[r-t] Foundational definitions (was Lead-based methods [was: Poll on consecutive blows in the same position])

[Don Morrison]

On Tue, Dec 30, 2014 at 3:52 AM, Mark Davies <mark at snowtiger.net> wrote:
> We are getting off track. The current discussion is interesting, but we
> should focus on the task in hand. Let us just concentrate on β methods, for
> the time being.

You're going to hate me (well, only if you don't already). There's no
reason most of the conversation can't continue to go forward
principally about β-methods, but as I'm still not certain we've
actually defined them, I'm afraid I'm going to be a pain and continue
trying to understand exactly what β-methods are or are not, which
necessarily also requires understanding a bit about non-β methods. If
we're going to sew a pair of trousers* for an animal, we do need to
know how many legs it has.

My weird concern du jour (which has only come to mind because of this
whole definitional exercise, as I had implicitly assumed it without
thinking about it) is:

Does a method at stage N have to be able to be applicable to any
possible row of stage N? It's another one of those "doh, of course it
does, oh, wait, maybe not, maybe it's just 'cause It Always Has?" kind
of things. Could you have some kind of incredibly weird method that
works for most maximus rows, but doesn't for some of them, presumably
because it depends upon a "rule" that cannot be meaningfully applied
to some rows? Unless someone comes up with a theorem to the contrary,
I think you probably can.

It is pretty natural to make this bizarrity go away for α-methods at
the same time as they must become finite if we do that (since it's
going to involve a description of all N! possible starting rows
anyway), but I think we do want to be sure to be explicit about it.

What other implicit properties of well behaved methods have we
neglected to think about explicitly?

Also, a reminder: if, as seems to be becoming the case, we do want to
adopt this whole α- and β-method scheme, we will eventually need to
come up with proper, descriptive replacements for α and β, that do not
involve implicit value judgements. No rush, it's fine to leave things
as they are for now, let's just be sure not to forget about it.



* I suppose a set of trousers for a quadruped isn't really a "pair",
is it?




-- 
Don Morrison <dfm at ringing.org>
"Science...is not just a matter of making mistakes, but of making
mistakes in public. Making mistakes for all to see, in the hopes
of getting the others to help with the corrections."
                       -- Daniel Dennett, _Darwin's Dangerous Idea_