Does Newtonian space provide identity to quantum systems?

Abstract: Physics is not just mathematics. This seems trivial, but poses difficult and interesting questions. In this paper we analyse a particular discrepancy between non-relativistic quantum mechanics (QM) and `classical' (Newtonian) space and time (NST). We also suggest, but not discuss, the case of the relativistic QM. In this work, we are more concerned with the notion of space and its mathematical representation. The mathematics entails that any two spatially separated objects are necessarily \ita{different}, which implies that they are \ita{discernible} (in classical logic, identity is defined by means of indiscernibility) --- we say that the space is $T_2$, or "Hausdorff". But when enters QM, sometimes the systems need to be taken as \ita{completely indistinguishable}, so that there is no way to tell which system is which, and this holds even in the case of fermions. But in the NST setting, it seems that we can always give an \ita{identity} to them by means of their individuation, which seems to be contra the physical situation, where individuation (isolation) does not entail identity (as we argue in this paper). Here we discuss this topic by considering a case study (that of two potentially infinite wells) and conclude that, taking into account the quantum case, that is, when physics enter the discussion, even NST cannot be used to say that the systems do have identity. This case study seems to be relevant for a more detailed discussion on the interplay between physical theories (such as quantum theory) and their underlying mathematics (and logic), in a simple way apparently never realized before.