As I said in an earlier post, the whole thing is a figment of the linear way we write (and speak, for that matter). Products are over unordered sets and any ordering is purely irrelevant. On Wed, 7 Feb 2001, Vaughan Pratt wrote: ...
I confess to some confusion as to what Charles is insisting is inevitable here. A binary product in C is a limit of a diagram 1+1->C (1+1 the two-object discrete category), and 1+1 has two automorphisms. This much and its mathematical consequences are surely inevitable.
But woven into Charles' argument is what Bill has called the "totally arbitrary singleton operation of Peano." It appears implicitly at the beginning when Charles names the projections, and then (after an indirect reference to the automorphisms of the binary product) more explicitly when he collects the names as a set.
Surely anyone insisting on names like 1 and 2 or red and blue for the projections of binary product is backsliding into the ZFvN tarpit of spurious rigidified membership. If this backsliding really is inevitable as Charles seems to be saying, how does one reconcile this with Bill's view of "rigidified membership" as "mathematically spurious"?
Must mathematics accept the spurious, in this or any other case?
Vaughan