EXCERPT Drawing on the four maxims of Goethe’s that we have placed as epigraphs, we would like to explain, here, the general focus of this Synthetic Philosophy of Contemporary Mathematics. The four terms in the title have, for us, certain well-defined orientations: ‘synthetic’ points to the connective, relational environment of mathematical creation and to a veiled reality that contrasts with invention; ‘philosophy’, to the reflective exercise of reason on reason itself; ‘contemporary’, to the space of knowledge elaborated, broadly speaking, between 1950 and today; ‘mathematics’, to the broad scope of arithmetical, algebraic, geometrical and topological constructions – going beyond merely logical or set-theoretical registers…