 ]
+
[ 0 0; 0 1 ]
[      p_↑ α_↑α_↓^∗; α_↓α_↑^∗      p_↓ ]
[ 0 0; 0 1 ]

=
[ p_↑   0;   0 p_↓ ]
= [ 1/2   0;   0 1/2 ]
=ρ(  Δψ).



The two versions of S=U give us two versions of finite discrete probability
theory where: #1) U is the sample space or #2) u_U is the
sample outcome.



  * The #1 classical version is the usual version which in this case is
like flipping a fair coin and getting head or tails with equal probability.















    < g r a p h i c s >








Figure 10: Outcome set for classical coin-flipping trial.





  2. The #2 quantum version starts with the indefinite entity
u_U=⊞{  u_i∈ U}, `the (indefinite)
outcome', and a trial renders it into one of the definite outcomes u_i
with some probability p_i so that u_U could be represented by the
density matrix ρ(  U) where (  ρ(  U)
)  _jk=√(p_jp_k). In this case, this is like a coin
u_{  H,T} with the difference between heads or tails
rendered indefinite or blobbed out, and the trial results in it sharpening to
definitely heads or definitely tails with equal probability.















    < g r a p h i c s >








Figure 11: `the outcome state' for quantum coin-flipping trial.



Experimentally, it is not possible to distinguish between the #1 and #2
versions by σ-measurements. But in QM the two states
ρ(  Δψ) and ρ(  ψ) can
be distinguished by measuring other observables like spin along a different
axis <cit.>. Thus we know in QM which version is
the superposition (pure) state |ψ⟩ =
[ α_↑; α_↓ ]; it is the #2 blob-state ρ(  ψ).



§ CONCLUSIONS


Quantum mechanics texts usually mention several interpretations such as the
Copenhagen, many-worlds, or hidden-variables interpretations. Now that we have
established a bridge from abstraction in mathematics to indefinite states in
QM, we may (for fun) cross the bridge in the opposite direction. For instance,
in the many-worlds (or many-minds) interpretation, 1∈π_0(
A)  ≅


ℤ would refer to a different specific coordinatized "once clockwise around the
hole" path in each different world (or mind).

Shimony, however, suggests the Literal or Objective Indefiniteness
Interpretation–which we have seen is suggested by the mathematics itself.


But the mathematical formalism ... suggests a philosophical
interpretation of quantum mechanics which I shall call "the Literal
Interpretation." ...This is the interpretation resulting from taking the
formalism of quantum mechanics literally, as giving a representation of
physical properties themselves, rather than of human knowledge of them, and by
taking this representation to be complete. <cit.>


We have approached QM by starting with the logical situation of a universe U
of distinct entities. Given a property S(  x) on U, we can
associate with it:



  * the #1 abstract object S={  u_i∈ U|S(  u_i)
}, the set of S(  x)-entities, or


  * the #2 abstract object u_S=⊞{  u_i∈ U|S(
u_i)  } which is the abstract entity expressing the
properties common to the S(  x)-entities but "abstracting away
from," "rendering indefinite," "cohering together," or "blobbing out" the
differences between those entities.


We argued that the mathematical formalisms of incidence matrices and then
density matrices can be used to formalize the two representations:



  * #1 representation as In(  Δ S) or
ρ(  Δψ); and


  * #2 representation as In(  S× S) or
ρ(  ψ).


This dove-tailed precisely into usual density-matrix treatment in QM
of quantum states |ψ⟩ as ρ(  ψ) which, as suggested by Shimony, can be interpreted as objectively
indefinite states.

Yet since the ancient Greeks, we have the #2 Platonic notion of the
abstract paradigm-universal `the S-entity', definite on what is
common to the members of a set S and indefinite on where they differ, so the
connection that may help to better understand quantum mechanics is:



The paradigm u_S, `the S-entity' represented by
In(  S× S)  the superposition state
ψ represented by the density matrix ρ(  ψ).



This recalls Whitehead's quip that Western philosophy is "a series
of footnotes to Plato." <cit.>

99                                                                                               


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