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fom
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12/4/12


when indecomposability is decomposable
Posted:
Feb 16, 2013 12:02 AM


I own only a handful of books on constructive mathematics, and, those are of recent acquisition. They are from the Russian school  Markov and Sanin.
In the introduction of Sanin's "Constructive Real Numbers and Function Spaces" there is a footnote with the following explanation of constructive objects:
"By constructive objects are meant objects which are the results of processes of construction realizable on the following basis: one assumes that the objects which figure in the given study as indecomposable initial objects are clearly described; one assumes, given a list of rules of formation of new objects from previously constructed ones, which in the given study plays the role of a description of the admissible steps of constructive processes; one assumes that the processes of construction are carried out in discrete steps, where the choice of each s succeeding step is arbitrary, within the limits determined by the list of already constructed objects and the set of those rules of formation of new objects which can actually be applied to already constructed objects."
When one invokes the axiom,
Ax(x=x)
by
a=a
there is an ontological interpretation of the sign of equality corresponding with the sense of indecomposability. Perhaps one could quibble over the difference between "ontological invariance" and "indecomposability." I would probably lose that argument since I am not particularly adept in debates.
Now, in the "list of rules of formation of new objects from prevously constructed ones" there is the notion of "definite description". In "Word and Object" Quine goes to great pains just to formulate an argument that eliminates names because whenever a name might be needed, a description may be used to introduce it.
Of course, mathematicians generally do not know of description theory. But, it is certainly discussed in metamathematics, and, I believe, it is discussed in the manner in which I tend to view it. But, I may be wrong.
Tarski really dodges the issue in "The Concept of Truth in Formalized Languages." However, in "Some Methodological Investigations on the Definability of Concepts" his statements and usage seem to correlate with my understanding, Section 1 of the paper begins with:
"The problems to be discussed in this article concern the specific terms of any deductive theory.
"Let 'a' be some extralogical constant and B any set of such constants. Every sentence of the form:
(1) Ax(x=a <> W(x;b',b'',...))
where 'W(x;b',b'',...)' stands for any sentential function which contains 'x' as the only real variable, and in which no extralogical constants other than 'b',b'',...' of the set B occur, will be called a *possible definition* or simply a *definition of the term 'a' by means of the terms of the set B*. We shall say that the term 'a' *is definable by means of the terms of the set B on the basis of the set X of sentences*, if 'a' and all terms of B occur in the sentences of the set X and if at the same time at least one possible definition of the term 'a' by means of the terms of B is derivable from the sentences of x."
The topic of my post is concerned with the syntax of the sentence
(1) Ax(x=a <> W(x;b',b'',...))
in which the sign of equality and the sign of logical equivalence act in coordination with one another.
One can entangle their respective meanings even more profoundly with the oftenquoted position expressed by Quine that "identity is eliminable." That is, when one presupposes the ontological interpretation that gives rise to the necessity of
 (x=y > Az(zex <> zey))
and takes as a contextual axiom,
(Az(zex <> zey) > x=y) 
one has properly distinguished contexts. But, to use those two pieces of syntax to treat set theory with a signature <e> rather than <=,e> yields an axiom such as
AxAy(x=y <> Az(zex <> zey)) 
Observe that in this form, the properties of an equivalence relation derive from the fact that logical equivalence is reflexive and symmetric by truth table semantics, and transitive relative to the tautologous formula from propositional logic,
 ((p<>q) <> ((p>q) /\ (p<q))
where, by tautologous I mean its truth table evaluation. Quine is clear concerning this meaning:
"The term 'tautology' is taken from Wittgenstein. The present notion of tautologous statements, as those true by virtue solely of truthfunctional composition, seems to agree with his usage;[...]"
So, while foundational investigations before Wittgenstein were necessarily axiomatic, the introduction of truth tables changed the situation.
As for the sign of equality, Tarski managed to incorporate the conjunctive propositional syntax into a quantificational context with the axiom:
AxAy(x=y <> Ez(x=z /\ z=y)) 
Here, the existential operator brings the investigation back to model theory, names (or constants (extralogical constants if one takes parameters as being the same as constants)) and definite descriptions.
In this last expression, the sign of identity is expressing its own transitivity through the symmetry of the truth functional semantics of conjunction.
===================
To return to the stated topic of the post, the problem with all of the above is that logical equivalence (hereafter, LEQ) is decomposable where the ontological notion of identity is not.
Even when authors like Quine and Carnap may have noticed that truth table representations were subject to permutable representations, they ignored it.
There are six column vectors which may, in various combinations, represent LEQ.
TTTFFF TFFTTF FFTFTT FTFTFT
I am grateful to everyone on sci.logic and sci.math who may have helped me to sort some of this out.
===================
To find a ground, both Kant and Frege looked to "contradiction." In Kant's case, it took the form of interpreting the modal notion "impossibility." In Frege's case, it took the form of individuating a class having no members. One would think such a class is indecomposable.
In the theory of orthocomplemented lattices, there is a notion of orthogonality defined by:
Two elements a, b of an ortholattice are said to be *orthogonal* if and only if a<=b'. For such elements we write a__b.
One consequence of this definition is that
Ax(x__x > x=0)
so, the situation above involving six column vectors may be treated as a single system satisfying that theorem relative to representation in terms of the unique selforthogonal Latin square on 6 symbols:
134625 625134 463512 512463 246351 351246
To do this, however, one needs a category of named objects that can be thought of in terms of that ontological invariance associated with indecomposability.
That means seeking ways, other than truth tables, to distinguish the 6 elements.
I will not try to explain my method since it is a matter of senselessly comparing syntax.
The results, if they may be called that, are simple.
The "indecomposable ground" is the fact that any 2edge coloring of the complete graph on 6 symbols must contain at least one monochromatic triangle.
That graph has 15 edges. So, by focusing on constructions that isolated LEQ from the other 15 basic Boolean switching functions, my method grounded partitions whose cardinal arithmetic is given by
16=15+1
6=5+1
Another issue  being that this concerned itself with logic  is bivalence and the law of excluded middle with respect to unary negation. But, unary negation is eliminable. What seemed required was to find an assymmetry by which one complete connective was a syntactic substitute for unary negation and the other complete connective was the semantic ground.
This search landed me in the algebra of miniquaternions, although I could not have recognized it until I had built the construction up to where it involved 91point projective planes.
One standard set of complete connectives would be {>,} which I call IMP and NOT. If my investigations can be seen to have established the following relations,
{FIX,LET,LEQ,XOR,DENY,FLIP}{IMP,NAND,NOR}
then those two sets may be given the algebra of a nearfield of order 9. The assignments
IMP=0 NOR=1 NAND=1
describe the real elements of the near field.
If one is willing to entertain the ability to designate nearfield algebras to this set, then choosing any element of the nonreal elements, say LEQ
{FIX,LET,XOR,DENY,FLIP,{LEQ}}{IMP,NAND,NOR}
permits all nine elements to be representable in forms such as
a+(b*LEQ)
where a and b are from the set of real elements.
There is another odd algebra that arises based on the attempt to isolate XOR from the 5set given above.
The objects FIX,LET,DENY,FLIP are distinguished in that they are invariant with respect to DeMorgan conjugation. It should be obvious how they will be assigned in the ortholattice L12 near expression (29) in the link,
http://www.clas.ufl.edu/users/jzeman/quantumlogic/generalized_normal_logic.htm
All that is important is that
c=XOR
Although one might consider interpreting the bottom of the lattice with a 6th symbol, its position in the larger framework means that it should be left null. Consider this an application of Leibniz' law.
But, the algebraic system that goes with these five objects is obtained using the alternating group on 5 symbols. Once again, the assignments are made to convey a sense of relation through purely syntactic construction  therefore, senselessly.
G=LET H=XOR I=FLIP J=FIX K=DENY
The algebra is the icosian group. Provided this link works, you should be able to see the relationship from the assignments above:
http://books.google.com/books?id=upYwZ6cQumoC&pg=PA207&lpg=PA207&dq=icosians+leech++lattice&source=bl&ots=_L1P4YjbCT&sig=xMmrxwlFUT09KI266toGazHZROs&hl=en&sa=X&ei=kwkfUdq9CSQ2AXTlYGIDg&ved=0CEUQ6AEwAg#v=onepage&q=icosians%20leech%20%20lattice&f=false
When looking at the icosians, observe that they are defined with respect to what would be a single field extension to the rationals if the coefficients were the rational numbers.
I point this out because statements such as
"infinity is a necessary assumption for identity"
reflect the fact that some representation of a process of quantization is needed to
======================================
I made a claim suggesting that I was a set theorist (although, I neglected to say nonprofessional) who thought about his work in terms of quantum mechanics. Here is a paper whose introduction mentions these geometries in terms of entangled qubits
http://arxiv.org/abs/1002.4287
As for the impression that I could not possibly "know" all of this math. I cannot and never will know it the way you know your subjects. I am simply a guy who was interested in a hard problem in set theory and followed his nose.
My mathematics, being impredicative is not a "set theory". Impredicativity is fractal.
My mathematics, admitting a null class is not a "mereology". Leibniz law needs a halt.
If anyone is interested, here are the sentences again...
...and, thanks.
===========================================
We take the consequences of the following as the basic theory.
It's signature is given by
<<M, M>, <c, 2>, <e, 2>>
with models interpreted coherently according to
M=V()
in the extended signature
<<M, M>, <c, 2>, <e, 2>, <EQ, 2>, <=, 2>, <V, 0>, <null, 0>, <set, 1>, <S, 1>, <P, 1>>
Definition of proper part: AxAy(xcy <> (Az(ycz > xcz) /\ Ez(xcz /\ ycz)))
Lesniewski's First Axiom is provable: AxAy(xcy > ycx)
Lesniewski's Second Axiom is provable: AxAyAz((xcy /\ ycz) > xcz)
Lesniewski's First Theorem is provable: Ax(xcx)
Definition of membership: AxAy(xey <> (Az(ycz > xez) /\ Ez(xez /\ ycz)))
Definition of grammatical equivalence: AxAy(xEQy <> (Az(xcz <> ycz) /\ Az(zcx <> zcy) /\ Az(xez <> yez) /\ Az(zex <> zey)))
Definition of object identity: AxAy(x=y <> Az(xez <> yez))
Assumption that equivalence of mereological filters imply equivalence of neighborhood filters: AxAy(Az(xcz <> ycz) > Az(xez <> yez))
Assumption that objectual inclusion implies mereological covering: AxAy(Az(zex > zey) > Az(ycz > xcz))
Assumption that proper parts are collectible: AxEyAz(zey <> zcx)
That proper parts imply objectual inclusion is provable: AxAy(xcy > Az(zex > zey))
That mereological covering implies objectual inclusion is provable: AxAy(Az(ycz > xcz) > Az(zex > zey))
That proper parts are expressible in terms of object extension is provable: AxAy(xcy <> (Az(zex > zey) /\ Ez(zey /\ zex)))
That object inclusion implies mereological inclusion is provable: AxAy(Az(zex > zey) > Az(zcx > zcy))
That equivalent neighborhood filters imply equivalent mereological filters is provable: AxAy(Az(xez > yez) > Az(xcz > ycz))
That grammatical equivalence is expressible in terms of mereological filters is provable: AxAy(xEQy <> Az(xcz <> ycz)
That grammatical equivalence is expressible in terms of object extension is provable: AxAy(xEQy <> Az(zex <> zey))
That grammatical equivalence is expressible in terms of neighborhood filters is provable: AxAy(xEQy <> Az(xez <> yez))
That grammatical equivalence is equivalent to object identity is provable: AxAy(xEQy <> x=y)
Assumption of Aquinian individuation: AxAy((Az(ycz > xez) /\ Ez(xez /\ ycz)) > Az((xez /\ ycz) > (Ew(xew /\ wcy) \/ Aw(zcw > ycw))))
Assumption of singletons via pairing: AxAy((Ez(xcz) /\ Ez(ycz)) > EwAz(zew > (z=x \/ z=y)))
Definition of top: Ax(x=V() <> Ay((ycx <> y=x)))
Assumption of top: ExAy((ycx <> y=x))
Assumption of almost universality: Ax(Ey(xcy) > Ey(xey))
Definition of set: Ax(set(x) <> Ey(xcy))
Definition of bottom: Ax(x=null() <> Ay((xcy <> x=y)))
Assumption of bottom: ExAy((xcy <> x=y))
Assumption of context separation (regularity/foundation): Ax(Ey(ycx) > Ey(yex /\ Ez(zex /\ zey)))
Assumption of arbitrary unions: AxEy(Az(zey <> Ew(wex /\ zew)) /\ (Ez(xcz) > Ez(ycz)))
Assumption of arbitrary intersection: AxEy(Az(zey <> Aw(wex > zew)) /\ (Ez(zcx) > Ez(ycz)))
Definition of power function: AxAy(x=P(y) <> (Ez(ycz) /\ Az(zex <> (zcy \/ z=y))))
Assumption of power set: Ax(Ey(xcy) > Ey(Az(zey <> (zcx \/ z=x)) /\ Ez(ycz)))
Definition of successor function: AxAy(x=S(y) <> (Ez(ycz) /\ Az(zex <> (zey \/ z=y))))
Assumption of successor set: Ax(Ey(xcy) > Ey(Az(zey <> (zex \/ z=x)) /\ Ez(ycz)))
Assumption of transfinite cardinalities: Ex(Ey(xcy) /\ null()cx /\ Ay(ycx > Ez(zcx /\ ycz)))
Let the restricted quantifier
Ap[pEQp]
be interpreted as
Ap[pEQp](phi(p)) <> Ap(pEQp /\ phi(p))
Then for each n and each wellformed formula phi(y, p_0, ..., p_n),
assume
================= Ap_n[p_nEQp_n]...Ap_0[p_0EQp_0] AxAy( Ew(ycw) > (Ez((Ew(zcw) /\ (yez <> (yex /\ phi(y, p_0, ..., p_n))))) <> Ew(xcw)) ) =================
and assume
================= Ap_n[p_nEQp_n]...Ap_0[p_0EQp_0] ( AxAyAz( ( ((Ew(xcw) /\ Ew(ycw)) /\ (phi(x,y, p_0, ..., p_n)) /\ ((Ew(xcw) /\ Ew(zcw)) /\ (phi(x,z, p_0, ..., p_n)) ) > (y=z) ) > AxAy( Ew(ycw) > (Ez((Ew(zcw) /\ (yez <> Ew(wex /\ phi(z,w, p_0, ..., p_n))))) <> Ew(xcw)) ) ) =================



