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Th
.
BY THE
10 lVui~ ~
SAC
Mathematical '^
sciences.
.
A Report
COMMITTEE OIL SUPPORT OF RESEARCH IN THE
MATHEMATICAL SCIEN CES
OF THE
NATIONAL RESEARCH COUNCIL
FOR THE
COMMITTEE ON SCIENCE AND PUBLIC POLICY
NATIONAI, ACADEMY OF SCIENCES
PUBLICATION 1681
National Academy of Sciences
WASHINGTON, D.C.
1968
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Available from
Printing and Publishing Once
National Academy of Sciences
2101 Constitution Avenue
Washington, D.C. 20418
Library of Congress Catalog Card Number 6841402

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February 2, 1968
Dr. Fred crick Seitz, Presid ent
Na t lo na l A cad emy o f Scien ces
2101 Constitution Avenue
Washington, D.C. 20418
Dear Dr. Seitz:
The Committee on Science and Public Policy takes pleasure in
forwarding to you the report of the Committee on Support of
Research in the Mathematical Sciences. Our committee has
reviewed this report at several stages in its development, and we
feel that it presents a clear and cogent case for the support of tile
mathematical sciences and a good picture of the present state of
both research and ed ucation in mathematics and related d isciplines.
The extraordinary rate at which classical mathematical theorems
have fallen before the onslaught of young American mathematicians
in recent years is impressive, as is the leading position acquired
by the United States in recent times.
The report calls attention to the penetration of mathematics
and mathematical modes of thought into many new areas of
scholarship and the resultant great increase in the relative
enrollment in undergraduate mathematics courses, especially at
the upper levels. It also brings out the increasing importance of
mathematical methods in industry and government, the rapid
growth in employment of mathematical scientists outside the
universities, and the pace of growth of mathematical sciences
relative to that of all other disciplines.
The report calls attention to the special problem of computer
science in universities, brought about by its high cost relative
to the rest of mathematical research and education and by the fact
that it is both a scientific and an engineering discipline, which
has made it difficult to fit into the academic structure. Meaningful
research in computer science often requires rather large-scale
support for complex projects of a partly developmental nature
that do not always lend themselves to the project-grant type of
support typical in pure mathematics and some other scientific
disciplines. The development of computer science only as a
. · .

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by-product of the application of computer techniques in other
fields often results in failure to develop a distinctive body of
theory and technique in computer science in its own right.
The report identifies special problems concerning mathematical
statistics and physical (i.e., classical applied) mathematics. In
physical mathematics the problem is not the need for rapid growth
of financial support, as in computer science, but for a place i'n the
academic structure that does not tie it too closely to a particular
set of applications. The authors of this report feel that there
are too few distinguished applied-mathematics groups in American
universities, and' that special encouragement is needed for
building such groups. Whatever place applied mathematical
sciences find in the academic structure, however, it is essential to
maintain a strong and continuing interaction between these
sciences and pure mathematics an interaction in which ideas and
people must move in both directions.
An outstanding feature of this report in comparison with some
of the earlier disciplinary reports is the attention it devotes to
questions of education, not only for mathematical scientists but
also for the many users of mathematics.
The Committee on Support of Research in Mathematical
Sciences is to be congratulated not only for its impressive work in
the preparation of the present volume but also for the valuable
contributions of two supplementary volumes produced under its
aegis. One of these, the detailed report of the Committee's Panel
on Und ergrad uate Ed ucation, analyzes the uniquely d emand ing
special problems of American undergraduate education in the
mathematical sciences and makes many suggestions for dealing
with these problems. The other, a collection of twenty-two essays
by distinguished mathematical scientists, brings together, in a
form accessible to a wide range of readers, expositions of the
achievements, traditions, and prospects in a rich variety
of mathematical fields.
Sincerely,
HARVEY BROOKS, Chairman
Committee on Science and Public Policy
1V

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THIS REPORT on the mathematical sciences is another in the series
of comprehensive survey reports on major fields of science pre-
pared under the aegis of the Academy's Committee on Science and
Public Policy.
With the growing importance of the mathematical sciences, both
in their own right and through their penetration of other sciences
and of technology, it has become clear that a report on mathe-
matics should be added to the survey reports. Thus the Committee
on Support of Research in the Mathematical Sciences was appointed
to undertake the preparation of such a report under the chairman-
ship of Professor Lipman Bers.
I am pleased to express gratitude to the Committee for its dedi-
cated work. I trust the findings and recommendations set forth here
will prove useful to those with responsibilities in the continuing
support of American science.
FREDERICK SEITZ, President
National Academy of Sciences
v

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Committee on Science ant! Public Policy
HARVEY BROOKS, Harvard University, Chairman
W. O. BAKER, Bell Telephone Laboratories
PRESTON E. CLOUD, JR., University of California, Los Angeles
HARRY EAGLE, Albert Einstein College of Medicine
CARL ECKART, University of California, San Diego
HERBERT FRIEDMAN, U.S. Naval Research Laboratory
RALPH W. GERARD, University of California, Irvine
J. G. HARRAR, The Rockefeller Foundation
G. E. HUTCHINSON, Yale University
MARK G. INGHRAM, University of Chicago
W. D. MCELROY, The Johns Hopkins University
GEORGE A. MILLER, Rockefeller University
KENNETH B. RAPER, University of Wisconsin
HERBERT A. SIMON, Carnegie-Mellon University
CYRIL STANLEY SMITH, Massachusetts Institute of Technology
ROBERT E. GREEN, National Academy of Sciences, Executive Secretary
V1

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Committee on Support of Research in the
Mathematical Sciences
LIPMAN BERS, Columbia University, Chairman
T. W. ANDERSON, Columbia University
R. H. BING, University of Wisconsin
HENDRIK W. BODE, Bell Telephone Laboratories
R. P. DILWORTH, California Institute of Technology
GEORGE E. FORSYTHE, Stanford University
MARK KAC, Rockefeller University
C. C. LIN, Massachusetts Institute of Technology
JOHN W. TUKEY, Princeton University
F. J. WEY1,, National Academy of Sciences
HASSLER WHITNEY, Institute for Advanced Study
C. N. YANG, State University of New York at Stony Brook
TRUMAN BOTTS, University of Virginia, Executive Director
V~l

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Foreworc!
The Committee on Support of Research in the Mathematical
Sciences (Cosa~Msj was appointed by the Division of Mathematical
Sciences of the National Research Council at the instigation of the
Committee on Science and Public Policy of the National Academy
of Sciences. Our task was to assess the present status and the pro-
jected future needs, especially fiscal needs, of the mathematical
sciences. It was clear to us from the very beginning of our work
that our report would have to differ somewhat in structure from
the corresponding reports for other disciplines that had already
appeared.
Though mathematics provides the common language for all sci-
ences, we realize that even scientific readers of our report, let alone
nonscientists, may feel that they are not adequately informed about
what mathematical research, especially modern mathematical
research, consists of. Similarly, even professional mathematicians,
or scientists who customarily use mathematics in their work, may
be unaware of the manifold applications of mathematics in various
sciences and technologies, especially the new applications influenced
by the computer revolution.
To provide additional background of factual information con-
cerning the mathematical sciences, we are supplementing our report
with a collection of essays, written by distinguished authors on
various topics in mathematics, in the applied mathematical sciences,
and in the applications of mathematics. With three exceptions,
which are reprints, these essays were written expressly for this col-
lection. They are intended not only for the nonmathematical
scientist but also for the scientifically oriented layman.
1X

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Mathematics pervades our whole educational system. As a matter
of fact, we believe that the mathematical community has no obli-
gations more important than those concerned with education, the
most critical area being collegiate education. We have, therefore,
included in our report questions of policy regarding higher edu-
cation. Our Panel on Undergraduate Education has carried out an
intensive study of this area; its report is presented in a separate
volume of our report.
Simultaneously with our activities, the Conference Board of
the Mathematical Sciences has been carrying out a survey of research
and education in mathematics, and its Survey Committee has agreed
to act as a fact-finding agency for our Committee. The Conference
Board Survey Committee's report will contain a wealth of factual
and statistical material pertaining to the matters discussed in our
report. We take this opportunity to express our gratitude to the
Survey Committee and to the Ford Fo~n~1~ion which Nor
their work.
^ ~-erred ~
The activities of our Committee have been financed mainly by
a grant from the National Science Foundation. This has been sup-
plemented by smaller grants from the Sloan Foundation, the Con-
ference Board of the Mathematical Sciences, the American Mathe-
matical Society, the Association for Computing Machinery, the
Association for Symbolic Logic, the Institute of Mathematical
Statistics, the Mathematical Association of America, the National
Council of Teachers of Mathematics, the Operations Research
Society of America, and the Society for Industrial and Applied
Mathematics. Columbia University has generously provided us with
once space and many auxiliary services. To all these organizations
we express our thanks.
We are deeply indebted to the authors of the essays, to the chair-
men and members of our panels, and to the many other individuals
who have contributed their time and expertise to our undertaking.
January 20, 1968
x
LIPMAN BERS
Chairman, Committee on Support
of Research in the
Mathematical Sciences