[19 February 2013]

For reasons not worth going into (it’s a long story) I was once involved in a discussion of how to tell, given a sequence of objects of a given kind and two descriptions (or variable references) denoting items X and Y in the sequence, whether X and Y are identical or distinct. (This topic came up during a discussion of whether we needed clearer notions of identity conditions for this class of objects.)

One of my interlocutors suggested that we did not need to specify any identity conditions at all. Since X and Y were presented to us in a sequence, we could always tell whether X and Y are the same or different by looking at their position in the sequence. If X and Y are both at the same position in the sequence, then X must be identical to Y, and if X and Y are at different positions in the sequence, then X and Y must be distinct from each other.

Now, the first claim (if X and Y are at the same position in the sequence, then X = Y) is obviously true. And the second (if X and Y are at different positions in the sequence, then X ≠ Y) is manifestly false, though sadly my interlocutor never stood still long enough for me to point this out. Let X be “the Fibonacci number F₁” and Y be “the Fibonacci number F₂“. These occur at different positions (1 and 2) in the Fibonacci series, but both expressions denote the integer 1. So we cannot safely infer, from the fact that X and Y identify things at different positions in a sequence, that X and Y identify different things.

As I discovered the other day, Frege turns out to have a word to say on this topic, too. I wish I had had this quotation ready to hand during that discussion.

… die Stelle in der Reihe kann nicht der Grund des Unterscheidens der Gegenstände sein, weil diese schon irgendworan unterschieden sein müssen, um in einer Reihe geordnet werden zu können.

… their positions in the series cannot be the basis on which we distinguish the objects, since they must already have been distinguished somehow or other, for us to have been able to arrange them in a series.

(Gottlob Frege, Die Grundlagen der Arithmetic, 1884; rpt Stuttgart: Reclam, 1987; tr. by J. L. Austin as The Foundations of arithmetic 1950, rpt. Evanston, Illinois: Northwestern University Press, 1980, § 42.)