Mathematical Knowledge and Pattern Cognition

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. (One might do so by developing a theory of meaning and truth, which together with observations from the sociology of mathematics would imply that mathematical knowledge exists. Mathematicians do seem to make knowledge claims, so all one needs is a theory which shows that here at least appearances are real.) 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. Granted the existence of mathematical knowledge, the major problem it poses is that it is a case of the acquisition of objective knowledge and belief with no apparent interaction with an external subject matter. This phenomenon or apparent phenomenon sets mathematics apart from the rest of science and grants a great deal of initial plausibility to the view that mathematics is a priori. On the other hand, the same data make it seem implausible that the epistemology of mathematics is a special case of the general epistemology of science. There are a variety of ways to react to this situation. One can, as Frege and Godel have, posit a special faculty which enables us to perceive external but abstract mathematical objects.1 Or like Brouwer and