>> If Norm's article had not been very rude, which it was, I think the >> response would have been different. > >I would describe it as provocative rather than rude. He had a very >forceful style, but his language stayed professional.
It might be professional in *tone*, but what it is saying has no content other than to be insulting. I've just reread the paper, and page after page contains *no* mathematical or logical reasoning, it simply makes one unsupported, insulting claim after another.
* modern mathematics doesn't make sense
* mathematicians don't properly understand the content of high school mathematics
* modern mathematics is like a religious cult
* the definition of a real number as an equivalence class of Cauchy sequences is a joke
Can you point out a *single* substantive claim in the entire paper?
The problem here is that he is mixing up two completely different "complaints" about modern mathematics: (1) His ontological/metaphysical misgivings about the axiom of infinity, and (2) the nonrigorous way that *elementary* mathematics is taught. There really is no connection between these two complaints, as far as I see. He certainly gives no coherent argument for the connection.
He complains that high school mathematics teachers can't readily give a clear answer to such questions as "what is a fraction?", "what is a real number?", "what is a function?", "is a polynomial a kind of function?". That's probably true, but what that reflects is lack of consensus about the best *pedagogical* approach to explaining these topics, not because there is any lack of rigor in the modern mathematical understanding of them.
What he's really pointing to is the size of the gap between mathematical foundations and the practice of elementary mathematics. Learning how to work with real numbers, to add them, divide them, etc. is very far removed from consideration of reals as "equivalence classes of Cauchy sequences of rationals". What exactly a real *is* is almost irrelevant.
Perhaps that means that the reals should be introduced in an *operational* way, by saying something like a real number is something that you can add, subtract, multiply, etc. But that approach is *more* axiomatic than the set-theoretic approach. I don't see how it is more foundational.
I really don't see that Norm's complaints in this paper have any coherence at all. If someone wants to defend this paper, how about quoting a *single* paragraph that is insightful?