Mathematical Knowledge: Motley and Complexity of Proof

Modern mathematics is, to my mind, a complex edifice based on conceptual constructions. The subject has undergone something like a biological evolution, an opportunistic one, to the point that the current subject matter, methods, and procedures would be patently unrecognizable a century, certainly two centuries, ago. What has been called “classical mathematics” has indeed seen its day. With its richness, variety, and complexity any discussion of the nature of modern mathematics cannot but accede to the primacy of its history and practice. As I see it, the applicability of mathematics may be a driving motivation, but in the end mathematics is autonomous. Mathematics is in a broad sense self-generating and self-authenticating, and alone competent to address issues of its correctness and authority. What brings us mathematical knowledge? The carriers of mathematical knowledge are proofs, more generally arguments and constructions, as embedded in larger contexts. Mathematicians and teachers of higher mathematics know this, but it should be said. Issues about competence and intuition can be raised as well as factors of knowledge involving the general dissemination of analogical or inductive reasoning or the specific conveyance of methods, approaches or ways of thinking. But in the end, what can be directly conveyed as knowledge are proofs. Mathematical knowledge does not consist of theorem statements, and does not consist of more and more “epistemic access”, somehow, to “abstract objects” and their workings. Moreover, mathematical knowledge extends not so much into the statements, but back into the means, methods and definitions of mathematics, sometimes even to axioms. Of course, statements are significant as encapsulations or