foundationalism and anti-foundationalism

[Stephen G Simpson]

Some quick reactions to Anand's comments.  I may be shooting from the
hip, but here goes.

 > Some of these developments can also be considered "foundational" in
 > the same way as Cartesian geometry was; for example, the penetration of
 > cohomology into many areas of pure math., the Weil conjectures, various
 > Lang conjectures (stating amazing conjectural relationships between
 > geometry and arithmetic),..... 

I don't agree with this remark.  My view is that Cartesian analytic
geometry can be considered "foundational" (in a limited sense) because
it shows that some geometric concepts can be redefined in terms of
some algebraic concepts, thereby moving the algebraic concepts closer
to the base of the conceptual hierarchy (see
www.math.psu.edu/simpson/Foundations.html).  This foundational aspect
makes Cartesian analytic geometry of general intellectual interest,
i.e., of interest to all educated human beings.  How can you say that
the penetration of cohomology is foundational in this sense?  I think
cohomology is and always will be a technical tool, by and for pure
math researchers, not a fundamental concept.  Am I missing something?
Same question regarding the Weil conjectures, the Lang conjectures,
etc.  It's interesting stuff, but I don't see why anyone would call it
"foundational."  If I'm missing something, please tell me what I'm
missing.

 > 3) For better or worse the "axiomatic" paradigm of math. activity (i.e.
 > considering mathematics as the derivation of theorems from axioms) is now
 > out of fashion. 

Really?  I wasn't fully aware of this change in fashion.  Is it part
of some bizarre postmodernist or deconstructionist fad?  Continental
philosophy, structuralism, primitivism, Levy-Strauss, Angst, etc.?  My
own view is that the logical and axiomatic approach to science
(i.e. Posterior Analytics -- Aristotle) is of eternal value.  Indeed,
it is part and parcel of what we mean by `science.'  Fads and fashions
will not change this.  Call me an Aristotelean, if you like.

 > It is worth trying to understand why these changes of fashion come
 > about and how serious they are, as they impact strongly on logic.

I'm not so sure it's worth trying to understand how these changes of
fashion (deconstructionism, anti-foundationalism, etc.) came about.
People can go off in all kinds of strange directions.  If you follow
every weird trend or fad, you will never do any serious work.  Life is
finite, so let's make the most of it.

Maybe John Burgess can help me here.  John, is the anti-foundational
trend connected with deconstructionism, continental philosophy, etc.?

-- Steve