Set theory was a case for Klein where this theoretical development was too fresh, and not yet accomplished and even further from having matured to the point of having induced an intra-disciplinary process of integration and restructuration. The concepts of set theory did not (yet) provide new elements for mathematics—hence Klein’s polemic against Friedrich Meyer’s schoolbook of 1885 who’s intention had been, in fact, to use set theory as new elements for teaching arithmetic and algebra (see Klein 2016a, b, p. 289, note 181). Meyer, mathematics teacher at a Gymnasium in Halle and friend of Cantor, introduced there the notions of set theory—not yet fully developed then by Cantor—as foundations for the number concept. Klein had sharply criticised this schoolbook in his first edition, but softened his critique in subsequent editions. In Klein’s times, mathematics had not achieved the level of architecture established by Bourbaki—and hence not of “modern math”.
Given that set theory has been almost identified with modernism in mathematics, I need to comment somewhat on the book by Herbert Mehrtens: Modeme—Sprache—Mathematik (1990), where he models Göttingen mathematics as bi-polar: Hilbert representing “modernism” and Klein representing “counter-modernism”. I was always critical of this book and Mehrtens’ assessments, since he misrepresents both mathematicians: Hilbert was not that theoretician and formalist who freely created abstract theories whom Mehrtens compared with the artists of that time as no longer bound by any claim to represent reality. And depicting Hilbert as “anti-intuitive” (1996, p. 521) deeply misunderstands Hilbert’s vision and practice of mathematics. Klein is, on the other hand, denounced by Mehrtens to be tied to reality and to intuition largely because German Nazi mathematicians later abused these notions.
Yet, one has to admit that Klein showed scepticism and reservation regarding set theory and axiomatics. On the one hand, he praised the progress in function theory brought about by Cantor’s new theories:
The investigations of George Cantor, the founder of this theory, had their beginning precisely in considerations concerning the existence of transcendental numbers. They permit one to view this matter in an entirely new light.
On the other hand, Klein warned against the abstractness of set theory. Thus, he showed misgivings when he spoke of the “modern” function concept launched by Cantor:
In connection with this, there has arisen, finally, a still more far-reaching entirely modern generalisation of the function concept. Up to this time, a function was thought of as always defined at every point in the continuum made up of all the real or complex values of x, or at least at every point in an entire interval or region. But since recently the concept of sets, created by Georg Cantor, has made its way more and more to the foreground, in which the continuum of all x is only an obvious example of a “set” of points. From this new standpoint functions are being considered, which are defined only for the points x of some arbitrary set, so that in general y is called a function of x when to every element of a set x of things (numbers or points) there corresponds an element of a set y. (Klein 2016a, b, p. 220)
Clearly, this abstract function concept was not at all adapted for Klein’s curricular reform programme with a function concept as its kernel, which could interrelate analysis and geometry. His misgivings were even stronger concerning his doubts as to whether all this might have applications:
Let me point out at once a difference between this newest development and the older one. The concepts considered under headings 1. to 5. have arisen and have been developed with reference primarily to applications in nature. We need only think of the title of Fourier’s work! But the newer investigations mentioned in 6. and 7. are the result purely of the drive for mathematical research, which does not care for the needs of exploring the laws of nature, and the results have indeed found as yet no direct application. The optimist will think, of course, that the time for such application is bound to come. (ibid.)
Given Klein’s intense plea for applications, one should remark, furthermore, that he not only alerted, in the first volume in the context of the emergence of set theory, against pushing a formalist programme for the foundations of mathematics too far, but he also had taken up the issue again in volume III of his Elementarmathematik, advising against searching for the New only for the sake of doing it:
Provided that a deep epistemological need exists, which will be satisfied by the study of a new problem, then it is justified to study it; but if one does it only to do something new, then the extension is not desirable. (Klein 2016a, p. 157)
Klein did even not exempt Hilbert from his critical scepticism: he commented upon Hilbert’s research on the foundations of arithmetic to establish the consistency of operating with numbers:
Obviously one can then operate with a, b, c,…, precisely as one ordinarily does with actual numbers. (Klein 2016a, b, p. 14)
The tendency to crowd intuition completely off the field and to attain to really pure logical investigations seems to me not completely realisable. It seems to me that one must retain a remainder, albeit a minimum, of intuition. (ibid., p. 15)
Klein added, however, a cautious remark that he did not want to criticise Hilbert, albeit in a rather implicit manner:
I have felt obliged to go into detail here very carefully, in as much as misunderstandings occur so often at this point, because people simply overlook the existence of the second problem. This is by no means the case with Hilbert himself, and neither disagreements nor agreements based on such an assumption can hold. (ibid., p. 16)
The second problem, which Klein is emphasising here, was put by him as the epistemological aspect of the task of justification of arithmetic—and his intention was to say that it should not be overlooked when researching upon the logical aspect of justification of arithmetic.