Mathematically, Rosen says, "the trick is to consider what happens on
certain /sets/ of phases of that underlying particulate system, rather
than on individual phases, and to express this in a /probabilistic/
language.... The Second Law thus asserts that /a closed system cannot
autonomously tend to an organized state/. Or, contrapositively, /a
system autonomously tending to an organized state cannot be closed/."
In 5D, "The Concept of Function", Rosen warns that this "has overtones
of end, purpose, finality, [though] nowadays it is considered again
acceptable to talk of function, because natural selection and the
adaptive evolutionary processes it generates can, it is supposed, be
invoked to exorcise any finalistic demons." Rosen talks about how
through "willful, active intervention" and observation, "the =
discrepancy
between the two /systems/ [before and after] defines the concept of
/component/; the discrepancy between the /behaviors/ defines the
/function/ of the component."
In 5E, "On the Strategy of Relational Modeling", Rosen continues to
emphasize the distinction between relational and traditional
reductionist approaches by saying: "I can epitomize a reductionist
approach ... as follows: throw away the organization and keep the
underlying matter." But he emphasizes, "there is nothing in the
relational strategy that is unphysical ... The organization of a =
natural
system ... is at least as much a part of its material reality as the
specific particles that constitute it at a given time, perhaps indeed
more so."
5F, "The Component", talks about how "the component may be thought of =
as
the /particle of function/"; and 5G discusses "Systems from Components"
via mappings, circuits, compositions and the "abstract block diagrams"
Rosen uses to describe these. In 5H, "Entailment in Relational =
Systems",
Rosen claims to "have made a gentle and natural progression from
/organization/ to /function/ to /component/, and thence back to
organization."=20
He describes how "inner entailments are of a restricted kind; only
/elements of sets/ can be entailed thereby." Whereas, "outer =
entailments
allow us to entail (in fact, to /construct/) mappings from mappings in
the category, and sets from sets." Here, he says, "we must invoke a
category axiom, the one that allows composition of mappings [and] gives
us an inferential rule F of precisely the form F(f,g) =3D gf [such =
that]
gf can be regarded as an /effect/, the pair (f,g) is its material =
cause,
and the inferential rule F is its efficient cause.... But it is crucial
to keep in mind that this inferential rule F, which governs the
entailment of mappings from mappings, is not /in/ the category; /it =
does
not encode a component/".
02/09/97. _Life Itself_ 5I introduces final causation & functional
entailment: "by invoking the concept of design ... we are talking about
finality. This is alright when we talk about machines as human
fabrications, but it is manifestly not alright to consider organisms in
such terms.... The traditional argument is that the produce of an
evolutionary process give the /appearance/ of design but without any of
the finalistic implications of design". Final causation is inconsistent
in the Newtonian picture, but not in the relational picture where "a
component is entailed by its function"; but, Rosen hastens to add, this
"does not connote anything vitalistic or transphysical in material
nature".
In 5J, on "Augmented Block Diagrams", Rosen claims that it is possible
to entail mappings because the mappings from one set to another are
themselves a set AND also a mapping. He is building a case for =
recursive
relations between system components as the basis of biology, vs. the
recursive chronicling of separate components that is the basis of
mechanics.
03/15/97. In 5K of _Life Itself_, on finality & entailment in 'abstract
block diagrams', Rosen argues that "finality is allied to the notion of
possibility, while the other causal categories involve necessity". He
shows how, with relational models, "there can be many distinct,
different ways of entailing the effect f(a) (i.e., of executing the
function), between which final causation cannot distinguish".
Finally, in 5L (p143), Rosen describes the Theory of Categories that he
has been leading up to: "The Theory of Categories arose out of topology
[and] the Cauchy concept of convergence ... a mapping is continuous if
it commutes with the operation of taking limits of sequences; if the
image of a limit is the limit of the images.... convergence and
continuity are local concepts ... the traditional figures of the
geometer are not local objects.... The obvious way to connect the two
... is to tie the notion of congruence (a global notion) to that of
continuity (a local one) ... we need continuous mappings (to preserve
the local structure) that are also invertible (so as to form a group)."
"In a word, we are led to the homeomorphisms, which turn a topological
space into a geometry in the sense of Klein.... If we wish to retain =
the
original geometric flavor of the Klein Program and treat individual
topological spaces as 'figures', we can see ourselves being led toward =
a
category of all topological spaces.... There is thus the problem of
trying to decide whether two particular topological spaces are
homeomorphic or not. This is the Classification Problem ... One way to
approach it is ... in terms of invariants, with numbers or other =
objects
that are constant on the equivalence classes.... The trouble is that
nonhomeomorphic spaces may also get the same number ... Most of what is
presently available in these directions goes back to Poincare ... it
embodies a very syntactical approach to topology."
Rosen then describes the progression from points, zero-simplexes, to
line segments, one-simplexes, to chains, two simplexes, to arrays of
such simplexes, complexes, and continues: "the boundary operation,
appropriately generalized, is a homomorphism between groups of chains =
of
successive dimensions ... if S1, S2 are two complexes, and f: S1 -> S =
is
a continuous mapping between them, there is a corresponding group
homomorphism Hn(f): Hn(S1):-> Hn(S2) for each n [nth homology group]; =
if
f is a homeomorphism, then Hn(f) is an isomorphism.... These Hn are
examples of functors. They basically associate groups Hn(S) with
complexes S ... It is most important to observe that such factors do
not, in general, associate or encode elements (points) of S with
elements of Hn(S).... In fact, all we retain about S is that some of =
its
points lie on boundaries that separate others; i.e., a relational
property. Indeed, the relation between S and the Hn(s) is what we
earlier termed a metaphorical one.... And it is important to note that
the metaphor from the Hn(S) to S ... runs in the opposite direction as
the functor itself (i.e., from spaces to groups). That is, we decode =
the
range of the functor to its domain."=20
"Category Theory itself embodies this picture but divorces it =
completely
from any specific referents. Thus, in place of the topological spaces
and groups of algebraic topology, we have unspecified objects; in place
of continuous mappings and group homomorphisms, we have correspondingly
unspecified morphisms. In fact, we can even dispense with the objects
and proceed entirely on the basis of morphisms alone. We only require
that morphisms can be composed, just as mappings can, and that this
composition is associative.... We can take functors as objects of a new
category, with natural transformations providing the morphisms between
them. And we can keep doing this, iterating this procedure, so that the
functors and natural transformations arising at any given level become
the objects and morphisms for the next. In a certain sense, then,
Category Theory can talk about itself, or describe itself, in ways more
nearly akin to natural languages than to the formal systems that
normally constitute mathematics.... It can be used as a kind of
transducer, to move ideas and methods from one part of mathematics to
another."