```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 theessential pieces are there.It is difficult for me to explain.As an undergraduate in 1996, I felt I had a solutionto GCH.  I wrote axioms to eliminate the axiom ofextension from ZF.  One of the axioms failed becauseof a simple oversight.  But, I have resolved thatproblem.Because of the difficulties in even getting adiscussion of the mathematics of those axioms, Iwas driven to investigate matters much more closely.I now know that I had stumbled on a Lesniewskianapproach to foundations.  I also know that Leibnizianand Aristotelian logic is oppositely directed from theextensional Scholastic logic.Neither the intensional nor the extensional approachesyields an appropriate set-theoretic foundation.  One requires an approach which is easily seen to be the duality of projectivegeometry in hindsight.  That duality captures both approachessimultaneously, but requires additional axiomatic assertions(such as pairing) to implement the principle of identityof indiscernibles.Driven further, however, the question of identityleads to a much deeper understanding of the influenceof projective geometry in the foundational investigationsof the late nineteenth century.  The very truth tablefor logical equivalence is little more than a specificlabeling of a trivial affine plane on six lines.Building on this, I discovered that one may treat thecomplete system of 16 truth functions as the affinepoints of a 21 point projective plane.  Negation, DeMorganconjugation and contraposition are merely geometricprojectivities with negation taking the line at infinityas its axis.  Thus, negation is "invisible" to thesystem of sixteen truth functions and becomes a "unaryconnective".But, what is more interesting is that the 16 pointsof 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 andpoints (as typically found elementary texts on projectiveplanes).  I soon realized that the structure of extensionsfor propositional logic to quantificational logic ormodal logics was isomorphic with the line labels.One of the truth functions--namely, constant falsity--isexchanged for a symbol not among the 16 point symbols.Constant falsity names the line at infinity and thenames for the other 20 lines fit into an ortholatticehaving 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 orKant used contradiction to ground their systems (non-existenceand impossibility, respectively).  The remaining 15truth function symbols are positioned exactly as theyshould be for their usual configuration.  And, theortholattice is an amalgam so that the remaining 4 pointsrelate to one another as a "square of opposition".Moreover, that amalgam distinguishes preciselyone atom of the 16 element lattice.  It is naturalto recognize that as the NOR locus, thus relatingthe entire structure to the analysis of logicalnegation relative to the complete connectives.By the nature of amalgams, one can take each quantificationalvariable as corresponding with a four-fold structure.  Thiskind of thinking is evident in Tarski's cylindricalalgebras where the quantifiers are indexed coherentlywith 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 bitof it.  And this is perfectly consistent withCarnap's description of logical syntax as beinglittle more than geometric form.During the last week, I have figured out howto "lift" the classical structure into thefree orthomodular logic on 2 generators.  Thisrequired some very specific group manipulationsand a {96,20,4}-difference set construction.I have an "alphabet" for quantum logic.So, why should I believe a word of logicism?  Thisgeometry is outside of logic or circular with it.In one case, logicism is restrictive and in theother it is just hypocritical.
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