FOM: foundations/applications clarification.

[marker@math.uic.edu]

Dear Harvey:

 

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.

 -Dave Marker