Presto! Digitization: Part I From NKS Number Theory to "XORbitant" Semantics, by way of Cayley-Dickson Process and Zero-Divisor-based "Representations" Robert de Marrais
[email protected] 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), emerg- ing from studies of synchronization since the Millennium – 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 2N-ions derived from Imaginaries by Cayley-Dickson Process (CDP) – can model such “enriched” nodules. Its (bit-string XOR-ing vs. matrix-multiplication- based) operations unfold into "representations" of the objects traditionally linked with “partitions of Nullity”: Singularities. The route from this “local” level of CT to fractals and Chaos, via CDP extensions to 2N-ions for growing N, will involve us in graphics of higher-dimensional “carry-bit overflow,” as manifest in the mandala-like patterns show- ing up in “emanation tables” (the rough equivalent, for ZD's, of group theorists’ Cayley Tables). More concretely, this route will model semantic propagation – “meaning” itself. I'll lead into this with a quote about "Hjelmslev's Net" (which I'll claim is the CDP man- qué) from a famous postmodern text deeply engaged with the philosophy of future mathe- matics, Deleuze and Guattari’s A Thousand Plateaus (where “rhizosphere” imagery arose).