Subject: categories: Re: fundamental theorem of algebra

From: "Reinhard Boerger" <[email-masked]>
Date: Mon, 03 Apr 2006 10:21:19 +0200

Hello,

Vaughan Pratt wrote:

> 3.  It is certainly not necessary to prove A before B merely because B
> depends on A; indeed one common-sense practice when proving a
> two-lemma proof is to get the easier lemma out of the way first, even
> if it depends on the harder one.  Is it kosher to truncate such a
> proof after the first lemma (or in this case the final result), call
> it an exposition, and point to the literature for the second lemma?

Indeed, should we expect non-mathematicians "believe in" results like  the Jordan
curve theorem, which is very easy to see and very hard to prove? Maybe they do not
appreciate a proof of something that looks completely obvious. Or, another veryeasy
thing: If somebody walks from A to B and somebody walks fron B to A at the same
time on the same route, they will me meet somewhere, even if they don't walk with
constant speed or if they stop somewhwere. Rigorously, this is essentiaally equivalent
to the intermediate value theorem, which is not too hard to prove, but nevertheless
not trivial.

I think there are better subjects to illustrate what mathematics is about. Some months
about I was asked to contribute to a calender, in which people working in different
scientific subjects give an insight in their work and their subjects. I contributed the
party theorem that a (simple) finite graph with at n>1 vertices has two vertices of the
same dergree. The proof just uses the pidgeon-hole principle and the observation
that a vertex of degree zero and a vetex of degree n-1 cannot both exist. If course, I
avoided mathematical terminology not known to the general public and spoke of
guests of a party, where some shake hands with each other and some don't. I think
this example may give a flavour of what a proof is, but does not bother them with
formalisms which are not necessay here (but somewhere else). Of course, category
theoty usrather abstract and can hardly be explained to non-mathematians.

One remark ti the fundamental theory of algebra. From Harald Holman I learned a
proof based on a simple ideas, which can be made rigorous quite easily: Since a non-
constant complex polynomial is large outside a sufficiently large circle, its modulus
must have a (local am global) minimum inside the circle (by compactness); we can
shift it into zero (by translation) and assume the minimum is attained in zero. Since
the polynomial is not constant, it is of the form a_0+a_m*z^m+higher terms, a_m
different from 0. If a_0=0, the polynomial has a zero in 0, and we are done, so
assume that a_0 is not  0. Then there exists a complex number w with w^m=-
a_m/a_0; this follows from the polar coordinate representation, which is taught in
calculus courses. Then for sufficienly every positive real h<1, the modulus of
a_0+a_m*(hw)^m is smaller than the modulus of a_0; if we choose h small enough,
the modulus of value of the polynomial at hw is also smaller than a_0, because the
higher terms can be neglected. This contradicts our assumption that the modulus
attains a minimum in 0.

                                                                              Greetings
                                                                              Reinhard