Klein’s growing influence can only be understood by examining the way in which he worked, which David Hilbert (1862–1943) once described as selfless and always in the interest of the matter at hand.
The young Felix Klein internalized, from his upbringing and early education, a strong work ethic, which he maintained throughout his life. Stemming from a family of Westphalian tradesmen and farmers, his father had risen high through the ranks of the Prussian civil service and had impressed upon his children such virtues as unwavering discipline and thriftiness. That such lessons continued to be imparted throughout Klein’s time at secondary school is evident from his following recollection: “We learned to work and keep on working” (Klein 1923). The essay that Klein wrote for his Abitur contains the following sentence, with a reference to Psalm 90:10: “Indeed, if a life has become valuable, it has done so, as the Psalmist says, on account of labor and toil” [Gymnasium Düsseldorf]. This creed increasingly defined his daily approach to work.
Whereas, in his younger years, Klein was known to meet up with colleagues and hike in the mountains, and although he continued take walks with colleagues and with his family into old age, over time he refrained, on account of his health, more and more from participating in pleasantries unrelated to his work. He devoted every possible minute to pursuing his research and to helping his (male and female) doctoral students and post-doctoral researchers, from Germany and abroad, advance their own work. To this end, he met with each of them on a regular basis. The number of projects and positions that he took on reduced his free time to such an extent that his supportive wife was able to remark that they could hardly ever spend their wedding anniversary or birthdays together because priority was always given to his duties at the university. This tendency to overwork took its toll. After a long stay in a sanatorium, Klein retired early at the age of sixty-three. Even in retirement, however, he remained highly active. He gave lectures on the history of mathematics, made contributions to the theory of relativity, and continued to exert influence over hiring decisions, the formation of new committees, and book projects, among them his own collected works (Klein 1921–1923). Collaborators and colleagues would visit him at home where, though confined to a wheelchair, he refused to waste any time.
Klein was aware that he could not work without cooperation, and this pertained to both his scientific and organizational undertakings. On October 1, 1876, for instance, he wrote the following words to Adolph Mayer: “It is a truly unfortunate scenario: When, as on this vacation, I only have myself to consult, then I am unable to complete anything of value. […] I need scholarly exchange, and I have been yearning for the beginning of the semester for some time now” (quoted from Tobies and Rowe 1990, p. 76). Already accustomed, while studying under Plücker, to developing new ideas through discussion, he had carried on this practice while working with his second teacher, Clebsch. Clebsch’s ability to find connections between distinct areas of mathematics that had hitherto been examined in isolation became a point of departure for Klein’s own research methods.
During his time studying in Berlin, Klein cooperated with the Austrian mathematician Otto Stolz (1842–1905) to develop the idea of combining non-Euclidian geometry with the projective metric devised by the British mathematician Arthur Cayley (1821–1895). With Ludwig Kiepert (1846–1934), a student of Karl Weierstraß (1815–1897), Klein made his first attempt to delve into the theory of elliptic functions. His most fruitful collaboration, however, was with the aforementioned Sophus Lie. They supported one another, published together, and maintained an intensive mathematical correspondence. Klein, moreover, went out his way to promote Lie’s career (see Rowe 1989; Stubhaug 2002). Even though they came to disagree over certain matters later in life, Klein took these differences in stride and, in 1897, even endorsed Lie’s candidacy to receive the inaugural Lobatschewski Prize (see Klein, GMA 1923).
Beginning in 1874, Klein also enjoyed a strong collaborative relationship with Paul Gordan, who had likewise studied under Clebsch. Both Lie and Gordan found it difficult to formulate their own texts, and so Klein was often asked to help them by editing their writing and systematizing their ideas. By recording their thoughts, he immersed himself in them and expanded his own knowledge. Through his discussions with Gordan, and on the basis of the latter’s knowledge of algebra, Klein entered into a wide—ranging field of research. Working together with students and colleagues at home and abroad, he combined the methods of projective geometry, invariant theory, equation theory, differential equations, elliptic functions, minimal surfaces, and number theory, thus categorizing various types of modular equations.
Klein applied this cooperative approach wherever and whenever he worked, vacations and research trips included. Even if not every mathematician from within Klein’s sphere in Leipzig and Göttingen was willing to collaborate with him, everyone who sought his advice benefited from it. Here there is not enough space to list all of these beneficiaries. Prominent examples include Robert Fricke (1861–1930) and Arnold Sommerfeld (1868–1951), who edited books based on Klein’s lectures and took his ideas in their own creative directions. Another mathematician worthy of mention is David Hilbert, who profited in Königsberg from the tutelage of Klein’s student Adolf Hurwitz (1859–1919) and earned his doctoral degree under the supervision of Klein’s student Lindemann, who was mentioned above. Klein personally supported Hilbert beginning with the latter’s first research stay in Leipzig (1885/86); he recommended Hilbert to travel to Paris, maintained a correspondence with him (see Frei 1985), and secured a professorship for him in Göttingen (1895). There they conducted several research seminars together, and Hilbert, despite many enticing invitations to leave, remained Klein’s colleague at that university.
Klein’s skill at cooperating was also reflected in his activities as an editor: for the aforementioned Mathematische Annalen; for the Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (B. G. Teubner, 1898–1935), which appeared in an expanded (and partially incomplete) French edition (see Tobies 1994; Gispert 1999); for the project Kultur der Gegenwart (see Tobies 2008); and for the Abhandlungen über den mathematischen Unterricht in Deutschland, veranlasst durch die Internationale Mathematische Unterrichtskommission (5 vols., B. G. Teubner, 1909–1916). Klein was able to connect a great number of people who collaborated on these projects.
Ever since Klein’s years at the Technical College in Munich (1875–80), engineers and business leaders also numbered among his collaborative partners. While a number of engineers and technical scientists in the 1890s were initiating an anti-mathematics movement (Hensel et al. 1989), Klein was able to keep things in balance. In 1895, he joined the Association of German Engineers (Verein deutscher Ingenieure) as a mathematician; and, regarding mathematical instruction, he instituted a more applications-oriented curriculum that included actuarial mathematics and teacher training in applied mathematics. In order to finance the construction of new facilities in Göttingen, Klein followed the American model and sought funding from industry. His solution, which was novel in Germany at the time, was the Göttingen Association for the Promotion of Applied Physics and Mathematics (Göttinger Vereinigung zur Förderung der angewandten Physik und Mathematik). Initially founded exclusively for applied physics in 1898 and extended to include mathematics in 1900, this organization brought together Göttingen’s professors of mathematics, physics, astronomy, and chemistry with approximately fifty financially powerful representatives of German industry. In this way, Klein convinced industrial leaders that one of their goals should be to improve the application-oriented education of future teachers. The Ministry of Culture supported this initiative by introducing a new set of examinations—developed by Klein—that, for the first time, included the field of applied mathematics (1898). This, in turn, provided the impetus for establishing new institutes and professorships for applied mathematics, technical mechanics, applied electricity research, physical chemistry, and geophysics (see Tobies 1991, 2002, 2012, ch. 2.3). With these developments in mind, Klein began to shift the focus of his teaching more and more toward applications and questions of pedagogy. In his seminars, he no longer only cooperated with Hilbert and others on teaching “pure” mathematics but rather also with newly hired professors and lecturers to teach applied fields as well mathematical didactics (see [Protocols]).
From the beginning, Klein’s approach was distinguished by its internationality. He profited early on from the international networks of his teachers Plücker and Clebsch, and he came away with the general impression “that we restrict ourselves to a level that is far too narrow if we neglect to foster and revitalize our international connections” (a letter to M. Noether dated April 26, 1896; quoted from Tobies and Rowe 1990, p. 36). Klein lived by these words even when the officials at the Prussian Ministry of Culture did not yet value such things: “We have no need for French or English mathematics,” or so the ministry responded in 1870 when, at his father’s prompting, he sought a recommendation for his first trip abroad (see Klein 1923).
Proficient in French since his school days and an eager learner of English, Klein developed his own broad network of academic contacts beginning with his first research trips to France (1870), Great Britain (1873), and Italy (1874). This served his research approach well, which was to become familiar with and integrate as many areas of mathematics as possible, and it also benefited the Mathematische Annalen, for which he sought the best international contributions in order to surpass in prestige the competing Journal für die reine und angewandte Mathematik (Crelle’s Journal), which was edited by mathematicians based in Berlin. His international network also helped to the extent that many of his contacts sent students and young scientists to attend his courses. Even while Klein was in Erlangen, Scandinavian students (Bäcklund, Holst) came to study with him at the recommendation of Lie; while in Munich, he was visited by several Italian colleagues, and after his second trip to Italy (1878), young Italian mathematicians (Gregorio Ricci-Curbastro, Luigi Bianchi) came to study under him (see [Protocols], vol. 1; Coen 2012). Gaston Darboux (1842–1917), with whom Klein had corresponded even before his first trip to Paris and with whom he had collaborated on the review journal Bulletin des sciences mathématiques et astronomiques, sent young French mathematicians to work with him both in Leipzig and in Göttingen. Darboux was the first person to commission a translation of one of Klein’s works into a foreign language—Sur la géométrie dite non euclidienne (1871)—and they would go on to work together for many years, work that included their participation on prize committees, teaching committees, and bibliographies (Tobies 2016).
During Klein’s first semester in Leipzig (1880/81), the following international students (among others) came to work with him: Georges Brunel (1856–1900), recommended by Darboux; the Englishman Arthur Bucheim (1859–1888), who had been educated at Oxford by Henry John Stephen Smith (1826–1883); Guiseppe Veronese (1854–1917), at the instigation of Luigi Cremona (1830–1903); and Irving W. Stringham (1849–1917), who had already earned a doctoral degree under James Joseph Sylvester (1814–1897) at Johns Hopkins University in Baltimore. Under Klein’s direction, they produced findings that were published in the Mathematische Annalen (Veronese in 1881 and 1882, Brunel in 1882) or in the American Journal of Mathematics (Stringham in 1881). To Daniel Coit Gilman, the president of Johns Hopkins, Stringham wrote enthusiastic letters about Klein’s critical abilities and about the international nature of his seminars. When Stringham’s former teacher Sylvester left his position in Baltimore, Klein was invited in 1883 to be his successor. Klein declined the offer for financial reasons, which was itself a sign of his international reputation. Ever since his time in Leipzig, Klein also made conscious efforts to enhance his relations with Russian and other Eastern European mathematicians. Wishing to foster exchange, he would always request his students from these areas to provide him with an overview of the institutions there, their staff, and their research trends.
In Göttingen, and thus back under the purview of the Prussian Ministry of Culture, Klein had to decline an invitation in 1889 to work as a visiting professor at Clark University in Worcester, Massachusetts (USA) because the Ministry did not approve ([UBG] Ms. F. Klein I, B 4). After securing his position, however, he ultimately travelled in 1893 with the official endorsement of the Ministry to Chicago for the World’s Fair, which included an educational exhibit and which was being held in conjunction with a mathematics conference. While there, Klein gave twelve presentations on the latest findings in mathematics. He spoke about the work of Clebsch and Sophus Lie, algebraic functions, the theory of functions and geometry, pure and applied mathematics and their relation, the transcendence of the numbers e and π, ideal numbers, the solution of higher algebraic equations, hyperelliptic and Abelian functions, non-Euclidean geometry, and the study of mathematics at Göttingen (Klein 1894). In his talks, Klein gave particular weight to his own recent findings and to those of his students and collaborators, thus waging a successful publicity campaign for studying at the University of Göttingen (see Parshall and Rowe 1994). With these lectures, which were later translated into French at the instigation of Charles Hermite (1822–1901), Klein did much to increase his international profile.
During the 1890s, Hermite occasioned additional translations of Klein’s work (on geometric number theory, the hypergeometric function, etc.), most of which appeared in the Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, which was then edited by Charles-Ange Laisant (1841–1920). Hermite gushed that Klein was “like a new Joshua in the Promised Land” (comme un nouveau Josué dans la terre promise) and nominated him, in 1897, to become a corresponding member of the Académie des Sciences in Paris (Tobies 2016). By this time, Klein was already a member of numerous other academies in Germany, Italy, Great Britain, Russia, and the United States. When, in 1899, Laisant and the Swiss mathematician Henri Fehr (1870–1954) founded the journal L’Enseignement mathématique, Klein was made a member of its Comité de Patronage, which consisted of twenty mathematicians from sixteen countries. As the first international journal devoted to mathematical education, it published several reports concerning educational reforms, including essays by Klein (in French translation). Fehr reviewed Klein’s books for the journal, among them his Elementarmathematik vom höheren Standpunkte aus (“Elementary Mathematics from an Advanced Standpoint,” as the work would be known in English).
L’Enseignement mathématique became the official organ of the International Commission on the Teaching of Mathematics, which was founded in 1908 at the Fourth International Congress of Mathematicians in Rome. Klein’s election to the board of this commission, which took place despite his absence from the conference, was a testament to his international reputation (see Coray et al. 2003). As president of this commission (from 1908 to 1920), Klein initiated regular conferences and publications devoted to the development of mathematical education not only in Germany but in all of the countries involved.
Felix Klein followed a principle of universality. When asked to characterize his efforts, he himself spoke about his universal program. As a young researcher, he wanted to familiarize himself with all branches of mathematics and to contribute to each of them in his own work, an approach that gave rise to his principles of transference (Übertragungsprinzipien) and his “mixture” of mathematical methods. Inspired by Clebsch, he also attempted from quite early on to bring together people with different areas of mathematical expertise in an effort to overcome disciplinary divides (see Tobies and Volkert 1998). This end was likewise served by his large-scale undertaking of the Encyklopädie der mathematischen Wissenschaften, for which he recruited international experts to provide an overview of all of mathematics and its applications (Tobies 1994). Klein’s participation in the preparations for the International Catalogue of Scientific Literature (1902–21), which was directed by the Royal Society of London, can also be interpreted in this way.
Klein’s universal program not only involved supporting and advancing new and marginal disciplines. He applied his universal approach to teaching as well. He promoted talented scholars regardless of their nationality, religion, or gender. Although a university professor, he was deeply interested in improving and fostering mathematical and scientific education from kindergarten onward. In this regard, Klein operated according to one of the guiding pedagogical mottos of the nineteenth century: “Teach everything to everyone.”