David E. Rowe
Mathematical knowledge has long been regarded as essentially stable and, hence, rooted in a world of ideas only superficially affected by historical forces. This general viewpoint has profoundly influenced the historiography of mathematics, which until recently has focused primarily on internal de- velopments and related epistemological issues. Standard historical accounts have concentrated heavily on the end products of mathematical research:
theorems, solutions to problems, and the technical difficulties that had to be mastered before a well-posed question could be answered. This kind of ap- proach inevitably suggests a cumulative picture of mathematical knowledge that tells us little about how such knowledge was gained, refined, codified, or transmitted. Moreover, the purported permanence and stability of mathe- matical knowledge begs some obvious questions with regard to accessibility – known to whom and by what means? Issues of this kind have seldom been addressed in historical studies of mathematics, which often treat priority dis- putes among mathematicians as merely a matter of “who got there first.”
By implication, such studies suggest that mathematical truths reside in a Platonic realm independent of human activity, and that mathematical find- ings, once discovered and set down in print, can later be retrieved at will.
If this fairly pervasive view of the epistemological status of mathematical as- sertions were substantially correct, then presumably mathematical knowledge and the activities that lead to its acquisition ought to be sharply distinguished from their counterparts in the natural sciences. Recent research, however, has begun to undercut this once-unquestioned canon of scholarship in the history of mathematics. At the same time, mathematicians and philosophers alike have come increasingly to appreciate that, far from being immune to the vicissitudes of historical change, mathematical knowledge depends on nu- merous contextual factors that have dramatically affected the meanings and significance attached to it. Reaching such a contextualized understanding of mathematical knowledge, however, implies taking into account the variety of activitiesthat produce it, an approach that necessarily deflects attention from
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the finished products as such – including the “great works of the masters” – in order to make sense of the broader realms of “mathematical experience.”
As Joan Richards has observed, historians of mathematics have resisted many of the trends and ignored most of the issues that have preoccupied historians of science in recent decades.1A wide gulf continues to separate traditional
“internalist” historians of mathematics from those who, like Richards, favor studies directed at how and why mathematicians in a particular culture at- tach meaning to their work. On the other hand, an actor-oriented, realistic approach that takes mathematical ideas and their concrete contexts seriously offers a way to bridge the gulf that divides these two camps. Such an approach can take many forms and guises, but all share the premise that the type of knowledge mathematicians have produced has depended heavily on cultural, political, and institutional factors that shaped the various environments in which they have worked.
Texts and Contexts
In his influentialProofs and Refutations,the philosopher Imre Lakatos offered an alternative to the standard notion that the inventory of mathematical knowledge merely accumulates through a collective process of discovery.2 While historians have not been tempted to adopt this Lakatosian model whole cloth, its dialectical flavor has proven attractive even if its program of rational reconstruction has not. For similar reasons, the possibility of adapting T. S. Kuhn’s ideas to account for significant shifts in research trends has been debated among historians and philosophers of mathematics, although no clear consensus has emerged from these discussions. Advocates of such an approach have tried to argue that contrary to the standard cumulative picture, revolutionary changes and major paradigm shifts dotake place in the history of mathematics. Thus, Joseph Dauben pointed to the case of Cantorian set theory – which overturned the foundations of real analysis and laid the groundwork for modern algebra, topology, and stochastics – as the most recent major mathematical revolution. Judith Grabiner made a similar argument by drawing on the case of Cauchy’s reformulation of the conceptual foundations of the calculus, whereas Ivor Grattan-Guinness has described the tumultuous mathematical activity in postrevolutionary France in terms of convolutions rather than revolutions, arguing that his term
1 Joan L. Richards, “The History of Mathematics and ‘L’esprit humain’: A Critical Reappraisal,” in Constructing Knowledge in the History of Science,ed. Arnold Thackray,Osiris,10(1995),122–35.
An important exception is Herbert Mehrtens,Moderne-Sprache-Mathematik(Frankfurt: Suhrkamp, 1990), a provocative global study of mathematical modernity that focuses especially on fundamental tensions within the German mathematical community.
2 Imre Lakatos,Proofs and Refutations,ed. John Worrall and Elie Zahar (Cambridge: Cambridge University Press,1976).
Mathematical Schools, Communities, and Networks 115 better captures the complex interplay of social, intellectual, and institutional forces.3
Most of the discussions pertaining to revolutions in mathematics have ap- proached the topic from the rather narrow standpoint of intellectual history.
Advocates of this approach might well argue that if scientific ideas can be viewed `a la Kuhn in the context of competing paradigms, then why not treat volatile situations like the advent of non-Euclidean geometry in the nine- teenth century similarly? Nevertheless, it cannot be overlooked that compar- atively little attention has been paid to other components of a Kuhnian-style analysis. Research trends, in particular, need to be carefully scrutinized be- fore the roles of the historical actors – the mathematicians, their allies and critics – can be clearly understood. Contextualizing their work and ideas means, among other things, identifying those mainstream areas of research that captivated contemporary interests: the types of problems they hoped to solve, the techniques available to tackle those problems, the prestige that mathematicians attached to various fields of research, and the status of math- ematical research in the local environments and larger scientific communi- ties in which higher mathematics was pursued. In short, a host of issues pertaining to “normal mathematics” as seen in the actual research practices typical of a given period need to be thoroughly investigated.4Perhaps then the time will be ripe to look more carefully at the issue of “revolutions” in mathematics.
This is not meant to imply that the conditions that shape mathematical ac- tivity deserve higher priority than the knowledge that ensued from it. On the contrary, the concrete forms in which mathematical work has been conveyed pose an ongoing challenge to historians. Enduring intellectual traditions have centered traditionally on paradigmatic texts, such as Euclid’s Elements and Newton’sPrincipia. After Newton, works of a comprehensive character con- tinued to be produced, but with the exception of P. S. Laplace’s (1749–1827) M´ecanique c´eleste,such synthetic treatments were necessarily more limited in their scope. Thus, C. F. Gauss’s (1777–1855)Disquisitiones arithmeticae(1801) gave the first broad presentation of number theory, whereas Camille Jordan’s (1838–1922)Trait´e des substitutions et les ´equations alg´ebriques(1870) did the same for group theory.
No field of mathematical research is likely to endure for long without the presence of a recognized paradigmatic text that distills the fundamental results
3 Joseph W. Dauben, “Conceptual Revolutions and the History of Mathematics: Two Studies in the Growth of Knowledge,” inRevolutions in Mathematics,ed. Donald Gillies (Oxford: Clarendon Press,1992), pp.49–71; Judith V. Grabiner, “Is Mathematical Truth Time-Dependent?” inNew Directions in the Philosophy of Mathematics,ed. Thomas Tymoczko (Boston: Birkh¨auser,1985), pp.201–14; Ivor Grattan-Guinness,Convolutions in French Mathematics,1800–1840(Science Net- works, vols.2–4) (Basel: Birkh¨auser,1990).
4 For a model study exemplifying how this can be done for the case of topology, see Moritz Epple,Die Entstehung der Knotentheorie: Kontexte und Konstruktionen einer modernen mathematischen Theorie (Braunschweig: Vieweg,1999).
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and techniques vital to the subject. Euler’sIntroductio in analysin infinitorum (1748) fulfilled this function for those who wished to learn what became the standard version of the calculus in the eighteenth century. Throughout the nineteenth century, new literary genres emerged in conjunction with the vastly expanded educational aims of the period. French textbooks set the stan- dard throughout the century, most of them developed from lecture courses of- fered at the Ecole Polytechnique and other institutions that cultivated higher mathematics. A. L. Cauchy’s (1789–1859)Cours d’analyse(1821), the first in a long series of French textbooks on the calculus that bore the same title, gave the first modern presentation based on the limit concept. In the United States, S. F. Lacroix’s (1765–1843)Trait´e du calcul diff´erentiel et du calcul int´egralwas introduced at West Point, displacing more elementary British texts. When E. E. Kummer (1810–1893) and K. Weierstrass (1815–1897) founded the Berlin Seminar in1860, the first books they acquired for its library were, with the exception of Euler’s Latin calculus texts, all written in French.5
Most students in Germany, however, spent relatively little time studying published texts since the lecture courses they attended reflected the whims of the professors who taught them. By the nineteenth century, the old- fashionedVor-Lesungen,where a gray-bearded scholar stood at his lectern and read from a text while his auditors struggled to stay awake, had largely disappeared. The Vorlesungen of the new era, though generally based on a written text,were delivered in a style that granted considerable latitude to spontaneous thought and verbal expression. Some Dozenten commit- ted the contents of their written texts to memory, while others improvised their presentations along the way. But however varied their individual ap- proaches may have been, the modern Vorlesung represented a new didac- tic form that strongly underscored the importance of oral communication in mathematics. It also led to a new genre of written text in mathemat- ics: the (usually) authorized lecture notes based on the courses offered by (often) distinguished university mathematicians, a tradition that began with C. G. J. Jacobi (1804–1851). Thus, to learn Weierstrassian analysis, one could either go to Berlin and take notes in a crowded lecture hall or else try to get one’s hands on someone else’sAusarbeitungof the master’s presentations.
Printed versions, like the textbook of Adolf Hurwitz and Richard Courant, only appeared much later. Monographic studies of a more systematic nature continued to play an important role, but the growing importance of special- ized research journals, coupled with institutional innovations that fostered close ties between teaching and scholarship, served to undermine the once- dominant position of standard monographs. This trend reached its apex in G¨ottingen, where from1895to1914the lecture courses of Felix Klein (1849–
1925) and David Hilbert (1862–1943) attracted talented students from around the world.
5 Kurt-R. Biermann,Die Mathematik und ihre Dozenten an der Berliner Universit¨at,1810–1933(Berlin:
Akademie Verlag,1988), p.106.
Mathematical Schools, Communities, and Networks 117 Hilbert’s intense personality left a deep imprint on the atmosphere in G¨ottingen, where mathematicians mingled with astronomers and physicists in an era when all three disciplines interacted as never before. Yet Hilbert exerted a similarly strong influence through his literary production. As the author of two landmark texts, he helped inaugurate a modern style of math- ematics that eventually came to dominate many aspects of twentieth-century research and education. Hilbert’sZahlbericht,which appeared in1897, assim- ilated and extended many of the principal results from the German tradition in number theory that had begun with Gauss. Just two years later, he pub- lishedGrundlagen der Geometrie,a booklet that eventually passed through twelve editions. By refashioning the axiomatic basis of Euclidean geometry, Hilbert established a new paradigm not only for geometrical research but also for foundations of mathematics in general. Three decades later, inspired by the work of Emmy Noether (1882–1935) and Emil Artin (1898–1962), B. L. van der Waerden’s (1903–1993) Moderne Algebra(1931) gave the first holistic presentation of algebra based on the notion of algebraic structures.
As Leo Corry has shown, van der Waerden’s text served as a model for one of the century’s most ambitious enterprises: the attempt by Nicolas Bourbaki, the pseudonym of a (primarily French) mathematical collective, to develop a theory of mathematical structures rich enough to provide a synthetic frame- work for the main body of modern mathematical knowledge.6
Yet even this monumental effort, which left a deep mark on mathematics in Europe and the United States in the period from roughly1950to1980, eventually lost much of its former allure. Since then, mathematicians have made an unprecedented effort to communicate the gist of their work to larger audiences. Relying increasingly on expository articles and informal oral presentations to present their findings, many have shown no reluctance to convey new theorems and results accompanied by only the vaguest of hints formally justifying their claims. This quite recent trend reflects a growing desire among mathematicians for new venues and styles of discourse that make it easier for them to spread their ideas without having to suffer from the strictures imposed by traditional print culture as defined by the style of Bourbaki. Since the1980s, some have even begun to question openly whether the ethos of rigor and formalized presentation so characteristic of the modern style makes any sense in the era of computer graphics. The historical roots of this dilemma, however, lie far deeper.
Shifting Modes of Production and Communication
Looking from the outside in, no careful observer could fail to notice the strik- ing changes that have affected the ways mathematicians have practiced their
6 Leo Corry,Modern Algebra and the Rise of Mathematical Structures(Science Networks,16) (Boston:
Birkh¨auser,1996).
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craft over the course of the last two centuries. Long before the advent of the electronic age and the information superhighway, a profound transformation took place in the dominant modes of communication used by mathemati- cians, and this shift, in turn, has had strong repercussions not only for the conduct of mathematical research but also for the character of the enterprise as a whole. Stated in a nutshell, this change has meant the loss of hegemony of the written word and the emergence of a new style of research in which mathematical ideas and norms are primarily conveyed orally. As a key con- comitant to this process, mathematical practices have increasingly come to be understood as group endeavors rather than activities pursued by a handful of geniuses working in splendid isolation. When seen against the backdrop of the early modern period, this striking shift – preceding the more recent and familiar electronic revolution – from written to oral modes of commu- nication in mathematics appears, at least in part, as a natural outgrowth of broader transformations that affected scientific institutions, networks, and discourse.
Working in the earlier era of scientific academies dominated by royal pa- trons, the leading practitioners of the age – from Newton, Leibniz, and Euler, to Lagrange, and even afterward Gauss – understood the activity of doing and communicating mathematics almost exclusively in terms of the symbols they put on paper. Epistolary exchanges – often mediated by correspondents such as H. Oldenburg and M. Mersenne – served as the main vehicle for conveying unpublished results. To the extent that the leading figures did any teaching at all, their courses revealed little about recent research-level mathe- matics, nor were they expected to do so. The savant mathematician wrote for his peers, a tiny elite. As fellow members of academies and scientific societies during the seventeenth and eighteenth centuries, European mathematicians and natural philosophers interacted closely. By the end of this period, how- ever, polymaths like J. H. Lambert had become rare birds, as the technical demands required to master the works of Euler and Lagrange were imposing.
Still, higher mathematics had never been accessible to more than a handful of experts, and to learn more than the basics one generally had to seek out a master: Just as Leibniz sought out Huygens in Paris, so Euler turned to Johann Bernoulli in Basel. Mathematical tutors remained the heart and soul of Cambridge mathematical education throughout the nineteenth century, a throwback to an earlier, more personalized approach.
After1800, mathematical affairs on the Continent underwent rapid trans- formation in the wake of the French Revolution, which sparked a series of profound political and social changes that reconfigured European science as well as its institutions of higher education. Enlightenment ideals of social progress based on the harnessing of scientific and technological knowledge animated the educational reforms of the period, which unleashed an unprece- dented explosion of scientific activity in Paris during the Napoleonic period.
Such mathematicians as Lazare Carnot, M. J. A. Condorcet, Gaspard Monge
Mathematical Schools, Communities, and Networks 119 (1746–1818), Joseph Fourier (1768–1830), and even the aged J. L. Lagrange (1736–1813) played a prominent part throughout. If leading figures like Carnot and Monge rallied to the Revolution’s cause during the years of peril, most later directed their energies to scientific rather than political causes. Monge fell from power along with his beloved emperor, but Laplace managed to find favor with each passing regime. Only the staunch Bourbon sympathizer Cauchy, the most prolific writer of the century, found the new regime of Louis Philippe so distasteful that he felt compelled to leave France. Teach- ing and research remained largely distinct activities, but the once-isolated academicians were thrust into a new role: to train the nation’s technocratic elite, a task that set a premium on their ability to convey mathematical ideas clearly.
In the German states, particularly in Prussia, a strong impulse arose to counter the rationalism and utilitarianism associated with the French Enlight- enment tradition. To a large extent, modern research institutions emerged as an unintended by-product of this Prussian attempt to meet the challenge posed by France. Drawing on a Protestant work ethic and sense of duty so central to Prussian military and civilian life, scholarship(Wissenschaft)gained a deeper, quasi-religious meaning as a calling. Somewhat ironically, this re- action was coupled with a neohumanist approach to scholarship that proved highly conducive to the formation of modern research schools – in con- trast to the “school learning” that continued to dominate at many European universities throughout the eighteenth century. Against the background of Romanticism, neohumanist values based on a revival of classical Greek and Latin authors permeated German learning from the founding of Berlin Uni- versity in1810up until the emergence of the Second Empire in1871.7
German scientists seldom faced the problem of having to justify their work to theological and political authorities. In their own tiny spheres of activity, scholars reigned supreme, while in exchange for this token status of freedom, they were expected to offer unconditional and enthusiastic allegiance to the state. Such were the terms of the implicit contract that bound the German professoriate to honor king andKaiser. In return, they enjoyed the privileges of limited academic freedom and disciplinary autonomy, along with a social status that enabled them to hobnob with military officers and aristocrats.
If French savant mathematicians bore responsibility for training a new pro- fession of technocrats, German professors mainly taught future Gymnasien teachers, a position that carried considerable social prestige itself. Indeed, a number of Germany’s leading mathematicians, including Kummer and Weierstrass, began their careers as Gymnasien teachers, but scores of others who never dreamt of university careers published respectable work in leading academic journals.
7 See Lewis Pyenson,Neohumanism and the Persistence of Pure Mathematics in Wilhelmian Germany (Philadelphia: American Philosophical Society,1983).