"And just as formalization in mathematics believed that everything could
be formalized /without loss/, so that all truth could be recaptured in
terms of syntax alone, so particle mechanics came to believe that every
material behavior could be ... reduced to purely syntactical sequences
of configurations in an underlying system of particles.... the form
taken by mechanical description ... has become the universal currency
for describing systems of any kind."
4B. "Chronicles." Rosen says: "I will begin by stepping back a bit, by
supposing that we do not yet have a notion of state at our disposal....
All the self can see is a sequence of percepts, ordered by its
subjective sense of time.... Thus, the result of the self looking at its
ambience is only a tabulation; a list of what is seen, indexed by when
it is seen. Such a list we shall call a /chronicle/."
"But the self is not merely an observer.... So, along with any time
series ... comes the urge to extend it into the future and into the
past, to extrapolate it, to predict and postdict ... we seek to extract
from the structure of the list itself something that will already entail
those entries that are yet to come or those that have come before." But,
Rosen notes, "all it can find this way are properties of the list, and
not in general of what the list represents." Such context-free samples
are doomed to "lose information precisely because they are samples, and
they also add irrelevant information (noise), which pertains to the
sampling process itself." (p70)
4C. "Recursive Chronicles." Showing that "there are an infinite number
of different formulas that will all produce the given chronicle", Rosen
concludes that "a list thus /can never entail a formula/. The reason
being that "in the present case, the independent variable (instants) is
simply a set of labels, unconnected causally or in any other way with
the values of f(n) that they label.... What we do want is ... entailment
between the /values/ f(n), and in fact, we want entailment that does not
involve the independent variable n at all ... what we earlier called
/contagion/."
4D. "Recursion: Some General Features". Rosen looks at a few examples of
recursive chronicles and the mapping, T, and its iterates that generate
them, mentioning groups, trajectories and orbits. Rosen notes that these
functions have many nice properties as long as they are invertible. Then
he concludes: "causality manifests itself /only/ through a sequence of
state transitions ... this is the basic picture that permeates all of
contemporary science. If there is something wrong with the picture, or
especially if there is something missing from it, then the root of the
trouble lies already here."
01/26/97. 4E. "On Taylor's Theorem", 4F. "Recursion and Constraints",
and 4G. "Coping With Nonrecursiveness: Recursion and Constraint in Sets
of Chronicles" covers 12 pages of mathematical technicalities that
basically describe how Taylor's theorem [ f(t0 + h) = f(t0) + hf'(t0) +
(h^2/2!)f"(t0) + .... ] "provides a glimpse at how we can make a
chronicle, which is itself not recursive, act as if it were by embedding
it in a larger set of chronicles." These are the successive velocity,
acceleration, etc. derivatives used in differential equations "to
convert synchronic information ... into diachronic information."
4H. "Newton's Laws". Rosen declares: "As I shall now show, Newton's laws
are at heart nothing but Taylor's theorem, raised now to a universal
principle via tacit but extremely restrictive limitations on encoding
and decoding of events in the ambience."
"It cannot be stressed too strongly that the Newtonian particle is a
/formal object/.... Aside from the parameters with which it is endowed
(e.g., its mass) ... its /only/ attribute at any instant is its position
in space. Thus, there is only one question to ask about it: namely,
where is it? The answers constitute a chronicle, consisting of positions
in space, labeled by corresponding instants of time.... The upshot is
that, although indeed a (differentiable) function entails its own
derivative /as a function/, its /value/ at any instant entails nothing
about the value of the derivatives at that instant."
"So far we have been looking at the single particle that constitutes our
(natural) system. Let us now notice that the same act that singled out
that particle from the ambience also /simultaneously/ specifies the
system environment.... But if the state of an individual particle is at
this point already an infinite object, to try to characterize a 'state'
of this environment is literally unthinkable. Newton did not even
attempt this. Instead, he took a completely different approach, embodied
in the concept of /force/. The environment was to be characterized ...
entirely through its effects on the system. Specifically, we
characterize the environment of a particle through its effect on change
of state."
"Newton chose an ingenious device ... he noted that /at the purely
formal level/, and /only/ at that level, we are free to make the
environment of a Newtonian particle anything we want.... The substance
of Newton's First Law is to specify what a particle does in such an
empty environment. He mandated that such a particle cannot
accelerate.... In a formal environment that is not empty, and is thus
the seat of /forces/ impinging on the particle, the effects of these
forces on the motion of the particle is thus tied to its acceleration.
The way in which force and acceleration are explicitly tied together
constitutes Newton's Second Law, which consists of two separate parts.
In the first part ... Newton mandates that the effect of any environment
forces on a particle, at any instant, are proportional to the particle's
acceleration at that instant ... The second part of Newton's Second Law
says essentially [that] from the particle's point of view, any forces F
it experiences from its environment /are already functions of its own
states/ ... The upshot of Newton's Second Law is to effectively collapse
the state of the particle, which is an infinite set of variables, /down
to only two/ of them, down to position and velocity alone."
"Newton's Laws thus serve to transmute the initial dualism between
system and environment into a new dualism, that between /phase (states
and forces/, or between /states and dynamical laws.... in the Newtonian
picture, a particle is shoved from one state to another through the
external agencies to which it must react. But when we go to systems of
several formal particles, the situation is different. Each particle
forms part of the environments of all the others." "This", Rosen says,
"requires a new ingredient ... Hence, the separate stipulation of, for
example, the Law of Universal Gravitation..." Rosen concludes: "just as
in the single-particle case, the system-environment dualism becomes
transmuted into a dualism between phases and forces. It is a little more
complicated now ... the recursion rules governing change of states or
phases of the system become /relative/ ... additional chronicles must be
stipulated which pertain to the environment."
02/08/97. 4I. "On Entailment in Physics: Cause & Effect". Rosen seeks to
show that "Newtonian mechanics as a formalism manifests a surprisingly
weak inferential structure" that also restricts any causal entailments
that may be derived from it. He examines how Newton's Second Law says
that "the way in which acceleration depends on phase is a measure of the
force impressed on the particles of a system by its environment" and,
thus, "the relations themselves constitute an encoding of environment
and not system. Thus, in the Newtonian picture, systems get states;
environments do not (and cannot); environments rather become identified
with dynamical laws."
"This is a fateful situation ... we have automatically placed beyond the
province of causality anything that does not encode directly into a
state-transition sequence. Such things have become /acausal/, out of the
reach of entailment in the formalism, and hence /in principle
undecodable/ from the formalism." (p102) From this, Rosen declares: "I
shall define a natural system N to be a mechanism if it possesses the
properties I have just articulated: namely, (1) it has a largest model,
consisting of a set of states, and a recursion rule entailing subsequent
state from present state; and (2) every other model of it can be
obtained from the largest one by formal means."
4J. "Quantum Mechanics, Open Systems, and Related Matters", Rosen claims
that "until the advent of quantum ideas, physicists did not in fact
think much about causality." He even quotes Bertrand Russell as stating:
"The reason why physics has ceased to look for causes is that, in fact,
there are no such things." The problem, Rosen says, is that "the
Heisenberg commutation relations said that classical phase could no
longer even be defined at the quantum level, let alone be recursive.
But, as was quickly realized, giving up the notion of phase did not mean
giving up the notion of state. It merely required an encoding of that
notion into a more complicated mathematical or formal object (wave
function) ... governed now by Schrodinger's equation (or its
equivalents) rather than by Newton's Second Law."
The key point for Rosen here is that "the new encoding ... of wave
functions ... could be related only in a statistical way to the old,
classical encoding". Rosen then notes that Newtonian mechanics does not
deal well with open systems, those where there is "turnover" in the
'particles' inhabiting a "distinguished region of space". In open
systems, as in quantum systems, he continues, "the 'state' was now to be
a density of some kind, and the 'velocity', or rate of change of state,
was now specified, not in terms of force, but solely by bookkeeping ...
the difficulty is precisely in assigning to such a situation a notion of
state that is recursive.... In every case, the strategy is then to
regard the 'open system' as an underlying closed system /plus
something/." Rosen concludes this section (p 107) by warning that "it is
precisely in recoding the concept of state that the transition from
classical to quantum mechanics manifests itself."
(to be continued -
Rosen is leading up to a discussion of relational biology in ch.5 and
Analytic and Synthetic Models in ch.6)
Mark Crosby