The Motives behind Cantor's Set Theory – Physical, Biological, and Philosophical Questions

Abstract

The celebrated “creation” of transfinite set theory by Georg Cantor has been studied in detail by historians of mathematics. However, it has generally been overlooked that his research program cannot be adequately explained as an outgrowth of the mainstream mathematics of his day. We review the main extra-mathematical motivations behind Cantor's very novel research, giving particular attention to a key contribution, the Grundlagen (Foundations of a general theory of sets) of 1883, where those motives are articulated in some detail. Evidence from other publications and correspondence is pulled out to provide clarification and a detailed interpretation of those ideas and their impact upon Cantor's research. Throughout the paper, a special effort is made to place Cantor's scientific undertakings within the context of developments in German science and philosophy at the time (philosophers such as Trendelenburg and Lotze, scientists like Weber, Riemann, Vogt, Haeckel), and to reflect on the German intellectual atmosphere during the nineteenth century.

The Critics and Contributions of Mathematical Philosophy in Hong Kong Secondary Education

“…In addition, the set of derived points is also denumerable. In this case, Cantor observed that there was a link between his results from 1874 and the continuum [9]. This indicated that he was interested in -the Labyrinth of infinity and the continuum‖.…”

“…He also introduced the concept of derived sets (exceptional sets of point P). Derived sets later became a very important tool for both theories of real functions and integration [9]. Cantor also showed that there are some infinite sets of points which are not relevant to the representation question of real functions.…”

“…That is, if we modify the normal ANN into a causal-dependency one, there will be another kind of analysis(like the one in Bayesian network or the causal dependency one) for us to explore. 8,9 This will be my final version of the generalised HKLam theory.…”

Theological Underpinnings of the Modern Philosophy of Mathematics.

“…In this very context, it should be remembered that Cantor's theory of transfinite numbers, which was held in high esteem by Hilbert and was praised by him as "the most admirable flower of the mathematical intellect and in general one of the highest achievements of purely rational human activity (die bewundernswerteste Blüte mathematischen Geistes undüberhaupt eine der höchsten Leistungen rein verstandesmäßiger menschlicher Tätigkeit)" (Hilbert, 1926, S. 167;van Heijenoort, 1967, p. 373), was intensively underpinned by theological and religious considerations, at least in the case of its creator. On the one hand, the theory of transfinite numbers is a characteristic example of new mathematics, but on the other, for Georg Cantor, his mathematics remained closely tied both to theology and natural philosophy (Ferreirós, 2004).…”

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“…The Cantor set was discovered in 1875 by the British mathematician Henry John Stephen Smith [5] and studied and introduced for the first time in 1883 by the German mathematician Georg Cantor [6][7][8]. The Cantor set acquires a very important role in many branches of mathematics, mainly in set theory and fractal theory [9,10].…”