Charles Wells and Steve Vickers both made the point that, for example, in a record of type (height in inches age, in years) you have to be able to identify the height entry, and the age entry, but you do not have to identify either one as the "first" entry or the "second". Yet, as Charles points out, the usual ways of identifying the two entries all carry a culturally-canonical ordering--as we say "one, two" usually in that order and "left, right" most often in that order. As he puts it:
However, in practice the index set is {0,1}, {1,2} or {x,y} (the latter in analytic geometry). All of these are canonically totally ordered in our culture, so inevitably binary products do have an order in practice.
Then Vaughan Pratt weighs in, in his rhinocerean way. (For those who do not read FOM, or at least Ionesco, Vaughan had a terrific post on FOM about categorists as kind of rhinoceros, and by the end he began transforming into one.)
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"?
I think Charles is not tarred by this pit. The Peano/ZFvN idea is to say that, given 0 and 1, and some one among all the two-element sets is actually the set {0,1}. Charles is merely saying that when we pick a two element set, and name its elements, we tend to use names with specific (helpful or spurious) connotations. The naming here is "local", a choice of how to talk about two objects we assme we have. It involves no idea that the two element set has any objective features making it the set of 0 and 1. best, Colin