FOM: poll
Vaughan Pratt
pratt at cs.Stanford.EDU
Mon Feb 2 13:55:04 EST 1998
From: kanovei at wminf2.math.uni-wuppertal.de (Kanovei)
>4) what is said above is so elementary that any attempt
>to play this into a foundational system is ridiculous.
Are you saying that the choice or design of one's foundations should be
based on technically deeper issues, or that the situation is so elementary
as to admit of only one possible interpretation?
If the former, there has been some sentiment on this list in favor of
foundations that take elementary considerations into account. I'm in
favor of this myself, and would argue more strongly against category
theoretic foundations if I felt they did not start from elementary
intuitions.
For me the geometric notion of morphism as a 1-cell, or path between
two points, is very elementary and understandable.
2-cells are almost as simple, in striking contrast to natural
transformations, the canonical entity that 2-cells model. This is why
2-categories are so important to categorical foundations. This is not
an uncontroversial stand even among category users: Dana Scott with some
drama put up a slide asserting "2-categories suck" at a 1987 conference of
category theorists in Boulder, though a couple of days later he allowed
that he was starting to see some virtue in them.
If the latter (that there is only one "right" interpretation of such
an elementary point), you are wedded to a viewpoint in which, since
Cantor, we have been so thoroughly indoctrinated that to question it is
tantamount to questioning the whole mathematical enterprise. This is
an understandable viewpoint given the intensity of the indoctrination,
but that does not make it any less narrow.
Taking the continuum as primitive is a viable alternative to taking
resolvability of entities into their atomic parts or elements as
primitive.
Vaughan Pratt
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