### Distinct yet Identical

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Mathematically "equivalence" can be defined as usual as in the case I.
shown below.

Physically though we can define "equivalence" in terms of quantum mechanic's
notion of identical particles as in case II.

The notion of distinct yet identical particles is more difficult understand
conceptually as is shown in case III.

We can use the terms 'metaphor' or 'analog' to define this notion of 'equivalence'.

We can employ metaphor to show an example of this definition of "equivalance".
For instance, we can say that:

"an individual in society is considered distinct,
and yet under the law they are (ideally speaking)
equivalent".

And this metaphor between the distinct yet identical particles
and the individual in society, I will call a 'strong metaphor'
or a 'literal metaphor', because at the fundamental level there
is no distinction between the assignment of:

A <- distinct yet identical particles
B <- distinct yet equal citizens

and saying that these are 'equivalent' in the sense of case III.
That A is distinct from B is obvious, and yet they are related by
'metaphor'. Or we can also say they are 'correlated'.

How does the relationship by metaphor affect our definition of
distinctness ? If the metaphor is very very strong, the
case III. interpretation degenerates into the case I. interpretation,
and if the metaphor is very very weak the case III. interpretation
degenerates into the case II. interpretation of 'equivalence'.

It is essential that the case III. interpretation be considered
carefully in the context of degrees of orthogonality, interference,
and aliasing. All of these are related effects based on levels
of resolution, dispersion, distinguishment,... etc.

Logic makes connections between causually dependant variables,
and analogic makes connections between non-causually dependant
(independant) variables.

When we employ logic, it is wholely contained within some causual system.
That system may be analogous to another system. So we may employ
analogy and say:

"If 'A' is analogous to 'B', then perhaps if I know how
'A' works I can use that information to understand better
how 'B' works."

Logic is employed subsequent to this theoretical step of making
an analogy. The logic is used within 'A' or within 'B' to make
the causual connection and analogic is used between 'A' and 'B'
to make non-causual relations or 'functional relations'.

In the same manner we can say if two statistical distributions
are the same for two systems 'A' and 'B', there may not be a
causual connection between these systems but any understanding
of what leads to that distribution in 'A' may be useful in
understanding how the same distribution is generated by 'B'
from it's dependant subsystems.

Theory, employs analogical thinking more than empircism, which
employs logical thinking and the relationship of these two
modes of thinking combined form the basis of rational thinking.

(one may look into Vaughan Pratt's [Knuth TAOCP] work on Chu spaces or
contact Stephan Paul King. I find it difficult to understand
what these people are doing but I sense they are in general
addressing these ideas in a more conventional manner. There

Now, it may be noted that the above description of the rational
process can be subverted from its usual implementation in the
pursuit of knowledge.

Suppose for instance, that we have someone who is engaged in
some field of study 'A' who realizes that another person's work
in a totally different field 'B', is analogous to what he is doing.

Suppose further that the person in the field 'B' has made a significant
contribution to her field and proven some logical (causual) connectives
between many subsystems in the field 'B' and received significant
awards for doing so.

The person studying in field 'A' may, through the recognition of analogs,
be able to replicate all the work of the person studying in the field 'B'
by establishing the same logical connections in the subsystem of 'A' as
have analogs in the subsystems of 'B' (to the extent this is possible).
The person studying in field 'A' may then "burn the analogical bridge"
and take full credit for his discovery in the context of 'A' without
making any reference to the work done by the other person in 'B'.
The 'A' person may then reap significant awards in his field as we
they are analogous awards to the 'B' person's awards.

This is essentially a holistic plagiarism and it is very likely that
many of our so called "greatest thinkers" in past history commited
this act either consciously or unconsciously.

It would take a holistic detective[1] to peer backwards through history
and try to determine who commited such acts. The process by which such
trangressions were uncovered would not be mearly useful for judicial
purposes but would be very instructive as well in the analysis of
information in quantum physics.

It should be noted that many skilled logicians satired their logic-
in life and in fictions. Doyle[2] became a "spiritualist" in later life
and Carroll's[3] works in fiction, are far better known than his books
on logic. And Houdini was indeed a master at hiding causual connections.

It might be possible to get away with saying that case I. is 'classical',
case II. is special relativistic (a deterministic connection between
A and B even though A and B are treated as distinct and obeying the same
rules) and case III. is quantum mechanical (a non-deterministic connective
provides the link between A and B).

Case I.   Classical - Absolute definition of equivalence. A=B
and A _is_ B

Case II.  Relativistic - relative, differential definition of
equivalence A=B and A is not the same as B.
A deterministic connection between A and B exists
(A can be transformed to B)

Case III. Quantum mechanical- a correlation exists between A and B.
A=B to the extent that A is correlated with B.
A is typically complementary to B and the act of correlation
is not necessaryily commutative.

Case III. degenerates into Case II. if A and B are
completely correlated (dependant) and Case II degenerates
into Case I if A and B are indistinguishable
(absolutely equivalent)

So these three seem to fold nicely into each other.

(relativity theory and quantum mechanics are not
currently well connected in theory. The field
of "quantum gravity" is attempting to do this
which seems to be trying to relate Case II to Case III
more rigourously, that is, we cannot always know if
in Case III whether A is completely correlated with
B in the sense of Case II or whether they are merely
somewhat correlated. Case III may be Case II in disguise
or it may actually be that A and B are independant
or partially dependant in Case III.)

Const MIN=0.0; Const MAX=1.0;

Type A,B,CORRELATE(),SUBDIVIDE(),TRANSFORM(): ABSTRACT

Procedure case3(A,B);  (* NON-DETERMINISTIC *)
Begin
x:=CORRELATE(A,B);
if x=MAX then    (* dependant; a deterministic connection
should exist *)
case2(A,B);
halt;
Else
if x=MIN then          (* independant *)
writeln(A,' and ',B,' are unrelated.');
halt;
Else
Begin
writeln(A,' and ',B,'are analogous to the extent:',x);
writeln('Trying to determine why...');
case3(SUBDIVIDE(A),SUBDIVIDE(B));
End;
End;

Procedure case2(A,B); (* DETERMINISTIC *)
Begin
(* try to find a deterministic function mapping/transforming
A to B. *)
y:=TRANSFORM(A,B);
(* if y is the identity transform then
writeln(A,' is classically equivalent to' B); (* case 1 *)
else
if y is the inverse transform then
writeln(A,' is the inverse of ',B); (* case 2 *)
else ... &c.
*)
End;

The above may look familiar. It is somewhat similar to
computer graphics programs that compute smooth minimal hulls
of surface intersections and probably is very similar
to many AI techniques I am unaware of.

It is not at all complete. For instance, just looking at
some of the cases Aristotle brings up in regards to
complementarity and commutivity etc. the above could be
enhanced by including these and other senarios.

As a model for 'intelligent' thinking we might interpret
case3 and case2 procedures as 'analogizing' and 'logizing'
respectively.

But case3 includes case2 as a subset, and case2 includes case1
as a subset.

If a problem cannot be reduced by case3 there is the halting problem.
It may be necessary to define MIN more practically, to avoid asymptotes
(like the "renormalization problem" in quantum physics,
but inherent uncertainties like Heisenberg's, seem to set
a useful limit).

As a model for the mind, suppose someone is not very good
at doing y:=TRANSFORM(A,B); we might label them as leaning towards
seeing everything as analogous. We might call them spiritual
or poetic or artistic or right-brainers, or sinistral...

If they are good at y:=TRANSFORM(A,B), they may do this instinctively.
In which case they can show you "y" but not how they derived it.
We might call these people instinctive, or extroverts. They are
able to climb mountains and ride skateboards with instinctive ease
as a kind of 'intelligent' behaviour.

If they are bad at doing x:=CORRELATE(A,B) we might tend to call them
anal(derogatively), or scientific, or not artistically inclined,
or left-brainers,... etc, etc.

A higher-function of 'rationality' might be used to balance
CORRELATE() and TRANSFORM() or at least to coordinate their
use. We speak of a "well balanced" individual when they can
analogize and logize effectively or optimally.

CORRELATE() degenerates into TRANSFORM() in some cases, and
in others not. So 'balance' is not really a good word for
what seems to be happening there.

[1]  Dirk Gently's Holistic Detective Agency, Douglas Adams
[2]  The Edge of the Unknown, Sir Arthur Conan Doyle
[3]  Lewis Carroll

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