  These are the notes written for the talk given at the workshop Rethinking
foundations of physics 2016. In section 2, a derivation of the the quantum
formalism starting from propositional calculus (quantum logic) is reviewed,
pointing out which are the basic requirements that lead to the use of Hilbert
spaces. In section 3, a similar analysis is done following for the
reconstruction of quantum theory using an operational approach. In both cases,
non-commutativity plays a crucial role. Finally, in section 4a toy model which
try to motivate non-commutativity is proposed. Despite this last section is
interesting to read, the analysis performed there is not complete. This toy
model will be re-formulated in a rigorous way (and extended) in future works.
