Diving into Math with Emmy Noether

Some 100 years ago a notice appeared in the journal of the German Mathematical Society that read: “Dr. Emmy Noether has habilitated as a lecturer in mathematics at Gottingen University.” This quiet announcement was actually the resounding final chord in a long struggle that went on for four years and only ended on June 4, 1919, when Noether joined the Gottingen faculty.

Hilbert’s Early Career

David Hilbert’s remarkable career falls into two clearly distinct periods: the quiet Konigsberg phase, which spanned the period from his birth on 23 January 1862 to that of his full maturity as one of Germany’s leading mathematicians, followed by the tumultuous Gottingen years. The latter began with his appointment in Gottingen in 1895 and ended with his death on 14 February 1943 when Nazi Germany had already entered its death throes. It would be difficult to exaggerate the contrast between these two phases, just as it remains difficult to picture life in Germany before the onset of the two world wars that so decisively shaped the course of twentieth century history.

Otto Blumenthal: Hilberts Lebensgeschichte

Mehrere Mitarbeiter nahmen an der Veroffentlichung von Hilberts Gesammelten Abhandlungen (Hilbert 1932, 1933, 1935) teil. Der dritte und letzte Band enthalt zwei Essays, geschrieben von seinen ehemaligen Assistenten Ernst Hellinger und Paul Bernays: Das erste ist eine breitere Darstellung uber Hilberts Beitrage zur Integralgleichungstheorie (Hilbert 1935, 94–155), wahrend das zweite Hilberts Arbeiten zur Grundlegung der Mathematik (Hilbert 1935, 196–215) gewidmet war. Dieser dritte Band schloss mit dem in diesem Kapitel abgedruckten biographischen Aufsatz aus der Feder von Otto Blumenthal (Blumenthal 1935).

Is (Was) Mathematics an Art or a Science?

If you teach in a department like mine, the answer to this timeless question may actually carry consequences that seriously affect the resources your program will have available to teach mathematics in the future. In Mainz, no one is likely to protest that mathematics has long been counted as part of the Naturwissenschaften (natural sciences). If it were part of the Geisteswissenschaften (humanities), this would probably have serious budgetary implications. Of course most mathematics departments are now facing a far more immediate and pressing issue, one that can perhaps be boiled down to a different question: is mathematics closer to (a) an art form or (b) a form of computer science? If yo…

On the Background to Hilbert’s Paris Lecture “Mathematical Problems”

Much has been written about the famous lecture on “Mathematical Problems” (Hilbert 1901) that David Hilbert delivered at the Second International Congress of Mathematicians, which took place in Paris during the summer of 1900 (Alexandrov 1979; Browder 1976). Not that the event itself evoked such great interest, nor have many writers paid particularly close attention to what Hilbert had to say on that occasion. What mattered – both for the text and the larger context – came afterward. Mathematicians remember ICM II and Hilbert’s role in it for just one reason: this was the occasion when he unveiled a famous list of 23 problems, a challenge to those who wished to make names for themselves in …

Emmy Noether: a Portrait

“I always went my own way in teaching and research,” Emmy Noether once wrote toward the end of her life.

Introduction to Part IV

When looking at the early development of relativity theory, one finds an astonishing number of contributions by mathematicians, some of which deeply influenced the work of leading theoretical physicists. Within the context of special relativity, Hermann Minkowski’s writings come immediately to mind (Walter 2008). Klein and Hilbert followed Minkowski’s ideas from their infancy, and both pursued some of their consequences after the latter’s premature death in January 1909. Two other figures with close ties to Gottingen, Max Born and Arnold Sommerfeld, were both instrumental in elaborating Minkowski’s 4-dimensional approach for physicists (Walter 2007). Born had been Minkowski’s assistant for …

On the myriad mathematical traditions of ancient greece

To exert one’s historical imagination is to plunge into delicate deliberations that involve personal judgments and tastes. Historians can and do argue like lawyers, but their arguments are often made on behalf of a picture of the past, and these historical images obviously change over time. Why should the history of mathematics be any different? When we imagine the world of ancient Greek mathematics, the works of Euclid (Heath 1926), Archimedes (Heath 1897b), and Apollonius (Heath 1897a) easily spring to mind. Throughout most of the twentieth century, our dominant image of Greek mathematical traditions has been shaped by the high standards of rigor and creative achievement that are purporte…

Felix Klein, Adolf Hurwitz, and the “jewish question” in german academia

Introduction to Part V

The shock of defeat at the end of the First World War left many German academics dumbfounded and numb. Even Hilbert, an outspoken internationalist, was deeply disillusioned by the chaos and instability that plagued the early Weimar years. Already during the war, political differences widened the gulf that had already formed within the Gottingen Philosophical Faculty, whose conservative members felt they were constantly being provoked by the “Hilbert faction.” The controversy over Emmy Noether’s candidacy to habilitate in 1915, mentioned in the introduction to Part IV, was only one of many such instances. Others were even more serious, as when Hilbert and his pacifist friends were accused of…

On Gauss and Gaussian Legends: A Quiz

For the last few years, students in my history of mathematics course have been required to do a bit of research on the web. Each of them chooses from a list of specially chosen questions designed to make them ponder whether the information they find on standard internet sites is solidly grounded and clearly sourced, or whether subsequent research (pursued in such unlikely places as the local university library) might lead a person to doubt what one reads online. The idea here is not to push for a definitive answer; in many cases, this would be a hopeless undertaking anyway. Instead, I ask students merely to report on what they found and how they went about tracking down the information cite…

Max and Emmy Noether: Mathematics in Erlangen

Until 1933, most of Emmy Noether’s life was spent in two middle-sized cities: Erlangen, her birthplace, and Gottingen, where she began her mathematical career.

Das Haupttheorem der finiten Mengen

Wir betrachten eine beliebige Menge M. Sei μ die M zugrunde liegende abzahlbar unendliche Menge der endlichen (gehemmten und ungehemmten) Wahlfolgen Fsn1…n2, wo s und die nυ die fur die betreffende Wahlfolge der Reihe nach gewahlten naturlichen Zahlen vorstellen, und wobei wir ohne Einschrankung der Tragweite des Beweises von ungehemmten, beendigten Wahlfolgen Abstand nehmen konnen.

Klein, Hurwitz, and the “Jewish Question” in German Academia

Mathematicians love to tell stories about people they once knew or perhaps only heard about. If the story happens to sound believable, others are apt to repeat it, possibly embellishing on the original tale. Such mathematical folklore occasionally finds its way into print, and once it does, readers are apt to take such stories at face value, lending them additional credibility. Occasionally, though, alleged facts come under scrutiny, and established stories are exposed as fiction. Yet even when someone comes along with decisive evidence refuting an earlier account it can easily happen that the original story just refuses to die.

Professor in Aachen (1905–1910)

Blumenthals ehemaliger Gottinger Lehrer Arnold Sommerfeld vertrat spater die Mechanik an der Technischen Hochschule Aachen. Auf seine Initiative wurde Blumenthal 1905 nach Aachen berufen. Dort musste Blumenthal sich bald neu orientieren, da er keine fruheren Erfahrungen von den Verhaltnissen an einer technischen Hochschule gemacht hat. Viele Technikprofessoren in Aachen betrachteten Sommerfeld und Blumenthal als Frontkampfer fur eine Kampagne Felix Kleins, der die technischen Hochschulen Preusens reformieren wollte. Diese Bestrebungen Kleins intensivierte die schon bestehenden Spannungen in der damaligen Hochschulpolitik, wie aus mehreren Briefen Blumenthals hervorgeht, zumal er in Aachen d…

Noether’s Early Contributions to Modern Algebra

As described in preceding chapters, Noether’s work on invariant theory broke new ground that led the Gottingen mathematicians, but first and foremost Hilbert, to invite her to habilitate there.

Klein, Mittag-Leffler, and the Klein-Poincaré Correspondence of 1881–1882

If a modern-day Plutarch were to set out to write the “Parallel lives” of some famous modern-day mathematicians, he could hardly do better than to begin with the German, Felix Klein (1849–1925), and the Swede, Gosta Mittag-Leffler (1846–1927). Both lived in an age ripe with possibilities for the mathematics profession and, like few of their contemporaries, they seized upon these new opportunities whenever and however they arose. Even when their chances for success looked dismal, they forged ahead, winning over the skeptics as they did so. Although accomplished and prolific researchers (Klein’s work has even enjoyed the appellation “great”), they owed much of their success to their talents a…

Models as Research Tools: Plücker, Klein, and Kummer Surfaces

In the late summer of 1869, 20-year-old Felix Klein made his way to Berlin, where he planned to attend the renowned seminar founded by Ernst Eduard Kummer and Karl Weierstrass. Klein had already taken his doctorate in Bonn and he would soon be recognized as a leading expert on line geometry, a new approach to 3-space launched by his mentor in Bonn, Julius Plucker. Just before Plucker died in 1868, he entrusted Klein to complete the classic monograph, Neue Geometrie des Raumes gegrundet auf die Betrachtung der geraden Linie als Raumelement. Overall responsibility for this project fell to Alfred Clebsch in Gottingen, which was how Klein first came to the prestigious Georgia Augusta. There he …

Blumenthals Würdigung von Schwarzschild, 1917

Nach dem Tod Schwarzschilds schrieb Otto Blumenthal kurze Kommentare zu den Arbeiten in seinem Nachlass. Diese benutzte er als Grundlage fur seinen wissenschaftlichen Nachruf auf Schwarzschild, der im Jahr 1917 im Jahresbericht der Deutschen Mathematiker-Vereinigung erschienen ist. Der Text dieses bedeutenden Nachrufs wird in diesem Kapitel neu abgedruckt.

REVOLUTIONS IN MATHEMATICS

Hauptbegriffe über reelle Funktionen einer Veränderlichen

Wir legen dem Folgenden die Cartesische Ebene mit x- und y-Axe zugrunde und definieren auf diesen Axen besondere Punktspecies x bezw. y, die wir als Punktkerne bezeichnen.

From Königsberg to Göttingen: a sketch of Hilbert’s early career

Book Review

Deine Sonia: A Reading from a Burned Letter by Reinhard Bölling, Translated by D. E. Rowe

It was in January 1990. Finally, just two months after the Berlin Wall had fallen, I had the opportunity to spend a few days at the Mittag-Leffler Institute in Djursholm, a small town just northeast of Stockholm. The palatial villa that today houses the Institute is the former home of Gosta Mittag-Leffler (1846–1927), and on entering its doorway I felt as if I had taken a step back into the world in which he lived. For me, the Institute’s single greatest attraction lay in its archival holdings, and particularly the extensive correspondence that linked Mittag-Leffler with many of the era’s leading mathematicians. A former student of Karl Weierstrass (1815–1897), Mittag-Leffler sought to pres…

Otto Neugebauer and the Göttingen Approach to History of the Exact Sciences

Otto Neugebauer (1899–1990) was, for many, an enigmatic personality. Trained as a mathematician in Graz, Munich, and Gottingen, he had not yet completed his doctoral research when in 1924 Harald Bohr, brother of the famous physicist, invited him to Copenhagen to work together on Bohr’s new theory of almost periodic functions. Quite by chance, Bohr asked Neugebauer to write a review of T. Eric Peet’s recently published edition of the Rhind Papyrus (Neugebauer 1925). In the course of doing so, Neugebauer became utterly intrigued by Egyptian methods for calculating fractions as sums of unit fractions (e.g. 3/5 = 1/3 + 1/5 + 1/15). When he returned to Gottingen, he wrote his dissertation on thi…

Book Review

Mathematik, Wissenschaft und Sprache

Mathematik, Wissenschaft und Sprache bilden die Hauptfunktionen der Aktivitat der Menschheit, mittels deren sie die Natur beherrscht und in ihrer Mitte die Ordnung aufrecht erhalt. Diese Funktionen finden ihren Ursprung in drei Wirkungsformen des Willens zum Leben des einzelnen Menschen: 1. die mathematische Betrachtung, 2. die mathematische Abstraktion und 3. die Willensauferlegung durch Laute.

On projecting the future and assessing the past—the 1946 princeton bicentennial conference

Der Kongress in Bologna: Hilbert vs. Brouwer (1927–1928)

Die Teilnahme deutscher Mathematiker an dem Internationalen Mathematikerkongress in Bologna blieb bis zur Eroffnung eine heis umstrittene Frage. Eine offizielle Einladung an die Deutsche Mathematiker-Vereinigung (DMV) fuhrte zu einer Debatte innerhalb des Ausschusses, an der Otto Blumenthal direkt beteiligt war. Viele deutsche Mathematiker wollten sich voll und ganz von der Union Mathematique Internationale (UMI) distanzieren, weil sie nach wie vor die seit 1919 bestehende Boykottpolitik vertrat.

Verfolgung und Ausgrenzung (1930–1942)

Bei den Rektoratswahlen 1930 kandidierte Otto Blumenthal fur das Amt des Rektors der TH Aachen. Die beiden folgenden Briefe von Heinrich Brandt an seinen ehemaligen Aachener Kollegen Erich Trefftz zeigen noch einmal deutlich, wie sehr, und zwar schon lange vor 1933, die akademische Atmosphare von Antisemitismus durchdrungen war (siehe dazu auch Band I, Abschnitt 1.8 und 4.2).

Emmy Noether’s Triumphal Years

When Emmy Noether returned from the September 1929 conference in Prague – where she and Hasse surely spoke about their mutual mathematical interests – she belatedly answered a postcard he had sent here.

Grundlagen aus der Theorie der Punktmengen

Wir denken uns in einer Ebene ein rechtwinkliges Koordinatenkreuz Oxy gezeichnet und zerlegen die Ebene in Quadrate κ1 mit der Seitenlange 1, deren Eckpunte ν ganzzahlige Koordinaten besitzen. Jedes dieser Quadrate κ1 zerlegen wir in vier Kongruente, homothetische Teilquadrate κ2 von der Seitenlange \( \frac{1}{2} = 2^{1-2}\) und definieren, in dieser Weise fortfahrend, Quadrate κ3, κ4, . . . . Unter einem Quadrat κ oder κ-Quadrat verstehen wir dann ein Quadrat κν mit willkurlichen Index ν.

Die Annalen-Krise: Hilbert vs. Brouwer (1928–1929)

Vermutlich erfuhr Blumenthal zum ersten Mal von Brouwer selbst, dass Caratheodory einen Versuch unternommen hatte, Hilberts drastische Mitteilung vom 25. Oktober irgendwie zu mildern. In einem personlichen Gesprach mit Brouwer in Laren wollte der Grieche ihm erklaren, dass aufgrund Hilberts Gesundheitszustand eine ruhige Diskussion mit ihm uber diese Thematik unmoglich sei. Die Anregung zu diesem Besuch, der am 30.

Who Linked Hegel’s Philosophy with the History of Mathematics?

Standard histories of mathematics are filled with names, dates, and results, but seldom do we find much attention paid to the contexts in which mathematics was made or past achievements recorded. Yet by widening the net, one can easily retrieve many interesting examples that reveal how mathematicians thought about these matters and much else besides. This column deals with one such person – whose identity readers are hereby challenged to uncover – in order to illustrate in a particularly striking way the potential confluence of mathematical and philosophical ideas. The sources to which I allude below are all in print and readily accessible, so I have reason to hope that these hints will lea…

Personal Reflections on Dirk Jan Struik By Joseph W. Dauben

Dirk Jan Struik, who taught for many years at the Massachusetts Institute of Technology and died on 21 October 2000 at the age of 106, was a distinguished mathematician and influential teacher. He was also widely known as a leading Marxist scholar and social activist. His early work on vector and tensor analysis, undertaken together with Jan Arnoldus Schouten, helped impart new mathematical techniques needed to master Einstein’s general theory of relativity. This collaboration lasted for over 20 years, but by the end of the 1930s, Struik came to realize that the heyday of the Ricci calculus had passed. After the Second World War, having now entered his 50s, he gave up mathematical research …

Making Mathematics in an Oral Culture: Göttingen in the Era of Klein and Hilbert

This essay takes a close look at specially selected features of the Göttingen mathematical culture during the period 1895–1920. Drawing heavily on personal accounts and archival resources, it describes the changing roles played by Felix Klein and David Hilbert, as Göttingen's two senior mathematicians, within a fast-growing community that attracted an impressive number of young talents. Within the course of these twenty-five years Göttingen exerted a profound impact on mathematics and physics throughout the world. Many factors contributed to the creation of a special atmosphere that served as a model for several other important centers for mathematical research. Göttingen exemplified a dyna…

Exil und Deportation (1939–1944)

Am 19. Oktober 1939 konnten die Blumenthals ihr vorubergehendes Quartier im Haus Zuilenveld verlassen und in eine richtige Wohnung nach Delft umziehen. Ihre Situation am Ende des Jahres beschrieb Otto Blumenthal in einer Silvesterkarte an David und Kathe Hilbert.

Blumenthal über Hilbert zu seinem 60. Geburtstag

David Hilbert feierte am 23. Januar 1922 seinen 60. Geburtstag. Richard Courant engagierte sich stark bei den Vorbereitungen eines stimmungsvollen Festes, zu welchem weit uber hundert Gaste kamen, u.a. Ferdinand Springer aus Berlin. Ursprunglich plante auch Einstein dabei zu sein, zumal er brennendes Interesse hatte, seine Deutung des neuen Ergebnisses bei dem Einstein-Geiger-Kanalstrahlenexperiment mit den Gottingern Physikern Max Born und James Franck zu diskutieren.

From Graz to Göttingen: Neugebauer’s Early Intellectual Journey

Otto Neugebauer’s early academic career was marked by a series of transitions. His interests shifted from physics to mathematics, and finally to the history of ancient mathematics and exact sciences. Yet even from his early years in Graz, Neugebauer was strongly attracted to the mathematical culture of Gottingen. When he arrived there in 1922, he quickly established a strong personal friendship with Richard Courant, the newly appointed Director of the Mathematics Institute. Neugebauer and Courant worked together closely up until 1933, when the Nazi government decimated the Gottingen scientific community. In this essay, Neugebauer’s historical work and his vision for a new approach to the st…

Debating Relativistic Cosmology, 1917–1924

Physical astronomy as we know it today matured during the latter half of the twentieth century. It was preceded by a period Jean Eisenstaedt has dubbed the “low water mark” in general relativity (GR), covering roughly the period 1925 to 1955 (Eisenstaedt 1988b). Starting in the 1960s, however, a series of startling developments helped pave the way for what has since been called the “renaissance of general relativity,” which suddenly took on great significance for astrophysics and cosmology. In the days of Einstein and Eddington, one could imagine a gravitational field so strong that it would produce a black hole, a true space–time singularity. People talked about such things, but hardly any…

Ein Leben für die Mathematik: Otto Blumenthal (1919–1944)

Otto Blumenthals mathematische Karriere fing sehr vielversprechend an, vor allem in Hinblick auf die vielen Anregungen und breiten Kenntnisse, die er wahrend seiner Jahre in Gottingen gewann. Nach seinem Studium und einer relativ kurzen Zeit als Privatdozent dort wurde er schon 1905 an die TH Aachen berufen. Er war damals noch nicht 30 Jahre alt, und wie viele andere junge Mathematiker strebte er eine Professur an einer deutschen Universitat an.

Introduction to Part VI

This last set of essays leaves the terrain of Gottingen, even though its tradition still looms large in the background. Here I present an assortment of mathematical people, some of whom are likely to be known to many readers. The scene now shifts somewhat abruptly to figures in North America during the last century, though usually with an eye cast toward their links with Europe. Curiosity about the roots of research mathematics in the United States – an important chapter in the larger story of American higher education – was a major factor that influenced my early interest in the German universities, particularly Gottingen’s Georgia Augusta. One can hardly exaggerate the strength of that un…

Neue Spannungen in den Nachkriegsjahren (1919–1923)

Im Fruhjahr 1919 nahm Hilbert als Gutachter an einem Berufungsverfahren fur eine Professur in Bern teil. Es stellte sich dabei als schwierig heraus, einen geeigneten Kandidaten zu gewinnen. Dies brachte Hilbert auf die Idee, sich selbst als Kandidat fur die Stelle ins Spiel zu bringen.

Konflikte in der Annalen-Redaktion (1924–1927)

Erich Trefftz, der nach dem Krieg Blumenthals Kollege in Aachen wurde, ging schon 1922 nach Dresden, wo er eine Professur an der Technischen Hochschule annahm. Zwei Jahre danach verstarb der Dresdner Professor fur Technische Mechanik, Karl Wieghardt. Bei einem Besuch von Trefftz in Aachen erfuhr Blumenthal, dass Theodore von Karman auf Platz eins der Berufungsliste fur die Nachfolge Wieghardts stand.

Memories and Legacies of Emmy Noether

Those who knew Emmy Noether best were her fellow Germans in exile, in particular her former colleague in Gottingen, Hermann Weyl.

Der Gegenstand der intuitionistischen Mathematik: Spezies, Punkte und Räume. Das Kontinuum

Zwei Mengenelemente heissen gleich oder identisch, wenn man sicher ist, dass fur jedes n die n-te Wahl fur beide Elemente dasselbe Zeichen erzeugt, und verschieden wenn die Unmoglichkeit ihrer Gleichheit feststeht, d.h. wenn man Sicherheit hat, dass sich im Laufe ihrer Erzeugung nie ihre Gleichheit wird beweisen lassen. Die Identitat mit einem beliebigen Elemente der Menge M, bzw. mit dem Mengenelement e, werden wir als die Mengenspezies, oder kurz als die Menge, M, bzw. als die Elementspezies oder kurz als das Element e bezeichnen.

Hilbert’s Legacy: Projecting the Future and Assessing the Past at the 1946 Princeton Bicentennial Conference

Saunders Mac Lane on Solomon Lefschetz (Mac Lane 1989, 220): In 1940 when he was writing his second book on topology, [Lefschetz] sent drafts of one section up to Whitney and Mac Lane at Harvard. The drafts were incorrect, we wrote back saying so – and every day for the next seven or eight days we received a new message from Lefschetz, with a new proposed version. It is no wonder that the local ditty about Lefschetz ran as follows: Here’s to Lefschetz, Solomon L Ir-re-pres-si-ble as hell When he’s at last beneath the sod He’ll then begin to heckle God.

Review††Edited by Adrian Rice and Antoni MaletAll books, monographs, journal articles, and other publications (including films and other multisensory materials) relating to the history of mathematics are abstracted in the Abstracts Department. The Reviews Department prints extended reviews of selected publications.Materials for review, except books, should be sent to the Abstracts Editor, Sloan Despeaux, Western Carolina University, Cullowhee, NC 28723, USA. Books in English for review should be sent to Adrian Rice, Department of Mathematics, Randolph-Macon College, Ashland, VA 23005-5505, USA. Books in other languages for review should be sent to Antoni Malet, Universitat Pompeu Fabra, Department of Humanities, Ramon Trias Farga 25–27, Barcelona, 8005, Spain.Most reviews are solicited. However, colleagues wishing to review a book are invited to make their wishes known to the appropriate Book Review Editor. (Requests to review books written in the English language should be sent to Adrian Rice at the above address; requests to review books written in other languages should be sent to Antoni Malet at the above address.) We also welcome retrospective reviews of older books. Colleagues interested in writing such reviews should consult first with the appropriate Book Review Editor (as indicated above, according to the language in which the book is written) to avoid duplication.

Hilbert’s early career: Encounters with allies and rivals

It seems to me that the mathematicians of today understand each other far too little and that they do not take an intense enough interest in one another. They also seem to know—so far as I can judge—too little of our classical authors (Klassiker); many, moreover, spend much effort working on dead ends. “ David Hilbert to Felix Klein, 24 July 1890

Brouwer als Mitglied der Annalen-Redaktion (1919–1922)

Seit Juli 1914 gehorte Brouwer zur Annalen-Redaktion (siehe seinen Brief an Klein vom 10. Juli in Kap. 7, Band I). Wahrend des Krieges gab es allerdings nur wenig fur ihn zu tun, da Blumenthal im Kriegsdienst arbeitete. Klein und Hilbert haben in dieser Zeit mehrere Aufsatze uber die Relativitatstheorie geschrieben, aber sie erschienen nicht in den Annalen, sondern in den Gottinger Nachrichten. Nach Ende des Krieges konnten die Gottinger, aber vor allem Blumenthal zusammen mit Springer, die Zeitschrift wieder beleben.

Historical Events in the Background of Hilbert’s Seventh Paris Problem

David Hilbert’s lecture, “Mathematical Problems,” [Hilbert 1900] delivered in Paris in 1900 at the Second International Congress of Mathematicians, has long been recognized as marking a milestone in the history of mathematics. Certainly for Hilbert himself, this marked the single greatest event and a true turning point in his storied career. When historians and mathematicians have written about the so-called Hilbert problems, they have usually looked forward into the twentieth century, sometimes by viewing their resolution as markers for mathematical progress.

Stellensuche im Ausland (1933–1938)

Die „Entfernung vom Lehramt“ traf Otto Blumenthal, wie er in seinem oben zitierten Brief vom 20. Dezember 1933 an Arnold Sommerfeld schrieb, hart, weil er „sehr gern unterrichtet und dem Unterricht viel zu viel Zeit und Kraft geschenkt“ habe (Abschnitt 9.2). Er vermisste seine Lehrtatigkeit sehr und suchte deshalb intensiv nach einer entsprechenden neuen Stelle, die er naturlich nur im Ausland finden konnte.

Emmy Noether in Bryn Mawr

In the annals of higher education for women, two elite colleges were particularly important for mathematics: Girton College, in Cambridge, England and Bryn Mawr College, near Philadelphia, Pennsylvania.

Hermann Minkowski’s Cologne Lecture, “Raum und Zeit”

A century ago, David Hilbert stepped to the podium at a special meeting of the Gottingen Academy of Sciences to recall the achievements of his close friend Hermann Minkowski. Just one month earlier, on 12 January 1909, the 43-year-old Minkowski had died unexpectedly after suffering a ruptured appendix, leaving those close to him in a state of shock. None was more deeply affected than Hilbert, whose memorial lecture (Hilbert 1910) reflects the deep sense of personal loss he felt at that time:

Einstein Studies, volume 11: A retrospective review

Freundschaft mit Schwarzschild (1909–1916)

Die Briefe in diesem Kapitel dokumentieren Blumenthals Freundschaft mit Karl Schwarzschild in den Jahren 1909 bis 1916, aber vor allem wahrend der Kriegszeit. Blumenthal konnte relativ fruh, dank dem Intervenieren seines Freundes Schwarzschild, eine Stelle in Hannover bekommen, und zwar als Leiter der dortigen Militar-Wetterwarte. Schwarzschild selbst hatte viel weniger Gluck; er starb im Mai 1916 an einer grausamen Hautkrankheit. In seinem Testament nannte er Blumenthal als einen von vier Wissenschaftlern, die seine hinterlassenen Manuskripte durchschauen durften. Blumenthals Kommentare zu diesen Arbeiten konnte er dazu verwenden, um einen Nachruf auf Schwarzschild zu schreiben, dem er vor…

Hermann weyl, the reluctant revolutionary

“Brouwer – that is the revolution!” – with these words from his manifesto “On the New Foundations Crisis in Mathematics” (Weyl 1921) Hermann Weyl jumped headlong into ongoing debates concerning the foundations of set theory and analysis. His decision to do so was not taken lightly, knowing that this dramatic gesture was bound to have immense repercussions not only for him, but for many others within the fragile and politically fragmented European mathematical community. Weyl felt sure that modern mathematics was going to undergo massive changes in the near future. By proclaiming a “new” foundations crisis, he implicitly acknowledged that revolutions had transformed mathematics in the past, …

Coxeter on People and Polytopes

H. S. M. Coxeter, known to his friends as Donald, was not only a remarkable mathematician. He also enriched our historical understanding of how classical geometry helped inspire what has sometimes been called the nineteenth-century’s non-Euclidean revolution (Fig. 35.1). Coxeter was no revolutionary, and the non-Euclidean revolution was already part of history by the time he arrived on the scene. What he did experience was the dramatic aftershock in physics. Countless popular and semi-popular books were written during the early 1920s expounding the new theory of space and time propounded in Einstein’s general theory of relativity. General relativity and subsequent efforts to unite gravitati…

Mini-Workshop: Max Dehn: his Life, Work, and Influence

Poincaré Week in Göttingen, 22–28 April 1909

When Paul Wolfskehl died in 1906, his will established a prize for the first mathematician who could supply a proof of Fermat’s Last Theorem, or give a counterexample refuting it. The interest from this prize money was later used to bring world-renowned mathematicians to Gottingen to deliver a series of lectures. Hilbert was apparently very pleased with this arrangement, and once jested that the only thing that kept him from proving Fermat’s famous conjecture was the thought of killing the goose that laid these golden eggs.

Brouwer und die Annalen (1911–1918)

Die Briefe in diesem Kapitel wurden in der Zeit geschrieben, bevor Brouwer in die Redaktion der Annalen eintrat. Sie dokumentieren u.a., wie Blumenthal zunehmend in den Kontroversen um Brouwer verwickelt wurde. In dem kurzen Zeitraum von 1910 bis 1913 veroffentlichte Brouwer seine revolutionaren topologischen Aufsatze in den Annalen. Ruckblickend urteilte Brouwer hieruber „viele meiner Arbeiten hatte ich ohne [Blumenthal] nicht geschrieben“. Da eingehende Erlauterungen entschieden zu weit fuhren wurden, werden in den Kommentaren zu diesen Briefen nur gewisse Hauptpunkte erklart, die fur ein Verstandnis der wichtigsten Streitfragen unerlasslich sind.

Emmy Noether’s Long Struggle to Habilitate in Göttingen

Doctoral degrees have a long prehistory, but the modern Ph.D. first arose as part of an educational reform launched at the German universities. Over the course of the nineteenth century, this degree came to be awarded not merely to those who displayed a command of established knowledge in an academic field.

History of Mathematics: A Global Cultural Approach

The Old Guard Under a New Order: K. O. Friedrichs Meets Felix Klein

Constance Reid’s recent tribute to K. O. Friedrichs (Reid 1983) undoubtedly brought back fond memories to those who knew the man and his many achievements (see also Reid 1986). It was a great pleasure for me to interview Friedrichs in January 1982, only about a year before his death. He was already in very delicate health. His wife Nellie (see Biegel 2012) was kind enough to arrange the interview, but warned me beforehand that her husband tired rather easily and was somewhat hard of hearing. Nevertheless, he was extremely forthcoming in discussing his early career with me, and quick to dismiss some of my faulty misconceptions regarding Gottingen mathematics in the 1920s, which was the main …

Otto Neugebauer’s Vision for Rewriting the History of Ancient Mathematics

Les historiens des mathematiques ont longtemps exalte les realisations des anciens Grecs, symbolisees par un seul nom, Euclide d’Alexandrie. Les treize livres qui composent ses Elements occupent, dans les mathematiques grecques, une place comparable au Parthenon dans sa tradition architecturale. L’appreciation du classicisme grec a en outre ete renforcee par l’ideal de la geometrie euclidienne, un style qui a persiste jusqu’en plein xixe siecle. Il a fallu attendre les premieres decennies du xxe siecle pour qu’emerge une nouvelle image des mathematiques anciennes, proposee par les recherches pionnieres d’Otto Neugebauer sur les mathematiques egyptiennes et surtout mesopotamiennes. Bien que …

Dozent in Göttingen und Marburg (1897–1905)

Blumenthals Briefe an Karl Schwarzschild und spater an Hilbert beleuchten die damalige Atmosphare in Gottingen, wo er bis 1898 studiert hat, wie auch in Marburg, wo er kurze Zeit die Mathematik an einer kleineren Universitat vertrat. Seine Freundschaft mit Schwarzschild ging sogar auf ihrer gemeinsamen Jugendzeit in Frankfurt zuruck. Als dieser 1901 nach Gottingen berufen wurde, befand sich Blumenthal im Zentrum einer besonders lebendigen wissenschaftlichen Umgebung, die seine spatere Karriere stark pragen wurde.

Brouwer und die Dimensionstheorie (1923–1924)

Im September 1923 fand die Jahrestagung der Deutschen Mathematiker-Vereinigung (DMV) in Marburg statt, bei welchem Anlass Paul Urysohn (1898–1924) einen Vortrag uber seine neue Dimensionstheorie hielt. Urysohn studierte vorher Brouwers fruhere Arbeit „Uber den naturlichen Dimensionsbegriff“ (Brouwer 1913a) und fand dabei, dass die Argumentation nicht fehlerfrei war. Urysohn erwahnte dies auf der Marburger Tagung, woraufhin Brouwer ihn ansprach, um eine schriftliche Erklarung zu bitten. Zu dieser Zeit war Otto Blumenthal Vorsitzender der DMV und Ludwig Bieberbach Schriftfuhrer ihrer Jahresberichte.

Noether’s International School in Modern Algebra

Pavel Alexandrov and Heinz Hopf met for the first time in Gottingen in the spring of 1926, soon after Alexandrov departed from Blaricum. Hopf had recently taken his doctorate in Berlin under Ludwig Bieberbach and Erhard Schmidt, and his research interests differed sharply from Alexandrov’s work in general topology.

Transforming Tradition: Richard Courant in Göttingen

Richard Courant had a knack for being at the right place at the right time. He came to Gottingen in 1907, just when Hilbert and Minkowski were delving into fast-breaking developments in electron theory. There he joined three other students who also came from Breslau: Otto Toeplitz, Ernst Hellinger, and Max Born, all three, like him, from a German Jewish background. Toeplitz was their natural intellectual leader, in part because his father was an Oberlehrer at the Breslau Gymnasium (Muller-Stach 2014). Courant was five or six years younger than the others; he was sociable and ambitious, but also far poorer than they (Reid 1976, 8–13).

Segre, Klein, and the Theory of Quadratic Line Complexes

Two of C. Segre’s earliest papers, (Segre 1883a) and (Segre 1884), dealt with the classification of quadratic line complexes, a central topic in line geometry. These papers, the first written together with Gino Loria, were submitted to Felix Klein in 1883 for publication in Mathematische Annalen. Together with the two lengthier works that comprise Segre’s dissertation, (Segre 1883b) and (Segre 1883c), they took up and completed a topic that Klein had worked on a decade earlier (when he was known primarily as an expert on line geometry). Using similar ideas, but a new and freer approach to higher-dimensional geometry, Segre not only refined and widened this earlier work but also gave it a ne…

Max von laue’s role in the relativity revolution

Whereas countless studies have been devoted to Einstein’s work on relativity, the contributions of several other major protagonists have received comparatively little attention. Within the immediate German context, no single figure played a more important role in developing the consequences of the special theory of relativity (SR) than Max von Laue (1879–1960). Although remembered today mainly for his discovery of x-ray diffraction in 1912 – an achievement for which he was awarded the Nobel Prize – Laue’s accomplishments in promoting the theory of relativity were of crucial importance. They began early, well before most physicists even knew anything about a mysterious Swiss theoretician nam…

Intuitionistische Kritik an einigen elementaren Theoremen

Zahlen wir nun aber die zwischen 0 und 1 gelegenen irreduziblen endlichen Dualbruche (ausschliesslich 0 und 1) in ublicher Weise durch eine Fundamentalreihe δ1, δ2 . . . ab, verstehen wir unter gn(x) die Funktion, die fur x = δn den Wert \( 2^{-n}\) besitzt, fur x = 0 sowie fur x = 1 verschwindet, wahrend sie sowohl zwischen x = 0 und x = δn wie zwischen x = δn und x = 1 linear verlauft, und setzen wir fn(x) = gn(x) fur n = k1, sonst fn(x) = 0, so besitzt die volle Funktion des Einheitskontinuums \(f(x)= \sum\limits_{n = 1}^{\infty } {f_{n} (x)} \) kein Maximum, womit der Existenzsatz des Maximums hinfallig geworden ist.

The Calm Before the Storm: Hilbert’s Early Views on Foundations

In recent years there has been a growing interest among historians and philosophers of mathematics in the history of logic, set theory, and foundations.1 This trend has led to a major reassessment of early work undertaken in these fields, particularly when seen in the light of motivations that animated the leading actors. The present volume may thus be seen as a reflection of this renewed fascination with the work of Hilbert, Brouwer, Weyl, Bernays, and others, an interest that stems in part from the desire to understand the historical and intellectual context that inspired their investigations. With regard to Hilbert, it has been my contention for some time that his stance in the acrimonio…

Einstein and Twentieth-Century Politics: ‘A Salutary Moral Influence’

Episodes in the Berlin-Göttingen Rivalry, 1870-1930

One of the more striking features in the development of higher mathematics at the German universities during the nineteenth century was the prominent role played by various rival centers. Among these, Berlin and Gottingen stood out as the two leading institutions for the study of research-level mathematics. By the 1870s they were attracting an impressive array of aspiring talent not only from within the German states but also from numerous other countries as well. The rivalry between these two dynamos has long been legendary, yet little has been written about the sources of the conflicts that arose or the substantive issues behind them. Here I hope to shed some light on this theme by recall…

Emigration (1938–1939)

In der sogenannten Reichspogromnacht vom 9./10. November 1938 wurde in Aachen, wie in vielen anderen deutschen Stadten auch, die Synagoge niedergebrannt, Geschafte wurden geplundert, und uber 70 Aachener Burger wurden verhaftet und in die Konzentrationslager Buchenwald und Sachsenhausen verschleppt. Unter ihnen war auch ein Sohn von Ludwig Hopf, der sich fur seinen Vater ausgegeben hatte (Muller-Arends 1995, 213). Otto Blumenthal blieb unbehelligt.

The Mathematicians’ Happy Hunting Ground: Einstein’s General Theory of Relativity

There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had comparatively little, general relativity theory very considerable, influence, above all upon the development of a general scheme for differential geometry. —Hermann Weyl, “Relativity as a Stimulus to Mathematical Research,” pp. 536–537.

Cast out of her Country

When Pavel Alexandrov wrote about his last visit to Gottingen some four decades later, he could hardly look past the traumatic events that were to follow in the wake of his departure.

Einstein and Relativity: What Price Fame?

ArgumentEinstein's initial fame came in late 1919 with a dramatic breakthrough in his general theory of relativity. Through a remarkable confluence of events and circumstances, the mass media soon projected an image of the photogenic physicist as a bold new revolutionary thinker. With his theory of relativity Einstein had overthrown outworn ideas about space and time dating back to Newton's day, no small feat. While downplaying his reputation as a revolutionary, Einstein proved he was well cast for the role of mild-mannered scientific genius. Yet fame demanded its price. Surrounded by social and economic unrest in Berlin, he was caught between two worlds, one struggling to be born, another …

Analyse des Kontinuums

Bei der klassischen Auffassung, welche das Kontinuum als vollstandig geordnet betrachtete, stellen sich als wesentlichen Eigenschaften dieser Spezies heraus, dass sie uberall dicht, in sich dicht, zusammenhangend und kompakt war; wir wollen untersuchen, inwiefern sich diese Eigenschaften nach passender Modifizierung in der intuitionistischen Theorie aufrecht erhalten lassen.

Three Letters from Sophus Lie to Felix Klein on Mathematics in Paris

Sophus Lie and Felix Klein first met in 1869 as students in Berlin. They soon became daily companions and spent the spring of 1870 together in Paris where they met the French mathematicians Michel Chasles, Gaston Darboux, and Camille Jordan. Jordan had just published his classic Traite des substitutions, and the two foreigners read it avidly. Mathematics has not been the same since, for it has often been said – and not altogether unjustly – that from this moment on they made group theory their common property: Lie taking the continuous groups and Klein those that were discontinuous. It should not be overlooked, on the other hand, that this observation was first made by Klein himself in the …

Blumenthal als Redakteur (1904–1914)

In diesem Kapitel geht es hauptsachlich um Blumenthals Tatigkeit als Redakteur von 1904 bis zum Ausbruch des Krieges. In diesem Zeitraum, als die Hauptverantwortung fur die Annalen auf seinen Schultern lag, vollzog sich ein groser Wandel in den Wissenschaften, begleitet von einem starken Zuwachs in der mathematischen Produktion. Trotzdem lief in der Vorkriegszeit Blumenthals Arbeit als geschaftsfuhrender Leiter relativ reibungslos ab. Er musste sich aber sehr bemuhen, einerseits den Betrieb in Gang zu halten und anderseits gleichzeitig abzusichern, dass die eingereichten Arbeiten von hoher Qualitat waren.

Einstein’s gravitational field equations and the bianchi identities

In his highly acclaimed biography of Einstein, Abraham Pais gave a fairly detailed analysis of the many difficulties his hero had to overcome in November 1915 before he finally arrived at generally covariant equations for gravitation (Pais 1982, 250–261).

Remembering an Era: Roger Penrose’s Paper on “Gravitational Collapse: The Role of General Relativity”

Back in the 1960s, Einstein’s theory of general relativity re-emerged as a field of important research activity. Much of the impetus behind this resurgence came from powerful new mathematical ideas that Roger Penrose and Stephen Hawking applied to prove general singularity theorems for global space-time structures. Their results stirred the imaginations of astrophysicists and gave relativistic cosmology an entirely new research agenda. A decade later, black holes and the big hang model were on the tongues of nearly everyone who followed recent trends in science. As popular expositions dealing with quasars, pulsars, and the geometry of black holes began to appear in magazines and textbooks, …

Blumenthal und die Mathematischen Annalen (1876–1918)

In diesem Essay werden mehrere wichtige Themen im Leben von Otto Blumenthal geschildert. Pragend fur seine Entwicklung war die tiefe Freundschaft mit dem beruhmten Astronom Karl Schwarzschild, eine Beziehung, welche schon aus ihrer Schulzeit in Frankfurt zuruckging, sowie die Erfahrungen seiner Gottinger Zeit, aus der lebenslange freundschaftlichen Verbindungen zu David Hilbert, Felix Klein und Arnold Sommerfeld entstanden sind.

Historische Stellung des Intuitionismus

Der Intuitionismus hat seine historische Stellung im Rahmen der Geschichte der Anschauung erstens uber den Ursprung der mathematischen Exaktheit; zweitens uber die Umgrenzung der als sinnvoll zu betrachtenden Mathematik [1]. In dieser Geschichte sind hauptsachlich drei Perioden zu unterscheiden.

On the Reception of Grassmann’s Work in Germany during the 1870’s

It has often been remarked that Grassmann’s mathematics was not widely appreciated during his lifetime. Although awareness of the dimensions of his achievements began to spread in the early 1870’s, even in Germany relatively few mathematicians appear to have been well acquainted with either the original 1844 edition of Grassmann’s Ausdehnungslehre or the mathematically more accessible edition of 1862. The main reasons for this weak and rather delayed reception have been described often enough---Grassmann’s isolated working environment and his nearly impenetrable language---but there are a number of related aspects that still deserve closer consideration.1

Euclidean geometry and physical space

It takes a good deal of historical imagination to picture the kinds of debates that accompanied the slow process, which ultimately led to the acceptance of non-Euclidean geometries little more than a century ago. The difficulty stems mainly from our tendency to think of geometry as a branch of pure mathematics rather than as a science with deep empirical roots, the oldest natural science so to speak. For many of us, there is a natural tendency to think of geometry in idealized, Platonic terms. So to gain a sense of how late nineteenth-century authorities debated over the true geometry of physical space, it may help to remember the etymological roots of geometry: “geo” plus “metria” literall…

An Enchanted Era Remembered: Interview with Dirk Jan Struik

Dirk J. Struik was born in Rotterdam in 1894, where he attended the Hogere Burger School from 1906–1911 before entering Leiden University. At Leiden he studied algebra and analysis with J. C. Kluyver, geometry with P. Zeeman, and physics under Paul Ehrenfest. After a brief stint as a high school teacher at Alkmaar, he spent seven years at Delft as the assistant to J. A. Schouten, one of the founders of tensor analysis. Their collaboration led to Struik’s dissertation, Grundziige der mehrdimensionalen Differentialgeometrie in direkter Darstellung, published by Springer in 1922, and numerous other works in the years to follow.