Kolmogorov and the General Theory of Problems

This essay is our modest contribution to a volume in honor of our dear friend and fellow logician Peter Schroeder-Heister. The objective of the article is to reexamine Kolmogorov’s problem interpretation for intuitionistic logic and the basics of a general theory of problems. The task is developed by first examining the interpretation and presenting a new elucidation of it through Reduction Semantics. Next, in view of Kolmogorov’s intentions concerning his problem interpretation, Reduction Semantics is employed in an brief epistemological analysis of Euclidean Geometry and its construction problems. Finally, on the basis of the previous steps, some theses are raised concerning intuitionistic logical constants and concerning proofs and hypotheses in Euclid’s Elements.