Foundations and Foundationalism

[Harvey Friedman]

Reply to Chow
https://mail.google.com/mail/u/0/#all/WhctKKXXJxWwpDfGTWlvBndlqCmSmVnHWFrjDScmGgXzbcprSkQCWMFBTXkhnKfgvLHsPBg

Tim writes:

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.

*************

I agree pretty much with what Tim has written above there.

Furthermore, I am not sure that we have much of a substantial disagreement
at all.

Let me start with a weak form of what I am saying. CERTAIN PURPOSES, what
Tim calls foundationalism and I rather think of as philosophical coherence,
is by far the MOST POWERFUL AND DECISIVE way of achieving those selected
purposes.

I never meant to deny that there are worthy kinds of foundational thinking
that doesn't amount to being foundationalist. In fact, the major
philosophically coherent advances are normally preceded by considerable
periods of mere foundational thinking.

This points to a possible disagreement with Tim.

Are there foundational advances that cannot be greatly improved by some
supporting philosophically coherent advance?

This quesiton has several aspects.

One is that maybe we can do some sort of associated
philosophically coherent development - but this will not be a serious
addition to the original foundational advance.
Another is that maybe there is no such associated philosophicaly coherent
development to be had. That additions in that form doesn't exist.
Another is that maybe there is no such that we are going to find with
reasonable effort, and that strong efforts will not be worthwhile.

I believe that all major foundational advances start off without being not
philosophically coherent, and then, perhaps after long periods of time,
become strengthened by philosophically coherent reworkings and enhancements.

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?

Whereas there is some to be found of a foundational nature concerning all
of these, if would be a huge breakthru if we had any philosophically
coherent (foundationalist) treatments of such.

Harvey Friedman
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