John: So circular definitions are permissible if and only if you can show there is a unique pair with the given relation.
Thats very interesting. How do you prove you have a unique pair? Do you know an example of such a circular definition that is popular or obvious? For example, I would define sqrt(49) as 7, but I would never define 7 as the sqrt(49). I always thought definitions should reduce the complexity of understanding. 7 seems simpler than sqrt(49) a number is simpler than a functional lookup. And when you get to a minimum complexity, you state it as an axiom, or throw in the towel. Ive never heard of circular logic as being an acceptable final state; that is, when you encounter one, it implies theres more work to do. Rob _____ From: [email protected] [mailto:[email protected]] On Behalf Of John Kennison Sent: Wednesday, April 29, 2009 8:03 AM To: The Friday Morning Applied Complexity Coffee Group Subject: Re: [FRIAM] emergence, again As a practicing mathematician, my understanding is that it is permissible to define anything by a property if and only if you can prove there exists a unique thing with that property. For example, you cannot define sqrt(49) as an integer whose square is 49 since there are two such integers. Nor could you define sqrt(-1) as the real number whose square is 1 as there is no such real number (and you cant define sqrt(-1) as the complex number whose square is 1 as there are two such complex numbers, i and i.) So circular definitions (where A is defined in terms of B and B in terms of A) are permissible if and only if you can show there is a unique pair (A,B) with the given relation. On 4/29/09 9:21 AM, "glen e. p. ropella" <[email protected]> wrote: Thus spake Nicholas Thompson circa 04/28/2009 08:33 PM: > let a, b, and c > constitute macro-entity E and let the behavior of E. be controled by the > properties and intereactions of a, b and c. Now, let one of the behaviors > of E to control the behavior of a, b, or c. Is there a problem here? There's no problem with it. It's called an impredicative definition, which basically means the application of a universal quantifier (e.g. "for all") over a set as a part of the definition of the members of that set. (IIRC, of course... ;-) Here's a quote from Barwise and Moss' "Vicious Circles" that may address the "problem" you've heard "philosophers" talk about: "In certain circles, it has been thought that there is a conflict between circular phenomena, on the one hand, and mathematical rigor, on the other. This belief rests on two assumptions. One is that anything mathematically rigorous must be reducible to set theory. The other assumption is that the only coherent conception of set precludes circularity. As a result of these two assumptions, it is not uncommon to hear circular analyses of philosophical, linguistic, or computational phenomena attacked on the grounds that they conflict with one of the basic axioms of mathematics. But both assumptions are mistaken and the attack is groundless." -- glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
