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Hi
Pete,
I really like your
phrasing:
[PVG]
"In my terminology, we create
the syntactic structure that defines the relationships among the
semantic referents.".
I have a somewhat
different take on the N-body system problem. It comes mainly from Rosen's
mentions of inertial vs gravitational aspects in chapter 1 and p. 109 of
Essays.
In the
Newtonian view, where fundamental particles are featureless, if we had 3
particles as the 3 referents, being pushed around by forces from the
environment, then syntax alone would be entirely adequate to define
the dynamical relationships among the referents. And not only would it be
syntax, but computable syntax (algorithm) at that. And so the system would be
simple. Even if the particles had certain qualities, such as inertial
mass, which do not depend on anything else about the system, the system
would remain simple. So, just having 3 or more semantic referents is, in my
view, not sufficient to cause a system to be complex. Newtonian mechanics
readily deals with large numbers of particles with inertial
mass.
But once particles
are endowed with the ability to generate forces, such as gravity, then
the relationship between such particles is no longer purely syntactic. Instead,
the system now has, aside from the referents to the bodies,
additional referents in the form of dependencies of each relationship
upon the other relationships in the system. So, one relationship refers to
another relationship which, in turn, refers back to the first relationship.
Voila - classic loops of reference.
In a 2-body
system, there is only one relationship between the two bodies, and hence, no
loop of reference. But for N-body systems, where N>2, the loops of reference,
or impredicativities, cannot be avoided. And so, the systems are
complex. And the system fails to be modelable in a purely predicative
syntactic form, which is required for a model to be algorithmic. As such, these
systems possess noncomputable models.
It also becomes
immediately clear why such systems cannot be solved reductionistically.
Attempting to do so breaks the loops of reference in the very act of
partitioning, say, a 3-body system into a 2-body and a 1-body system. So, the
problems solved for the separate 2- and 1-body cases cannot address the problem
of the loops that are found only in the whole 3-body system.
With regard
to:
[PVG]
Granted, I picked a simple system to make it easy to
illustrate the point. Had I picked a three-body system rather than a two-body
system, the syntax would have been quite a bit more complicated, requiring me
to treat two of the bodies -- say, m1 and m2 -- as a
single, composite "virtual mass" that interacts with m3 , and then
consider m2 and m3 as a composite virtual mass that
interacts with m1, and so on... and then run the calculations
iteratively to generate my results within certain specified ranges of values
for the variables. It's more complicated, but there is still no indeterminacy
in the syntax.
The procedure
described is, of course, an example of a simulation, and is not a
model. This can be easily seen by trying to put the inferential
entailment structure of such a procedure into congruence with the causal
entailment structure of an 3-body system. In the 3-body system, all 3 bodies
move simultaneously - the causal forces apply to each other simultaneously. This
is entirely different than the stepwise grouping of two masses, which then
entails a total force on the third mass, and then running it for the other
two groupings. Such simulations are useful, but they only mimic
behavior of the system. They do not embody the same entailment
structure as a model would. So, lack of precision is a pragmatic issue
as far as successful mimicry of behavior is concerned, but precision
itself is not a criterion for deciding simplicity vs. complexity or
model vs. simulation.
I believe it
would be possible to create a purely syntactic, but necessarily noncomputable
(because of those loops), model of the 3-body relationships (or even
better, for a generalized N-body system) using something along the lines of a
relational model, as exemplified by the (M,R)-system.
Regards,
Tim
Hi John K., Tim, et al:
----snip-----
~As RR (via Tim) correctly
points out, we impute the relations to the qualities we observe.
In my terminology, we create the syntactic
structure that defines the relationships among the semantic
referents. I agree with RR's assertion that the relations (the
"syntactic relationships") are mental constructs. Since the "syntactic
relationships" we create are constructs of the human mind, we can define them
to any degree of precision we wish, and philosophers & theologians are
unabashed in doing just that. In science, however, we're accountable to more
sharply defined constraints; our [syntactic relationships among semantic referents] =
[models] are
subject to empirical corroboration. The requirement for corroboration in the
"material world" is universal in any discipline that purports to be a
"science".
In general, syntax is not the principal epistemological
limitation on the precision (and therefore the functionality) of our models.
The history of science is replete with examples of apparent roadblocks in our
ability to organize percepts/semantic referents into a coherent syntactic
structure that defines the nature of their relationship to each other. In more
familiar terminology, nature frequently stumps us in our attempt to create
theoretical models. Yet, time and time again, we develop ever more ingenious
syntactic statements that sidestep such apparent obstacles -- the
"mathematical tricks" that allow us to bypass the roadblocks -- with the
result that new theoretical models emerge... models that ultimately prove to
be empirically corroborable.
Neither science nor philosophy knows
"why" the capability of the human mind to think mathematically turns out to be
reflected in the syntactic relationships among the semantic referents we
perceive -- or as some would say, "in the structure of the universe" -- but
the fact that the human mind has that capability is indisputable. For the sake
of staying on point, I'd prefer to avoid the philosophical implications of
that congruence in this discussion, except to say that it seems to me to be a
remarkable coincidence. As Paul Davies has observed, it almost seems fishy.
'Nuff said.
What we've learned from even our most successful physical
theories is that the epistemological limitations on the
precision/functionality of our models are principally imposed by semantic
constraints, not syntactic ones. That is, limitations in our knowledge
structures derive from our inability to specify or describe "the things we are
talking about" -- i.e., the semantic referents themselves --
with infinite precision, not necessarily the relationships between those
things.
For example, we have no difficulty with syntactic imprecision
in the following statement:
... the well-known inverse
square relationship for the gravitational force of attraction in a two-body
system. When we make predictions using that statement (in a classical,
non-relativistic context) , those predictions can be robustly corroborated by
observations that fit our predictions to any degree of precision of which our
measuring instruments are capable.
But measurement is a
semantic process, not a syntactic one; that is, measurement is
the process by which we define the quantities represented by the variables in
the inverse square relation. It's the process by which we associate symbols
with their referents, and therein lies the limitation on precision. We cannot
measure the masses or their separation distances to infinite precision. Hence,
it is our inability to specify or define those semantic
referents that imposes the limitations on precision, not the syntactic
structure that defines the relationship between the semantic elements.
The syntactic elements in the inverse square statement above are the
mathematical symbols:
- the "=" sign;
- the scalar multiplication operators "•" between
G, m1, and
m2; and,
- the division operator " / ".
The operations specified by those symbols are precisely
defined with zero uncertainty, hence there is zero non-specificity in those syntactic
elements.
Granted, I picked a simple system to make it easy to
illustrate the point. Had I picked a three-body system rather than a two-body
system, the syntax would have been quite a bit more complicated, requiring me
to treat two of the bodies -- say, m1 and m2 -- as a
single, composite "virtual mass" that interacts with m3 , and then
consider m2 and m3 as a composite virtual mass that
interacts with m1, and so on... and then run the calculations
iteratively to generate my results within certain specified ranges of values
for the variables. It's more complicated, but there is still no indeterminacy
in the syntax.
But is it a fair example, since I've picked a
relatively simple system -- even with the three-body system? Yes, it's fair...
because JJK's statement was unequivocal:
...there is no absolutely
precise syntactic statement of anything (e.g., Rosen's theory) that can be
made. I'm assuming that the statement was
intended to apply universally, irrespective of whether any given "syntactic statement" is being made about a
simple system or a complex system.
What may be surprising is that the
three-body system is NOT a simple system -- at least not in the sense that it
is simulable. We cannot create an algorithmic representation of the system
that is a more concise model of the system, which model enables us to predict
the future states of the system with greater accuracy than simply letting the
system operate and observing those future states in real time.
This is
not a particularly recent revelation. As Poincaré demonstrated a century ago,
the system is non-integrable, a characteristic that Prigogine has since shown
is the basis of the complexity that engenders "life
itself." (His words!) The dynamics of the system are complex in
that they are increasingly unpredictable with increasing ∆t
(system run-time) relative to any given starting time
t0, because of an essentially infinite
sensitivity to the precise values of the initial conditions. How would we
determine those values? By measuring them, and we can't measure them to
infinite precision. So all of the imprecision in our ability to define the
system state at any given future time t is, once again, a result
of a limitation on semantic precision, not on syntactic
precision.
That's not to say that we are able to cough up the same high
degree of syntactic precision that we can attain in simple systems in our
statements about systems that are vastly more complex, even if their
complexity arises only because they comprise a much higher number of elements,
components, sub-systems, etc. There are obviously some very significant
challenges involved in modeling such complex systems, as RR so convincingly
observed. Many of the syntactic "tricks" that have been so successful in the
domain of relatively simple physical systems are utterly powerless to model
the dynamics of even the most primitive self-organizing systems, let alone the
vastly more complex, principally self-referential ones.
It remains a
mystery to me how anyone can deny the overwhelming inapplicability of
context-independent, principally syntactic models as adequate descriptions of
complex systems. I happen to place a great deal of weight on Gödel's
Incompleteness Theorem as an epistemological limitation on the sufficiency of
purely syntactic models to describe all natural systems, but whether or not
one shares my appreciation for the profound implications of Gödel's work is
really quite beside the point. As I've demonstrated with the example above,
even where an absolutely precise syntactic relationship can be established as
a model of system dynamics, there is already a fundamental semantic
limitation in the model, which limitation renders the system complex.
Essentially, all it takes to enable the transition from simple natural
dynamical systems to complex natural dynamical systems is the multiplicity of
system elements (i.e., semantic referents) beyond the number
2.
If all it takes to engender complexity is 3 semantic elements, RR's
conclusion that simple systems are the non-generic ones is
inescapable.
Best regards,
PVG
© 2003 Peter V. Giansante -- All rights reserved.
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