pillay at math.uiuc.edu
Thu Sep 25 12:17:39 EDT 1997
I found your article interesting.
Let me make a couple of maybe overly polemical comments:
1) The development of Foundations of Math (Frege-Russel-Hilbert-Godel...)
was closely related to foundational problems coming out of the math. of the
time and in particular to the figure of Hilbert, and this gave it its life.
Current work on Foundations should similarly be informed by the mathematics
of today, although not in a dogmatic fashion.
2) There has been enormous abstract development of math in past 60-70
years. 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),..... One has the feeling that we should make the
effort to have a "position" on these and other developments, and see to
what extent the amazing techical development of different parts of math.
logic can be relevant. (Of course there is a long tradition in logic of
doing exactly this.)
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. It is worth trying to understand why these changes of
fashion come about and how serious they are, as they impact strongly on
logic. Similarly for other changes of taste, such as the down-grading of
general topology compared to algebraic and geometric topology.
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