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