categories: Inevitability of ordering products

[Charles Wells <charles@freude.com>]

Vaughn Pratt made some valid points about my earlier remarks on the
inevitably of binary product projections being ordered.  For the most part,
I agree with him (but see below), since my (unclearly made) point was that
it is inevitable in our current mathematical culture, not that it was
mathematically inevitable.  

However, I am stuck on one point:  Sometimes one needs to refer to one of
the projections, and that involves giving the projections names.  I
mentioned "red" and "blue" as examples of names that do not introduce a
spurious ordering. But in practice, we must occasionally give names.  This
is not only for computation, either.  For example, one sometimes needs to
assume that an n-ary operation factors through one of the projections, and
then deduce consequences from that (Peter Johnstone did something like that
in his study of varieties that are ccc's).  In the proof one must give a
name to the projection it factors through.

So I argue that naming the projections is sometimes a practical necessity,
and given current mathematical habits the names are likely to have some
intrinsic (culturally intrinsic!) ordering.  But we could use red and blue.
 Or vanilla and chocolate.




Charles Wells, 105 South Cedar St., Oberlin, Ohio 44074, USA.
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