Re: Intuitionism's Limits

[categories <cat-dist@mta.ca>]

Date: Wed, 05 Mar 1997 22:36:19 -0800
From: Vaughan Pratt <pratt@cs.Stanford.EDU>


	Date: Tue, 4 Mar 1997 18:19:51 +1030 (CST)
	From: William James <wjames@arts.adelaide.edu.au>
	(I grant you the original question would have been more
	recognisable given better use of language: "...philosophies of
	constructive mathematics and *of* category theory...")

Your original question was "Which view should dominate?", where "the
category theoretic view" was one of your options.  (You had several
questions but this one seemed the most central.)

If you are asking whether the primary expression of structure should be
in terms of relations between elements or transformations of objects,
then I would answer this as follows.

The analogous question for physics is whether energy and matter consist
of particles or waves.  The consensus in physics today is that both
energy and matter can be viewed more or less equally accurately, if not
equally insightfully, as either particles or waves.  Which offers more
insight depends on the circumstances.

The corresponding position for mathematics would be that structure can
be expressed more or less equally well in elementary or
transformational terms, and that which approach gives more insight
depends on the circumstances.

The extent to which this is not the consensus in mathematics today is
less a reflection on either approach than on the conceptual health of
mathematics relative to that of physics.

Vaughan Pratt