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Re: defn of impredicativity (was Re: goals and language?)

Calvin Ostrum wrote:
On 8/10/05, glen e. p. ropella <***> wrote: 
The reason I sustained a conversation with Torkel even after he provided
his personal definition of "impredicativity" was because he seems to
think that type of circularity is completely different from the other
types of circularity talked about in Kercel, Kline, B&M, etc.  And I
don't understand that difference.


I don't think it is accurate to describe the definition
Torkel provided as "his personal definition".   I believe it
is simply the standard definition that is used in formal logic.

It is accurate, regardless of how many people share his definition. Further, I suspect there are plenty of logicians and philosophers who periodically expand their usage to more than the rigid "standard" definition. So, it is a personality trait of Torkel's that he is more fastidious about his usage.

And the difference is quite clear.  They are not even applied to
the same thing (as explained before).  It is definitions that are
impredicative, but sets that are not well-founded, such as the
set a containing only a, for example.   Impredicative definitions
will never specify sets such as this as long as there is something
that entails the axiom of foundation among the axiom set.

And, as has been explained before, definitions are objects. So, what you're saying is that some objects are impredicative and others are not. In any case, I think it's completely reasonable to extend the use of the term, as RR, Kercel, and Kline do, especially since we're speaking English, which is a living language.

Heck, if we all froze up every time someone abuses a term or applies it in a new domain, our knowledge would never advance.

This is important to
me because, as a modeler, models can be maps between things (as tight as
real functions -- M:Input -> Output -- or as loose as simile) and things
that can be mapped by other models.  In particular, when dealing with
multi-level or multi-scale models.


I don't see what this means.  You would have to put it in more precise
terms.

I don't buy that. If that were true, you would never be able to read a novel and understand what it means. [grin] Precision is not necessary for the communication of meaning. Now, you may not understand it for other reasons... like you don't have the same grounding for the term "model" or "function" or something.... But, it's not the lack of _precision_ that prevents you from understanding the above.

The same can be said of much of RR's work. He, like any good practitioner (as I understand it he was a practitioner before he became a philosopher), regularly crosses between the extremes of rigor and vernacular. This crossing can muddy the waters somewhat. But, it is also necessary when working in an interdisciplinary context, which RR was (and we are).

The basic lambda calculus is untyped. So we have the issue that in a model for the lambda calculus, functions can be applied to themselves.
In fact, the only thing you can apply a function to is a function of the
very same type. This results in a "circular" sort of "domain equation"

D == [D -> D]

A great deal of research has been done in solving such domain equations to
specify the set of objects D. Non-well-founded set theories are not necessary,
although I think they can indeed be very helpful. But there is nothing
"non-computable" or otherwise mysterious about these objects (in fact,
the lambda calculus was invented by Alonzo Church as one way of specifying the class of computable functions, and here they are, functions
that are intrinsically "circular" in some way or other. Perhaps this could be
a suggestive metaphor, or even more)

Well, I'm not fixated on "paradoxes" or the non-computable. I'm more concerned with iteration and recursion (and "closure", though I've been cautioned that the term "closure" is a show-stopper just like the term "impredicativity" because people will jump in and argue about nothing if the term is used). And the lambda calculus is excellent for studying those issues. The only thing unsatisfying about it is the difficulty in representing practical, engineering-oriented systems. My goal, as a modeler, is to create models that will aid me in manipulating real systems (like, a sick patient). The lambda calculus is too abstract to help with many aspects of those models. I at least need concurrency and some form of individuality.

In this context, I do need some formal way to specify "self-production" and "self representation". I can formulate such things without worrying about the assumptions underlying such a formulation (like the AFA). However, I will _eventually_ approach formal models that can be automatically _evaluated_. And that evaluation will include verification techniques like automated theorem provers. When I get to that point, I will need to have a good understanding of what these things mean formally.

It's my belief that RR (along with people like von Neumann, Turing, Varela, Langton, etc... all the people trying to clarify the machine-organism distinction) can help... in spite of (or because of) his interdisciplinary works.

--
glen e. p. ropella              =><=                Hail Eris!
H: 503-630-4505                       http://ropella.net/~gepr
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