Optima and Equilibria

By Way of Warning As in ordinary language, metaphors may be used in mathematics to explain a given phenomenon by associating it with another which is ( or is considered to be) more familiar. It is this sense of familiarity, whether individual or collective, innate or acquired by education, which enables one to convince oneself that one has understood the phenomenon in question. Contrary to popular opinion, mathematics is not simply a richer or more precise language. Mathematical reasoning is a separate faculty possessed by all human brains, just like the ability to compose or listen to music, to paint or look at paintings, to believe in and follow cultural or moral codes, etc. But it is impossible (and dangerous) to compare these various faculties within a hierarchical framework; in particular, one cannot speak of the superiority of the language of mathematics. Naturally, the construction of mathematical metaphors requires the autonomous development of the discipline to provide theories which may be substituted for or associated with the phenomena to be explained. This is the domain of pure mathematics. The construction of the mathematical corpus obeys its own logic, like that of literature, music or art. In all these domains, an aesthetic satisfaction is at once the objective of the creative activity and a signal which enables one to recognise successful works. (Likewise, in all these domains, fashionable phenomena - reflecting social consensus - are used to develop aesthetic criteria).