I’m reading through Ferreiros’s Labyrinth of Thought. It is so far a decent history of the early development of set theory, beginning with Riemann and Dedekind.

Ferreiros poses, and answers, a question I’ve wondered about. He asks, why did some 19th century mathematicians consider set theory to be a part of logic? Both Dedekind and Frege make arguments that depend on moving from a concept to a set. Ferreiros argues that Riemann does too in his work on manifolds. Ferreiros’s answer is that it is a matter of upbringing. All three, but especially Riemann, were educated in an environment in which logic was conceived of very differently. It was logic in the quasi-Aristotelian, quasi-Kantian form that included doctrines about judgments and formation of concepts. One of the doctrines of general ideas, cited from the Port Royal logic, was that each idea has an extension and a conception, or intension. The intension is the general idea’s attributes or marks and the extension is the things to which it applies. In traditional logic, one would often talk about extensions of concepts as a matter of logic, which spawned a school of so-called formalists, so it was a natural move to talk about the set extension of a concept when the notion became available. 

(A quick digression.) Ferreiros notes that the first explicit formulation of a comprehension axiom was in Frege’s Grundgesetze in 1893. I was rather surprised that it was so late.

If we assume that Ferreiros is right in his theory about why 19th century mathematicians thought that set theory was logic, then there is a further question about why the logic of that period took the ascribed view towards intension and extension. I think that the answer to that is a development of logic coming out of Aristotle. Aristotle’s logic dealt with propositions composed of so-called terms and a term connective. These terms were similar to more modern concepts, although in the logic they were treated extensionally for evaluating arguments. Propositional logic, in the modern sense, were developed roughly contemporaneously by the Stoics, but it was not slow to be incorporated into Aristotelian thinking. I suspect that tracing the development of the notion of term through the middle ages would explain why 19th century logic was the way it was. One might not find the suggestion to trace a concept through the history of logic to be particularly illuminating. My Aristotle is shaky, at best, but I think the outlines of the important ideas can be found in the Aristotelian corpus on logic.

This explanation might work for Dedekind and Riemann, but I’m doubtful that it works for Frege. He was at pains to reject a lot of previous logical doctrine that it is surprising he would incorporate this without much comment in his own views. It helps explain Frege’s ridiculous footnote in the Foundations that he assumes it is understood what is meant by the extension of a concept.

Advertisements