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6 Examples of Mathematics in Use Many years ago Auguste Cowte claimed that a science is a science only insofar as it is mathematical. The mathematization of physical science has been going on for centuries, that of the life and be- havioral sciences for a shorter time. Engineering, which is a tech- nology based on physical science, has always used mathematics as an essential tool. The mathematization of a wide variety of other tech- nologies is in process. Accordingly, an exhaustive review of the penetration of mathematics into various areas of human endeavor would require volumes. In this chapter we describe a few typical ex- amples: physics a science completely mathematized almost from its very inception; engineering design a fully mathematized tech- nology; mathematics in the newer environmental sciences specifi- cally, numerical weather prediction; economics in which the penetration of mathematics is about a hundred years old; the tech- nology of management and operations in which mathematization is a World War II development. Clearly we have omitted many important examples. See, for some specific instances, the essays by Cohen on mathematics in biology, by Lederberg on some uses of mathematics in chemistry, and by Harris on mathematical linguistics.7 The degree of mathematization, the sophistication of mathemati- cal tools used, and the lasting intellectual value so far achieved by the use of mathematics vary widely from field to field, as we shall comment at the close of this chapter. 101

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102 ~ he State of the Mathematical Sciences MATHEMATICS AND PHYSICS Physics is an experimental science concerned with the material world around us. The aim, as physicists define it today, is to describe and to correlate the multitude of experimental phenomena in terms of theoretical concepts formulated in the language of mathematics. Why natural phenomena should be describable in the language of mathematics is a matter of controversy. (For instance, E. Wigner en- titled a lecture "The Unreasonable Effectiveness of Mathematics in the Natural Sciences.") Yet it is indisputable, and indeed usually taken for granted, that natural phenomena have been so described with brilliant success. Because physics deals with quantitative measurements, mathe- matics comes into physics naturally as an aid for computation and as a tool for the logical operations in theoretical developments. The traditional main branches of mathematics algebra, analysis, and geometry have been extensively used in many fields of research in physics in this way. As soon as computers were developed, physi- cists immediately began to use them to great advantage, to aid in data processing as well as to solve numerical problems. While mathematics plays an important role in physics in the man- ner just described, it plays, at the same time, a far more important role at a more fundamental level. In fact, mathematics supplies many of the basic concepts that physicists use to describe natural phenom- ena. For example, the abstract mathematical concept of noncommu- tative multiplication lies at the foundation of quantum mechanics. Non-Euclidean geometry is the very starting point of general relativ- ity. There are physicists who believe that analytic continuation is a mathematical concept needed to describe the physical principle of causality. As one reviews the development of physics through the centuries, starting from the early studies of astronomy and Newtonian me- chanics, proceeding through the nineteenth century formulation of electromagnetic phenomena and of the theory of heat and thermo- dyr~amics, and then to the modern development of relativity, quan- tum mechanics, and high-energy physics, one is struck with the in- creasingly abstract and sophisticated nature of the mathematical concepts that it was necessary to introduce for the description of natural phenomena. Such observation was undoubtedly behind the remark of the late British physicist, Jeans, that God is a mathema- tician. Some examples of the sophisticated mathematical concepts

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Examples of Mathematics in Use 103 that have been introduced into physics in recent years are found in the essays by Dyson and Wightman in reference 7. We quote from an article by the great physicist P. A. M. Dirac tProc. Roy. Soc., 133, 66 (1931~: The steady progress of physics requires for its theoretical formulation a mathematics that gets continually more advanced. This is only natural and to be expected. What, however, was not expected by the scientific workers of the last century was the particular form that the line of advancement o the mathematics would take, namely, it was expected that the mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundations and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time considered to be purely fictions of the mind and pastimes for logical thinkers, have been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and that ad- vance in physics is to be associated with a continual modification and gen- eralization of the axioms at the base of the mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation. Many physicists believe that the central problem they face today, namely the structure of atomic nuclei and their constituent parts (also known as high-energy physics), may well be solvable only upon the introduction of mathematical concepts not hitherto used in physics and perhaps as yet unknown to mathematicians. Be this as it may, it has been repeatedly demonstrated that a sense of form and an appreciation of elegance, abstraction, and generalization, which are the hallmarks of good mathematical development, are often also the characteristics of the new breakthroughs in physical insight. In fact, what one refers to as physical ideas often derive from properties of abstract mathematical concepts, which turn out to have wide- spread and deep-rooted applicability in natural phenomena. In re- viewing the interplay between mathematics and one branch of physics, M. J. Lighthill A. Roy. Aeronaut. Soc., 64, 375 (1960) ~ observed that an important task of mathematics is to generate new physical ideas, that is, . . . ideas which have been originated by mathematical investigation but which later become amenable to almost exclusively physical description, and whose properties, although first derived mathematically, become familiar and are commonly described in purely physical terms. The value of physical ideas in practical work, of course, is their elasticity. Provided that they are sound ideas, such as those thrown up as the genuinely appropriate physical

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104 The State of the Mathematical Sciences description of the mathematical solution of some well-defined class of prob- lem, they usually show a splendid capacity to stand up to distortion of the problem, and indeed to radical changes and complication in its conditions, and still give the right guidance about what needs to be done. It is worth remarking that this in fact holds for practically all appli- cations of mathematics. The relationship between physics and mathematics is by no means a one-way street. While physics uses mathematical concepts, mathe- matics draws inspiration and stimulation from the physicists' need for new mathematics. The invention of calculus, of differential geometry, of the ergodic theory, all represent mathematical develop- ments stimulated by physical problems. We quote from a recent report by physicists (reference 1 1, page 162~: Through centuries of intimate contact, theoretical physics and mathematics have interacted strongly to their mutual benefit. Theoretical physics uses the concepts developed in mathematics to formulate descriptions of natural phenomena. Mathematics, in turn, is stimulated in its direction of develop- ment by the problems posed by physics. In recent years, the influence of theoretical physics on the development of mathematics seems to have weak- ened. However, there are still many conspicuous examples of mathematical development influenced by physics: for example, the theory of unbounded operators, the representation theory of noncompact groups, and the theory of distributions. It would be rash to believe that natural phenomena would not again, in the future, as in the past, serve as the fountainhead for impor- tant directions of research in mathematics. The tendency for core mathematics as a whole to move away from physics may be an inevitable characteristic of the mathematics of this century. ~See, e.g., M. Stone, "The Revolution in Mathematics," American Mathematical Monthly, 68, 715 (1961~.] It would be detrimental to both disciplines if this tendency were allowed to take the extreme form of isolating the mathematicians and physicists from mutual intellectual contacts. To forestall such eventuality it is highly important that at the undergraduate, graduate, and post- graduate levels students be given the chance to be exposed to the exciting basic developments of each of the two disciplines. MATHEMATICAL SCIENCES IN ENGINEERING The uses of mathematics in engineering represent one of the best recognized as well as one of the most important manifestations of the

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Examples of Mathematics in Use 105 general mathematization of our culture. Whether one looks at rela- tively old fields, like civil and mechanical engineering, or relatively new ones, like nuclear technology or electronics, one finds a steady increase in the amount and sophistication of the mathematics used. Information theory is a well-known example. It involves very deep mathematical problems, but its dramatic development was stimu- lated by the practical need of communications or electronics engi- neers for a measure of the job of transmitting a given class of messages through a given medium. Modern developments in aero- dynamics depend on the desire to build aircraft and missiles flying at higher and higher speeds. It is found, as expected, that the more advanced the technology, the more sophisticated are the basic con- cepts involved, and the more they depend on mathematics. It is often impossible to understand the concepts used by engineers (even such basic ones as impedance matching, reduction of drag by interference, and subharmonic resonance) without using mathematics. One form that increased mathematization often takes is develop- ment of more precise theories to take advantage of concurrent ad- vances in other directions. For example, a margin of safety of 4:1 has frequently been used in structural design. Such a factor of safety would be utterly unworkable in the design of most missiles; the missiles would be too heavy to get off the ground. Margins of safety as low as 20 to 30 percent are sometimes used. To live with such factors one must have a much more accurate knowledge of aero- dynamic forces, natural modes of vibration in the body, and stress distribution than he would otherwise need. Of course, one must also have very good control of the materials and processes that make up the final structure. However, this merely illustrates the common fact that to take maximum advantage of improved technology, advances on the theoretical side and on the physical side must go hand in harld. New mathematical techniques, particularly in conjunction with computers, are frequently used also in situations where old-fashioned "hand" methods would be too slow or too laborious. For example, the analysis of mathematical models is extensively used in modern civil engineering-an analysis made possible by the existence o large-scale computing machines. The mathematical formulation often leads to linear programming. Prediction of satellite orbits, guided and controlled, is another activity involving careful mathe- matical formulation and extensive numerical calculations. Once a computational program is well designed, the work is routine and

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106 The State of the Mathematical Sciences repetitive, but the original formulation often involves deep insight, which can be acquired only by mathematical analysis. The design of gas turbines is another example of the use of computers. :Fluid mechanics, one of the best-established areas of applied mathematics, is important in several engineering fields. The noise produced by jet aircraft and the shock waves associated with super- sonic flight must be understood in detail before remedies and im- provements can be suggested and designed; combustion instability leads to critical problems in the development of rockets; and sys- tematic progress in minimizing the devastating effects of tornadoes cannot be expected until the understanding of atmospheric dynamics has greatly improved. The field of electronics and communications is particularly rich in the applications of mathematics. Some of these are like repre- sentative problems in other engineering disciplines. For example, lumped circuit theory, both linear and nonlinear, has many points of resemblance to similar areas in mechanics. For continuous media, many problems in electromagnetic field theory are at least broadly similar to typical problems in fluid mechanics or elasticity. The de- sign of radio antennas, for example, can be based on wave-interfer- ence effects similar to those that figure in airplane and ship design. More generally, the study of the propagation of radio signals in- volves a variety of special problems that have challenged such "Teat mathematical physicists as Sommerfeld and such Great mathema- ticians as Hermann Weyl. In recent years, still more complicated mathematical questions have arisen in the study of plasmas, a field of increasing importance in many areas of electrical engineering. The most fundamental applications of mathematics to electronics and communications, however, are found at the conceptual level. These normally stem from situations in which mathematics offers the best language in which to express both the original engineering problem and the final result. Such situations arise because electricity itself is an intangible that can hardly be described except through mathematics, and because electrical systems are frequently so exten- sive and complicated that considerations of mathematical regularity and simplicity must be paramount in laying them out. Perhaps the most elementary example is furnished by the mathe- matical concept of an impedance. This, though engineers seldom realize it, is strictly a mathematical artifice, a nonphysical "imagi- nary" quantity by means of which the real currents and voltages in which the engineer is finally interested can be calculated con

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Examples of Mathematics in Use 107 veniently. Its use becomes a necessity when we deal, as we frequently must, with circuits containing dozens or hundreds of elements. The use of transform methods, established in communications engineer- ing for many years, provides another example. No communication engineer could possibly deal with the variety of signals he must meet in practice without the help of the concept of a frequency spectrum. In more recent years, concepts from mathematical logic have turned out to be an important basis for switching and computer circuits. The best example, however, is probably furnished by information theory. The identification of communication engineering and mathe- matical logic as two systems that are both concerned largely with the manipulation of arbitrary symbols according to formal rules, made possible by information theory, tremendously broadened the horizon of the communication engineer and at a stroke opened to him vast areas of mathematics as a source of ideas for particular ciphers and coding schemes. It may happen that the same engineering objective may best be served first by one physical technique and then by another, generat- ing new mathematical problems as they evolve. An example is fur- nished by surveying, which goes back to classical times and may be regarded as the generator of both geometry and trigonometry. The basic techniques in this case were, of course, optical. With the inven- tion of other instruments such as the telescope, the sextant, and the chronometer a few centuries ago, and greater interest in accurate navigation, still more complicated problems of the same nature emerged. In modern times the basic technique is electronic, as rep- resented by radio networks like LORAN or by navigation satellites. Here the important questions turn on analysis of the coherence and other statistical characteristics of the signal. Still more complicated problems of statistical signal analysis occur in connection with the "surveying" of the solar system by means of radar signals bounced off the nearer planets. Almost the converse situation occurs when several branches of en- gineering may be involved in a single system. Here the fact that they can all be dealt with mathematically may be all that holds the situa- tion together. An example is furnished by missiles, which typically involve not only aerodynamics and structural mechanics, but chem- istry through the propulsion system and ablative coating and elec- tronics through the control and guidance equipment. They may all be involved if, for example, the missile is required to execute a vio- lent maneuver, and it is only the fact that they are all relatively well

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108 The State of the Mathematical Sciences understood mathematically that makes it possible to design the system as a whole. In general, modern engineering systems are too large, complex, and precisely integrated to be designed by empirical test. They must be thoroughly analyzed mathematically in advance of testing in order to obtain reasonable assurance of success. These diverse applications include many unifying mathematical threads, which deserve to be identified as areas of applied mathe- matics worth studying on their own account. Once this basic level has been reached, the knowledge gained applies to other branches of technology. Thus problems of transient loading in mechanical sys- tems benefit from techniques originally developed for communica- tions, and chemical and aeronautical engineers work on problems of blood flow and contribute to the advance of medical science. AN EXAMPLE FROM THE ENVIRONMENTAL SCIENCES: NUMERICAL WEATHER PREDICTION The environmental sciences include earth sciences, oceanography, atmospheric sciences, telecommunications sciences, and aeronomy. Some of the most difficult and sophisticated mathematical problems in the environmental sciences arise from efforts to study the at- mosphere and the oceans by means of mathematical models in the form of deterministic fluid systems. This leads to nonlinear partial differential equations subject to rather general boundary and initial conditions. The most successful attempts to treat these problems have involved the use of high-speed computers. Of special day-to-day importance are the numerical methods of weather prediction now in regular use. The National Meteorological Centers provides, on an operational round-the-clock basis, guidance material to all the national forecast services, as well as to foreign services, under the auspices of the United Nations. The guidance material consists of large-scale wind and weather patterns over the entire northern hemisphere. The basis for this is the approximate numerical solution on large electronic computers of hydrodynamic and thermodynamic partial differential equations constituting a mathematical model for the behavior of the atmosphere. # Of the Weather Bureau, Environmental Science Services Administration, U.S. Department of Commerce. The Center was established in 1954 in Suitland, Mary- land, expressly for numerical weather prediction.

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Examples of Mathematics in Use 109 The general idea of mathematical weather prediction dates from the first few years of the twentieth century. Detailed pioneering proposals and experiments in weather prediction through the ap- proximate solution of relevant hydrodynamic and thermodynamic equations go back to the British scientist L. F. Richardsoni2 in the early 1920's. Richardson's forecasts were not successful, the most fundamental reason for this being his violation of a then unknown stability criterion for numerical processes in the solution of partial differential equations (discovered in 1928 by Courant, Friedrichs, and Lewy). Another reason for the failure of Richardson's forecasts was in- su~cient data. It was not until the 1930's and early 1940's that em- pirical observations began to approach the frequency and detail needed for successful attempts at weather prediction on a mathe- matical basis. In particular, the new upper-air observational network developed during the 1930's elucidated the dynamics of the so-called jet stream, a great meandering river of air, five to eight miles high and hundreds of miles wide, which loops completely around the northern hemisphere at middle latitudes. The jet stream, together with important vorticity-conservation ideas in its mathematical modeling, has turned out to be the key to large-scale weather pre- diction in the northern hemisphere. A third difficulty that would almost certainly have defeated Rich- ardson's original 1922 proposals for numerical weather prediction was the lack of fatalities for high-speed computation on the intricate and massive scale needed. He had visualized a giant "weather fac- tory" staffed by an estimated 64,000 human computers busily en- gaged in obtaining approximate solutions for the appropriate partial differential equations of hydrodynamics and thermodynamics. Look- ing back, experts today feel it would hardly have been possible to organize such an operation to produce timely weather forecasts. This is, however, the kind of task for which the modern high-speed electronic computer is ideally adapted. The first such computers ap- peared in the late 1940's. {ohn Von Neumann played a vital role both in the logical design of these computers and in their first suc- cessful use in weather calculations in 1948. During the succeeding years, computers have become steadily more powerful, in speed, versatility, and capacity. Thus, weather calculations that required 24 hours in 1948 were being performed in five minutes by 1951. These and subsequent advances in computer design have been paralleled by advances in research on numerical

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110 The State of the Mathematical Sciences weather prediction. The relatively simple equations defining the first models for the atmosphere have been extensively refined and elabo- rated. The early models carried no detail of the vertical structure of the atmosphere, but rather orally a vertical average of its motion. By mid-1962, however, a system allowing for three vertical levels had become the principal operational model at the National Meteoro- logical Center. This in turn was replaced, in mid-1966, by a six-level model. In addition, over the past 20 years considerable advances have been achieved in the methods of numerical analysis used in weather prediction. Many aspects of the numerical techniques are still highly unsatisfactory, however, imposing severe limitations on the kinds of simulation that can be attempted. It is often difficult even to dis- tinguish between distortions introduced by the numerical methods and those resulting from deficiencies in the mathematical model. Thus the future will continue to present challenging and thorny problems in this field. MATHEMATICAL SCIENCES IN ECONOMICS The explicit penetration of nonelementary mathematics into eco- nomics began about a century ago with the introduction of rates of change in terms of marginal ratios and elasticities. For many decades, economists carried out derivations in words or graphical patterns rather than by formulas. These derivations did, of course, involve chains of symbolic reasoning and were thus intrinsically mathemat- ical, though not always recognized as such. The twentieth century has seen a rapidly increasing use of formulas and of mathematical results and theorems (see the essay by Klein in reference 7~. It has rather steadily been true that about half of the papers appearing in the principal economics journals would have been rejected ten years earlier as "too mathematical." The earlier uses of mathematics in economics centered around the use of the calculus as a means of describing interrelationships. The emphasis slowly changed to prob- lems of maximization or minimization, originally of smoothly vary- ing functions with no or few constraints. Problems dealing with the appropriate behavior that which ra- tional participants "ought," or clients should be advised, to exhibit -were intermingled from an early date with problems concerning the effects of ideal mechanisms, such as free competition, in distrib- uting goods and services. Both classes of problem involve optimizing

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Examples of Mathematics in Use allocation of resources. Today, the mathematics of optimized alloca- tion, control, and decision, discussed in Chapter 5, has many appli- cations in economics, and economists as well as mathematicians contribute to it. Alongside mathematical economics is the very active area of econometrics, in which statistical tools, many of them developed for the purpose, are essential. Here, too, both economists and statis- ticians are contributing to the development of new techniques and to fresh understandings of old ones. As in so many other areas, large parts of economics are being al- most revolutionized by the availability of modern computing systems to store and digest quantities of data and to solve complex problems either directly or by tentative approximation. Large areas of economics are now highly mathematized. For ex- ample, there are mathematical theorems concerning the existence of competitive economic equilibrium. The study of business cycles leads to systems of differential equations similar to those occurring in dy- namics of physical systems. Indeed, many of the most respected econ- omists are mathematically oriented. Key economists know as much of the details of modern control theory and what is known about the stability of nonlinear systems, to take two examples, as do all but the most specialized mathematicians. It is therefore not surprising that most members of the President's Council of Economic Advisers and its professional staff have been trained as mathematical economists. We recall also that l. M. Keynes, the father of modern economics, had been trained as a mathematician. Education in economics is now highly mathematically oriented. In most major departments in the United States, all economics PhD's are required to learn calculus, selected topics in advanced calculus, the elements of linear algebra and probability, statistical inference, and econometrics. As a foundation for everyone, this is an impressive array, especially in contrast to the situation one, two, or three dec- ades ago. In many leading graduate centers, additional mathematics courses are replacing the requirement of a second language for PhD candidates. MATHEMATICS IN FINANCE AND INSURANCE The vast majority of humanity uses elementary mathematics pri- marily in handling money. As a matter of fact, the reawakening of mathematics in Europe during the Renaissance coincides roughly

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112 The State of the Mathematical Sciences with the transition from a barter economy to a monetary economy. The universal spread of rudimentary mathematical literacy was a concomitant of the development of that economy. On a more sophisticated level, mathematics is used in insurance, in particular in life insurance, which dates back to the end of the sixteenth century. Statistical methods have been developed partly as a result of the needs of insurance companies. Mortality tables were among the first statistical tables published. The actuarial profession is a typical instance of a thoroughly mathematized technology. The flowering of mathematical statistics in Scandinavia during recent decades was certainly seeded by Scandinavian concern with the mathematics of insurance. One of the more interesting applications of mathematics in ac- tuarial work is the extraction of a set of mortality rates from ob- served data and the substitution of smoothly progressing rates for the irregular set extracted. This is the classical problem of gradua- tion which has been attacked in many ways over the years, most recently within the framework of a Bayesian approach. Among the early graduation methods employed by actuaries were various adaptations of curve-fitting. Later a number of linear com- pound formulas were devised to produce smoothly progressing rates, judged by the reduction of error in third differences. In recent years, the most widely used graduation formula has been one based on a difference equation, which represents a compromise between smooth- ness and closeness of fit. Within the last year, new graduation meth- ods have been developed as a problem in straightforward statistical estimation of a large set of mortality rates simultaneously, proceed- ing, however, from a prior distribution of the "true" rates grounded on personal probability and making use of Bayes' theorem. Another type of problem is illustrated by various elaborations of the theory of risk, which has been developed as a special case of the theory of stochastic processes. The so-called "collective risk theory" focuses attention on the distributions of total claims of an insurance company at the end of a specified period of time, so that a reasonable judgment might be made about appropriate limits of retention or bounds of acceptable adverse fluctuations. These are essential con . - S1C .eratlOIlS In relnSUranCe. Collective theory of risk was developed by a number of Scan- dinavian actuaries (notably F. Lundberg, H. Cramer, and C. O. Segerdahl) for investigating insurance company operations from a probabilistic viewpoint or, more realistically, the emergence of prof- its in a risk enterprise. The basic model considers the distribution of

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Examples of Mathematics in Use ~3 total claims of a risk enterprise as being composed of two elements - frequency and severity; the resulting distribution of total claims can be regarded as a stationary stochastic process with independent increments and as a compound Poisson process. In recent years, the fundamental assumptions of the theory, and hence its range of ap- plication, have been significantly enlarged by the use of more general probability models that allow for certain types of fluctuations in basic probabilities. The exact distribution of total claims of an in- surance company has been studied analytically for a variety of assumptions. Ingenious numerical approximations have been devel- oped, and more recent broader analytic-numerical studies of total claim distributions have been made, relying extensively on computer . slmu atlon. MATHEMATICS IN MANAGEMENT AND OPERATIONS During World War II, the use of simple mathematical models and mathematical thinking to study the conduct of military operations became a recognized art, as first scientists and later mathematicians, lawyers, and people with other backgrounds demonstrated its effec- tiveness. After the war, attempts to apply the same attitudes and approaches to business and industrial operations and management were pressed forward rather successfully. Combined with techniques and thinking drawn from, or suggested by, classical economics, this line of development has now led to an active field about whose names EIoward Raiffa of Harvard University has observed: Some names which are used more or less interchangeably are: Manage- ment Science, Operations Analysis, Operations Research, Decision Analysis, Systems Analysis, Cost-Benefit Analysis, Mathematical Programming (under certainty and under uncertainty), Decision and Control, Optimization Theory, Control Theory, Applied Mathematics II (Roman numeral I is reserved for mathematical physics and astronomy). Of course, researchers and practitioners in these areas could each argue persuasively that their title is most appropriate and that what they do is somewhat broader than what others do. Whatever the title, the flavor of what is done is the same, com- bining the use of numerical data about operating experience so characteristic of early military applications with mathematical models to provide guidance for managerial action and judgment. This field was created by scientists accustomed to the use of mathe

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114 The State of the Mathematical Sciences matics; both its spirit and its techniques have always been thor- oughly mathematical in character. This mathematical approach is steadily penetrating the practice of management and operation. A number of the leading schools of business administration have concluded that mathematics is important both as a tool and as a lan- guage for management, and that training for the professional class of managers should include a substantial dose of this field of many names. Therefore, calculus, linear algebra, and computer program- ming either must be prerequisite for entrance or must be taken early in the graduate training program. At a leading business school (Har- vard), which is not "mathematically oriented" and where no such requirements are imposed, about 75 percent of the entering students have at least two years of college mathematics, several elective courses requiring that degree of mathematical sophistication are given, and there is a sizable group of faculty members who have PhD degrees in mathematics or applied mathematics. This field is pervasively mathematized and computerized, but it is far from being strictly a mathematical science. The pattern of its problems is frequently described as formulating the problem, con- structing a mathematical model, deriving a solution from the model, testing the model and the solution, establishing control over the solution, and implementing the solution. Only one of the six steps is completely mathematical; the others involve the actual problem in an essential way. In these other steps, of course, there are many ap- plications, some of them crucial, of statistics and computer science. The mathematical step, especially when dealing with management rather than operational problems, often draws on concepts and re- sults Mom the field of optimized allocation, control, and decision. A good practitioner combines the characteristics of most profes- sional consulting and of most effective application of mathematics: abundant common sense, willingness to produce half-answers in a half-hour, recognition of his key roles as problem formulator and contributor to long-run profits (rather than as problem solver or researcher). Yet for all this, and in an alien environment, he must retain his skill as a mathematician. CONCLUDING REMARKS We have touched upon only some of the uses of mathematical meth- ods in disciplines outside of the mathematical sciences proper. The

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Examples of N7Tathen~atics in Use 115 number of such instances is steadily increasing, and the boundary lines between mathematical sciences and sciences that use mathe- matics are often difficult to draw. The increasing use of mathematical methods in the biological sciences was pointed out earlier in the section on The Mathematiza- tion of Culture (see page 3~; and the essay by Hirsh Cohen in reference 7 discusses in more detail a variety of biomedical applica- tions of mathematics. A 1967 compilation by Thrall et al.~4 pro- vides extensive further illustration of applications of mathematical models in biology. An important omission in our discussion is the burgeoning field of mathematical psychology. A comprehensive survey of this held can be found ire reference 13. Another important example of the pene- tration of mathematical methods into hitherto unmathematized areas is in the young science of mathematical linguistics, which ap- plies mathematical methods arid the mathematical way of thinking to the study of living languages. (See, for instance, the essay by Harris in reference 7.) All great goods spawn small evils. Every new and powerful tool is misused as well as used wisely. This leas true of printing and me- chanical power when these tools were new. Today, in any held of endeavor where mathematics or statistics or computing is new, there will be those who use these tools inadvisedly, as a means of persua sion when the evidence is incomplete or even incorrect, or as a means of "blessing" conclusions that do not deserve support. All fields now well mathematized or well statisticized or shell computerized have suffered through these difficulties. Those in process now, or to be in process in the near future, will have to suffer too. Such difficulties often slow down the incorporation of mathematics or statistics or computing into the heart of a new field of application. These delays are, we {ear, inevitable. The one antidote that has proved effective is an increased amount of mathematical or statistical or computing literacy for the majority of those who work in the field. This increase comes in two parts, separable but usually joined: on the one hand, enough literacy about the mathematics involved to understand the meaning, perhaps even the details, of the manipulations required; on the other, often even more important, art understanding of how mathematics or statistics or computing fits into actual problems in similar areas. This latter includes an appreciation of one of the skills of an effective user: the ability to be usually sound as to what must be taken into account in

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116 The State of the Mathematical Sciences formal or numerical manipulations, what can probably be neglected, and what is surely negligible. It is not easy to teach these things ex- plicitly; they are usually learned by experience in doing and thus come to depend on at least some facility with the manipulations concerned.