Speaking of mathematical proofs and techniques, I continue to be puzzled by the number of rejections of Joy Christian's framework by those who should know better.
The idea of a continuous function in analysis is neither new nor esoteric. What Christian has essentially done, mathematically, is to apply the function to a measurement domain suited to vector spaces, tractable to analytical continuation (geometric algebra). One finds, as previously discussed, that the conditions for continuity (the limit of a continuous function is a continuous function), are satisfied in Joy's framework. For f(x) continuous at x_0:
1. f g is continuous at x_0.
2. f - g is continuous at x_0.
3. fg is continuous at x_0.
4. f/g is continuous at x_0 if g(x_0) /= 0.
Then, in two variables, z = x^2 y^2/x^2 - y^2, discontinuous at the origin (initial condition), with limit 0 along the line x = y, length 1 along the x axis, and length - 1 along the y axis.
It should be quite obvious that the choice of topology that allows the complex part of the algebra its real expression (positive real measure values) has to satisfy:
1. Dimension > 3 (the real line compactified in a single point at infinity)
2. Nonzero torsion (exchange of signs for the two variables, in the limit)
Joy's framework satisfies all these conditions exactly, and the proof -- being constructive -- is much stronger than the nonconstructive proofs that support Bell's theorem arithmetic functions.
Tom
