foundations of mathematics
marker at math.uic.edu
Wed Oct 1 20:42:06 EDT 1997
You are missing the point.
I do not except your conclusion that the Frege-Hilbert-Godel line
is the only approach to the foundation of mathematics.
As you say yourself one of the basic problem in the foundations of any
subject is to understand the relationship between the primitive concepts.
Let's examine one of the areas you dismiss in your e-mail.
The Lang conjectures (and the mathematics around them),are trying to
elucidate one of the most important foundational questions the interplay
between Geometry and Arithmetic.
I get the impression that you feel that once Descarte reduced geometry
to algebra there was nothing left to do. This is far from the case.
To my mind there are many results which I consider foundations which
are clarifying the interaction:
Three examples (all of which I've mentioned before):
i) the classical theorem that a Desarguesian projective plane is
by a skew field
ii) Falting's theorem (the Mordell conjecture): If a complex curve has
more than one hole there are only finitely many rational points. (The rest
of the Lang conjectures are attempts to find versions of this for higher
iii) the Hrushovski-Zilber theorem giving a topological characterization
of the topological spaces arising from algebraic curves. (While this is
the capstone, I could add here the whole of "geometric model theory" which
is devoted to understanding the relationships between the combinatorial
geometry of forking, incidence geometry and algberaic structure--but this
is at a more abstract level)
Though mathematical all of these results are also foundational. They
clarify the connection between the fundamental mathematical concepts of
geometry and number.
Of the other areas you mention I might agree with you and not call the
Weil conjectures "foundations" in this sense. While I am much less
familiar with the Langalands program, from what I know I might consider it
"foundations". Please don't take my comments as implying that I don't
consider Harvey's work as very important work in the foundations of
mathematics. I do! I just feel that it is not the only important work.
I am also sympathetic with Anand's point about cohomology. While you are
right that cohomology is a mathematical tool with mathematical
applications, one might wonder about the meta-mathematical significance
of the pervasive role of cohomology in modern mathematics.
Finally let me give a more benign version of one of Baldwin's comments
about mathematicians not worried about foundational points. My guess is
that most mathematicians simply view the foundations as "done". ZFC
provides a strong enough system (which is probably consistent) to prove
any theorem which is currently accepted as proven. If forced any result
they want to prove can be traced back to a formal proof in ZFC, so there
is no need to worry about foundational questions and they can carry on.
What convincing argument do you give them that they should care that
their results can be proved in a much weaker system? I don't think
the answer that "ZFC might be inconsistent" is good enough.
(Below I include a copy of part of a note I sent to Harvey last
week giving my answers to the questions he initially asked which is
relevant to this point)
Let me give my answers to the questions you raised in your first message.
As for "reverse mathematics".
-While I find the fact that most theorems of undergraduate and basic
graduate mathematics end up equivalent (mod RCA_0) to one of five natural systems of
second order arithmetic quite fascinating, I find the individual results
finding the exact level of any particular theorem uninspiring. (Of course
that this is a somewhat inconsistent position as there would be no forest without
trees but that is how I feel.) Indeed at the moment I tend to find the
anomolies (Ramsey's theorem for pairs, the Hilbert basis theorem) more
the mounting evidence for the classification.
I am led to several questions about the program.
-First from a (unviersity wide) intellectual level: While the existence
a classificaton is interesting: a)Why is this true? What does this tell
us about mathematics. ( I mean what does it tell us about mathematical
knowledge and the activity of doing mathematics, rather than what
do we gain)
(These questions are rather like your question of "Why is it that all
natural r.e. sets are recursive or complete.")
b) How "natural" is the underlying assumption of RCA_0? [You want
to say that a logical system is equivalent to a "purely mathematical" system,
but the presence of RCA_0 makes the "purely mathematical" claim abit
-Second , what do we learn on a mathematical level.
a) Does knowing there is a general classification give us any new
b) Does knowing the exact strength of a particular theorem give us new
Possibly. Certainly the new proof needed to show that the strength of a
theorem was less than we might expect could well lead to new insights
(but this could also be true if we gave a new proof using lots of
sophisticated machinery in a richer theory).
We know we can also read off bounds on provably total functions.
While I find the general principle that model theoretic-algebraic proofs
of bounds tend to lead to primitive recursive bounds appealing, but in
mathematics you usually
either only need to know there are bounds or you need sharp
(doubly-exponential say) bounds and the knowledge that the bounds are
easily computable is not
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