Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.


fom
Posts:
1,968
Registered:
12/4/12


Re: distinguishability  in context, according to definitions
Posted:
Feb 11, 2013 11:53 AM


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From the unpublished "A Formal Description of Identity":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 firstorder 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 selfsimilar 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 nonorientability 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 designtheoretic 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 nonDesarguesian. 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 12dimensional informationtheoretic Golay code. Thus, the model reflects quantization by virtue of informationtheoretic 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.



