New Trends in the Foundations of Mathematics

A new emphasis on finitist methods and results in mathematics indicates a turn toward foundations happening under the slogans of HARD ANALYSIS and PROOF MINING. While previously non-constructive or infinitistic methods were thought (by an influential minority) to be philosophically defective, the revival of interest is caused by mathematical needs. Some of the central results needed development of new tools that turned out to be instances of well-known constructions of proof theory. The finitist trend is called "hard analysis" by T. Tao and contrasted with the ordinary or "soft" mathematical analysis. The latter works without any restrictions on the abstract notions and infinitistic methods. Proof mining initiated by G. Kreisel and developed by U. Kohlenbach applies proof theoretic tools to get essential strengthening of results proved in the mainstream mathematics. We present some examples.