Commentary on Labirynth of Thought. A History of Set Theory and its Role in Modern Mathematics

  August Crelle 1780 - 1855 𝔅𝔢𝔯𝔩𝔦𝔫 University,¹ founded 1810, soon turned into the most important university in Prussia and all the German-speaking countries. Gauss might have ended up being a professor there, since Alexander von Humboldt made two attempts to bring him to Berlin, in the 1800s and 1820s.² The already mentioned Martin Ohm became an extraordinary professor in 1824 (and was named "ordinary" professor in 1839, at the same time as Dirichlet ). But the situation in mathematics was not notable until 1828, when Humboldt and Crelle began to play an important role in their respective positions as court counselor and adviser on mathematics for the ministry. There was a tragically failed attempt to bring Abel to Berlin in 1829, the year of his death, but with Dirichlet and, from 1834, Jacob Stiener ,³ lectures of a high level began to be offered. Dirichlet is usually considered to have shaped modern-style mathematics lectures, and also established an informa...

  Felix Klein 1849 - 1925 𝔗𝔥𝔢 University of Göttingen was the most advanced German one in the late eighteenth-century [ McClelland 1980], and is famous in connection with mathematics, since it was here that Gauss worked in the early nineteenth-century, and by 1900 it had become a leading research center under Klein and Hilbert .¹ But the Göttingen of 1850 was quite different from such a center. The teaching of mathematics was far from advanced. Gauss was the professor of astronomy, and he was not attracted by the prospect of teaching poorly prepared and little interested students the basic elements of his preferred discipline. Thus, he only taught some lectures on a restricted field of applied mathematics, for instance on the method of least squares and on geodesy [ Dedekind 1876, 512; Lorey 1916, 82].²  𝔊𝔞𝔲𝔰𝔰 was a retiring man, and, strange as it may seem, he was particularly hard to approach for mathematicians, less so for astronomers and physicists. Therefore...

 ℑ𝔱 has frequently been written that the Aristotelian horror infiniti  reigned among scientists and mathematicians until Cantor's vigorous defense of the possibility and necessity of accepting it. Like other extreme statements, this one does not bear a historical test. At least in Germany, and perhaps here it makes much sense to speak of national differences, there was a noticeable tendency to accept the actual infinite. The philosophical atmosphere could hardly have been more favorable, and there were several attempts to develop mathematical theories of the infinite. We shall begin with philosophy, and then consider the views of some mathematicians.  𝔗𝔥𝔢 history of philosophical attitudes toward actual infinity in 19th-century Germany would be a long one. By the beginning of the century, during the time of idealism, the potential infinity of mathematics was called the "bad infinite" by Hegel and his followers. The implication was clear: there was a 'good' in...

 𝔗𝔥𝔢 great British empiricist Hume , in this epistemological works, emphasized the fallibility of empirical knowledge. Perhaps for this reason, one of Kant's basic presuppositions is that anything that is absolutely certain must not be empirical, but based solely upon the mental constitution of the subject. Necessary truth, including mathematical truth, must be a priori: truley mathematical propositions are always a priori, not empirical, judgements, since they involve necessity, which cannot be gained from experience.¹  David Hume 1711 – 1776 𝔄𝔰 we have seen, Kant regarded the world as a representation in the subject, partly determined by the impressions recieved from the "things in themselves," partly by a priori characteristics of the subject's sensitivity and conceptual abilities. As is well known space and time were not to be seen as traits of the external world, but as a priori forms of our sensitivity or intuition [Anschauung], that determine our repres...

𝔗𝔥𝔢𝔯𝔢 have always been many possible approaches to the philosophical problems raised by mathematics – empiricism, Platonic realism, intellectualism, intuitionism, formalism, and many other intermediate possibilities. From what we have seen, one many expect to find, among 19th-century German mathematicians, an influence of philosophical ideas and a greater speculative tendency then among their foreign colleagues. Interesting early examples that will not be discussed in detail here are those of Bolzano and Kummer . But more relevant for our purposes is the fact that the influence of philosophy seems to have led to an increase of intellectualist view points in 19th-century Germany .  Immanuel Kant 1724 — 1804 𝔎𝔞𝔫𝔱𝔦𝔞𝔫 philosophy has always been more congenial to scientists then idealism, so it is not surprising that during and after the acme of idealism it retained an importation status among them. A characteristic Kantian idea is that the subject (the philosophical I) en...