Chapter 11
AN IMAGE CONFLICT IN MATHEMATICS AFTER 1945^*

Amy Dahan Dalmedico

 

 

INTRODUCTION

Was World War II an important rupture for twentieth century mathematics? According to the traditional historiography of mathematics, it may seem surprising to assign a status that is a priori epistemological to such a political event, however major it may have been. No historical event has hitherto been submitted to such an appraisal. What does it mean to confront mathematics and its development with a precise historical moment that is above all characterized by its political dimensions? A special topic in the work of several historians of science, the period of the French Revolution has similarly been studied, mostly in regard to the transformation of the social and institutional conditions for scientific life. These historians have emphasized the importance of the constitution of a true scientific community and of related institutions set up by the Revolution. As for mathematical disciplines, however, the history of analysis from Lagrange to Cauchy is still often written without the Revolution being explicitly mentioned; in the case of geometry, given Monge's political role in the foundation of the École polytechnique where descriptive geometry was taught for the first time, the intimate link of its destiny with the revolutionary event has, on the contrary, been widely recognized.

Concerning World War II, historians of physics were the first to reorient their ways of writing the history of their field and to stop treating the war period as a parenthesis in the history of the development of various physical theories (particle physics, field theory, thermodynamics, statistical physics, etc.). On the contrary, they started to study mutations in the physicists’ practices and their new interactions with numerous partners (the military, engineers, scientists from other domains, politicians, etc.). In the introduction to the book Science in the Twentieth Century, J. Krige and D. Pestre have written: ‘whereas in the former case [WWI], most scientists returned to their laboratories once the conflict was over, now the intimate link between science, the state, and the civil management of civil society remained in place … Seen in terms of the relationship between science and the state, and the military in particular, World War II did not punctuate two periods of (relative) peace.’ In particular, these historians have stressed the crucial shift in the physicist's cultural role in American society. Other historians have moreover emphasized the pragmatic turn taken by scientific activity, henceforth seeking mainly operative, predictive results able to be applied, rather than the ideal of fundamental, coherent, knowledge of phenomena and laws of nature.

In a way, mathematicians did not escape this major historical movement: new branches arose, original problems were raised, new practices emerged, and the mathematician's figure as viewed by his peers was partly modified. In this chapter, I wish to show that an additional element specifically characterizes the field of mathematics and its community. Indeed, the Second World War initiated what I shall call an ‘image war’ or a ‘representation war’ concerning what mathematics was about, what it dealt with, and how. Over the course of the 1950s and 1960s, this ‘war’ was progressively developed until the balance of power began to shift perceptibly at the end of the 1970s and during the 1980s. This ‘war’ was focused mainly on the cleavage between pure and applied mathematics, and on the tacit hierarchy—of concepts as much as of values—informing these categories of ‘pure’ and ‘applied.’

At the turn of the twentieth century, all mathematicians obviously did not share the same practice nor the same conception of mathematics. As figureheads of the two most powerful national communities, Poincaré and Hilbert for example symbolized very distinct viewpoints. In his own words, Poincaré was above all concerned by ‘problems that arise by themselves, and not problems that we raise ourselves [des problèmes qui se posent, et non des problèmes qu'on se pose].’ By this, he sought to privilege important questions stemming from the natural sciences (dynamical systems, celestial mechanics), and to downplay artificial problems constructed by mathematicians. In order to solve these ‘objective’ problems, however, Poincaré forged indispensable abstract tools and founded entirely new disciplines such as topology and algebraic topology. Concerning Hilbert, his eminence derives from several achievements (invariant theory, number theory, geometry, Hilbert spaces, etc.); but, above all, he taught mathematicians how to think axiomatically. His enterprise concerning the axiomatic foundations of geometry soon started to represent an ideal for all mathematics. With the 23 problems of his celebrated talk in 1900, he suggested a vision and a choice about what was important and ‘deep’ in mathematics. Progressively, this vision and this choice would be transformed into a true ‘hierarchy’ of disciplines, and would for several decades position the trilogy—algebra, algebraic geometry, and number theory—at the pinnacle of mathematics. First constructed solely for algebra by Hilbert's students, and especially by E. Noether and B. van der Waerden, the axiomatic, structural image became the only prestigious one for the whole of modern mathematics. Nevertheless, Hilbert knew how to strike a deal with mathematicians opposed to him by taste and inclination, in particular Felix Klein.¹ The world's mathematical center until 1933, Göttingen united around its Faculty of Science several research institutes in physics, applied mathematics and mechanics, electrical technology, geophysics, etc. While in the first decades of the century the vogue for abstraction and the hegemony of Hilbert's school were on the rise in the European world of mathematics and in the United States, they were far from being dominant.

Until the 1930s, the American academic mathematicians’ milieu was strongly marked by a tendency towards abstraction and very few of its members interacted with the industrial world or showed an interest in technical questions. True, connections between mathematics and physics (and notably relativity theory) provided the restricted elite of the American Mathematical Society (Oswald Veblen, George G. Birkhoff, Marston Morse) with possible strategies and discourses for increasing its prestige and funding, of which the foundation of the Institute for Advanced Study at Princeton was the most outstanding example. But this ‘ideology'² was rejected by most of the professional mathematical community which remained inwardly focused, and uninterested in physics or technology. A few notable exceptions, however, deserve a mention: University of Wisconsin applied mathematician Warren Weaver who directed the Natural Science Division of the Rockefeller Foundation after 1934 and co-authored with Max Mason a book on the electromagnetic field; scientific director R.H. Kent at the Ballistic Research Laboratory at the Aberdeen Proving Ground after 1920; MIT professor Vannevar Bush, a reputed specialist in electric circuit theory and a pioneer in the application of advanced mathematical techniques to the transmission of energy, who collaborated with Norbert Wiener, and a few others. In a 1941 report, Thornton Fry, Mathematical Research Director at the Bell Telephone Laboratories, noted that the mathematical needs of industrial enterprises were essentially being fulfilled by physicists and engineers and that it was about time that applied mathematicians be trained directly. It will not come as a surprise that precisely these personalities would be prominent when science was mobilized for war.

THE WORLD WAR II RUPTURE

The very important immigration of German and Eastern European scientists into the United States, the rise of Nazism and the imminence of world conflict triggered a spectacular change of mind. The urgent need to train applied mathematicians became evident, and various institutional initiatives were taken: appointment of a ‘War Preparedness Committee’ by the AMS, organization of a summer school at Brown University addressing questions concerning partial differential equations and continuous medium mechanics, constitution of networks and small groups of applied mathematicians, such as the one around Richard Courant in New York City, etc. With the United States’ entry into the war, Vannevar Bush established an agency, the Applied Mathematics Panel (AMP) headed by Warren Weaver, whose role was intended to be that of a mathematicians’ organization providing other scientists engaged in collaboration with the military—or even the military itself—with mathematical help. Since a significant number of mathematicians were Jewish emigres obliged to flee Nazism, they immediately accepted to work in project-oriented research contributing to the war effort and to collaborate with the military. In addition to several government laboratories of the civil administration and the armed forces (National Bureau of Standards, Ballistic Laboratory at Aberdeen, the various laboratories of the Navy, etc.), important laboratories set up in university centers (including notably the Radiation Laboratory at MIT) became places of active cooperation. For two and a half years, Weaver's agency supervised an effort to mobilize nearly 300 people including very prestigious mathematicians (John von Neumann, Richard Courant, Jerzy Neyman, Garrett Birkhoff, Oswald Veblen, etc.), producing hundreds of technical reports, and spending over three million dollars. With a program of applied mechanics at Brown University (Richardson and Prager), with J. Neyman's statistical laboratory at Berkeley, with Richard Courant's New York University group tackling mainly submarine sound transmission and shock wave behavior, the AMP promoted an institutionalization of applied mathematics. Working by contracts, the panel established a practice which, persisting after the war, modified several mathematicians’ habits and states of mind by getting them much closer to other scientists and offering them a very wide spectrum of research topics. Joining talents and competences running from the pure to the applied, from fundamental research to operations and actions, teams on several occasions had the opportunity of collaborating. In the state of emergency, with the need for immediate results, the rigor and proof demanded by fundamental mathematics were often sacrificed to the benefit of numerical explorations, forms of experimentation or rough approximation hitherto considered to be the exclusive territory of physicists and engineers.

The Rise of New Fields: Partial Differential Equations, Probability and
Statistics, ‘Cyborg Sciences'

Three domains in particular would benefit from this powerful impulse. The first of them was the field of partial differential equations (PDE), numerical analysis, and algorithm theory. Questions of wave propagation and shock wave behavior in air and water were indeed central to the understanding of charges, rockets, and firing. From the Aeronautical Bureau, a demand for the conception of jet-propelled airplanes spurred numerous researches on gas dynamics, supersonic flow, explosion theory, etc. From Hermann Weyl to John von Neumann, via Courant, Karl Friedrichs, or van Kármán, a range of mathematicians from the purest to the most applied partook in the effort. Let us also mention Garrett Birkhoff's work at Harvard on submarine ballistics, as well as Saunders MacLane's, Hassler Whitney's and the Columbia group's work on air ballistics. Around Prager and Tamarkin at Brown University problems in continuous medium mechanics mobilized people like Feller, Bergman, Lipman Bers, etc. In 1944, the New York group prepared a classified textbook which prefigured Courant and Friedrichs's Supersonic Flow and Shock Waves published in 1948. Not allowed to remain theoretical and concerned only with solubility conditions, all of these studies were required to achieve numerical solutions because of the military and political context.

Secondly, probability theory and statistics witnessed a spectacular rise in the global architecture of mathematics. An important part of the research triggered by the aerial war dealt with the study of fragmentation: it relied on the probabilistic study of the damage caused by anti-aircraft fire on one airplane or a group of them. Combining theoretical analyses with statistical models, and problems of bombardment with the control of quality, Columbia statisticians in particular studied various firing configurations in air battles. These theoretical developments were picked up in Abraham Wald's postwar treatises on sequential analysis and statistics. In addition, understanding the statistical properties of noise and detection of a noisy signal were central to several of the undertakings of the Radiation Laboratory at MIT, where mathematicians— Nobert Wiener in the first place, but also Marc Kac and others— actively collaborated with physicists, engineers, and biologists. In the study of differential and integro-differential equations, the statistical approach was also developed within the framework of research on nuclear reactions undertaken for the Manhattan Project; S. Ulam and J. von Neumann studied Monte-Carlo simulations, as well as possible applications of probability theory to hydrodynamic computations.

Another field of research located at the intersection of operations research, game theory, and decision-making mathematics must be mentioned. During the conflict, operations research dealt with technological research, optimal strategies for convoys, optimal disposition of radar stations, etc. In the United States, Philip Morse's group at the Radiation Laboratory was the most important. He considered himself more as a ‘consumer’ of mathematics—and at least at the beginning a rather rudimentary kind of mathematics—than as a producer of new knowledge. Significantly, however, Morse was in 1947 invited to give the highly prestigious 21st Gibbs Lecture of the AMS, on which occasion he stressed the potentialities of operations research for the economy and industry. Together with the 1948 new edition of von Neumann and Morgenstern's famous Theory of Games and Economic Behavior, this conference launched the institutional development of decision-making mathematics.

Cooperation between mathematicians and the military or civil powers certainly did not evolve without tensions, rivalries, and conflicts. Ultimately some very talented mathematical minds, such as Nobert Wiener, whose expertise and suggestions were in high demand, were judged too eccentric by politicians or found it impossible to adapt to the constraints of guided research. Relations soured between military officers and a few mathematicians, even if deeply engaged in the effort: J. Neyman was loath to ‘sell’ them his most original theoretical statistical results; even J. von Neumann who was solicited on all sides tended to establish his own hierarchy of priorities. Examples of administrative inflexibility were moreover manifest: irritated by Byzantine squabbles within the Air Force, Weaver disappointedly renounced his ambition to coordinate the major tactical study of B-29s. Marston Morse, President of the AMS, and Marshall Stone, blamed Weaver and Bush for underestimating the potential role of mathematicians and for concentrating on physicists, chemists, and engineers. The ensouing resentment produced fractures in the mathematical community that would not be mended until long after the war. [Owens, 1989].

A New Paradigmatic Figure for Mathematicians: John von Neumann

In the course of the war, however, a new role model for mathematicians emerged—the figure of a mathematician acting outside academia, collaborating with other scientists (physicists, engineers, or economists), and integrated in complex social networks to which belonged also political and administrative decision-makers, businessmen, and military officers. Comparable to some physicists, a few mathematicians can be mentioned, who among others embodied this new figure: Wiener, Ulam, Goldstine, Garrett Birkhoff, etc. An extreme and paradigmatic symbol, however, was J. von Neumann. Socially engaged, he contributed to some strategic and technological decisions made by the United States, belonged to circles close to political power, and promoted the idea that the entire world was within the mathematician's domain of intervention. In particular, von Neumann's conceptions and practices led him to a new articulation of scientific disciplines, which shattered firmly established cleavages and hierarchies between the ‘pure’ and ‘applied.’ He blurred the accepted boundary between what belonged to mathematics and what did not, and, what would previously have been counted amongst topics in disciplines like as engineering science, the physical sciences, or economics.

Begun in the 1940s, von Neumann's most significant recomposition of interests concerned hydrodynamics, computer science, and numerical analysis. He judged hydrodynamics crucial to physics and mathematics but in need of new computational methods and means, whose development he then took on: under his impulse, the Electrical Computer Project (built at the Institute for Advanced Study) and the Numerical Meteorology Project were launched at Princeton. Von Neumann's field of intervention ranged from the theoretical level to practical implementation; his energy was invested as much in abstract analysis and the modeling of complex problems as it was in their technical implementation and the direction of the collective project necessary to their resolution. Von Neumann's scientific stature was so great, and his mathematical insight so unanimously recognized, that his engagement on the side of applied mathematics conferred on it a new dignity and prestige, which it had never enjoyed before.

THE NINETEEN-FIFTIES AND SIXTIES

In the aftermath of World War II, a redistribution of scientific forces occurred internationally. By the sheer size of its mathematical community, the breadth of its scientific coverage, and its dynamic university and research system, the United States became the foremost mathematical power in the world. Entirely new domains of intervention and enormous opportunities for interacting with more and more scientific and technological sectors were suddenly open to mathematicians. Testifying to increasing interest in applied mathematics, several organizations were established in the United States: The Association for Computing Machineries (1947), The Industrial Mathematical Society (1949), The Operations Research Society of America (1952), The Society for Industrial and Applied Mathematics (1952), The Institute for the Management Sciences (1953), and so on. In addition to university professors, they brought together people working in industry, government agencies, and the military. One should note, however, that the professionalization of applied mathematics would take place outside the American Mathematical Society.

Indeed on the academic level the mathematical community was reluctant to move into applied mathematical domains, notwithstanding the important stimulus they had received from the war. True, building on experience gained in the war, a few centers were developed in the United States and acquired excellent reputations in various applied domains: New York University's Courant Institute in the study of partial differential equations and numerical analysis; Brown University in the study of differential equations, dynamical systems, and control theory; Berkeley and Stanford in statistics, etc. At the research as well as educational levels, institutionalizing efforts in favor of applied mathematics were clearly successful in a few, well-circumscribed places. But, mere islands of resistance, they hardly affected the major tendencies in the country.

The case of Princeton University and the Institute for Advance Study, for example, would deserve a major study which is unfortunately still lacking.³ Indeed, enjoying a tradition of excellence in pure mathematics, the Institute under John von Neumann's lead was the first to be endowed with a computer. Distinguished in the 1950s by its open spirit, Princeton University's mathematics department welcomed young men wishing to explore new areas of mathematics: game theory with John Nash, experimental computer economics with Martin Shubik, artificial intelligence with Marvin Minsky, etc. As a symbol of this openness, two of the department chairs, Solomon Lefschetz and his student and successor Albert Tucker, who were both initially pure mathematicians specializing in algebraic topology, moved in original directions (Dahan [1994]). Dynamical systems with Lefschetz, dynamic programming with Tucker, mathematical logic and combinatorics as a way to solve computer science problems with Alonzo Church—there is a long list of new mathematical sectors in which Princeton mathematicians would play a crucial role.

Nevertheless, on the whole, the American, as well as international, mathematical communities witnessed a forceful return of, let us say, structural pure mathematics along with the correlative sidestepping of applied mathematics. Several mathematicians have explained this evolution as due solely to the internal dynamical evolution of mathematics. For them, abstraction and formalization constituted an obligatory passage point in the progress of most domains. And it hardly bothered them whether both of these processes went along with an outgrowth of the formal, algebraic side of theories and the relative self-closure of the professional mathematical community.

In my opinion, the internal development of mathematics alone cannot account for what was the outcome of situations arising, and choices made, on political, institutional, and intellectual levels. It cannot explain what resulted from the actors’ tacit value hierarchy and disciplinary representation. In the rest of this chapter, only the diffuse question of the cultural image of mathematics will be tackled.

THE CULTURAL IMAGE OF PURE MATHEMATICS

A characterization of the cultural image of pure mathematics in the 1950s and 1960s cannot be unique because images were not uniform; it cannot even be reduced to those produced either by American mathematicians or by Bourbaki's group and the French mathematical school. We must moreover keep in mind the properly French, vs. American, features of this image. Nevertheless, we think it is relevant to elucidate some of the main features of this image and understand how these features were inscribed in the spirit of the time. Some differences and nuances do exist but even the point where these nuances lie reveals main tendencies. In the 1980s, a shift in the dominant image goes along with very critical judgments in the mathematical community with respect to the preceding period. Even if occurring post facto, free expression of opinions helps to the historian to capture the cultural image of mathematics in earlier years.

American Mathematicians’ Rebellion against Utilitarian Ideology

In 1957, the president of the American Mathematical Society Marshall Stone was surprisingly invited to give the 31st Gibbs Lecture. Entitled ‘Mathematics and the Future of Science,’ his talk is an instructive introduction to American views about what mathematics should be and how it should be considered.⁴ His mathematical interest having always been directed toward the pure branches of the discipline, Stone shared with all mathematicians of the time—applied mathematicians or, let us say, ‘cyborg mathematicians’ (John von Neumann, Norbert Wiener, A. Wald, G. Dantzig, Philip Morse)—a tremendous faith in the growing importance of mathematical thought for the future of science. But while for von Neumann, this faith rested on the idea that mathematics could be applied to the whole world (nature, economics, society, human behavior, the brain, etc.), the possibility of making both deterministic and statistical predictions also being used to distinguish scientific from nonscientifìc disciplines, Stone, by contrast, expressed his faith in mathematics in a simple syllogism: ‘science is reasoning; reasoning is mathematics; and therefore, science IS mathematics.’ Hence, science was reduced to those disciplines in which mathematical reasoning played a predominant or crucial part. Fields such as geology or meteorology in which observation played an essential role could therefore hardly deserve to qualify as science! Opposing attitudes about applications of mathematics, Stone explained, resulted from a deep philosophical cleavage in American culture concerning the place of individuals in society. Due to the conflict between liberal and utilitarian conceptions of education, this cleavage opposed two standpoints: one valuing whatever developed the individual's intellectual and spiritual capabilities and the other prizing whatever worked or led to useful results.⁵ ‘For mathematics, which is at once the pure and untrammeled creation of the mind,’ Stone emphasized, ‘the adoption of a strictly utilitarian standard could lead only to disaster.’ No doubt he shared National Science Foundation director Alain Waterman's conception of mathematics according to which this discipline, ‘in a sense, bridges the gap, real or imaginary, which exists between the sciences and the humanities.’

According to Stone, an essential prerequisite for the extraordinary flowering of pure mathematics since 1900 was that ‘mathematicians have recognized and acted upon the fact that mathematics is not closely bound to the material world or to physical reality—if, indeed, it is bound at all!’ Hence, ‘utility alone,’ he declared, ‘is not a proper measure of value’ and could be, if strictly applied, dangerous and false as a measure. Pure mathematicians could not accept ‘reference to action’ as the sole criterion by which their work was to be judged. The sole use of the axiomatic method, however powerful or characteristic, could not capture the essence of this intellectual movement. Axiomatization had to be combined with an ideology—the desire to free mathematical theory from a dependence on physical necessity. Only this combination made possible ‘the dissection of mathematical concepts into their elementary components, the recombination of these components into new constructs of intrinsic interest, the critical evaluation of alternative approaches to the important mathematical theories, the unification of hitherto unconnected branches of mathematics—that best expresses the spirit of modern pure mathematics.’

In fact, a paradoxical situation existed in the US: the utilitarian, pragmatic bent of American ideology and culture coexisted with strong tendencies towards abstraction and generality in American mathematics. How can we account for this paradox? American mathematicians’ response to broad trends in the development of their discipline, Stone wrote, had been significantly affected by specifically American circumstances. The highly pragmatic character of industrialists and businessmen—an outgrowth of the pioneers’ experience—prevented mathematics from being called on to play anything but the most modest utilitarian role during the critical period during the late nineteenth century and first part of the twentieth century. By chance, the development of great mathematical centers in the US almost exactly coincided with the flowering of pure mathematics. From this cultural lag, a complete freedom accrued to American mathematicians. Neither strong academic traditions nor previous higher mathematics prevented them from enjoying a singular independence from utilitarian demands and the freedom to direct their efforts towards central themes in modern mathematics.

Many years later, Peter Lax ventured a similar analysis of what he called ‘the American tide of purity,’ to which he personally strongly objected:

The bold proposal to cut the lifeline between mathematics and the physical world was put forth only in the 20th century, mainly by the Bourbaki group. Besides being wrong headed, this raises profound philosophical problems about value judgments in mathematics. The question ‘What is good mathematics’ becomes a matter of a priori aesthetic judgment, and mathematics becomes an art form … Next to Bourbaki, the greatest champions of abstraction in mathematics came from the American community. This predilection for the abstract might very well have been a rebellion against the great tradition in the United States for the practical and pragmatic; the postwar vogue for Abstract Expressionism was another such rebellion. [Lax, 1986].

In this era of specialization, much of the best applied mathematics work was done by men who considered themselves not mathematicians at all but rather physicists, chemists, or biologists. According to pure mathematicians, a majority of scientists in America fell too easily into a pragmatic, utilitarian trap when dealing with mathematics. Ultimately, they generally considered it as a useful tool about which one needed to know no more than immediately useful features. In this way, communication was to a serious extent broken down between pure mathematics and many branches of applied mathematics. Moreover, mathematical instruction and training at the secondary level had been only slightly affected by the modernization of mathematics. The most serious obstacle to a modernization of the mathematical curriculum, Stone concluded, was the utilitarian spirit that pervaded secondary education.

Elitist Poetization of Pure Mathematics

Bourbaki's enterprise obviously had nothing to do with rebellion against a dominant utilitarian culture. For its first generation of members, the project was to reformulate mathematics on definitive foundations. As a new Euclid, Bourbaki would for thousands of years to come establish strict disciplinary standards. He was inscribed in a certain Zeitgeist of the interwar years: to lay down definitive solutions— some would even use the fateful expression ‘a final solution'—to theoretical questions: ‘There are good reasons to hate that sentence,’ Pierre Carder acknowledged, ‘but people thought that we could reach a final solution.’ In his historical notes, Bourbaki thus always expressed the conviction that mathematical developments were a series of stages unavoidably leading to the present, and ultimate, state—the axiomatic, structural conception—which would endure forever in the future [Corry, 1997].

This eternalism, let us say, also characterized the spirit of Hilbert's metamathematical program. Defeated by Gödel's work, this program was indeed intended to provide a ‘final solution to the consistency problem’ of arithmetic axioms.⁶ Clearly, this attitude was also shared by the Vienna Circle project, which aimed at putting an end to metaphysics and to base the scientific conception of the world on the foundations of logical language analysis.

This spirit is obviously insufficient to characterize the Bourbaki group completely after World War II. Moreover, Bourbaki itself exhibited a certain degree of heterogeneity, as is shown by Carder's list of four different generations in the group:

1°) The founding-fathers: Andre Weil, Henri Cartan, Claude Chevalley, Jean Delsarte, and Jean Dieudonné, the last of whom frequently expressed the definitiveness of solutions to certain questions, rigor for example, provided by modern mathematics;

2°) Those who joined the group during the Second World War or in its immediate aftermath, such as Laurent Schwartz, Jean-Pierre Serre, Jacques Koszul, Jacques Dixmier, Roger Godement, or Samuel Eilenberg. Among them, Serre emerged as the natural leader since he was the only one with a truly universal understanding of mathematics;

3°) Armand Borel, Alexander Grothendieck, Claude Bruhat, Pierre Cartier himself, Serge Lang, and John Tate: a generation which, according to Cartier, was ‘more and more pragmatic’ (or less and less dogmatic). Trained within the new axiomatic tradition, which obviously appeared excellent to them, these mathematicians may have felt that they had nothing to prove anymore.

4°) The next generation was formed by Grothendieck's pupils at the moment when himself left the group [Cartier, 1998].

Not only were there quarrels between different generations but also within them. Moreover, if the Bourbaki group was small and well-delimited, the séminaire Bourbaki was much more open, and it is quite impossible to determine precisely the exact borders of Bourbaki's following.

The cultural image promoted by Bourbaki was fundamentally elitist. For the most part, group members were exceptionally gifted individuals who enjoyed a broad humanist culture. Their interests were many: philosophy for Chevalley, classic Greek and Indian civilizations for Weil who knew Sanskrit, and so on. Most of them played musical instruments at an honorable level. In this vein, Andre Weil [1960] has described wonderfully the pleasure he derived from mathematics:

Nothing, as every mathematician knows, is more fecund than these obscure analogies, these troubling reflections from one theory to another, these furtive caresses, these unexplainable scrambles [brouilleries inexplicables], nothing gives more pleasure to the researcher. One day, illusions are dissipated; presentiments become certainties; twin theories reveal their common source before they fade away; as is taught by the Gita, knowledge is reached at the same time as indifference. Metaphysics has become mathematics, and is ready to form the topic of a treatise whose cold beauty would be incapable of moving us.⁷

Similarly, Laurent Schwartz has expressed in his autobiography the esthetic enjoyment of mathematical discovery [Schwartz, 1997]. In many respects, these mathematicians seemed closer to poets and literary creators than practical and experimental scientists.

They moreover considered that deep, important mathematics was produced by a very small number of individuals. True, lesser mathematicians could have a role to play in clearing the ground, in acting as sounding boxes, in teaching, etc. But only the production of significant results really counted.

The Bourbaki group recruited a few talented American mathematicians (Eilenberg, Tate, Lang) and many of its influential members spent extensive periods of time in the United States. While almost all spent at least one year there, Chevalley stayed from the end of WWII to 1955, and Weil and Armand Borel (who was Swiss) settled down indefinitely in Princeton. Between France and the United States, quite distant political and cultural contexts nonetheless fostered an axis around which the cultural image of pure mathematics was forged. In these two countries as well, this image was most clearly defined. Links were especially strong between both mathematical communities.

Bias Against Physics

In the 1920s, connections between mathematics and physics provided American mathematicians with opportunities to increase both their prestige and funding. At the time, mathematical physics and in particular relativity theory offered them the possibility, at least by means of rhetorical strategies, to escape their marginality, for instance in the creation of the Princeton Institute for Advanced Study [Butler Feffer, 1997]. Later, during the war years, mathematicians and theoretical physicists had several opportunities to collaborate. But, as already mentioned, the ideology of freedom from physical reality and independence from physics or other sciences established a tight barrier around pure mathematicians in the United States.

In France also, Bourbakists shared ‘a strong bias against physics’ [Cartier, 1998]. Visiting Göttingen in 1926, at the very moment when quantum theory was experiencing its strongest developments with Heisenberg's elaboration of matrix mechanics, Andre Weil seemed to have completely ignored quantum mechanics. In the obituary he and Chevalley wrote for Hermann Weyl, both Bourbakis failed to mention two of Weyl's major books on general relativity and quantum mechanics. R. Hermann wrote:

The most curious aspect of the Bourbaki story is that they started at the time of the initial flowering of quantum mechanics, reached full speed at the time when Einstein's geometric theory of gravitation was finally being understood, when elementary particle physics started its proliferation of accelerators and Lie groups across the countryside, and when many aspects of the ‘core’ mathematics with which they were concerned were being integrated into engineering and economics via system, control, and optimization theory and yet not a trace of such developments has ever appeared in their pages [1986, p. 32].

It seems obvious that several barriers were erected between mathematics and theoretical physics. Many pure mathematicians recognized that mathematical successes in the field of physics were impressive and spectacular, but in their opinion they were mainly based on nineteenth century mathematics. Although the geometrization of physics underwent crucial steps with general relativity, although Hermann Weyl exhibited the gauge invariance principle of physical laws, and although Elie Cartan studied from a mathematical standpoint the spinor notion, geometers generally resisted topics like spinor fields or differential operators acting on them. In 1954, two physicists working in the US, C.N. Yang and R. Mills suggested a model for the study of strong interactions in terms of a gauge theory with a very simple invariant group. But it was not until the 1970s that mathematicians began to recognize in the Yang-Mills theory an exact coincidence with modern differential geometry. A striking parallel between concepts elaborated separately by theoretical physicists and mathematicians was then established by Yang and Wu.⁸

The use of mathematical statistics has also been crucial in elementary particle physics, in the mechanisms governing heredity, in theoretical biology and genetics, in the social sciences, etc. Originally investigated by mathematicians S. Ulam and N. Metropolis for computer simulations of the H-bomb, Monte-Carlo methods have progressively acquired major theoretical importance. Notwithstanding its wide success, these methods drew very little attention from mathematicians.⁹ In Marshall Stone's opinion, all sciences embodying the essence of inductive reasoning were based on a single principle: ‘A sufficiently improbable event may be ignored.’ He added: ‘why the real world should be amenable to such a rule is a philosophical question no more—and no less—mysterious than the problem of why it should be amenable to logic’ These questions were but two aspects of the ultimate problem of the 'connection between mind and matter.’ The distinction between inductive and deductive reasoning offered no help in defining what was meant by science. In any case, he noticed, through measure theory and modern mathematical statistics, ‘virtually all the detailed procedures of inductive reasoning are deductive in character'.

This last explanation should be considered along with two more statements which have flourished since they were first made. The first is Wigner's expression about ‘the unreasonable effectiveness of mathematics’ [1960] and the second Bourbaki's passage on the ‘miraculous adaptation of mathematical structures to reality’ [Bourbaki, 1948, p. 46]. Here are three expressions of the same philosophical conviction, the same ideology which could be summarized in the same terms: world and nature can, and must, be described in abstract mathematical structures, an adequacy which reflects the deep connection between mind and matter. But this absolute conviction conferred to mathematicians the legitimacy to turn their back on concrete, real-world problems and to be exclusively concerned with the internal development of mathematics.

Monistic Unity of Mathematics

In discourses and texts published in the 1950s, two features characterized modern mathematics: (1) the necessary unity of mathematics; and (2) the axiomatic method which allowed this unity to be constructed. Bourbaki strongly advocated a structural point of view for this unity, and in accordance with Hilbert's views set theory was thought to provide the needed general framework. But, as several people have since proved, category theory—not set theory—is perhaps a more flexible tool for logical foundations. Pleading for an organic, rather than structural, unity of mathematics, Cartier for instance recently wrote: ‘Categories can offer a general philosophical foundation—that is the encyclopedic, or the taxonomic part—and a quite adequate tool to be used in mathematical situations. That set theory and structures are, by contrast, more rigid, can be seen by reading the final chapter in Bourbaki's set theory, with a monstruous endeavor to formulate categories without categories’ [1998]. Still, developed by Samuel Eilenberg, who belonged to the group, and by Saunders MacLane who was pretty close to it, category theory was by and large an offspring of Bourbaki's.

As was underscored by Leo Corry, the structural conception truly refers to a particular way of doing mathematics, to a tacit knowledge of the mathematicians’ daily practice, and to a hierarchical conception of the mathematical corpus, but this character can only be defined in non-formal terms in relation to a discourse bearing on mathematics. The notion of ‘mother-structure’ and the description of mathematics as a hierarchy of structures, he has noted, do not internally result from any theory, but merely appear in non-technical popularization articles (see Corry, [1992]). In my opinion, this conception of mathematics can be appropriately grasped through an examination of the way some treatises were conceived. Concerned with mathematical domains not primitive for an architectural conception, these treatises were at the crossroads of several branches and involved method or intuition transfers from one to another.

Let us focus on the exemplary case of homological algebra. In the 1940s and 1950s in several mathematical topics, including algebraic topology and algebraic geometry, new foundations and internal results were systematically developed. Homological algebra first emerged from algebraic topology, but its domain of application was soon extended to several other fields including algebraic geometry. Homological algebra deals with both the homology of algebraic systems and the algebraic aspects of homology theory. The first topic leads to homological and cohomological theories of groups of associative algebras and of Lie algebras; the second to exact sequences and spectral sequences and the manipulation of functors of chain complexes. This algebra had been qualified, if indulgently, as 'abstract nonsense' by the topologist Norman Steenrod [Lang, 1995, p. 340].

The first treatise on the topic was Cartan and Eilenberg's Homo-logical Algebra which intensely mobilized categories. It is interesting to read MacLane's review of this book: ‘The authors’ approach can best be described in philosophical terms and as monistic: everything is unified.’ MacLane explained this sentence with an example: ‘Consider for instance the homology of groups; in view of its application to class field theory and to topology, this topic is central in homological algebra. In this book the homology of groups appears as a special case of the homology of monoids, which in turn is a special case of the homology of supplemented algebras, again a case of the homology of augmented algebras, which is an instance of a torsion product, which at your choice is an instance of a derived functor or an iterated satellite functor'. [MacLane, 1956]. He went on: ‘Historically, each monistic doctrine is resolved by a subsequent pluralism. So it was here. When the authors started to write, it was true that all known cases of homology of algebraic systems (groups, algebras, and Lie algebras) could be neatly subsumed under the resolution, Tor and Ext pattern. When they finished writing this was no longer so—and this because of the authors’ own separate efforts elsewhere! … Perhaps Mathematics now moves so fast—and in part because of the vigorous unifying contributions such as of this book—that no unification of Mathematics can be up to date.’ MacLane also criticized the accumulation of brilliant, promising ideas whose usefulness was obvious to no one at the time. As a result, he believed that uninitiated readers could hardly hope to understand the book. This was, he concluded, an unfortunate confusion between research paper and treatise.

The same impression of dogmatic, monistic tonality is present in A. Mattuk's book review of Chevalley's Fundamental Concepts of Algebra, which Mattuk considered as ‘tight, unified, direct, severe … Relentlessly and uncompromisingly, it pursues its end.’ He added: ‘The unity is monolithic. Gone is the discursive rambling of previous texts. This one marches unswerving and to its own music … [The] book [is] abominably hard for a beginner, unreasonably hard, I should say’ [Mattuk, 1957]. The author Claude Chevalley had warned: ‘This is an exercise in rectitude of thought, of which it would be futile to disguise the austerity.’ Mattuk countered: ‘The voice that we hear resounding is that of an old Testament prophet, but the mental attitude is more like a tenth grade Latin teacher's wreaking with the old theory of formal disciplines.’ A few years earlier, Weil had devoted a long technical, enthusiastic review of Chevalley's Introduction to the Theory of Algebraic Functions of One Variable. His friend however had to recognize that: ‘It appears that the author has somewhat overstated his claims and has been too partial to the method dearest to his algebraic heart. Who would throw the first stone at him? It is rather with relief that one observes such signs of human frailty in this severely dehumanized book ‘ [Weil, 1951].

The reception of a book written by an American mathematician who was very close to Bourbaki provides us with a third example and enriches our global perception of their style in, and conception of, mathematics. Even among mathematicians working in the same pure, abstract domains, polemics pitted one against the other. In 1962, Serge Lang, who belonged to Bourbaki and shared many aspects of his spirit and conception of mathematics, published Diopbantine Geometry. A historical authority on the topic, Mordell reviewed it in a piece which became famous for its very negative reaction to mathematical trends in the 1950s and 1960s. Mordell wrote: ‘A general question that immediately suggests itself to a reader is what object an author has in mind when writing a book. Some have the true teacher's spirit or even a missionary spirit, wishing to introduce their subject to a wide circle of readers in the most attractive way … Lang is not such an author'. After having quoted Lang's declaration: ‘One writes an advanced monograph for one's self because one wants to give permanent form to one's vision of some beautiful part of mathematics, not otherwise accessible,’ Mordell went on:

Much of the book is practically unreadable unless one is familiar with among other Bourbaki, the author's books on algebraic geometry and Abelian varieties, and Weil's Foundations of Algebraic Geometry and is prepared occasionally to go to the original sources for proofs of some theorems needed in the present volume … The reader will require the patience of Job, the courage of Achilles, and the strength of Hercules to understand the proofs of some of the essential theorems. He will realize that some of the proofs will be above his head [Mordell, 1964].

A German mathematician deeply involved in number theory, Siegel, sent Mordell a letter of approval that is also worth quoting:

The whole style of the author contradicts the sense of simplicity and honesty which we admire in the works of the masters in number theory—Lagrange, Gauss, or on a smaller scale, Hardy [and] Landau … Unfortunately there are many ‘fellow-travelers’ who have disgraced a large part of algebra and function theory. However until now, number theory had not been touched. These people remind me of the impudent behavior of the national socialists who sang ‘ Wir werden weiter marschieren, bis alles in Scheiben zerfällt.’ I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction—as I call it: the theory of the empty set—cannot be stopped.¹⁰

In Lang's opinion, Mordell and Siegel were great mathematicians, but failed to understand the accomplishments of this period, in particular the relations between number theory and algebraic geometry which would later come to be known as number field case and function field. More profoundly they were missing the conceptual unification of topology, complex differential geometry, and algebraic geometry which went on in the seventies and beyond with the work of Alexander Grothendieck, and Pierre Deligne. According to Lang, this was the reason why algebraic geometry was largely at a dead end in England and Germany in spite of Atiyah's and Hirzebrook's work, and had been mainly developed in France and the United States.

The structural, monistic point of view had very important consequences in the way of presenting mathematics in rhetorical style, and in the extremely elitist conception of mathematical books or treatises. Moreover the required structural unity increased the dogmatic view of mathematics and arbitrarily excluded large branches of the discipline: applied mathematics whose method was clearly not axiomatic, combinatorics, concrete geometry, several branches of analysis among which differential equations, statistics, logic, etc. This vision strengthened a hierarchical conception of the mathematical corpus that privileged algebraic branches and internal mathematical dynamics.

Conception of Analysis: A Confrontational Matter

The domain of mathematical analysis is crucial. It is one of the oldest branch of mathematics, especially linked to the study of nature, physics, and engineering science. Various conceptions of analysis and what its teaching should be strongly opposed those of pure and applied mathematicians. In the 1940s and 1950s, the emphasis put by the former on functional analysis was absolutely enormous. For Bourbaki, this was justified by the general state of confusion in mathematics at the time. In fact, except for Laurent Schwartz, none of its members was really an analyst. Bourbaki labored towards a conception in which algebra, analysis, and topology would form a single unified domain giving rise to vast syntheses at increasing levels of abstraction. Traditional branches of analysis were considered bleak and limited in their ambitions. When he tackled nonlinear oscillations, Solomon Lefschetz noticed that differential equation theory was deemed the most boring topic possible. L. Carleson has described the reigning state of mind regarding classical analysis: ‘There was a period, in the 1940s and 1950s when classical analysis was considered dead and the hope for the future of analysis was considered to be in the abstract branches, specializing in generalization.’ Writing in 1978, he went on: ‘As is now apparent, the rumor of the death of classical analysis was greatly exaggerated and during the 1960s and 1970s the field has been one of the most successful in all mathematics’ [Carleson, 1978, p. 53].

About Coddington & Levinson's Theory of Differential Equations, which he very much appreciated, Richard Bellman (who had joined the Rand Corporation in the late 1940s) wrote:

It has become fashionable of late, in various mathematical centers, to present the fundamental tools of analysis, real and complex variable theory in an increasingly abstract manner to those most defense-less, namely fledgling graduate students. In the process, motivation for the introduction of new concepts has been on the whole by-passed as an atrophied relic of those early pioneer days when mathematicians were forced to consort with astronomers and physicists, and indeed, in some cases were indistinguishable from them … It is doubtful, whether this species can perpetuate itself. Most likely, mathematical education will continue along the same simple logical principles that have guided the greatest scientists of the past, from the simple to the complex, from the concrete to the abstract [Bellman, 1956].

This book review was a testimony to the irritation provoked by the fashion of the abstract mathematics, exclusively oriented towards itself. But abstraction was offered by pure mathematicians as the indispensable price one had to pay for achieving greatest theoretical simplification and deepest unification.

As mathematicians opposed to these trends observed, both constant distancing from concrete empirical sources and an increasing esthetic conception of mathematics always went together. Among scores of examples, let us mention two positions. In 1956, von Neumann wrote: ‘Mathematical ideas originate in empirics; once they are conceived, the subject begins to live a peculiar life of its own … As a mathematical discipline travels far from its empirical source, … it becomes more and more pure aestheticizing; … at a great distance from its empirical source, or after much abstract inbreeding, a mathematical subject is in danger of degeneration’ [1956]. Criticizing the representation of mathematics as artwork, Paul Halmos violently denounced what he called the substituting of ‘mathology to matophysics,’ i.e. the tendency to generate problems and research areas by perfecting the elegance and symmetry of axiomatic formulations at the expense of increasing power of action on the world and the analysis of broad fields of classical problems and mathematical physics [Halmos, 1968].

At a conference on the evolution of modern mathematics in 1970, lively polemics on abstraction and axiomatization focused on analysis. While for some mathematicians like Browder and Dieudonné, this attitude remained fully justified, others like Garrett Birkhoff held that it exhibited a tendency to function solely for itself and had become excessive. In the long run, the latter asked, was not the computer more important than functional analysis? Was not the Lax-Richtmyer theorem concerning linear differential equations one of the major theorems of numerical analysis? Was the most important thing in analysis, not so much Hilbert or Banach spaces, but the notion of norm leading to error measures? etc. At the time, such iconoclastic claims shocked pure mathematicians. [Birkhoff, 1975].

In summary, it can be said that two concurrent images of mathematics collided:

— on the one hand, the image of pure mathematics developed above all ‘in the honor of the human spirit', whose methodology par excellence was axiomatic and structural. Progressing via the internal dynamics of their problems at the interface of various branches, mathematics was a collection of deep theories produced by very few exceptional minds and constituted a structurally unified corpus to which one referred as artwork with a rhetoric of elegance and esthetics;

— on the other hand, the image of the mathematics which is applied, mathematics stemming from the study of nature, of technical problems, and of human affairs (war, weapon technology, statistics, economics, management, etc.). This was less prestigious mathematics, less rigorous mathematics as regards its approaches and methods (numerical analysis, approximations, modeling, etc.), less noble and less universal mathematics produced by the mass of the proletarians of science, and subordinate to social interests and conflicts.

In fact, the odds being clearly in their advantage until the end of the 1970s, pure mathematicians gave shape to this opposition. Applied mathematicians’ self-identity was defined negatively in relation to the former. Except for bastions like the Courant Institute, the California Institute of Technology and a few others, applied mathematics was often developed outside the university world, in engineering schools, foundations, military agencies, or industry (the Bell Laboratories). Albeit establishing their own professional organizations, applied mathematicians remained relatively on the margins of the international mathematics community.

IMAGE SHIFT IN THE 1980s

Over the course of the 1970s, the general landscape of mathematics was progressively modified as a result of new economic, technological, and cultural contexts in contemporary societies. In France, pure mathematics suffered from its elitist image. Paradoxically, since pure mathematicians always retained a leftist progressive political image, students’ rebellious protests in 1968 upset their status. Whole domains which had remained dormant for decades, were reclaimed, unknown fields of research, linked in particular with the computer, randomness, and experimental mathematics, were reopened: probability theory and Brownian motions, dynamical systems, fractals, combinatorics, code theory, etc. Widely used in the aeronautic, space, nuclear, and medical imagery industry, modeling and simulation called for a development of new mathematical methods.

Bringing Mathematics Down to Earth

In 1984, Edward D. David chaired an American committee which produced a report titled ‘Renewing U.S. Mathematics: Critical Resource for the Future.’¹² Looking at the explosion in the applications of mathematics due to the computer since the 1970s, the David report underscored the vital character of mathematics for science, technology, and contemporary mathematics. It had a major influence in increasing federal funding for mathematical research in subsequent years. Not surprisingly, this new conscience was very favorably welcomed by applied mathematicians [Lax, 1986].

A few years later, in a report titled ‘The Endless Frontier Meets Today's Realities,’ Richard H. Herman indicted the American mathematical community. The title of his report was a clear allusion to Vanevar Bush's 1945 ‘Science, the Endless Frontier.’ Herman noted that media and government accused mathematicians of lacking social responsibility. A loss of public confidence, he wrote, was now perceptible. Society needed to re-negotiate its contract with the scientific community. Mathematicians could not go on ‘cloning’ themselves, self-reproducing identically. While appeals to responsibility had periodically taken place in the past, mathematicians could not ignore, this time, the social demands which concerned them [Herman, 1993].

Troubled by their isolation and eager to improve their social image, mathematicians increasingly tried to put forward a more open image of mathematics entertaining multiple interactions with other disciplines, the world, and human needs. The American Mathematical Society widely echoed David's and Herman's reports. In France, a large conference on the future of mathematics ('Mathématiques à venir') jointly organized by the Société mathématique de France and the Société de mathématiques appliquées et industrielles (SMAI) was witness to the receding of the ‘tide of purity.’ The actors’ disciplinary ideological representations and the implicit philosophy sketched above also gave room to other representations privileging other values: ties with political power, capacity to earn money, entrepreneurial dynamism, pragmatic and operational character of results, etc. A mathematician was thus forced to note with nostalgia that ‘the psychological climate has changed, there is less of an emotional element, a sharp decline in the poetization of pure mathematics’ [Fomenko, 1986, p. 8].

While the increasingly pressing theme of mathematicians’ social responsibility tended to bring mathematics back to earth, questions emerged in the community, which a few years earlier would been deemed totally incongruous. While few mathematicians shared Vladimir Arnold's violently provocative pronouncements about the ‘criminal Bourbakizers and algebraizers of mathematics’ [Arnold, 1995], many more wished to counter the esoteric image of their discipline and the myth of its inaccessibility.

Debates about Proofs, New Results, and Mathematical Activity

A first issue concerned the status of demonstration in mathematical production. Demonstrations of a new type that were heavily dependent on the computer, such as the four-color theorem established by K. Appel and W. Haken, gave rise to questions on the nature of the proof in mathematics: how could a human mind grasp a demonstration that it could not follow since it filled nearly 400 pages and distinguished close to 1500 configurations by means of very long automatic procedures? Authors moreover acknowledged having found, and then corrected, dozens and dozens of errors: what kind of conviction could one therefore have that the theorem was thus proved? [Appel & Haken, 1986]. From computer proofs, the debate was soon extended to other demonstrations either quite long (e.g. the exhaustive classification of simple finite groups) or mobilizing an extremely complex architecture of conjectures stemming from various domains, as for example Edward Witten's work on knot theory and string theory. Since impossible to follow by a single reader, numerous proofs done without the help of a computer were susceptible to the very same critiques as Appel and Haken's. After all, countered Haken, if logicians accepted that an 'unassailable’ demonstration may be executable on a Turing machine, mathematicians could hardly demand higher certainty standards.

In 1990 a controversy shook the mathematical milieu. Initiated by an article of A. Jaffe and F. Quinn, the controversy was followed by a large number of replies to which the two mathematicians in turn reacted. Here, a different starting point concerned norms of rigorous, axiomatic-deductive demonstrations that mathematicians had to defend as regards new theoretical practices (speculative reasoning, physical intuitions, etc.) which were gaining ground as a result of theoretical physicists’ entry in the mathematical community [Jaffe & Quinn, 1990, p. 171–172].¹³ In fact, replies to this article, the majority of which were issued by prestigious pure mathematicians (W. Thurston, M. Atiyah, A. Borel, R. Thorn, etc.) extended beyond the sole point of rigor and concerned the values of contemporary mathematical activity in general. They illustrated that the cultural image of mathematics described above, if it has been accurately sketched, was widely considered as out of date and that the social and institutional loci in which these practices were inscribed had been extraordinarily diversified. A strict defense of standards linked to a previous cultural image of mathematics now seemed hopeless.

The computer's place was already prominent in mathematical research; evolution in the exploration of results and conjectures by means of this tool was irreversible. The founders of the journal Experimental Mathematics declared: mathematicians ‘are interested not only in theorems and proofs but also in the way in which they have been or can be reached … The role of the computer in suggesting conjectures and enriching our understanding of abstract concepts by means of examples and visualization is a healthy and welcome development.’¹⁴ A large number of conjectures emerging from extensive computer usage had sometimes been used for years before they could be rigorously proved, as was the case with the topological properties of the Lorenz attractor.

Similarly, W. Thurston has recently emphasized the multiple and collective aspects of mathematical activity. Adding that it should therefore not be judged on the sole criterion of new theorems obtained, he wrote:

Mathematical knowledge and understanding were embedded in the minds and in the social fabric of the community of people thinking about a particular topic … In any field, there is a strong social standard of validity and truth … One used to analyze the motivation to do mathematics in terms of a common currency that many mathematicians believe in: credit for theorems. This has a negative effect on mathematical progress. We must recognize and value a far broader range of activity … Soccer can serve as a metaphor. There might only be one or two goals during a soccer game made by one or two persons. That does not mean that the efforts of all the others are wasted. We do not judge players on a soccer team only by whether they personally make a goal; we judge a team by its function as a team [Thurston, 1994].

A ‘Sociological Turn'

The metaphor of the soccer team was a reference to the social which became surprisingly prominent under the mathematicians’ pen. Apart from Thurston and among many others, we may cite Vladimir Arnold: ‘The difference between pure and applied mathematics is social rather than scientific. A pure mathematician is paid for making mathematical discoveries. An applied mathematician is paid for the solution of given problems’ [Arnold, 1997]. Similarly Rene Thorn claimed that ‘rigor can be no more than a local and sociological criterion.’ When checking Andrew Wiles's proof of Fermat's theorem, several mathematicians recognized that the social and institutional dimensions of the confidence granted to some of them were at least as decisive as the rigor of the verification they could perform.

Even in the philosophy of mathematics, T. Tymoczo argued for a conception in which the notion of the mathematical community, rather than mere abstract isolated individuals, ought enter. Mathematics being a public, collective affair in which there was, says he, no commonly shared ideal of rationality and truth, the certainty of proofs in a theory today was laden with a collective dimension [Tymocko, 1986]. Thus, in practice, the rigor of the axiomatic-deductive character of mathematics functioned less as a reality than as an idealized myth. Promoted in the 1960s by philosophers and historians of science marching in Thomas Kuhn's steps, the ‘sociological turn’ was catching up with mathematicians who hitherto had seemed the least susceptible to it!

At the beginning of the twenty-first century, the value shift and hierarchy reversal occurring in mathematics is hardly independent of changes in the general image of science. While in the 1950s and 1960s, the first term in the oppositions mobilized in the ‘image wars’ in mathematics—pure vs. applied, abstract vs. concrete, structural vs. procedural, fundamental vs. useful, universal vs. specific, general vs. operative—had always been privileged and charged positively, in the subsequent period the second term was always emphasized [Dahan, forthcoming b].

While structure was the emblematic term of the 1960s, model has now taken its place. In the physical sciences, climatology, engineering science, economics, and the social sciences, the practice of model-building has gradually dominated the terrain. It is today absolutely massive and intricately bound up with numerical experimentation and simulation. In some parts of the mathematical community, this practice naturally gives rise to new concerns: those who study, with the help of the computer, supersonic flow dynamics, plamas in fusion, or shock waves, all those who model a nuclear reaction or a human heart in order to test an explosion velocity or the validity of an artificial heart, what theorems have they precisely and clearly proved? Can we consider that all these people share the same profession? Can such diverse mathematical practices still be inscribed in a unified domain? At the Berlin International Congress of Mathematicians in August 1998, the old opposition between the pure and the applied—still widely shared in the community—has been formulated in quite different terms: ‘mathematicians who build models versus those who prove theorems.’ [Mumford, [1998]. But the respect enjoyed by the former is now definitely at least as high as that of the latter.

NOTES

* Translated by David Aubin.

1. See D. Rowe's paper in this volume.

2. We use this term in the same sense as Loren Butler Feffer [1997); i.e.: we use it here without negative connotations, to encompass the prevailing values common to members of a group or of a scientific community.

3. About Princeton, several narratives are particularly lively. See (Nasar [1998], pp. 66–103; Mahoney [1997]).

4. Stone [1957]. Traditionally, Gibbs lecturers had been applied mathematicians.

5. Stone mentioned the report of the National Research Council's Committee on Training and Research in Applied Mathematics, on which he served; see F.J. Weyi [1955).

6. Hilbert's words are quoted and discussed by J-Y.Girard. See Nagel and al [1989], p. 155.

7. 'Rien n'est plus fécond, tous les mathématiciens le savent, que ces obscures analogies, ces troubles reflets d'une théorie à une autre, ces furtives caresses, ces brouilleries inexplicables; rien ne donne plus de plaisir au chercheur. Un jour vient où l'illusion se dissipe; le pressentiment se change en certitude; les théories jumelles révèlent leur source commune avant de disparaître; comme l'enseigne la Gita on atteint à la connaissance et à l'indifférence en même temps. La métaphysique est devenue mathématique, prête à former la matière d'un traité dont la beauté froide nesaurait plus nous émouvoir'.

8. See Wu & Yang [1975] This example is related by J.-P. Bourguignon [1989].

9. Dieudonné [1991] for example explicitly underlined that it would be a concession to computational fashion to consider Monte-Carlo methods and fast Fourier transforms as major mathematical results.

10. Siegel's letter seemed to have widely circulated at the time, but it has only been recently published by Serge Lang himself. See Lang [1995].

11. The first occurrence of this expression appeared in a letter from Jacobi to Legendre, July 2, 1830. Jacobi, Gesammelte Werke, vol. 1, Berlin 1884, p. 454. It reappeared in 1943 in André Weil, ‘L'avenir des mathématiques,’ Les grands courants de la pensée mathématique, ed. F. Le Lionnais (Paris: Blanchard, 1948). It has then been often picked up by various people in the 1960s and 1970s until it became the title of one of Jean Dieudonné's book.

12. David was President of Edward David Inc., and served as science advisor to President Nixon. See David [1985].

13. The whole debate has been published in the July 1993 and April 1994 issues of the Bulletin of the American Mathematical Society.

14. This journal was founded in 1991, see Epstein & Levy [1995].

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