FOM: an application of reverse math to core math
Stephen G Simpson
simpson at math.psu.edu
Tue Sep 14 18:18:11 EDT 1999
In his extremely long and angry FOM posting of 2 Aug 1999 12:54:07,
Soare demanded that reverse mathematics justify itself in terms of
applications to computer science and core math.
Before answering Soare's question, I would like to point out once more
that the question itself is based on a severe misunderstanding of
reverse mathematics. Soare is overlooking the fact that reverse
mathematics does not need to justify itself *indirectly*, in terms of
such applications. That is because reverse mathematics *directly*
addresses key foundational problems such as ``What are the appropriate
axioms for mathematics?'' Such problems are of obvious general
intellectual interest. The primary justification of reverse
mathematics is as a direct attack on foundational problems of this
kind. For a fuller discussion, see my posting
FOM: what is the value of reverse mathematics?
of 5 Aug 1999 20:39:49.
Nevertheless, despite the fact that Soare's question is inappropriate
and misguided, I want to mention that there is at least one
interesting application of reverse mathematics to algebraic geometry.
It is discussed in Harvey Friedman's posting
FOM: 40:Enormous Integers in Algebraic Geometry
of 17 May 1999 11:07:03. In this posting, Friedman used my 1988
reverse mathematics result concerning the Hilbert Basis Theorem
(``every ideal in a polynomial ring is finitely generated'') to obtain
upper and lower bounds for a certain mathematically interesting
function which arises in connection with descending chains of complex
algebraic varieties. See Theorem 1 of Friedman's posting. It turns
out that the function in question grows like the Ackermann function.
Interesting estimates for particular values are also obtained.
There is also an interesting connection to computational algebra.
Although I didn't mention it in my 1988 paper, the techniques there
are closely related to what are called Groebner bases. (The basis
obtained in the proof of lemma 3.4 is precisely the Groebner basis of
the given ideal.) Over the last few decades, Groebner bases have
revolutionized the computational aspects of algebraic geometry. See
for instance the 1993 monograph by Becker and Weispfenning.
Once again, I want to stress that such applications and connections
are not the primary purpose of reverse mathematics. But occasionally
it is nice to have such applications and connections to point to.
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