𝟒. 𝔗𝔥𝔢 𝔊𝔬𝔱𝔱𝔦𝔫𝔤𝔢𝔫 𝔊𝔯𝔬𝔲𝔭

 

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, although Riemann and Dedekind established some contact with him, and the second was his doctoral student, one should not expect much influence from the direct contact with him. Rather, it was through his writings that both were highly influenced and became his followers. 

𝔗𝔥𝔢 teaching of mathematics proper was in the hands of a rather unimportant Ulrich and of Stern, who was already been named. The later was a specialist in number theory, with a difficult carrer because of being a Jew, although he seems to have been a good professor [Lorrey 1916, 81-82]. Ulrich and Stern founded in 1850 a mathematico-physical seminar for the training of Gymnasium teachers, also led by Wilhelm Weber and another physics professor. To sum up, despite the presence of the "princeps mathematicorum," Göttingen was still a rather traditional university as far as mathematics is concerned, university lectures were not adjusted to the high level of contemporary research, since the characteristically German combination of teaching and research had not yet arrived. Dedekind says that the teaching was perfectly sufficient for the purpose of preparing students for the Gymnasium entrance examination, but quiet insufficient for a more through study [Lorey 1916, 82]. Projective geometry, advanced topics in number theory and algebra, the theory of elliptic functions, and mathematical physics were not available [ibid]. 

Wilhelm Eduard Weber 1804 - 1891

𝔗𝔥𝔢 situation was different with physics. Wilhelm Weber had began to incorporate Göttingen to the modernizing trend: 

𝔚𝔢𝔟𝔢𝔯’𝔰 extensive lecture course on experimental physics, in two semesters, made the most profound impression on me. The strict separation between the fundamental facts, discovered by means of the simplest experiments, and the hypotheses linked with them by the thinking human mind, afforded an unmatchable model of the truly scientific research, as I had never known until then. In particular, the development of electricity theory had an enormously stimulating effect....³ 

𝔗𝔥𝔲𝔰, Weber was teaching on his own research topics, which brought new standards of precision to electrical experimentation, and provided a first-rate theoretical contribution with his unification of electrostatics, electrodynamics and induction.⁴ This came after his collaboration with Gauss on terrestrial magnetism in the 1830s, which, by the way, was the occasion for Gauss's research on potential theory. Interestingly, Weber's methodology, as described by Dedekind, coincides essentially with Riemann's in his frame work in geometry [Riemann 1854]. 

𝔗𝔥𝔢 mathematico-physical seminar soon went beyond its initial purpose of training secondary-school teachers, to assume the function of affording a better laboratory training. Both Dedekind and Riemann participated in it [Lorey 1916, 81], although it was the latter who got more involved in physical practices, eventually becoming Weber's teaching assistance around 1854 [Dedekind 1876, 512-13, 15]. Riemann's collaboration with Weber was the background for his many efforts to establish a unified treatment of the laws of nature. A manuscript note that must come from this or a later period, indicating Riemann's research topics, mentions his great research on Abelian and other transcendental functions, and on the integration of partial differential equations, and goes on: 

𝔐𝔶 main work deals with a new conception of the known natural laws – expression of them by means of different basic notions — which would make it possible to employ the experimental data on the interactions between heats, light, magnetism and electricity, in order to investigate their interrelation.⁵ 

𝔗𝔥𝔦𝔰 was also consequential for his mathematical work. Among experts on Riemann it is commonplace that, in his mind, ideas of a physical origin found a natural development in pure mathematics, and conversely [Bottazzini 1977, 30]. 

𝔗𝔥𝔢 combination of teaching and research came to Göttingen mathematics after Gauss's death. In order to maintain the university's renown in mathematical research, his professor ship was divided into one for pure mathematics, and through the good offices of Weber, Dirichlet received and accepted the call.⁶ This "opened up a new era for mathematical studies at Göttingen," not because Dirichlet established new organizational arrangements (as Clebsch and Klein later), but because his brilliant lectures went all the way up to the research frontier. As Dedekind said, "through his teaching, as well as frequent conversations...., he turned me into a new man. In this way he had an enlivening influence on his many students...."⁷ 

𝔗𝔥𝔢 presence of Dirichlet, Riemann and Dedekind at Göttingen, and the courses they tought from 1855 onward, turned Göttingen into one of the most important mathematical centers, to be compared only with Berlin and Paris. The three mathematicians had common traits in that they all followed lines initiated by Gauss, and promoted an abstract, conceptual vision of mathematics. They valued each other very much. In 1852, one year after the death of Jacobi, Riemann wrote that Dirichlet was, with Gauss, the greatest mathematician alive [Butzer 1987, 58]. Dirichlet also regarded Riemann as the most promising young mathematician in Germany. And after Gauss's death, in 1856, Dedekind wrote that Riemann was, after or even with Dirichlet, the most profound living mathematician [Scharlau 1981, 37]. Within the group, Dedekind had the role of the student who, even as a privatdozent, enjoyed for the first time the opportunity of a first-level mathematical education. Both Dirichlet and Riemann left him important posthumous duties. 

𝔚𝔦𝔱𝔥 regard to research fields, those of Riemann and Dedekind were quiet far apart, since the former excelled in function theory and mathematical physics,  while the latter devoted himself to algebra and number theory. In a sense, they divided among themselves the topics studied by Dirichlet, who was the most universal of them all. (This is understandable, since there was a growing trend towards specialization all along the century.) But, underlying these differences, it is possible to find meeting points at a deeper level, that of theoretical and methodological preferences. This, and the net of influences they exerted on each other, makes it justifiable to speak about a group with common traits.⁸ 

Augustin Louis Cauchy 1789 – 1857 

𝔄𝔩𝔱𝔥𝔬𝔲𝔤𝔥 the point will become clearer in what follows (particularly chapters II, III, V), it is possible to give some examples at this point. The conceptual approach to mathematics is clear, for instance, in Cauchy, when he basis his treatment of analysis [Cauchy 1821] upon the notion of a continuous function, where continuity is defined independently of the analytical expressions which may represent the function.⁹ Such a viewpoint is taken further by Dirichlet when, in a paper on Fourier series [1837, 135-36], he purposes to take a function to be any abstractly defined, perhaps arbitrary correlation between numerical values.¹⁰ Already in 1829 he had given the famous example of the function f(𝒙) = 0 for rational 𝒙, f(𝒙) = 1 for irrational 𝒙 [Dirichlet 1829, 169], which he wrongly took to be a function not representable by an analytical expression, and a non-integerable function. Riemann takes up Dirichlet's abstract notion of function in his function-theoretical thesis [Riemann 1851, 3] where he also makes reference to his teacher's work on the representability of piecewise continuous functions by means of Fourier series (see chap. V). This became the subject of his Habilitation thesis [Riemann 1854b], where we find the famous definition of the Riemann Integral, a definition that finally consolidated the abstract notion of function, since it opened up the study of discontinuous real functions. Lastly, Dirichlet's notion of function was given its most general expression when Dedekind [1888, 348] defined, for the first time, the notion of mapping within a set-theoretical setting. 

𝔗𝔥𝔢 emergence of the conceptual viewpoint was mentioned by Dirichlet in a frequently quoted passage in his obituary of Jacobi (see also Eisenstein's description in §1). According to him, there is an ever more prominent tendency in recent analysis "to put thoughts in the place of calculations."¹¹ It is interesting, though, to consider the context of Dirichlet's quotation. He was calling attention to the fact that, in spite of that tendency, there were fields in which calculations preserved their legitimecy, and that Jacobi had obtained admirable results in this way. Also in the work of Dirichlet it is possible to find a clever combination of new thoughts, which sometimes are extremely simple (i.e., the box principle in number theory), with complex analytical calculations. One has the impression that it was above all Riemann who brought the conceptual trend to a new level, and his friend Dedekind followed his footsteps. Among other reasons to think so, we may consider the fact that the approach promoted by Riemann and Dedekind was quite specific in comparison with those of other Dirichlet students, such as Heine, Lipschitz or Eisenstein, not to mention Kronecker. Dedekind [1876, 512] tells us that, around 1848, Riemann discussed that with his friend Eisenstein the issue of the introduction of complex numbers in the theory of functions, but they were of completely different opinions. Eisenstein favoured the focus on "formal calculation," while Riemann saw the essential definition of an analytic function in the Cauchy–Riemann partial differential equations (see below). 

ℑ𝔫 this connection it is telling that Dedekind's most committed methodological statements consistently refers to Riemann's function theory as a model. For Dedekind always regarded himself as a disciple of Dirichlet. In 1859 when he had already left Göttingen, he wrote to his family that he owed to Dirichlet more than to any other man [Scharlau 1981, 47]. His debt to Dirichlet was particularly important in connection with his general formation as a mathematician, with number theory, and with the issue of mathematical rigor [Haubrich 1995, ch. 5]. Nevertheless, when it came to mathematical methodology and the conceptual approach, the main name he mentioned was Riemann. The following text, written in 1895, is interesting enough to warrant full quotation. Dedekind therein defended his approach to the foundations of algebaric number theory, in contrast to those of Kronecker and Hurwitz. This gives the occasion for a passage in which he opens his "mathematical heart" [Dugac 1976, 283]: 

𝔉𝔦𝔯𝔰𝔱 i will remember a beautiful passage in Disquisitions Arithmaticae, which already in my youth made the most profound impression upon me. In art. 76 Gauss relates that Wilson's theorm was first made known by Waring, and goes on: "But niether could approve it, and Warning confesses the proof seems the more difficult, as one cannot imagine any notation that could express a prime number. – In our opinion, however such truths should be extracted from concepts rather than notations." In these last words, if they are taken in the most general sense, we find the expression of a great scientific thought, the decision for the inner in contrast to the outer. This contrast comes up again in almost all fields of mathematics; it suffices to think about function theory, about the Riemannian definition of functions by means of characteristic inner properties, from which the outer forms of representation arise with necessity. But also in the much more limited and simple field of ideal theory both directions are effective …¹² 

𝔄𝔱 this point, Dedekind mentions that he has always set himself such requirements, and refer to a passage of an 1876 paper where he again takes Riemann's function theory as a model, and states that a theory "founded upon calculation would not offer the greatest degree of perfection."¹³ A similar idea is found in a letter to Lipschitz written the same year: 

𝔐𝔶 efforts in the theory of numbers are directed toward basing the investigation, not on accidental forms of representation (or expressions), but on simple basic notions, and thereby – though this comparison may perhaps seem pretentious – to attain in this field something similar to what Riemann did in the field of function theory.¹⁴ 

𝔗𝔥𝔢 relevant details of Riemann's function theory will be mentioned in chap. II, while those concerning Dedekind's ideal theory can can be found in chap. III. At this point, however, we can give some simple examples which will be instructive, since they enable us to contrast the Göttingen abstract approach with the viewpoint adopted at Berlin (studied in the next section). Riemann sought, in his theory, to found global, abstract ways of determining complex functions by means of minimal sets of independent data [Riemann 1857, 97]. As he had written in his 1851 dissertation: 

𝔄 theory of those functions [algebraic, circular or exponential, elliptical and Abelian] on the basis of the foundations here established would determine the configuration of the function (i.e., its value for each value of the argument) independently of any definition by means of operations [analytical expressions]. Therefore one would add, to the general notion of the function of a complex variable, only those characteristics that are necessary for determining the function, and only then would one go over to the different expressions that the function can be given.¹⁵ 

ℜ𝔦𝔢𝔪𝔞𝔫𝔫 starts with what is general and invariant, and from it the many possible analytical expressions for the function would be derived. He thus needed a general definition of analytic function which, as we have already mentioned, he found in the Cauchy-Riemann equations [Riemann 1851, 5-6]: 

𝔚𝔥𝔢𝔯𝔢 𝒖 and 𝒗 are the real and complex parts of f(𝒙÷𝒊𝒚).¹⁶ This is a perfect example of his preference for simple fundamental properties as a basis for the developmemt of the theory, which contrasts nicely with Weierstrass's definition of analytic functions (next section). Further aspects of his theory, such as the introduction of Riemann surfaces, were consistent with that preference. 

𝔖𝔦𝔪𝔦𝔩𝔞𝔯𝔩𝔶 Dedekind preferred to use basic notions of an abstract character, such as his fields and ideals. In this, more basic case, avoidance of forms of representation led to reliance on sets endowed with an algebraic structure. A field [Körper] was for him any subset of ℂ that is closed with respect to the four basic algebraic operations [Dedekind 1871, 224]. An idea was a set of algebraic integers which is characterized by two simple properties that can be stated in terms of sums, differences and products of algebraic integers [op.cit., 251]. As in the case of Riemann the decision to the base the theory upon such basic notions deviated sharply from established tradition, since it was then customary to focus on the numbers themselves, not on sets of numbers. Besides, it was customary in number theory to use 'forms,' or algebraic equations, freely in the development of the theory, but Dedekind avoided that completely since it meant reliance on expressions. His main reason for preferring abstract notions was their generality and lack of arbitrariness, which meant that they immediately conveyed what was "invariant" (his term) in the object defined, be it a field or an ideal. Once again, this definition of the basic objects, and the further development of the theory, is in sharp contrast to Kronecker's (next section). 

𝔄𝔱 this point it should be clear that Riemann and Dedekind brought foreward the conceptual tendencies that could be found, more or less clearly, in the work of Gauss, Cauchy and Dirichlet. In doing so, however, they effected a clear turn in the meaning of 'conceptual' which would be extremely consequential for mathematics, since their work had an enormous impact upon 20th-century function theory and most general appropriate setting, in such a way that "outter forms of representation" were avoided, new basic objects were chosen, and the definition of the characteristic "inner properties" of these objects (i.e., a fundamental concept) was placed at the very beginning of the theory. We may refer to this particular brand of the conceptual viewpoint as abstract conceptual. Thus, on of the characteristic traits of modern mathematics, which is frequently called its 'abstract' viewpoint, can be traced back to Riemann and Dedekind. 

𝔗𝔥𝔢 emergence of a research school following the abstract-conceptual approach to mathematics, promoted by Riemann and Dedekind was hindered by a confluence of events, Riemann, who succeeded Dirichlet at Göttingen in 1859, incapacitated since 1862 due to a lung illness, spending most of the time in Italy and dying in 1866 at forty. Dedekind was slow in publishing original research, so that he was offered almost no university position until 1870, and afterwards he consciously chose to remain at the Technical school in his birthplace Braunschweig. After Riemann's death, a great tradition of mathematicians consolidated at Göttingen with Clebsch, Fuchs and Schwarz, but these men came and went in rapid succession. It was only with the arrival of Klein in the 1880s, and later with Hilbert, the something that could be compared with a school was firmly established.¹⁷ The diffusion of the methodological viewpoint favored by Riemann and Dedekind was difficult and slow since the only available means, given the circumstances was their published work. Besides this, only Dedekind's rich correspondence with such mathematicians as Heinrich Weber, Forbenius and Cantor could have been instrumental in spreading that standpoint. It is worth mentioning that it might be quite interesting to study H. Weber as a key figure in the diffusion of the conceptual approach.¹⁸ 

1. Klein came to Göttingen in 1886, strongly supported by the minister, with the purpose of building a center that could be compared with Berlin. He remained until 1913, while Hilbert came in 1895, until 1930. See [Rowe 1989]. 

2. Lorey quotes in a full letter from Dedekind with reminiscence from this Göttingen time. 

3. Dedekind in [Lorey 1916, 82]: "hat mir die über zwei Semester vertheilte grosse Vorlesung von Weber über Experimentalphysik den tiefsten Eindruck gemacht; die strenge Scheidung zwischen den durch die einfachsten Versuche erkannten fundamentalen Tatsachen und den durch den menschlichen denkenden Geist daran geknüpften Hypothesen gab ein unübertreffliches Vorbild wahrhaft wissenschaftlicher Forschung, wie ich es bis dahin noch niemals kennen gelernt hatte, und namentlich war der Aufbau der Elektrizitätslehre von grossartiger begeistrender Wirkung... "

4. See [Jungnickel & Mc Commach 1986, 138-48]. Weber's main work was his 'Elektrodynamische Massbestimmungen' of 1846.

5. [Riemann 1892, 507; emphasis added]: "Meine Hauptarbeit betrifft eine neue Auffassung der bekannten Naturgesetze — Ausdruck derselben mittelst anderer Grundbegriffe — wodurch die Benutzung der experimentellen Data über die Wechselwirkung zwischen Wärme, licht, Magnetismus und Elektricität zur Erforschung ihers Zusammenhangs möglich würde."

6. Dedekind in [Lorey 1916, 82]. See [Jungnickel & McCommach 1986, 170 - 72]. Dirichlet used the call to try getting freed from the heavy teaching at the military academy, but the Prussian ministry reacted too slowly. 

7. [Lorey 1916, 82-83]: "… womit für das mathematische Studium in Göttingen eine neue Zeit anbrach, … er hat durch seine Lehre, wie durch häufige Gespräche in persönlichen Verkehr, der sich nach und nach immer vertrauter gestaltete einen neuen Menschen aus mir gemacht. So wirkte er belebend auf seine zahlrieche Schüler ein.... " see also [Scharlau 1981,  35ff]. 

8. In this case, I avoid the word "school" because there was not a relevant production of advanced students, that would later become research mathematicians (see below). 

9. Euler understood by "functiones continuae" those that correspondend to a single analytical expression throughout. See [Youschkevitch 1976]. 

10. There has been some debate whether Dirichlet ever thought about applying this concepts to functions more 'arbitrary' than piecwise continuous functions, but it is an expository paper published in a physics journal. It seems plausible to me that he entertained the abstract notion of function, but thought that in mathematics there is no need to consider highly arbitrary functions – except as counter-examples. 

11. [Dirichlet 1889, vol 2, 245]: "wenn es die immer mehr hervortretende Tendenz der neueren Analysis ist, gedanken an die Stelle der Rechnung zu setzen…"

12. [Dedekind 1930, vol. 2, 54–55]: "Ich erinnere zunächst an eine schöne Stello der Disquisitions Arithmeticae, die schon in meiner Jugend den tiefsten Eindruck auf micht gemacht hat. Im Art. 76 berichtet Gauss, dass der Wilsonsche Satz zuerst von Waring bekanntgematch ist, und fährt fort: sed neuter demonstrari potuit, et cel. Waring fatetur demonstrationem eo difficiliorem videri, quod nulla notatio fingi possit, quae numerum primum exprimat. – At nostro quidem judicio hujusmodi veritates ex notionibus quam ex notionibus hauriri debebant. – In diesen letzen Worten liegt, wenn sie im allgemeinsten Sinne genommen werden, der Auspruch eines grossen Wissenschaften gedankens, die Entscheidung für das innerliche im Gegensatz zu dem Äusserlichen. Dieser Gegensatz wiederholt sich auch in der Mathematik auf fast allen Gebieten: man denke nur an die Funkitionentheorie, an Riemann's Definition der Functionen durch innerliche charakteristische Eigenschaften, aus welchen die äusserlichen Darstellungsformen mit Notwendigkeit entspringen. Aber auch auf dem bei weitem enger begrenzten und einfacheren Gebiete der Idealtheorie kommen beide Richtungen zur Geltung… "

13. [Dedekind 1930, vol. 3, 296]: "une telle théorie, fondée sur le calcul, n'offrirait pas encore, ce me semble, le plus haut degré de perfection; il est preférable, comme dans la thèorie moderne des fonctions… "

14. Dedekind to Lipschitz, June 1876, in [Dedekind 1930/32, vol. 3, 468]: "Mein Streben in der Zahlentheorie geht dahin, die Forschung nicht auf zufällige Darstellungsformen oder Ausdrücke sondern auf einfache Grundbegriffe zu stützen und hierdurch – wenn diese Vergleichung auch vielliecht anmassend klingen mag – auf diesem Gebiete etwas Ähnliches zu erreichen, wie Riemann auf dem Gebiete der Functionentheorie."

15. [Riemann 1892, 38-39]: "Eine Theorie dieser Functionen auf den hier gellieferten Grundlagen würde die Gestaltung der Function (d.h. ihren Werth für jeden Werth ihres Arguments) unabhängig von einer Bestimmungswiese derselben durch Grössenoperationen festlegten, indem zu den allgemeinen Begriffe einer Function nothwendigen Merkmale hinzugefüget würden, und dan erst zu den verschiedenen Ausdrücken deren die function fähig ist übergehen."

16. The reason for the name is that Cauchy had already recognized that a complex function is analytic if and only if it is differentiable, although he did not used the differential equations as a definition. 

17. Klein tried to bring farther the tradition of Gauss and Riemann, but he conceived for himself a role that was much broader and more ambitious than that of a school leader, and the impressive number of mathematicians who studied with Hilbert may not constitute a school, strictly speaking. For details and nuances concerning this period, the reader should consult [Rowe 1989]. 

18. He was strongly influenced by Riemann's work and by his collaboration with Dedekind (chap. III), and led an active academic career in Köingsberg (where he counted Hilbert among his students), Göttingen and Strassburg, among other places. He was also extremely influential through several important text books. 

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By Sohail Tahir