Mathematics – Rings and Algebras
Scientific paper
2006-03-13
Mathematics
Rings and Algebras
Expanded version of talk presented Oct. 29, 2005, at Midwest NKS Conference, "Systems Based on Numbers," U. of Indiana at Bloo
Scientific paper
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.)
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