What is a Mathematical Structure of Conscious Experience?

Abstract

In consciousness science, several promising approaches have been developed for how to represent conscious experience in terms of mathematical spaces and structures. What is missing, however, is an explicit definition of what a 'mathematical structure of conscious experience' is. Here, we propose such a definition. This definition provides a link between the abstract formal entities of mathematics and the concreta of conscious experience; it complements recent approaches that study quality spaces, qualia spaces or phenomenal spaces; it provides a general method to identify and investigate structures of conscious experience; and it may serve as a framework to unify the various approaches from different fields. We hope that ultimately this work provides a basis for developing a common formal language to study consciousness.

Sources of richness and ineffability for phenomenally conscious states

“…While we will not focus on the implementation details of how conscious experiences might relate to neural processes, intuitively, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathbf{s_t}$\end{document} is some mathematical object (e.g. a vector of real numbers) representing one state in an abstract space of possible experiences (for background on what it means to formalize \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathbf{s_t}$\end{document} as a mathematical object and current approaches, see Kleiner (2020) ; Kleiner and Ludwig (2023) ). Subsequently, information theory gives us the ability to reason about the relative richness and ineffability of conscious experience based on the computation graph, without needing implementation details of the functions.…”

On the non-uniqueness problem in integrated information theory

“…Such an approach requires experimental evidence as a means to justify the specific choices that were made in translating phenomenological axioms into precise mathematics. There is a growing body of literature discussing how experiments may be used to assess different theories of consciousness Doerig et al (2020) and how the mathematical structure of the theory, in general, must be consistent with what is known experimentally about conscious percepts Kleiner and Ludwig (2023) . Assessing the impact of non-uniqueness, in all of its forms, is a crucial part of this research paradigm.…”

Mathematized Phenomenology and the Science of Consciousness

“…also the "Scott topology"; nLab authors ( 2023)). 7 Such a topology could ground a version of the "unity of consciousness" as follows (Kleiner & Ludwig, 2024;Prentner, 2019): Assume that any conscious experience is represented by closed sets of the topology on X (and vice versa). Then it follows that any union of such sets is itself a closed set and stands, by assumption, for a conscious experience.…”