[r-t] Hudson stuff

[Richard Smith]

Graham John wrote:

> How many ringing problems have actually been solved by the direct application
> of group theory as opposed to original analysis and thought?

I think this is rather unfair -- you make it sound like
"direct application of group theory" and "original analysis
and thought" are mutually exclusive.

Group theory is just one tool that can be used to help
understand composition.  I would say one of the main
advantages of apply group theory to ringing is that it
provides a standard vocabulary for describing some of the
more sophisticated ideas that crop up when trying to analyse
certain problem ringing.

For instance, when, earlier in this thread, I described how
Hudson methods worked, I made quite heavy use of terms like
subgroup, coset and transversal.  This was quite unnecessary
-- I could have described from first principles how it all
worked.  However, this wouldn't have changed the fact that
half a lead of Hudson Delight *is* a transversal of the
cosets of Hudson's group.

By using the standard group-theoretic terminology, people
who are familiar with it should be able to follow the
discussion more easily.  And those not familiar with the
terminology ought to be able to read up on it if they are
sufficiently interested.  There are plenty of books and web
sites on the subject, and the majority will make a far
better job that I would make of describing, for example,
what a coset is.

What ever we might like to think, the mathematics behind
ringing *is* the mathematics of group theory and graph
theory.  Simply avoiding the standard terminology and
notation doesn't change this fact.

You talk about "original analysis and thought", but very
frequently this is just an application of group theory, even
if we don't refer to it as such.  For instance, I might
wonder whether I can get a true bobs-only 90 of Grandsire
Doubles.  I expect it's obvious to most people on this list
that you cannot because Grandsire is pure doubles and the
bob is also a double change.  But *why* does this mean you
can't get an bobs-only 90?  How do we know that it isn't
possible to find some sequence of double changes that yields
an odd-parity change?  In short, what is parity?

And to answer these questions, we are inevitably using basic
ideas from group theory, even if we refer to it as "original
analysis and thought".

But by avoiding the terminology of group theory we shut
ourselves off from the vast body of mathematical literature
and we are forced to rediscover everything ourselves.


I'm sorry this has turned into something of a rant, but (as
you can probably tell) it is something I feel strongly
about.  I'm about to send an email to the list about the use
of singles in treble-dodging minor.  Although I have tried
to avoid using too much mathematical terminology, it is has
still ended up quite mathematical -- more so than any of my
previous posts.  However, I'm not going to apologise for
this, even though I know it will put many people off reading
it.  There may well be a better, less mathematical way of
saying what I want to say, but I can't see it.  Perhaps
someone else will, in which case, I hope they'll share it.

Richard