categories: Re: Why binary products are ordered

[Michael Barr <barr@barrs.org>]

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
>