Hard Analysis: new life of old ideas

New emphasis on finitist methods and results in mathematics indicates a turn in 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 close analogs of well-known tools of proof theory.