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14 Tasks and Needs It is inevitable that in considering the tasks and the needs of the mathematical sciences we treat the various disciplines separately. We hope that this way of presentation will not obscure the in- tellectual ties between the mathematical sciences. To preserve these ties is itself an essential need of the mathematical community. Over and above the specific needs enumerated below, there is a general need of the mathematical sciences, and of sciences in general, for an atmosphere in which they can flourish. This involves not only the availability of financial support but also a broad understanding of and appreciation for intellectual endeavor, a willingness to sup- port theoretical work, and an uncompromising commitment to in- tellectual freedom. THE CORE Research within the core is primarily concerned with identifying, extending, and refining the critical concepts that serve to organize mathematical thought and with the precise analysis of the inter- relations of these concepts. It is often the concepts and theorems of the core areas that breathe life into the applications of mathematics. Support for research in the core subjects thus represents capital in- vestment in mathematics. As an illustration, we mention that electrical engineers, computer scientists, and others are now demanding that the undergraduate curriculum in mathematics should include a sizable amount of so- called discrete mathematics. This includes such topics as the theory of algorithms, lattice theory, graph theory, and other parts of com 197

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198 Conclusions binatorial mathematics. All these topics grew up in or around algebra at a time when it seemed that, for the most part, only "con- tinuous mathematics," that is, mathematical analysis, was of concern to applications. Up to now, most mathematical ideas and techniques have been developed within the core, though often as a response to challenging problems coming from outside mathematics. There is every reason to believe that the process will continue. The central core of mathe- matics is also the laboratory in which progress in mathematical sciences is ultimately unified and integrated. Finally, it provides an unequaled training ground for mathematicians in all fields. Many leaders in the new applied areas of mathematical sciences received their training in the rigorous atmosphere of established pure mathe- matics. Many future leaders in these fields, as well as leaders in fields not yet initiated, may also develop in this way. It is our opinion that maintaining the health and vigor of research in core mathematics is essential for the health of all the mathe- matical sciences and, indeed, for our scientific life as a whole. It would be tragic if a misdirected drive for immediate applicability or a dominant criterion of social usefulness should be permitted to de- stroy or weaken the position of unchallenged leadership in mathe- matics now occupied by this country. It is useful to examine how much support has been directed to the core areas of mathematics and how effective it has been. Classify- ing research by subject matter presents far less of a problem than classification as pure or applied. Still, boundaries between subjects are not sharp, and it is not always possible to identify the subject of a research project by looking at its title. Probability theory, for ex- ample, presents a particular problem, because a great deal of the support for research in probability has been administered in con- nection with statistics. Altogether the federal government seems to have spent approxi- mately $15 million on core-area research in fiscal year 1966. With some reservations concerning the accuracy of our classification, it appears that the Mathematical Sciences Section of the National Sci- eIlce Foundation (NSF) obligated about $8 million to the support of core research in fiscal year 1966 out of a total obligation of $14.9 million. The three Department of Defense (DOD) research o~ces- the Army Research Once, the Once of Naval Research, and the Air Force Once of Scientific Research have each devoted about 40 percent of their mathematical research budgets to core subjects. For fiscal year 1966 this came to $5.8 million for core research. Although

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Tasks and Needs 199 a much larger sum ($124.9 million) was reported as obligated in fiscal year 1966 by various federal agencies to research in the mathe- matical sciences, very little appears to have been spent on core re- search except through NSF and the three DOD offices. The National Aeronautics and Space Administration, for example, spent $91,000 on core research out of a total of more than 57 million for mathe- matical research. No doubt the largest unaccounted federal contri- bution to the core areas came through the research done by full-time employees of the government and of private research institutes such as RAND and the Institute for Defense Analyses, which are largely supported by the government. There can be no doubt that American mathematics has moved into a position of world leadership during the period in which federal support of science has grown. But it would be difficult to decide to what extent this is due to research support. We can, however, inquire whether the money does or does not go for the support of those mathematicians who are actually producing the research that justi- fies the country's claim to leadership. A check of three leading Amer- ican mathematical-research journals (~American Journal of Mathe- matics, Annals of Mathematics, and Journal of Mathematics and Mechanics) for 1966 shows that 133 of the 221 papers (about 60 percent) published carry a footnote acknowledging federal support of the research reported. A similar count in a journal devoted to short papers (Proceedings of the American Mathematical Society) showed that 49 of 113 papers (about 40 percent) in two issues acknowledged federal support. These figures show that federal money has been involved in a truly significant portion of the mathe- matical research being done in this country. It appears that the support has been effective, since great progress has been made toward the goals that might reasonably have been set 10 or 15 years ago. There has been astonishing progress within the discipline itself; a number of long-standing problems have been solved that a dozen years ago seemed almost unapproachable. More- over, this activity In the core seems to have had the desired indirect effects. Techniques and concepts invented to solve core problems are finding applications in other areas, interest in mathematics has increased, and the number of mathematicians has grown signifi cantly. While some local dissatisfactions exist, as they inevitably must, there is every reason to be pleased with support policies as they have affected the core areas until now. Although there has been a generally satisfactory pattern of support in the past, the signs for the future are disquieting. The gross budget

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200 Conclusions for mathematical sciences in the four agencies that support core mathematics was fixed for fiscal year 1967 at about the same size as that for fiscal year 1966. Coupled with the new rules, which allow a larger proportion of direct costs to be charged as overhead, a net decrease in funds for core research in mathematics is to be expected at a time when both the per capita cost of research and the number of qualified investigators have increased. If funding does not show an increase commensurate with the net growth of the nation's mathematical activity, we must expect that the forces that have linked the expansion of core research to the general expansion of mathematical competence will work in reverse. Needless to say, at a time when the demand for mathematically trained personnel is expanding in every facet of our society, such a reversal would be regrettable. We may be simply unable to meet our needs in the areas of general science, education, technology, technical management, and defense. In an area in which the effects of support on major national goals are only indirect, it is extremely difficult to judge the optimum levels of support. The core areas in the mathematical sciences have been making very good progress in recent years on support, which has beers increasing at about 18 percent per arlnum. No one can say that this rate is necessary to achieve our goals, but it would seem im- prudent to allow the growth to fall significantly below this level at a time when, according to all estimates, further expansion is necessary. A specific need of core mathematics is the preservation of the es- tablished great centers. Their importance should not be obscured by the recognized need to develop new centers of research and edu- cation. A great center of mathematics requires a certain critical num- ber, never precisely determined, of first-rate mathematicians. It must attract some of tile best students. At such a place there is a spirit of high tension, an awareness of the important problems, and a willing- ness to explore new approaches and to accept new ideas, coupled with a sense of history and uncompromisingly high standards. # A famous example of a great mathematical center was the University of Got- tingen in Germany. Gottingen was the world capital of mathematics until 1933. Its fame dates from the times of Gauss and Riemann, but it became a great school only later, primarily under the intellectual leadership of Felix Klein and Hilbert. Almost every leading European mathematician spent some time of his life at Gottingen; it was also a place of pilgrimage for many American mathematicians. The destruction of mathematics in Gottingen by the Nazis, however, shows how fragile a scientific center is. During the two decades after the end of World War II, several first-rate mathematicians have appeared in Germany, but Gottingen, and Germany as a whole, have not regained their former stature in mathematics.

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Tasks and Needs 201 At present there are more first-rate mathematical centers in the United States than in the rest of the world. Nowhere else, with the exception of Moscow and Paris is there such concentration of lead ing mathematicians in various fields as in the five to ten leading American universities. Princeton, the Cambridge area, the New York area, the Bay area in California, and Chicago are well recognized as exceptionally strong centers of mathematical life. There are also several strong centers at the great state universities. (The recent rat- ing by the American Council of Education,32 which lists nine "dis- tinguished" and 16 "strong" departments of mathematics reflects well the general feeling within the mathematical community.) The strongest mathematical centers play a triple role. They attract some of the most talented undergraduates and in most cases give them an excellent education. In comparison with PhD deg~ee-grant- ing universities generally, they produce a disproportionately large number of PhD's in mathematics, and an even larger percentage of the really outstanding ones. (This is documented in Appendix E.) They also serve as centers for postdoctoral education, either as a major function, as in the case of the Institute for Advanced Study, or, more incidentally, through special research instructorships, ter- minating assistant professorships, and visiting professorships. Also, most of the outstanding research in mathematical sciences is done at the great centers.. PHYSICAL MATHEMATICS An essential task of physical mathematics (sometimes called classical applied mathematics) is to strengthen the intimate interplay be- tween mathematics and the physical sciences for the benefit of both. This is closely connected with teaching and research in the universi- ties, which should provide the appropriate setting in the form of faculty positions or even departments. Although physical mathe # One should always remember, however, the many exceptional cases. A single outstanding teacher in a place located far from established centers may succeed in educating a large number of research mathematicians. R. L. Moore in Texas is the striking example. A brilliant investigator may make historic discoveries while isolated from a major mathematical community S. Lefschetz during his Kansas stay, for instance. An important discipline may develop entirely at a sec- ondary place. For example, the recent theory of trigonometric series was mainly created by a large number of workers in a large number of places other than the major centers. See also S. Lefschetz,"A Page of Mathematical Autobiography," Bull. Amer. Math. Soc., 74, 854 (1968~.

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202 Conclusions malice is primarily concerned with mathematical physics in general and mechanics in particular, the same spirit can be applied to other mathematical sciences, such as mathematical economics, mathe- matical biology, and the basic parts of computer science. It is in a comprehensive approach, with a view toward the interplay among the major lines of research, that one can see the most rewards. One may expect that mathematization of new areas in science and tech- nology will result, and that new problems within mathematics itself will be uncovered as well. Thus, the contribution of the physical mathematician to society extends from the very practical aspects of engineering to engineering science, to basic science, and to the stimulation of pure mathematics itself. Like core mathematics, physical mathematics needs the support of federal funds that will grow or ~ in proportion to the growth of the number of workers in the field. The physical mathematician must, however, be concerned with both the mathematical and the scientific aspects of his work; that is, he must bring his theories to the point of comparison with experiments and observations from time to time. This will usually result in a somewhat wider scope of activities (such as extensive numerical computations) and therefore a greater need of funds. In exceptional circumstances, it might even lead to large- scale computing operations. (One such example is Van Neumann's development of numerical weather forecasting.) Unlike certain other countries, especially Great Britain, the United States lacks a long tradition in physical mathematics. For this reason too, the current level of federal support of this field is not so high as it should be. For a few years, therefore, it may need to grow at a higher rate than the average for all mathematical sci- ences. There should be a conscious policy of federal support for the research of new faculty members in physical mathematics appointed at the various universities. Industrial organizations can also provide support for physical mathematicians in the form of research and consultant groups. There Is great potential value in the flexibility and breadth of knowledge that the physical mathematician brings to his approach to a number of scientific and technological problems. Some indus- trial organizations and government laboratories already have excel- lent mathematical-research groups. Hopefully, for the future we may look forward to the continuation and expansion of such groups.

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Tasks and Needs STATISTICS 203 Statisticians with research training are needed to fill very different places in our society. First, there is a need for mathematically trained and competent people to develop statistical techniques and evaluate them. Such persons should have an appreciation of how statistical methods are used and what the objectives of statistical applications are. Other statisticians with research training as well as some mathe- matical competence have great opportunities in positions aca- demic, industrial, governmental-that involve consulting on the problems of other workers. Many of the great advances have been and still are being made by statisticians of this type. Statisticians who can take operational responsibilities in connection with surveys and other large data-gathering operations governmental, industrial, or nonprofit are also badly needed. In their turn, some of them have made basic contributions. Many statisticians operate as individuals within organizations of quite diverse kinds. There are still great needs for routine work, particularly at the lower levels of survey organizations and in the organizations that spot statisticians singly here and there, but more and more of this is now being done by computers. Statisticians are thus increasingly concerned with innovation. When a problem is posed to him, a sta- tistical consultant may decide that some standard technique can be used; even this often requires innovation in subject-matter concepts. The innovation is more likely, however, to take the form of a newly modified technique. As a consequence, research training is more widely needed than it might seem to be at first sight. There are demands for new and better formulations of what users of statistics are trying to do, for better and more detailed mathe- matical treatments of many problems, for the re-examination and modification of many results and techniques to make them more closely matched to the situations in which they are in fact used, for innovation in the form of new techniques, some for wholly new purposes, and [or innovation in how modern computing systems are to be put to work (as well as in how their results are to be presented to humans), to name some specific areas. The demands for innova- tion, coupled with strong demands for many more people with re- search training to work in a variety of ways, continue to stress research and research training in statistics.

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204 Conclusions These demands go well beyond what statisticians are presently able to cope with. The weakest link is the number of students quali- fied and motivated to enter graduate study. We have discussed this problem in the section on Statistics (page 156~. Continued expansion of research support is of course essential. As the supply of candidates for graduate study improves, research support may well grow at a somewhat greater rate than in the mathematical sciences proper. On some campuses, the independent existence of both statistics and computer science poses problems of cooperation and mutual stimulation. On all campuses, making adequate use of modern com- puting systems in teaching and learning statistics at all levels is of great and growing importance. In this activity, two crucial aspects are: (1) the invention and provision of programming systems that make it easy to do what statisticians have always done to data and, as soon as may be, what statisticians should do to data now that they have modern computing systems, and (2) financing the use of col- lege and university computers by these students. Most teaching in statistics is "service" teaching, in which the guidelines of the Pierce report,3 once followed, should provide the needed financing of computer time. Courses predominantly taken by statistics majors, or by potential graduate students in statistics, are likely to require considerably greater contact with modern comput- ing systems, involving correspondingly increased costs. Means for financing such costs are also urgent. Invention and adaptation of programming systems to meet these needs more effectively is likely to involve elements of research in at least three fields: computer science, statistics, and communication. Support of such research on a selective but diversified basis will be of importance. As in the case of other frontier fields that are both mathematical sciences and something else, competition for fellowships may be viewed as likely to be unfair to statistics candidates unless statis- ticians are unusually well represented on the selecting group, and unless these statisticians are sensitive to the extramathematical abili- ties of the candidates as well as to their mathematical abilities. Changes in practice that make it clear that there are no inequities of this sort would be very worth while. Separate panels for statistics candidates have been suggested; the adequacy of separate panels combining all applied fields of mathematical science where, as noted in the section on Applied Mathematics (page 149), attitudes are de

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Tasks and Needs 205 sirably different than in core mathematics, probably deserves care- ful discussion. COMPUTER SCIENCE Three related aspects of computer science make it very likely that unusually rapid payoff will result from a heavy investment in re- search during the next few years. First, it is a field of strikingly rapid growth in which significant theoretical advances are still being made. Second, because organized complexity is becoming a way of life in the United States, arid because computers are well adapted to deal- ing with such complexity, the solution of many important problems awaits research in computer science. Indeed, there are problems of information processing, vital to many citizens, for which computing systems hold the only foreseeable means of solution. Some striking examples are air and automobile traffic control, management science, information-retrieval problems, and nationwide credit and popula- tion records. Of special national concern are applications in medical science, space science, oceanography, weather prediction, and air- defense problems. However, the solution of these and other impor- tant problems will come only after extensive research in computer science; building machines is not enough. It has been recognized for several years that the design of efficient programs is even more im- portant than the design of machines. Third, the computerization of many applications involves large sums of money, so that an increase in effectiveness of even, say, five percent can mean a big saving in just two or three years an un- usually rapid payoff for research given that the current annual cost of the federal government's acquisition and operation of computers is in the neighborhood of $? billion. There is a critical undersupply of leaders in computer science, which has been described at greater length in Chapter 9. Large sums of money go into supplying computing machines to universities and businesses. It is perhaps not widely realized that money for research in computer science is not provided with anywhere near such gen- erosity. We note that the Rosser reported sets forth certain dollar figures for the support of research in computer science. We feel that these figures should be regarded as minimal. Because of the great needs for both research itself and for training, for leadership through

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206 Conclusions research experience, every effort should be made to support as many good proposals as possible for research in computer science. While the contributions of industrial research centers are very substantial, it appears likely that for the long-term development of computer science a more academic setting is needed. In many of its activities, university work will be related to the practical software activities of manufacturers in much the same way that work in major engineering schools is presently related to conventional industrial activity. That is, it will deal with aspects of the subject that have the greatest generality and permit some degree of conceptualization. Research problems in computer science vary widely in character. Many follow the typical pattern of academic research, while a smaller number, including certain very important problems that should be tackled in university environments, require a combination of the academic concern with fundamentals and the industrial ca- pability for organizing substantial group efforts. The shortage of trained research and teaching staff for academic computer sciences, however it may be structured within the uni- versity, is acute. New, badly needed research people must come from varied backgrounds, aIld the opportunities for sponsored research should include particularly support for young persons of imaginative and experimental bent. The most acute shortage is for research per- sons with expert knowledge of systems programming. We find that industry and nonprofit research laboratories have a very substantial number of strong research persons in computer sci- ence, including some of the very best. As a result, a much larger frac- tion of the key research in computer science is done in industry than for the other mathematical sciences. The reasons are clear: (1) non- academic salaries in these fields, especially for persons just graduat- ing from a university, have been very much higher than those in academic institutions; (2) until the establishment of computer-sci- ence departments, there was little or no recognition of the need for teaching of the nonnumerical aspects of computer science in uni- versities; (3) some of the best researchers in the field have not taken PhD degrees and thus appear less desirable to a university than to an industrial laboratory. It is likely that a substantial number of these persons could be attracted to university teaching and research positions, if these posi- tions were made sufficiently desirable. This would impose a severe strain on university salary scales, a strain that will be uncomfortable but should be faced. Computing has arrived so fast on the national scene that very

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Tasks and Needs 207 little preparation has been made for the organization and financing of research in computer science, instruction in the field, federal financing of fellowships and traineeships, and other matters that have had time to evolve for the older sciences. For this reason it has not been easy to find the agencies with clear responsibility and budget for financing research and education in computer science, though this situation appears to be changing. Research and education In computer science are substantially more expensive than for most of the mathematical sciences, mainly because they involve access to sophisticated computer systems. We estimate that the over-all annual cost per senior research investi- gator will be about the same in computer science as in physics, usually ranging from $35,000 to $160,000, with an average near $60,000. The situation is complex because, for economy's sake, the . computer systems involved should be integrated with an operating computer center. (Such a center, appropriate to a large university or other organization, is likely to have an initial cost of many mil- lions of dollars.) Moreover, the availability of such computing re sources for research and education is usually quite dependent on fundamental accounting policy decisions of the federal government. Some of the issues here are outlined in Appendix D. The funding of urgently needed space for research in computer science is proving unusually difficult. Money for buildings is always difficult to find in a university, and building projects often have to wait several years to acquire enough priority to demand financial backing. Computer science's very rapid growth makes such delays exceedingly serious. OPERATIONS RESEARCH AND MANAGEMENT SCIENCE With the aid of the computer, mathematical methods have been penetrating into every form of human activity. Those that concern the science of decision-making and its application comprise opera lions research or management science. Essentially, mathematical models are constructed of parts of or sometimes entire industrial and governmental systems, and computers are used to determine optimal schedules or plans. In recent years such applications have grown rapidly. One measure of the practical importance is the cost of com- puting for linear-programming optimization. This is estimated to be $5 million per year. Development of new computer programs

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208 Conclusions based on established theory runs about $500,000 per new computer system. It is estimated that industry will require 30,000 trained workers in management science in the next few years. Present financial support of mana~ement-.scienc~ ~timn 0 ~ ~.^ Is geared to only a fraction of this need. Research and training is di- rected toward analysis of the mathematical structures of typical systems to be optimized. This has stimulated such mathemn~i~1 air ~;~1 :~ ~ _ ~ 1~ ~ clpllnes as linear, nonlinear, and integer programming; network. graph, and matroid theory; queueing, stochastic processes reliabil- ity theory; dynamic programming and control theory Graduate programs have expanded rapidly. An example is the PhD program in operations research at Stanford, which has grown from zero to 50 graduate students in the last six years. Graduate programs are pres- ently limited by the amount of funds available for f~ellow~hinc epoch ~ v ^ ^ ~ ~ v ~ ~ ~ ~ ~ ~ A _ _ ~ _ 1 1 _ ~ ~ _ _ _ ng teilowshlps, and research assistantships. There is a need for greatly increased support of postdoctoral fellows and at least partial support of people in industry and in government who wish to further their knowledge by spending one or two years at a university. There is need of money for experimentation on computers, par- ticularly for testing out new proposals for large-scale system op . . tlmlzatlon. OTHER AREAS Most of the smaller specialized areas, such as mathematical genetics; mathematical psychology; transmission, modulation, and coding theory; and mathematical economics, have been at least reasonably well supported as parts of the respective fields of application in these examples: genetics, psychology, engineering, and economics. This has come about because the value of such work to the field of appli- cation has been relatively clear. Since the value of such mathematical areas usually becomes more rather than less clear, research support in these areas will probably increase at a substantial rate, as it should. Moreover, since workers in these areas are usually integrated into academic departments concerned with the discipline of applica- tion, a fair number of students with appropriate interests are likely to come in contact with the fields. Thus, graduate-student recruit- ment is likely to be fairly satisfactory. Two problems may prove important in individual areas. Depart- ments of mathematics may not adequately recognize the need, at

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Tasks and Needs 209 both undergraduate and graduate levels, for new "service courses" of a less usual character. (In certain of these fields, calculus, and the analysis that grows out of it, plays a relatively minor role. Early training in various areas of "discrete mathematics" may be far more important.) Proper evaluation of all facets of the ability of fellow- ship candidates requires special care. In other such areas, particularly but not exclusively in those at an early stage of development, other problems may arise. Two seem worthy of notice here: Mathematics may serve as an unwarranted sanction for such projects as the writing of a book on "The Mathe- matics of X" by a man who gives no evidence of being a good mathe- matician and who has had no real contact with modern work in X. The converse problem arises when either the mathematics applied or the problem to which it is applied appears too simple, although real gain can come from the application. We are fortunate that this was not the case, for example, in the recent application of mathematical thinking to the description of all possible kinship systems satisfying certain axioms, recently carried out independently by both anthro- pologically and more mathematically oriented workers. Instances of things appearing too simple are, of course, but specific instances of work that "falls through the crack" by appearing insufficiently interesting both to those working in the discipline and to mathe . - matlaans. No simple prescription can be given to meet these problems. Joint evaluation, of both research projects and fellowship applications, by mathematicians and by workers in the discipline can be of great help, but it is vitally important that those from each of these inter- ests be clearly perceptive of those elements of the other sort of activ- ity that are least natural in their own sort. Mechanisms that already exist can be of great help in wisely channeling support into new areas and interdisciplinary connections, where intelligent chance-taking is very important for future progress. The Social Science Research Council has a long history of sparking and nurturing important developments in all the behavioral sciences and can be counted on to do well in making selections in all the less highly exploratory areas. The Mathematical Social Sciences Board, though in existence for only three years, has already had very great influence on the rate of progress in the more highly exploratory areas. Both need increased support in their areas of special com- petence. In individual universities and colleges, much can be done if indi

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210 Conc/~3foni vldua1 departments in any of the frontier mathematica1 herds me encouraged to spread ~ little outside their nominal boundaries not only cooperating with Workers concerned in various special arena but becoming involved in loins research or serving as inida1 homes, Folly or in park for mathematically oriented faculty not as well housed in their disciplinary departments