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

[glen e. p. ropella]

Mark P. Line writes:
 > So the limits of theory are the limits of compression?

Basically, yes.  Theoretical constructs are the compression of
observed data.  The iterative flow is as follows: observation, pattern
recognition, theory/hypothesis generation, testing.  And like all
chicken-or-egg problems, it's not a given that observation is really
the starting point.  But, that's a side issue.

Thus, a theory, in itself, is not sufficient for navigation through
the ambience.  It is only a part of the iterative process.  But, the
theory, by itself, does represent the compression of the observed
data.  Now, like all compressions, some way (algorithm) for
uncompressing the data is necessary.  This is where formal systems,
simulations, logic, etc. play a significant role.

But, in essence, the theory is encoded data and the previous
observations should be deducible from the theory in a given context.
If that weren't the case, then the whole process of hypothesis
formation/refutation would not work.  An untested hypothesis in a
given theoretical framework is simply an extrapolation or
interpolation of the theory over data points from which it wasn't
derived.

 > I'll go along with that to the extent that we can assume, in a
 > particular instance, that a (useful) higher formalization might exist.
 > In all other cases (and in the former cases, pending the necessary
 > epiphany), I'm happy to just crank the simulation.

Agreed!  I happen to be a fence straddler with respect to the
tug-of-war between theory and practice.  So, sometimes I wear my "i
just work here" hat and sometimes I wear my "i'm on a mission from
God" hat.  [grin]

David Sumpter writes:
 > I think formal systems must have an external influence since they were made 
 > by 
 > `us' with our limited view of the world (i.e. I'm not a Platoist - I'm not 
 > sure 
 > why if God were to give us one idea it would be mathematics!). However, this 
 > doesn't mean that Godel's therorems find the inacurracies in what we made 
 > up. It 
 > simply points out another feature of the thing we created. As must have been 
 > said through a hail of controversy over and over again - Godels theorem is 
 > simply a liar's paradox.

Hmmmm.  "Simply a liar's paradox?"  I don't think so.  (And believe
me, I realize that I'm disagreeing not only with David, but with some
of the greatest thinkers in the world.  But should that stop me?  Heh,
heh, what do you think?)  The liar's paradox is basically presented by
the unparsable nature of the statement, "This statement is false."
What this provides is an English statement to which a truth value
cannot be deduced.  I.e. I can't understand this sentence by saying
things like, "If the statement is true, then it is false," or "If the
statement is false, then it is true."  But, as a statement in the
English language, the sentence is *easily* understood as "The person
presenting the sentence is casting doubt on his own credibility."
It's simply pointing out that human language is not logic.

Now, you might respond to what I just said by saying that one can
present the "liar's paradox" in logic.  But, when this is attempted,
you have to muck up your words so badly that you have to resort to
mathematics in order to clearly state what you mean.  What I'm saying
here is that the liar's paradox is an *analogy* to undecidability in
formal systems.  It's useful in explaining what Goedel's theorem means
to logic; but, it really is wrong to say that the incompleteness
theorem is simply a liar's paradox.  (However, it *could* be
appropriate to say that the liar's paradox is an example of the
incompleteness theorem.... But, I would object to that, as well.)

 > We created formal systems in an attempt to describe the world. They wern't 
 > made 
 > for answering the question of `what does it all mean?'. Glen seems to be 
 > suggesting Rosen is trying to use them to answer that question, all be it by 
 > reducing doubt to zero.

This is also wrong, but much wronger than the earlier comment.  I
would never claim that Rosen is trying to attach meaning to formal
systems.  Rosen states quite clearly, over and over again, that a
model is a *relationship* between a formal system and a real system.
There's no way in Hell that I could miss such an abject
statement. [grin]

What I *am* saying, however, is that Rosen has identified an aspect of
formal systems that fails to model real systems closely enough to make
it useful.... And that is that they cannot manage "final cause."
Hence, Rosen goes about augmenting formal systems with this idea of
recursivity and the ratio of unentailed elements to entailed elements.
He means to *approach* finality by successive iterations.

This does not mean, in any way shape or form, that he's trying to
establish God as an infinite iteration of entailment.  [grin] That's
patently silly.

 > I've always thought this is impossible for a horribly 
 > naive philosophical reason: I can never be sure that Glen exists or anything 
 > exists AND it is also impossible to attach a probability his existance. 
 > Meeting 
 > didn't make him any more likely to exist as neither do his long emails 
 > [grin]. 
 > But I have to get on with my life looking at the cause and effect of Glen's 
 > life 
 > and I may want to use a formal system to analyse this. None of this removes 
 > the 
 > fact that there is always something I may never know AND I will never be 
 > able to 
 > assess the probability of my being wrong.

What you're talking about, here, *is* relevant to modeling, just maybe
not relevant to the use of formal systems in modeling.  (Note that
I've slipped out of Rosen's technical meaning of the word "model.")
Modeling is inherently dependent of our *opinion* or understanding of
what's going on OUT THERE in the real world.  Rosen establishes
himself as either a realist or as giving in to realism by positing his
tech defn of the word "model" as a relationship between a formal
system of logic and a *real system.* But, even when one is a realist
at bottom, the admission that interpretation of sensory data and
control of the feedback between our effectors and our sensors will be
present.

This is where autopoiesis and Husserlian phenomenology begin to play
critical roles.

 > I'm sorry if I brought what was meant to be a scientific discussion down to 
 > an 
 > `up in the skys' thing but it seemed to be going that way.
 > 
 > I'm probably (though without an assessed degree) wrong,

[grin] Well, I certainly didn't see that particular trajectory coming
about.... But, then again, I'm notorious for running with blinders on.
I can start out talking about breakfast cereal and end up talking
about semiotics pretty easily.  I just worry a little about the other
members' tolerance.  [sigh] Oh well.

Paul Johnson writes:
 > I'm glad to read this and i've got Rosen on my list of things to learn
 > more about.

I sincerely recommend him.  There are seemingly some flaws ... but,
then again, show me an unflawed presentation of an idea, and I'd
probably burst!

==============
With the next bit, I couldn't figure out where to cut it so that it
came up peanuts... So, I included the whole enchilada.
==============

 > >From Glen's message:
 > 
 > > 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.
 > 
 > I've got to say that your comment about simulation misses the point, at
 > least as far as I can see in political science.  We can formalize models,
 > but there are no deductions because of their complexity. You can write
 > some utility function for each of N people, you can specify how much
 > information they have, how their behaviors affect each other.  The
 > question is whether there are any meaningful deductions--comparative
 > statics or dynamics--and the model does not beget such generalizations
 > without imagining its performance in simulation, under a broad number of
 > initial conditions, for example. 
 > 
 > Hence, the simulation is a "hypothesis generating" experience, where our
 > beliefs about the effects of parameters on variables can be investigated. 
 > 
 > This is especially important where we want to understand the relative
 > importance of individual traits and system-level structures (called
 > political insitutions).  In data, the effects of culture and
 > institutions are confounded.  In a simulation, you can fix the
 > institutional setting or the cultural setting to "mix and match" to find
 > out about various effects.  Since they won't let me redesign the federal
 > government every few years, simulation is the only real way to theorize
 > about the effect of changes in constitutions/procedures/rules.
 > 
 > Paul Johnson
Roger M. Burkhart writes:
 > Well, let's get our yucks where we can.  You seem to be equating
 > formalization with the compression of theory.  Formalization just
 > defines a new system that we can talk about with some control over what
 > went it.  Formalization implies nothing about analyzability; it's just
 > the way that we define a (more-or-less) deterministic machine so that
 > we have some artifact of our own to study as opposed to any we might
 > never have understood in the first place.  The formalized system may
 > still have no wedge of analyzability that's any better than simulation.
 > If the system is complex enough, we may still want to just watch it for
 > a while before having any clue of the planes where we could try to pry
 > it apart, assuming we're so disposed.  Paul Johnson also raises cases
 > (e.g., from political science) in which no system other than the
 > artificial, formalized one may be available.

Paul and Roger,

You seem to be echoing me precisely.  I know it may not seem that way,
[grin] but you are.  What I intended by my remark about simulation is
that the purpose of it is to provide *data* on systems that don't seem
analyzable (regardless of whether they *are* or are not analyzable).

A simulation should provide us with data about a system, be it a real
one that we're modeling or a toy one.  With this data, we *might* be
able to step back and find patterns with which to generate hypotheses.
Then we take these hypotheses and test them.  After they're tested,
they take on a truth value, false, possibly true, or undecided.  At
the very least, a collection of affirmed (notprovenfalse) hypotheses
about a system *is* a theory.

That collection of almost-true hypotheses (theorems) comprises a
compression of the system (along with the initial conditions used and
the protocol for the experiments performed).  I.e. one no longer has
to state the entire system in order to make a true statement....

E.g. I no longer have to state, "Swarm. Heatbug. HeatbugModelSwarm.
HeatbugObservrSwarm. Default inputs. Therefore, clustering
phenomenon."  I can now state "There can exist a clustering phenomenon
as a result of locally controlled agents in an interactive diffusive
space."  The first statement is *huge* ... greater than 20,000 lines
of objective C code and a process of execution steps.  But the latter
is just a sentence.  Granted, as with *all* theory, it uses jargonal
words that must be understood in order to decompress it into either
the same workable model or some new hypothesis.  But, the latter
statement is a compressed version of the former.  It is a theorem.

Now, as to Roger's comment about my seeming equation between
formalization and theory... I understand how one might think that I'm
doing this.  But, I'm not.  A formalization is, basically, a framework
that will apply to a *class* of systems.  Now, this class of systems
can contain just the empty set, one system, a finite number of
systems, or a transfinite or infinite number of systems.

In the case where the class consists of just the empty set, who cares?
[grin] In the case where it consists of just one system, then the
formalism is useful for transmitting statements and ideas about the
system, and that's about all it's useful for.  But, in the case where
the class consists of a bunch of real systems, not only is the
formalism useful for transmitting ideas about those systems, but it
can be useful for comparing systems and making statements about *all*
or *some* of the systems in the class.

When one can begin making statements about subsets of the class (or
the whole set), then those statements comprise a theory.

And this theory doesn't require the systems to have the property of
"analyzability."  All they need to have is some common properties
about which we can make statements.

Now, to move on, I would again posit that in order to develop a theory
of Alife, we need a formalization of Alife systems.... i.e. some
formal system that is common to all or, at least, large subsets of the
set of systems we call Alife systems.  The existence of this formalism
(and our access to that formalism) are what is required for us to make
useful and meaningful statements about Alife.  It's not that the
formalism *is* the theory, just that the formalism is necessary (and
not sufficient) for the theory.

I realize that some people would argue about the way I'm using the
word "theory," here.  It could be argued that "theory" is developed
from the specific to the general.  And so, statements about heatbugs
or AnasaziWorld that don't apply to any other system, can comprise
system-specific theory.  E.g. Freudian theory was a theory of mind and
nobody demanded that he also develop his theory to apply to frogs as
well as humans before they started calling it a theory.  But, most
people will admit that many of the theories of mind we have today are
*better* than Freud's because they apply to more systems than Freud's
did.  They're more useful because they apply in more cases.  And I
would even go so far as to posit that they are based on more
abstracted formalisms.  I wouldn't really criticize Freud for
developing his theory, though.  I would just suggest that looking at
the general case is a useful thing to do and can sometimes help avoid
years of chasing a red herring.

 > I agree that we need better formalizations, including "open," expandable
 > ones that give a better chance of modeling life, thought, and complex
 > dynamics, but the purpose is not to make them any more generally
 > analyzable.  A likely outcome is to make them only more contextual and
 > bound by culture and history, and so perhaps more ultimately interesting
 > than those that don't even resist a simple compression.

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])

glen
p.s. Sorry for the length!  Feel free to tell me to "shut up."
Oblique comments about "length" aren't quite good enough.  I'm from
Texas...[grin] meaning you have to hit me with a ton o' bricks to get
your point across.
-- 
{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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