Non-foundationalist foundational thinking

[Timothy Y. Chow]

Harvey Friedman wrote:

> Tim's notion of "foundational thinking" is "the type of thinking that 
> mathematicians engage in when they work on the "foundations of X" where 
> X is some specific subfield of mathematics". This is *NOT* a good 
> example of what I have in mind when I refer to "foundational thinking". 
> This is because a large component of that Tim is referring to involves 
> issues of exposition and simplification and effectiveness in presenting 
> proofs, and various in house issues mathematicians deal with when they 
> are involved in the practical presentation of mathematics to make the 
> exposition clearer and more readily understandable.

There is certainly an element of "routine housekeeping" involved in 
"foundations of X," but I think that what I have in mind overlaps a lot 
with what you have in mind.  This may become clearer below, with the 
discussion of examples.

> "But there is more to the story than reverse mathematics, at least as 
> the subject is usually thought of.  When I say "what is really needed" I 
> mean to refer to what practitioners intuitively feel is essential, and 
> not just to strict logical implication."
>
> Now I challenge Tim to create a significant new subfield of mathematics 
> out of his obvious point. If Tim can set up a new subfield dealing with 
> this obvious point across mathematics, then I might be the first to want 
> to Simpsonize this by jumping on it with abandon.

I am not proposing a significant new subfield of mathematics in your 
sense.  But that is a feature and not a bug.  I am suggesting that what 
I've been calling "foundational thinking" is not a new category in the 
American Mathematical Society Mathematics Subject Classification (MSC); 
rather, it is a *methodology* that can be widely applied.

In fact, you seem to be thinking along similar lines when you talk about 
playing classical piano.  Suppose a critic were to say, "Playing classical 
piano is an esoteric subfield of music.  It is not of general intellectual 
interest.  Only a tiny handful of people in the world care about it." 
That may be true as far as it goes, but the general intellectual interest 
lies not so much in the technicalities of classical music or pianos or 
human physiology; the general intellectual interest lies in the wide 
applicability of a particular "foundational" mode of thinking and approach 
to intellectual subject matter.  Moreover, the distinction between 
"foundational" and "foundationalist" is applicable here.  Foundational 
thinking is applicable, and can result in significant new ideas, without 
having to develop a comprehensive theory of all of music theory and the 
human body via logical reduction to a small number of basic concepts.  It 
suffices to circumscribe a suitable subfield of knowledge and analyze 
that.

Examples from mathematics of "foundations of X" that have been discussed a 
lot in the past include X = algebraic geometry and X = algebraic topology. 
For variety, let me take a contemporary example that has not been 
discussed as much from a philosophical viewpoint: the condensed 
mathematics of Clausen and Scholze.  Here are some lecture notes that give 
some idea of what this subject is about.

https://www.math.uni-bonn.de/people/scholze/Condensed.pdf

There is something distinctly foundational about condensed mathematics. 
The goal is not the proof of any particular flashy conjecture.  Instead, 
one takes a step back and asks a very basic question: how should one study 
structures that carry both an algebraic structure and a topology? 
Examining this question with a critical eye reveals certain blemishes in 
the existing theory that may not be apparent to a casual observer.  For 
example, topological abelian groups do not form an abelian category. 
This is the sort of thing that a "problem-oriented" mathematician might 
look at and shrug: "So what?"  Why develop a theory to address such 
blemishes with no specific big conjecture in mind?

The solution offered by Clausen and Scholze similarly has a foundational 
flavor.  Rather than address each of the perceived blemishes in a 
piecemeal fashion, they attack the root of the problem and offer a new 
definition: a "condensed set/ring/group/..." is a sheaf of 
sets/rings/groups/... on the pro-etale site of a point.  Never mind for 
now what exactly that means; the crux of the matter is that the solution 
is a fundamental redefinition of all the basic objects in question. 
(Scholze, by the way, has said that what he cares about most are 
definitions.  "The essential difficulty in writing 'Etale cohomology of 
diamonds' was (by far) not giving the proofs, but finding the 
definitions.")

In case there is any doubt about the value of these new definitions, they 
have already had some striking applications in complex geometry.

https://people.mpim-bonn.mpg.de/scholze/Complex.pdf

It has long been known that certain results in complex geometry that have 
a strongly algebraic flavor, and which you might think could be proved 
algebraically, have so far been provable only via "transcendental" (i.e., 
analytic) methods.  A remarkable byproduct of condensed mathematics is 
that it yields new and much more algebraic proofs of many of these 
results.

Getting back to the main topic of "foundational thinking," let us first 
note that the condensed mathematics project is not "foundationalist." 
Although the very first section is entitled "condensed sets," and 
condensed sets play a fundamental role in the theory, the claim isn't 
being made that we should throw conventional set theory in the trash can 
and start all over with "condensed sets" instead.

Second, I'm not claiming that condensed mathematics is a new MSC number, 
akin to 03B30 (reverse mathematics) that Friedman should "Simpsonize" (to 
use his term).  MSC numbers are not the point.  The point is that there is 
a general methodology of foundational thinking that is potentially 
applicable to virtually any area.  The point isn't to create a new MSC 
number for "foundational thinking" but to educate all mathematicians in 
foundational thinking so that they can go off and revolutionize their own 
subjects and generate their own new MSC numbers.

I predict that Friedman will not disagree with anything I've said here, 
but will attempt to dismiss it all as "obvious" and that something 
non-obvious needs to be said before we can be "productive."  However, I 
can guarantee that what I've said here is far from obvious to to most 
mathematicians (or there would be more foundational thinkers out there). 
Even those who are reading what I'm saying here and agree with it, may not 
recognize it as being consonant with Friedman's views.  If I'm right about 
that, then more work needs to be done to explain to people just what 
"foundational thinking" is, how it differs from "foundationalist 
thinking," and how "foundational thinking" is of general intellectual 
interest.

Tim