[FOM] foundations and model theory II
John T. Baldwin
jbaldwin at uic.edu
Wed Jul 9 22:08:24 EDT 2003
Baldwin:
One of the important discoveries of the middle 20th century is the
futility of a trying to find a general foundations of mathematics.
Friedman:
I have tried to figure out just how to read this sentence so that it is
even partly true. This has proved difficult.
Baldwin: As I replied to Shipman, the intent of this rather extravagant
remark is that the foundational enterprise has
separated itself from much of what mathematicians do.
Baldwin in another place
>
> The general explanation by model theorists of the Borel Tits Theorem
> is a more
> direct contribution to the foundations of algebraic geometry and group
> representations than results about measurable cardinals.
> (See Poizat: Stable groups Chapter 4.)
To examine just what you mean by "foundations" here, could you explain
further? Is there information of a "foundational" character about
algebraic geometry and group representations that comes from this work
in model theory that the algebraic geometers and group representation
theorists understand and did not know?
Baldwin replies. This will be the extent of my reply to the general
question. Marker's note explained the importance of Chow's theorem in
terms of
the difference beween algebraic and transcendental methods. That was
pretty convincing as a foundational issue to the model theorists then
and now.
Let me continue briefly on the other example. Algebraic groups (of an
algbebraically closed field) are defined in a fairly elaborate way as
groups that admit certain kinds of represnetations.
The Hrushovski-Weil theorem gives a simple definition: groups definable
in an algebraically closed field. (For this this to be satisfying you
have to be comfortable
with definablitity; few algebraists are now. But we are surprised from
time to time to walk down the hall and hear the algebraic geometers
tossing around words
like `inducition on quantifiers'. So the sociological answer may
change.) This allows a conceptual proof of the Borel Tits theorem. In
particular, one which does
not involve extensive case analsyis as all algebraic proofs do.
I strongly recommend Poizat's book -- now available in English. The
introductions to the various topics are not technical and will give you
a feel for the issues.
The foundational character is in the analysis of the FUNDAMENTAL
CONCEPTS OF THE SUBJECT UNDER STUDY -which are groups, group morphisms
etc. not sets and orderings.
More information about the FOM
mailing list