FOM: Categorical pseudofoundations
Harvey Friedman
friedman at math.ohio-state.edu
Sat Jan 31 09:01:44 EST 1998
I thought I would take Sol's advice and lay off of this topic for a while.
I find this convenient since I am on to something else that will be
announced in the fom. However, I was delighted to see Robert Black 5:34PM
1/3198. This is a very thoughtful and clear statement of the kind of points
I was trying to make, and it is surely refreshing to see someone else make
them so well. This may show that there is great virtue in my waiting to see
if the vast subscription list (now at 295!) will yield new, articulate, and
sensible voices.
I will later get into an evaluation of the axioms McLarty presented in
12:46PM 1/30/98 with further comparisons with set theoretic axioms, after
Tennant 3:50PM 1/30/98 is answered.
I would like to extend the comparison: graph theory, general topology, and
category theory. Nobody calls graph theory foundations of computer science,
yet it is the language in which almost (local) foundations of computer
science is cast - e.g., for circuits, networks, etcetera. Similarly, nobody
calls general topology foundations of geometry even though it is used in
the formulation of almost everything in modern geometry and topology -
e.g., its general definitions of (topological) spaces, bases, subbases,
compactness, metric spaces, closures, interiors, boundaries, connectedness,
connected components, paths, arcs, etcetera. The same kind of thing is true
about category/topos theory according to algebraic number theorists and
algebraic geometers that I talk to. Graph theorists and general topologists
are completely content with the obvious and standard set theoretic
foundations of graph theory and general topology. So is Grothendieck. Why
aren't category/topos theorists? Very very peculiar.
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