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

[glen e. p. ropella]

Barry McMullin writes:
 > Me too - we may need a break from this...

Well, I took a break over the weekend and I'm ready to gingerly 
step back in.  Besides, I often find that I need discussions like
this to keep the rest of my life from running me over like a large
Mack truck on a long Texas highway.  [budumph]

 > > Invariant composition necessarily implies an invariant (or at least
 > > finite possibilities of) organization at *some* level.  
 > [...]
 > > For example, we can put together a system of particles in several
 > > different ways without varying the particles themselves, which gives
 > > us variant organization and invariant composition.  But, the
 > > organization(s) used to plug those particles together is not infinite. 
 > 
 > As I say - lost.  I don't know what force you want to apply to 
 > "infinite" here, but we could easily have an invariant composition
 > of an *infinity* of elements, which would presumably (?) yield an
 > infinity of organisations; even if the number of elements are finite,
 > any continuous valued "relationship" would yield (another kind of)
 > infinity of potential organisations.  We're surely taking across
 > each other in some way here, but I'm not sure there is any
 > substantial point at issue...

Yes, I'm going to try to bring it back together.  I think there is
a substantial point, here.  Whether or not that point is critically
related to our discussion is another matter. [grin]

The basic categories you just outlined are:
 I. infinity of elements
    a. finite relationships between elements
    b. infinity of relationships between elements
II. finite # elements
    a. finite relationships
    b. infinity of relationships

But, there's a third category where there are *no* elements at all.
I.e. the composition does not bottom out at atoms or the particle
zoo.  In this framework, the composition is really this imagined 
thing that we come up with to accomodate the organization.

It's akin to talking about the difference between the set of integers
and the set of reals.  Between any two real (or rational, even) numbers,
there is an infinite set of numbers.  But between any two integers 
there is only a finite set of integers.  In this sense, I would say
that the "composition" of the reals is arbitrary, whereas the "composition"
of the integers is not.  My reason for saying this is because when you
think about things like the "composition of the reals," no matter how
silly the idea might first seem, you can end up in interminably long
arguments about the concept of a "point."  These problems do not arise
when talking about what the integers consist of because it's obvious
that one can use apples and oranges and there is no need for silly 
things like "points."

When considering sets of arbitrary composition, no matter how you 
cut it, it comes up goo.

 > > But, you are arguing for *variable* composition where I'm arguing for
 > > *arbitrary* composition.... I.e. my case includes your case because
 > > not only can the type and amount of the composite pieces of my goo
 > > change, but the fact that they are pieces at all can change (e.g. a
 > > pi-meson can decay into a muon and another pion at any time).
 > 
 > In my mind that is certainly an interesting further wrinkle.  But my
 > point (there is one buried there somewhere) is that even with
 > "variable" composition, we can already point
 > at this funny class of "systems" ("objects", whatever) like gliders
 > that are not "dynamical systems", and that I (for one) have no idea of 
 > "appropriate" formalisations for.

OK, on to more important things.  I agree that there don't seem to 
exist "appropriate" formalizations for this funny class of "systems."
But, if you ensure that the larger context always accompanies the 
discussion, I think we can find appropriate formalizations.  I.e. 
with the GoL, when we consider the *whole* system that I listed 
before, including the human, the computer, etc, then I think we have
a chance of talking about what a "glider" is.

If we exclude any part of that entire system, including the final
cause to which Rosen provides a mechanism, then there will be an
incompleteness to the formalization.

In other words, you seem to be asking if we can find a formalism
for *just* the gliders and the eaters, etc, without having to 
back off and describe the entire system.  I don't think that's
possible.  Again, what we need for an Alife formalism will have
to be explanatorily pervasive all the way from the emergent Swarm
down to the last transistor.  Of course, we can do some scaffolding
with respect to languages and automated theorem provers over the 
syntax of programming languages.  But, that level is being worked
on already.

What we need to work out is the specification of Alife in the context
that has already been set by solid state physics, Goedel, and the
myriad of works being done in computer science.  To trash that context
would be a mortal sin.  Whatever is lacking in that context is what we
should focus on.  And I happen to think that one of the things lacking
is finality... or the purpose for the glider's existence.  This is
critical if we ever intend to develop a generic method for the
"design" of agents to perform specific functions.

 > >  > Yup, that's where I'm at (albeit wary of a creeping inductivism)...
 > > 
 > > I don't think there's really anything "creeping" about this.  It is at
 > > least analogous to induction, if not synonymous with it.  But, if we
 > > don't allow induction, then we're in trouble.  What we need to avoid
 > > is inappropriate generalization.
 > 
 > OK, coming swiftly out of the closet, I point out that I side
 > strongly with Karl Popper on the issue of induction (briefly, 
 > "logical" induction simply doesn't exist; "psychological" induction 
 > does, is a fascinating subject, but is outside the scope of the 
 > current conversation). I don't mention this in order to start an
 > induction thread (though I sometimes enjoy philosophy too), but
 > only to explain my unconsciously cryptic allusion to "creeping 
 > inductivism".  Unbelievable as it sounds, I sometimes forget 
 > that, in rejecting induction, I'm currently in a very very tiny 
 > minority  <shrug>.  For what it's worth, I agree that not allowing
 > induction means "we're in trouble" - I just don't think that 
 > particular trouble is avoidable, so we have to deal with it.  But that
 > is going way off topic (though I'll happily continue - out of band -
 > if anyone wants).

Allow me to try to outline "logical induction" so that we can verify
that we're speaking the same language.  Logical induction is the
process by which an inference is derived from some assumed set of
statements to apply to a set of statements containing the original
assumed statements.  Popper claims that "logical induction" is invalid
where the assumed set of statements are considered "data" observed in
the environment.  He claims this because "data" doesn't exist.
Rather, "data" is an artifact our minds use to reference the end
product of a long coupling process between our sensory/motor/neural
system and the outside world.

How's that?

If that is adequate, then let me go on and state that it's irrelevant.
[grin]  Since we're dealing with the realm of formal systems, it is
safe to assume that logical induction works.  I'm not claiming that 
it "exists" (I am actually a "conceptualist" and I lean toward 
"intuitionism."  That puts me far, far, from the "platonists.") or 
that *we* do it in our digestion of the ambience.

But, when I admit my data as true axioms, the transduction from 
the ambience into my formal system is complete and I no longer 
need worry about Popper's objection.

And I'd also like to point out that the induction that occurs in
processes like "mathematical induction" are fundamentally different
from the induction (if we may call it that [grin]) that occurs between
the ambience and our mental models of the world.  Mathematical
induction is a property of the iterative operation, "successor."  It
is well-defined and it *is* iterative.  I haven't satisfied myself
that Rosen's scheme is well-defined; but, I'm positive that it is
iterative and is unrelated to "worlding" or the transduction of the
ambience into models of the world.  He sticks, exclusively, to the
world of modeling.  He only goes outside that world when giving our
reasons, as humans, for partaking in the exercise of modeling.

Now that I've said all that... I've never read any of Popper's 
works.  I have "The Self And It's Brain", which I intend to read
...., well, soon.  But, I could easily be wrong about any of this.
I would be worried if someone stated that we couldn't continue 
trying to decide if Alife systems are formalizable without taking
the necessary time out to study Popper's idea to it's fullest.
But, I could accept that if necessary.

 > > And if the human didn't exist, then the computer wouldn't exist and
 > > the visualization mechanism wouldn't exist and, therefore, the glider
 > > wouldn't exist.  However, if the human exists at time t0, the computer
 > > at time t1, the visualization mechanism at time t2, and the CA at time
 > > t3, then the glider would certainly exist at time t4.  This is an
 > > imprecise way of stating that the system is not really time-dependent.
 > > But it is causally dependent.
 > 
 > So you grant that the gliders *will* still exist at t4, even if, 
 > by then, the humans and the visualization mechanism have been 
 > trashed?  Am I understanding correctly?

No.  The existence of the glider depends on the visualization 
mechanism at the same level of causality as the CA.  I.e. the 
glider is directly dependent on the CA and the VisMec in the diagram
I drew: 
             +------------------------+
             v                        |
         +--------+                 +-----+
         |Computer|----+ +----------|Human|<-------+ ?
         +--------+    | |          +-----+
             |         | |            |
             | +-------|-|------------+
             | |       | |
             v v       v v
          +------+   +----+
          |VisMec|   | CA |
          +------+   +----+
              |        |
              |    +---+
              |    |
              v    v
            +--------+
            | Glider |
            +--------+

The human, however, has a kind of latent causality that doesn't
require continual synchronization.  And, so, the glider would
exist if the human were trashed, but not the computer, the VisMec,
or the CA.  Unfortunately, I don't think these diagrams are 
powerful enough to express concurrency and dependency richly 
enough to show these higher order relationships.

 > [Incidentally: the "would certainly exist" seems too strong: many
 > GoL configurations do *not* yield any gliders...]

True, but I'm assuming the "interesting" case.

 > > This is fundamentally different from the "God is in the Quad" problem
 > > because the glider is artificial/synthetic/man-made.
 > 
 > Ya got me: what is the "God is in the Quad" problem?

Here's one of the references.  I have no clue about the truth of
where the limericks come from.  I read it in a book, but have since
forgotten which one.  Thank god for the net, eh?

----------------------------------------------------------------------
From:   address@hidden (Carol Keene)
Sender: address@hidden
Reply-to:       address@hidden
To:     address@hidden (Members of the list)
Date: 95-05-31 10:56:02 EDT

The reply to Knox's limerick was written by Leslie Paul, according to 
James Collins in his _God in Modern Philosophy_.  Collins cites Paul's 
_The English Philosophers_ as his source.  The version of the limerick 
which Collins quotes from Paul is as follows: 

   There was a young man who said "God
   Must think it exceedingly odd
   If he finds that this tree
   Continues to be
   When there's no one about in the Quad."

Dear Sir,
   Your astonishment's odd:
   I am always about in the Quad
   And that's why the tree
   Will continue to be, 
   Since observed by
       Yours faithfully,
                        GOD.


Carol A. Keene
Southern Illinois University at Edwardsville
-----------------------------------------------------------------------

 > Cheers to that: but if others on this list are getting tired of it,
 > please feel free to tell us (preferably by private email, not via 
 > the list).

Agreed!  I'll gladly quit if it gets annoying to anyone.
I also intend to get back to Roger's and David's seemingly short-lived
discussion on formalizing models.  But, I have limited time.

 > Thanks for sticking with it Glen,

[heh] Like I could stop even if I wanted to.  [grin]  It's you, Roger,
Chris, Mark, Paul, David, and James, I should be thanking.

glen
-- 
{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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