The Mathematics in Society Project (MISP) began in 1980 as an international association of mathematics educators in three continents. The main purpose of MISP is the writing of innovative secondary school mathematics courses based on a new conception of mathematics itself. The starting point for MISP was the fact that mathematics in school is found to be difficult and unpleasant by the great majority of pupils, but mathematics in society is widely diffused and used implicitly by most people. MISP therefore sees mathematics as a ‘living body’ representing all its uses (implicit and explicit) in society, in contrast to the ‘skeleton’ concept of mathematics which has led to such failure in schools. The gradual development of this new conception is informally analysed in the style of Kuhn and Lakatos.