|
Thanks for your list and this opportunity to discuss
Rosen's ideas!
Is it possible that my name (Boris Saulnier) appears
in the "from" email field, instead of cybob10?
I'm glad I can speak with you about this largest model
question as I think it is a keystone in Rosen argumentation.
I go on with Judith and Tim's answers to my first post.
*** (1) Models as sets or as mappings ***
Tim, concerning your first point, let's collect what Rosen
says:
a) "I shall call any _expression_ of a set S as a
cartesian product of quotient sets an analysis of S". (LI, p.162)
b) "I will call the mapping f:S -->X an observable of
S" (p.156)
c) "Every family of such observables gives rise to an
analysis" (p.164) (the initial set S is represented in terms of the
cartesian product of the spectra of observables. Let's remark here that the
observables mus be non linked).
d) "Every such analysis gives us a model"(p.164) : here we
might think that a model is an anlysis.
e) "The totality of analytic models can be identified with
the totality of equivalence relations on S, or the totality of sets of
observables" : here Rosen insists more on the observable side of the
model.
So it is not so clear if a model is defined as an analysis
or as a set of non linked observables. But I agree it should be possible to go
reversly from one to the other (this is the "trick" I was speaking about in my
first post).
Then Tim could you help me in having a better
understanding of the second part of your 1st point, when you say :
>"Each model (analytic or synthetic) will have
relations between observables which will place that model as members
>in the set of relations specified by a member
of either the direct product or the direct sum (or, of course,
where direct
> product/direct sum overlap, of
both)."
My interpretation of this is that it will much more
fruitfull to see models as mappings, because it is the organization of relations
between all the models that will really count.
*** (2) Largest model of a mechanism
***
Let's come to the 2nd point.
I'm convinced by Tim's formulation, but we may still
have a step to clarify :
a) If the set of all (inequivalent) simulable
models Mi is countably infinite then we can list the countable infinite
sequence of finite programs Ui corresponding to these models, then
construct M' as the intersection of all the Mi's (related question : is this
infinite intersection also a model?), consider the associated program U', and
see that this U' must necessarily be (already) among the Ui's, so that we can
not have an countably infinite sequence of inequivalent models.
b) Then we need to proove that
"having no largest model" implies "having a countably infinite set of
inequivalent models".
That's easy, because all the models are simulable, and
because we have a 1-1 correspondance between models and programs, and because
the set of finite programs is countably infinite.
Nevertheless, don't you think we
still need to proove that the infinite intersection of models is still a
model?
Boris
----- Original Message -----
Sent: Saturday, December 11,
2004 3:49 PM
Subject: Re: Is it sure that
any mechanism has a largest model? (LI,8C, p.205)
Hi
Boris,
Welcome to the
list. :)
1) I disagree
that "models are sets (cartesian products or direct sums)". Instead, the
totality of analytical models of a system arises from
a Cartesian product (of the spectra of the family of observables)[LI
164], and similarly, the totality of synthetic models of a
system arises from a corresponding direct sum. Each model (analytic
or synthetic) will have relations between observables which will place that
model as members in the set of relations specified by a member of
either the direct product or the direct sum (or, of course,
where direct product/direct sum overlap, of
both).
2) The argument
on LI 205 regarding the largest model for a mechanism. I believe this argument
is a kind of Cantor-like diagonal argument. He begins by hypothesizing
the negative: that there is no largest model for a mechanism and therefore
there are a countably infinite number of inequivalent simulable
models. Each Mi thus has a corresponding and unique program associated
with it, since each Mi is simulable by hypothesis. Then, he proposes
taking the intersection of all the Mi, which will result in yet another M'
which must be inequivalent to all the other Mi. And this M' must
also have a corresponding and unique program (again, because by
hypothesis, all these models are simulable). But, if the set of Mi is already
a countably infinite set, then M' must already a member of {Mi}, and
therefore the program for M' must also be a member of the corresponding set of
programs for {Mi}. Therefore M' cannot be distinct, nor can the
program for M' be distinct, and therefore the hypothesis fails, and the
opposite must be true: there is a largest model for a
mechanism.
Regards,
Tim
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
|