The problem of the existence of mathematical entities is the subject of lively discussions. Realists defend the independence and autonomy of mathematical objects, while antirealists point to their dependence and conventionality. The problem of the existence of mathematical objects is also strongly linked to the problem of mathematical cognition: do we recognize mathematical truths in special acts of intuition, as some realists claim, or do we create mathematical knowledge only by building appropriate formal systems – as some anti‑realists imagine? In this article we present the K. Gödel’s and W.V. Quine’s realistic stances and comment on them from the perspective of Roman Ingarden’s phenomenology. We point out the role that Gödel attributed to his mathematical intuition, and then we present the process of eidetic intuition in Ingarden’s perspective (indicating Gödel’s and Ingarden’s common points of view). We also argue that Ingarden’s rich ontology could contribute in a significant way to the debates currently taking place in the mainstream philosophy of mathematics.

In this paper I take up one of the fundamental themes of ontology concerning the proper understanding of such ontological objects as ideal qualities, properties (features) and tropes. These objects, i.e. properties, qualities and tropes, help us understand more fully what an object in itself (substance, being, object) happens to be. The aim of this work is to present Ingarden’s position on this subject, but also to present a certain new formal solution that uses tools of topological ontology. The background for the problems here is to be found in the works of Aristotle, Ch. Wolff and N. Hartmann.