Re: Refute an alleged proof of the Riemann Hypothesis was Re: Refute a proof of the Riemann Hypothesis, round #4921

On Wed, 18 Jun 2008 21:19:38 -0700 (PDT), mike3 <mike4ty4@xxxxxxxxx>
wrote:

On Jun 18, 7:52 pm, Gerry Myerson <ge...@xxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
In article <erh6k.49$yg7.10@edtnps82>,
 "Larry Hammick" <larryhamm...@xxxxxxxxx> wrote:

One time in university calculus, a test required us to prove something about
continuity. I used the letter epsilon where the textbooks conventionally use
delta, and delta where they use epsilon. The advanced student who marked the
paper gave me a zero on that question, apparently thinking that I had no
idea what I was doing. The episode taught me lesson, not about mathematics,
but about academia.
[...]

I think the lesson you learned, or ought to have learned, was not so
much about academia as about how important conventions are as an aid to
understanding.


Although I do think giving a "zero" on the question is too severe if
that was his only error.

What error? If things are as he says there was no error at all, and
he should get full credit. There's nothing incorrect about proving
that f is continuous at 0 by saying

Suppose delta > 0. Let epsilon = ______. Assume that
0 < |x| < epsilon. Then ________, so that |f(x) - f(0)| < delta. QED.

It's a very bad idea to write the proof that way, of course, but
there's nothing erroneous about it. (And realizing that it's
correct is important - the delta in one place often becomes
what corresponds to "epsilon" in another place...)

(And of course students often omit important words in a proof,
evidently thinking that since they see these words in _every_
proof that means they can just be regarded as implicit, instead
of realizing that the fact that they see these words in every proof
shows that they really need to be there. In particular students
often omit the initial "Suppose epsilon > 0", evidently thinking
that epsilon _must_ be an arbitrary positive number because
it's called "epsilon". If he wrote it swapping the usual roles
of epsilon and delta and _also_ omitting the key quantifiers
then it's certainly wrong.)

One should get a little more than that if one
understood the concept correctly even if having not memorized the
convenetions. Conventions are good but are they more important than
understanding the concept? I don't think so.

David C. Ullrich
.

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