All through the ages, the spectrum of educational practice has known two extremes. At the one end we have the guilds, in which knowledge is kept as a well-guarded secret, and for that reason is never formulated explicitly; the apprentice joins a master for seven meagre years and absorbs the craft by osmosis, so to speak. At the other end we have the university, where the students listen to the professor, who tries to formulate his knowledge and the key elements of his abilities as explicitly as possible, thus bringing it all out in the public domain.
Along this scale, mathematics occupies a curious, double position. While mathematical results are published and taught quite openly —to the extent that many mathematical curricula are very much “knowledge-oriented"—, how to do mathematics is hardly taught explicitly. Mathematical methodology is not a topic of explicit concern and, as we shall see in a moment, when mathematicians feel threatened, they do so in their capacity of members of their guild. In passing we note that it is not so much that they are unwilling to teach how to do mathematics, but that they are unable to teach it, not knowing how they do it themselves.