David E. Rowe

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 acrimonious foundations debates of the 1920s has tended to reinforce a quite misleading picture of his actual views on the nature of mathematical knowledge. In particular, I have argued that the main tenets of Hilbert’s “formalist program” of the 1920s represent only a portion of his mature “philosophy” of mathematics, as should be readily apparent from the views Hilbert set forth in his 1919-20 lectures “Natur and mathematisches Erkennen” [Hilbert 1992].2

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).

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, even uprooting the entire edifice of mathematical knowledge. At the same time he drew a parallel with the “ancient” foundations crisis commonly believed to have been occasioned by the discovery of incommensurable magnitudes, a finding that overturned the Pythagorean worldview that was based on the doctrine “all is Number.” In the wake of the Great War that changed European life forever, the Zeitgeist appeared ripe for something similar, but even deeper and more pervasive.

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 named Einstein (Fig. 19.1).

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 recalling some episodes that tell us a good deal about the competing forces that animated these two centers. Most of the information I will draw on concerns events from the last three decades of the nineteenth century. But to understand these it will be helpful to begin with a few remarks about the overall development of mathematics in Germany, so I will proceed from the general to the specific. In fact, we can gain an overview of several of the more famous names in German mathematics simply by listing some of the better-known figures who held academic positions in Gottingen or Berlin. As an added bonus, this leads to a very useful tripartite periodization:

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

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.

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 gravitation with electromagnetism in a global field theory gave research in differential geometry a tremendous new impetus. Geometry became entwined with physics as never before, and higher-dimensional geometric spaces soon abounded as mathematicians grew accustomed not just to four-dimensional space-times but to the mysteries of Hilbert space and its infinite-dimensional progeny.

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

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 as lecturers. Indeed, as teachers they exerted a strong influence on the younger generation of mathematicians in their respective countries. Klein’s German students included such prominent figures as Ferdinand Lindemann, Walther von Dyck, Adolf Hurwitz, Robert Fricke, Philipp Furtwangler, and Arnold Sommerfeld. His influence on North American mathematicians was, if anything, even stronger, as will be briefly described in this essay. Since Stockholm’s Hogskola, founded in 1878, could hardly compete with the much older universities where Klein taught, most notably Leipzig and Gottingen, Mittag-Leffler was clearly not in a position to draw large numbers of doctoral students. Nevertheless, he attracted several, four of whom left their mark on modern mathematics: Edvard Phragmen, Ivar Bendixson, Helge von Koch, and Ivar Fredholm. (Stubhaug 2010)