Re: Category Theory and Rosen - some clarifications (i hope 8-))

[glen e. p. ropella]

Roger M. Burkhart writes:
 > > Agreed.  I'm not suggesting that the purpose of formalization of Alife
 > > is to make Alife more analyzable.  But, I *do* want to be able to make
 > > statements about Alife systems.  Not only that, I want to be able to
 > > state *theorems* of Alife systems.  (Namely, theorems of
 > > controllability and observability. [grin])
 > 
 > Does your [grin] mean that you don't really expect these systems to submit
 > to your conditions of controllability and observability?

Well, no, actually.  The "tone" modifier in that paragraph was
intended to refer back to a discussion we had on this list awhile back
about whether or not an Alife system could be designed with
prescriptive goals in mind rather than designed in an experimental
wait-and-see fashion.  But, that's another discussion.

 > I don't see that
 > you can have any such expectation until you first narrow things down to
 > something more specific than "Alife systems."

This is an excellent point.  First off, "Alife systems" is not very
well defined.  I realize that we can get several peoples' views on
what the underlying properties of Alife (living?) systems are and
practically none of them will be comensurate.  And that can lead to an
opinion like yours where we'd reasonably have to pick some sub-domain
of Alife to study in order to make useful statements.  And this is
exactly what I think Rosen has done.  In fact, I can't recall now if
he even mentions Alife...

Anyway, the class of systems he's picked might be summed up with some
phrase like "open systems" or "relational systems," where the
fundamental criterion for membership is that a system have many
causally interrelated pieces.

When *I* talk about Alife, however, I usually intend to focus simply
on "iterated systems."  This is, I think, a little more broad than
what Rosen would consider.  The fundamental criterion for membership
in this set is that a system must be iterated in order to make useful
statements about it.  I.e. we have a big bag of "stuff" (could be
particles, could be patterns of interaction, etc) whose behavior we
can't predict; so we iterate it either forward in time or via some
other successive operator.  Then we take measurements on it and try to
extract information from it.

The reason that finding a formalism for Alife is difficult is because
the formalism relies on the set of fundamental criteria chosen.  And
the reason I'm excited about Rosen's idea (not necessarily his method
of implementation) is because it speaks directly to "iterated
systems."  And that gives me hope that an underlying formalism can be
found for *all* of "iterated systems" rather than just open or
relational systems.

In other words, yes, you're right that "Alife systems" is too broad to
work with.  But, that's not because it really is too broad to
formalize; it's because it's definition is murky and variant.

 > The class of systems we're talking about is, literally and figuratively,
 > everything under the sun -- any sufficiently rich, self-organizing system
 > that is driven thermodynamically.  These systems can generate novelty
 > without limit.  For the totally general, unrestricted class of "Alife
 > systems" you can't rule out the ability of the system to outsmart or
 > confound any theorem you attempt to establish.
 > 
 > You who live out there in coyote-land ought to know about tricksters.
 > (For non-North Americans, Coyote is the trickster god of native American
 > mythology, and they're also the feral canines that yip and yelp during the
 > night around Santa Fe, especially in the foothills around the Institute.)
 > If a trickster happens to end up inside your system, none of the blunt
 > instruments at your disposal will be enough to pin it down.
 > 
 > To fast-forward the image beyond metaphor, Goedel showed that a trickster
 > lives in most any constructive system you care to pick.  There are
 > computability bounds on how quickly a system might calculate or develop a
 > response, but beyond that I question the feasibility of theorems on an
 > utterly unrestricted class of evolving, adaptive systems.  The first thing
 > you've got to do before you even start theorizing is to somehow be more
 > specific about the kind of system you've got.  (Maybe the No Free Lunch
 > theorem is enough to rescue us from General Systems Theory.)  We must
 > observe and theorize about specific classes of systems that we either
 > specify and build ourselves (literally as simulations or abstractly as
 > mathematics), or in which we actually live.

Well, I'm going to have to disagree with the idea that we're talking
about *everything* under the sun. [grin] There are some systems that
are subject to algorithmic compression (i.e. we can predict what they
will do before/without iterating them).  I would even submit that
there are some "nonlinear" systems that are predictable in this sense.
But, of course, the number of systems where this is true is
pathetically small compared to the others.  So, in the limit, we are
talking about everything under the sun.

Good metaphor, by the way.

As for using what Goedel showed us in a more generic context, that
paranoia is a useful thing even in mathematics [grin] (trust no one,
not even your old friend logic), I would have to say that we can only
fail.  And it is hoped that this failure will be instructive.

What I want to do and suspect needs to be done, is to find statements
that can be made about iterated systems.  We have some useful
statements about some specific classes of iterated systems already,
from the successor/predecessor ideas of basic number theory, to
Taylor's Theorem, to iterated function systems.  In all these special
cases, though, the operator is well-defined.  In
successor/predecessor, it's addition.  In Taylor's theorem, it's the
derivative, which is based on the concept of the limit.  In IFSes,
it's the contraction mapping, which is based on the topological
concept of distance.

So each useful statement about where an iterative application of an
operator might take us, is very peculiar to the system being examined.
And *all* are based on the specific properties of the operater being
used for iteration.  Fine.  But, somehow and somewhere, at the bottom
of all these special cases where iteration is the key element that
allows us to move forward, it is my guess that we can find general
properties for what iteration means and why it works.

OK, look.  I'm not going anywhere with this, really, and so I should
stop.  But, let me leave you with the question of, "Why is iteration
so fundamental to every mathematical 'truth' and every powerful
operational solution to a problem we find in the world
(e.g. mathematical induction, recursivity/composability in computer
science, Frederick Taylor and Henry Ford in industry, repetition in
skill-based tasks, etc.)?"

With that in mind, take one more look at what Rosen outlined in trying
to *approach* complete entailment in formal systems through causal
iteration.  My intuition tells me that this idea could give
constructivist mathematics a whole new impetus.

glen
p.s. Just in case you're wondering.... I feel like one of those
lunatics on the steps of the courthouse that ramble on and on for days
while the pedestrians walk by and stare.  [grin] Or, maybe, it's more
like being a country singer, singing about how my wife left me, my dog
died, and my truck dun give out, but singing about it in a fern bar in
NYC.
-- 
{glen e. p. ropella <address@hidden> |  Send lawyers, guns, and money!  }
{Hive Drone, SFI Swarm Project         |            Hail Eris!            }
{http://www.trail.com/~gepr/home.html  |               =><=               }


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