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Re: Turing machines and tape length
- From: Tim Gwinn <***>
- Date: Wed, 29 Dec 2004 09:09:32 -0500
Howard,
It seems to come
down to a difference in what we each consider "commute" to mean, or the
requirements for a modelling relation to "commute". For me, I take
Rosen's stance that it involves bringing the entailment structures
into congruence via the encoding/decoding. Indeed, I agree that the
encoding/decoding are creative acts. However, in Rosen's view, in order for a
modelling relation to commute, there is the following requirement placed on
those creative acts:
"The only
condition on them is that they bring the two entailment structures into
congruence -- that is, they satisfy the commutativity condition, which I have
written as [1 = 2 + 3 + 4]". [EL p. 159]
In your
post, you say: "As Hertz emphasized, the only limit
on a model is the commutation condition." This "commutation
condition" seems to be a different kind of requirement than the one Rosen uses.
As I read Hertz, it seemed to me to pretty much mirror Rosen, albeit in a less
precise manner; but, I think we must be reading it differently. Can you explain
what you mean by the "commutation condition" on a modelling relation - what kind
of requirement does this place on the modelling relation?
Regards,
Tim
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?