_ . DOCUMENT RESUME ED 031 411 SE 007 174 The Mathematical Sciences:A Report. National Academy of Sciences,Washngton, D.C. Pub Date 68 Note-265p. Available from-National Academyof Sciences, 2101 Constitution Avenue, Washington,D.C. 20418 EDRS Price MF-$1.00HC-$13.35 Descriptors-*ConferenceReports,*Educational Finance, *Mathematical Applications, *EdUcationalPhilosophy,GraduateStudy, *Mathematics Educamn,Research, Teacher Education, Presented Undergraduate Study are position statements forthe support of themathematical and a descriptionof the presentstate of both research sciences mathematics and related and educationin disciplines. The reportcalls attention to thepenetration of mathematics and mathematicalmodes of thouaht and the resultant into many newareas of scholarship increase in the relative enrollmentin undergradqate mathematics courses, especially at iheupper levels. Also identified mathematical methods is the increasing importance of in industry and government,the rapid growthin employment of mathematicalscientistsoutsidetheuniversities, mathematical and thepace of growth of sciences related to that of allother disciplines.Considerable attention is devoted to questions ofeducation, not only for users of mathematics. (RP) scientists but also for themany ......,T05=,. , , . .141..li,,TVA,Sitt kirk The Mathematical Sciences: A Report BY THE COMMITTEE ON SUPPORT OF RESEARCH IN THE MATHEMATICAL SCIENCES OF THE NATIONAL RESEARCH COUNCIL FOR THE COMMITTEE ON SCIENCE AND PUBLIC POLICY NATIONAL ACADEMY OF SCIENCES PUBLICATION 1681 National Academy of Sciences WASHINGTON, D.C. 1968 Available from Printing and Publishing Office National Academy of Sciences 2101 Constitution Avenue Washington, D.C. 20418 Library of Congress Catalog Card Number 68-61402 Z1, February 2, 1968 Dr. Frederick Seitz, President National Academy of Sciences 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 the mathematical sciences and a good picture of the present state of both research and education in mathematics and related disciplines. The ex traordinary 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 111 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 in 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 mathematicsan 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 pro Xu-ced under its aegis. One of these, the detailed report of the Committee's Panel on Undergraduate Education, analyzes the uniquely demanding 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, ina form accessible to a wide range of readers, expositions of the achievements, traditions, and prospects ina rich variety of mathematical fields. Sincerely, HARVEY BROOKS, Chairman Committee on Science and Public Policy iv 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 Committeeon Science and Public Policy HARVEY BROOKS, Harvard University, Chairman w. 0. 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 vi I CommitteeonSupport 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. WEYL, National Academy of Sciences HASSLER WHITNEY, Institute for Advanced Study C. N. YANG, State University of New York at Stony Brook TRUMAN Burrs, University of Virginia, Executive Director vii Foreword The Committee on Support of Research in the Mathematical Sciences (cosRms) 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 mathematicalresearch,especiallymodern 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