C Y <[EMAIL PROTECTED]> writes: | --- Bill Page <[EMAIL PROTECTED]> wrote: | | > The point of category theory as a foundation for mathematics | > is that a lot of mathematics can and should be done long before | > it becomes necessary to define what is meant by "set". | | I have seen a few comments to the fact that it should be possible, in | theory, to describe virtually all of mathematics within the framework | of category theory. If this is true, to me that makes it not only the | obvious foundational choice for The Axiom Library but the essential | one.
I've been trying to stay out of this debate... In computational mathematics, we have computation which brings in computer science. Basing math implementation on set theory is, in both conceptual and practical points of view, like using OO design in the sense that everything derives from a universal Object type. That has been tried many times with failure -- but that won't stop people from trying again. On the other hand, Category Theory (or the Theory of Empty Set), does not require set -- you can do things with small and large categories. It only cares about *forms and structures*. From an implementation point of view, it means that you don't need to require all computational objects to derive from a single universal base. That gives flexibility for composition -- something much harder and clumsy with OO paradigms (all incarnations that have been tried so fart). You can failures of OO thinking in the current library in forms of the curious Abelian Monoids that are not Monoids. No undergrade in math will get away with that. But, Axiom apparently does :-( So from my perspective (*Computational Math*), the question is not on what Math is based, but on what computation is based and which framework allows for easy of composition and extension. -- Gaby _______________________________________________ Axiom-developer mailing list [email protected] http://lists.nongnu.org/mailman/listinfo/axiom-developer
