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

On Fri, 20 Jun 2008 05:41:51 -0700 (PDT), gerry@xxxxxxxxxxxxxx wrote:

On Jun 20, 9:38 pm, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Wed, 18 Jun 2008 21:19:38 -0700 (PDT), mike3 <mike4...@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.

Well, let's take this to its logical conclusion and suppose he had
submitted his homework in Bulgarian. The difference between taking
the time and mental effort to translate from the Bulgarian to English
and taking the time and effort to translate from unconventional
English to conventional (mathematical) English is a difference of
quantity, not quality. Either way, you're asking the marker, who is
probably woefully underpaid and has her own life to get on with, to
put extra effort into a thankless job. Writing in Bulgarian, swapping
deltas & epsilons, they are both erroneous, if part of the purpose of
the course (and, thus, the assignment) is to instill a little of
the local mathematical culture.

This is more a wild extrapolation than a logical conclusion.
There was an implicit requirement that the assignment be
submitted in English. There's simply nothing incorrect about
swapping "delta" and "epsilon", as long as it's done correctly.
("Done correctly": We don't know whether the required
quantifiers were included. If not then of course the solution
was wrong as written. But if the quantifiers _were_ included
then the solution as written was much _better_ than what
one often gets from students, where the letters have their
traditional roles but the quantifiers are omitted.

I tend to suspect that the quantifiers were in fact omitted,
since so many students omit them so often. If so then
_that_ is in my opinion a valid reason for a 0 on the problem.
In my class when students submit things using "epsilon"
and "delta" in the traditional way but omitting the
quantifiers they get a 0, with a note saying that I have
no idea what they mean by "epsilon"...)

You omitted two points I made: (i) Swapping the letters this
way _is_ a very bad idea (ii) Regardless, it's very important
to realize that there's nothing sacred about the choice of
letters.

For example, say we're proving that f o g is continuous
at x if g is continuous at x and f is continuous at g(x) :

Suppose epsilon > 0. Choose delta_1>0
such that |f(x) - f(y)| < epsilon if |x - y| < delta_1.
(*) Now choose delta > 0 so that |f(y) - f(g(x))| < delta_1
if |y - g(x)| < delta. Etc.

In (*) it's important to realize that we _can_ substitute
something other than "epsilon" in the definition of
"f is continuous at". In my experience teaching this
stuff students often do think there's something sacred
about the choice of letters, and hence they're unable
to come up with the proof of this utterly trivial fact.

And that means that they don't understand what the
definition really means, which is a bad thing.
David C. Ullrich
.

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