Forgive my laziness. I attempted to direct to a link with extraneous information rather than simply restate myself.
The only necessary background is that I am highly motivated by understanding the nature of identity and equivalence classes in mathematical foundations. That has led to discerning certain structures for which distributive laws do not necessarily hold. Once recognized, Lawvere's discussion of the category of pointed sets as an example of a category in which the distributive laws fail helped me to crystallize some of my views.
In that context, one sometimes sees a comparison between Fregean number classes and von Neumann ordinals as a definition of number class from which the von Neumann ordinal is a representative. Although this had never been my motivation, it reflects the sense of 'number-as-object' as 'number-as-pointed-class'.
What follows is the relevant excerpt from that link
At the time, the most I could mumble about had been the idea of an "equivalence attachment". My description at that time had simply been that every equivalence class must have distinguished element. So, if
was an equivalence class, it would have an internal organization corresponding with
or, more accurately,
a*=*b a*=*c a*=*d : :
where '*=*' is intended to reflect the relation with the distinguished element within the class.
Before commenting on this further, I want to observe that an underlying "claim" in metamathematical research is that standard mathematical practice has been "represented". It is not that metamathematics does not stand on its own as an interesting and relevant discipline. Rather, its subject matter is often interpreted relative to beliefs concerning mathematics that are not mathematical. So, what is or is not being "represented" will come into play here.
As obnoxious as my participation on these newsgroups may have been, it is because of that participation that I have been able to work out that the idea concerning equivalence classes mentioned above involves mathematics in which the distributive law does not necessarily hold or,at least, is not two-sided.
My first indication of this had been when I found the paper,
In this paper, Pavicic and Megill determine that the canonical algebraic model for the classical logic of Principia Mathematica is a model in which the usual distributive laws are not necessary. They did so by analyzing equivalences in algebraic models. In my mind, it supported the idea that there were, in fact, foundational issues associated with identity, although it would be occurring at an even lower level than I had originally envisioned. As, I had already been considering the eliminability of negation using the complete connectives, it redirected some of my efforts to considering logical equivalence (the biconditional) more closely.
Another indication came recently when my geometric constructions involving finite projective geometries led me to consider 91-point projective planes. There are four of these planes. One is Desarguesian and three are not Desarguesian. These planes are generated from near-fields on nine elements. Although one of the planes is generated from a Galois field, the others correspond with an algebraic system called miniquaternions.
The paper by Pavicic and Megill is motivated by investigations into quantum logic. My reasons for investigating 91-point planes is precisely because of a construction intended to relate the free Boolean lattice on 2 generators to the free orthmodular lattice on 2 generators in order to understand how the "benzene ring" (the ortholattice O_6) is central to both models. This construction involves an additional structures closely associated with studies of quantum physics. What I believe accounts for this is another instance where distributive laws are not necessarily two-sided.
This instance comes from automata theory. One of the more recent issues is mobile communication. In his book describing the pi calculus,
Robin Milner shows how classical language equivalence with regard to regular sets (where regular sets are built up by concatenation -- the system obeys semigroup associativity and has two-sided distributivity) fails to distinguish between deterministic and non-deterministic automata. He uses this to motivate his definitions of simulation and bisimulation.
In that illustration, it is precisely because of an application of one of the distributive laws that the two different automata become identified.
So, while my investigations have led toward mathematics which is of interest to those who investigate quantum logics, it seems that this arises from a relationship between non-deterministic automata and the distributive laws.
Returning to the discussion of equivalence relations with a distinguished element, such equivalence classes would be pointed sets.
Session 28 of "Conceptual Mathematics" by Lawvere and Schanuel is included precisely for the purpose of giving an example of a non-distributive category. That category is the category of pointed sets.
Wikipedia does not have much to say, but in the article on pointed spaces, you will see that quotients of pointed spaces map distinguished elements to distinguished elements as would any maps in the category of pointed sets
Thus, at least I know the relationship of my original idea to various pieces of standard mathematics. I did not know that before.
But why should equivalence classes be thought of as pointed sets?
Since I do not feel bound to logicism or the metamathematics that is used to enforce that philosophy on others, I would immediately direct attention to the non-Boolean partition lattices and topological quotient maps as the general intuition grounding this claim.
However, that would not be acceptable.
The "acceptable" answer is that metamathematics has chosen to ignore a standard practice in its "representations".
When we teach the arithmetic of fractions, we speak of one particular fraction being in lowest terms.
When we speak of the operation inverse to the squaring of real numbers, we speak of a principal square root.
When we speak of periodic functions, we speak of principal branches.
When we speak of well-formed formulas, we speak of normal forms.
When we speak of equations for a line in analytical geometry, we speak of a general equation of a line.
In other words, it is standard mathematical practice to deal with a multiplicity by trying to find a criterion for selecting a distinguished element in order to have a system with single-valued reference. It is true that these are just stipulations. But,these stipulations are done with specific intent as a matter of dealing with multply-valued collections that need to be given a single-valued reference.
Constructions like Dedekind cuts make that practice a difficult thing to visualize. But, like the axiom of choice, one does not need to visualize it if one recognizes that it is a matter of practice that should be "represented" (For those of us who respect model theory, the axiom of choice is necessary in order that the sign of equality can be represented relationally).
Perhaps it is not necessary that every equivalence class be such. However, in the case of logic and quotient models, one forms equivalence classes of terms in order to form term models. It is certainly not unreasonable to suggest that the application of logical systems depends on the presupposition of a system of canonical names that would then characterize those equivalence classes as pointed sets.
Does this make me right? Right and wrong have no place here. It is just a different viewpoint that leads to a different opinion concerning the foundation of mathematics.
As for "private language" or "this is not mathematics", there are practical difficulties as to what is good mathematics when one is investigating a problem that is independent of known axiom systems. One cannot be a formalist, and, one cannot simply comply with 'received knowledge'. For that, however, I am sorry.
In any case, it has been difficult. I was a good student. I had good prospects. This took that all away. And, in fact, much more than that.