On 14 Jul 2006 07:22:57 -0700, firstname.lastname@example.org (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
True. Beliefs rarely make sense.
>* mathematicians don't properly understand the content of > high school mathematics > >* modern mathematics is like a religious cult
Beliefs invariably evolve into cults. That's where cults come from.
>* 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?