Entwicklungslinien in der Mathematik im 20. Jahrhundert. (Trends in the mathematics of the 20th century) (Q1821767)

Every mathematician, more or less, was once asked to speak about the present state of his art for some official academic occasion. This is what happened to the author and is left written in the present paper. Our mathematician decided to deal with the subject first from a ''sociological'' point of view (population of mathematicians). Later he exhibits some more technical problems. In the first pages, mathematical schools are described from 1900 onwards. The author likes superlatives (''herkulische Mathematikerpersönlichkeiten'', ''Supergipfel'', etc.), but it is clear that his description can only be sketchy. He insists on the domination from the two powers, USA and USSR, but explains particularly the case of Hungary and Israël. Afterwards he describes some tasks realized by mathematicians during this century: problems of classification (3 pages of drawings classifying knots, taken from Reidemeister back in 1932, almost without an explanation, but it looks nice), then proceeds to discuss the axiomatic method from a philosophical point of view, and ends with a brief description of the use of mathematics outside its realm (an entry like theology is rather surprising, even when ''justified'' by studies on logic from 1200 onwards). A very long bibliography enriches this paper: surprisingly \textit{J. Dieudonné}'s relevant book Panorama des mathématiques bourbachiques [Paris 1977, translation into English (New York 1982; Zbl 0482.00003)] is not mentioned, but the author apparently restricts his choice to books in English or German. (Only one Russian title is quoted from Lobatchevskij, one in Latin from Bolyai and one in French from Poincaré.