Subject: categories: Why binary products are ordered

From: Charles Wells <[email-masked]>
Date: Mon, 29 Jan 2001 10:18:52 -0800

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.

Charles Wells, 105 South Cedar St., Oberlin, Ohio 44074, USA.
email: [email-masked]. 
home phone: 440 774 1926.  
professional website: http://www.cwru.edu/artsci/math/wells/home.html
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