Mathematical Foundations of Consciousness

TL;DR

This paper develops a mathematical framework based on non-well-founded set theory and axioms like the Anti-foundation Axiom to formalize the foundations of consciousness and experience, linking neural networks to set-theoretic models.

Contribution

It introduces a new axiomatic system with consciousness operators, including the Russell operator, to mathematically model consciousness and its neural correlates.

Findings

Neural networks can be modeled as non-well-founded graphs generating sets with Platonic aspects.

Consciousness operators are characterized within a set-theoretic framework.

The supervenience of consciousness on neural correlates is formalized using these operators.

Abstract

We employ the Zermelo-Fraenkel Axioms that characterize sets as mathematical primitives. The Anti-foundation Axiom plays a significant role in our development, since among other of its features, its replacement for the Axiom of Foundation in the Zermelo-Fraenkel Axioms motivates Platonic interpretations. These interpretations also depend on such allied notions for sets as pictures, graphs, decorations, labelings and various mappings that we use. A syntax and semantics of operators acting on sets is developed. Such features enable construction of a theory of non-well-founded sets that we use to frame mathematical foundations of consciousness. To do this we introduce a supplementary axiomatic system that characterizes experience and consciousness as primitives. The new axioms proceed through characterization of so- called consciousness operators. The Russell operator plays a central role and is shown to be one example of a consciousness operator. Neural networks supply striking examples of non-well-founded graphs the decorations of which generate associated sets, each with a Platonic aspect. Employing our foundations, we show how the supervening of consciousness on its neural correlates in the brain enables the framing of a theory of consciousness by applying appropriate consciousness operators to the generated sets in question.