Ontology and mathematics in quantum field theories and in quantum mechanics

A comparison between the use of mathematics made in classical field theories and in quantum mechanics is presented. It is claimed that Hilbert’s space formalism does not provide a spatio-temporal representation of physical events. The picture of the electromagnetic field as an entity which is real in itself, a wave without support, fostered by the emergence of special relativity is seen as a first step toward a gradual flight form intuition into a purely mathematical representation of the external world. Contrary to this trend, it is maintained, the introduction of fibre bundle formalism in recent theoretical physics lends to the classical notion of field a new spatio-temporal intuitiveness which was clearly foreshadowed in the Kantian and Meinongian analysis of the notion of magnitude. Contrary to what happens in quantum mechanics, it is further maintained, mathematics plays a truly explicative role in general relativity, without any loss of spatio-temporal intuitiveness.