categories: Re: Why binary products are ordered

[Vaughan Pratt <pratt@cs.stanford.edu>]

(Just back from Australia, where I was able to combine my mum's 90th
birthday with the annual Australian Computer Society meeting which by
pure luck turned out to be only a 20 minute drive away, in the process
running into Bob Walters and Mike Johnson as well as some other friends
I hadn't seen for ages.  The following arrived right on Mum's birthday.)

>From: Charles Wells <charles@freude.com>
>
>One might say that the ordering of binary products, with a first projection
>and a second projection, is spurious but inevitable.  
>
>The two components of a binary product must be distinguished, as Colin
>McLarty explained, but they must be allowed to be isomorphic.  The usual
>way we handle a situation like this in mathematics is to index them, in
>this case by a two-element set.  One could use {red,blue}.  As far as I
>know, in Western culture, this set has no canonical ordering, but
>nevertheless one knows that there is a redth component and a blueth
>component and they might or might not be different objects.  
>
>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.

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