Mathematics at the university of Göttingen

The development of mathematics in Göttingen

Erwin Neuenschwander and Hans-Wilhelm Burmann, translated by Eugene Chislenko

At the founding of the university, only five chairs were planned in the Philosophical Faculty. Its first mathematician, appointed in 1735, was Johann Andreas von Segner, who had studied philosophy, mathematics and medicine in Jena and had practiced as a physician before starting his academic career. Apart from mathematics, Segner was responsible for the lectures in physics, wrote numerous textbooks on both subjects, and occasionally lectured on chemistry. His prolific activity soon brought Segner a reputation as one of the best living mathematicians in Germany. Friedrich Penther, originally hired as overseer of the academic buildings in Göttingen, was appointed professor for mathematical topics in areas such as economics, architecture, and mapmaking. After Penther's death his place was occupied by the well-known astronomer and mapmaker Tobias Mayer. Already before Mayer's arrival in Göttingen, Segner had campaigned for the establishment of an astronomical observatory, which he and Mayer then shared for some time; but as competition over the observatory grew and the authorities in Hanover generally supported Mayer, Segner left Göttingen in 1755 to take over the chair vacated by Christian Wolff at the University of Halle.

Segner's successor was Abraham Gotthelf Kästner, who worked in Göttingen for almost 50 years. When not busy with mathematics, he worked especially in physics and astronomy, as well as in chemistry, botany, anatomy, philosophy, literature, and many other areas, and served as director of the observatory after Mayer's death. In one of his now classic aphorisms, Georg Christoph Lichtenberg called Kästner "an encyclopaedic dictionary". One of his many influential books was the four-volume Anfangsgründe der Mathematik, or "starting points of mathematics" arranged by field and treating mechanics and hydrodynamics in addition to arithmetic, geometry, analysis, algebra, and applied mathematics. In Kästner's time Albrecht Ludwig Friedrich Meister and Mayer's brother-in-law Georg Moritz Lowitz taught applied mathematics, Lichtenberg and Johann Christian Polykarp Erxleben taught physics, and Karl Felix Seyffer taught astronomy. After Kästners death no suitable replacement could be found, and for a time there was no towering personality of the kind the University of Göttingen would find some years later in Gauss. At the same time, Hanover experienced a political crisis, submitting to Prussian and then French occupation.

Carl Friedrich Gauss came from Braunschweig, or Brunswick, where he spent his youth. Thanks to a scholarship from the duke of Braunschweig, who recognized and supported his talent, he went to Göttingen in 1795 to study mathematics there. He came into particularly close contact with Seyffer, the professor of astronomy at the time, and thought highly of Lichtenberg, but was less impressed by Kästner and the other mathematics lecturers. Other German mathematicians, however, may not have been able to meet his standards either. In 1798 Gauss returned to Braunschweig, where he worked on his magnum opus Disquisitiones Arithmeticae and received a doctorate in 1799 in Helmstedt. With the publication of the Disquisitiones and the sighting of the dwarf planet Ceres based on his computations in 1801, Gauss became one of the most prominent astronomers and mathematicians in Europe. He received a salary increase from the Duke of Braunschweig and an offer to direct the observatory in St. Petersburg, upon which Göttingen Professor Wilhelm Olbers stepped into negotiations with the University of Göttingen to do as much as possible to keep Gauss in Germany. Among Gauss's main reasons for accepting the resulting appointment in 1807 were the Hanover government's plan to establish a new observatory in Göttingen and the relatively light obligations which he had to take over within the university. After the completion of the new observatory in 1816, Gauss moved in as its director and lived there until his death in 1855.

Gauss was the leading mathematician of his time, and many regard him simply as the greatest mathematician. At 18 he discovered that the regular seventeen-sided polygon can be constructed with a ruler and compass, the first major discovery in this area in 2000 years. His Disquisitiones Arithmeticae provided the foundations for modern number theory. His methods for the calculation of planetary motion have still not been essentially improved upon even today. Just how far Gauss was ahead of his contemporaries in complex analysis, and especially in his insights about non-Euclidean geometry, became clear only with the publication of his posthumous works, letters, and scientific diary. His interest in geodesy - he started a project to survey the lands of the Kingdom of Hanover in 1818 and carried its main workload for some years - led him to the investigation of curved surfaces, culminating in the Theorema Egregium and other results developed further by Riemann. Together with William Weber he investigated electricity and magnetism, leading to the invention of the electrical telegraph as a byproduct of their research. These are only a few of the highlights of Gauss's enormous output. He made crucial discoveries in almost all areas of mathematics, and shaped the entire field with his demands for mathematical rigor.

Gauss seems to have had an aversion to lecturing at least in his earlier years, and appeared content when his lectures were cancelled due to a lack of students. In Gauss's time, basic mathematics lectures were conducted largely by Thibaut, mentioned above, and later by George Karl Justus Ulrich and Moritz Abraham Stern. Thibaut usually covered pure mathematics, differential equations and integrals, and finite analysis, while Ulrich lectured on solid geometry and trigonometry, applied geometry, mechanics and civil architecture. Thibaut was known as the best lecturer in Göttingen, and because of his perfected rhetorical style was even compared with Goethe.

The death of Gauss left the University of Göttingen with a gap that could not easily be filled. The directorship of the observatory remained vacant for several years and was administered intermittently by Weber until 1868, when it was finally divided between William Klinkerfues and Ernst Schering. In the mathematical areas a worthy successor was found in Peter Gustav Lejeune Dirichlet, who at that time enjoyed the highest reputation among German mathematicians. Dirichlet's work united two currents, number theory in the tradition of Gauss and applied mathematics from the French school, as for example in his theorem on prime numbers in arithmetic progressions together with its beautiful proof, and in the extraordinarily fruitful Dirichlet principle. Dirichlet was an inspiring teacher, and his lectures on number theory, published after his early death by Richard Dedekind, remained the standard work for decades.

Dirichlet's successor was Bernhard Riemann, who had studied first in Göttingen and then under Dirichlet in Berlin and obtained his doctorate and Habilitation under Gauss in 1851 and 1853. It was a great moment in mathematics when the young Riemann delivered his Habilitation lecture "On the Hypotheses that Lie at the Foundations of Geometry" in front of Gauss; Gauss was deeply impressed by it and, according to Weber, had the highest expectations about him, and Riemannian geometry would later supply the mathematical framework for Einstein's relativity theory. Riemann's innovative ideas were understood in their full depth only gradually, but their effects were lasting and provide mathematics with essential stimulation even today. He set the theory of analytic functions on a firm foundation and gave it a new dimension with the Riemann mapping theorem, which he proved with the help of the Dirichlet principle. The notion of Riemann surface shed light on the preceding half century's investigations of algebraic curves, unified and standardized them and opened the way for many future developments in algebraic geometry and topology. In an eight-page work on number theory. Riemann provided the key to problems of prime number distribution. Hilbert remarked in a lecture (W.S. 1896/97, p. 264) that "only very rarely has an essay of such shortness, sharpness and genius flowed from the pen of a human being, as did this masterwork of one of the greatest spirits of our science." The Riemann hypothesis, whose demonstration would greatly extend our knowledge of the distribution of prime numbers, has defied all attempts at proof and refutation until today.

Unfortunately, Riemann, like Dirichlet, was prevented from working for a very long time as a full professor in Göttingen, since he contracted tuberculosis just three years after his appointment and then spent much of his remaining time at health resorts in Italy. After Riemann came Alfred Clebsch, who also died a few years after his appointment, and then Lazarus Fuchs, who one year later followed a call to Heidelberg. For a brief period Göttingen mathematics once again lacked a long-term leading figure, and ceded supremacy to Berlin, just as the nearby capital Hanover ceded much of its political power to Berlin around this time.

In 1850, still during the lifetime of Gauss, the Hanover government established the Mathematics-Physics Seminar, from which today's Mathematics Institute emerged in 1922. Its purpose was, according to the statutes, the training of teachers for mathematics and physics instruction at institutions of higher learning as well as the general elevation of the study of the mathematical and physical sciences. The members committed themselves for approximately 4-6 hours weekly to participating in the activities of the mathematics and physics departments, which included conducting lectures and exercise sessions. They were also provided with opportunities for hands-on experience in the natural sciences and later in astronomy. The university's board of trustees gave each semester's most outstanding members prizes in the form of scholarships, in addition to providing the seminar with a small budget to cover its expenses. At first the number of participants was relatively small, their number wavering around 15. Among the first members were Dedekind, Riemann and Schering, the latter two working also as assistants who helped new members to get oriented and keep up the pace of the exercises. In the first years the leadership of the seminar rotated among Listing, Stern, Ulrich and Weber, while the full professors Gauss, Dirichlet and Riemann did not participate in the seminar. Later this would change, and all full professors of mathematics, physics and astronomy would take part in the direction of the seminar.

After the departure of Fuchs in 1875, Hermann Amandus Schwarz was appointed from Zurich. Schwarz organized his lectures according to a well worked out study plan and arranged them into two cycles. The first, introductory cycle covered differential and integral calculus, analytic geometry, surfaces of second degree, curved surfaces and double curvature, and synthetic geometry. The second, simultaneously running cycle covered analytic functions, elliptical functions, minimal surfaces, the hypergeometric series, and other areas of function theory. Schwarz, supported by Stern, spearheaded the creation of a circulating library for the Mathematics-Physics Seminar in 1878, and it remained under his administration until his departure for Berlin. With the death of Ulrich he also took over the collection of mathematical instruments and models that had grown out of the old model and machine room. Thibaut had bequeathed a collection of geodesic instruments to the model collection in 1832, and the expanded model collection was put on display starting in 1865 in a roomy hall of the auditorium building at Weender Gate. Over the years the model collection underwent many changes, as old or no longer used models and instruments were often transferred to other institutes and private collectors to make way for new acquisitions. After the takeover by Schwarz, and above all in the time of Felix Klein, the collection was modernized with the help of large subsidies and developed systematically for the then expanding instruction in representative geometry and geodesy. One can admire the over 500 models today in the upper lobby of the Mathematics Institute, whose planning provided specifically for exhibition space for the collection.

Felix Klein, the great organizer of Göttingen mathematics and physics, studied in Bonn, Göttingen, Berlin and Paris, particularly with Plücker and Clebsch, and received his Habilitation in Göttingen in 1871. In 1872 he was appointed full professor at Erlangen, from where he went to Munich in 1875, to Leipzig in 1880, and finally back to Göttingen upon Stern's retirement in 1886. In his 1872 Erlangen Program Klein used the notion of group to formulate a classification principle for geometry that had a lasting effect on geometrical thought. He saw himself as a developer of the brilliant ideas of Riemann, whose geometrical core he worked out further and brought into the investigation of model functions and automorphic functions.

Klein's main objectives apart from pure mathematics were the reinforcement of the connections between mathematics, the natural sciences and technology and the restructuring of education in mathematics and science from the earliest grades to the university. Both of these goals shaped much of his time in Göttingen. Already in his appointment negotiations, Klein advocated the creation of a mathematical reading room with a reference library like the one he had organized in Leipzig. His request was approved by the time he came to Göttingen, and immediately after his arrival the Reading Room of the Mathematics-Physics Seminar was opened. At that time it was in Auditorium No. 20 on the upper floor of the auditorium building at Weender Gate, directly beside the model collection and the lecture rooms of the Mathematics-Physics Seminar. At first its furnishing was relatively modest, with 20 working places and a library of about 500 volumes around 1890.

With Schwarz's departure for Berlin as successor to Weierstrass in 1892 and the subsequent departure of Heinrich Weber, Klein had the freedom to reorganize mathematical instruction in Göttingen according to his wishes. The small seminar library developed by Schwarz was combined with the library of the mathematics reading room and put under Klein's direction. Klein introduced term fees for the gradually expanding student body and applied for 3000 Marks for the library's further development, a sum five times the annual budget for mathematics. He also took over the collection of mathematical instruments and models and obtained funding for a special assistant for it, and took part in the establishment of the Mathematical Society and the first edition of study plans, distributed to students free of charge upon matriculation.

At the same time, Klein worked on his second main goal: strengthening the ties between mathematics, the natural sciences and technology. Gathering together interested professors and industry leaders, he created the Göttingen Association for the Advancement of Applied Physics and Mathematics in 1898. This association raised over 200.000 Marks over the next ten years to support these sciences in Göttingen, allowing for the establishment of numerous new buildings and institutes. At the 10-year anniversary celebration, Klein could report that since the Association's inception the Institutes for Applied Electricity, Applied Mathematics and Mechanics and the Geophysics Institute had been founded, and the number of professors in mathematics and physics had doubled. This led, among other things, to the appointment of Hermann Theordor Simon, Carl Runge, Ludwig Prandtl and Emil Wiechert.

The number of mathematics students and reading room users was constantly rising, and space shortages and the move of the Physics Institute to Bunsenstrasse left the mathematics and physics facilities scattered across the city. Klein felt a growing need for a new Mathematics Institute. By 1911 he believed this dream almost attained, as the Göttingen Association had allocated 200,000 Marks for the purpose and its chairman had purchased a suitable property on Bunsenstrasse in direct proximity to the physics institutes. But the realization of Klein's plans was delayed by the First World War and later by the struggling economy and spiralling inflation, and one of his great projects did not come to fruition in his time.

In the meantime, however, Klein continued to further mathematics in many other ways. He served for several years as chairman of the International Commission on Mathematical Instruction and oversaw several series of publications associated with the Commission. He served as editor of the leading mathematics journal Mathematische Annalen and the monumental Encyclopedia of the Mathematical Sciences and their Applications. Among his greatest victories for Göttingen was his success, thanks to his excellent relations with Prussian Ministry Director Friedrich Althoff, in drawing David Hilbert to Göttingen in 1895 and in keeping him despite repeated calls to other universities. After one such competing offer in 1902, Hilbert and Klein convinced Althoff to create a third mathematics professorship in Göttingen, which was then occupied first by Hermann Minkowski and then by Edmund Landau. Thus Göttingen mathematics rose to first place among German universities even in the number of professors.

The mathematical center of Göttingen at that time was undoubtedly David Hilbert, who shaped 20th century mathematics as no one else has. Hermann Weyl wrote in a letter in 1927: "The spirit in which we do mathematics, we received from him." In his universality - every few years he turned to a completely new sphere of activity - he is comparable only with Gauss. He ended the classical era of invariant theory with his new "transcendental" methods. His unification of algebraic number theory, commonly known as the Zahlbericht or &qot;report on numbers" was for at least a half century the classic work for everyone concerned with the field. His Foundations of Geometry inaugurated the axiomatic method. He introduced a method for the justification of the Dirichlet principle which now belongs to the essential tools of analysis. In his investigations of integrals he recognized the necessity of treating infinite-dimensional areas - one speaks today of Hilbert spaces - and he developed a spectral theory that provided the foundations for quantum theory and gave rise to functional analysis, a powerful branch of modern mathematics. His program for the proof of the consistency of mathematics could not attain its goal, as Kurt Gödel showed in 1933, but it directed attention to mathematical models of calculating machines and to the theory of formal languages, which today form the basis of computer science and computer engineering. In his famous lecture at the Second International Congress of Mathematicians in Paris in 1900, Hilbert put forth a now classic list of 23 unsolved problems in a range of areas of mathematics, setting the agenda for much of 20th century research. The overall course of his lectures is of tremendous span, covering all areas of pure mathematics and extending to physical topics such as mechanics, electromagnetic oscillations, and relativity theory, and to philosophical topics such as the nature of mathematical knowledge. He attracted many students, and among his 69 graduate students were eminent figures such as Otto Blumenthal, Richard Courant, and Hermann Weyl.

The cooperation between Hilbert and Minkowski was very close and fruitful, and their friendship dated from their youth in Königsberg. Minkowski had come to widespread attention at age 18 when he received the Grand Prix des Sciences Mathématiques from the Paris Academy of Sciences for the solution of a prize problem about quadratic forms, a topic to which he returned throughout his life. His most characteristic creation is the geometry of numbers, a method linked with number-theoretical problems and still fruitful today. In the last years of his life - he died suddenly of appendicitis in 1909, at the age of 44 - he was intensely occupied with Einstein's new special relativity theory, which now owes to Minkowski the idea of combining space and time into a four-dimensional continuum. His notes were worked over and published by Max Born, paving the way for Einstein's general relativity theory. The gap left by Minkowski's death was filled by Edmund Landau, an outstanding representative of analytic number theory, whose handbook on the theory of the distribution of prime numbers was for a long time a standard work in this area.

In 1904, Carl Runge was appointed to a newly created chair for applied mathematics, the first professorship in this field in Germany. Runge had a substantial influence on numerical analysis; the Runge-Kutta methods play a central role in approximations to the solution of differential equations. Runge's overriding goal was to make mathematics useful for natural science and technology, and he led the Institute for Applied Mathematics and Mechanics together with Prandtl. Meanwhile, Felix Bernstein of the Seminar for Insurance Science became the leading figure in the new Institute for Mathematical Statistics, created in 1918 with support from the Göttingen Association. Beginning with pure mathematics, which he enriched with substantial contributions, Bernstein later turned above all to problems in medical statistics. He also provided advisory assistance to the government of the Weimar Republic in connection with government loans.

Under Klein, Hilbert, Minkowski, Landau and Runge, together with the astronomer Karl Schwarzschild and the physicists Ludwig Prandtl, Peter Debye and Emil Wiechert, there formed the creative atmosphere, described vividly by Max Born, Harald Bohr and Richard Courant, which attracted scientists and students from all over the world and made Göttingen the Mecca of mathematics. This atmosphere outlasted the First World War, later to be brought to a sudden end by the Nazis in 1933. One can be impressed simply by a selection of the Privatdozenten or private lecturers in mathematics between 1895 to 1933: Arnold Sommerfeld, Ernst Zermelo, Otto Blumenthal, Gustav Herglotz, Constantin Carathéodory, Erich Hecke and finally, in 1920, Richard Courant.

Courant's sphere of activity was mathematical physics, and he always advocated the opening of mathematics to applications in the spirit of his predecessor Felix Klein, following Klein's footsteps as organizing leader of Göttingen mathematics. After the Mathematical-Scientific Faculty separated from the Philosophical Faculty in 1922, Courant oversaw the incorporation of the mathematics reading room and the collection of mathematical instruments and models into the Mathematics Institute. Courant, Hilbert and Landau became directors of the Institute, and Courant took over the main workload of management and kept it until 1933. Runge's department for applied mathematics was incorporated into the Institute in 1925, soon after his retirement, leading to a correspondence with the Ministry about the Mathematics-Physics Seminar and a subsequent transfer of all important mathematics-related functions to the Mathematics Institute. Gustav Herglotz, more of a pure mathematician, was appointed to Runge's chair, and only in the late 1960s did applied mathematics in Göttingen once again receive its own representative.

Courant possessed a great talent in handling the budgets he administered, and he succeeded, as did Klein, in securing private funds for the Institute. It is to Courant's credit that Klein's idea of a new institute building became a reality only four years after Klein's death, as Courant developed contacts through Niels and Harald Bohr with the Rockefeller Foundation, convincing them to donate US$350,000 for the building. The expansive, well-designed building, built around a large reading hall and reference library, was completed in 1929, and presents a visible sign of the esteem of Göttingen mathematics worldwide. Otto Neugebauer, at that time assistant at the Mathematics Institute and Courant's right hand, played a large role in the planning of the building, and built up the library into one of the best mathematical libraries in the world.

Emmy Noether, one of the leading modern mathematicians, came to Göttingen in 1915 to work with Klein and Hilbert on problems in relativity theory. She stayed until 1933, when she was one of the first to lose their position under the new National Socialist regime. Her first attempt at Habilitation failed because of the private lecturer regulations at that time, according to which only male applicants could be accepted. Hilbert worked around this refusal, by announcing her lectures under his own name "with the support of Ms. Dr. E. Noether". She received a Habilitation in 1919 after a change in the political climate, and in 1922 she was given the position "unofficial special professor", followed later by a teaching position in algebra which secured her a modest income. Noether gave algebra a theoretical structure that today it cannot do without. As Hermann Weyl came to Göttingen in 1930, she formed, according to him, the strongest center of mathematical activity.

Hilbert retired in 1930, and his successor was his most important student, Hermann Weyl. Weyl's interests were just as broadly varied as those of his teacher, ranging from almost all areas of pure mathematics to relativity theory and quantum mechanics and to intuitionistic logic and related philosophical questions. His original ideas and new points of view on old problems continue to surprise again and again. His monograph The Concept of a Riemann Surface was for decades a model of mathematical concept formation. Others often said of his book The Theory of Groups and Quantum Mechanics that it taught the physicists group theory and the mathematicians quantum mechanics. In its preface Weyl remarks: "I can never resist, in this drama of mathematics and physics - which inseminate each other in the dark, but so gladly deny and misjudge each other face-to-face - the role of the (as I amply learn, often unwanted) messenger." Weyl's investigations in differential geometry moved away from Riemann and built the edifice of Weyl geometry. He also introduced the concept of a gauge, which now plays an important role in gauge theories in physics. Though Weyl had spent many of his formative years in Göttingen, he only stayed for three years as full professor, and accepted an offer from Princeton in 1933.

Once the National Socialists seized power, the laws against Jews, political pressure, and defamation and boycott campaigns ensured an exodus of scientists that hit mathematics particularly hard. Of the full professors, only Herglotz remained in office, and most of the special professors, private lecturers and assistants were driven out. Hilbert, when asked in 1934 by Minister Rust whether the Mathematics Institute had suffered from the departure of the Jews and friends of Jews, replied: "Suffered? It hasn't suffered, Mr. Minister. It doesn't exist any more!" Attempts at a revival by Helmut Hasse and Carl Ludwig Siegel foundered under hostile political conditions and the breaking out of the Second World War.

Translated loosely from Erwin Neuenschwander und Hans-Wilhelm Burmann „Die Entwicklung der Mathematik an der Universität Göttingen“