[FOM] Foundational Challenge

[Timothy Y. Chow]

M. Katz wrote:

> It has been assumed for about 70 years that in homological algebra and 
> certain subfields of algebraic geometry, category theory is a more 
> appropriate framework than set theory.
[...]
> Moreover, attempts by set theorists to challenge the conclusion that 
> category theory is a more appropriate foundation in these fields can 
> only lead to such attempts being discredited.
[...]
> To retain credibility, arguments in favor of ZFC or such as foundation 
> should be grounded in a choice of field where such an argument is 
> proposed, and preferably made by experts in such fields together with 
> set theorists.
[...]
> However, I think that ostrich insistence on ZFC as universal foundation 
> for all of mathematics is no longer credible, and may have the opposite 
> effect as far as grants are concerned.

I think that Maddy's terminology, and her distinctions between different 
roles that a "foundation" for mathematics might play, are helpful here.

Also, as I've said before on FOM, I think that it's important to 
distinguish between "foundation of X" where X is some subfield of 
mathematics, and "foundation of mathematics."

In the 20th century, there was a shift from varieties to schemes in 
algebraic geometry.  This is regarded by practitioners of algebraic 
geometry as a change in the "foundations of algebraic geometry." 
Categories, of course, play an essential role in scheme-theoretic 
foundations (and its descendants).

In my opinion, the shift from varieties to schemes should *not* be 
regarded as a shift from "set theory" to "category theory" in the 
"foundations of algebraic geometry," much less in the "foundations of 
mathematics."  When practitioners think about the "foundations of X," they 
are generally *not* thinking about drilling down to the foundations of all 
of mathematics, or trying to build up the entire theory from absolutely 
primitive philosophical concepts.  They are drilling down only as far as 
they need to for the purposes of furthering their research in X.  In 
Maddy's language, they are concerned with Essential Guidance.  They 
couldn't care less about whether their new foundations provide a Generous 
Arena that is generous enough to encompass some other branch of 
mathematics that they don't care about.  They also couldn't care less 
about providing an adequate Metamathematical Corral, unless for some 
reason metamathematical concerns are directly relevant to their next 
research paper in X.  Conversely, if someone comes along and insists on 
defining categories in terms of sets, the general attitude will be, "Fine, 
do whatever you want, as long as I can have my categories and reason with 
them the way I want to reason with them."  Turf wars between category 
theory and set theory are of little interest to them, again, unless their 
next paper actually requires them to pay attention to such niceties.

This kind of attitude toward "foundations of X" can, for the most part, 
coexist peacefully with a variety of views of the "foundations of 
mathematics."  In particular, I don't share M. Katz's pessimistic view 
about set theory getting discredited if someone insists on set theory as a 
foundation of mathematics.  As I said above, if you want to insist on 
using set theory as a Generous Arena for *all* mathematics, and you claim 
that your set-theoretic framework can support everything that the 
algebraic geometer wants to say about categories and topoi and schemes and 
so forth, the algebraic geometer is unlikely to complain.  Just don't 
commit the faux pas of saying that you're proposing set theory as a 
"foundation for algebraic geometry" in a way that suggests that they're 
going to have to abandon their talk of categories and whatnot.

I think that the real clashes come not when some particular subfield X 
makes some assertion about the "foundations of X" but when a broader claim 
is made, about something (say, category theory) being a complete 
substitute for set theory for the foundations of *mathematics*.  If and 
when such a claim is made, then I think one can rightly ask if the 
alternative foundation provides not just better Essential Guidance for 
some subfield X, but is also well equipped to shoulder the burden of other 
tasks that a foundation for all mathematics is often expected to 
shoulder---a Generous Arena and Shared Standard for *all* of mathematics, 
and a Metamathematical Corral that gives a clear explanation for why we 
need not worry about logical paradoxes arising.

The vibe that I sometimes get from proponents of alternative foundations 
is that they don't see the need for Generous Arena and Metamathematical 
Corral.  They may talk about a foundation for "mathematics" but secretly 
they really only care about their own subfield X, and so it doesn't bother 
them that their proposed Arena is awkward for other mathematicians.  They 
also, by and large, don't care too much about metamathematics, and I think 
that part of this is that from their point of view, mathematics is 
*obviously* sound, so worrying about its soundness is silly philosophical 
nonsense.  The feeling that mathematics is obviously sound is what, in my 
opinion, underlies Penrose's notorious arguments about Goedel's theorems. 
I also think that it explains Voevodsky's notorious lecture on whether PA 
might be inconsistent.  In the last part of that lecture, he talks about 
"reliable" mathematics in a way that, to me at least, betrays his feeling 
that the mathematics that *he* is interested in is obviously "reliable" 
and so there is surely some way to formalize that obvious fact.

A cavalier attitude toward Generous Arena and Metamathematical Corral is 
one thing that annoys those with a traditional view of foundations, who 
feel that the hard-won insights of the early 20th century are being 
slighted.  So again, I don't agree with M. Katz that ZFC as a universal 
foundation for all mathematics is not credible, if we understand that 
claim rightly.  The claim isn't that for every subfield X of mathematics, 
we must explicitly use raw set theory for the "foundations of X" and 
eschew any defined notions.  The claim, rather, is that set theory is 
still the most convincing candidate when it comes to Generous Arena, 
Shared Standard, and Metamathematical Corral for mathematics considered as 
a whole.  In my view, this remains a highly credible claim, and it is not 
threatened by the success of any particular "foundation of X" that one 
might devise for particular subfields X.

Tim