The aim of the present article is to make the notion of an ontology of fields mathematically rigorous. The conclusion will be that couching an ontology in terms of mathematical bundles and cross‐sections both captures many important intuitions of conventional ontologies, including the universal‐particular paradigm, the connection of universals and their ‘instantiations’, and the notion of ‘possibility’, and makes possible the framing of ontologies without ‘substrata’, bare particulars, and primitive particularizers.

The paper ends with an argument that says: necessarily, if there are finitely spatially extended particulars, then there are monadic universals. Before that, in order to characterize the distinction between particulars and universals, Roman Ingarden’s notions of existential moments and modes (ways) of being are presented, and a new pair of such existential moments is introduced: multiplicity–monadicity. Also, it is argued that there are not only real universals, but instances of universals (tropes) and fictional universals too.

About the ontology of the Polish philosopher Roman Ingarden, as presented in his treatise 'The Controversy about the Existence of the World'.

Theism does not require substance ontology but is compatible with an alternative ontology which I call stuff ontology.