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Abstract submitted for the 1st Annual Web Conference of the International Society for the Philosophy of the Sciences of the Mind, November 24-24, December 1-2, 2023.

Mathematical Evidence for a Cognitive Framework of Four Levels of Knowledge

In category theory, the Yoneda embedding is an isomorphism {$L\cong R$} of two functors, {$L=\textrm{Hom}(A,B)$} and {$R=\textrm{Hom}(\textrm{Hom}(B,\_),\textrm{Hom}(A,\_))$}, where {$A$} is fixed and {$B$} is variable. Each of these functors expresses objects (such as {$A,B,X$}) and morphisms (such as {$A\overset{u}{\rightarrow}B, B\overset{f}{\rightarrow}X$}) in a category {$\textbf{C}$} in terms of sets and set functions in the category {$\textbf{Set}$}. [1] This isomorphism defines a bijection {$A\overset{u}\rightarrow B\Leftrightarrow\theta^{A\overset{u}\rightarrow B}$}, which can be understood as four facts {$W_0,W_1,W_2,W_3$} about a morphism {$A\overset{u}{\rightarrow}B$}.

{$$\begin{matrix} W_1\equiv\;\;\;\; A\overset{u}{\rightarrow}B\in\textrm{Hom}(A,B) \\ W_2\equiv\;\;\;\; \theta^{A\overset{u}{\rightarrow}B}\in\textrm{Hom}(\textrm{Hom}(B,\_),\textrm{Hom}(A,\_)) \\ \end{matrix}$$}

{$W_1$} asserts What is {$A\overset{u}{\rightarrow}B$}. It is a morphism from {$A$} to {$B$}, an element in the set {$\textrm{Hom}(A,B)$}, which is the output of the functor {$L=\textrm{Hom}(A,\_)$} applied to {$B$}. This is one way of thinking about a functor: "A functor is a picture of one category in another category." [2]

{$W_2$} asserts How is {$A\overset{u}{\rightarrow}B$}. It is the action {$\theta^{A\overset{u}{\rightarrow}B}$}, the natural transformation which prepends {$A\overset{u}{\rightarrow}B$} as a puzzle piece that was missing, thereby converting morphisms {$\textrm{Hom}(B,X)$} to morphisms {$\textrm{Hom}(A,X)$}, in toto. This is a different way of thinking: "A good way to think of a functor is as a kind of construction." [2]

What and How are understood within the familiar world of {$\textbf{Set}$}. What speaks of What is, How deals with What is not. What is one assertion {$A\overset{u}{\rightarrow}B$}, How is the answering {$A\overset{u}{\rightarrow}B$} of all questions {$B\overset{f}{\rightarrow}X$}. What understands {$A\overset{u}{\rightarrow}B$} in the context of something (itself), How understands {$A\overset{u}{\rightarrow}B$} in the context of anything (morphisms {$B\overset{f}{\rightarrow}X$} for a given object {$X$}).

{$$\begin{matrix} W_0\equiv & \theta^{A\overset{u}{\rightarrow}B}_B(B\overset{\textrm{id}_B}{\rightarrow}B)=A\overset{u}{\rightarrow}B\overset{\textrm{id}_B}{\rightarrow}B \\ W_3\equiv & A\overset{u}{\rightarrow}B\overset{f}{\rightarrow}X=\theta^{A\overset{u}{\rightarrow}B}_X(B\overset{f}{\rightarrow}X) \\ \end{matrix}$$}

{$W_0:W_2\rightarrow W_1$} and {$W_3:W_1\rightarrow W_2$} establish the natural transformations {$\theta^{A\overset{u}\rightarrow B}\Rightarrow A\overset{u}{\rightarrow}B$} and {$A\overset{u}\rightarrow B\Rightarrow\theta^{A\overset{u}\rightarrow B}$}, respectively, which constitute the bijection.

{$W_0$} asserts Whether {$A\overset{u}{\rightarrow}B$} is. This refers to the application of {$\theta^{A\overset{u}{\rightarrow}B}$} to the do-nothing action {$B\overset{\textrm{id}_B}{\rightarrow}B$} by prepending and thereby yielding {$A\overset{u}{\rightarrow}B$}. This has been called the Yoneda-y principle: "the one thing we know we have in any world of homsets is the identity, and all the Yoneda functors and natural transformations are acting by composition on one side or the other." [3]

{$W_3$} asserts Why {$A\overset{u}{\rightarrow}B$} is. This refers to the way that {$A\overset{u}{\rightarrow}B$} grounds {$\theta^{A\overset{u}{\rightarrow}B}$}, whereby extending {$A\overset{u}{\rightarrow}B$} variously with {$B\overset{f}{\rightarrow}X$} defines {$\theta^{A\overset{u}{\rightarrow}B}_X(B\overset{f}{\rightarrow}X)$}, the extraction of the essence, thus the unifying, comprehensive view upon all of the relationships. This is the Yoneda perspective. "Mathematical objects are completely determined by their relationships to other objects." [4]

Whether and Why are understood within the underlying world {$C$}. Whether understands {$A\overset{u}{\rightarrow}B$} in the context of nothing (the do-nothing action {$B\overset{\textrm{id}_B}{\rightarrow}B$}) and Why understands {$A\overset{u}{\rightarrow}B$} in the context of everything (morphisms {$B\overset{f}{\rightarrow}X$} for all objects {$X$}).

This interpretation offers a model for active inference, where sensations (What) and actions (How) are related by internal states (Why) and hidden, external states (Whether). [5] Moreover, the identification with four levels of knowledge allows a form of active inference in the human brain to be interpreted in terms of human experience.

Specifically, these levels of knowledge suggest a dialogue between two minds, an intuitive, sensory, unconscious mind that knows the answers and a rational, conceptual mind that does not know, thus asks questions. In Kahneman and Tversky's terminology, these would be fast System 1 and slow System 2, respectively. [6] From the vantage point not of implementations but of requirements, it is meaningful that there be a dialogue between knowing and not knowing, respectively, What and How, which may be typically championed in the brain by right and left hemispheres, and in society, by female and male genders.

Hypothetically, a third mind, final System 3, consciousness, works to balance System 1 and System 2, What and How, so that predictive conceptual models accord with intuitions, yielding consonance or simply peace. Subsequently, our emotional life can be the language that System 1 speaks to System 2. In response, System 2 may rework its conceptual model. System 3 would determine when to impose that new model as satisfactory, at which point the internal, soft-wired Why of our burgeoning emotional life would be replaced by the external, hard-wired Whether of our successfully implemented updated conceptual model. Here the external fact, in the world, assumed pragmatically, is that the unconscious System 1 and the conscious System 2 are in accord. Thus consciousness, System 3, functions rhythmically.

This internal view is asymmetrical as the internal witness appears and disappears. Our minds readily shift from How to What and consequently from Why to Whether. Such a conclusion can be drawn by investigating how these levels of knowledge {$(W_0,W_1,W_2,W_3)$} are documented by various thinkers, including Plato (ignorance, false opinion, true opinion, wisdom), Aristotle (final, formal, efficient, material explanations) and Peirce (object, icon, index, symbol). The Yoneda embedding, thus interpreted, is mathematical evidence for a cognitive framework (Whether, What, How, Why) that can be attributed to the perception-action loop of active inference, can be sought by neuroscientific means, can be observed in social discourse, can be identified with introspected perspectives, and can be leveraged to validate further such identifications, relating brain and mind.

References

[1] E.Riehl. Category Theory in Context. Dover Publications. 2016. https ://math.jhu.edu/~eriehl/context.pdf

[2] S.Awodey. Category theory foundations 3.0. Oregon Programming Languages Summer School. July 16-28, 2012. (25:30-28:30) https ://youtu.be/G9d8cY9_v2E?t=1531

[3] E.Cheng. The Joy of Abstraction: An Exploration of Math, Category Theory, and Life. Cambridge University Press. 2022. Pg.361.

[4] T-D.Bradley. The Yoneda Perspective. Math3ma. August 30, 2017. https ://www.math3ma.com/blog/the-yoneda-perspective

[5] T. Parr, G. Pezzulo, K. Friston. Active Inference: The Free Energy Principle in Mind, Brain, and Behavior. The MIT Press. 2022.

[6] D.Kahneman. Thinking, Fast and Slow. Farrar, Straus and Giroux. 2013.

Key words

Category theory, Yoneda lemma, active inference, cognitive framework, epistemology