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- Lie The Mathematical Sciences in Society~s Service For wise policy decisions, the tasks and needs of the mathematical sciences. discussed in the previous chapter, must be viewed in proper context. A major element of this context is the whole process by which the mathematical sciences contribute to society's ends. To understand this in full detail would be extremely difficult probably impossible but a general overview can be given in reasonable space and with some clarity. THE MATHEMATICAL POPULATION The most significant fact about the people and institutions that employ the mathematical sciences, at one level or another, in our society is their number and diversity. Exact figures on the number of our people who have had at least two years of high school mathe- matics might be hard to find. Rough but probably adequate estima- tion leads to a figure of perhaps one fourth of the nation's adult population. With the increasing complexity of society's mech- anisms, institutions, and interrelations, leading inevitably to greater public education, this fraction is in the process of slowly moving # According to reference 44, 47.7 percent of the U.S. population over 14 years of age has completed four years of high school and 70.5 percent has completed one through three years of high school. Also, 17.6 percent has completed one through three years of college and 8.1 percent of the population over 14 years of age comprises college graduates. 211

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212 Conclusions upward. It may well reach one third, one half, and even two thirds as the years pass. A fundamental education in mathematics extends through high school and about two years of college. Perhaps 8 to 10 million of our people have this foundation, which required 9 to 11 years of study of mathematics. Not all these people make regular use of what they have learned, but a large fraction of those engaged in physical sci- ence and engineering do. Altogether, the number who use 11 to 13 years of mathematics, at least occasionally, probably runs between 1 and 2 million. Now we look at those who work in mathematical sciences in some way or another. The two largest groups are somewhat over 100,000 high school teachers of mathematics and 200,000 computer pro- grammers, to which we should add the roughly 50,000 members of professional societies dealing with mathematical research, college teaching of mathematics, statistics, computer science, operations re- search, and management science all told, approximately 350,000 to 400,000 people. Another group that needs specific notice includes many workers in agricultural, biological, and medical research, many concerned with production or marketing in industry, most research psycholo- gists, and many workers in other fields of behavioral sciences. Some 50,000 to 100,000 such people use statistical methods in their pro- fessional work. Many look actively to research in statistical method- ology to provide better tools for their use. Finally, toward the tip of the pyramid there are about 7,000 PhD's in the mathematical sciences, among whom over 1,000 are active innovators. This relatively small group bears the responsibility for practically all research in mathematical sciences, for all research education, and for directing much of the college-level education. The fraction of our population contained in any of these groups can confidently be expected to increase. Society's increasing com- plexity and the increasing complexity of its individual mechanisms and institutions will assure this. To estimate the rates of increase, however, is exceedingly difficult, and the figures we now give are only a very rough guess. It seems that the number of people with two years of high school mathematics increases at about 4 percent a year. The pool of those with two or more years of college mathe- matics grows perhaps 8 percent a year-somewhat more rapidly- while pools of those who use college mathematics may grow as much as 12 percent a year. The numbers of those who use mathematical

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The Mathematical Sciences in Society's Service 213 science as a main professional component Stow at diverse annual rates: high school teachers of mathematics at perhaps 5 percent, computer programmers at a rousing 30 percent, professionals at perhaps 14 percent, overall at, say, something over 10 percent. The annual rate of increase of new PhD's over recent years has been 18 percent; as already stated, we believe that active investigators are increasing at about the same percentage Per year. The greater rates of increase at the upper layers of the pyramid are inevitable consequences of the increasing subtlety and complex- ity of society's demands. An 18 percent annual increase at the re- search level is no more than would be expected from the other parts of the picture. We cannot be sure whether the recent rates of growth will or will not continue for the near future. The rate of educating scientifi- cally trained people in each specialty is controlled by students' choices and available facilities. These factors are constantly chang- ing. Over the post-Sputnik decade, enrollments in advanced under- graduate courses in mathematical science grew rapidly, but with a possible tendency to flatten out. This apparent flattening out may be a short-time fluctuation, may represent limited facilities in terms of faculty, may reflect the absence of undergraduate programs in applied fields, or may be due to a partial reorientation of student values from technical and scientific to social concerns. Graduate enrollments in mathematical science, particularly at research-trainir~g levels, are still growing actively, but their future behavior is equally uncertain. If research training in mathematical science continues to expand rapidly, it will be either because of a continuing atmosphere of general public approval or because we shall have opened the way to graduate work to a wider variety of students by removing social obstacles, by establishing a greater di- versity in undergraduate programs, or by broadening understanding and knowledge of mathematical sciences among all college teachers of mathematics. All these reasons for continuing growth are signifi- cant and appropriate. If they produce a continuing expansion, the nation has important tasks for all those who will receive research . training as a consequence. ~ At the same period the engineering numbers flattened out, physics showed moderately strong increases followed by flattening, geology suffered a severe de- cline followed by a partial recovery, and biology grew, first moderately then more rapidly.

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914 MATHEMATICAL STRATEGY Conclusions To assess the significance of research in mathematical sciences from the national point of view, it is important to remember that, as already described in Chapter 3, the strategy of research in mathe- matics is rather different from that in other sciences. In nuclear or high-energy physics, for instance, a few problem areas are regarded as crucial at any given time. These are confronted in great force by many people. Substantial numbers of groups of reasonable size, each necessarily well supported by machines of various kinds, attack the same problem. This strategy has brought rapid progress to these areas of physics, probably in part because of the relative narrowness of their research objectives. Even physics in general aims at understanding only one universe, under only one system of laws. Mathematical research, seen from society's perspective, has a broader objective: the full development of concepts, results, and methods of symbolic reasoning that will apply to as many as pos- sible of mankind's diverse problems, includin~vitally but far from exclusively the problems arising from the progress of physics. The development of concepts and theories motivated by the needs of mathematics itself must be part of this objective since experience shows that these may well become crucially important for applica- tions. As a result, mathematics must contribute to understanding diverse situations under widely different systems of laws. Mathematical sciences must eventually travel many roads. All past experience, from the dawn of history to recent times, teaches us that the ultimate applicability of a mathematical concept or technique can hardly ever be predicted, that only quite short-range forecasts can be trusted, and that calls for massive effort at one point, at the expense of efforts at other points, should usually be resisted. In this situation, mathematical sciences proceed by a large num- ber of small independent research efforts, often conducted by single individuals or by small groups of men. A large variety of problems is attacked simultaneously. The choice of problems to work on, as in all sciences, is one of the things that determines the success or failure of an investigator. But the mathematician and the mathe- matical scientist have, and need, great freedom in making these choices. This strategy has proved successful. It involves dispersal of forces and active work in many seemingly disconnected fields. Thus there

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The Mathematical Sciences in Society's Service 215 have been repeated periods of apparent overspecialization when mathematicians seemed to be drawing too far from one another (most recently in the 1930's and 1940's). Every time this has hap- pened, however, an apparently inherent unity of mathematics has shown itself again through the appearance of new and more gen- eral concepts and approaches that, as in the present decade, have restored to mathematics much more in the way of unity than had seemed possible a few decades earlier. TRANSFER TIMES When the British steel industry was Denationalized in 1967, the new chairman warned the British people not to expect too much too soon, saying that, in a "capital-intensive" industry, only slow change could be expected. Our nation's system of mathematical service is an institution in which change must be even slower. Indeed, this is a "training-intensive" institution; its greatest investment is in people with years of training. The building of steel plants can be greatly accelerated, but no large group of people can be given 10, or 15, or even 20 years of continuous training in appreciably less than that number of years. As a language, used for communication both with others and with oneself, mathematics shares with the language of words the need for a long and arduous apprenticeship. Including mathemati- cal training in elementary and secondary school, a college graduate majoring in mathematics has typically studied mathematics about three times as long as a college graduate majoring in a science has studied his or her science. The proper time scale for thinking about our society's system of mathematical science is not merely long, as it must be for all the sciences, but very long. From society's viewpoint, the largest reason for supporting self- motivated research in mathematical sciences is the continuing im- pact of the resulting innovations, first as immediate mathematical applications and then more broadly. How fast ought society to ex- pect the results of innovation to be transferred? Surely not in a day or a week or a month. But in one year, or three, or ten? We have stressed the differences between the strategy of most mathematical research and that of the other sciences and tech- nologies. The individual character of the work and the difficulties of forecasting where progress will prove most important have led to

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216 Conclusions a spreading out of attention over a wide variety of problems. (Im- portant mathematical problems, like many of those posed by David Hilbert in a celebrated 1900 address, are often under attack for several decades before their final solution.) As a consequence of its implicit strategy of pressing ahead wherever it seems that reasonably valuable ground can be gained, mathematical science sometimes prepares the way very far in advance. That important uses should follow discovery by decades, often by several decades, should neither be a cause for surprise nor a reason for criticism. It is a state of affairs intrinsic in an efficient use of human resources and the facilities and money that support them. A bare trace of our progress in inner-directed research will come to use in one year. A little more will come in three years. A sub- stantial fraction of what will contribute outside mathematical sci- ences will have begun to make its contribution in 10 years, but only a substantial fraction and only as a beginning. We will do well to do whatever we reasonably can to speed up the process of transfer to use, for there are real gains to be had if we can, but we dare not delude ourselves that great gains in speed can be had by some drastic rearrangement of activity and interest. The mathematical strategy of widespread attack in small parties is weI1 adapted to both the subject and the demands for innovation laid upon it by the diverse needs of society: long delays in transfer to use are an inevitable consequence of this strategy. In planning support of mathematical sciences, especially support of inner- directed research, we must take the long view if our programs are to contribute to the demands that society will make at times spread through the future. We do many other things today with a hope of social gain over decades. The elementary and high school education of youth is to be of value to them and to society, not just for a decade or two, but for four or five or even six decades. All our affairs cannot be con- ducted in the way a field of corn is tended, plowed under this year but reseeded for next year. Innovation in science is more like an apple tree; 10 to 25 years are needed for the crop to return, many- fold, the effort of planting, grafting, and cultivation. Mathematical sciences call for time scales even longer than do other sciences.

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The Mathematical Sciences in Society's Service 217 EMERGENCIES There is an exception to the usual need for long transfer times typical in the mathematical sciences the use of creative mathe- maticians during emergencies. World War II generated many technical emergencies. Mathe- maticians usually involved in inner-directed research responded to many calls: how to conduct antisubmarine warfare, what principles to use in fighter and bomber gunsights, and many questions in ballistics, radar, atomic weapons, and cryptography. The crisis was clear; insight, knowledge, and skill were freely mobilized. Many concrete problems whose solution was explicitly demanded were attacked powerfully and effectively by mathematicians, precisely because of their professional ability and training in thinking into the heart of a problem and seizing on its essentials. And the free- wheeling instinct of the self-motivated researcher played its part. Another emergency arose slowly and imperceptibly in mathemati- cal education, primarily in elementary schols and high schools. The urgent need for reform became apparent about 15 years ago. Leader- ship in meeting this emergency came in a large measure from uni- versity and college mathematicians active in inner-directed research. (There are various opinions in the mathematical community about the success of the reform movement thus far; there is no dispute about the necessity of curriculum reform.) When the next major change in mathematical education comes, leadership will again have to be drawn from those concerned with inner-directed research. Mathematical scientists who give most of their time to inner- directed research are an important national resource in emergencies. This sort of resource serves special purposes in an emergency, but it cannot be used for these purposes steadily. This type of value has to be a by-product. As long as emergencies continue to arise at the usual rate, society can count on this resource created by basic mathe- matical research as one sort of return from its investments in re- search training in mathematical science. It can do this, however, only by using research-trained people quite differently most of the time. THE LEVEL OF INVESTMENT We are now in a position to ask whether the total investment in basic mathematical research has been at a reasonable level. Let us

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218 Conclusions follow our discussion of Level and Sources of Support (see page 163) and assume federal support of basic research in the universities as $35 million in 1966. At an 18 percent rate of growth, the total through all the past would be about six times as large as the present annual amount, amounting to $200 million. Allowance for non- federal support and for slower rates of growth in the past might raise this to $250 million. If we believe in compound interest, and insist on totaling up present values of all sums spent in the past, this figure would be roughly $300 million. Thus, if we want a total investment picture, we can say that the whole U.S. investment in research in the mathematical sciences to date is about as much as we will spend as initial capital investment on the new super-large "atom-smasher" now approved. When we look at this investment in mathematical research as contributing in diverse and important ways to the effectiveness of the whole na- tional system of mathematical service, where some of these ways have begun, others are beginning, and others will start at times spread forward through decades, and where most contributions will continue for a long time, our investment seems conservative and cautious, perhaps disproportionately small. GROWTH CANNOT BE FOREVER An 18 percent a year increase means doubling every four years. A 10 percent annual increase means doubling in less than 10 years. Such doubling cannot continue indefinitely. Not only mathematical science but all science and all technologies with growing research sectors must face the need for an ultimate tapering off. Neither the fraction of gross national product that can be devoted to research nor the number of people potentially capable of becoming research investigators can increase indefinitely. At the moment, the need for the innovations of mathematical- science research is large and growing. These needs would requir continuing rapid expansion for the near future, even without the necessity of continuing the mathematical teaching of nonprofes- sionals. But what of the day when tapering off of growth becomes appropriate? What will be the environment, the pressures, the appropriate adjustments? Education above the high school level is in transition. The clas- sical division between undergraduate and graduate work is more and more clearly seen to be at an inappropriate place. We may be

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The Mathematical Sciences in Society's Service 219 moving toward a three-part scheme of scientific education, consist- ing of (1) the present freshman and sophomore years, (~) the last two undergraduate years and graduate study through the compre- hensive examination, and (3) thesis research and postdoctoral train- ing. It will be after this pattern has appeared more clearly that we shall have to face declining rates of increase in supported research. At present, five sixths of new PhD's follow career patterns that do not lead to federally supported research. If our needs for profes- sionally trained personnel were to cease to rise, adjustments would be fairly simple. If, as seems more likely, the future demand for professionally trained personnel increases much more than in pros portion to the amount of sponsored research that the nation can afford, there will have to be major readjustments. Direct contact with research leadership contributes greatly to education in mathematical sciences below the research level. In the future, research leaders may not be able to contribute much time to this activity. Once this situation arises, forms of mass communi- cation, both television and film, will become important in the process, and there will have to be real innovations designed to pro- vide face-to-face contact with many students. Inevitably, the selection of a principal investigator for research support will focus increasingly on the number and quality of his PhD candidates, as well as on the number and quality of his own contributions. As a consequence, it will become clearer that academic tenure exists primarily for teaching rather than for research sup- port. Such times, when they come to each science, will be times of change indeed. There are many reasons, however, why they will not come to all sciences together. The bigger the initial budget, the sooner its doubling will bring it to a point where things must be done. The larger the system of activities that research in a science supports, the larger the total the nation can wisely spend on it. Today, academic mathematical research receives a relatively small amount of support less than three cents out of every dollar spent by the federal government for research at universities. Yet this mathematical research is the leading wedge of a very large national effort. Thus, reduction of growth rate in mathematical research ought, wisely and advisedly, to take place only after similar reduc- tions have been made in many other fields. Thus the mathematical sciences can expect much guidance by the time they are faced with making their own adjustments.