Dialectics and Modern Science
by Sinek Docchi of Studies in Dialectics

September 1976


"Composed during the late 1970s, the booklet-length essay, "Dialectics and Modern Science", still evinces an easy optimism, carried forward from the proto-revolutionary global ferment of the 1960s.

It envisions a coming global "Socialist Renaissance".

Though conceived in a thoroughly contra-Stalinist, contra-Leninist sense — rejecting state-capitalist pseudo-socialism — this vision still had to pass through the historical "Valley of the Shadow of Death" of the apparent eclipse of Marxian Socialism in the onslaught of resurgent "conservatism", and in the collapse of the [pseudo-]"Soviet" Union, and its Eastern European, pseudo-socialist colonies, that ravaged the 1980s and 1990s.

Those cataclysmic and agonizing events had to transpire before anything like a true "Socialist Renaissance" could return to history's agenda.

This unfinished essay does have the merit of locating immanent movements, within mathematics and within the other sciences of the twentieth century, toward a mathematics and general science adequate to Marxian, dialectical reason.

This essay also has the merit of locating the connexion between nonlinear differential equations — unsolvable in general for reductionist mathematics — and the '''self-reflexivity''' of '''auto-dynamical dialectical-ness''', and of locating other groundings of an immanent critique of reductionist mathematical logic, mathematics, and science in general. Moreover, all of this is seen as portending the emergence of an explicit, ideographic, algorithmic logic, mathematics, and science of Marxian dialectics itself.

This essay locates these groundings principally:

  1. in the "reflexive" paradoxes of formal logic;

  2. in the set-theoretical paradox of the '''self-reflexive''', self-developing "Set Of All Sets";

  3. in the very "unsolvability" of the '''self-reflexive''' nonlinear total and partial integro-differential equations of mathematical physics, biology, ecology, and economics, within the present, reductionist-atomistic forms of these [ideology-permeated] sciences;

  4. in the still unresolved mathematics of "retarded potentials", "self-forces", and "re-normalization" in sub-atomic physics.

This essay also saw, in the Musean hypernumbers, a potential for a new kind of "analytical", "closed-from" solutions of nonlinear differential equations — solutions, at least, for those integro-differential equations whose "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 and predicting their state-space trajectories.

Our research — our Studies in Dialectics — have, by today, moved far beyond these beginnings, but our beginnings they remain."

Sinek Docchi

Table of Contents (PDF Release)

  • Pages i – ii
    Dialectics and Modern Science Title Page
    Dialectics and Modern Science Table of Contents

  • Pages 1 – 47
    0. Prologue: Recent Developments at the Forefronts of Science
    (i) Marx's Reflexive Functions
    (ii) Reflexiveness and Dialecticalness
    (iii) On the Special Notation Embedded in Marx's Writings
    (iv) The Method of Immanent Critique

  • Pages 48 – 90
    (i) The Internal Contradictions of Set Logic
    (ii) Leibniz's Dream

  • Pages 91 – 104
    (iii) George Boole's Calculus of Conception: Logic as the Arithmetic of Unity
    The Mathematics of Logic; An Algebra of Thought?
    The Existential Calculus: Algebraic Aspects
    Arithmetic Aspects

  • Pages 105 – 116
    The Calculus in Action
    The "Fundamental Law of Thought"
    Formal Logic as Linear Logic
    The "Proper Fractions" of Logic
    Formal Logic as a "Statics"
    Some "Uninterpretable" Propositions
    Boolean Dialectics?

  • Pages 117 – 128
    Laws of Still Higher Degree?
    Uninterpretables and Paradoxicals
    Paradox, Impredicativity, Nonlinearity
    Qualities of Unity & of Universe
    Logical Evolution and Evolutionary Logic
    The Veritical Calculus
    'Booleanization' of the Paradoxes
    Heterodoxy Of Our Solution

  • Pages 129 – 140
    Interpretation of the Solution
    The Method of 'Circuit-ization'
    Reflexion and Refluxion; Logical Oscillators
    Justification («Cherchez la suject»)
    The Other Root

  • Pages 141 – 143
    Filling-In Logical Space

  • Pages 144 – 184
    (iv) "Imaginary Roots": G. Spencer Brown's Solution of the Paradoxes in his Laws Of Form A 'Nootkaesque' Notation
    The Computation of Distinctions
    Self-Referentials and "Imaginary" Values
    Insolubilia: Infinite Descriptions?
    Infinite Nonlinearity of Varela's "reverse-G"
    Transcendence and Infinite Degree; The Logic of the Distinction Operator, Denoted "d"
    "Functions of Themselves"
    Function versus Function Values
    Three Modalities of Reflexive Function
    Temporalized Reflexiveness; d-Waves
    Principles of Operatorial Ideography

  • Summary of Findings 30 summary theses on Dialectics and Modern Science [14 pages]

Table of Contents (fragmentary HTML Release)

  • 0.  Introduction (Recent Developments at the Forefronts of Science)
    1.  Dialectics and Marxian Theory