Category theory and Rosen

[glen e. p. ropella]

Chris Landauer writes:
 > there is no consensus of opinion, and what follows is strictly my own - the
 > fact that rosen seems to feel the need to _demonstrate_ that biological
 > systems are not formal means that he does not understand what formal systems
 > are, despite his use of the words

I hate to disagree with you... [grin] But, I think Rosen has a
defendable definition of formal systems: 1) an alphabet, 2) a set of
axioms, and 3) a set of rules for generating expressions. I believe
these are sufficient for generating the minimal set of criterion for
the systems (languages, L) considered by Goedel, which was:

1) A denumerable set E whose elements are called the *expressions* of L.
2) A subset S of E whose elements are called the *sentences* of L.
3) A subset P of S whose elements are called the *provable* sentences of L.
4) A subset R of S whose elements are called the *refutable* sentences of L.
5) A set H of expressions whose elements are called the *predicates* of L.
6) A Goedel numbering function.
7) A set T of sentences whose elements are called the *true* sentences of L.

  --basically taken from "Goedel's INcompleteness Theorems" by 
    Raymond M. Sullivan

And I think his point is not that biological systems are not formal
systems.  I think, rather, that he is trying to develop a formal
system that (to use his technical term) is a model for biological
systems.  And he is trying to do this via his ideas of causal
entailment, which he instantiates using Category theory.

All this, I accept.... if not agree with.  But, it's my suspicion that
his model... his formalization will be flawed because it won't map to
biological systems or Alife.

 > formal systems are by definition separated from external influences -
 > everything you can do in them is within them - biological systems, chemical
 > systems, and any other *real world* system is not so limited - there is
 > leakage from the context all the time (in fact, that is one unpredictable
 > source of variation that is exploited continually by biological systems in
 > evolution)

I'm not sure I understand this.  For the first part, I would disagree 
with the statement that formal systems are, by definition, separated
from external influences.  It seems to be an implication of Goedel's
Incompleteness Theorem is that there necessarily exist undecidable
sentences in any formal system.... in other words, a formal system
requires external influence.  Now before you run off accusing me of
inappropriate extrapolation, let me explain.

Rosen is trying to say that the reason physics (and science, in general)
*can't* answer the question of "Why does phenomenon X occur?" is because
it is based on a formal system, namely, Newtonian mechanics.  Since it
is based on a formal system, it cannot provide resolution to Aristotle's
fourth type of cause, "final cause."  Formalisms satisfy the first three:
material cause -- input, efficient cause -- operation/manipulation, and
formal cause -- proof/algorithm.  The final cause is left to God.

This horse has been completely pummeled in the exploration of scientific
"explanation" or "description."  And it's why there is still a field
of philosophy called "metaphysics."

So, it's not that Rosen misunderstands what a formal system is, he's
trying to find a formal system that plugs the hole of final cause.
At first glance this would seem silly, since it's been proven that 
formal systems can't plug that hole.  But, he's re-discovered an 
extraordinary thing... the concept of the limit.  He knows he can't
devise a formal system that incorporates final cause, so he's going
to define a *ratio* of "unentailed elements" to "entailed elements"
and then he's going to force that ratio *towards* zero.

In so doing, he achieves a kind of approximation to reality that is
remarkably akin to the what Cantor, Liebniz, and Newton did.

 > ok, the diagrams are a useful shorthand, sort of like type declarations, but
 > they don't actually _do_ anything - especially concerns of independence and
 > order are important for real systems, because there are very few "point
 > events"; almost every activity is a process of some kind
 > 
 > of course, as long as you stay within the realm of "abstract block diagrams",
 > and don't confuse the model with the phenomenon being modeled, rosen is
 > perfectly all right; he can compose the diagrams at will - however, he cannot
 > then go and claim correspondence between the composed diagram and the large
 > system in the real world containing the smaller systems that correspond to 
 > the
 > smaller diagrams -

I do have grave worries about his concepts of mapping and his use of
Category theory.  But, you have to remember what we're doing, here, in
order to appreciate what the diagrams are for.  They aren't really
intended be close models of processes.  They're intended to be close
models of relationships.  There's a whole barrel of monkeys left to
move the formal set of relationships into meaningful statements about
processes.  And that's even if we accept his formalism for
relationships (which I don't, yet).

But, all in all, I think this is a good start to providing a formalism
for Alife, even if the fundamental objection I see so far is
defensible, which is that the basis for Rosen's system seems to assume
analyzability (linearity).  After all, if he's describing a linear
formalism for relational biology, we can just about scrap it without
doing any research, eh? [grin] But, even if he is, the concept of
using recursion to bootstrap final causation from material, efficient,
and formal causation is a good one.

glen



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