Newton. The chief domains of this classical mathematics are analysis in its various branches, algebra, number theory and geometry. The wellknown paradoxes in set theory discovered at the end of nineteenth century have led to an entirely new understanding of the foundations of mathematics. The so-called platonic viewpoint, which looks at numbers as entities existing independently of the human mind, is now abandoned, numbers are now considered as existing only if they can be constructed in a number of finite steps. This new viewpoint led to many difficulties. The classical disciplines have to be rebuilt. This was only possible by an exact analysis of mathematical thinking; the methods of mathematical logic were developed, by them it was possible to give a proof of the non-contradictory nature of the classical disciplines of mathematics. But this new understanding of the foundations of mathematics is only one side of modern mathematics. The axiomatic method, first developed in geometry, led to a new classification of mathematical notions. The basic disciplines of modern mathematics are algebra, theory of ordered sets and topology. This is shown by an analysis of the wellknown concept of a real number. By reducing every mathematical concept to a few simple basic notions, the so-called elementary structures, the whole of mathematics now appears as a unity, a hierarchy of structures, wherein the classical disciplines and many new branches of mathematics finds their places. A group of French mathematicians, under the name of Bourbaki, are now writing a systematic exposition of the elements of mathematics from this new viewpoint.
Redaktion: Gabriele Dörflinger
Zur Inhaltsübersicht Historia Mathematica Heidelbergensis Homo Heidelbergensis