I am not really satisfied with Dr Aloisius' answser
(concerning proof in LI, p.205, section 8C) (cf copy below).
As a matter of fact, why should the program for M should
be the set of all Mi programs?
The (semi)order relation in the lattice of models does
not force any relation between corresponding programs.
For that we would need a result like : "if M1<M2 then
the program u1 for M1 is part of u2" or something alike.
I'd rather stick to the solution we've been discussing
with Tim.
Still is to be determined if the intersection of all
models is still a model.
---
Now a remark : in the definition of "mechanism" and
"machine" Rosen relies heavily on the "simulation" notion.
Simulable mappings are those which are "definable by an
algorithm" (LI, p192).
Therefore most of mappings thar are definable in maths
are not simulable. We can not work in continuous sets anymore if we are
restricted to simulable mappings. And we loose also most of physics. We loose
chaos. We loose statistical mechanics, Q mechs...
So I do not understand really why, as a result of his
talk "On states and recursivity in mechanisms", Rosen says (LI,p212,
8H), :
"As we have seen a contemporary physicist will feel very
much at home in the world of mechanisms".
[...] Indeed the claim that () every natural system is a
mechanism is the sole support of contemporary physic's claim to
universality.
---
For me, an important idea is the encoding/deconding
diagram : it allows us to speak about "inferential
structures", and the possible correspondance between a natural system N and
Formal system F if diagram commutes (hertz condition, thanks Howard, that
helps!)
Then comes the idea that physics, looking at a natural system, tries to find a unique formalism as a
model of it. That is : physics tries to rely on a unique efficent cause (e.g.
: newtonian force) and does not look for an explanation of this cause. And the
role of symetries in contemporary physics may not really change this
fact.
Instead, relational modeling says that the network of
models is important, and even more, NECESSARY. This leaves room for biology :
its "function" notion, the degenerate relation between structure and function,
the multiple levels of organization, "multicausality".
From this point of view, if a "continuous Turing
machine" could be imagined, Rosen's distinction between software and hardware
would still work, and let us understand that, through simulation, inferential
entailments, which play the role of efficient causes, become
material causes. And that makes a clear distinction between simulation and the
modeling relation, because the modeling relation "respects" inferential
structures.
Boris
*************
Here is Aloisius answer :
>The important thing to remember is that a Turing machine must halt
after
> FINITE number of steps. So Rosen's argument was that
this model M,
>which is strictly larger than any of the Mi, was forced
to have a
>program that was an INFINITE set (all the Mi programs).
This was the
>part that was "clearly impossible": M cannot be a Turing
machine AND
>have an infinite set as its program at the same
time.
> What Rosen meant was that it was impossible for an
infinite set to be (the
>representation of) a Turing
machine.