FOM: Chow's lemma?
cxm7 at po.cwru.edu
Mon Nov 3 11:16:38 EST 1997
Hilbert and Brouwer both thought issues like Chow's lemma were
paradigms of foundational problems. Half of Brouwer's dissertation (titled
"On the foundations of mathematics") studies basic conceptions of space,
using variations of Brouwer's proof that 1- and 2-dimensional continuous
groups are analytic. Hilbert had earlier urged such foundations for geometry
(see his 1902 essay "On the foundations of geometry"). The work made Brouwer
internationally famous and won him Hilbert's respect. I assume the
comparison to Chow's lemma is obvious.
If a man on the street had asked Brouwer or Hilbert in 1907 about a
weakened Koenig's lemma, both men would have thought it an excellent
question. But they also thought that relating spatiality to function theory
was crucial foundational thinking.
This is "foundations" in Steve's sense: it is unavoidable in any
"more-or-less systematic analysis of the most basic or fundamental" facts of
geometry. It deals with concepts "low in the heirarchy" in that you run into
them as soon as you think seriously about coordinate geometry in relation to
point-line geometry. In the case of Chow's lemma: as soon as you think
seriously about curves defined by polynomials. Indeed Descartes DID run up
against it in his GEOMETRY--though he was no closer to a clear statement of
Chow's lemma than Fermat was to a clear statement of the Matjesevich result.
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