Mathematical Foundations of Consciousness
Willard L. Miranker, Gregg J. Zuckerman
arXiv Preprint Archive October 23, 2008 via arXiv
Summary
AI-generated from the abstractThe Zermelo-Fraenkel axioms, especially the Anti-foundation Axiom instead of the standard Axiom of Foundation, allow sets that can contain themselves, supporting Platonic interpretations. Using graphs, decorations, and labelings, a syntax and semantics of operators acting on these non-well-founded sets is developed. This framework is extended with new axioms that treat experience and consciousness as primitives, introducing consciousness operators, with the Russell operator as a central example. Neural networks provide non-well-founded graphs whose decorations generate sets with Platonic aspects. Applying consciousness operators to these sets shows how consciousness can supervene on its neural correlates, framing a theory of consciousness.
Study at a glance
| Characteristics | Theoretical or philosophical paper Peer reviewed |
|---|---|
| Keywords | Consciousness theory Mathematical logic math.lo Neuroscience Set theory Cognitive modeling |
| Key finding | Proposes that consciousness can be mathematically modeled by applying consciousness operators to non-well-founded sets generated from neural network graphs. |
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.