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1
Mathematics and Society
The recent history of mathematics has been like that of science
generally. The number of investigators has grown rapidly and so
have the pace and quality of the research. An intellectual leadership
that used to reside in Western Europe is now shared by Americans
and others.
Today, society, especially through other sciences and technologies,
makes unprecedented demands on mathematics and on the com-
munity of mathematical scientists. The purpose of this report is to
identify the major demands, to assess the capabilities of the mathe-
matical community to satisfy them, and to propose measures for
preserving and extending these capabilities.
THE MATHEMATIZATION OF CULTURE
Mathematics has long played a central role in the intellectual and
technological history of mankind. Yet this statement hardly begins
to convey or account for the current explosive penetration of
mathematical methods into other disciplines, amounting to a virtual
"mathematization of culture."
Mathematics can be described as the art of symbolic reasoning.
Mathematics is present wherever chains of manipulations of ab-
stract symbols are used; such chains may occur in the mind of a
human being, as marks on paper, or in an electronic computer.
Symbolic reasoning appears to have been first used in connection
with counting. For this reason, mathematics is sometimes described,
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Representative terms from entire chapter:
mathematical methods
4
Summary
though not completely accurately, as the science of numbers. In
fact, it turns out that all symbolic reasoning may be reduced to
manipulation of whole numbers; and it is this fact that makes the
digital computer into the universal tool it is.
Mathematics as we know it originated more than 2,000 years
ago warn the ~reeks. They transformed into a deductive science
the collection of facts and procedures about numbers and geometric
figures known to the older civilizations of Egypt and Babylon. The
Greeks applied mathematics only to astronomy and statics. The
possibility of applying it to other sciences, in particular to dynamics,
was discovered in the sixteenth and seventeenth centuries. This dis-
covery revolutionized mathematics; it led to the creation of calculus
and thereby made modern physical science and technology possible.
The development of the physical sciences continues to use mathe-
matical techniques and concepts. So do the new technologies based
on the discoveries of the physical sciences. Furthermore, these sci-
ences and technologies use mathematical techniques of ever-increas-
ing sophistication, so that the mounting role of physical sciences
and technologies in the contemporary world is a dominant aspect
of the mathematization of culture. It is, however, not the only
aspect. Mathematical methods are penetrating into fields of knowl-
edge that have been essentially shielded from mathematics until
not long ago; for instance, the life sciences. The twentieth-century
penetration of mathematical methods into the biological sciences
has come about in several ways, perhaps most importantly through
the increasing study of biological phenomena by the methods of
chemical physics. The development of statistics, which the needs
of the biological sciences helped to stimulate, has led to the exten-
sive field of biostatistics. This has connections with mathematical
genetics, which has evolved out of the celebrated Mendelian laws
of inheritance, and with mathematical ecology, which is concerned
with such interactions as competition for food or the feeding of
one species on another. Differential-equation models for the con-
duction of signals along nerve fibers have had notable success.
Computer simulation of functions of living organisms is just one
of the ways in which computers are becoming increasingly im-
portant in the biological sciences.
Mathematics is penetrating the social and behavioral sciences,
too, and even traditionally humanistic areas. Mathematical eco-
nomics is now a central part of economics. The field of econometrics
has grown out of applications of probability and statistics in eco
. . . ~_ _
Mathematics ant! Society
nomics; and statistical techniques are important in anthropology,
sociology, political science, and psychology. Analysis of mathemati-
cal models for various social phenomena has been greatly aided by
computer simulation and data processing. The mathematical view-
point has even found new application in linguistics.
The above remarks refer to the mathematization of various
academic disciplines. Mathematics is also becoming an indispensable
tool in the world of government, industry, and business. The terms
`'operations research" and "management science" are among those
describing the rapidly growing use of mathematical methods to
solve problems which arise in managing complicated systems involv-
ing the movement and allocation of goods and services.
Computers have extended the possibility of applying mathemati-
cal methods to a degree that would have seemed fantastic a short
time ago. Computer science, which deals with manifold problems
of building and utilizing computers, contains, among other things,
very important mathematical components.
THE NEED FOR MATHEMATICALLY TRAINED PEOPLE
The mathematization of our society brings with it an increasing
need for people able to understand and use mathematics. This need
manifests itself at various levels.
We need people who can teach mathematics in grade school in
a way that will not create a permanent psychological block against
mathematics in so many of our fellow citizens. We need people who
can understand a simple formula, read a graph, interpret a state-
ment about probability. Indeed, all citizens should have these
skills. We need people who are able to teach mathematics in high
school and cope with the necessarily changing curriculum. We need
people who know what computers can do, and also what they can-
not do. We need computer programmers who can work with under-
standing and efficiency. We need engineers, physicists, chemists,
geologists, astronomers, biologists, physicians, economists, sociol-
ogists, and psychologists who possess the mathematical tools used
today in their respective disciplines and who have the mathematical
literacy for learning the new skills that will be needed tomorrow.
Equally we need, though in smaller numbers of course, people in
these fields who are able to use mathematical tools creatively and
if necessary to modify existing mathematical methods. The numbers
6
Summary
are hard to estimate, but it is clear that our society needs many more
mathematically literate and educated people than are available now.
For instance, computer programmers are already more numerous
than high school teachers of mathematics, and their numbers will
. .
continue to increase.
These demands call for massive amounts of mathematical edu
cation at all levels and so produce a mounting pressure on the
mathematical community. To do all this training calls for a larger
body of mathematics teachers. It is difficult to expand the supply
rapidly enough. Since teachers of intermediate-level mathematics
must themselves be trained by people of higher competence, these
pressures also quickly transmit themselves to the relatively small
community of mathematical scientists who do research. Many of the
needs of mathematics for support come from a need for balanced
growth meeting all these requirements.
Since we recognize a rising level of mathematical literacy as a
national objective, and since the community of mathematical sci-
entists bears the primary responsibility for attaining this objective,
our report cannot separate problems of research from problems of
education, including those of undergraduate education. (See both
Part III and the report) ~ of our Panel on Undergraduate Edu
cation.)
APPLIED MATHEMATICAL SCIENCES
There are now four major areas in the mathematical sciences that
have particularly direct and important relationships with other
sciences and technologies: computer science, operations research,
statistics, and physical mathematics (classical applied mathematics).
We shall ordinarily refer to these as applied mathematical sciences.
For statistics and computer science there is a more accurate term,
partly mathematical sciences, which we shall sometimes use in
recognition of the individual character of these fields and their
strong extramathematical components.
These four major areas must each have special attention if we
are to come close to meeting national needs. At the same time,
there is a need for general support of applied mathematical sciences
in a way, not closely tied to particular applications, that will encour
~ Superscript numbers refer to the list of references at the end of the report.
Mathematics and Society
7
age creative interaction between mathematics, science, and tech-
nology, and among the various applied mathematical sciences
themselves.
The sciences and technologies associated with the computer-
whether concerned with the nature of information and language,
the simulation of cognitive processes, the computer programs that
bring individual problems of all kinds to today's computing systems,
the software programs that convert cold hardware into a complex
computing system, or the hardware itself face an intense and grow-
ing challenge.
The field sometimes labeled operations research is now growing
rapidly, though not so explosively as computer science. Its emphasis
today is on solving problems of allocation (routing problems and
scheduling problems are two major types) and on a broad class of
operational applications of probability (inventory management and
improving the service of queues and waiting lines, for example).
Again there are national needs both for a substantial body of people
who can apply the techniques effectively and for a leadership that
can innovate, reshape, and transform.
The field of statistics and data analysis is older and more firmly
established than the two just described. Yet there is still a shortage
of statisticians who can bring mathematical techniques and insights
to diverse applications. The development of computer techniques
is also having a strong impact on this field.
Physical mathematics, also called classical applied mathematics,
has evolved into various modern forms. In its traditional form, it
emphasized the mathematics essential to classical physics and the
established fields of engineering. Even more it emphasized the
evolution of the mathematical models under study. Nowadays, the
concepts of physics have been expanded to include the well-
established aspects of quantum mechanics and the theory of rela-
tivity. New developments apply to an ever greater variety of
subject-matter fields, and we must now look once again toward a
closer collaboration and mutual stimulation between mathematics
and all the other sciences.
Alongside the four main applied mathematical sciences there are
still newer areas of application where no self-identifying community
of mathematical scientists yet exists areas of central importance
to a variety of national objectives of great and growing concern.
The interplay between mathematics and sciences and the mutual
stimulation, cooperation, and transfer of ideas among applied
8
Summary
mathematicians working in diverse areas all suggest that the applied
mathematical sciences, because of their common features, constitute
an area of study worthy of support in its own right.
CORE MATHEMATICS
The foundation of the manifold mathematical activities just dis-
cussed is the central core of mathematics the traditional dis-
ciplines of logic, number theory, algebra, geometry, and analysis
that have been the domains of the so-called "pure mathematician."
The relationship between the core and applied areas is not one-
sided; many of the essential ideas and concepts in the central core
can be traced ultimately to problems arising outside of mathematics
itself. In the central core, mathematical ideas and techniques, no
matter what their origin, are analyzed, generalized, codified, and
transformed into tools of wide applicability.
In assessing the importance of the core, one should keep in mind
that there is always an interplay and exchange of ideas between
so-called "pure" mathematics, that is, mathematics pursued pri-
marily for intrinsic intellectual and aesthetic reasons, and so-called
"applied" mathematics, that is, mathematics consciously used as a
tool for understanding various aspects of nature.
Thus geometry, literally "earth measurement," originated as an
applied art, presumably in the Nile delta. The Greeks transformed
it into a pure deductive science, the prototype of pure mathematics.
Among the geometric objects studied by the Greeks were the curves
(ellipses, parabolas, hyperbolas) obtained by intersecting a cone
with a plane. These "conic sections," though they may have been
discovered by observing sundials, were then of interest to the pure
mathematicians alone. Today conic sections are working tools of
engineers, physicists, and astronomers. On the other hand, calculus,
which was developed by Newton as a mathematical tool for studying
the motions of physical bodies, is also the foundation of a large
part of modern "pure" mathematics.
The most spectacular uses of core mathematics are its direct
applications in science and technology. Remarkably enough, it is
impossible to predict which parts of mathematics will turn out to
be important in other fields. We have one guide: Widely useful
mathematics, for the most part, has proved to be also the kind that
mathematicians earlier characterized as "profound" or "beautiful."
Mathematics and Society
9
Important mathematical ideas have also been generated by people
who were not professional mathematicians.
The time lag of 2,000 years between the invention of the conic
sections and their applications in astronomy is, of course, not typi-
cal of recent developments. But the unexpected character of the
application is typical. The theory of Lie groups, for instance, was
developed for many years because of its intrinsic mathematical
interest. It seems now to be the natural way of describing sym-
metries in elementary-particle physics. The theory of analytic func-
tions of several complex variables has been undergoing a dramatic
development during the last two decades. The experts in this theory
were quite surprised to discover its usefulness in quantum field
theory.
We stress once more the totally unpredictable nature of such
applications. It is not the motivation of the mathematician who
creates a new theory that determines its future relevance to other
fields of knowledge. In particular, one should not be repelled by
the seemingly frivolous origins of many mathematical theories. A
puzzle about the seven bridges in Kcinigsberg led to the theory of
graphs, a basic mathematical tool of computer science, and in-
directly influenced the development of topology. A question raised
by a professional gambler led Pascal and Fermat to the theory of
probability.
But the application of a particular mathematical result or a
specific mathematical concept is not the only way in which core
mathematics is used. The total impact of mathematics on science
and technology is more difficult to document but probably even
more important. In all such applications of mathematics (model
building and mathematical reasoning about models, statistical
analysis, the use of computers), the investigators will use some of
the concepts, methods, and results developed by core mathe-
maticians. In a typical case, however, they will not find in the
storehouse of core mathematics the precise tools they need but will
rather have to develop those tools either alone or in cooperation
with mathematicians. How successful they will be depends to a
large extent on the general status of mathematics in the country, on
the level of mathematical knowledge among the people involved,
and on the number and quality of mathematically trained people.
All this depends ultimately on a healthy and vigorous develop-
ment within the central core of mathematics. We are convinced that
without this one cannot have efficient use of mathematical methods
10
Summary
in science and technology, imaginative mathematization of new
fields, or spirited and effective teaching of mathematics at all levels.
The central core of mathematics is not static. It is now under
going rapid and in many ways revolutionary development. Many old
and famous problems are being solved. The traditional boundaries
between different mathematical subfields are disappearing. New
unifying ideas are applied with great success. Though dynamic, the
central core of mathematics preserves historical continuity and un-
compromisirlgly high standards.
THE POSITION OF THE UNITED STATES
IN MATHEMATICS
At the beginning of this century, mathematical research activity in
this country was chiefly concentrated in a very few centers. Note-
worthy was the center at The University of Chicago under the
leadership of E. H. Moore, himself trained in Europe. Among
Moore's illustrious students were Oswald Veblen, G. D. Birkhoff,
and Leonard Dickson, each a major figure in mathematics. In the
interval between World Wars I and II, mathematics in the United
States became somewhat more important relative to world mathe
matics.
Political developments in Europe in the 1930's led many Euro-
pean mathematicians to seek refuge in the United States and to
become active members of the American mathematical community.
This greatly stimulated mathematical research activity in this
country. The Institute for Advanced Study in Princeton became a
world center of mathematical research. Until after the Second
World War, however, financial support for mathematical activity
was extremely limited, and only a handful of undergraduates seri-
ously considered careers in research mathematics.
During World War II, the relevance of mathematics to the tech-
nological might of the nation and the critical shortage of mathe-
matically trained people became apparent. After the war, the
mathematical sciences became for the first time a concern of the
federal government, initially through the research departments of
defense agencies and then also through the National Science
Foundation. While the influx of federal money into the support
of mathematics was very modest compared with the funds poured
Mathematics and Society
11
into such expensive fields as high-energy physics, its effect on the
scientific life of the United States was stupendous.
Before World War II, the United States was a consumer of
mathematics and mathematical talent. Now the United States is
universally recognized as the leading producer of these. Moreover,
graduate education in mathematical sciences at major centers in
this country is far superior to that in all but two or three centers
in the rest of the world. Some more specific indicators of the posi-
tion of the United States in the mathematical world are given
below.
1. International congresses of mathematicians, meeting at roughly
four-year intervals, were inaugurated around the turn of the cen-
tury. The earliest such congress, in 1897 in Zurich, was attended by
only about 200, whereas at the most recent congress in 1966 in
Moscow, attendance was approximately 4,300. An invitation to a
mathematician to address an international congress evidences world-
wide recognition of his contributions to mathematical discovery at
the very highest level.
During the first four international congresses, those of 1897,
1900, 1904, and 1908, there were 26 invited addresses, of which only
one was by an American. During the four most recent congresses,
those of 1954, 1958, 1962, and 1966, there were 274 invited ad-
dresses, of which 96, more than one third, were by U.S. mathe
maticians.
2. The Fields Medals, for recognition of distinguished achieve-
ment by younger mathematicians anywhere in the world, were
established in 1932 bY the late Professor T. C. Fields. Beginning
with the Oslo congress of 1936, two Fields Medals have been
awarded at each international congress of mathematicians, except
that most recently at the Moscow congress of 1966 four were
awarded. In all, 14 Fields Medals have been awarded, with the
distribution of medalists by country as follows: France, four; the
United States, four; England, two; Finland, Japan, Norway, and
Sweden, one each.
Of the 12 Fields Medalists since 1945, three have been Americans
trained ir1 America: Paul Cohen (1966), John Milnor (1962), and
Stephen Smale (1966~. Three others are long-time residents of the
# Simon Newcomb, who addressed the 1908 congress in Rome on the history and
present status of the theory of lunar motion. Participants at this congress num-
bered 535, of whom 16 were from the United States.
12
Summary
United States and should now be considered to be members of the
U.S. mathematical community. Four or five others have been and
are frequent visitors to the United States, each having spent at
least one academic year here. Some of these are active collaborators
with various American mathematicians.
3. English has recently become the dominant language in world
mathematical circles. For instance, in the German journal, Mathe-
matische Annalen, the percentage of papers in English rose from
approximately 5 percent in the mid-1930's to nearly 20 percent in
the mid-1950's, and to 55 percent in the mid-1960's.
4. In some representative issues of three of the leading mathe-
matical journals of Europe, Acta Mathematica
(Sweden), Com-
mentarii 3Iathematici Helvetici (Switzerlandj, and Mathematische
Annalen (Germany), the following percentages of references to
papers in U.S. journals were found:
Acta
YEAR
Math.
Math.
Ann.
935
950
965
3%
12%
42%
12%
24%
25%
A%
4%
25%
5. There has been a significant increase in the number of foreign
mathematicians visiting in this country. Figures assembled by the
American Mathematical Society show that, in 1956, 73 foreign
mathematicians spent at least a semester at a U.S. university; in
1960, the number came to 144; in 1965 there were 199 such visitors.
In addition, other foreign mathematicians made briefer or more
casual visits or more lengthy stays outside universities.
