Well, I finished reading this paper by Landauer and Bellman (LB).
Unfortunately, as far as I am concerned, it is perhaps useful only
insofar as it exemplifies ways in which a newcomer to Rosen might
misunderstand him.
LB take as their point of departure the (M,R)-system model:
"We start with the description from [Essays on Life Itself] of a
model of living systems, his (M,R)-systems, since it gives the
clearest description of what Rosen intended to do. Then *we explain
the equational distinction* from [Life Itself], and discuss its
implications. Finally, we describe our example solutions to the
equations from three areas of Mathematics, to show that *equational
distinction* does not work."[p. 60, bold added]
At this point, LB have already erred. LB have mistakenly interpreted the
/relational mappings/ in the (M,R)-system diagram as /equations/ (see
bolded in quote above). They spend p. 61 describing how the "equational
formulation" is wrong. They clearly do not understand the entire concept
of relational models. For example, here is their supposedly damning
conclusion of this discussion:
"Since metabolism and repair change an organism, we should actually
expect the function /f/ that comes out of /p(b)/ to be different
that the one that goes into /b(f)/ and transforms /a/ to /b/, since
the process is actually unfolded in time (this is the "helical"
argument). This modeling approach, however, projects all of time
into a single point." [p. 61]
That is, they argue that the specific /f/ that is acting as the function
of metabolism cannot be the same /f /is being output by the repair
process at a given moment in time. Hence, the logic behind the
model must be faulty. However, relational models are precisely
/atemporal/ representations of relations. (This is why chapter 5 of
/*Life Itself*/ is conspicuously entitled "Entailment Without States:
Relational Biology".) Near the beginning of chapter 5, Rosen clearly states:
"On the formal side, we shall see that the inferential structure
characteristic of relational biology is much richer than, and at the
same time very different from, the formalisms we have considered
heretofore. Our systems are assigned /no states/, /no environments/,
/and there is no recursion/." [LI p. 109, ital. orig.]
This is later reiterated. After describing the basics of relational
models, Rosen contrasts the relational approach with the Newtonian,
state-based approach:
"In the relational approach, on the other hand, the situation is
quite different. As I have developed it so far, there is no time
parameter, no states, no state transition sequences. There are only
components (mappings), and the organizations, the abstract block
diagrams, which can be built from them." [LI p. 134]
Thus, it is entirely erroneous for LB to interpret relational models,
such as the (M,R)-system, as an "equational formulation". They do not
address their argument to a relational model, but instead to what they
think is a set of equations. Therefore, their main argument and their
main point is entirely invalid.
LB demonstrates another lack of understanding of Rosen in sec 2.6 of
their paper after summarizing their main argument, when they say:
"More importantly, this [Rosen's approach] is probably not the right
approach. After all, claims that complexity is non-computable
contradict claims that formal systems are computable, since
computability is only defined for formal systems, and cannot be
proven or even properly defined for non-formal systems. There are
many such confusions in the literature." [p. 65]
Again, by misinterpreting relational models as predicative equations,
LB mistakenly asserts that these models are thus computable (by virtue
of being formal systems). They further suggest that Rosen is claiming
that non-computability is a property of the organism, rather than of the
formal model, and so Rosen must be confused, since it is rightly
nonsensical to speak of computability of non-formal systems (such as
organisms). On the contrary, it is LB who is confused. Rosen is quite
clear that computability criteria apply to the /models/, not to the
organism:
"I call a material system with only computable models a /simple
system/ or /mechanism/. A system that is not simple in this sense, I
call /complex/. A complex system must possess noncomputable models."
[EL p. 325]
In their next section, LB spend 3 pages discussion analytic vs.
synthetic models. They state at the beginning of the section that "we
describe some other difficulties with the Theoretical Biology program as
stated...". [p. 66] It is unclear why LB perform this exercise, since,
in fact, they end up restating the same conclusion that Rosen makes in
ch. 6 of /*Life Itself.*/ Namely, that there are generically far more
analytic models than synthetic models to a system.
As best I can discern, LB seem to provide this argument because they
mistakenly think Rosen is arguing that a "largest model" is the
universal case: that analytic models are always only the inverse of
synthetic models. Nothing could be more wrong. Rosen remarks near the
beginning of his chapter on analytic and synthetic models:
"In a sense, it is the thrust of this entire work that this
hypothesis of analysis = synthesis must be dropped. Above all, it
must be dropped if we are to do biology, and hence a fortiori, it
must be dropped if we are to do physics. By dropping it, we enter a
new realm of system, which I call /complex/, and which in certain
sense needs to have no synthetic models at all. The distinction
between relational and Newtonian models of natural systems will
become crucial here, because as we shall see, the former extend to
the realm of complex systems, while the latter cannot." [LI p. 154]
LB clearly do not realize the consequences of dropping the "analytic =
synthesis" hypothesis (consequences which lead directly to Rosennean
complexity), even though they (apparently unwittingly) agree with Rosen
that such a hypothesis is unwarranted by purely formal considerations.
All in all, I find that this paper by Landauer and Bellman fails to
comprehend the fundamentals of Rosen's relational modeling concepts, the
import of Rosen's discussion of analytic vs. synthetic models, Rosen's
use of computability criteria of models, as well as Rosen's concept of
complexity. Their paper misinterprets these concepts and their arguments
are directed against those misinterpretations. As such, their paper
offers no legitimate argument and thus no damage to Rosen's actual
concepts; but by its gross misstatements, this paper does a disservice
to Rosen's work.
Regards,
Tim