FOM: cat as f.o.m.?

[Harvey Friedman]

Comments on Mclarty 3:38PM 2/19/98, which is his reply to Riis:

>Friedman and Simpson and others keep saying category theory
>is like that--set theory "in bad notation". But they are
>wrong and Cartier knows it.

Friedman and Simpson and others keep saying category theory *as foundations
of mathematics* is like set theory "in bad notation." This fits right in
with  conventional wisdom, in that almost everybody who uses category
theory views it as being ultimately founded in set theory - not any kind of
independent foundation for mathematics. This is the indefensible point that
is being adhered to. It is a combination of a misunderstanding of the
notion of foundation of mathematics, and a desire to undermine the vital
importance of this notion. Genuine f.o.m. is a special case of a wider
notion of *foundation of subjects*. Therefore, one should not do violence
to this notion by indiscriminately calling useful organizational schemes
"foundations."

I could imagine that category theory/topos theory could play a meaningful
role in genuine f.o.m. if, say, the project that I mentioned earlier on the
fom in my 12:21AM 2/13/98 items 7 (and 8,9) could be appropriately carried
out.

>How much is too much? But experience keeps showing that great
>amounts of category theory are not too much.

A very famous core mathematician I often converse with who likes
categorical formulations, says that one has too much "general nonsense"
(the affectionate word for category theory) when that "general nonsense"
becomes nontrivial. It then gets in the way.

>I like this way of putting it--mathematicians discovered something
>that was not at all obvious at the start, and is still denied by
>many. Categorical tools make it natural to neglect much and focus
>on a little, in a way that actually works on things like a 300
>year old question in Diophantine equations, and that cuts so deep
>with such generality that it becomes foundational.

The use of algebra - groups, rings, modules, and fields, etcetera, in
myriad contexts is a much more impressive set of tools that "make it
natural to neglect much and focus on a little, in a way that actually works
on things like a 300 year old question in Diophantine equations," and much,
much more. Yet the algebraists do not attempt to elevate these notions to
the status of foundations of mathematics in any accepted sense of the word.
Similar remarks apply to general topology and graph theory.

Again, these algebraic notions *could* play a meaningful role in genuine
f.o.m. if one could appropriately carry out a project like this: Give a
general theory of what kinds of algebraic structures are "important" and
classify them. E.g., why groups? as opposed to other things.