pillay at math.uiuc.edu
Thu Oct 23 01:57:22 EDT 1997
1) This integer versus rational points discussion begins to be a little
surreal. But let's just remark that the various Lang conjectures can be
considered as being about the qualitative and quantitive properties of
INTEGER solutions to systems of homogeneous polynomial equations with
integer coefficients, namely integral points of projective varieties. Now
the passage to projective space (namely compactifying) should be something
very "basic" and of "general intellectual interest", especially from the
point of view of the "hierarchy of concepts". One is closing off, bounding,
fixing, defining, the situation.
2) As the discussion proceeds it is interesting to see how Simpson's (I
hope not Harvey's) foundational view unfolds. As in many cases, the
original "rhetoric" or "manifesto" turns out to be in direct contradiction
with the reality. Steve's "absolute objectivism" turns out to be "absolute
subjectivism": what is "basic", or of "general intellectual interest" etc.
amounts just to what Steve happens to understand.
3) I repeat what I said in my original message (to the smaller group). I
firmly believe that if foundations of mathematics or metamathematics or
mathematical logic is to be a living subject with a future, it has to be
informed by current mathematics and the concepts which it has thrown up.
This is a view which (I believe) was always espoused by the "model theory
of fields" tradition, but which I have eventually come to accept. Note I
just say "informed", so I am not saying that one should ALWAYS be concerned
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