This paper is concerned with the genesis of mathematical knowledge. While some philosophers might argue that mathematics has no real subject matter and thus is not a body of knowledge, I will not try to dissuade them directly. I shall not attempt such a refutation because it seems clear to me that mathematicians do know such things as the Mean Value Theorem, The Fundamental Theorem of Arithmetic, Godel's Theorems, etc. Moreover, this is much more evident to me than any philosophical (...) view of mathematics I know of — including my own. So I am going to take mathematics as my starting point. (shrink)

A standard illustration' of this situation in this: let M~ be a theory of mechanics employing mass points as basic objects and let Mz be similar yet with only extended objects as its primitive elements. Let M> postulate that mass points come only in dense collections. Granted reasonable assumptions about the further details of Mq and M2, we can define the extended objects of Mz in M~ as dense sets of mass points whereas the latter can be defined in Mz (...) as nested sets of the former. Both theories offer equivalent descriptions of the world in the sense that any claim of Mq can be systematically translated into a theorem of Mz and conversely. (shrink)