Boyko Bantchev <boyk...@gmail.com> wrote: >> Definitions do not eliminate ambiguity. > Speaking of formal definitions, I would say they ought to. There is > not much use of defining formally and ambiguously.
There is a formal definition for "formal". It does not mean "unambiguous". In fact, it turns out that we can formally prove that a finite formal definition will always be ambiguous when applied to a set whose formal description is larger than the formal description of the definition itself. (I'm not formally stating that, though.) "Formal" means that we can manipulate objects without caring about their meaning, only caring about their __form__. The only way to make a formal definition unambiguous is to restrict its applicability to a finite area, and to describe that entire area. Unfortunately, since mathematics is infinite and space is very big, someone will probably find a way to find an area that exactly fits the FORM of your formal definition, but is in a different context. Fortunately, the resulting surprises can sometimes be pleasant -- as when Wildberger elaborated the ancient Greek rational trigonometry and discovered that all of the classical results (Euler's line, 9-point circle, etc) hold for triangles in hyperbolic and elliptic geometries (a fact that does not hold for modern trigonometry!) -- and then he found that the results for the three geometries applied to the same triangle have a simple, symmetric interrelationship. -Wm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
