Mathematical entities without objects:: On the realism in mathematics and a possible mathematization of the (non)Platonism: Does Platonism dissolve in mathematics?

International audienceRepetition, Seriality, Temporality are among concepts that undoubtedly refer, in mathematics, to the famous questioning on realism. Repetition highlights identity and non-identity, seriality questions the possibility of decomposition of sequences into identical ones, and temporality addresses the phenomenon of successive actions. The three of them are inherited from, among others, natural sciences or philosophy: in other words real situations. Naively, they don’t belong to the core of mathematics, but rather to their domain of applications: in principle there is no intrinsic time in mathematics, and identity (together with the strongly related concepts of repetition and seriality) is supposed to be defined once forever in the mathematical concept of definition.But less naively, let us say that their pertinence in mathematics is definitively tied to the general pertinence of realism in mathematics.There is a temptation, in mathematics, to fall into the somehow Platonist alternative “everything is real"/”nothing is real". Between these two extreme positions, we would like to propose, by looking at three significant examples in recent mathematics, the possibility of having two levels of realism: the upper one, the one of entities, and the ground one, the one of objects (these two denomination being taken purely as definitions whose main objective is to create a distinction). The upper level (entities) is more the one of “operations", mathematics in action, dynamics of mathematics including temporality, though the ground floor (objects) is more dedicated to culturally well-defined objects, inherited from our perception of physical or real world, where, e.g., repetition makes sense.We will show in the three examples studied (taken in analysis, geometry and the theory of dynamical systems) that the upper level is wider than the ground one, therefore highlighting the possibility of having in mathematics entities without underlying objects. This way, we will also show how identity (and therefore repetition and seriality) takes off in order to join the upper floor, the one of action and temporality.Finally we will discuss how these situations incorporate in their own core the Platonist alternative mentioned earlier, and how - in fine – mathematical Platonism has a vocation for becoming a true branch of mathematics