Haecceities and Mathematical Structuralism

MCMP – Philosophy of Mathematics

Christopher Menzel (Texas A&M University) gives a talk at the MCMP Colloquium (18 June, 2014) titled "Haecceities and Mathematical Structuralism". Abstract: It is well-known that some earlier versions of mathematical structuralism (notably from Resnik and Shapiro) appeared to be committed to a rather strong form of the Identity of Indiscernibles (II) that is falsified by the existence of structures like the complex field that admit of non-trivial automorphisms, or symmetries. In light of more recent work (notably, by MacBride, Ketland, Shapiro, Ladyman, and Leitgeb and Ladyman), it is widely accepted that the mathematical structuralist is not committed to II and that, in fact, the principle can be rejected outright on robustly structuralist grounds. I accept a qualified form of this view but I don't think the issue is as cut and dried as it might appear. In a 2007 Analysis article, José Bermúdez suggests that a strong version of II is still in play for the structuralist that can meet the challenge of non-trivial symmetries. The key to the proposal (as I will interpret it) lies in allowing identity properties, or haecceities, like being identical to c (for an arbitrary complex number c, say) to count as structural properties. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as properly structural and, in some circumstances, can serve as legitimate properties for discerning otherwise indiscernible “positions” in structures. Drawing on the model theoretic concept of an expansion, I base my argument on a notion of discernibility rooted intuitively in “underlying structure”. This notion turns out to be equivalent to a notion of discernibility identified in some previous studies but proves useful in focusing when haecceities can legitimately be invoked and why Bermúdez's proposed version of II falls short of a fully satisfactory structuralist principle.