The uniqueness of the parallel lines is independent from the analogous statement on parallel planes and the usual further axioms of three-dimensional affine geometry. MSC: 51A15, 03F65.

On the occasion of the 150th birthday of Georg Cantor (1845–1918), the founder of the theory of sets, the development of the logical foundations of this theory is described as a sequence of catastrophes and of trials to save it. Presently, most mathematicians agree that the set theory exactly defines the subject of mathematics, i.e., any subject is a mathematical one if it may be defined in the language (i.e., in the notions) of set theory. Hence the nature of formal (...) definitions plays an important role within the logical foundations of mathematics. Its study is also helpful to answer the question of how it is possible that the set theory as a universal new ontology for the subject of mathematics (as people hoped around 1900) totally failed but nevertheless the language of set theory is successful in all the mathematical practice. (shrink)

Starting from the thesis that the history of mathematics, for saving its duration and support as a scientific discipline, has to look for the needs and problems of contemporary mathematics, six main problems of contemporary mathematics are listed (concerning partly its social state) and subsequently, 16 questions about the history of mathematics. Moreover, some analogues between the work-sharing of the historians of mathematics and that of the scientists are stated.