Objects and Arrows

Giandomenico Sica (ed.) What is Category Theory? (Milan: Polimetrica)

Giandomenico Sica’s What is Category Theory? is a collection of papers on category theory which should contribute to the philosophical community’s engagement with this fascinating region of mathematical investigation. In addition to papers of direct interest to philosophers of mathematics, the volume contains some introductory discussion of category theory, and some discussions of applications. While there are several technically difficult papers, the volume as a whole is reasonably accessible. While category theory is central to contemporary mathematics and theoretical computer science, there seems to be a relatively modest (though often quite excellent) philosophical literature on the subject. While most philosophers are interested in category theory insofar as it pertains to questions in the foundations of mathematics, it may also have important applications in philosophy of physics, metaphysics, and philosophy of language. Projects like Sica’s will encourage more interest I think.

There is an increasing number of introductory texts which can provide philosophers a technical acquaintance with category theory (See for example Lawvere and Schanuel 1997; Awodey 2006; Peirce 1991) and at least one excellent historical account of the place of category theory in the development of formal philosophy in the twentieth century.  I’m thinking of:

Awodey, S. and Reck, E. (2002) “Completeness and Categoricity II. Twentieth-Century Metalogic to Twenty-first-Century Semantics”, History and Philosophy of Logic, 23, 2, 77–94.

Category theory is a branch of mathematics which has been used to provide an axiomatic treatment of the commonalities between distinct areas of mathematical inquiry. The first presentations of category theory arose out of algebraic topology and specifically with Samuel Eilenberg’s observation that Sanders MacLane’s calculations on a specific case of a group extension coincided precisely with Norman Steenrod’s calculation of the homology of a solenoid. (This is discussed in interesting detail in the piece by Jean Pierre Marquis in the Sica volume). Eilenberg and MacLane’s interest in making sense of this coincidence across apparently distinct areas of mathematical inquiry gave rise to their development of category theory.

I’ll write some more about this soon.