In article <firstname.lastname@example.org>, email@example.com (Randy Poe) wrote: > On Wed, 17 Jan 2001 01:57:50 GMT, firstname.lastname@example.org wrote: > > >For example, I might be asked to justify the following step > >(I will assume that I am working in the field of complex numbers, > >which you have refused to do for some reason): > > The reason is that if he admits that saying "x and y are integers" is > insufficient to discuss the behavior of (x+sqrt(-1)y) here, he'd have > to admit it's insufficient in the FLT proof.
But, nothing will be lost for James Harris if he would admit that he is working in the field of complex numbers, and a lot would be gained since he could use all the theorems proved for complex numbers.
However, this would only serve to make his statements to be meaningful and allow him to define things like "mod" and "fractional". He would still need to specify a subring of the complex numbers, since the complex numbers contain "too many" numbers for what he wants to do.
James Harris has admitted that he is working in at least two distinct rings: ring of integers and ring of polynomials. His statements in the proof also imply that he is working in the ring of complex numbers and the ring of symbolic expressions (like sqrt(x^2+y^2)).
My first impressions were that he was working in just a subring of the complex numbers. When he claimed that he was also working in the ring of polynomials, I thought that would not be possible since he is using the equation x^5+y^5 = z^5, which is not true in a polynomial ring. But, he nicely got out of that problem by eliminating the z and claiming that he is also working in the ring of symbolic expressions. This clarification has helped a lot. But, now he is going to the other extreme of rejecting these known mathematical rings and he is trying to create his own rings from scratch, which will demand even more explanations and proofs than what was originally required.