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

[glen e. p. ropella]

Chris Landauer writes:
 > Perhaps I was remembering the conference discussions and not the book -
 > clearly, i will have to read it again

Yeah, I have the reverse problem... I get to read all these
cool ideas in books, but I never make it to any conferences
so I have nobody to discuss things with. [grin]

 > formal systems and models:

It's very true that many are guilty of thinking that the
map is the territory.  But, I don't think Rosen is guilty
of it.  At worst, he's guilty bad analogies.

 > go:del's theorem:
 > 
 > go:del's theorems do not exactly show that you require external influences in
 > a formal system; they show that, because the formal system allows no external
 > influences, you need to be _outside_ the system to prove some true statements
 > (so if you have a goal of proving any true statements, then you are 
 > interested
 > in something which is outside the formal system) - so here i think we are not
 > disagreeing exactly; we just have different interests in the systems

Yeah, I thought you might object to my extrapolation.  But,
I will still hold to it.  The way you word it, "you need to 
be _outside_ the system to prove some true statement" sounds
funky or wrong to me.  I think a better way to say it is that
some true statements are not provable, period, from inside or
outside (because there is no "true" or "false" outside the 
system).

That's why I choose to think that any formal system must be
*augmented* by some outside mechanism in order to justify
the assignment of a truth value to those unprovable statements.
(Note that I use the word "justify" rather than prove.  That's
an attempt to imply that the formal system, wherein "truth"
is defined, is only part of the picture.)

Of course, all of this relies on the law of the excluded middle...
We could just back off and use a tertiary system.  [grin]

 > for some years now, we have been trying to stimulate the mathematical
 > community to come up with some new kinds of mathematics, precisely in this
 > area (among others): mathematics is the most concrete form of reasoning we
 > have, because you have to define everything you use up front - go:del's
 > theorem shows that that approach has limitations, and we are looking for some
 > new ideas - for example, linguistic arguments allow new information to be
 > brought in after the reasoning process starts, and sometimes completely 
 > change
 > the nature of the entities under discussion - one of the things we are trying
 > to get is a combination of the reliability of mathematical reasoning and the
 > expressive power of natural language (yes i know this is real hard 8-))

But, you do realize that Rosen has found an addition to present
mathematics that attempts to get around the limitations Goedel showed
*without* trashing all that we've achieved with formal systems, right?
I understand the desire to find a "new" mathematics not susceptible to
these limitations like the linguistics you mention; but, you should
spend some time examining what Rosen proposes because if we can
establish it in a rigorous way, we don't have to reinvent math to
continue.

After all, part of the beauty of mathematics is it's stability.  A
proof made in geometry will hold in topology.  We don't necessarily
want to gain the expressibility of something like natural language at
the expense of that stability.  Rosen's idea simply adds to
mathematics rather than trashing and reinventing.

 > models of Alife:
 > 
 > i am very intrigued by these questions; i just wish the math were done better
 > in the treatments (one way to do it better is to keep to the story, define 
 > the
 > important qualities, and not try to choose a formal systems and write down
 > formulas; many authors mess up at that point because there aren't enough of
 > the right ideas yet to make the models formal in a useful way)
 > 
 > there are several other models that look quite appropriate for various 
 > aspects
 > of Alife

I don't think I understand what you're implying is so objectionable
about Rosen's math.  I know what my objection is.... but, it
seems to me that he is sticking to the point.  And that point is
an augmentation of formal systems to approximate open systems
entailment.  It's just *one* aspect that needs to be dealt with
before complex systems can be handled in a theoretically coherent
way... But, I don't think Rosen is claiming that it's the whole
enchilada, anyway.  He's just addressing one of the limitations.

 > - the actor model in computing (in which all interaction is via
 > messages, but of course physics has been moving that way for a long time with
 > its particles that carry the various forces), a kind of strange alternative
 > model due to ben goertzel (which is really the same as the actor model, i
 > think, just described differently **oh, and beware the math; it isn't very
 > well done**),

Yuck!  I realize this isn't the main point of what you just
said.... But, I absolutely hate the idea of particles with
attributes.  It is only useful in statistical systems, where
all the particles are approximately the same.

It fails utterly in any other context.  But, this is an old
problem, as well.  Reworded, you can't describe any "object"
with a finite list of attributes.  This is the original problem
that made me think Autopoiesis was so attractive... because
it distinguished between the "organization" of an object and
the "structure" of an object.  The reason the particle view
doesn't work, according to autopoiesis, is because the way in
which an object is viewed is dependent on a cooperative
coupling between the observer and the object.  This coupling
is continuously changeable so it isn't constrained to a finite
list of object attributes, indeed, it would defy such a list.

To go even a little further, I would say that even a system
based on message passing would fail.  I say this because the 
message formulation (syntactics), the message semantics, and
the communication channels are all *mutable* in life systems.
So, even the restriction that an object adhere to a communication
protocol is too much of a restriction.  Now, having said that,
one can probably "simulate" co-evolving syntax, semantics, and
channels via a low enough level of message passing.  For instance,
in Swarm our message passing consists of [id message: args].
If we can perform our couplings by consensus decisions over the
messages, then could manage.  But, this doesn't happen yet.
(I pretend like I'm working on a Swarm app that will do this.
[grin])

 > but the important thing is to define the qualities any
 > appropriate model must have, and only then try to define formal systems that
 > have enough of those properties to be interesting - i think this is a
 > completely doable project that is well suited to this mailing list

In the short term, yes.  But, for the long term, I think we 
want a formalization that will handle all systems of this
type.  Part of the problem is "what type is that?"  But, that's
basically what we're here for, to find the holes in our present
tools that make them ill-suited to modeling alife systems. Then
once we have such a formalism, we should be able to make provable
statements about the entire class of systems.

Sure, simulation is *the* tool; but, it's inadequate.  Scientific
theory is about compression.  The idea that alife models *must* be
simulated in order to be studied runs against the grain of any decent
theorist.  It's as bad as saying, "you won't know whether or not this
number is computable until your turing machine actually stops."
[yuck]  Simulation is a stepping stone that we must have to provide
us with enough data about these systems to abstract to the higher
formalization.

glen
p.s. Speaking of the mailing list... [grin] ... we've gone a bit
astray.  I wonder, now, if this is the appropriate place for this
discussion.  Any opinions besides mine and Chris'?
-- 
{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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