Re: Is it sure that any mechanism has a largest model? (LI,8C, p.205)

[John Kineman <***>]

Judith,
This strikes me as the correct explanation. It also defines a logical 
paradox. The very concept of "mechanism" presumes a finite largest model 
that is therefore a reality, and yet we also understand any mechanistic 
theory to be an approximation to reality. Both cannot be true. Hence 
either the entire universe is a mechanism, and there is no complexity, 
or there are no real mechanisms, all of them are abstractions.
JK.

Judith Rosen wrote:

> 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 ----- 
>  From: Cybob10
>  To: ***
>  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
>
>