Re: Turing machines and tape length

[Howard Pattee <***>]

Tim, 

The way I interpret your question I would have to answer: No, I do not
agree. The exact wording here is important. When you use "formal
model" it can be ambiguous because the phrase can refer to only one
side or to the whole modeling relation. To disambiguate I suggest "a
formalism interpreted or encoded as a model." I hope you accept this
clarification. As I expressed it in the last post: "I would
emphasize that no formalism is a model until it is interpreted, which
requires the encoding/decoding process." Or to quote Rosen:
"Formalisms become science, as we have seen, precisely when their
elements are endowed with referents.  .  . Indeed, they acquire
their external referents by virtue of the specific encoding and decoding
arrow that have been mandated" (LI, p. 98). 

The reason the formalism does not limit the model is because all
interpretation is limited only   by our imagination. As I
expressed it: "The important point is that coding and interpretation
are unentailed by both Nature and the formalism." Rosen is also
clear on this point: "the encoding and decoding arrows were
themselves unentailed " (LI, p.61).

I think your question implies entailment: " . . .do you not also
agree that the model is likewise limited to being capable of modelling
only aspects of the natural world which meet the limits of Church's
Thesis?" There must be some kind of intrinsic entailment between the
formalism and the encoding if it is a limit on the model. I believe that
interpretation is really a free creative act (and I think Rosen said that
somewhere too). As Hertz emphasized, the only limit on a model is the
commutation condition.

One example comes to mind. So-called pseudo-random numbers are
deterministically generated by feedback shift registers (i.e, the
sequence ultimately repeats). Nevertheless this formal deterministic
computation is often interpreted as random in some models because for all
observable purposes of the model these numbers "behave as if"
they were random. In other words the actual formal computational
determinism does not limit the capability of the model to represent a
complete indeterminism. 

This type of interpretation of formalism is not uncommon, and it also
shows that any limit or lack of limit we choose from a formalism on a
model is not generalizable, but must be determined empirically according
to the requirements of the model.

Howard
    
> TG:
> If what it means to be an "empirical model" is to be a formal
> model in a commuting modelling relation with some natural system, then if
> that formal model is a member of the class of computable models, then do
> you not also agree that that model is likewise limited to being capable
> of modelling only aspects of the natural world which meet the limits of
> Church's Thesis?