FOM: E. Noether on cat as f.o.m.?

[Colin McLarty]

Harvey Friedman wrote:

        Saying my view of category theory as not only an organizing
principle for math but a foundation

>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."

        You also do violence to "foundations of subjects" if you ignore the
subjects they are to found, in favor of some fixed notion taken as defining
"foundations". I come back to this below.


>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.

        Yes. The main virtue of category theory is to make as much math
trivial as possible. Of course one person's triviality is another's vast
complication or irrelevant sidetrack. So it remains a matter of taste and
judgment.  

>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.

        Emmy Noether tried to make it a foundation. She called this "set
theoretic foundations for group theory" and makes it clear this is not the
(long familiar) approach to groups as sets with operations. For her, a
"purely set-theoretic" approach would "proceed independently of any
operations" on elements, and take subgroups and quotient groups as the basic
units (Noether, E. 1927: "Abstrakter Aufbau der Idealtheorie in
algebraischen Zahl- und Funktionenkörpern", Mathematische Annalen 96,
pp.26-91, quoting p.46. In her Gesammelte Abhandlungen 493-528). 

        The basic tool would be the homomorphism theorem on induced
homomorphisms between quotients--what we now call the universal property of
quotients. Her "set theory" is our "category theory".

        The effort failed, but she did make it, and not from any desire to
defame "foundations" (as attributed to me above). She did it precisely to
bring out what is essential to any more-or-less systematic analysis of the
most basic or fundamental concepts of Abelian groups and their uses. The
effort failed because the descriptions she sought could not stand on their
own. But today a description of the category of Abelian groups can stand on
its own (pace Feferman's repeated but undefended claims to the contrary).
The features it relies on are the focus of most practical work with Abelian
groups. So I claim it is "foundational" in just the sense Steve Simpson has
posted.

Colin