Re: meanings of model

[Howard Pattee <***>]

Tim, you asked:
I am unclear what you mean by "physical laws" in your post. To me, "physical
laws" are just those formal theories and their mathematical formulations
that we humans create. In your rephrasing of my question, it sounds like you
equate the phrase "physical laws" with something like "effective processes
of nature". Is that the case?

I answered:
> I usually mean by physical laws the natural laws that I imagine exist whether
> or not I have created a model of these laws.

HP: What I was trying to distinguish was your use of "physical laws" as 
what we humans create in the models on the right side of the modeling 
diagram from my use of "physical laws' as Rosen's "natural causes" on the 
left side. Failing to make this distinction leads to misunderstanding.

> >HP: Also Constructivists, like most computer modelers, see no reason why the
> > formal concept of Turing computability should limit how we actually write
> > programs.
>
> TG: The limits of Turing computability are inherent in any program that runs
> on a computer. (Obviously excepting an analog computer, which is an entirely
> different meaning of "computer".

HP: What I mean is that Turing computability is a formal concept that does 
not enter into normal programming because it has never been a serious 
limit. Programmers have written billions of lines of correct and useful 
code without knowing Turing existed.

> >HP: Many programs for physical and biological models are not even
> > algorithms because they do not halt by themselves.
>
> TG: This is rather vague. I have no idea what kind of programs you are
> talking about, so I don't know if you are talking about successive numerical
> approximation routines or something else. I disagree that they are not
> algorithms -- of course they are algorithms: they are running on a computer,
> aren't they?

HP: There are several definitions of algorithm. A common definition of 
algorithm is a finite set of program steps that leads to an answer. That 
means the computation must halt when the answer is reached. If the computer 
never halts how would you know when you have an answer?

> >HP: In fact, strictly
> > speaking, our computers are not Turing-equivalent. They are finite memory
> > and finite state machines, and all theorems on computability
> > depend on the
> > syntax of infinite sets.
>
> TG: The limitations of a universal Turing machine will apply equally (or a
> fortiori) in a finite Turing machine.

HP: Our finite machines do not in general compute exact functions, but only 
approximate answers. These are good enough for scientific models that are 
only approximations anyway. That is not good enough for formal proofs so 
they are not Turing-equivalent and formal Turing proofs about limitations 
say nothing about approximate answers.

> TG: Any natural system which possesses closed loops of entailment cannot be
> modeled solely by a computable model.

HP: Closed loops of entailment may be a useful abstract concept. I do not 
know how you can empirically recognize closed loops of entailment by their 
inability to be modeled because the proof that they cannot be modeled by a 
Turing-equivalent computer is again a strictly formal proof that says 
nothing about how one can construct an approximate model.

Howard

>