My favorite example is the following definition for a pair of differentiable functions, s and c, from the reals to the reals. They are defined so that:
s' = c, c' =- s, s(0) = 1-c(0), c(0) = 1+s(0) Of course, the last two conditions just say that s(0) = 0 and c(0) = 1. The only possible pair with these properties are s(x) = sin x and c(x0 = cos x. One waty to see this is to note that s and c are both solutions of y'' + y = 0 and the only known solutions of that equation are of the form Asin(x) + Bcos x where A,B are arbitrary constants. But even if calculus had been developed before trig, one can still prove there exists a unique pair of solutions to the conditions defining s and c by appealing to quite general existence and uniqueness theorems for differential equations. In fact, I understand that some exotic but useful functions are defined by differential equations, sometimes for pairs of functions (I am a category theorist, not a differential equations theorist, but I have had courses in that subject, which are still a dim memory) ________________________________________ From: [email protected] [[email protected]] On Behalf Of Robert Howard [[email protected]] Sent: Wednesday, April 29, 2009 12:04 PM To: 'The Friday Morning Applied Complexity Coffee Group' Subject: Re: [FRIAM] emergence, again John: So circular definitions are permissible if and only if you can show there is a unique pair with the given relation. That’s 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. I’ve never heard of circular logic as being an acceptable final state; that is, when you encounter one, it implies there’s 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 can’t 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
