[r-t] Re: Proofs

[Richard Smith]

Robin Woolley wrote:

> My wife, who, to save her from ennui, is currently doing
> an M.Math course at the Open U (M336) confirms my recollection from 30 years
> ago that the convention (then and now) is that 'ab' is 'a operating on b'
> or, if you like as I say sotto voce when I write it down, 'a of b'.

My experience is that both conventions appear in the
mathematical literature.

> In fact, the whole analysis is just as valid the other way round, C.inv(A),
> except that it might be less natural when starting from first principles.

I'm not sure whether you're saying that if you chose the
other multiplication convention, so that you wrote C.inv(A)
instead of inv(A).C while still refering to the same thing;
if so, then of course you are right.  But that's not what
I'd understood Richard's comments to mean.  He said:

> Since your convention is that "ca" means "a first, then c", it might
> be better to define X by
>
>    X(A,C) = C .inv(A)

... which sounds, to me, like Richard is using the same
multiplication convention as you.  And if so, this is a
completely different quantity to your X(A,C).

Let's go back a step or two.  What are we actually trying to
study?  It's the falseness between two leads of a method.
We want a succinct way of deciding whether a lead starting
from lead head A is false against the lead starting with
lead head B.

It's intuitively obvious that if A is false against B, then
AC is false against BC, for any row, C.  (As the rest of
this thread has used the convention that AB means A
operating on B, i.e. B transposed by A, I have stuck with
this convention.)  Multiplication on the right simply
corresponds to relabeling the bells; in the language used in
Brian Price's "The Composition of Peals in Parts", it's a
transfigure rather than a transposition.  Simply relabeling
the bells can't change whether the leads are false against
each other.

Now, your definition X(A,C) = inv(C).A gives us a lead head
that is false against the lead starting from rounds. We can
see this because row A of the lead starting
from rounds can be written A.1 (rounds transposed by A), and
row C of the lead starting X(A,C) can be written C.X(A,C)
(that is, the lead head X(A,C) transposed by C).  As
C.X(A,C) = C.inv(C).A = A, it's clear that the lead head
X(A,C) is false against the lead starting from rounds.

With Richard's definition, X(A,C) = C.inv(A), this is no
longer true.


If we wanted to tackle this in a slightly more rigorous
manner, we'd perhaps start by defining the falseness table
more fundamentally in terms of the set of rows, M, in a
lead of a method begining with rounds.

  F(M) = { f : Mf intersection M  !=  {} }

i.e. the set of lead heads, f, that are false against the
lead starting with rounds.

(I'm using the notation Mf as in my previous emails to mean
each elements of M multiplied by f.  Written more formally,
Mf = { mf : m in M }. )

It's relatively straighforward to get from this definition
to your expression for X(A,C).

Richard