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l ~
Applied Mathematical Sciences
Fields of mathematical science outside the core are of various kinds.
Both computer science and statistics have dual sources of identity
and intellectual force, only one of which is mathematical; hence
they are more accurately described as partly mathematical sciences.
These two already apply to almost as wide a variety of activities as
does mathematics itself.
Computer science is both a mathematical science and something
else. In its present form it could not exist without mathematics. But
it would be utterly unproductive without a piece of machinery-
the computer, both as we have access to it at the moment and as
we can envisage it in the future.
Modern statistics could not operate without mathematics, espe-
cially without the theory of probability. Equally, it could not exist
without the challenge of inference in the face of uncertainty and the
stimulus of the quantitative aspects of the scientific method. Like
computer science, statistics is both a rr~athematical science and
something else.
At the other extreme are herds mathematical economics, mathe-
matical psychology, mathematical linguistics, among others that
deal with the mathematical aspects of rather specific areas.
At an intermediate level of breadth and identity stand fields
that have derived their nature from some area of application,
usually a quite broad one. These fields have initially chosen their
tools to meet the application area's problems but have grown and
developed far enough to have a mathematical character of their
own, thereby gaining a certain independence of existence and
84
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85
breadth of application they could not otherwise have had. The
outstanding example is physical mathematics, or classical applied
mathematics, which has grown out of the mathematics of classical
physics to a point where it deals with a wide variety of applications
of mathematical analysis. A similarly intermediate field is today
growing rapidly in an area stimulated by applications to pure and
applied economics and to management and the conduct of oper-
ations of all kinds business, governmental, and military. As yet
there is little agreement about this held's name, but its concentra-
tion on problems of optimizing any or all of allocation, control
and decision is so evident that we can label it accordingly.
FLUID DYNAMICS: AN EXAMPLE OF
PHYSICAL MATHEMATICS
To illustrate the nature of physical mathematics (classical applied
mathematics), we shall describe in some detail one of its typical
and central subdivisions the mathematical theory of fluid motion.
The description will be followed by a few general comments on
related areas.
Both liquids and gases (neutral or ionized) are fluids. The study
of fluid motion is, strictly speaking, a part of physics. It has been
left, however, for many decades now, largely in the care of applied
mathematicians, engineers, astrophysicists, and geophysicists (in-
cluding meteorologists and oceanographers). Applied mathematics
has found in the study of fluid motion some of its most spectacular
triumphs and also has suffered some of its, hopefully temporary,
notable defeats.
Attempts to describe the motion of fluids mathematically are
almost as old as mathematical analysis. The early history of fluid
dynamics is associated with names such as Bernoulli, Euler, and
D'Alembert. Classical fluid dynamics led to some of the most im-
portant developments in mathematics and to techniques and ideas
that have found many other applications. Riemann contributed to
the basic theory of propagation of strong compression waves. His
ideas led to the development of the notion of "characteristics,"
which greatly influenced the theory of partial differential equations.
Much of Hadamard's work was concerned with wave propagation.
All this work laid the basis for many of the recent developments in
the theory of supersonic flow arid shock waves, which occur on the
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86
The State of the Mathematical Sciences
earth (sonic boom associated with supersonic aircraft) as well as
in the stars (novas and supernovas).
Much of the classical fluid dynamics of incompressible flow
seemed originally to be completely useless as a model for physical
reality. For instance, D'Alembert proved that a body moving
through fluid experiences no drag. Of course, D'Alembert's deriva-
tion neglected viscosity (the tendency of one part of a fluid to pull
adjacent parts along with it); but his result was contrary to experi-
ence even in cases where viscosity effects seemed to be very insig
.
Decant.
Fluid-flow theories applicable to physical reality appeared only
in the nineteenth and early twentieth centuries, largely as a response
to the needs of aeronautics. A major achievement was Prandtl's
theory of the boundary layer, which showed in which cases one can
use classical fluid dynamics to obtain physically valid results, and
how in such cases the viscosity effects are limited to a small layer of
fluid next to a moving body (whence the name "boundary layered.
Boundary-layer theory led, by the way, to important developments
in the general theory of differential equations.
Compressibility of air was neglected in aerodynamics that proved
sufficient for the engineering needs of the period up to World War
II. Interest in high-speed flight required that compressibility be
taken into account and that the essentially nonlinear theory of gas
flow be faced. While certain mathematical points in that theory are
still unexplained, today sufficient theoretical understanding is avail-
able for practical needs. The "sound barrier" turned out to be
largely imaginary, but the high altitudes achieved in rocket flight
necessitated the creation of a new type of fluid dynamics, the dy-
namics of rarefied gases. This theory contributed to the successful
solution of the re-entry problem for earth satellites.
A young and very active discipline of modern fluid dynamics is
magnetohydrodynamics, which describes the interaction between
electromagnetic fields and conducting liquids or ionized gases
(plasmas). We can hope that this subject will contribute as much
to the future course of mathematics as did the individual subjects
of electromagnetic theory, fluid dynamics, and statistical mechanics.
Magnetohydrodynamics has applications to astrophysics (stellar
atmospheres and interstellar matter), to the search for a controlled
thermonuclear reaction, and to a promising use of ionized-gas
"armatures" in generating electric power. The striking spiral struc-
ture of manY galaxies (including our own) can now be understood
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87
in terms of magnetohydrodynamics and statistical dynamics of
stellar systems. Here the very old subject of self-gravitating fluids,
studied by MacLaurin, Jacobi, Riemann, and Dedekind in another
connection, plays a predominant role.
The behavior of stellar systems and rarefied plasmas has been
known to be governed by a type of "collisionless Boltzmann
equation" and to exhibit "collective behavior." On a still larger
scale, the distribution of matter in the universe is often treated in
terms of fluids governed by Einstein's general theory of relativity.
The question of the existence of gravitational radiation offers chal-
lenging nonlinear partial differential equations begging for solu-
tion.
In recent years, there has been a great upsurge of the study of
fluid mechanics for the purpose of describing phenomena in geo-
physics (meteorology and oceanography), where the rotation of the
earth often plays an essential role. Much of the work was done by
applied mathematicians; Von Neumann was among the pioneers in
the use of numerical methods for weather prediction (see An
Example from the Environmental Sciences: Numerical Weather
Prediction, page 108~. Problems of hydrodynamic stability, on which
work is still flourishing, occur in a variety of ways.
Yet some problems in fluid dynamics, which are almost as old as
the science itself, still resist all efforts at mathematical treatment.
An outstanding example is turbulence, a phenomenon easily ob-
served by anybody watching the disintegration of the smoke jet
from a factory chimney or from a cigarette. Although some of the
best minds in physics and mathematics have applied themselves to
this problem and have achieved some partial successes, there is at
the present little hope for a satisfactory solution in the near future.
It is now clear that the existing studies and the still unresolved
problems both relate to the broader area of nonlinear random
processes.
At the same time, the theory of the fundamental partial differ
ential equations governing the motion of incompressible viscous
flow has not been fully developed. It remains a challenging and
interesting problem to pure mathematicians and a source of much
current stimulation for work.
As mentioned above, there is a similarity between the behavior of
stellar systems and that of rarefied plasmas in exhibiting collective
behavior. The similarity is not only in the basic ideas but also, sur-
prisingly, in the details. Concepts used in fluid mechanics are also
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The State of the Mathematical Sciences
useful when continuum mechanics deals with other material-
elastic solids, plastic solids, and bodies exhibiting both flow and
elastic properties. The study of the basic laws of continuum
mechanics depends heavily on the concepts of "covariance" (i.e.,
the assertion that physical laws would remain in the same form no
matter what observer is making the description).
Indeed, the concepts and methods that originate in one branch
of applied mathematics are often applicable to other branches. Once
the theory is put into mathematical form, even in a form not yet
deserving serious interest of core mathematics, it could be of basic
importance to other branches of applied mathematics. Equilibrium,
stability and instability, wave motion, linear versus nonlinear proc-
esses, reversible versus irreversible processes, entropy and one
can name many more are certainly concepts whose applications
are general. The mathematical theory of economics makes use of
several of them. The new science of information theory, with its
fascinating coding theorems and important applications to practical
problems in communication engineering, also uses the concept of
entropy.
These observations demonstrate that much of classical applied
mathematics has a tendency to merge into a coherent whole across
traditional scientific disciplines, adding to the strength and useful-
ness of the entire area of activity.
STATISTICS
Statisticians easily recognize other statisticians as such, though they
are often surprised by the fields in which they work and the tools
that they use. It is now almost three quarters of a century since
Karl Pearson turned from graphical mechanics to the application
of algebra and differential equations to problems arising in the
analysis of experimental and observational data. The mathemati-
cal theory of probability, now considered part of core mathematics,
has applications to the "adjustment of data," first in surveying and
astronomy and then in many other fields, going back to the work
of Gauss and Laplace a century earlier.
A variety of traditions have merged in contemporary statistics,
not only the application of the theory of probability and the more
quantitative aspects of scientific method but the numerical skills
of the economist and the meteorologist and the data-handling skills
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89
that have been served by one form or another of automation since
the invention by Hollerith of the punched-card machine in the last
century. Today, statisticians are concerned with how it would seem
we might well make inferences, how we do make inferences, and
how we might and do collect, process
variety of fields.
and digest data in a wide
Karl Pearson's willingness to attack the dimensions of crabs, the
inheritance of intelligence, and the mechanisms of evolution with
mathematical techniques, however, signaled the opening of a new
era, one in which new problems and mathematical theories have
shared responsibility for intellectual stimulation, and where in
one field after another, the statistician has done much to introduce
workers in that field to new ideas and new ways of thinking as well
as to new techniques. The basic ideas of quantitative scientific
method, of uncertainty, and of elementary mathematical models
have been brought into many fields by statisticians. Agriculture,
biology, and medicine; insurance and actuarial techniques; indus-
trial production and market research; psychology, sociology, and
anthropology; all have benefited.
Once statistics was thought of as confined to the study and
manipulation of large masses of data. Such problems are still among
the most challenging, but the problems of doing a good job of con-
verting three or five or ten numbers into suitably qualified conclu-
sions are today at least equally important. Individuals, corporate
bodies, governments, scientists, engineers, economists all use a
wide variety of statistical techniques to assess "what the data say"
and thus to guide them in their deliberations and actions. Wherever
the evidence is less than complete or bears only indirectly on the
point at issue in truth, almost everywhere-there is a place for
statistics in untangling the data and assessing the conclusion.
Karl Pearson, "Student" (W. S. Gossett), and R. A. Fisher took
leading parts in the early development of a mathematical statistics
of wide application. During the years between the two world wars,
new statistical methods and new methods for arranging experiments
made major contributions to the development of agricultural tech-
nology. (One of the three leaders of the first American seminar on
R. A. Fisher's analysis of variance was later president of the Pioneer
Hybrid Corn Breeding Company.) Statistical techniques made our
agricultural progress possible, while the needs of agricultural ex-
perimentation called whole areas of statistics into existence.
Beginning in the early 1920's, farsighted men at Bell Telephone
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The State of the Mathematical Sciences
Laboratories recognized the importance of statistics' potential in
industrial production. Walter Shewhart founded and developed the
theory of statistical quality control, and Harold Dodge led in the
development of sampling schemes for inspecting mass-produced
items. Both of these techniques are sometimes given major credit
for significant contributions to the success of the Allied production
effort in World War II. In particular, statistical quality control was
a major factor in the early success of the plutonium-production
process at Hanford. After the war, the application of statistical
techniques to the experimental improvement of industrial produc-
tion processes made outstanding progress, particularly in connection
with chemical processes.
Stimulated by Student and Shewhart, i. Neyman and E. S. Pear-
son developed the first of the modern mathematical theories of
formally optimized inference in the latter 1930's. During the same
years and in the decade that followed, the mathematical theory of
sampling developed to meet the needs of studying large groups of
people, farms, business firms, and many other things by observing
relatively small samples. The sample survey thus became effective,
efficient, and irreplaceable.
Wartime needs for even more efficient inspection led Abraham
Wald, once a pure mathematician, in one sleepless night, to the
basic theory of sequential analysis and seeded his later work on
statistical decision theory (to which a brief introduction is given in
the essay by Kiefer in reference 7~. The needs of a variety of prob-
lems in geophysics (waves on the ocean, earthquakes, gusts that
disturb airplanes) and in engineering (e.g., tracking radars) de-
manded new tools to isolate phenomena tagged by their frequencies
of oscillation. Statistical spectrum analysis developed rapidly,
spreading into a wide range of applications, including recent con-
gressional inquiries into the proper adjustment of economic series,
such as unemployment, for the season of the year.
The implications of psychological and educational testing have
often been discussed and will be discussed many times again,
demonstrating the ever-increasing importance of these techniques
in our lives. The selection of test material and the study of indi-
vidual arid interrelated behavior of tests has had to be carried out
in a statistical way, demanding and receiving the development of
new statistical techniques.
Qualitative studies of a national economy and of the economic
environment of single firms share the difficulties of (1) complexities
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91
requiring explicit treatment of many variables and (~) limitation
of available data almost to the point where there are not enough
to obtain meaningful answers. As a result, econometricians have
developed, and have stimulated others to develop, another broad
area of statistical technique econometrics. For instance, important
issues in the recent Federal Communications Commission hearing
about telephone rates came down to questions of statistical tech-
nique and statistical appropriateness.
Most areas of biological and medical research have come to de-
pend on statistical techniques. Experimenters routinely use
methods that grew up in agricultural research, supplemented by
special statistical techniques appropriate to the effectiveness of
drugs and poisons. The study of epidemic and occupational disease
calls on other techniques and poses very difficult problems.
The study of medical and surgical performance is an even more
difficult field. The recent national study of one of the anesthetics
most used for surgery fortunately showed certain specific suspicions
to be unfounded. Yet it turned up suggestive differences between
postoperative death rates that may prove to be associated with the
anesthetics used. These apparent differences could not have been
seen without the use of statistical techniques not in existence when
the study began; the untangling of their true nature and causes
will require both better statistical techniques and better data, each
of which stimulates the other.
The arrival of modern electronic computation opened many
new opportunities in almost every field of statistics. Routine sea-
sonal adjustment of economic time series no longer calls for an
expert; a dollar's worth of computer time does it. The periods of
the natural vibrations of the earth would never have been measured
without computer processing, and we would have known much less
about the earth's interior. Yet most of the large-scale challenges
involved in harnessing modern computing systems to the effective
analysis of data have hardly been tackled.
lust as it is still a methodological science, a computational sci-
ence, and a behavioral science, statistics continues to be a mathe-
matical science. Its health depends on effective exchanges of results
and problems with workers in a wide variety of fields. The increas-
ing variety of applications has increased the needs for the inven-
tion of new techniques appropriate to new problems. Throughout
the areas of statistical technique that have been developed over the
last four or five decades there are great needs for modernization.
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The State of the Mathematical Sciences
There is a need to develop probabilistic models for new phenomena,
to study the properties of these models, and to investigate statistical
procedures appropriate for them. Most formal optimization of
statistical methods has been carried out for classes of distributions
that are too narrow (normal or Gaussian) or too broad (all con-
tinuous distributions) to correspond satisfactorily to Practice. The
:~ . r .1 . . ~
~c o~ ~e computer nor one on what is noccibl~ half ~1~, an
what is reasonable in statistical techniques has been only barely
thought about and only in very special cases. Communicating results
to people can be much more effective once we understand how to
use computer-produced graphs and pictures. Adequate treatment
for the nonexperimental observations of behavioral science and
medicine requires rethinking of the ideas whose formalizations
underlie statistical techniques. These are but four of many direc-
tions in which progress is being demanded.
'-In ^^ ~ 7 ~^ ~ ~ ^~ ~ ~ ~ ~1 ~ ~ 1_ ~ I at # ~ J I I
COMPUTER SCIENCE
Automatic high-speed computing is scarcely 20 years old. Neverthe-
less, it already constitutes one of the nation's most valuable scientific
and economic resources.
Until automatic electronic digital computers first became com-
mercially available around 1950, most persons took at least 10 sec-
onds to multiply two moderately large numbers. With today's elec-
tronic computers, the same multiplication can be done nearly
10,000,000 times faster. In comparison, the speedup in travel be-
tween walking and going by jet is by a factor of approximately 100,
while that in communication between sound and radio waves is by
a factor of approximately 1,000,000. These changes in rates of
transportation and communication have completely remade the
world. The even greater speedups in information processing are
remaking our world again. It will be many years before our capacity
to exploit the new computers fully will catch up with even their
present capabilities. Meanwhile, rapid advances in speed and in the
capacity of computers to store information continue.
The development of computing is accelerating the penetration
of mathematics into old and new fields of human endeavor. In fact,
the original purpose for developing computers was the numerical
solution of mathematical problems. The success achieved in this
direction has been phenomenal. Calculations that required years
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93
of effort by teams of people can now be done in a matter of minutes.
Mathematical problems that were thought inaccessible to effective
numerical treatment are now solved as a matter of routine. During
any rocket launch, for instance, dozens of nonlinear ordinary
differential equations are being solved in a completely automatic
manner.
Numerical analysis, the branch of mathematics concerned with
the invention and evaluation of methods for mathematical calcula-
tion, has been revitalized by the advent of the computer but has so
far been unable to keep pace with technological developments. (See
Davis' essay in reference 7.) Computers vastly increased the im-
portance of parts of numerical analysis and made others completely
obsolete. Carrying out a stupendous number of arithmetic oper-
ations in solving a single problem introduced a host of difficulties
associated with the limited precision and range of computer num-
bers. (See the essay by Forsythe in reference 7.) Meeting these diffi-
culties has led to an array of new and powerful algorithms for solv-
ing mathematical problems.
It is important to realize the variety of fields in which computing
has become an important tool. One of these is mathematics, but this
is a relatively minor fraction of the total volume of computing done
today. Others include experimental and theoretical physics, business-
data processing, economic planning, library work, engineering de-
sign (from transportation systems to computers themselves), edu-
cation, inventory management, police operations, space science,
musical performance, and content analyses of documents. One can
speak without reservations of a computerization of our culture that
is already broader, though less deep, than its mathematization.
The modern computer can do many things. The logical possibil-
ity of building a universal computing machine that in principle can
do anything any computing machine can do was first recognized, as
a theorem in mathematical logic, by A. M. Turing. This discovery
influenced the work of the mathematician Von Neumann, who
contributed the crucially important suggestion that computers
should store their instructions together with their data.
The practical importance of computing to our society can be
judged from the direct annual cost to the federal government of
acquiring and operating electronic computers in 1967. It was in the
range of $2 billion. (Source: The New York Times, 25 January 1966,
p. 21, quoting the President's budget message.)
The near-universality of a modern computer means that it can
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The State of the Mathematical Sciences
perform almost any manipulation of symbols. The breathtaking
future perspectives to which this may someday lead are described
in an essay by J. T. Schwartz on nro.snertc nit r~mn,~t~r criP=rp 7
TO _~.:_ 1~
--- ~~~-r~ ~. ...
,~ Purr, computers can be used for direct simulation of
complicated technological, natural, and social phenomena, where
computer instructions play the role of formulas in setting up a
formal model. Such models are as truly mathematical as those ex-
pressed in formulas. Lists of computer instructions can be long and
interrelated in complicated ways and still be executable. If many
details must be treated, examples of behavior can be obtained by
direct simulation when general results cannot be obtained from
similarly complicated lists of mathematical formulas. Examples of
problems that have been treated by direct simulation include high-
way traffic control, the design of telephone networks, and the design
of concrete shields for nuclear reactors.
lust as zoology is the study of animals and animal behavior, so
computer science is the study of computers and computation. Three
aspects are currently of outstanding importance:
1. The design and analysis of computer hardware the com-
ponents and total electronic and mechanical systems that comprise
computers;
2. The design and analysis of computer software- the basic
languages and resident programs essential to convert bare hardware
into productive computing systems, including control programs,
compilers, and time-sharing executives;
3. The methodology of solving problems with computers- those
techniques that are common to solving broad classes of problems, as
opposed to the preparation of individual programs to solve single
problems. One of these techniques is the appropriate representation
of complex information.
Today, computer science is, among other things, a mathematical
science. As Professor Allen Newell describes it in his memorandum
to COSRIMS:
Computer science shares with mathematics a concern with formalism and a
concern with the manipulation of symbols. It also shares with mathematics
the role of handmaiden to all of science and technology. It shares with elec-
trical engineering the concern with the design and construction of informa-
tion processing systems that accomplish ends.
It shares with all of engineering a concern with the process of design,
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95
considered as an intellectual endeavor. It shares with linguistics a concern
with language and communication. It shares with psychology a special con-
cern with forms of information processing that result in intelligent behavior,
broadly viewed. It shares with the library sciences a concern with how to
store and retrieve large amounts of information, either as documents or as
facts. It has both theoretical aspects, as in the study of automata, and ex-
perimental ones, as in the discovery of new types of systems through pro-
gramming them and exploring their behavior. All of these shared problems
with other parts of science and technology imply that the future status of
computer science is still indeterminate. It may permanently become one of
the mathematical sciences. It may become an autonomous science, such as
geology. The result is genuinely ire doubt, because "science" is a social con-
struct, shaped as much by social forces as by anything intrinsic to its subject
matter.
Lying as it does across the research fields of mathematics, elec-
tronic engineering, linguistics, psychology, among others, computer
science, though still somewhat formless and unfocused, is also an
independent entity. Both government and universities tend to view
it as such, as is evidenced by dozens of university computer science
departments and a separate Office for Computing Activities within
the National Science Foundation. As a mathematical science, com-
puter science emphasizes the constructive, problem-solving, algo-
rithmic aspect of mathematics, in contrast with the structure aspects,
often emphasized in core mathematics. As a young field that must
respond to soaring demands for knowledge at all levels of speciali-
zation, computer science can use the cooperation of mathematicians
in creating a solid body of knowledge.
Computer science is at once abstract and pragmatic. The focus on
actual computing systems introduces the pragmatic component; the
central questions are economic ones like the relations among speed,
accuracy, and cost of a proposed computation, and the hardware and
software organization required. The often better understood
theoretical questions of existence and computability provide an
important conceptual basis for the study of the more pressing ques-
tions. (And these in turn have led to new mathematical research in
such questions as degrees of finite computability.) On the other
hand, computer science deals with information in an abstract way.
The meanings of symbols and numbers may change from appli-
cation to application, just as they do in every application of mathe-
matics. Thus computer science shares the main goal of mathe-
matics to create a basic structure in terms of inherently defined
concepts that is not bound to any particular application. Computer
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Representative terms from entire chapter:
statistical techniques
96
The State of the Mathematical Sciences
scientists have barely begun the creation of such a basic structure
and are still mainly concerned with exploring what computers can
and cannot do economically.
So far the most mathematical components of computer science
are numerical analysis and the theory of programming languages.
The first computers were programmed in machine language, a
language that might be likened to microsyllables of human speech.
Typical of machine language is an instruction saying in effect: "Add
the number in cell 4565 to the number in arithmetic register A, and
leave the answer in register A." The solution of substantial mathe-
matical problems in terms of such instructions expended many man-
months of tedious, repetitive human labor. After some years, it was
realized that computers, with their infinite capacity for carrying out
details, could themselves be instructed to translate an algebraic
7'r ~7 are / ~ rr ~ ; ~ ~ ~ ~ ~ _~: _ ~ 1 ~ ~ ~ ~ _ _ ~, ~ ~ .
'ww<~ ^~ `~ACL~11111~ `"ll~"a~C. I 111~.~; Ilil`~l;ln n,~ln
Applied Mathematical Sciences
97
require the combined skills of mathematicians, operations analysts,
computer scientists, and engineers, since extremely complex prob-
lems are involved. This is a field of very active research in computer
science today.
In addition to the reasonably successful methods developed for
numerical computation and programming languages, algebraic
(nonnumerical) methods for some classes of problems are now
emerging. Methods for handling pictorial data have been formalized
for large classes of pictures. There still remains an almost unlimited
array of topics that are not being treated systematically. This is the
frontier of computer science. It includes a large number of non-
numerical problems, as well as such problems as executive control
(scheduling, allocation, handling, interruptions) and simulation of
intelligent behavior.
OPTIMIZED ALLOCATION, CONTROL, AND DECISIONS
The needs of economic theory for results about allocation, those of
management practice for approaches to decisions (which can usually
be cast in terms of allocation), and those of engineering operation
for patterns of control (whether in trajectory choice or the feedback
control of production machinery) all lead to a single interrelated
complex of problems, concepts, and results. In almost every case,
these problems can be described in terms of maximizing or mini
. . . . . . . . . . .
mung be., optimizing- some criterion subject to constraints on
the variables.
Separately developed traditions are now being merged into a
coherent body of concepts and results. Workers in the field have
come to recognize its unity as more important than its diversity.
The needs of economists, managers, and engineers for explicit guid-
ance in choosing answers, and for as many explicit answers as pos-
sible, have anchored this area close to the frontier. There is interest
in a procedure that in principle describes a solution, but ordinarily
only as a step toward a feasible procedure for finding actual solu-
tions. Since feasibility is a matter of computation, exactly what
computing tools are available is vitally important. This whole area
is therefore intimately bound up with computer systems and with
machine arithmetic.
An interesting mathematical development in this area was Von
Neumann's and Morgenstern's "theory of games and economic be
98
The State of the Mathematical Sciences
havior," which tackled the strategic and mathematical problems of
.CouS optimization of rewards by two or more name Dlaver.c
or economic competitors.
A ~;~ ^1 ~ 4_ _ .. 1
~ . ,
~ 1 J
~ IIlalOr unread reran with linear nro~nmmin~ th'` Nil ^'
V - ~ _ 1 ~-Act ~ ~-~^ ~ ~-51A VA
optimizing a linear function of many variables when subject to very
many simple linear inequalities. Linear programming problems
were first systematically attacked in the 1930's, in Russia by
Kantorovich, whose results escaped notice for many years, and in
the United States at about the same time by Koopmans and Hitch-
cock. The held's growth in the United States was greatly stimulated
~ n A ~ 1 ~ . · ~ v
1n 1Y9/ by L)antzlgrs discovery of a method that made noccibl~ the
~1~ ~:~ ~r ~ ~ ~ 1_ _ ~
OWl"Ll~ll Ql Icy complex problems with then available com
puter facilities. A burst of incll]strin1 ~nnlir~timnc crier, c~'h^~
A ~
both the practical and the theoretical boundaries of the subject.
Nonlinear programming was introduced, in which the function to
be maximized was generalized from a linear function of the given
variables, and the constraining inequalities were also generalized.
The fact that many practical problems involve units that cannot be
subdivided (how many dresses can be cut from a bolt of cloth?)
now demanded, and received, an effective theory of integer program
mlng.
~ r~ r~~~~ ~O~V14 ~ LO ~ L~11~t
A third line of development involved problems of simultaneous
allocations: How can n persons be best assigned to n jobs? How can
many factories best ship their products to many customers? Some al-
location problems have proved to be intrinsically very difficult: What
pattern of warehouse locations will minimize shipping costs? What
Is one shortest route passing through every one of a list of cities?
_ _ . 1_ _ _ 1~
Treated directly, such problems have a characteristic irreducible
complexity, since the entire solution must be examined simultane-
ously.
~v
~7 ~
Certain types of these problems have fortunately been found to
be reducible to problems of linear and nonlinear programming and
have thus become much more easily soluble. Others have yielded to
special algorithms, and the remainder pose serious problems for the
techniques of combinatorial analysis and the capacities of modern
computing systems.
A fourth line of development involved dynamic programming or
the theory of multistage decision processes, which treats problems in
which a number of decisions are made over a period of time. The
maintenance of inventories in the face of uncertain demand, where
Applied Mathematical Sciences
99
there must be repeated decisions of whether and how much to order,
poses a typical problem, as do the scheduling of production, equip-
ment replacement, the conversion of stored water to power, and a
wide variety of queuing processes.
As these lines of development continued, perspectives broadened,
and the field matured, linear and nonlinear programming and
dynamic programming fused together into mathematical program-
m~ng.
A fifth line of development, carried on vigorously both in the
Soviet Union and in the United States, involved the optimum con-
trol of systems reasonably thought of as tracing a path. When
choosing a rocket trajectory that will require least use of fuel, the
path is physical. When deciding how the temperatures in a chemical
reaction vessel should change to maximize production of the desired
chemical, the path is symbolic. The general mathematical problem
is the same, and the same mathematical results are applicable. In
many ways, these problems of control theory are continuous-time
analogs of the problems of dynamic programming. Both lines of
development have contributed ideas to one another, with control
theory more frequently taking the lead, since continuous time-
with a decision every instant requires much more sophisticated
mathematical techniques and much deeper mathematical results.
Allocation is the immediate end of the theory of games and a
consequence of most applications of mathematical programming.
Control is the aim of control theory and many applications of
dynamic programming. Decisions have to be made in every case.
Optimization of some prescribed function is the common thread
that links all these lines of development together. Opinions differ
on whether the area thus described has already become a clearly
identifiable field of mathematical science. If not yet, it soon will
become one.
OTHER AREAS
The fields of mathematical science outside the core will continue
to make increasing contributions. They will require deepening
mathematical education and broadening mathematical literacy
among their own workers. This places a great responsibility on uni-
versity and college faculty in the mathematical sciences. They must
100
The State of the Mathematical Sciences
provide instruction in core mathematics, including its own latest
and most useful concepts, approaches, and results, which alone will
strain their capacities. In addition, they must provide instruction
that identifies and illustrates the multifaceted role of mathematics
in our society.
