On Fri, 19 Jan 2001 00:22:04 GMT, firstname.lastname@example.org wrote:
>In article <942neb$dd5$1@nntp.Stanford.EDU>, > Michael Hochster <michael@rgmiller.Stanford.EDU> wrote: >> >> >> : (Sort of like if AB = 0, A or B = 0. These people are saying that >must >> : be proven, and that it is a "gap" in my proof that I don't do so.) >> >> : If so, I'd like them to say that is their position here and we can >see >> : if we can't work that one out. >> >> Yes, that is my position. I would like an explanation of why >> it is true that if AB = 0, then A = 0 or B = 0. I grant that >> this statement is true when A and B are integers. However, >> I would like you to verify it when A and B are funny things >> like x + sqrt(-1)y and x - sqrt(-1)y (x, y integers). >> > >Hey, I've already seen the post where someone says you guys proved that >AB = 0, when A = 0, or B = 0 by using x^2 + y^2 = 0.
It's hard to believe you can be so obtuse without trying deliberately.
What the proof uses is that x^2 + y^2 is not zero for all reals x, y if either x or y is nonzero. Can you see why that is not a good thing to rely on in a proof that x^2 + y^2 is not zero for all nonzero integers x, y?
> >I concede that one could debate the question of whether or not there >might exist some objects in an infinite ring that could be nonzero and >multiply times each other to give 0. After all, it's trivally done in >a finite ring.
Then perhaps you can be made to understand that mathematical theorems are built up from a few small starting axioms. A theorem is built on results you have already established. When you read a mathematics book, results in Chapter 1 are not proved by using theorems in Chapter 12 that were proved by using theorems in Chapter 1. That would not be a proof.
Have you ever taken even a single math course? It's not a question of whether we BELIEVE it's true, it's a question of whether it can be proved by previously-established results.