Axiomatic world theory: An overview the general theory of evolution in brief

The world, its many subsystems and all their theories, starting with logic, can be reduced to two related functions: a combinatorial system generator and a hamiltonian system organizer. These can be derived, in turn, from an Axiom of Lawfulness, the expansion being guided by pseudo‐category and pseudo‐functor analysis to produce an axiomatic theory of the world or general theory of evolution. Specifically, world evolution is generated by a constrained combinatorial world generator, F:G_(X), deduced from two related axioms: I. The Axiom of World Lawfulness and II. The Axiom of World Constraint Constants, c = c₁, c₂, of primordial physical combinatee (substance), c₁, and physical combinator (motion), c₂.

Axiom I postulates a lawful analysis by an analyzer adhering to appropriate coordinate systems, CS, of a lawful analysand obeying a conservation law, X = X. The analysand consists of a base combinatee (the set and elements), X = {x₁, x₂,… x_n}, and a base combinator, namely, the universal Boolean operator, NOR = NOT + OR. Base combinatee and combinator both have attributes of quantity combinatorially generated by NOR operating on the universal number, 1, and of quality generated by NOR operating on the universal dimensions, MLT (mass, length, time), including the null sets.

Axiom II fixes the base constants, c, = c₁, c₂, thereby converting X to material substance using c₁ and NOR to material motion using c₂. This comprehensive, quality and quantity‐competent foundational science is called Universal Combinatorics. Its elements comprise the logical alphabet or metavector, A = {c, 1, MLT; X, NOR}, where c is obtained from the remaining terms. These give: (1) the attributive pseudo‐functor, F = P_(c,1,MLT), where P is the power set of the indicated attributes, and (2) the logic generator, G_(X), where G = NOR_(NOR). F then maps G_(X) into world evolution, F:G_(X) → world evolution, as follows:

Expanding the abstract generator, F₁:G,_(x), with world constants eliminated, i.e., c = 0, generates Universal Grammar consisting of (1) the substantive content of the abstract science chain running from linguistic grammar to mathematics and logic and (2) a comprehensive epistemology equivalent to an explicit theory of the strategic aspects of the scientific method, including a universal hamiltonian theory structure informally related to a mathematical category. The four epistemological theorems are:

1. I. The Combinatorial System Generator, F:G_(X), (read as “The attributive functor, F, maps the logic generator, G_(X), into world theory” or “The world is an attributive combinatorial function of logic").

2. II. The Hamiltonian System Operating Theorem, h (an abstract theory‐category structure).

3. III. The System Stability Theorem, PI?, where PI is the extremal Performance Index or controlling law.

4. IV. The Intersystem Abstraction Ranking Theorem given by the Attributive Functor/ Function, F.

F₂ admits the world constants, c > 0, to materialize the grammar generator, G_(X), to an homologous concrete Euler combinatorial physical wave generator, namely, the superstring equation of quantum theory, E_(NI) = A(σ,τ), where E is the permutational function, NI, is the set of nonintegers and the solution is the dual amplitude, σ,τ. Expanding generates the elementary particles of nonadaptive physics and, by inference, the substantive content of Universal Physics consisting of three additional primary systems comprising the world, where a primary system is defined as one having a distinct but derivative extremal controlling law:

1. I. Nonadaptive physics and chemistry (harmonic hamiltonian wave systems) : Minimize Action, subject to conservation constraints.

2. II. Adaptive physics or biology (membrane bound duplicating polymer‐copolymer hamiltonian systems) : Maximize Survival, subject to energy constraints.

3. III. Sentient physics or sociopsychology (neuromatrix hamiltonian systems) : Maximize subjective Happiness, subject to survival constraints and

4. IV. Representational physics or language (a symbolic combinatorial routine): Maximizes the Information Gain, subject to happiness constraints.

The world can then be viewed as a perpetual superfluid computer implicitly using the epistemology of Axiom I as a world program to process the physical data base, c > 0, of Axiom II into world evolution. After evolving through Systems I and II, mankind, i.e., System III, evolves as an internal metacomputer which makes the combinatorial program explicit and uses it to put all four primary systems in standard hamiltonian theory (pseudo‐category) format and terminology. This can be viewed as a generalization of the Darwinian variation‐and‐selection theme in which combinatorial‐variation is recursively hamiltonian‐selected thereby incrementing world logic and logic constraints on successive primary systems. Because Universal Physics and Universal Grammar are functor‐related homologous concrete and abstract combinatorial pseudocategories, related by a pseudo‐functor, thus, differing only in the presence and absence, respectively, of the World Constants, c ≥ 0, they constitute, ipso facto, Universal Science (Formal Philosophy, World Evolution, World Unification, Explicit Theory of Everything, ETOE, or Axiomatic World Theory).

QED: Because intricate verified predictions, ranging from particles to personality types, mental disorders, political parties and the abstract sciences, result from a system which is merely expanding to fill its possibility set, it is concluded that the world is lawful and that this means it is an object deterministic but not fully analytically determinable combinatorial system. In the object domain, the world is system‐number complete at four. Dually, in the analytical codomain, understanding of it is approximately complete, as measured by a world information gain function. Hence, the dualistic, analysand‐analyzer world program is finite and has dualistic completion criteria, as required of an involuted program.