Subject: categories: Not getting it wrong

From: Ross Street <[email-masked]>
Date: Fri, 18 Nov 2005 11:20:25 +1100

I took the liberty of circulating to our Math Dept Andrej Bauer's
message which included the example

>>
>> Limit[((1 + 4 x^2)^(1/4) - (1 + 5 x^2)^(1/5))/(a^(-x^2/2) - Cos[x]),
>>       x -> 0]

In retaliation my colleague Alf van der Poorten commented as below.
--Ross

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-
Dear Ross,

I've long said that "mathematics ain't about getting it right but
about not getting it wrong". Our students of course think that we
teach them methods/algorithms/recipes for getting it right ---
indeed, we often give them partial reward just for seeming to have
started to attempt to use some reasonable way that might possibly
have led them to a correct answer. In that context looking it up or
having a machine look it up (or asking/copying from a wise friend) is
a perfectly fine recipe for maybe getting it right.

The trick is verifying that it is indeed very likely right (or,
occasionally, actually \emph{is} right) and much better yet, noticing
that it is wrong or that there are grounds for suspicion warranting
their consulting a dinkum mathematician to get the good oil.

By the way, the particular limit cited didn't tempt me towards error
at all: Almost automatically I scribbled $(1+4x^2)^{1/4}=1+x^2+\ldots
\,$, and so on, realising a line or so later that I had better warm
up those $\ldots\,$ to $-\frac32x^4+\ldots\,$, and so on. I hadn't
realised that I was checking whether some zero is isolated and am
sorry to learn that I might have been.

Ideally our students will learn from us to see when they've surely
got it wrong and, more subtly, when they might well possibly have got
it wrong ("Yup, it does indeed work for $a=1$ but, oops! Why is there
an unspecified $a$ at all? I'd better phone a mathematician to ask
--- hoping she does something sensible such as reminding me of Taylor
expansions rather than waffling about isolated zeros.").
Unfortunately, as always, distinguishing right from wrong requires
quite a bit of knowledge (both of what's right and of what isn't)
that students are often slow in acquiring.
------------------------

Baseball and Quantum Physics
Umpire: "Some is balls and some is strikes, but until I calls 'em
they ain't nothing."

Centre for Number Theory Research    1 Bimbil Place, Killara NSW
2071  +61 2 9416 6026

---

From: Andrej Bauer <[email-masked]>
Date: Fri, 18 Nov 2005 15:55:10 +0100

> In retaliation my colleague Alf van der Poorten commented as below.

I don't quite understand who is retaliating for what in the posted comment.

Even though this is not strictly speaking category theory, I will take
the further liberty to tell you the whole story.

The above limit cropped up when I told my students in "Computer Science
I" to bring examples of problems they do in Analysis and Algebra. The
idea was to solve them using Mathematica. Someone bought the above
limit, except that instead of the parameter 'a' he had the constant 'e'
(base of natural logarithm). Now, as it happens, in Mathematica this
constant is written 'E' instead of 'e'. The student of course typed in
'e' which Mathematica took as a "free parameter" and gave the "generic"
answer 0, while the correct answer was 6. It took a while to explain the
whole mess. I suspect the only thing the students learned was that (a)
the limit is stupid, (b) Mathematica is stupid and (c) the teacher is a
pedantic fanatic.

By the way, to see what is going on in the above limit, draw a family of
functions for various values of 'a' approaching and passing through a=e.
All will be clear as you see two "waves" passing through each other.

I guess I am trying to point out that current Computer Alegbra Systems
are very tricky to use _correctly_. In Mathematica's defense it should
be said that it also found a couple of errors where the Analsyis teacher
did not (one involved cancelling "a common factor of b" between b^2 and
sqrt(b^2) for a real parameter b).

Andrej