𝓐. ℑ𝔫𝔰𝔱𝔦𝔱𝔲𝔱𝔦𝔬𝔫𝔞𝔩 𝔞𝔫𝔡 ℑ𝔫𝔱𝔢𝔩𝔩𝔢𝔠𝔱𝔲𝔞𝔩 ℭ𝔬𝔫𝔱𝔢𝔵𝔱 𝔦𝔫 𝔊𝔢𝔯𝔪𝔞𝔫 𝔪𝔞𝔱𝔥𝔢𝔪𝔞𝔱𝔦𝔠𝔰

ℑ𝔱 is a common characteristic of the various attempts to integrate the totality of mathematics into a coherent whole ― whether we think of Plato, of Descartes, or of Leibniz, of arithmetization, or of the logistics of the nineteenth–century ― that they have all been made in connection with a philosophical system, more or less wide in scope; always starting from a priori views concerning the relations of mathematics with the two fold universe of the external world of thought.¹ 

ℑ𝔫 order to understand the growth of mathematical knowledge it is sometimes important to identify and consider the role played by schools of mathematical thought. Such a school usually possesses an underlying philosophy by which i mean a set of attitude towards mathematics. The members of a school tend to share common views on what kind of mathematics is worth pursuing or more generally, on the manner in which, or the spirit in which, one should investigate mathematical problems.² 

 𝔄𝔰 will become clear in the body of the present work, a certain trend with in 19th-century German mathematics, the so called conceptual approach, seems to have been strongly associated with the rise of set theory. Therefore, it seems convenient to start by analyzing two different 'mathematical styles,' those that reigend in Göttingen and Berlin immediately after 1855. Such will be the topic of §§4 and 5. The reason for that particular selection of institutions is simple: the main figures in the first two parts of the book are Riemann, Dedekind and Cantor. Riemann and Dedekind studied at Göttingen, where they began their teaching career, while Cantor took his mathematical training from Berlin. It will turn out that the conceptual approach was present at both universities, but in different varieties, that we will identify as an abstract and a formal variety. The abstract conceptual approach that could be found at Göttingen promoted the self-theoretical orientation strongly. 

ℑ𝔫 order to better understand what the conceptual viewpoint meant, it is necessary to overview, however briefly, a number of trends in the foundations of mathematics that were influential in Germany during the first half of the century. As we have seen, an important part of the rise of set theory was the emergence of new language that expressed a novel understanding of mathematical objects ― the language of sets and the idea that set constitute the foundation of mathematics. This raised some questions that had already been discussed by the Greeks: what kind of existence do mathematics, and the famous issue of the actual infinite. These were philosophical questions, that the German mathematicians understood as such.³ In fact, those questions are not the exclusive domain of mathematics, and there is some evidence that the wider intellectual and philosophical atmosphere in Germany had some impact upon the mathematical discussion. Our discussion of these topics in §§ 1 to 3 will note aim to be complete, but just to clarify and make plausible this last thesis. We will observe that there has been some misunderstanding concerning such matters, and particularly the question of the infinite, since the actual infinite was not rejected by all German mathematicians as of 1800 or 1850. 

𝟏. 𝔐𝔞𝔱𝔥𝔢𝔪𝔞𝔱𝔦𝔠𝔰 𝔞𝔱 𝔱𝔥𝔢 ℜ𝔢𝔣𝔬𝔯𝔪𝔢𝔡 𝔊𝔢𝔯𝔪𝔞𝔫 𝔘𝔫𝔦𝔳𝔢𝔯𝔰𝔦𝔱𝔦𝔢𝔰

𝔗𝔥𝔢 German scientific community seems to have been somewhat peculair, within the context of the international panorama in the nineteenth-century. One has the impression that a certain intellectual atmosphere marked or conditioned German approaches to the sciences, including mathematics.⁴ That can be better understood by taking into account the institutional context of mathematical research, which emerge from the educational reform that was undertaken after the Napoleonic invasion. Sub research as we will analyze throghout his book was clearly placed with in the university context, but the 19th-century German universities were the result of a complex and radical transformation that began some time around 1810 (see[McClelland 1980]). 

Napolean Invasion in German States 1813

𝔗𝔥𝔢 traumatic Napoleonic invasion had the effect of making clear the need for important transformations in order to elevate the political, economical, military and scientific situation in Germany to the level of France. To some extent, the Germans explained the defeat to themselves by pointing to the high level of scientific education enjoyed by the French officers. That high level was a consequence of the educational reform undertaken in France after the revolution, particularly the creation of the Parisian Ècole Polytechnique in 1794. Here, a higher scientific education, including the calculus, was for the first time regularly available to students. The German states undertook parallel reforms, but some of them, especially in Prussia, did not simply copy the French model. On the contrary, they tried to forge their own peculiar model.⁵ While in France the universities had been abolished as an outmoded medieval institution, in Germany there were very some precedents for a reformed university, adapted to the cultural standards to the enlightenment, as was the case of Göttingen University in Hannover [McClelland 1980]. 

𝔗𝔥𝔢 reforming impulse which came out of the Napoleonic merged in a natural way with the educational aspirations of the German Enlightenment. In the second half of the eighteen–century, this had given rise to so‐called neohumanism, a movement that reacted against rationalistic viewpoints, aspiring to an integral formation [Bildung] of the individual, a kind of board, harmonic education not guided by utilitarian aims. Such ideals were strongly fostered by historians and philologists like F.A Wolf at the Prussian University in Halle, who is regarded as founder of the new 'scientific' philology and of the university seminar. Wolf devoted his efforts to turning history and philology, what he called the "science of antiquity" [Altertumswissenschaft], into a true "system", to "unify it into an organic whole and elevate it to the dignity of a well-ordered.... science."⁶ This was not done simply for professional or academic reasons, but emboided a characteristic educational ideal. As Goethe said to Eckemann in 1827,

𝔄 nobel man in whose soul God has put the capacity for future greatness of character and has intellect will develop must splendidly through the acquaintance and the intimate intercourse with the lofty characters of Greek and Roman antiquity.⁷ 

Neohumanist professors went beyond the traditional role of a university teacher, namely the transmission of well-established knowledge, to its expansion by means of criticism and research, Through the institution of the seminar, selected groups of students were thought how to do research by themselves, and research came to be seen as an indispensable ingredient of teaching. One further important aspect of this movement was its association with the late eighteenth-century struggle of the Wissenchaften (philosophy, history, philology, mathematics, etc.) for a recognition of equality with, or even superiority over, law, medicine and theology ― the traditional 'higher' university faculties. 

𝔗𝔥𝔢 German reform of higher education is normally traced back to the founding of Berlin university in 1810. The Philosophical Faculty, housing the sciences, was given the task of preparing Gymnasium (secondary school) professors; who were required by the Prussian state to pass an examination where their knowledge of philology, history and mathematics would be tested. Thus, the Philosophical Faculty was established as an equal to the professional faculties, where mathematics found a sounder institutional frame.⁸ One should paused to consider what adaptation to the miliei of the Philosophical Faculty may have meant. Far from being mere practitioners, as frequently happen in the past, the mathematicians now became part of a small élite of university professors, and precisely within the context of the humanities, where the ideas of "living the sciences" (W.von Humboldt) and pure knowledge were strong. The implications of this move had been beautifully expressed by Schiller in a short poem that was well known at the time, Archimedes and the Apprentice. 

𝔗𝔬 Archimedes came an eager-to-learn youngster; Intimate me, he said to him, into the divine art, That such magnificent fruits gave the fatherland, And the city walls protected from the sambuca, Divine you call the art! She is, the sage replied, But so she was, my son before she served the State. If you want fruits, those a mortal can also begt, He who woos the Goddess, seek in her not the maid.⁹ 

𝔗𝔥𝔦𝔰 cultural orientation helps to explain the 19th-century tendency toward pure mathematics, which was particularly noticeable in Germany.¹⁰ 

University of Göttingen, Germany. 

𝔅𝔶 the early century, the Philosophical Facility was dominated by history, philology and philosophy, then living its golden age in Germany. The sciences did not enjoy strong support, a situation that was badly felt by physicists (see [Jungnickel & McCommach 1986]) Mathematics was in a better situation, since it was regarded as central for educational purposes, given its assumed relation with the training of logical and reasoning abilities. But, even so, the mathematics curriculum was elementary and rarely included the calculus [op.cit., 6–7]. It was only gradually that university professors raised their standards, partily in response to the French model, partly in imitation of the philologists, and partly as a result of the better mathematical level of students coming out of the reformed Gymnasia. The teaching of research topics started in the late 1820s, particularly with the courses and seminars offered by Jacobi at Königsberg and, somewhat later, by Dirichlet at Berlin, both Prussian universities. 

University of Berlin, Germany. 

𝔉𝔯𝔬𝔪 about 1830 there were periods of tension between the interests of the scientists and those of the humanists, that can be followed up to the early twentieth-century (see[Pyenson 1983]). But, even so, the context of the Philosophical Faculty, and the neohumanist idea of Bildung, promoted an atmosphere in which the 'two cultures' were not separated. Throughout the century, we find scientists interested in philosophy and philosophers interested in the sciences. In fact, the unity between philosophy and sciences was repeatedly promulgated, for instance by the idealist philosopher Schelling, who originated the tradition of Naturphilosophie. Much has been written about the influence of this movement on German science, but here i would like to warn against simplifying assumptions. Many historians tend to identify neohumanism and this with philosophy generally. But philosophy in Germany was not by any means, identical with idealism, and many new humainists opposed the Romantic trends, including idealistic philosophy. For instance, the above-mention Wolf was much closer to Kant then to the idealists in philosophical matters [Paulsen 1896/1897, vol 2, 212-214]. Among philosophers, early 19th-century followers of Kant who remained close to the sciences and opposed the idealism of Schelling and Hegel included Fries and Herbart, whom Riemann regarded as his master in philosophy.¹¹ 

𝔗𝔥𝔲𝔰, rather than paying attention to peculairities of the idealists, it would be more useful to analyze the ideas they shared even with their detractors, since these marked the development of German science after the anti-idealist reaction of the 1830s and 40s. It is my contention that much of the special ways of German scientists can be explained by their adaptation to the context of Philosophical Faculty, and by the fluid intellectual contact they established with the philosophers. Among the particular orientations that were promoted in the process are the preference for a strictly theoretical orientation, the concentration on narrowly defined specialties or branches of mathematics, and in many cases a close attention to the philosophical presuppositions of the advocated theories.¹² The main authors studies in the present work afford good examples of such traits, and their philosophical preferences will be analyzed briefly (see especially §§11.1-2,VII.5, VIII.I-2 and 8). 

𝔉𝔯𝔬𝔪 the 1820s, there was a clear scientific renaissance in Germany [Klein 1926, Vol I, 17]. Peculiar idiosyncratic approaches began to be abandoned, and closer attention was paid to foreign ideas. A professional scientific community began to emerge and fight for new standards in higher education. In 1822 the Naturphilosoph Lorenz Oken founded the Deutscher Naturforscher und Ärzte [German Association of Natural Scientists and Physicians] which from the 1828 meeting at Berlin would become dominated by scientists opposed to the ideas of Naturphilosophie. This move was related to influence of Alexander von Humboldt, the famous travler and naturalist of neohumanist allegiances, who played an important role in promoting the sciences in his position as a court counselor in Prussia. In connection with mathematics, an other important event was the founding in 1826 of the Journal ƒür die reine und angewandte Mathematik [Journal of pure and applied mathematics], directed by the engineer and high level civil servant A. L. Crelle, who had always been keenly interested in mathematical research. Thanks to the availability of original material by such men as Abel, Jacobi, Dirichlet, and Stiener, Crelle's undertaking was a great success and acted as a binding agent for the emerging community of mathematicians.¹³ 

Alexander von Humboldt 1769-1859

𝔗𝔥𝔢 careers and ideas of Jacobi, Kummer and others might be used in order to show the impact neohumanism and philosophy had upon German mathematicians.¹⁴ In discussing the role of Jacobi in the German scientific renaissance and the establishment of the first important research school in mathematics, the Köingsberg school, Klein writes: 

ℑ𝔣 we know ask about the spirit that characterizes this whole development we can in short say: it is a scientifically-oriented neohumanism, which regards as its aim the inexorably strict cultivation of pure science, and in search of that aim establishes a specialized higher culture, with a splendor never seen before, through a concentrated effort of all its powers.¹⁵ 

𝔗𝔥𝔢 cases of Grassmann, Riemann and Cantor underscore the freedom and new possibilities of thoght that the fluid exchange of ideas between philosopheds and mathematicians could sometimes promote. But of course, any institutional arrangement has its pros and cons. Dirichlet's career reminds us of the negative effects that neohumanist standards sometimes had: it will also serve to clarify several aspects of the situation in Germany at the time.¹⁶ 

ℑ𝔫 1822 Gustav Lejeune Dirichlet, son of the town postmaster at Düren, made the wise decision to take advantage from his family's connections and study mathematics at Paris, not an any German university. There he attended free lectures at the Collège de France and the Facultè des Sciences, and the subsequently he became a key figure in the transmission of the French tradition of analysis and mathematical physics to Germany. Early success with a paper on indeterminate equations of degree five, send to the académie des Sciences in 1825 and printed in the Recueil des Mémoires des Savans étrangers,¹⁷ made him known in Parisian scientific circles. Dirichlet enter the circle around Fourier then "secrétaire prepétuel" of the Académie and established contact with A. von Humboldt, who would promote his career in Prussia. Thanks to the support of Humboldt, Dirichlet was awarded an honorary Ph. D from Bonn University in 1827, became privatdozent at Breslau against faculty opposition, and was named extraordinary professor in 1828. Humboldt was again instrumental in bringing his young friend to Berlin, where he was forming plains for a crearion of an important scientific center in imitation of the École Polytechnique. Dirichlet became teacher of mathematics at the military academy, shortly thereafter Privatdozent at the University, and in 1831 he was appointed extraordinary professor. His brilliant career continued with further papers on number theory, and with a famous 1829 article on the convergence of Fourier series (see Chap. V). In 1832, when only twenty seven, he became a member of the Berlin Academy of Sciences, and in 1839 he was promoted to an ordinary professorship. 

𝔅𝔲𝔱 such a quick career had not conformed to all the rules then in force, and Dirichlet subsequently face some difficulties. Since he had not studied at a German university, nor even completed his education at the Gymnasium [Schubring 1984, 56-57], Dirichlet lacked some of the knowledge required by neohumanist curricula. Although it had been possible to avoid most of the consequences by means of an expeditious honoris causa, he had not satisfied a formality on the occasion of his qualification. [Habilitation] as a privatdozent at Breslau. Aspirants were required to submit a second thesis, written in Latin, and to defend it in an open discussion [Disputation] with faculty members, also to be conducted in Latin.¹⁸ Dirichlet did not master the spoken language and was relieved from the Disputation, but this had the effect that in 1839 the Berlin faculty would not grant him full right as a professor until he compiled with that formality. That only happen in 1851; in the meantime, Berlin's most influential mathematics professor could not take part, as a voting member, in Ph. D. and habilitation proceedings.¹⁹ 

𝔇𝔦𝔯𝔦𝔠𝔥𝔩𝔢𝔱 did not creat a school in the strict sense, but among those who were his students and felt strongly influenced by him we find such important names as Hiene, Einstein, Kronecker, Christoffel, Lipschitz, Riemann and Dedkind. His contributions to number theory, following and making available Gauss's work on Fourier analysis, multiple integrals, potential theory and mathematical physics, were all of fundamental importance. Moreover, in all of these fields he carried further the rogorization of mathematics. A famous passage in a letter from Jacobi to Humboldt says that it is only Dirichlet, not Gauss, Cauchy or Jacobi, who knows "what a completely rigorous mathematical proof is". Jacobi goes on to mention the other specialty of Dirichlet, analytic number theory, and writes that "he has chosen to devote himself mainly to those subjects which offer the greatest difficulties".²⁰

𝔄𝔠𝔠𝔬𝔯𝔡𝔦𝔫𝔤 to Einstein, Gauss, Jacobi and Dirichlet started a new style of argument in mathematics, which avoids, "long and involved calculation and deductions" in favor of the following "birilliant expedient:" "it comprehends a whole area [of mathematical truths] in a single main idea, and in one stroks presents the final result with utmost elegance," in such a way that "one can see the true nature of the whole theory, the essential inner machinery and wheel-work"²¹ This has frequently been called the conceptual approach to mathematics, and will be of our concern in §§ 4 and 5. Some aspects of Dirichlet's work that are of consequence for the present study, will be mentioned in the following chapters. 

1.  Nicolas Bourbaki [1950, § 1]. 

2. Hawkins [1981, 234]. 

3. Riemann [1854, 255] regarded his discussions of the motion of manifold as 'philosophical'. Just like Kronecker [1887, 251] his analysis of number. Cantor [1883] entered into the philosophical arena in order to justify his introduction of transfinite numbers, while Dedkind [1888, 336] felt the need to make clear that is 'logical' analysis of number did not presuppose any particular "Philosophical or Mathematical" knowledge. 

4. As is well known, the meaning of the German word Wissenschaft is different from that of its English counterpart 'science:' it refers to any academic discipline, including history, philosophy, etc. When I write about the sciences, I mean Naturwissenschaften ― physics, chemistry, biology, etc. It is important to observe that mathematics was sometime treated as being closer to the humanities in early 19th-century Germany. 

5. Prussian university reform was undertaken during French occupation, and there was a conscious attempt to establish clear counter parts of French cultural orientations: the university would be an exponent of German Kultur as opposed to French civilization, and in particular the notion of Bildung is opposed to a more utilitarian Ausbildung [formation or instruction]. See [Ringer 1969]. 

6. Quoted in [Paulsen 1896/97, vol. 2, 209]: "zu einem organischen Ganzen zu vereinigen und zu der Würde einer wohlgeordneten philosophisch-historischen Wissenschaft emporzuheben"

7. As translated in [Jungnickel & McCommach 1986, 4]. 

8. The import of such institutional changes for mathematical result has been carefully studied by Schubring [1983]. 

9. Schiller, Die Horen (1795), in [Schiller 1989, 280]: "Archimedes und der schüler / Zu Archimedes kam ein wissbegieriger Jüngling:/ Wehe mich, sprach er zu ihm, ein in die göttliche kust,/ Die so herrliche Früchte dem Vaterlande getragen,/ Und die Mauem der Stadet vor der Sambuca beschützt,/ Göttlich nennst du die Kunst! Sie ist's, versetzte der Wiese, / Aber das war sie, mein sohn, eh sie dem Staat noch gedient./ Willst du nur Früchte, die kann auch eine Sterbliche zeugen,/ Wer um die Göttin freyt, suche in ihr nicht das Weib." Quoted in [Hoffman, vol. 5 624]: Jacobi wrote a parody of this poem, that is quoted in [Kronecker 1887, 252]. The 'sambuca' was a war machine used by the Romans against Syracuse, Archimedes' fatherland. 

10. Compare [Scharlau 1981]. I hardly need to make explicit that i am not making a case for a reductionistic explanation: the contextual factor will probably be only one among several others. 

11. On Hegel, Fries, Herbart and mathematics, see [König 1990]. 

12. This had negative effects insofar as it implied lack of attention to applied topics and to interconnections between branches, etc. See [Rowe 1989], where the fight against some implications of neohumainism around 1900 is discussed; see especially [186-87, 190]. 

13. One should indicate, as David Rowe has urge me to do, that the emergence of a mathematical community was not easy, due to the depression of professors, their adscription to different universities (with remnants of a guild mentality) and states, etc. The Deutsche Mathematiker Vereinigung was not easy to launch even in 1890.

14. See [Jahnke 1991], and on Kummer [Bekemeier 1987, 196-203]. 

15. [Klein 1926, Vol I, 114]: "Fragen wir nun nach dem Giest, der diese ganze Entwicklungträgt, so könen wir kurz sagen: es ist der naturwissenschaftlich gerichtete Neohumainismus, der in der unerbittlich strengen pflege der renien Wissenschaft sein ziel sieht und durch einseitige Anspannug aller kräfte auf dies Ziel hin eine spezialfachliche Hochkultur von zuvor nicht gekannter Blüte erreicht."

16. No full biography of Dirichlet has yet been written, the best is still Kummer's obituary (in [Dirichelt 1897, 311-44]). Important archival material can be found in [Biemann 1959]. 

17. Legendre, who acted as a reviewer of the paper, was able to use Dirichlet's results's for a proof of Fermat's Last Theorem of the exponent. 

18. The feature of the Habilitation varied from place to place, but was similar in Breslau and Berlin. 

19. On this issue, see [Biemann 1959, 21-29]. 

20. Letter of 1846, in [Pieper 1897, 99]: "Er allein, nicht ich, nicht Cauchy, nicht Gauss weiss, was ein volkommen strenger mathematischer Beweis ist, sondern wir kennen es erst von ihm.… D[irichlet] hat es vorgezogen, sich haupt sächlich mit solchen Gegenständan zu beschäftigen, welche die grösten Schwierigkeiten darbieten; darum liegen seine Arbeiten nicht so auf der breiten Heerstrasse der Wissenschaft und haben daher, wenn auch grosse Anerkenunng, doch nicht alle die gefunden, welche sie verdienen."

21. As translated in [Wussing 1969, 270], where this passage from Einstein's autobiography written when he was 20, is quoted in full. 

اُردو ترجمہ Urdu Translation 

Next Blog 

Previous Blog

By Sohail Tahir