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The Community of Mathematics HISTORICAL PERSPECTIVES Starting with the first expression of rational thought, mathematical thinking influenced the development of our culture in general and was decisive in the creation of science and technology. The inven- tion and development of the infinitesimal calculus preceded the eighteenth century age of enlightenment; these mathematical ideas and techniques made possible the beginning of the industrial age. The many mathematicians engaged in the development of analysis extended the scope of mathematics itself and provided tools for solving problems as diverse as those of the motions of dynamical systems, the flow of liquids and gases, problems involving the strength of materials, and, late in the nineteenth century, the theory of electrical and magnetic phenomena. The definitive formulation of this latter area by Maxwell required still more advanced mathe- matics. Alongside these developments in analysis, general mathemati- cal ideas in algebra and geometry found their important role in foundations of physical theories. The non-Euclidean geometries and more abstract spaces became the mathematical precursors of new physical theories the theory of relativity, for example. Certain abstract mathematical constructions constitute a prelude to formu- lations of quantum theory. By the end of the nineteenth century and at the beginning of the twentieth century a vast mathematical apparatus was being used not only for dealing with problems in astronomy and in physics and in many branches of engineering, but also in sciences like chemistry, which are built upon theories in 45

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46 The State of the Mathematical Sciences physics and involve an increasing use of mathematics on all levels from the elementary to the most abstract. Because of the long, deep, and essential connection between the growth of mathematics and that of physics, mathematics is often classed with the physical sciences; its impact on civilization until very recently was almost exclusively through its applications to astron- omy, physics, chemistry, and engineering. Yet mathematics has features that distinguish it from all experimental sciences and perhaps from all other sciences. One is the deep roots that mathematics has in the past. Some of the problems on which mathematicians work today go back several thousand years. Much of contemporary mathematics is a direct continuation of the work begun by the ancient Greeks. A more basic difference between mathematics and experimental sciences, includ- ing even the most theoretical branches, lies in the extent to which mathematical research originates within the body of mathematics itself. Physicists, chemists, biologists, and psychologists are more or less directly concerned with observable phenomena. Mathematicians often receive the seed of their problems from the outside world, sometimes through other sciences, but once a mathematical problem or concept has been formulated, it acquires, as it were, a life of its own. In a simplified way mathematics consists of abstractions of real situations, abstractions of abstractions of real situations, and so on. It is surprising but true that these abstractions of abstractions often turn out to further our knowledge and control of the world in which we live. During and after the Second World War, mathematical thinking penetrated the sciences and the development and organization of technology at an ever-accelerating rate. Probability theory has played an increasing role not only in the natural sciences but in- creasingly in all work dealing with problems of organization, in economics, and in sociology. In the last decades, problems of life sciences in biology especially have begun to require mathemati- cal formulations. Dramatic recent discoveries in molecular biology open still newer prospects for the use of mathematical ideas and models in the understanding of life processes. This increasing rate of mathematization is further accelerated by the development and use of automatic computers. New fields of technology, made possible through the exploitation of the discoveries in physics of this century, require sophisticated use of mathematics as an absolutely necessary condition. The enor

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The Community of Mathematics 47 mous progress in electronics that has changed the entire pattern of communication including information theory, network synthesis, feedback theor~would have been unthinkable without all the mathematics that goes into the development of basic concepts and into their application. The technology of nuclear energy and the conquest of space are being made possible only through the advent of the automatic computer, a development which in turn was predi- cated upon all the work in foundations of mathematics and mathe- matical logic, along with the technology of electronics. Mathematics is involved in the problems of communication- through the telephone, the radio, the transmission of pictures- and, more recently, in the exponentially growing problems of collec- tion, classification, and transmission of data in general. Physical communication and transportation, on the ground and especially in the air, require increasingly a study of organization and tra~c- control problems of mathematical character. It is no exaggeration to say, therefore, that the fundamental problems of national life depend now, more than ever before, upon the existence and the further growth of the mathematical sciences and upon the con- tinuing activities of able people skilled in their use. The developments sketched above amount to a mathematization of our culture. The remarkable growth in this mathematization is easily documented. For instance, an authoritative report on applied mathematics submitted to President Roosevelt in 19408 estimated that in the future industry might require as many as 10 mathe- matics PhD's per year. By contrast, the figures of new mathematics PhD's entering industry in 1966 has been estimated at 200. Several new disciplines, like mathematical linguistics, hardly existed 25 years ago. Less dramatic but not less significant is the increase in mathematical sophistication in all fields of engineering. In all engineering journals, 50 percent or more of the papers in recent issues would probably have been rejected by the editors as too mathematical ten or fifteen years ago. The preceding lines are from W. Prager's memorandum to COSRIMS. In this same memorandum he mites that: Twenty years ago, the typical mathematical preparation of the future engi- neer consisted of a review course in trigonometry and analytic geometry and a formal course in calculus aimed at proficiency in handling routine prob- lems. Today, the review course has disappeared from the curriculum while

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48 The State of the Mathematical Sciences the emphasis in the second course has shifted from problem solving to the understanding of basic concepts. A fairly rigorous course on ordinary differ- ential equations has been added to the typical curriculum, and students are encouraged to take as many mathematical electives as can be fitted into their program. Foremost among these are vector and tensor analysis, complex variables, transform methods, variational calculus, probability and statistics, and numerical analysis. Mathematics was, of course, always a traditional part of the edu- cation of an engineer. But now it is becoming an essential part of the education of biologists, psychologists, economists, and many others. Therefore, the first obligation the mathematical community has toward society is in the realm of education. We consider these problems in Part III of this report. Another novel situation is the greatly increased number of people doing mathematical research. Mathematical sciences have shared with all sciences the almost explosive increase in the number of investigators. For instance, Mathematical Reviews, an international abstracting journal, reviewed approximately 2,000 papers in 1940 and approximately 15,000 in 1966. The number of mathematical journals published throughout the world has increased correspond- ingly. These figures in themselves would be of little significance had the increase in quantity been paid for by a drop in quality. While objective studies of quality in mathematical or indeed any research are difficult, it is the considered opinion of this Committee that both the average quality of published research and the quality of the best work today are comparable with what they were in the 1920's or in the decade preceding World War I. THE NATURE OF MATHEMATICS The essential features of mathematical search and discovery have remained the same for centuries. So have its main sources of in spiration: the external world and its own internal structure. Mathematics done for its own sake is traditionally designated as "pure mathematics" and mathematical investigations aimed at in- creasing our understanding of the world are classified as "applied mathematics." Such a division of mathematics into pure and ap- plied, however, is difficult to maintain; the origin of many important mathematical ideas can be ultimately traced to applications, and,

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The Community of Mathematics 49 on the other hand, mathematics created for its own sake often turns out to be important for applications. The mode of origin of a mathematical concept or technique may be only very loosely correlated with its ultimate applications. Fourier considered mathematics as a tool for describing nature. But the impact of "Fourier series," crucially important as these series are in physics and engineering, has been particularly felt in some of the "purest" branches of mathematics. Cayley, on the other hand, believed that matrices, which he invented, would never be applied to anything useful (and was happy about it). They are now an everyday working tool of engineers, physicists, economists, and . statisticians. Geometry originated in practical problems of land measurement. It was developed by the Greeks into an axiomatic system and inten- sively studied for its own sake. Among Euclid's axioms, there was one (the parallel axiom) that was less intuitively evident than the others. The realization that this axiom could not be logically de- rived from the remaining ones came only after centuries of struggle and led to the discovery of non-Euclidean geometries (see Coxeter's essay in reference 7~. At the time of their birth, these geometries were the answer to a purely logical problem, and even their creators did not dare to attribute to them any counterparts in the real world. But Riemann already suspected that the physical universe in the large may be non-Euclidean, and his intuition was brilliantly con- firmed by Einstein's theory of general relativity, a theory that still provides mathematicians with challenging problems. These examples show the futility of attempting to draw dividing lines between pure and applied mathematics. The difference is not so much in subject matter as in original motivation. Also the name "pure mathematics" is unfortunate since it implies a monastic aloof- ness from the world at large and an isolation from its scientific, technological, and social concerns. Such an aloofness may be char- acteristic of some mathematicians. It is certainly not characteristic of mathematics as a collective intellectual endeavor. In fact, many of the greatest mathematicians have attacked, with equal vigor, enjoyment, and success problems posed by nature and problems arising from mathematics itself. For example, Hermann Weyl con- tributed in equal measure to the theory of groups as a pure mathe- matical discipline and to the effective uses of this theory in the theoretical constructions of atomic physics.

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50 The State of the Mathematical Sciences We, therefore, prefer not to use the term "pure mathematics," but replace it by the more descriptive term "core mathematics." The core comprises those highly developed subdivisions of mathe- matics that have been and are being investigated primarily for their own sake. The term "core mathematics" also reminds us of the central position of so-called "pure" mathematics with respect to all mathematical sciences. Intellectual curiosity and intellectual excitement are the main motivating forces behind research in all mathematical sciences, as they are in all sciences in general. The thrill of recognizing a pattern in a seemingly chaotic situation and of reducing a large number of apparently unrelated phenomena to a single simple principle are again characteristic of all sciences; but in the mathematical sciences, and especially in core mathematics, the part played by such con- siderations is so predominant that some mathematicians consider mathematics to be as much an art as a science. Mathematics is typically done by individuals rather than by groups; yet it is a collective effort. This is seen, for instance, in the fact that important discoveries are often made by several people working independently in different parts of the world. Personal contact among mathematicians in the same field is often an essen- tial condition for successful work. But the actual work is almost always done by one man thinking about a problem. Very few mathematical papers have three authors and practically none have more. Since mathematics is one of the intellectual activities in which one man unaided, unsupervised, and undirected can make im- portant contributions, it continues to attract some of the brightest young people. One striking feature of mathematical research during the past decade is the confluence of various mathematical disciplines. Dur- ing the years around World War II the tendency toward speciali- zation appeared serious. It was feared that the boundaries between various fields of core mathematics would solidify so fast that mathe- maticians of different specialties would be unable to talk to each other. We see now that these fears were not justified. Recent devel- opments have led to an interpenetration of fields, which will be illustrated in Chapter 4. Indeed many mathematicians believe that this is a golden age of mathematics. They point out that many famous problems that have baffled mathematicians of the past are being solved now, that mathe- matics is being applied in an ever-increasing number of other dis

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The Community of Mathematics 51 ciplines, and that young men entering upon research careers (and their advisers) have no difficulty in identifying interesting problems. CORE AND APPLICATIONS "The similarities are essential, but the differences are vitall" Progress in the central core of mathematics proper, where most of the essential tools and new concepts are hammered out, has long and in large measure depended upon abstraction, upon recogniz- ing that seemingly very different things have enough in common to be governed by common considerations, by an abstractly formu- lated mathematical structure. History shows that the worker in the core accomplishes more if he acts, and often feels, as if things ab- stractly equivalent were really the same. In the applied mathematical sciences' however, it is often neces- sary to take almost the opposite attitude to that which is so helpful ire the core. Here the important aspects are specific (sometimes quite concrete, sometimes more abstract from the viewpoint of some particular subject-matter field), rather than mathematically abstract. Indeed, the mathematical abstraction is almost always approximate, and it is usually vital to admit, at the proper time, the approximate character of the answers. For the applications of mathematics to grow at a rate com- mensurate with the demands of society, they must continue to receive an ever-growing supply of new results, new concepts, and new approaches from the central core. Applied areas would ulti- mately starve without what is made available by the core. Less directly and obviously, but just as inevitably, the core would lose much stimulation and implicit guidance, and indeed its develop- ment would be badly distorted and stunted, if it were cut off from applications. It is essential to recognize both the existence and com- plexity of these relations between the core and applied areas and the importance of maintaining communication between them. We shall return to this question. Those intellectual efforts coming half way between core mathe- matics, on the one hand, and the sciences and technologies to which mathematics is applied, on the other, are often referred to as "applied mathematics." This is accurate only if it is understood that these efforts include not only the traditional parts of mathematics concerned with applications in physics, chemistry, and engineering

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52 The State of the Mathematical Sciences (which we shall refer to as physical mathematics or classical applied mathematics), but also statistics, computer science, the gen- eral field of optimized allocation, control, and decision-making, and fields such as mathematical biology. Each of these fields has a char- acter of its own, but they have in common a spirit that is well described in reference 9: . . . mathematics is "applied' if conceived in a spirit of ready cooperation with sister sciences in the grand endeavor of comprehending our environ- ment, making sense of ourselves, and bringing order into their interaction. Like all good mathematics, good applied mathematics is original and imagi- native in the invention and use of its concepts and in its tentative modeling constructions. Its chief distinction from pure mathematics, which shows self-motivated progress along dimly discernible natural paths of growth towards intellectually satisfying goals, lies in adding to this conceptual activity a deep concern for the world of outer experience and a ready in- terest in problems beyond the confines of mathematics. In common with the core mathematician, many applied mathe- maticians are interested in the stimulation of the development of new mathematics but with primary emphasis on those aspects strongly and directly motivated by scientific, technological, or sociological problems. In common with the theoretical scientist, the applied mathematician seeks knowledge and understanding of facts and phenomena through the use of mathematical methods. The applied mathematician is most typically engaged in the formulation, analysis, and interpretation of mathematical models: in other words, (a) the formulation of scientific, technological, or practical problems in terms of abstract mathematical models; (b) the solution of the resultant mathematical problems, which may consist of detailed answers or new techniques; and (c) the discus- sion, interpretation, and evaluation of the results of this analysis, especially in relation to the correspondence between the abstract model and the real world. In different areas of the applied and partly mathematical sciences, the character of these models may be quite different. In computer science, for example, they some- times take the form of a detailed computer program. The final goal of these efforts is the creation of ideas, concepts, and methods that are of basic significance to the subject in question. As mentioned above, these efforts may lead to the creation of new mathematical ideas and theories. When thinking of "applications" and "mathematics" together, the nonmathematician tends to think of an immediately applicable

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The Community of Mathematics 53 solution to his own pressing problem. The applied mathematical sciences do indeed contribute in such immediate ways, but they also contribute throughout a range, of greater or less immediacy, stretching to the very edge of core mathematics. All these varied kinds of contribution are important, not least those that have the same pure-science character as core mathematics. CRITICISMS AND TENSIONS The highly exciting achievements within core mathematics and explosive penetration of mathematics into other fields are generally recognized. At the same time, criticism of mathematicians by other scientists as well as criticisms within the mathematical community itself are far from uncommon. Certain of these criticisms may be summarized as follows. It is said that during the recent past mathe- maticians have alienated themselves from the mainstream of scien- tific development. One thing modern mathematicians tend to overlook is that the giants of former days were all actively interested in physics as well as mathematics. This is the first generation for which this is no longer true. From a letter by a prominent physicist It is also claimed that what contemporary pure mathematicians do is of interest only to themselves and most, if not all, of it will never be used in any other discipline. While all sciences require sophisti- cated mathematical tools and present interesting and challenging mathematical questions, mathematicians are said to be ignorant of and not interested in these questions. This state of affairs is alleged to be particularly and perhaps specifically characteristic of Ameri- can mathematicians. Has American mathematics become so remote that scientists will have to turn to other schools of mathematics "Russia?] for help in the future? From the same letter by the physicist This criticism is not new. Mathematicians have long been criticized for being too abstract and too remote from the needs of science. Thus as knowledgeable and distinguished a mathematician as Felix Klein considered the Hamilton-Jacobi theory of dynamics as a play- thing of no use for physics (F. Klein, Vorlesungen uber die Entwick

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54 The State of the Mathematical Sciences lung der ~lathematik im 19. Jah~hundert, I, Springer, Berlin, 1926, p. 207~. Yet the Hamilton-lacobi theory proved to be the basis of quantum mechanics. Even the greatest of physicists may err in their evaluation of mathematics. Einstein was at first reluctant to accept Minkowski's four-dimensional geometry as the proper description of special relativity, and several years later he was reluctant to express his physical ideas in the language of tensors. The history of science has shown time and time again that it is impossible to predict what mathematical theories will turn out to be useful outside of pure mathematics. This point can hardly be stressed too much. Planning the over-all emphasis of mathematics, either for the progress of mathematics as an end in itself or for its application to other sciences and tech- nologies, must fail lamentably. There is a process of natural selec- tion in mathematics. For instance, abstraction for abstraction's sake rots away rapidly if it leads to no new intellectual progress; abstrac- tion that only appears to be for abstraction's sake often proves to be of vital importance. Unconstrained opportunities to succeed or fail; broad diversity, both of individual activities and organizational patterns; ease of change, especially in giving up old patterns these are the characteristics that have tended to allow the mathematical sciences to make progress. The claim that "the giants of former days were all actively inter- ested in physics as well as mathematics" is illusory. To be sure, Gauss, Riemann, and Poincare were actively interested in physics. But one can easily think of first-rank mathematicians of the nine- teenth century whose interests were exclusively mathematical. Among them there were several whose work, unbeknown to them- selves, turned out to be of vital importance for other sciences (for instance, Frobenius, who founded the theory of group representa- tions). There have been contemporary mathematicians with inter- ests as catholic as those of the greatest men of the nineteenth cen- tury for example, Hermann Weyl, Wiener, and Von Neumann or Gelfand and Kolmogorov but it is unreasonable to expect more than a few such men in a generation. Still, the charge that mathematicians have separated themselves from their sister disciplines like physics and astronomy reflects a true state of affairs. Specialization is the inevitable price of the increase of knowledge. One cannot expect many future scientists to combine mathematical and physical achievements in the manner of Newton or Gauss. Yet we must not allow ourselves or our stu

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The Community of Mathematics 55 dents to become intimidated by the "explosion of knowledge." 'The problem facing us is mainly that of communication and, by impli- cation, that of education. IMPROVING COMMUNICATIONS In the mathematical sciences, as in most other sciences, the explo- sive growth of activities has had an important side effect. The problems of communication between scientists in various disciplines, between the creators of new theories and the users of these theories, between investigators and teachers, between teachers at various levels, between working scientists and scientific administrators, and last but not least between the scientific community and the citizenry at large have become difficult and require a conscious effort at solutions. The volume of scientific literature has increased so much that the average working mathematician cannot even scan all the review journals. This results in the paradoxical situation that while many more printed pages of mathematics appear than ever before, mathe- maticians must depend to a large extent on personal contacts and personal correspondence in order to keep abreast of the current developments in their specialties. (This explains why conferences and symposia are an indispensable part of contemporary scientific research.) The ultimate solution may come from radical new methods for storing, retrieving, and dispensing information. Per- haps journals, books, and libraries as we know them will be replaced by other techniques. This, however, will not happen at once. A different problem is that of communication between various branches of the mathematical community. Here the difficulties are not technical but psychological and intellectual. In the central area of core mathematics any threat to the exchange of ideas and results between subdisciplines has been overcome through recent unifying tendencies. On the other hand, this danger is acute as far as the interfaces between core mathematics and the applied and partly mathematical sciences are concerned and even more so be- tween the mathematical sciences as a whole and the various users of mathematics. What is needed is a conscious effort by highly qualified people to overcome this "communication gap" by specially written books and articles, by interdisciplinary conferences and courses, and by other appropriate means. If this endeavor is to be successful,

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56 The State of the Mathematical Sciences leadership must come from the profession, specifically from indi- viduals. Financial support, where necessary, will also have to be found. The problems of communication and publication have recently been considered in more detail by the American Mathematical Society's ad hoc Committee on Information Exchange and Publica- tion in Mathematics. By special permission we have reproduced that committee's final report in Appendix B. Its recommendations are addressed to the American Mathematical Society but may provide more general guidance. During the past decade, the mathematical sciences have made strenuous and successful efforts to establish communications be- tween the investigators and teachers of mathematics at various levels. The various programs for curriculum reform in high schools and elementary schools involve joint work by university mathe- maticians and teachers. The manifold activities of the Committee on the Undergraduate Program in Mathematics (CUPM) establish a link between graduate schools and colleges. We do not believe that any of these groups has already found definitive solutions to the problems of mathematics education. Valuable steps have already been taken, however, and such work should be continued and supported. The problem of making mathematics understandable to the edu- cated layman continues to be almost insurmountable. A Greek tyrant who wanted a rapid explanation of mathematics was told that there was no royal road to this science. This remains true. It is equally true, however, that the mathematical community has a duty to do all it can to make its work understandable to those who want to understand it. The collection of essays7 to be published in conjunction with our report is a modest contribution in this direc- tion.