> ...no professional mathematician appeals to axiomatic set theory or formalistic logic in their thinking nor in their publications.
None of them. Not even the ones whose specialize in axiomatic set theory or formal logic.
And I think Wayne is right (mostly) for a change. Euclidean geometry *is* a good place to learn what proof is. Though I disagree with him about how many students actually *did* learn much about proof in the "good old days". There were about 25 of us in my high school geometry course. Maybe three of us got it. (And the teacher wasn't one of them.)
That being said, I also think that there are other areas where students can learn what proofs are, how to read them (a step often omitted in the rush to teach what proof is), and how to write them. But it is a mistake---a big one---to try to be too formal or too rigorous at the outset.
- --Lou Talman Department of Mathematical & Computer Sciences Metropolitan State College of Denver