Re: WHY ARE WE CONCERNED? I

[selinger@mathstat.dal.ca (Peter Selinger)]

> F W Lawvere wrote:
> > WHY ARE WE CONCERNED? I
> >
> > 	"Dumbing down" is an attack not only on school children and on
> > undergraduates, but also one taking measured aim at colleagues in
> > adjacent fields and at the general public. The general public is
> > thirsty for genuinely informational articles to replace the
> > science fiction gruel served constantly by journals like the
> > Scientific American and the New York Times "Science" section.
> > Those journals have never published anything resembling a
> > mathematical proof and hence have rarely actually explained any
> > scientific subject in a usable way.

jim stasheff wrote:
>
> a math proof is hardly necessary to explain a scientific subject in a
> usable way.
>
> now for a mathematical subject a math proof is sometimes but not always
> necessary

I agree with Bill that the prevailing style of expository writing,
especially in newspapers, is often of poor quality. It would be nice
if such articles more often gave a glimpse into the nature of
research, rather than serving, as Bill puts it, entertainment.

However, I disagree on the role of proofs in expository writing.
Clearly, proofs are central in mathematics. But to say that
mathematics is only about proofs is a bit like saying that dentistry
is only about clinical research. Of course, the research is important,
and most of us who have root canals are very glad that it is being
done. However, I would like to believe that mathematics is ultimately
about solving problems that *matter*, and the reason they matter often
has nothing to do with their proofs.

I am of course not advocating replacing proofs by conjecture. I am
only speaking of expository writing, where I believe it is often more
important to explain the results than their proofs. And sometimes, it
can even be justified to give an "approximate" proof, i.e., a proof
idea, or even an "approximate" definition, if it is stated clearly
that there has been some simplification.

The poor state of mathematical exposition is not confined to articles
about mathematics. The following quote, from an ordinary new article
in yesterday's Times, send my logic-circuits spinning:

 French lawmakers, for example, gave preliminary support this month to
 a measure that would require the company to open the iPod to play
 music purchased from any online music service; currently, songs
 purchased from iTunes can be played only on iPods.

 New York Times, 2006/03/29, "Apple vs. Apple in Dispute Over Trademark"

This is of course not a logical contradiction; but I would be very
surprised if it is what the writer really meant to say. Sadly, most
readers probably won't know the difference one way or the other.

-- Peter