When I can find the time to put it here, you will find my following, already completed, research into the foundations of mathematics:
The Axiomatic Structure of Boolean Algebras
Using the notation presented by G Spencer-Brown in Laws of Form,
a complete boolean algebra is demonstrated from a single initial-equation.
It is demonstrated that this can be achieved using any one of exactly 5 distinct
nontrivial single initial-equations.
The five single initial-equations and their consequences are stated.
Using the same notation a complete boolean algebra can factored into a maximum
of 6 independent operations.
The 6 operations, and equations representing them, are presented with an analysis
of the 64 algebras obtained by asserting an incomplete set.
Transliterations of these results into traditional notations are provided.
The Axiomatic Structure of the Propositional Calculus
Using the same notation, a complete propositional calculus is demonstrated
from a single initial-proposition.
It is demonstrated that this can be achieved using either of exactly 2 distinct
nontrivial single initial-propositions.
The two single initial-propositions and their consequences are stated.
Using the same notation a complete propositional calculus can factored into
a maximum of 12 independent operations.
The 12 operations, and propositions representing them, are presented.
Quantum logic is shown to be a calculus with one of the 12 operations omitted.
Transliterations of these results into traditional notations are provided.
A brief commentary on how this work relates to Wittgenstein's Tractatus Logico-Philosophicus
is also provided.
Re-entrant Boolean Structures, Information Waves and the
Causal Interpretation of Quantum Mechanics
A discussion of how recursive boolean structures relate to the causal, ontological,
pilot-wave interpretation of quantum mechanics originally proposed by Louis
de Broglie and more recently developed by David Bohm, Basil Hiley and Jean-Pierre
Vigier.
You can email me at graham@ellsbury.com
Copyright © Graham Ellsbury 2001