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.

According to Kurt Gödel, Bertrand Russell misinterpreted the incompleteness theorem, but did it in ʻa very interesting manner’. To understand what he meant we need to consider their attitudes to defining truth. Even more revealing is the discussion of two fundamental approaches to logic: one is universalistic, and assumed by both Russell and Gödel, and the other is model‑theoretical, Alfred Tarski’s style. It turns out that a misleading or erroneous interpretation can be interesting, as it reveals something fundamental. William Byers claims that truly great ideas in mathematics and about mathematics are in a way false, as they lead to errors, but at the same time they can help to make advances in math. Logicism provides a good example. In addition it may be mentioned that when Russell argued in its favor, he committed a logical fallacy.

Celem artykułu jest przedstawienie oryginalnych poglądów Stanisława Lema dotyczących relacji między filozofią a nauką oraz nauką i filozofią. Wybrane uwagi Lema poprzedzone są przybliżeniem światopoglądu pisarza oraz nakreśleniem jego stosunku do nauki. Na uwagi Lema składają się wybrane opinie tego myśliciela dotyczące następującej problematyki: (1) zorientowania scjentycznego cywilizacji Zachodu, (2) filozofii jako propozycji opisu świa-ta, (3) problemu „matematyczności” świata, (4) specyficznej relacji filozofii (epistemologii) do technologii, oraz (5) Lemowskiej interpretacji twierdzeń Kurta Gödla.