Mathematical Theories

Mathematics can apply to consciousness in two ways. The first approach involves methods, models, and simulations that are increasingly rigorous and sophisticated, describing and explaining essential features and mechanisms of conscious experience, primarily its structure, level, content, and dynamics (Labh, 2024).

Here mathematics supports various headline theories. Integrated Information Theory relies on a mathematical determination of consciousness. Friston's Free-Energy Principle formalizes and optimizes the representational capacities of physical/brain systems. Hoffman's Conscious Realism (Idealism) utilizes a mathematical formulation of consciousness.

The second approach posits deep claims that mathematical structures form the foundations of consciousness, much as mathematical structures form the foundations of quantum mechanics. In a sense, the first way, clear and common, is epistemological; the second, highly speculative, is ontological.

Mathematics as Ontology

As for mathematics as ontology, Max Tegmark has the entire universe, all reality, as a fundamental mathematical structure (Tegmark, 2014a). Roger Penrose has the Platonic world of perfect forms as primary such that physical and mental worlds are its “shadows.” We “perceive mathematical truths directly,” Penrose says, in that “whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts” (Penrose, 1996). Both visions, certainly controversial, would be consistent with mathematical constructions of consciousness, suggesting that consciousness is “made of” mathematics.

Initiatives to link the abstract formal entities of mathematics, on the one hand, and the concrete of conscious experience, on the other hand, have proliferated, the challenge being to “represent conscious experience in terms of mathematical spaces and structures.” But what is “a mathematical structure of conscious experience?” (Kleiner & Ludwig, 2023).

Mathematical Structures of Conscious Experience

Mathematicians Johannes Kleiner and Tim Ludwig seek a general method to identify and investigate structures of conscious experience—quality, qualia, or phenomenal spaces—to perhaps serve as a framework to unify approaches from different fields. Their prime criterion is that for a mathematical structure to be literally of conscious experience, rather than merely a tool to describe conscious experience, “there must be something in conscious experience that corresponds to that structure.”

In simple terms, they say, such a mathematical structure consists of two building blocks: the first brings in one or more sets called the ‘domains’ of the structure, where the elements of sets correspond to aspects of conscious experiences. The second are relations or functions that are defined on the domains. The authors claim that this definition does not rely on any specific conception or aspects of conscious experience. Rather, it can work with any theory of consciousness in that “every conscious experience comes with a set of aspects,” whether holistic, irreducible approaches to qualia and phenomenal properties or theories built on atomistic conceptions of consciousness such as multiple mind modules (Kleiner & Ludwig, 2023).

Duan’s Bug Theory

Mathematician Yucong Duan proposes a mathematically based “bug” theory of consciousness in that, with respect to consciousness, a bug is “not only a limitation in information processing, but also an illusion that leads human beings to create abstract and complete semantics and use them as tools” (Duan & Gong, 2024a).

He calls mathematics “the language of consciousness” required to find patterns, periodicity, relevance and other characteristics in consciousness, to reveal causal relationships and interactions among them, and to understand the structure, dynamics and functions of consciousness. For example, “dynamic system theory can describe the evolution track and stable state of consciousness, and information theory can quantify the information flow and entropy value in consciousness, thus revealing the dynamic characteristics and information processing mechanism of consciousness.”

Moreover, Fourier transform can “decompose complex consciousness signals into simple frequency components and reveal the laws and mechanisms of consciousness activities through frequency domain analysis, filtering and time-frequency analysis”—combining to yield “new perspectives of consciousness regularities.” Duan does recognize the limitations of mathematics (Duan & Gong, 2024b).