Moscow Philosophy of Mathematics Seminar’s fifth Collection of papers is devot- ed to mathematical proof as to one of pivotal philosophical problems under discussion. This book may be of interest to philosophers and historians of mathematics, as well as to logicians, mathematicians, psychologists, postgraduate and PhD students in all areas of mathematics and natural sciences.

We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (and the cotensor product associative). Besides, for a number of technical reasons we mostly have to assume that the basic ring has a finite homological dimension (no such assumptions about the coring and semialgebra are made). In the final chapters we construct model category structures on the categories of complexes of semi(contra)modules, and develop relative nonhomogeneous Koszul duality theory for filtered semialgebras and quasi-differential corings. Our motivating examples come from the semi-infinite cohomology theory. Comparison with the semi-infinite (co)homology of Tate Lie algebras and graded associative algebras is established in appendices; an application to the correspondence between Tate Harish-Chandra modules with complementary central charges is worked out; and the semi-infinite homology of a locally compact topological group relative to an open profinite subgroup is defined.

Modes of matematical applications in different areas of activity connected with the cycles of mathematics development.

Moscow Philosophy of Mathematics Seminar’s fifth Collection of papers is devoted to mathematical proof as to one of pivotal philosophical problems under discussion. This book may be of interest to philosophers and historians of mathematics, as well as to logicians, mathematicians, psychologists, postgraduate and PhD students in all areas of mathematics and natural sciences.

The following paper considers the debate on disciplinary boundaries of logic in German philosophy of the early 19th century. It is supposed to distinguish four competing views on understanding of the logical knowledge. The analysis of the controversy enables to adjust the location of the Hegelian idea of the "Science of Logic," project and to clarify the historical context of the emergence of formal logic as a discipline.