Presto! Digitization, Part I: From NKS Number Theory to "XORbitant" Semantics, by way of Cayley-Dickson Process and Zero-Divisor-based "Representations"

Abstract

The objects of the great Nonlinear Revolutions - Catastrophes and Chaos in the 1960s-70s (henceforth, CT); and, small-world and scale-free Network Theory (NT), emerging quite recently - will be spliced together by a New Kind of Number Theory, focused on digitizations (i.e., binary strings). NT nodes then become feature-rich representations (nodules in a "rhizosphere") of CT contents. The "Box-Kite" formalism of zero-divisors (ZD's) - first showing in the 16-D Sedenions, then in all higher 2^N-ions derived from Imaginaries by Cayley-Dickson Process (CDP) - can model such "enriched" nodules. Its (bit-string XOR-ing vs. matrix-multiplying) operations unfold into "representations" of the objects linked in CT with "partitions of Nullity": Singularities. The route from here to fractals and Chaos, via CDP extensions to 2^N-ions for growing N, will involve us in graphics of higher-dimensional "carry-bit overflow," as manifest in the mandala-like patterns of "emanation tables" (the rough equivalent, for ZD's, of group theorists' Cayley Tables). I'll lead into this with a quote about "Hjelmslev's Net" (which I'll claim is the CDP manque') from a famous postmodern text, Deleuze and Guattari's "A Thousand Plateaus" (where "rhizosphere" imagery arose). With strong assists from the CT-based structuralism of Jean Petitot, Algirdas Greimas's "Semiotic Square" will show us how to explicitly link CT to semiotic foundations via ZD "representations," while the infinite-dimensional ZD meta-fractal or "Sky" where Box-Kites fly - first appearing in the 32-D Pathions and incorporating the higher 2^N-ions - will provide sufficient lebensraum for Levi-Strauss's "Canonical Formula" of mythwork to unfurl in. (These results serve to extend my NKS 2004 paper, available at the Wolfram Science website.)