Mathematical Infinity “in prospettiva” and Spaces of Possibilities

The Italian Renaissance painters invented the first symbolic form for the mathematical concept of ''actual'' infinity : the projective point of the perspective. This was the result of a long debate on infinity, in philosophy, since Aristotle, and in theology, during the Middle Age, and had a major fall out in this early geometrization of space. In particular, this invention set the grounds for the construction of the infinite spaces of all possible physical dynamics. In spite of the diversity of the theories in physics (Classical, Relativistic, Quantum … theories), an a priori ''phase space'', as a generalization of the infinite Newtonian space-time, is a core presupposition of each theoretical frame. The problem will be posed of the pertinence of this construction as for historical sciences, biology in particular, where the “space of all possible paths” (the list of all phenotypes and their phylogenetic history) cannot be pre-given.