From: Crosby_M (Crosby_M@bls.gov)
Date: Thu Apr 03 1997 - 12:03:02 MST
03/28/97. Ch.6 of Robert Rosen's _Life Itself_ discusses 'Analytic and
Synthetic Models': "Modeling ... is the art of bringing entailment
structures into congruence.... the essence of art is that, at root, it
rests on the unentailed, on the intuitive leap. I have stressed
repeatedly that the encodings and decodings on which modeling relations
depend are themselves unentailed. Thus, theoretical scientists must be
more artists than craftsmen; Natural Law assures them only that their
art is not in vain, but it in itself provides not the slightest clue how
to go about it".
6A. 'Modeling Relations': "An examination of modeling within mathematics
reveals two quite different approaches to modeling. One of them, which I
shall call /analytic/, is intimately tied to the idea of direct
(Cartesian) product ... the notion of efficient cause. The other, which
I shall call /synthetic/, is tied rather to the idea of /direct sum/ and
to the notion of material cause.... Only in very special situations do
they happen to be equivalent in any sense, and these special situations
all inherently involve some kind of linearity.... In fact, the
difference between direct sum and direct product, between synthetic and
analytic models, is also closely allied to the difference between
/syntactic/ and /semantic/.... from this viewpoint, Godel's
Incompleteness Theorem is an assertion that direct product [does not
equal] direct sum (more specifically, direct product is 'bigger' ... we
shall see further that an assumption of coincidence between analysis and
synthesis leads directly to the concept of /mechanism/ ... In a sense,
it is the thrust of this entire work that this hypothesis of analysis =
synthesis must be dropped.... By dropping it, we enter a new realm of
system, which I call /complex/ ... 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."
6B. 'Some Preliminaries: Equivalence Relations': "At one level,
equivalence relations can be looked upon as a vehicle for constructing
new sets (quotient sets) from given ones.... the relation R ... allows
us to ... first, find in S/R the equivalence class to which s belongs,
and then, locate s within this class. A property of s that depends only
on its class ... will be called a /general property/ of s, relative to
R. On the other hand, a property of s that depends on where it is within
its class ... will be called a special property.... We can think of the
set S/R of equivalence classes as a /base space/. Over each point in
this space, there is an equivalence class, the /fiber/ consisting of all
elements of S that project onto this point.... the base space here is
the habitat of the general properties under R; the fibers embody the
special properties.... I will henceforth call the mapping f: S -> X an
/observable/ of S; the totality of values f(S) of f will be called its
/spectrum/.... When we write the expression 'f(s)', we get the
impression that f, the mapping, is fixed, and s, its argument, is
variable in S. But the role of argument and mapping are formally
interchangeable.... given S, the set R(S) of all equivalence relations
on S is determined. It is in fact a certain set of subsets of 2^SxS. The
usual operations of union, intersection, and complementation are
available in R(S) ... R(S) has an enormous amount of internal structure;
it is a /lattice/.... The symbol R here ... is in fact an example of a
functor, which at this level operates, not on individual systems, but on
whole collections (categories) of them... we know that there is a close
relation between H(S,*R*), the set of all observables, and R(S), the set
of all equivalence relations on S. In fact, if f is any observable, we
know we can associate with it a definite equivalence relation Rf in R(S)
... if one observable f tells us something about the elements of S, then
two observables (f,g) should tell us more.... two elements s, s' are
equivalent under Rfg if neither f nor g can tell them apart. The
equivalence classes of Rfg are thus the intersections of all the
equivalence classes of Rf with all the equivalence classes of Rg. In
other words, Rfg has /more classes/ than either Rf or Rg alone.... In
formal terms, it is said that Rfg is a /refinement/ of Rf and of Rg."
In case, like me, that wasn't all immediately intuitive to you, you can
take it as providing a mathematical argument for the notion that 'the
whole is greater than the sum of its parts'. Or, if that phrase seems
simplistic or bothers you, that different functions can be observed from
different 'angles' or levels of the same physical system.
03/28/97. (The next few chapters of _Life Itself_ describe simulation,
mechanisms (Turing machines) and a relational theory of machines,
rounding out Rosen's mathematical case for a ontological distinction
between mechanisms and organisms. 03/28/97.)
03/30/97. 6C & D discuss the set-theoretic details of analytic vs
synthetic models: "The whole purpose of [synthetic] direct sums is
precisely to create a situation in which encoding is itself entailed....
For in this situation, and in it alone, we may replace art by craft."
These models are equipped with a natural order, but it is one which the
model builder provides through his own indexing.
The remainder of ch.6 is very mathematical but basically claims that:
"synthetic models are an embodiment of pure syntactics, whereas analytic
models are inherently semantic." Rosen concludes: "Every synthetic model
is an analytic model ... but, there generally exist analytic models that
are not synthetic models." The reductionist approach, says Rosen,
mandates that analytic and synthetic models coincide; but, he claims,
this situation is limited only to mechanisms and is totally inadequate
for the explanation of organisms.
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