Foundations and Foundationalism

[Timothy Y. Chow]

Harvey Friedman wrote:
> This points to a possible disagreement with Tim.
>
> Are there foundational advances that cannot be greatly improved by some 
> supporting philosophically coherent advance?

It is true that I am rather pessimistic about the prospects of replicating 
the success of foundationalism in mathematics in other arenas.  That 
mathematics is an outlier has been apparent for a very long time.  Ever 
since Euclid, it has been clear---in rough outline if not in full 
detail---that foundationalism (or something close to it) in mathematics 
ought to be possible.  As for foundationalism in any other philosophical 
arena, the entire history of Western philosophy can be regarded as a huge 
mountain of heuristic evidence that it is at best a very distant hope, and 
quite likely a wild goose chase.

That is not to say that what I have been calling foundational thinking is 
worthless in philosophy; it's just that I think that we can only 
realistically expect "local" success and not "global" success.

> For example, at this point we do not have any philosophically coherent
> understanding of, for example,
>
> 1. what is mathematics?
> 2. given 1, what is algebraic? what is geometric? what is combinatorial?
> what is set theoretic?
> 3. given 1,2 what is an important question or conjecture, or definition or
> theorem?
> 4. what is theory building and what is problem solving?

These questions are close enough to mathematical questions that it is 
perhaps not futile to hope that the success of foundationalism in 
mathematics could extend to cover them, but I remain doubtful.  For #2, I 
think that our best bet for making progress in the foreseeable future is 
to pursue "foundations of X" for the X in question.

Anyone who has followed the FOM mailing list closely for a long time will 
know that the archives are full of vehement rejections of overblown claims 
that advances in the "foundations of X" entail a complete overhaul of the 
foundations of mathematics.  There is certainly a place for calling out 
overblown claims for what they are, and I have done my share of that, but 
at the same time, we should be wary of throwing out the baby with the 
bathwater.  Higher topos theory, homotopy type theory, condensed 
mathematics, etc., are not just esoteric advances in esoteric branches of 
mathematics; some very powerful foundational thinking has gone into their 
creation, and they provide fresh insight into fundamental questions such 
as "what is algebraic topology?" or "what is the difference between 
algebra and analysis?"  If one is really interested in arriving at 
definitive answers to such questions, then paying close attention to the 
foundational work being doing in these areas is a must.

Incidentally, for those who are not already aware of it, there was a very 
interesting discussion in the comments section of Michael Harris's blog 
some years ago, in which (among other things) Jacob Lurie expressed doubts 
about whether homotopy type theory would be of any value in answering the 
types of foundational questions that he (Lurie) was interested in.

https://mathematicswithoutapologies.wordpress.com/2015/05/13/univalent-foundations-no-comment/

The discussion illustrates that even a seemingly narrow question like 
"what is homotopy theory?" is far from having a settled answer.

Tim