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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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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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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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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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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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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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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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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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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.
Representative terms from entire chapter:
weather prediction
