Category Theory and the Participatory Ontology

1,000 words

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“It is all unfolding. Hah!” – Andrew Zuckerman, Director of Operations at the Qualia Research Institute

Various strands of philosophy and cognitive science have entrenched the notion that the mind is fundamentally separate from the world. Cartesian dualism, which influenced philosophy of mind for centuries, claimed that the nature of the mind is categorically different from that of the physical universe. The essence of the former is the act of thinking, which is immaterial, whereas the essence of the latter is extension in the three spatial dimensions (length, width, and depth), a characteristic that only material phenomena can have. Representationalism, an idea that was foundational to cognitive science, believes that the intrinsic nature of the mind is to make internal representations of the external world. Conscious awareness is in here, whereas the world is out there.

Cognitive scientist Marvin Minsky frames the distinction between the mind and certain physical processes in a novel way. He suggests that the physical processes in machinery are defined by the transformation of inputs into outputs (this property can be generalized to other natural phenomena); on the other hand, brains “use processes that change themselves – and this means we cannot separate such processes from the products they produce. In particular, brains make memories, which change the ways we’ll subsequently think. The principal activities of brains are making changes in themselves.” 

However, as Francisco Varela, Eleanor Rosch, and Evan Thompson insightfully point out in their book The Embodied Mind, there is no reason to think that this characteristic property of continuous self-modification doesn’t extend to phenomena in the world as well. Self-organizing systems, which are pervasive in nature and govern everything from reaction-diffusion processes in chemistry to free-market economics, are essentially systems that modify themselves. In a colony of ants, for instance, an individual ant will react to the concentrations of pheromone produced by its neighbors, which will then produce a global pattern of activity that constrains the behavior of the individual ant. Thus, in self-organizing systems, the environment is not separate from the individual who is acting within it, but rather engages in a reciprocal cycle of mutual co-determination; the environment transforms the individual, and in turn the individual transforms the environment. In this sense, the individual and the environment are entangled in a single process of self-modification. While the science of self-organizing systems has not aimed, for the most part, to explain the mind, it nonetheless challenges a lengthy tradition in Western thought of assuming a fundamental distinction between consciousness and the world, between “in here” and “out there.” The science of self-organization points towards a new metaphysics, which I call “the participatory ontology.”

Arguably, this tradition has deeply influenced all fields of study, including those that attempt to depict the world in purely “objective” terms and appear to shy away from any metaphysical presuppositions about our relationship with it. Indeed, Varela et al argue that the study of mathematics is rooted in our tendency to divide the internal from the external. As I wrote in the previous blogpost, early 20th-century mathematicians attempted to ground all of mathematics in set theory, which stems from one basic question: which mathematical objects are inside the set, and which are outside? [1] Varela et al state that “the logic of sets” originates from our “bodily experience,” whose “structural elements are ‘interior, boundary, exterior’” and whose “basic logic is ‘inside or outside.’” In other words, our cognitive structures predispose us to perceive a boundary between ourselves and the world, and these structures inform our set-theoretic conception of mathematics, in which the fundamental structure is one whose purpose is to divide the objects contained inside it from the objects that are outside it. [2]

Clearly, the participatory ontology demands a radical revision of our entire mathematical apparatus. What sort of mathematics would be implied by this ontology? I propose category theory as the answer. A category is defined as an aggregate of objects and mappings between them (known formally as “morphisms”), which have to obey a series of axioms. Importantly, according to philosopher of math Jean-Pierre Marquis, “objects play a secondary role and could be entirely omitted from the definition.” That is, category theory is a theory of the mappings between objects rather than the objects themselves. While a set can be constructed in such a way that there are no relations between the contained objects, there must be a mapping between every pair of objects in a category, at least according to one definition. In fact, as Marquis states, category theory was invented precisely in order to elucidate the connections between the various mathematical structures.

As I noted in the last blogpost, there is a tendency in philosophy to think of mathematical objects as structures that exist objectively in an abstract universe, independently of the humans who refer to and work with them. This notion reinforces the view that there is a fundamental separation between the (human) self and the physical universe. However, category theory acknowledges the deep interrelationship between the self and mathematical structures, despite their seeming objectivity. Category theory characterizes mathematical objects “up to isomorphism,” meaning that it is concerned with the equivalence classes between objects rather than the intrinsic nature of those objects. As Marquis writes, “There is no such thing, for instance, as the natural numbers. However, it can be argued that there is such a thing as the concept of natural numbers. Indeed, the concept of natural numbers can be given unambiguously … but what this concept refers to in specific cases depends on the context in which it is interpreted, e.g., the category of sets or a topos of sheaves over a topological space.” The mathematician determines the definition of a mathematical object that is most relevant to his aims, and this definition consequently determines the results that he will derive in his work. Thus, in category theory, the self (mathematician) and world (mathematical objects) are entangled in a process of mutual co-determination.

[1] Thank you to Dario de Janeiro, Oxford DPhil student in computer science, for describing set theory in this way. 

[2] This discussion also raises the fascinating question of whether mathematics is discovered or invented. This essay assumes the latter viewpoint. For more on this topic, check out this wonderful interview of Stephen Wolfram.

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