Daryl McCullough wrote: > Stephen Montgomery-Smith says... > > >>Gene Ward Smith wrote: > > >>>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. > > > Are we talking about > http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf > > 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?
As I have said elsewhere, I think his remarks about the natural numbers are spot on. It could be argued that his claims are not "substantiated" but then again his claims are philosophical in nature and are not really substantiatable in the sense that a mathematical proposition or a scientific paper is. I sort of see where he is coming from with the real numbers. I completely disagree with him about his views on elementary education, but it is a professional disagreement, worthy of civilized discussion, and does not put him in the crackpot category.
> 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.
I agree with you here.
> > 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.
Again, I agree.
> > 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?
As I said above, I really liked his remarks about prime factorization (even if I think I would disagree about how to solve the problem). But in these kinds of discussions, insightful is perhaps in the eyes of the beholder.