Creating New Concepts in Mathematics: Freedom and Limitations, Zbigniew Semadeni

Описание: The celebrated dictum of Georg Cantor “The very essence of mathematics lies precisely in its freedom” (1879) expressed the idea that in mathematics one can freely introduce new notions (which may, however, be abandoned if found unfruitful or inconvenient). Cantor opposed the efforts of Kronecker who demanded to restrict the objects of mathematics to constructive, finitistic ones only. It is clear that the freedom of mathematics is limited by logical constraints. A mathematician trying to prove a theorem knows the feeling of an invisible wall which blocks the intended argument. Moreover, new concepts must be consistent with earlier ones and must not lead to contradiction or ambiguity. Jan Łukasiewicz (1910) distinguished constructive notions from reconstructive ones, i.e., empirical. He referred (with reservation) to Dedekind’s statement that the constructed notions are free creation of the human mind and pointed out that a consequence of our “creation” of those notions is the spontaneous emergence of countless relations which no more depend on our will. The purpose of the talk is to discuss the question of freedom vs. limitation in the development of mathematics combined some traditional philosophical issues. We will look for criteria to distinguish between concepts which are natural follow-ups of the previous ones and concepts which were opening new ways of thought and reasoning, with particular attention to the rise of category theory.

Krakow Methodological Conference: 2019 is financed from the funds of the Minister of Science and Higher Education allocated to the dissemination of science [761/P-DUN/2019].
Krakow Methodological Conference: 2019 - zadanie finansowane w ramach umowy 761/P-DUN/2019 ze środków Ministra Nauki i Szkolnictwa Wyższego przeznaczonych na działalność upowszechniającą naukę.

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