Identity, Isomorphism

Many strands of Western thought — from Augustinian theology to Cartesianism to set theory — have suffered from overly strong notions of what amounts to a privileged, originary, self-evident, contentful Identity of things. (There are also many significant exceptions. With their emphasis on distinctions of form, Plato and Aristotle only needed a weak identity. Spinoza’s emphasis on relations; Leibniz’s identity of indiscernibles; Hume’s dispersive empiricism; and Kant’s critical perspective are all closer to Plato and Aristotle in this regard. Hegel makes identity derivative from a Difference associated with Aristotelian contrariety or Brandomian material incompatibility. Nietzsche, Wittgenstein, and many 20th century continentals explicitly criticized the overly strong concept.)

21st century mathematics has seen tremendously exciting new work on foundations that bears on this question. Homotopy type theory very strongly suggests among other things that the identity needed to develop all of mathematics is no stronger than isomorphism. This provides a formal justification of the common practical attitude of mathematicians that isomorphic structures can be substituted for one another in a proof by an acceptable “abuse of notation”.

More generally, type theory and category theory provide an independent basis in contemporary mathematics for reaffirming the priority of form as difference over identity. I am tempted to say that they exemplify a kind of inferentialism in mathematics. (To those who say mathematics holds no lessons for philosophy, I would say that generalization disregards the specific character of these developments. nLab, the website for higher category theory, even has a page on Hegel’s logic as a modal type theory that explicitly refers to Brandom’s interpretation of Hegel!)