Thus spake John Kennison circa 04/29/2009 08:02 AM: > 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.
I think that's right. The relevant axiom is the anti-foundation axiom (AFA). And I believe it can be proven that the AFA means that every system of equations has a unique solution. (If we assume the foundation axiom instead, as in the "usual" math, that's not true.) But the AFA can be stated as: a set can contain itself as its only element, which is the simplest form of the circularity issue broached by Nick. -- 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
