Date: Jan 30, 2013 8:32 AM Author: fom Subject: Re: Matheology § 203 On 1/30/2013 5:29 AM, WM wrote

> On 30 Jan., 12:02, fom <fomJ...@nyms.net> wrote:

>

>> As for those "logical considerations," I mean that

>> one can develop a hierarchy of definitions that

>> depend on actual infinity. To say that mathematics

>> is "logical" is to concede to such a framework. I

>> do not believe that mathematics is logical at all.

>

> That is a very surprising statement. Why do you think so?

My other response is very disjointed, although the

essential pieces are there.

It is difficult for me to explain.

As an undergraduate in 1996, I felt I had a solution

to GCH. I wrote axioms to eliminate the axiom of

extension from ZF. One of the axioms failed because

of a simple oversight. But, I have resolved that

problem.

Because of the difficulties in even getting a

discussion of the mathematics of those axioms, I

was driven to investigate matters much more closely.

I now know that I had stumbled on a Lesniewskian

approach to foundations. I also know that Leibnizian

and Aristotelian logic is oppositely directed from the

extensional Scholastic logic.

Neither the intensional nor the extensional approaches

yields an appropriate set-theoretic foundation. One requires an

approach which is easily seen to be the duality of projective

geometry in hindsight. That duality captures both approaches

simultaneously, but requires additional axiomatic assertions

(such as pairing) to implement the principle of identity

of indiscernibles.

Driven further, however, the question of identity

leads to a much deeper understanding of the influence

of projective geometry in the foundational investigations

of the late nineteenth century. The very truth table

for logical equivalence is little more than a specific

labeling of a trivial affine plane on six lines.

Building on this, I discovered that one may treat the

complete system of 16 truth functions as the affine

points of a 21 point projective plane. Negation, DeMorgan

conjugation and contraposition are merely geometric

projectivities with negation taking the line at infinity

as its axis. Thus, negation is "invisible" to the

system of sixteen truth functions and becomes a "unary

connective".

But, what is more interesting is that the 16 points

of the affine plane form 20 lines.

Using a difference set labelling for the projective plane,

I was able to use the same labels for both lines and

points (as typically found elementary texts on projective

planes). I soon realized that the structure of extensions

for propositional logic to quantificational logic or

modal logics was isomorphic with the line labels.

One of the truth functions--namely, constant falsity--is

exchanged for a symbol not among the 16 point symbols.

Constant falsity names the line at infinity and the

names for the other 20 lines fit into an ortholattice

having a 16 element Boolean sublattice.

The one odd symbol is the bottom of the lattice and

"grounds" the structure in the same sense that Frege or

Kant used contradiction to ground their systems (non-existence

and impossibility, respectively). The remaining 15

truth function symbols are positioned exactly as they

should be for their usual configuration. And, the

ortholattice is an amalgam so that the remaining 4 points

relate to one another as a "square of opposition".

Moreover, that amalgam distinguishes precisely

one atom of the 16 element lattice. It is natural

to recognize that as the NOR locus, thus relating

the entire structure to the analysis of logical

negation relative to the complete connectives.

By the nature of amalgams, one can take each quantificational

variable as corresponding with a four-fold structure. This

kind of thinking is evident in Tarski's cylindrical

algebras where the quantifiers are indexed coherently

with variables (so Ex is not E and x, but E_x(x), E_y(y),

and so on.

It is all geometry and topology. Every last bit

of it. And this is perfectly consistent with

Carnap's description of logical syntax as being

little more than geometric form.

During the last week, I have figured out how

to "lift" the classical structure into the

free orthomodular logic on 2 generators. This

required some very specific group manipulations

and a {96,20,4}-difference set construction.

I have an "alphabet" for quantum logic.

So, why should I believe a word of logicism? This

geometry is outside of logic or circular with it.

In one case, logicism is restrictive and in the

other it is just hypocritical.