FOM: applied model theory; what is "foundational"?
Stephen G Simpson
simpson at math.psu.edu
Wed Oct 22 03:09:00 EDT 1997
Dave Marker writes:
> You clearly have no idea what is going on in model theory.
That may be, but I still think it's more reasonable to refer to the
viewpoint outlined in Lou's points 2-8 as "applied model theory",
rather than "model theory" as Anand proposes. The term "model
theory", thanks to Tarski, has a much wider scope, especially when you
consider Lou's admonition that the G"odel incompleteness phenomenon is
to be avoided. Therefore, I intend to continue referring to Lou's 2-8
viewpoint as "applied model theory", until somebody gives me a good
reason to call it something else.
> I find your ignorance quite shocking.
And I too have been shocked by a number of things. Still, I think
it's best to concentrate on scientific issues rather than invective.
> A key insight is that one can often recognize algebraic structure
> from the combinatorial geometry of model theoretic dependence
> (forking).
...
> I also completely agree with Anand that these results are
> telling us something about how mathematics works.
...
> I agree with Anand that they are foundational.
Under what definition of "foundational"? It seems to me that these
forking insights are of interest only internally, within pure
mathematics. To me that's a pretty good tipoff that they aren't what
I would call foundational, in the sense of f.o.m. But maybe your
unarticulated notion of "foundational" is completely different from my
concept expressed in
www.math.psu.edu/simpson/Hierarchy.html
Would you and/or Anand care to explain your notion of "foundational"?
Dave writes:
> I see. The real test for whether a result is foundation is if it
> makes good cocktail party conversation.
That's not a correct summary of what I said. My test for whether
something is foundational is, how much does it focus on the most basic
concepts (in terms of the hierarchy of concepts). General scientific
interest and intelligibility is a significant byproduct of this, but
not the essence of it.
> I expect that the intelligent layman would have an easier time
> understanding the statement of Falting's theorem than
> Matiyasevich's.
Could you please remind me what Faltings' theorem says? Please try to
phrase it in terms of the most basic concepts possible. Pretend I'm
an ignorant layman; that should be easy.
-- Steve
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