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Hi
JohnK,
I was thinking
along a somewhat different line than you, regarding my blunder. But your point
is well taken. I realized that I was committing a fallacious bit of logic
as follows:
If a subset of system X can model system Y, then X is a model
of Y.
The conclusion is not a valid one.
Regarding
simple/Turing-computable/mechanism/etc....
"I define
a system to be simple if all its models are computable, or simulable."
[EL 303]
"We shall
say that a natural system N is a mechanism if and only if all of its
models are simulable." [LI 203]
So, by these two
statements, simple systems could be either formal or natural systems
meeting certain criteria, whereas mechanisms refers specifically
to natural systems meeting the same criteria. However, at several
points [LI 186, EL 292, 303, 325, etc.] Rosen also indicates that
simple system is a synonym for mechanism, which would seem to
imply that simple systems refer only to natural
systems.
"A system is called complex if it has a nonsimulable
model." [EL 306]
As for complex
systems, he seems to make no restriction to it being applicable to either
only formal or only natural systems. Rosen refers to Number Theory as a
purely formal system that is complex [EL 293], and also, of course, to organisms
as being complex [EL 269, LI 280]. So, complex seems to be able to be
applied to either system type.
The conditional
criteria for simple, mechanism, and complex is a
condition on the models of the system, not on the system itself. Since
formal systems as well as natural systems can have models that meet the
respective criteria, the condition on the models does not, by itself, limit
us to either only formal or only natural systems as being able to be categorized
as simple or complex. Mechanism, however, always seems, in his writing, to
refer to a natural system only.
"We shall
further say that a natural system N is a machine if and only if it is a
mechanism, such that at least one of its models is a mathematical machine."
[LI 203]
So, a
machine is a mechanism with an additional restriction. This
restriction can be alternately worded as: the mechanism can
perform the task of simulation [LI 185]. Simulation, in turn,
involves the concepts of algorithm, program, hardware and software; it
roughly means that an entailment structure of something else (call it the
"original system") can be fed as software input into the simulator (which
utilizes its own entailment structure as defined by its
own "hardware"). In other words, the entailment structure of the original
system is not put into correspondence with the entailment structure of the
simulator (i.e., the "hardware"), as would happen in a Modeling Relation;
instead the entailment structure (material, formal and efficient cause) of the
original system is converted entirely into material and formal cause for the
simulator hardware.
Since machines are
defined as a subset of mechanisms, then machine therefore seems to
refer to certain kinds of natural systems. However, just to note, Rosen
also refers in passing to machines as being systems in either
formal or material form. [LI 185, 194]
So, from all
this, my view is:
1) to consider
simple and complex to both be able to refer to either formal
or natural systems whose models meet the respective criteria.
2) to consider
mechanisms as being only those simple systems which are also natural
systems.
3) to consider
machines as being only those simple systems which are also natural
systems and which further meet the criteria of having at least one model that is
already a mathematical machine (i.e., it is capable of performing
simulation).
Simulable, or Turing-computable, refers to
the condition on the models, not on the original system. (Based on
Natural Law, we do not "know" systems in the material world directly,
instead we comprehend them via making models of them; therefore, any
categorization of what we have comprehended can only be done so by
categorizing our resulting models. [LI 203]) It further implies that the models
in question are formal system models, since it is only of formal systems that we
can sensibly ask if they are Turing-computable or not (that is, we can't
sensibly ask if a tree or a rock is Turing-computable; we can only ask this
of formal models).
[A copy of the
original 1936 Turing paper, defining a Turing machine and "computable", can be
found here:
I don't pretend to
have studied too much of the formal argument in it.]
Turing-computability, as a condition, imparts at least
two major restrictions on the formal models meeting that condition. First, the
models must be entirely syntactic since a Turing machine is only a
symbol manipulator - it has no capacity for handling semantic
referents. Second, Turing-computability imposes limits on the type
of entailment structures that the model can possess. In short, the formal models
must essentially be able to be codified into a program for the
Turing-machine.
The nature of
these restrictions are such that there are close parallels between: 1) the
state-based nature of the simulator and the state-based nature of Newtonian
mechanics, 2) the recursive nature of algorithm in the simulator and
recursive nature of Newtonian mechanics, 3) the requirement of pure syntax in
the simulator and the requirement of featureless particles in Newtonian
mechanics, 4) the "largest model" being decomposable into (and re-assembleable
from) atomic "smallest models" and the reductionism inherent in Newtonian
mechanics. The upshot of these kinds of parallels (I think there are more but
thats all that comes to mind) is that the range of systems that are describable
in Newtonian mechanics formalism are essentially the same as those systems whose
models are Turing-computable. This makes Turing-computability an eminently ideal
condition to place on models for the purpose of distinguishing simple systems
from complex systems. Insofar as these kinds of parallels extend to
state-based relativistic mechanics and quantum mechanics, those systems of
mechanics are likewise limited to describing mechanisms. Further,
insofar as any known computing machine can be represented by a
universal Turing-machine means that the criteria of
Turing-computability has the broader implication of telling us
which systems (via their models) are, in any real sense, computable, and which
are not.
Regards,
Tim
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