𝟓. 𝔗𝔥𝔢 𝔅𝔢𝔯𝔩𝔦𝔫 𝔖𝔠𝔥𝔬𝔬𝔩

 

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 informal seminar with selected groups of students. As we have seen, however, there was the grotesque situation that that he could not take part in doctorates and habilitationen until 1851. The situation for mathematics became even better in 1844, when Jacobi came from Köingsberg to Berlin as a member of the academy. The academy of sciences had been the most important scientific center in Berlin upto 1800, and it was a notable support for the new university, since its members enjoyed the right to impart lectures. This possibility was exploited by Jacobi during his Berlin period, as it was by other mathematicians before and after.⁴ 

Niels Henrik Abel 1802 - 1829

𝔅𝔢𝔯𝔩𝔦𝔫 had thus turned into an ever more important center for mathematics since about 1830. But the situation became even better and the actors changed completely in the 1850s, following the death of Jacobi in 1851 and Dirichlet's transfer to Göttingen in 1855. That same year, the specialist in number theory Ernst Eduard Kummer became Dirichlet's successor, on his proposal. Kummer regarded himself as a disciple of Dirichlet, although he never attended one of his lectures. He had studied mathematics and philosophy at Halle in the late 1820s, become a corresponding member of the Berlin academy in 1839, through Dirichlet's proposal, on the basis of important work on the hypergeomatric series. By then, he was still a Gymnasium teacher, but in 1842 he became professor at Breslau on the recommendation of Jacobi and Dirichlet. In this decade he began his path-breaking work on ideal numbers, that will be mentioned in chapter III; in the 1860s he made important contributions to geometry.⁵ For our purposes, it is important to emphasize that Kummer stressed more the formal and calculational aspect of mathematics, that its conceptual side (see [Haubrich 1999]). One may conjecture that this would have been different, had he been a real student of Dirichlet. 

𝔄 devoted teacher Kummer became the driving force behind the new institutional arrangements at Berlin, which fully implemented the characteristically German combination of teaching and research. The institution of the seminar created by the neohumanist philologists, had been adapted to the natural sciences at some places, as was the case with the famous Köingsburg mathematico-physical seminar of Jacobi and Neumann, established in 1834. Kummer and his colleague Weierstrass were the first to create a seminar devoted to pure mathematics. This happened in 1861 and was, together with the high quality and novelty of Weierstrass's lectures, the reason for the immense appeal that Berlin exerted on young mathematicians throughout the world in the following decades. Karl Weierstrass, a completely unknown Gymnasium teacher, became a rising star after the publication of his first paper on Abelian functions in 1854. Until then, his mathematical education had been uncommon: he was basically self-taught, although he spent sometime at an Academy in Munster, where the influence of his teacher Gudermann was notable. Interestingly, Gudermann was a follower of the combinatorial tradition (see [Manning 1975), and some aspects of Weierstrass's work - particularly his eternal reliance on infinity series – are reminiscent of that tradition.⁶ Weierstrass was also deeply influenced by the work of Jacobi and Abel on elliptic functions. In 1854 he recieved an honoris causa from Köingsberg, and Kummer began to care for his becoming a professor at Berlin. In 1856 he was appointed to Berlin's Industrial Institute (later the Technical School), while rejecting offers from Austrian universities; months later Kummer had obtained  him a position as extraordinary professor, and full membership in the Academy. But it was not until 1864 that he became full professor and abandoned the Industrial Institute. 

Kummer and Weierstrass went in close personal and scientific contact with Leopold Kronecker, a wealthy man who lived privately at Berlin from 1855, and in 1861 also became member of the Academy with the right to teach. Kronecker knew Kummer from his Gymnasium years, when the latter simulated his mathematical interests; he studied at the university of Berlin, where he took his doctorate in 1845, but also spent two smesters at Breslau, again with Kummer. In this way, Kronecker recieved a strong influence from both Dirichlet and Kummer, but some aspects of his work, particularly his interest in algorithms and effective calculation, bring him closer to the latter. In 1881, on the occasion of the 50th anniversary of Kummer's doctorate, he said that Kummer had provided the "most essential portion" of "my mathematical, indeed ... my intellectual life."⁷ In 1883 he became a professor, after his teacher retired, but already in the 1860s and 70s he had gained an ever more influential position on university affairs, in Berlin and elsewhere in Prussia – and, after 1871, the Empire. 

The triumvirate turned Berlin into a world-renowned center. Weierstrass and Kummer establish a coordinated, biennial structure of the university courses, which was in effect from 1864 to 1883. Kummer covered fundamental well established subjects such as elementary number theory, analytical geometry, the theory of surfaces, and mechanics, leaving his research topics for the seminar. His clear lectures were followed by as many as 250 students, an impressive number that Weierstrass eventually equaled. Weierstrass, like Kronecker, normally lectured on advanced research topics: analytic functions, elliptical and Abelian functions, and calculus of variations. With his sense for rigorous logical development and systematization, he arranged his lectures so that he could build on what he had already proven, and thus he hardly cited other sources. Kronecker, on the other hand was a demanding and less careful teacher, who had few students and lectured on the theory of algebraic functions, number theory, determinants and integrals. 

𝔗𝔬 some extent, there was a continuity between this and the previous generation, since Dirichlet's words on Jacobi can also be applied on lectures of Weierstrass and Kronecker (and to Kummer's seminar activities): 

ℑ𝔱 was not his style to transmit again the closed and the traditional; his courses moved totally outside the limits of textbooks, and dealt only with those parts of the discipline in which he worked creatively himself....⁸ 

𝔖𝔲𝔠𝔥 was the atmosphere in which Georg Cantor recieved his education as a mathematician, from 1863 to 1869. By this time, Berlin was reaching the hieght of its power, being led by a harmonious group of mathematicians who offered the most advanced course of studies in Germany. Cantor earned his doctorate in 1867 under Kummer, and his 1869 Habilitation was also on a topic in number theory enjoying the guidance of Kummer and Kronecker. Soon, however, he would devoted himself to the theory of trigonometric series, and the influence of Weierstrass who expressed more openly and clearly their preferences. They shared a number of fundamental ideas, especially in the early period, up to about 1870. Both were stern partisans of the conception that mathematics must be rigorously developed starting from purely arithmetical notions. This was an idea that they shared with Dirichlet (see[Dedekind 1888, 338]), an idea that could also be found earlier in the work of Ohm (§2).⁹ Nevertheless, the arithmetizing legacy of Dirichlet was taken up in diverging ways: in Dedekind it was tinged with abstract connonations, which express themselves in a determined acceptance of the set-theoretical standpoint; in Weierstrass it was adapted to the formal conceptual viewpoint - accepting the irrational numbers, but emphasizing the point of view of infinity series (see below and § IV.2); in the case of Kronecker, arithmetization came to be understood more radically, meaning reduction to the natural numbers, but without the use of any infinitary means, be the series or sets. This difference of opinion between Weierstrass and Kronecker began to show up around 1870,and led to a growing estrangement in the late 1870s and, above all, the, 1880s.¹⁰ 

𝔚𝔢𝔦𝔢𝔯𝔰𝔱𝔯𝔞𝔰𝔰 and Kronecker also shared a dissatisfaction with 'generalist' viewpoints in mathematics. They took pains to carefully consider the variety of particular cases that can show up in any mathematical topic. This can be seen in the attention that Weierstrass and his diciples paid to 'anomalous' functions – i.e., the famous examples of nowhere differentiable continuous functions. This feature has been considered in detail by Hawkins [1981] in connection with the work of Frobenius and Killing, both members of the Berlin school. Hawkins take as a model Weierstrass's theory of elementary divisors. It had been common to deal with issues in algebraic analysis in a so-called 'general' way, as if there were no singular types of situations for particular values for the arguments. Weierstrass's theory of elementary divisors showed how to deal with those issues in a detailed way, paying attention to all possible special cases, as Kronecker said in an 1874 paper inspired by that theory, 

ℑ𝔱 is common — especially in algebraic questions – to encounter essentially new difficulties when one breaks away from those cases which are customarily designated as general. As soon as one penetrates beneath the surface of the so-called singularities, the real difficulties of the investigation are usually first encountered but, at the same time, also the wealth of new viewpoints and phenomena contained in its depth.¹¹ 

𝔒𝔯, as he said in 1870, "all those general theorems have their hideout, where they are no longer valid."¹² This philosophy of paying a close attention to special cases may have reinforced constructive tendencies in the Berlin school. 

𝔗𝔥𝔲𝔰 the new group of Berlin professors pursued an approach to mathematics that is quite different from the abstract conceptual one, that we have seen in association with the names of Riemann and Dedekind. The Berlin standpoint can also be called 'conceptual,' particularly in the case of Weierstrass, who followed in the tradition of Cauchy and Dirichlet. But it did not share what we have called the 'abstract' turn, typical of Riemann. The differences between the Göttingen group and the Berlin school became evident when we consider the Berlin analogues of the basic notions, employed by Riemann and Dedekind, that we mentioned in the previous section. In his theory of analytic functions, Weierstrass defined them as those functions which are locally representable by power series. This allowed him to base the theory upon clear arithmetical notions, and to elaborate, its first rigorous treatment. A necessary prerequisite was the principle of analytic continuation, which made it possible to 'reconstruct' the entire function from its local power‐series representation. Weierstrass was able to establish that principle, thus creating a method for generating, from a given local representation or "element", a chain of new "analytic elements" defining the entire function.¹³ 

𝔒𝔣 course, this is what Riemann would have called an approach which starts from "forms of representation" or "expressions," precisely what he was trying to avoid. Weiestrass, on the other hand, although admitting that Riemann's definition was essentially equivalent to his, criticized its reliance upon the notion of differentiable real function. This was not satisfactory because, at the time, the class of differentiable real functions could not be precisely delimited (see [Pincherle 1880, 317-318]). Weiestrass's approach reduced the whole issue to representability by means of a perfectly specific class of series. Actually, it is clear in his work an interest in defining whole classes of functions by mean of representability theorms using well-known simple functions. An interesting example in his 1885 theorm on the representation of continuous functions, in a closed interval, by an absolutely and uniformly convergent series of polynomials [Weierstrass 1894/1927, vol. 3, 1 – 37]. This is quite obviously a constructive trait in Weierstrass's approach to analysis. 

ℜ𝔦𝔢𝔪𝔞𝔫𝔫, on the contrary, regarded differentiablity and the Cauchy-Riemann equations as perfectly precise conditions, in an abstract sense. His approach was superior in that it yielded a direct, global overview of the multi-valuedness of complex functions. It was inferior insofar as it raised problematic issues and was not easily to amend or treat with complete rigor. To sum up, in contrast to Riemann's preference for global, abstract characterizations, Weierstrass was in favor of a local, relatively constructive approach. Borrowing the then-frequent terms 'form' and 'formal,' we may refer to Weierstrass's viewpoint as a formal conceptual one. In calling it 'conceptual,' I wish to emphasize that, in spite of his preference for representability theorems, he was no strict constructivist. The conceptual element was clearly present in the notions Weierstrass established as the foundation of analysis, for instance his definition of the real number (chap. IV) and the Bolzano-Weierstrass theorem (below and §VI.4.2). 

𝔗𝔥𝔢 latter were elements that would come under severe criticism on the side of Kronecker. Although he and Weierstrass shared common points, they followed divergent lines of development, perhaps because of their dedication to such different fields. Kronecker came to advocate a radical arithmetization of the whole of mathematics, in the sense of acknowledging only the natural numbers and the algebra of polynomials, and requiring effective algorithms in connection with all the notions employed (see [Edwards 1989]). In a 1887 paper, Kronecker wrote: 

And I also belive that some day we will succeed in "arithmetizing" the whole content of these mathematical disciplines [algebra, analysis], i.e., in basing them exclusively upon the notion of number, taken in the most restricted sense, and thus in eliminating again the modifications and extensions of this notion [note: I mean here especially the addition of irrational and continuous magnitudes], which have mostly been motivated by applications to geometry and mechanics.¹⁴ 

𝔗𝔥𝔦𝔰 is a sharp criticism of the theories of irrational numbers purposed by Weierstrass, Cantor and Dedekind (§IV.2). Kronecker regarded them as meaningless since they went beyond what is algorithmically definable from the natural numbers, depending upon the actual infinite instead. But the differences had already began to emerge around 1870. In his Berlin lectures, Weierstrass employed the Bolzano-Weierstrass theorem as a key stone; its proof was based on the principle that, given an infinite sequence of closed intervals of ℝ, embedded on each other, at least one real number belongs to all of them. Around 1870, Kronecker started to attack this principle and the conclusions drawn from it, as we know from Schwarz's letters to Cantor. He regarded the Bolzano-Weierstrass principle as an "obvious sophism," and was convinced that it would be possible to define functions "that are so unreasonable" that, in spit of satisfying all the condition for the Bolzano-Weierstrass theorem, they would have "no upper limit."¹⁵ Schwarz and Cantor, however, were on the side of Weierstrass and defended his principle as indispensable for analysis. 

𝔎𝔯𝔬𝔫𝔢𝔠𝔨𝔢𝔯 can thus be regarded as the first constructivist, and it is only natural that his approach to algebraic number theory would be extremely different from Dedekind's. The latter preferred a radically abstract, infinitistic approach employing the notion of set. The former, not less radically, favored a constructivist, finitistic viewpoint. At the same time, both were extremely concerned about questions of mathematical rigor. The first basic notion that both needed, in the context of algebraic number theory, was that of field. Whereas Dedekind defined it extensionally, as a certian set closed for algebraic operations, Kronecker defined it algorithmically: "a domain of rationality" [Rationalitätsbereich] in the totality of all "magnitudes" that are rationally representable over ℕ by means of a finite set of generators Rʹ, Rʹʹ, Rʹʹʹ, ... Of course, that 'totality' is conceived as a potential infinity, not an actually existing set. Dedekind would have objected that Kronecker's definition is based on form of representation," replacing Rʹ, Rʹʹ, Rʹʹʹ, ... by other sets of generators (see [Lipschitz 1986, 59-60; Dedekind 1895]). 

𝔗𝔥𝔢 differences become even clearer when we consider the notions employed by both mathematicians in order to solve the main problem of algebraic number theory. In Kronecker's eyes, Dedekind's ideals were mere "symbols" and his approach was a "formal" one, since he did not grant the existence of infinite sets, mean while, Dedekind regarded them as "totally concrete objects" (see [Edwards, Neumann & Purket 1982, 61]), Instead of Ideals, i.e., sets of complex numbers with a certain structure, Kronecker relied on "divisors" that were not defined directly, but in association with certain "forms" or polynomials. Using modern language, we can say that, in order to study the integers of a certain field 𝐾, Kronecker relied on the ring of polynomials 𝐾 [𝒙, 𝒙ʹ, 𝒙ʹʹ, ...]. The variables 𝒙, 𝒙ʹ ... are introduced only formally and attention is focused on the coefficients. What is important is that those "forms" or polynomials make it possible to elaborate a method for the effective construction of the required divisors (see [Edwards 1980]). As Dedekind said, the introduction of polynomials in the development of algebraic number theory "always seems to me an auxiliary means that is foreign to the issue." and "muddies the purity of the theory," [Dedekind 1895, 53, 55]. If such a method affords interesting results, there must be a deeper reason that can be formulated in pure number-theoretical terms. 

𝔒𝔫 the basis of Weierstrassian analysis, Cantor found the possibility of developing a theory of sets and infinity, but this led him to pursue a more abstract viewpoint, trespassing the limits, set on mathematical research by the Berlin school. That provoked a strong negative reaction on the side of Kronecker, but even Weierstrass questioned Cantor's introduction of quantitative differences among infinite sets, and he never openly defended the work of his former student. As we shall see, Cantor's abstract turn can be related to the increasing influence of Riemann and Dedekind.¹⁶ 

1. The name was coined in the 19th-century, and it is interesting to consider that, by then, the word ''school'' frequently had a negative ring, connoting a one-sided orientation. In this case, it may have also referred to the extremely powerful position of the school in academic matters. 

2. His purpose was to make him a professor at a new Polytechnical School that he was attempting to launch, as happend with Dirichlet and Abel later. Dirichlet was the only one who actually came to Berlin, becoming a professor at the University after Humboldt's plans failed. See [Biermann 1973, 21‐27]. 

3. Steiner (see §3) was also a protegè of Humboldt and Crelle. He became an extra ordinary professor and member of the Berlin Academy of Sciences in. 1834.

4. University professors customarily were members of the Academy, Among the mathematicians, this was only false for M. Ohm [see Biermann 1973]. 

5. See Biermann's biography in [Gillispie 1981, vol 7] or [Biermann 1973]. 

6. It would require a careful analysis to ascertian the depth of this influence, but it is clear from the outset that Weierstrass's did not treat series in a purely formal way, as the combinatorialists, but rather viewed them in the 'conceptual' way of Abel and Cauchy (see [Jahnke 1987; 1991]). On the other hand, he was of the opinion that not all traits of the "combinatorial school" had been lost, as Hilbert recorded in his 1888 trip to Berlin (quoted in [Rowe 1995, 546). 

7. As quoted by Biermann in [Gillispie 1981, vol. 7, 523]. 

8. [Dirichlet 1889, vol 2, 245]: "Es war nicht seine Sache, Fertiges und Ueberliefertes von neuem zu überliefern, seine Vorlesungen bewegten sich sämtlich ausserhalb des Gebietes der Lehrbücher, und unfassten nur diejenigen Thiele der Wissenschaft, in denen er selbst schaffend aufgetreten war....  

9. It is likely that Ohm's ideas were influential on Weierstrass and Dedekind, in both cases before, 1855, i.e., before they established closer contact with leading mathematicians. This would explain similarities in their treatment of the elements of arithmetic (chapter IV). As we saw, Weierstrass may have been influenced by the combinatorial tradition, to which Ohm was close; Dedekind's teacher Stern was also influenced by Ohm and the combinaturalists, and his Habilitation lecture of 1854 is strongly reminiscent of Ohm (see also VII.1). 

10. See [Biermann, 1973], [Dugac 1973, 141 - 46, 161 - 63], by 1884, Kronecker was promising to show the "incorrectness" [Unrichtigkeit] of all those reasonings with which "so-called analysis" [die sogennate Analysis] works [Dauben 1979, 314]. Such commets affected Weierstrass strongly, and led him to fear that his mathematical style would disappear after his death, but he did not have the strenght to defend his viewpoint publicaly. 

11. As translated in [Hawkins 1981, 237]. 

12. [Meschkowski 1969, 68]: "All solche allgemeinen Sätz haben ihre Schlupfwinkel, wo sie nicht mehr gelten." But here he was referring to the Bolzano-Weierstrass theorm! 

13. On Weierstrass's theory, see [Dugac 1973] and [Botazzini 1986]. 

14. [Kronecker 1887, 253]: "Und ich glaube auch, dass es dereinst gellingen wird, den gesammten Inhalt aller dieser mathematischen Disziplinen zu 'arithmetisieren,' d.h. einzig und allein auf dem im engsten Sinne genommenen Zahlbegriff zu gründen, also die Modification und Erweiterungen dieses Begriffs [Ich meine hier namentlich die Hinzunahme der irrationalen sowie der continuirlichen Grösen] wieder abzustreifen, welche zumeist durch die Anwendungen auf die Geometrie und mechanik veranlasst worden ist." 

15. [Meschkowski 1967,68]: "Kronecker erklärte... die Bolzanoschen Schlüsse als offenbare Trugschlüsse "dass man Funktionen wird aufstellen können, die so unvernünftig sind, dass sie trotz des Zutreffens von Weierstrass ' voraussetzungen keine obere Grenze haben." see [op. cit., 239-40]. 

16. By the 1880s, as a result of theoretical differences mixed with personal difficulties, Cantor came to feel confronted with the Berlin school as a whole. In an 1895 letter to Hermite, he denied being a member of the school and stated that the mathematician he felt closer to was Dirichlet [Purkert & Ilgauds 1987, 195-96]. 

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By Sohail Tahir