Re: Evil in bicategories

[David Roberts <droberts@maths.adelaide.edu.au>]

Dear Jean,

You wrote:
> Maybe my english isn't so "beautiful", but in all cases where "evil" has
> been used, what is wrong with "wrong" instead?

I'm not so enamoured with the use of the word 'evil', but it seems to
be more entrenched than perhaps it was intended, namely as a joke.
Regardless of my personal convictions, I like to remain a mathematical
agnostic, so 'wrong' seems to me to be too strong. In my everyday
mathematical work I use choice and excluded middle and equality at
will, but I know that these foundational ideas ('evil' in categories,
constuctivism etc) exist and are useful and interesting.

Unfortunately I don't have any decent alternatives to offer, but the
philosophy boils down to, in my opinion, a structuralist view of
foundations (as opposed to the standard ZF with 'member of'
foundations) combined with (even a simple grasp of) type theory. The
former is essentially the 'sets are bags of points' approach and the
latter is the 'you can't ask: is \pi = (sin:R \to [-1,1])?' idea.
Perhaps readers of this list with the inclination and an eye for
nomenclature will suggest some words.

Toby Bartels calls categories where one is not allowed to test for
equality between arbitrary objects 'weak' and those where one can do
so 'strict' (most often these latter are internal categories in some
version of Set, or perhaps using universes). This reflects thinking
about higher categories (and completely exemplified by Makkai's
approach via FOLDS). This terminology takes the 'moral dimension' out
of talking about serious foundational ideas. But we don't have a word
that replaces 'evil' in this context that conveys the sort of mild
disdain for attempting to make the naive mistake of trying to ask if a
scalar is contained in a vector, as one can do in traditional
foundations (note that the answer depends on how one defines tuples).

That's my two cents, for what it's worth.
David


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]