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1 Mathematics and Society The recent history of mathematics has been like that of science generally. The number of investigators has grown rapidly and so have the pace and quality of the research. An intellectual leadership that used to reside in Western Europe is now shared by Americans and others. Today, society, especially through other sciences and technologies, makes unprecedented demands on mathematics and on the com- munity of mathematical scientists. The purpose of this report is to identify the major demands, to assess the capabilities of the mathe- matical community to satisfy them, and to propose measures for preserving and extending these capabilities. THE MATHEMATIZATION OF CULTURE Mathematics has long played a central role in the intellectual and technological history of mankind. Yet this statement hardly begins to convey or account for the current explosive penetration of mathematical methods into other disciplines, amounting to a virtual "mathematization of culture." Mathematics can be described as the art of symbolic reasoning. Mathematics is present wherever chains of manipulations of ab- stract symbols are used; such chains may occur in the mind of a human being, as marks on paper, or in an electronic computer. Symbolic reasoning appears to have been first used in connection with counting. For this reason, mathematics is sometimes described,

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4 Summary though not completely accurately, as the science of numbers. In fact, it turns out that all symbolic reasoning may be reduced to manipulation of whole numbers; and it is this fact that makes the digital computer into the universal tool it is. Mathematics as we know it originated more than 2,000 years ago warn the ~reeks. They transformed into a deductive science the collection of facts and procedures about numbers and geometric figures known to the older civilizations of Egypt and Babylon. The Greeks applied mathematics only to astronomy and statics. The possibility of applying it to other sciences, in particular to dynamics, was discovered in the sixteenth and seventeenth centuries. This dis- covery revolutionized mathematics; it led to the creation of calculus and thereby made modern physical science and technology possible. The development of the physical sciences continues to use mathe- matical techniques and concepts. So do the new technologies based on the discoveries of the physical sciences. Furthermore, these sci- ences and technologies use mathematical techniques of ever-increas- ing sophistication, so that the mounting role of physical sciences and technologies in the contemporary world is a dominant aspect of the mathematization of culture. It is, however, not the only aspect. Mathematical methods are penetrating into fields of knowl- edge that have been essentially shielded from mathematics until not long ago; for instance, the life sciences. The twentieth-century penetration of mathematical methods into the biological sciences has come about in several ways, perhaps most importantly through the increasing study of biological phenomena by the methods of chemical physics. The development of statistics, which the needs of the biological sciences helped to stimulate, has led to the exten- sive field of biostatistics. This has connections with mathematical genetics, which has evolved out of the celebrated Mendelian laws of inheritance, and with mathematical ecology, which is concerned with such interactions as competition for food or the feeding of one species on another. Differential-equation models for the con- duction of signals along nerve fibers have had notable success. Computer simulation of functions of living organisms is just one of the ways in which computers are becoming increasingly im- portant in the biological sciences. Mathematics is penetrating the social and behavioral sciences, too, and even traditionally humanistic areas. Mathematical eco- nomics is now a central part of economics. The field of econometrics has grown out of applications of probability and statistics in eco . . . ~_ _ OCR for page 3
Mathematics ant! Society nomics; and statistical techniques are important in anthropology, sociology, political science, and psychology. Analysis of mathemati- cal models for various social phenomena has been greatly aided by computer simulation and data processing. The mathematical view- point has even found new application in linguistics. The above remarks refer to the mathematization of various academic disciplines. Mathematics is also becoming an indispensable tool in the world of government, industry, and business. The terms `'operations research" and "management science" are among those describing the rapidly growing use of mathematical methods to solve problems which arise in managing complicated systems involv- ing the movement and allocation of goods and services. Computers have extended the possibility of applying mathemati- cal methods to a degree that would have seemed fantastic a short time ago. Computer science, which deals with manifold problems of building and utilizing computers, contains, among other things, very important mathematical components. THE NEED FOR MATHEMATICALLY TRAINED PEOPLE The mathematization of our society brings with it an increasing need for people able to understand and use mathematics. This need manifests itself at various levels. We need people who can teach mathematics in grade school in a way that will not create a permanent psychological block against mathematics in so many of our fellow citizens. We need people who can understand a simple formula, read a graph, interpret a state- ment about probability. Indeed, all citizens should have these skills. We need people who are able to teach mathematics in high school and cope with the necessarily changing curriculum. We need people who know what computers can do, and also what they can- not do. We need computer programmers who can work with under- standing and efficiency. We need engineers, physicists, chemists, geologists, astronomers, biologists, physicians, economists, sociol- ogists, and psychologists who possess the mathematical tools used today in their respective disciplines and who have the mathematical literacy for learning the new skills that will be needed tomorrow. Equally we need, though in smaller numbers of course, people in these fields who are able to use mathematical tools creatively and if necessary to modify existing mathematical methods. The numbers

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6 Summary are hard to estimate, but it is clear that our society needs many more mathematically literate and educated people than are available now. For instance, computer programmers are already more numerous than high school teachers of mathematics, and their numbers will . . continue to increase. These demands call for massive amounts of mathematical edu cation at all levels and so produce a mounting pressure on the mathematical community. To do all this training calls for a larger body of mathematics teachers. It is difficult to expand the supply rapidly enough. Since teachers of intermediate-level mathematics must themselves be trained by people of higher competence, these pressures also quickly transmit themselves to the relatively small community of mathematical scientists who do research. Many of the needs of mathematics for support come from a need for balanced growth meeting all these requirements. Since we recognize a rising level of mathematical literacy as a national objective, and since the community of mathematical sci- entists bears the primary responsibility for attaining this objective, our report cannot separate problems of research from problems of education, including those of undergraduate education. (See both Part III and the report) ~ of our Panel on Undergraduate Edu cation.) APPLIED MATHEMATICAL SCIENCES There are now four major areas in the mathematical sciences that have particularly direct and important relationships with other sciences and technologies: computer science, operations research, statistics, and physical mathematics (classical applied mathematics). We shall ordinarily refer to these as applied mathematical sciences. For statistics and computer science there is a more accurate term, partly mathematical sciences, which we shall sometimes use in recognition of the individual character of these fields and their strong extramathematical components. These four major areas must each have special attention if we are to come close to meeting national needs. At the same time, there is a need for general support of applied mathematical sciences in a way, not closely tied to particular applications, that will encour ~ Superscript numbers refer to the list of references at the end of the report.

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Mathematics and Society 7 age creative interaction between mathematics, science, and tech- nology, and among the various applied mathematical sciences themselves. The sciences and technologies associated with the computer- whether concerned with the nature of information and language, the simulation of cognitive processes, the computer programs that bring individual problems of all kinds to today's computing systems, the software programs that convert cold hardware into a complex computing system, or the hardware itself face an intense and grow- ing challenge. The field sometimes labeled operations research is now growing rapidly, though not so explosively as computer science. Its emphasis today is on solving problems of allocation (routing problems and scheduling problems are two major types) and on a broad class of operational applications of probability (inventory management and improving the service of queues and waiting lines, for example). Again there are national needs both for a substantial body of people who can apply the techniques effectively and for a leadership that can innovate, reshape, and transform. The field of statistics and data analysis is older and more firmly established than the two just described. Yet there is still a shortage of statisticians who can bring mathematical techniques and insights to diverse applications. The development of computer techniques is also having a strong impact on this field. Physical mathematics, also called classical applied mathematics, has evolved into various modern forms. In its traditional form, it emphasized the mathematics essential to classical physics and the established fields of engineering. Even more it emphasized the evolution of the mathematical models under study. Nowadays, the concepts of physics have been expanded to include the well- established aspects of quantum mechanics and the theory of rela- tivity. New developments apply to an ever greater variety of subject-matter fields, and we must now look once again toward a closer collaboration and mutual stimulation between mathematics and all the other sciences. Alongside the four main applied mathematical sciences there are still newer areas of application where no self-identifying community of mathematical scientists yet exists areas of central importance to a variety of national objectives of great and growing concern. The interplay between mathematics and sciences and the mutual stimulation, cooperation, and transfer of ideas among applied

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8 Summary mathematicians working in diverse areas all suggest that the applied mathematical sciences, because of their common features, constitute an area of study worthy of support in its own right. CORE MATHEMATICS The foundation of the manifold mathematical activities just dis- cussed is the central core of mathematics the traditional dis- ciplines of logic, number theory, algebra, geometry, and analysis that have been the domains of the so-called "pure mathematician." The relationship between the core and applied areas is not one- sided; many of the essential ideas and concepts in the central core can be traced ultimately to problems arising outside of mathematics itself. In the central core, mathematical ideas and techniques, no matter what their origin, are analyzed, generalized, codified, and transformed into tools of wide applicability. In assessing the importance of the core, one should keep in mind that there is always an interplay and exchange of ideas between so-called "pure" mathematics, that is, mathematics pursued pri- marily for intrinsic intellectual and aesthetic reasons, and so-called "applied" mathematics, that is, mathematics consciously used as a tool for understanding various aspects of nature. Thus geometry, literally "earth measurement," originated as an applied art, presumably in the Nile delta. The Greeks transformed it into a pure deductive science, the prototype of pure mathematics. Among the geometric objects studied by the Greeks were the curves (ellipses, parabolas, hyperbolas) obtained by intersecting a cone with a plane. These "conic sections," though they may have been discovered by observing sundials, were then of interest to the pure mathematicians alone. Today conic sections are working tools of engineers, physicists, and astronomers. On the other hand, calculus, which was developed by Newton as a mathematical tool for studying the motions of physical bodies, is also the foundation of a large part of modern "pure" mathematics. The most spectacular uses of core mathematics are its direct applications in science and technology. Remarkably enough, it is impossible to predict which parts of mathematics will turn out to be important in other fields. We have one guide: Widely useful mathematics, for the most part, has proved to be also the kind that mathematicians earlier characterized as "profound" or "beautiful."

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Mathematics and Society 9 Important mathematical ideas have also been generated by people who were not professional mathematicians. The time lag of 2,000 years between the invention of the conic sections and their applications in astronomy is, of course, not typi- cal of recent developments. But the unexpected character of the application is typical. The theory of Lie groups, for instance, was developed for many years because of its intrinsic mathematical interest. It seems now to be the natural way of describing sym- metries in elementary-particle physics. The theory of analytic func- tions of several complex variables has been undergoing a dramatic development during the last two decades. The experts in this theory were quite surprised to discover its usefulness in quantum field theory. We stress once more the totally unpredictable nature of such applications. It is not the motivation of the mathematician who creates a new theory that determines its future relevance to other fields of knowledge. In particular, one should not be repelled by the seemingly frivolous origins of many mathematical theories. A puzzle about the seven bridges in Kcinigsberg led to the theory of graphs, a basic mathematical tool of computer science, and in- directly influenced the development of topology. A question raised by a professional gambler led Pascal and Fermat to the theory of probability. But the application of a particular mathematical result or a specific mathematical concept is not the only way in which core mathematics is used. The total impact of mathematics on science and technology is more difficult to document but probably even more important. In all such applications of mathematics (model building and mathematical reasoning about models, statistical analysis, the use of computers), the investigators will use some of the concepts, methods, and results developed by core mathe- maticians. In a typical case, however, they will not find in the storehouse of core mathematics the precise tools they need but will rather have to develop those tools either alone or in cooperation with mathematicians. How successful they will be depends to a large extent on the general status of mathematics in the country, on the level of mathematical knowledge among the people involved, and on the number and quality of mathematically trained people. All this depends ultimately on a healthy and vigorous develop- ment within the central core of mathematics. We are convinced that without this one cannot have efficient use of mathematical methods

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10 Summary in science and technology, imaginative mathematization of new fields, or spirited and effective teaching of mathematics at all levels. The central core of mathematics is not static. It is now under going rapid and in many ways revolutionary development. Many old and famous problems are being solved. The traditional boundaries between different mathematical subfields are disappearing. New unifying ideas are applied with great success. Though dynamic, the central core of mathematics preserves historical continuity and un- compromisirlgly high standards. THE POSITION OF THE UNITED STATES IN MATHEMATICS At the beginning of this century, mathematical research activity in this country was chiefly concentrated in a very few centers. Note- worthy was the center at The University of Chicago under the leadership of E. H. Moore, himself trained in Europe. Among Moore's illustrious students were Oswald Veblen, G. D. Birkhoff, and Leonard Dickson, each a major figure in mathematics. In the interval between World Wars I and II, mathematics in the United States became somewhat more important relative to world mathe matics. Political developments in Europe in the 1930's led many Euro- pean mathematicians to seek refuge in the United States and to become active members of the American mathematical community. This greatly stimulated mathematical research activity in this country. The Institute for Advanced Study in Princeton became a world center of mathematical research. Until after the Second World War, however, financial support for mathematical activity was extremely limited, and only a handful of undergraduates seri- ously considered careers in research mathematics. During World War II, the relevance of mathematics to the tech- nological might of the nation and the critical shortage of mathe- matically trained people became apparent. After the war, the mathematical sciences became for the first time a concern of the federal government, initially through the research departments of defense agencies and then also through the National Science Foundation. While the influx of federal money into the support of mathematics was very modest compared with the funds poured

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Mathematics and Society 11 into such expensive fields as high-energy physics, its effect on the scientific life of the United States was stupendous. Before World War II, the United States was a consumer of mathematics and mathematical talent. Now the United States is universally recognized as the leading producer of these. Moreover, graduate education in mathematical sciences at major centers in this country is far superior to that in all but two or three centers in the rest of the world. Some more specific indicators of the posi- tion of the United States in the mathematical world are given below. 1. International congresses of mathematicians, meeting at roughly four-year intervals, were inaugurated around the turn of the cen- tury. The earliest such congress, in 1897 in Zurich, was attended by only about 200, whereas at the most recent congress in 1966 in Moscow, attendance was approximately 4,300. An invitation to a mathematician to address an international congress evidences world- wide recognition of his contributions to mathematical discovery at the very highest level. During the first four international congresses, those of 1897, 1900, 1904, and 1908, there were 26 invited addresses, of which only one was by an American. During the four most recent congresses, those of 1954, 1958, 1962, and 1966, there were 274 invited ad- dresses, of which 96, more than one third, were by U.S. mathe maticians. 2. The Fields Medals, for recognition of distinguished achieve- ment by younger mathematicians anywhere in the world, were established in 1932 bY the late Professor T. C. Fields. Beginning with the Oslo congress of 1936, two Fields Medals have been awarded at each international congress of mathematicians, except that most recently at the Moscow congress of 1966 four were awarded. In all, 14 Fields Medals have been awarded, with the distribution of medalists by country as follows: France, four; the United States, four; England, two; Finland, Japan, Norway, and Sweden, one each. Of the 12 Fields Medalists since 1945, three have been Americans trained ir1 America: Paul Cohen (1966), John Milnor (1962), and Stephen Smale (1966~. Three others are long-time residents of the # Simon Newcomb, who addressed the 1908 congress in Rome on the history and present status of the theory of lunar motion. Participants at this congress num- bered 535, of whom 16 were from the United States.

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12 Summary United States and should now be considered to be members of the U.S. mathematical community. Four or five others have been and are frequent visitors to the United States, each having spent at least one academic year here. Some of these are active collaborators with various American mathematicians. 3. English has recently become the dominant language in world mathematical circles. For instance, in the German journal, Mathe- matische Annalen, the percentage of papers in English rose from approximately 5 percent in the mid-1930's to nearly 20 percent in the mid-1950's, and to 55 percent in the mid-1960's. 4. In some representative issues of three of the leading mathe- matical journals of Europe, Acta Mathematica (Sweden), Com- mentarii 3Iathematici Helvetici (Switzerlandj, and Mathematische Annalen (Germany), the following percentages of references to papers in U.S. journals were found: Acta YEAR Math. Math. Ann. 935 950 965 3% 12% 42% 12% 24% 25% A% 4% 25% 5. There has been a significant increase in the number of foreign mathematicians visiting in this country. Figures assembled by the American Mathematical Society show that, in 1956, 73 foreign mathematicians spent at least a semester at a U.S. university; in 1960, the number came to 144; in 1965 there were 199 such visitors. In addition, other foreign mathematicians made briefer or more casual visits or more lengthy stays outside universities.