FOM: foundations/applications clarification.
marker at math.uic.edu
Wed Oct 22 17:32:33 EDT 1997
Your statement below is a misstatement of what I have been saying.
However, there is the contentious issue on the table that direct applications to
mathematics - of the kind with which there have been recent successes - is
somehow to be construed as genuine Foundations of Mathematics, in a
legitimate use of this term.
I view as "foundational" results which shed light on the nature of basic mathematical
objects and the underlying connections between them. To me, the basic foundational
questions are "Why does mathematics work?" and "How does mathematics work?". The
approach to this problem in the logical (ie. FOM) tradition is an important one (probably
the most significant and possibly the only systematic approach possible), but I also
believe that other mathematical results and programs also sometimes shed light on these
questions and deserve to be called "foundational".
The example I have given several times is understanding the interaction between
algebraic/number theoretic structure and geometric/topological structure. As an
(admittedly extreme) example of this I offered geometric model theory in general and the
Hrushovski-Zilber theorem in particular.
The main theme (much in the spirit of classical results in elementary geometry) is to show
that from simple combinatorial geometric or topological information one detects the
presence of underlying controlling algebraic structure.
In fact this result later had striking applications but it is the underlying principles, not the
mathematical applications that I view as "foundational".
I feel that our disagreements have been blown out of proportion. While your definition of
"Foundations of Mathematics" will always be a little too exclusive for my tastes. I fully
believe that the program you are advocating is an important one and I look forward to your
future essays on the issues raised in your first message today.
Just to clarify another possible misunderstanding: My message of last night, as I said in
the preface, was an attempt to explain Pillay's comments that the distinction between
"pure" and "applied" model theory is at this point a fairly artificial one. I was certainly not
offering model theory as a competing approach to the foundations of mathematics.
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