fom
Posts:
1,037
Registered:
12/4/12
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Re: Matheology § 203
Posted:
Jan 30, 2013 8:32 AM
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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.
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