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Hi Boris,
Since mathematics is not my forte, I've forwarded your message on
to one of my father's former PhD students, Dr. Aloisius Louie. I'll post
his response, when I receive it.
However, the crux of the matter in the passage you cited (Page 205,
Life, Itself) is this phrase:
Robert Rosen wrote: "Now let us suppose that N is a
mechanism (every model simulable) but that C(N) contains no largest model. Then
we can find an infinite sequence of increasingly refined models in
C(N)."
If N is a mechanism it would have a finite "largest
model" which is the sum of all the models we could make and which would
completely describe the system. If C(N) is an infinite sequence with no largest
model, one would have to conclude that the system N is not a
mechanism.
He used the word "suppose"... just as he says on the previous page
"In this context, we can see the true import of
additional suppositions, such as 'Every natural system is a
mechanism', or 'Every organism is a machine'. These are, of
course, part of the very fabric of which contemporary science is composed. We
are now in a position to assess them directly; this has been, in fact, my
intentention all along."
He was disproving the idea that every natural system is a
mechanism, or that an organism is a machine. He nailed down what a mechanism is
and then showed that if a system has an infinite category of models, it
isn't a mechanism.
Judith
My favorite discussion list (Independent-- Not part of Rosen
Enterprises): ***
----- Original Message -----
Sent: Saturday, December 11, 2004 7:54
AM
Subject: [ROSEN] Is it sure that any
mechanism has a largest model? (LI,8C, p.205)
I'd like to submit to Rosen readers what might
appear at first sight as a rather technical "detail". But it is not only a
math problem because it is the difference between relational models and
(analytic=synthetic) models that is put into question.
I shall begin with 2 remarks, and then come to my
point.
Remark 1: in LI, p.203, section 8B (Machines and
mechanisms), Rosen says "A natural system N is a mechanism if and only if all
of its models are simulable". But from what I
understoood, the notion of "simulability" (calculability) covers mappings.
Whereas models are sets (cartesian products or direct sums). As a consequence
the notion of simulability for a model is not straightforward.
However, it might be possible to see a set
A=\prod Ai (cartesian product of sets Ai), that is to say an analytic
model, as the identity mapping from to A to A, and say that A is
simulable if and only if this identity mapping is simulable. And in the case
where A is a synthetic model just consider the associated analytic model, and
apply the same "trick". So it should be ok.
Remark 2: p.205, section 8C (On the largest
model of a mechanism), we want to prove that if a natural system N is a
mechanism then it has a unique largest model. First we suppose that the
category C(N) of all models of N has no largest model. Rosen says "Thus we can find an infinite sequence of
increasingly refined models M1<M2<...<Mi<...". This is not
obvious, but seems to be true, if you give yourself the "choice axiom" (so
called in french) of set theory, and consider that it is always possible to construct the refinement of two
models.
But then, and here I come to my point, Rosen
says "We thus end up with a countable family of distinct programs, each of
which is a distinct word of finite length on a finite alphabet : this is
clearly impossible". This is where I don't understand. Consider for
instance the sequence of all binary programs (binary sequences)
ordered in lexicographic order
(0<1<00<01<10<10<000<001<010<010<011<100<....) :
this is a countable family of distinct programs, each of which is a distinct
word of finite length on a finite alphabet. So it seems possible to have such
a family.
Am I mistaking? If not, we loose the property
that the category C(N) of all models of a mechanism has a largest model, and
we loose the property that if N is a mechanism then analytic and synthetic
coincide in C(N) (conclusion 4, p.205, section 8B).
What do you think?
Boris
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