Mathematical objects, structures and proofs (introduction to the special issue)

The paper serves as an introduction to the issues discussed in the following articles. It raises the problem (challenge) of integration, according to which an adequate solution of a philosophical problem should simultaneously be an answer to both ontological and epistemological questions. This problem is described speculatively and by referring to P. Benacerraf’s dilemma. In addition, the problem is illustrated by comparing classical and intuitionistic mathematics and also through interpretation of the concept of computer proof. The paper demonstrates that adequate philosophy of mathematics must simultaneously take into account the ontological and epistemological aspects of mathematics and mathematical practice.