Stephen has a number of fine questions about Hal's paper
(way *too* many, really) and while I am still working on
what one or two questions I may pose, there is one of
Stephen's questions that perhaps I can answer:
> I am still worried about how a measure can exist over a set, collection,
> class, or whatever of computations! Does not the notion of a measure require
> the existence of a space where each point is an object of the class and the
> measure itself defines the similarity/difference between one object, here a
> computation, and some given other?
Here is what I would guess to be the UDist answer:
The notion of a measure *does* require the existence of a space, and a
space, by usual mathematics conventions does consist of points, as
you imply. But no, I would *not* say that a measure defined on this
space has anything to do with the similarity/difference between
points of the space. (You refer to these as "objects" which seems
like a land mine of confusion, at least to me :-)
I think that what is meant is very similar to the ordinary mathematical
definition of measure which you know. Each point is mapped to a real number
that sort of indicates its weight. (In math, a measure is defined on some
*subsets* of the space, but here only a single point at a time needs to
have a weight.)
So for two computations (in the UDist view), each may be thought of as a
point in the space metaphor, if you want. Suppose that computation X
has greater measure than computation Y. X might be, for example, a
calculation that proves 1,000,000,000,061 is prime. And Y might be
Lee Corbin. As it takes a much, much shorter program to give rise to
X than to Y (I say not with just a trace of pride), the measure of X
is greater than that of Y (unfortunately for me).
> What ontological status does "Computation Space" have?
The space of all computations is a subset of the space of all math
patterns, and we math Platonists regard them as exceedingly real.
P.S. Platonists != UDist-ers != "computationalists" != COMP