Well, I'd still suggest that saying "wrong, for exactly the same reason!" is an exaggeration, when one argument is incoherent because nothing's defined and the other argument is just the sort of thing mathematicians say all the time, where in fact all the details could easily be filled in.
OTOH you point out below that it's more circular than I'd realized, so you have a better point than I thought at first:
On 30 Dec 2000 01:20:29 GMT, firstname.lastname@example.org (Steven E. Landsburg) wrote:
> > >Regarding my assertion that James's FLT proof and his sum-of-squares >proof are wrong in essentially the same way, David Ullrich writes: > >> Actually there's a big difference: The proof that x^2 + y^2 <> 0 >> is perfectly correct. Suppose that x and y are reals not both >> zero. If x^2 + y^2 = 0 then it _does_ follow that >> >> (x+iy)(x-iy) = 0 > >But *why* does it follow? It follows because this equation takes place >in a particular ring (the Gaussian integers or the complex numbers, >as you prefer) which we can prove has a certain property (namely >the property of being an integral domain). > >And how do we know these rings are integral domains? We know it, >ultimately, because we know that -1 is not the square of any integer. >That's perilously close to the whole strength of what James is >trying to prove and hence perilously close to being circular. >Therefore I question the relevance of Prof. Ullrich's statement: > >> Seriously: Suppose hypothetically you knew all about the >> real and complex numbers as _fields_, but you were somehow >> unaware that the reals could be ordered, so you didn't >> know the result about sums of squares of reals, because >> the word "positive" didn't exist. If someone showed you >> exactly what James wrote you would accept it as a proof >> that the sum of the squares of two non-zero reals cannot >> vanish. > >The point is that it is not *possible* to "know about the complex >numbers as a field" without first knowing that -1 is not the square >of an integer.
My first reaction was "why not?". Then I thought about it. When we construct the complexes as ordered pairs of reals (because we're too ignorant to construct the complexes in a manner P L would approve of) we need to show that if x and y are reals not both zero then x+iy has an inverse. The obvious way to do this is to write down what the inverse _is_, but when we do that we're going to be dividing by x^2 + y^2.
So it's circular. (But it's not nonsense. And I still think that "what in the world does this have to do with the question it purports to be answering?" is a much more relevant reply than "this is wrong, for the same reason".)
> Without that knowledge, you cannot know that the >complex numbers form a field. > >So I continue to maintain that: > > a) James's sum-of-squares proof is wrong > b) It is wrong for exactly the same reason that his FLT proof is > wrong >and c) If James could be made to understand the reasons for a), then > he would understand most of what everybody has been trying to > explain to him.
I don't believe that James understands as little as he claims to. In particular when Winter gives a precisely analogous proof of an obviously false result I don't _really_ think that James is missing the point, and I don't think that he _really_ thinks he's explained the difference.