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fom
Posts:
1,035
Registered:
12/4/12
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Re: distinguishability - in context, according to definitions
Posted:
Feb 11, 2013 11:53 AM
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On 2/11/2013 4:06 AM, Shmuel (Seymour J.) Metz wrote:
> In <5sidnRpmsrGpz4rMnZ2dnUVZ_ridnZ2d@giganews.com>, on 02/10/2013
> at 01:54 AM, fom <fomJUNK@nyms.net> said:
>
>> Subject: distinguishability - in context, according to definitions
>
> Does this derive from the Hermeneutics of Quantum Gravity?
>
Should I assume this question is contemptuous?
I am not so smart about these things. The "logical
investigation" of the foundation for mathematics
is characterized by "linguistic analysis".
First, no. This post derives from having watched
the most amazing discussion about "distinguishability"
in binary trees which seemed to be at odds with
what I understood. The error had been mine because
I had interpreted "complete" in the topological
sense rather than in the sense of definition for a
type of binary tree. However, I still do not
think the participants whose usage had confused me
had been using the phrase according to standard
definitions. The post gave me a place to redirect
for a response concerning the hierarchy of defined
logical types.
As for what I suspect from your question, what follows
are some quotes from a paper that was rejected for
publication several years ago. If it was rejected
for technical reasons, it would be a surprise to me.
In notifying me of receipt, the Journal of Symbolic
Logic mistakenly addressed me as "Professor". My
notice of rejection came immediately after I informed
them of their error. They rudely and contemptuously
informed me that people without appropriate
qualifications should not attempt to publish in
academic journals.
I have regularly received that kind of response
from professionals no matter how politely and
respectfully I have approached them. I do respect
the circumstances of their situation and do
not make a nuisance.
From the unpublished "A Formal Description
of Identity":
===========
"This paper rejects acceptance of the axiom
of extensionality as being fundamental in
its assertion. Rather, it is the consequence
of a subtle language construction strategy
which is not generally applicable to other
first-order models.
"There are several issues here. First of all,
the implementation of identity as a 'logical'
symbol of the language fails to provide any
intuition as to how the identity predicate
differs from arbitrary equivalence relations.
"Clearly, the notion of an equivalence relation
generalizes the properties of the identity
predicate. However, conventional model theory
provides no foundation for the special properties
which distinguish identity from other notions
of equivalence."
==========
Having been generally unknowledgeable concerning
the genesis of much of modern set theory, I spoke
about "classes" in the sense of "every constituent
of a set is a set".
In the next quote, I should have used the term
"duality" where I wrote "complementarity"
==========
"This phenomenon may be referred to as complementarity
of representation. It does not occur for any object
type which is different from a class.
"Because of this unique mode of presentation,
it is possible to combine the predicates of a
class theory so that topological separation
properties constitute the basis for an identity
predicate. That is, a formal description of
the relationships between classes should assert
that any two 'dot' representations be separated
by some 'circle' representation"
"The first thing to recognize is that degrees
of freedom are necessary to implement complementarity
in the formal sense. That is, the 'dot' and 'circle'
representations must have related, but independent,
interpretations in the formal system.
"This is accomplished with the sentences
AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))
AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))
[...]
"Specifically, the objective here is to capture
topological relationships between the representations
presented in the Venn diagrams. These two sentences
manage the first step with an undifferentiated
parallel syntax that expresses the complementarity of
representation discussed above."
=============
Actually, there had actually been an error in the
paper which is now corrected (I believe -- never
peer reviewed).
But, look at the two sentences to which everyone
rejects as "circular". What they really are
are the smallest representatives (Length(wff))
of an infinite class of axioms obtained by
substitutions. My version of a foundation for
set theory has a self-similar grammatical form.
Given that possibility for a foundation to
"the language of science", here is one of
the possible interpretations for general
relativity,
http://en.wikipedia.org/wiki/Scale_relativity
Among other things, my views on the foundations
of mathematics permitted me to formulate a
notion of "Riemannian presupposition at a
double origin" based on an understanding
of logic in relation to the ortholattice
(may not display well)
....................................TRU....................................
............................./.../..//\..\.................................
......................../..../.../../....\...\.............................
.................../...../..../..../.........\.....\.......................
............../....../...../....../...............\......\.................
........./......./....../......../.....................\.......\...........
....../......./......./........./...........................\........\.....
.....IF......NAND.......IMP.....OR.........................ALL........NO...
..../.\.\..../.\.\..../..|.\..././\.\\..................../...\.....././...
.../...\./\......\./\....|./..\./..\...\...\...................../.........
../../..\...\.../...\./.\|...../.\..\....\............/....../...\../......
.//......\.../\.../....\.|.\../....\.\......\...\......../.................
LET.......XOR..FLIP....FIX..LEQ.....DENY........./.../............/\.......
.\\....../...\/...\..../.|./..\...././......./...\.......\.................
..\..\../.../...\.../.\./|.....\./../...../...../.............../....\.....
...\.../.\/....../.\/....|.\../.\../.../.../...........\.........\.........
....\././....\././....\..|./...\.\/.//.....................\./.......\.\...
.....NIF......AND......NIMP.....NOR........................OTHER......SOME.
......\.......\.......\.........\.........................../......../.....
.........\.......\......\........\...................../......./...........
..............\......\.....\......\.............../....../.................
...................\.....\....\....\........./...../.......................
........................\....\...\..\..../.../.............................
.............................\...\..\\/../.................................
................................NOT.NTRU....................................
You will find mention of
"One passes from one definition to the other by the
transformation dt <-> -dt (differential time reflection
invariance), which actually was an implicit discrete
symmetry of differentiable physics."
in
http://luth.obspm.fr/~luthier/nottale/arcasys03.pdf
This ortholattice corresponds with the kind
of orthologic derived from slit experiments.
In terms of my analysis, it derives from
considering the semantics of propositional
logic relative to the affine part of a
projective plane on 21 points. The sixteen
points of the affine part are arranged on
20 lines, and, by careful use of names,
the points of the ortholattice correspond
to the labels of the dual plane.
The double bottom in the diagram reflects
the one switching function that correspondingly
names the line at infinity. So, "NOT" is on the
line at infinity and "NTRU" names the line
at infinity.
The double origin as a Riemannian presupposition
derives from different considerations by which
an unoriented three dimensional origin is
represented using six line affine planes.
In the context of the presuppostion in four
dimensions, the double origin represents the
non-orientability of a toroidal surface --
itself represented using Karnaugh maps.
As for relations to quantum mechanics,
I just completed a logical alphabet for
the free orthomodular lattice on two
generators using a difference set
construction.
The construction reflects the quantum
duality of its letters relative to
certain design-theoretic constructions.
First, of the 96 elements in the
lattice, 5 are eliminated for each
given letter relative to an artifactual
name having multiplicity 5 in the
corresponding difference set.
The remaining 91 points form 4 distinct
projective planes, 3 of which are
non-Desarguesian. This reflects the
singlet state/triplet state sense
of quantum objects.
The difference set construction
reflects the axiom of regularity
in such a way that one component
copy of the free boolean lattice
on two generators contains no elements
of the difference set. This boolean
part contains both the given letter
and the last of the elements associated
with the artifactual name mentioned
above.
Now, paradigmatically, the quantum
duality can be thought of as a
Steiner Quadruple System on 14
elements in the semantics of
propositional logic by virtue
of the fact that for every logical
connective different from LEQ (iff)
and XOR the associated switching
function is linearly separable.
There are other reasons for accepting
this involving invariance under
DeMorgan conjugation.
In terms of the free orthomodular
lattice, one forms the Steiner
Quadruple System using the given
letter and the remaing artifactual
element relative to the component
Boolean lattice in which they
are situated.
In addition, one makes the same
sort of construction on the entire
91 element subset using a
S_5(2,5,89)
where every pair of elements of the
89 element set occurs in 5 blocks
of order 5.
Presumably, this construction can
be made to coincide with the Karnaugh
maps mentioned above via Curtis'
construction of the Miracle Octad
Generator. The elements of the
difference sets are dispersed over
5 boolean components in such a
way that each of 5 Karnaugh maps
has 4 distinguished elements. With
the Karnaugh maps embedded in
the MOG array, the entire construct
becomes interrelated with the
12-dimensional information-theoretic
Golay code. Thus, the model
reflects quantization by virtue
of information-theoretic means.
This is all geometry/syntax.
I would love to see it implemented
into a computer to see if a true
quantum logic or quantum computational
model could be discerned by
investigating group actions.
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