The article summarizes panel discussions led at the Polish Scientific Networks conference. It covers the topics of social and (un)social innovations, their sources, and applications, as well as the new approaches to the concept of the wisdom of the crowds (as opposed to swarm mentality). The article draws on academic research on trust and distrust, declining reliance on formal expertise and a turn against the science, and posttruth society phenomenon. The article concludes with observations about risk aversion in different cultures, to suggest some practical solutions in education programs, needed to address the challenges of the future.

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.