This is Chapter 8 of my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.
This essay deals with a connection between a relatively recent (1940s and 1950s) field of mathematics, category theory, and a hitherto vague notion of philosophical logic usually associated with Plato, the self-predicative universal or concrete universal. Consider the following example of “bad Platonic metaphysics.”
Given all the entities that have a certain property, there is one entity among them that exemplifies the property in an absolutely perfect and universal way. It is called the “concrete universal.” There is a relationship of “participation” or “resemblance” so that all the other entities that have the property “participate in” or “resemble” that perfect example, the concrete universal.
All of this and much more “bad metaphysics” turns out to be precisely modeled in category theory.
While category theory can be quite forbidding to the nonspecialist, simple examples can be used based on inclusion between sets [see Ellerman 1988 for a bit more category theory]. Given two sets A and B, consider the property of being a set X that is contained in A and is contained in B. In other words, the property is the property of being both a subset of A and a subset of B. In this example, the “participation” relation is the subset relation. There is a set, namely the intersection, meet, or overlap of A and B, denoted A∩B, that has the property (so it is a “concrete” instance of the property), and it is universal in the sense that any other set has the property if and only if it participates in the universal example:
concreteness: A∩B is a subset of both A and B, and
universality: X is a subset of A∩B if and only if X is contained in both A and B.
Thus the intersection A∩B is the concrete universal for the property of being a subset of A and a subset of B.
The idea of a concrete universal is frequently found in ordinary thought and language. For instance, we might say that some instance or example represented the “essence” of a property. It would be the “paradigm example” that sets the “standard” for all the other instances of the property. The main theme of the essay is the interpretation of category theory essentially as the mathematical theory of concrete universals.
In philosophical logic, these themes can be traced back to Plato’s theory of ideas, forms, or universals. The sets of set theory are often taken as the mathematical explication of Plato’s universals (i.e., the symbol ε for set membership was taken from the Greek word ειδη for Plato’s ideas). But in view of the set theoretical antinomies discovered at the turn of the century, we know that sets cannot be self-predicative (i.e., cannot belong to themselves). The set representing a property must always be more “abstract” than the entities having the property. Thus set theory is the theory of “abstract universals.” Yet there was a definite self-predicative strand in Plato’s theory of universals [see Malcolm 1991]. Some universals were also concrete ideal instances of the property (i.e., concrete universals). Set theory could not be the mathematical model for that type of self-predicative universal. We show that category theory is that mathematical theory, and also argue that this recognition throws some light back on the antinomies since they resulted from trying to use one mathematical theory for both abstract and concrete universals.
The Third Man Argument against self-predication in Platonic scholarship is that if “whiteness itself” is white alongside all other white objects, then there must be a “One over the Many” (a super whiteness) by virtue of which they are all white, and so on in an infinite regress. But with the rigorous modeling of concrete universals in category theory, we see that the flaw in the Third Man Argument is the assumption that the “One over the Many” is distinct from the “Many.” In the example cited above, the process of forming the “One over the Many” is the process of taking the union of all the sets with the property of being a subset of both A and B. But the “One” that was the result of taking this union, namely A∪B, was also one of the “Many” (one of the subsets of both A and B taken in the union).
The interpretation of category theory as the theory of concrete universals again raises the question of category theory’s relation to the foundation of mathematics. Lawvere and Tierney’s theory of topoi is an elegant category-theoretic generalization of set theory so it generalizes the set-theoretic foundations of mathematics in many new directions. We argue that category theory is also relevant to foundations in a different way, as the theory of concrete universals. Category theory provides the framework to identify the concrete universals in mathematics, the concrete instances of a mathematical property that exemplify the property is such a perfect and paradigmatic way that all other instances have the property by virtue of participating in the concrete universal.
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