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Gottfried Köthe
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Descartes,
Leibniz, and 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
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Historia Mathematica Heidelbergensis
Homo Heidelbergensis