Nonlinearity and Self-Reflexion
Mathematical Models of Self-Developing Process

by Sinek Docchi of Studies in Dialectics
August 1978


"Composed during the late 1970s, the booklet-length essay, "Nonlinearity and Self-Reflexion: Mathematical Models of Self-Developing Process", is an experiment in presenting a dialectical, immanent critique of modern mathematics, leading to a mathematics of dialectics, without employing any of the traditional dialectical or Marxian vocabulary, in the hopes of communicating past the blockages which that terminology triggers in those indoctrinated by corporate-capitalist mass media "popular" ideology, and by orthodox academic ideology, as well as by orthodox Lenino-Stalinoid and Lenino-Trotskyoid ideology — all of which are virulently anti-Marxian.

This essay has the primary merit of describing the connexion between nonlinear differential equations — unsolvable in general for reductionist mathematics — with the '''self-reflexivity''' of 'auto-dynamical' processes, and of the paradoxes of formal logic and set theory.

This essay motivates its search into alternative and higher concepts of "Number" via the very "unsolvability" of the '''self-reflexive''' nonlinear total and partial integro-differential equations of mathematical physics, biology, ecology, and economics, within the present, reductionistic-atomistic forms of these sciences, and in the revelations of the recent emergence of some analytical, closed-form, "exact" solutions for "weakly-nonlinear" partial differential equations — the "particle"-like "solitary wave", or "soliton" solutions.

This essay also saw, via the Musean hypernumbers — as a working hypothesis — a potential for a new kind of "analytical", "closed-from" solutions for nonlinear differential equations, called "hypernumber-valued self-reflexive functions" — solutions for, at least, those total, or "ordinary", integro-differential equations whose state-space "solution-geometries" are of the "limit-cycle", or asymptotically periodic, type.

It located this potential in the possibility that the "power-orbits" of the Musean hypernumbers might form the basis for a new, previously-unrecognized class of '''self-reflexive functions''' that would exactly solve such "limit-cycle" dynamical systems, precisely reconstructing their past and predicting their future state-space trajectories.

Our research — our Studies in Dialectics — have, by now, moved far beyond these beginnings, in this working hypothesis, and far beyond the limitations of the Musean species of hypernumbers.

Nevertheless, these beginnings may contain useful lessons for those pursuing related quests today."

Sinek Docchi

Table of Contents

  1. Pages i – v, 0
    Title Page
    Contents Outline
    Preface; Abstract
    The Nonlinearity Barrier
    Figure 0. Trajectory As Biography

  2. Pages 1 – 51
    INTRODUCTION — The Problem of Nonlinearity
    The Meaning of Non-Linearity
    The Historical Pattern of Mathematical Progress

  3. Pages 52 – 110
    LIMIT-CYCLES — State Space
    The State-Space Geometry of Linear and Non-Linear Solutions
    The Meaning of Limit-Cycles

  4. Pages II-0 – II-16
    APPENDIX II — CYCLONES AND SOLITONS (Hypernumbers and Partial Differential Equations)