|
TG: Holonomic constraints ('rigidity' is a typical holonomic constraints)
is clearly present in structural organization, such as the skeletal
components. In terms of an organism's functional organization that I was
concerned with, I am imagining that the constraints are largely, or all,
non-holonomic, such that the functional organization could be maximally
constrained so that it remains invariant, even when all dynamics are removed
(i.e., the cell frozen).
Even rigidity, as in skeletel structure, is context dependent. It
varies according to the context of time, as in where in life-cycle the
organism happens to be. In human childhood, for example, as bones are rapidly
growing, they are much softer and more flexible than in adulthood. In old age,
they are often brittle. (Perhaps this means that rigidity is also dependent on
the context of health.) In pregnancy, female adults structural rigidity alters
radically near the end of term, due to hormonal action. (This allows the bones
of the pelvic girdle to expand as the baby passes through during delivery. I
can tell you from personal experience, it's very strange to feel all your
joints getting loose during the third trimester!)
TG: I think we are using the term
'holonomic constraint' in different ways. I use it to indicate a situation where
the degrees-of-freedom are restricted in a particular way, as the term is used
in analytical mechanics. In the case of rigidity, if I take a bone and rotate
it, the whole bone rotates...all the degrees-of-freedom of the particles in the
bone are tied together such that specifying the overall position and
orientation of the bone in space also defines the position of all the particles
in that bone.
I do not
use context-independence/context-dependence as respective criteria
for holonomic/nonholonomic. I use the terms as they are defined in
analytical mechanics.
It is also not true that all dynamics are removed when cells are
frozen. They are simply slowed down. But frozen cells have a "shelf life",
even when the temperature is maintained.
TG: The interesting thing to me is that the (M,R)-system is a model of an
invariant functional organization, and therefore I further conjecture whether
realizing an (M,R)-system involves realizing a maximally constrained
functional organization?
How do you mean that organisms are "invariant functional
organizations"? Specifically, it's the word "invariant" I want to
understand your meaning of.
TG: I did not say that organisms
are "invariant functional organizations". But if the (M,R)-system is a
valid functional model of an organism, that functional organization in that
model as realized in the organism is invariant with respect to time and its
interactions with its environment. An organism would always be a realization of
an (M,R)-system (excepting damage, of course).
(I must also reiterate the need to include the words
"non-holonomically" with "constrained" if context changes the constraint
values. Otherwise the meaning is radically different. A "maximally
constrained system" would have to be a very limited subsystem within a
complex system or else a simple system-- like a toaster.)
TG: I use the term "maximally
constrained system" as Rosen defined it in the paper. His definition
entirely involves non-holonomic constraints. Therefore it seemed redundant
to specify it as "nonholonomic maximally constrained system". There is no
definition given for a maximally constrained system consisting of holonomic
constraints.
TG: I agree that functional organization cannot be equated, and probably
not even mapped, to structure. That is the "structure-function
complementarity" that he mentions in the paper and which I base my conjecture
on. This is why I say it "would be very difficult to describe in strictly
structural terms". (It may well be impossible, but I don't want to exclude a
priori such a possibility.)
Quite right. However, the structure of living organisms is
infinite in variety, but the organization is the same: It is how we recognize
them as "living organisms".
TG: Likewise, my comments do not
depend upon any particular structure. Rather, they depend upon particular
relationships between elements of structure such that they allow
realization of functions. My conjecture is that those relationships would
be enforced in physical terms by a nexus of nonholonomic
constraints.
When you kill an
organism, its structure can remain completely intact, but you have
destroyed the complex organization. Similarly, functions in a
living organism are not entirely the product of a relation between
structural aspects and other structural aspects. The functions of blood,
for example, are fulfilled in many different ways which utilize many
different means, depending on the evolutionary context of each species of
organism.
It seems to me that the notion of "function" could be viewed as
an exploitation of the degrees of freedom created by the action of
non-holonomic constraints on the system-- either from inside or from outside
the organization (or both, perhaps in various modes
of interaction...). It also seems to me that what would be defined as a
non-holonomic constraint on one aspect of system organization could also
simultaneously be playing many other roles within that organization. Some of
the roles may be functions, which are themselves "non-holonomically
constrained"-- perhaps even by the the first function's activity-- in a
closed entailment loop. In other words, one function's activity creates
or impacts the context that determines various non-holonomic constraint
values. Those constraints may or may not serve some functional value, in turn.
An example would be maintainance of body temperature in a
warm-blooded organism. There are constraints on the range of temperature that
will support continued organization (i.e., keep the organism alive), but those
constraints are different depending on on context... A bear in
hibernation would have different "normals", different ranges, than when it is
not hibernating... Illness or injury changes the context again, which impacts
the temperature constraints. Each of these contextual changes has effects on
the constraint values and, in turn, those effects may have
critical functional properties in the organization of the system. (Fever
and immune system function, perhaps.) Yet, temperature is not a
structural component.
It is worth noting that warm blooded animals have certain
survival benefits and options which are conferred upon them by the ability to
maintain a "constant" internal temperature. At the same time, the need to
maintain that temperature, at whatever ranges and levels the context dictates,
can also be regarded as a non-holonomic constraint on the
system.
Judith
|