[FOM] Foundational Challenge

[Harvey Friedman]

On Thu, Jan 16, 2020 at 1:05 PM <katzmik at macs.biu.ac.il> wrote:
>
> The "foundational challenge" thread is actually a pair of threads woven into
> one: (1) is ZCF the best framework for implementing proof checker packages;
> and (2) apart from computer verification, is ZFC the best "universal"
> framework for mathematics.

This is certainly true at least at present. And I believe that this
will continue to be the case.

> 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.

This is based on a confusion. Even if you look at MacClane's
Categories for a Working Mathematician, category theory is presented
as a development within set theory where a category is defined as a
set together with a set of "arrows", etcetera, This is just like a
normal development of a special mathematical area within the usual set
theoretic foundations of mathematics.

One can strain this set theoretic interpretation category theory by
having large categories and other conveniences, but these can be
handled by standard modified forms of set theory, or by simply
acknowledging that these conveniences are easily dispensable. E.g.,
the practitioners laugh at the idea that one needs anything like
Grothendieck universes to prove anything they really care about like
FLT.

>This is the position of
> leading practitioners in these fields, such as MacLane, based on their
> own work.  HIstorians like Corry have studied the question in detail,
> and showed in particular that the Bourbaki's slowness in accepting
> category theory as foundation resulted in serious shortcomings in some
> of their volumes.  I think it would be futile for people outside the
> field to challenge such a conclusion.  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.

Again, this is baed on a misunderstanding. I don't know of any serious
challenge to the idea that category theory is a very useful way -
maybe even an imperative way - to develop such mathematics. Not so for
some other areas of mathematics. But the category theory is easily
viewed as simply a normal straightforward set theoretic development as
MacClane made so clear. If you alternatively attempt to regard set
theory as a normal straightforward development within some categorical
foundation for mathematics, you do not get anything philosophically
coherent - at least not at this time.

That doesn't mean that anybody using category theory for these
purposes necessarily needs to care at all that it sits within et
theory. They only need to care if they want to have a philosophically
coherent foundation for what they are doing.

Of course, another reason to adopt a heavily set theoretic approach is
if one gets interested in the combinatorial mathematics that has now
been perfected over 53 years that I have been discussing in the New
Tangible Incompleteness series here on the FOM.

Harvey Friedman