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c)
Recommendations
Our recommendations aim at enabling the community of mathe-
matical sciences to do all it can toward meeting national needs.
Even if all these recommendations are implemented, inherent limi-
tations on the rate of growth of research and teaching activities in
the mathematical sciences will make it impossible to meet all these
needs fully. It is therefore necessary to move as vigorously as possible
toward the recommended goals.
The needs, outlined in the preceding chapter and throughout our
report, are an inevitable consequence of the growing complexity
of our society and its increasing dependence on a variety of science-
based technologies. These technologies, and the physical, biological,
and behavioral sciences that support them, are becoming more and
more mathematized. They utilize the ideas and techniques of core
mathematics and the methods and results of physical mathematics,
statistics, and other applied mathematical sciences. Most of these
technologies depend, often in a crucial way, on effective use of
electronic computers. Accordingly, society needs a rising level of
mathematical literacy and competence at all levels, a sufficient sup-
ply of mathematically trained people in many sciences and profes-
sions, and a sufficient number of qualified mathematical scientists.
# Added in proof: All our estimates and predictions about future numbers of
PhD's were formulated before the February 1968 issuance of new Selective Service
rules affecting graduate students. If these rules should result in a serious deple-
tion of the graduate student population, this would, of course, intensify the
predicted shortage of PhD's in the mathematical sciences.
13

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14
Summary
Research in the mathematical sciences serves a twofold social
function. The research workers create, develop, refine, and adapt
the mathematical tools needed now and in the future for the mani-
fold applications of mathematics and for the growth of the mathe-
matical sciences themselves. The same people, because of the insight
and stimulation acquired through active participation in research,
are the leaders of the whole educational effort of the mathematical
community. A thoughtful policy concerning support of research
has to be responsive to both aspects.
In these recommendations, as in the report as a whole, we have
treated research and education together. Research apprenticeship,
as carried out by postdoctoral fellows and research instructors and
by graduate students writing theses, is inseparable from research.
Education at the undergraduate and early graduate levels is so
intimately related on a long time scale- to research, that to con-
sider the continuing good of one without the other would be fool-
ish. From a national point of view, these are parts of one problem,
a problem that must be faced in its entirety.
Our recommendations, to be stated and discussed below, fall
under six main heads:
Improvement in the quality of education in the mathematical
sciences at the undergraduate level through expanded federal sup-
port, especially at key points. There is a growing shortage calf college
teachers in the mathematical sciences. ~ Faculty improvement is
essential; and specific kinds of support of early graduate work can
avoid losses, both of people and of opportunities for sound train-
ing. (See Recommendations 14 through 17 and the report) of our
Panel on Undergraduate Education.)
Maintenance of momentum in research, research apprenticeship,
and graduate education. This will require continuing growth in
federal support of these activities. Even if this is provided, the
mathematical community will riot grow fast enough to meet
national needs. Accordingly, recent slackening in federal support)
are a cause of deep concern to the mathematical community and
should, we {eel, be a matter of general concern. (See Recommenda-
tions 1 and 11 through 13.)
Support for the explosive :,rowth~ of computer science, especially
~ This is documented and discussed in Chapter 7 under Quality and Distribution
of Mathematical-Science Faculty (see page 127) as well as in the report) of our
Panel on Undergraduate Education.
-I- See reference 2, Volume XVI, Appendix C.

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Recommendations
15
as a field of research in its own right. (See Recommendations 2, 3,
and 19.)
Support of research and education in the applied mathematical
sciences as such, and not merely in connection with either mathe-
matics or the particular sciences that use mathematics. Such sup-
port will not be expensive, but a failure to seize today's oppor-
tunity would be costly. (See Recommendations 4, 18, 21, and 22.)
Agencies and mechanisms for federal support of research and
research apprenticeship in the mathematical sciences. (See Recom-
mendations 5 through 10.)
A continuing program of information-gathering about research
and education in the mathematical sciences. (See Recommendation
20.)
RESEARCH
The encouragement and support that the research effort in the
mathematical sciences has received during the last 20 years in the
United States have made this effort eminently successful by any
test. The record shows numerous and great intellectual achie~re-
ments as well as substantial material and social benefits resulting
directly from the endeavors thus set in motion. The United States
now holds a position of leadership in all mathematical sciences.
To prepare for future needs, the momentum of research should be
preserved.
Growth and Level
Federal support of research and research apprenticeship in the
mathematical sciences has developed and nurtured a process of
growth limited more by natural abilities and by individual prefer-
ences among fields than by available funds. Even if this momentum
continues, the mathematical community will not grow fast enough
to meet national needs.
1. We recommend that, as a national policy, federal support for
basic research and research apprenticeship in the mathematical
sciences and in each of their major subdivisions including the
areas of core mathematics-continue to grow in proportion to the
number of appropriately qualif ed investigators and graduate
students.

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16
Summary
DISCUSSION An analysis of the present level of support for research
in the mathematical sciences is presented in Chapter 10. An extraor-
dinary development of mathematical competence and prestige has
accompanied modest expenditures: a relatively small (approxi-
mately $15 million in 1966) annual investment by the National
Science Foundation in academic mathematical research, a larger
total investment in basic mathematical research at universities
(approximately $35 million in 1966) by all federal agencies, the
much larger government investment in both basic and applied
research in the mathematical sciences (approximately $125 million
in 1966~. Vastly larger investments (e.g., $2 billion estimated for
the purchase and utilization of computers by the government in
1967) are importantly affected in their effective use by research in
the mathematical sciences.
The ratio of the national investment in basic research to the
investment in fields of application is so small, and the benefits
from basic research so large, that the goal of programs for the
support of basic mathematical research, in core mathematics and
in the applied mathematical sciences, should be limited only by the
availability of high-quality investigators.
We estimate that at present about one out of every six PhD's in
the mathematical sciences is consistently active in research. Over
the past five years the number of PhD's has been growing at an
average rate of 18 percent per year; a rate of at least 10 percent is
projected for the next five years (see Chapter 8~. We believe that
throughout the next decade the increase of qualified investigators
in the mathematical sciences will lag substantially behind society's
need.
We consider that the level and rate of growth of support was
adequate a few years ago at least within the core areas. (In the
other mathematical sciences there has been a shortage of funds for
basic research not tied to specific applications. In computer science
the support for unrestricted basic research in the software area has
been quite inadequate.) At present the failure of current and pro-
jected budgets to provide expansion adequate to take account of
larger numbers of qualified mathematicians, advancing costs of
research, and advancing overhead rates is a matter of serious con-
cern. The above recommendation can be thought of as urging the
* For 1966 the $35 million spent in basic academic mathematical research was
2.4 percent of the total federal research expenditures in that year.

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Recommendations
17
maintenance of at least a natural rate of growth throughout the
mathematical sciences.
The Special Situation of Computer Science
National needs and developments in contemporary technology
require the stimulation of higher than natural rates of progress in
some areas ot the mathematical sciences. Of particular urgency at
this time are the requirements of computer science.
2. We recommend that at the national level special priority be
given to support of the expansion of research and graduate study
in computer science. Appropriate actions would include: special
support for developing and updating courses, support for research
during the academic year when needed, grants to departments to
cover costs of computer usage in research, special attention to needs
for space, and expansion of numbers of research assistantships and
traineeships to stretch the capacity of all departments of high
quality.
DISCUSSION The proliferation of high-speed electronic data-process-
ing equipment, combined with the rapidly expanding art of its use,
constitutes one of the newest and most dynamic forces affecting the
mathematical sciences. There is a critical shortage of research
leaders in computer science, and urgent steps are required to over-
come it as fast as possible. Electronic computers have evolved
so rapidly that in many areas they have become an integral part of
operations before there has been time for the research needed to
determine the best, or even fairly good, ways of using them. The
vast expenditures for computing in the operations of the federal
government alone mean that even modest improvements achievable
from research at relatively low cost will almost surely pay off in large
cost reductions and improvements in service within a very few
... . .
years. Despite the large sums available to finance computing service
on campus, the money available for research in computer science
has been, with the exception of a few spectacular projects, seriously
inadequate.
~#
The role of computers in higher education across the board has
recently been studied in the Pierce reports of the President's Sci

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18
Summary
ence Advisory Committee. The detailed impact of projections made
by this and similar studies on the requirements for research and
education in computer science itself is, however, most inadequately
understood. This matter obviously demands urgent attention.
3. We recommend a thorough study of the implications for
research and advanced education in computer science of an ade-
quate implementation of the recommend ations of the Pierce
report.3
Basic Research in Applied Mathematics
Research of sufficiently high quality is wisely supported as research
for its own sake. In at least two areas of applied mathematics-
physical mathematics (sometimes called classical applied mathe-
matics) and the mathematics underlying operations research and
modern economics there are growing communities of mathemati-
cal scientists whose efforts meet this criterion. Their basic research
should thus be supported partly for its own sake and as a field in
its own right, rather than solely because of its immediately per-
ceived contributions to particular fields of application.
4. We recommend that f ed eral su pport f or research and research
apprenticeship in high-quality basic applied mathematics be given
on the basis of intellectual worth (recognizing, of course, the over-
all importance of progress in applied mathematics to many sciences).
DISCUSSION There is a significant distinction between such applied
mathematical sciences as physical mathematics and the mathe-
matics underlying operations research and such partly mathemati-
cal sciences as statistics and computer science. Mathematics is
applied in all these disciplines, as it is in so many others. The partly
mathematical sciences gain their identity and their only partly
mathematical nature-from the existence of problems, not initially
mathematical, that run broadly through most fields of science and
technology. The applied mathematical sciences, like physical mathe-
matics and operations research, have had the en ect of uniting
mathematics with specific areas of application- an effect that will
not disappear. However, they have not developed a sufficiently
strong identity of their own. Much can be gained from the develop

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Recommendations
19
ment of such identities, the foundations for which already exist in
intellectually worthwhile research of high quality. Recent work,
characterized by the evolution of ever more appropriate mathe-
matical models, together with the evolution of mathematical tech-
niques,~ display clearly the comprehensive nature of the discipline
of applied mathematics. Support of such work for its own sake is
now clearly justified and need not interfere with other work in
these areas supported because of its more immediate usefulness.
Sources of Support
The major federal support of research and higher education in the
mathematical sciences comes from a variety of agencies, the most
important being the National Science Foundation and certain of
the mission-oriented agencies, as indicated in more detail in
Chapters 10 and 11. We believe that activities in the mathematical
sciences will continue to be relevant to the tasks of all these agencies,
and that all of them should continue to share in the future support
of these sciences.
5. We recommend a level of growth that will enable the National
Science Foundation to continue effectively in its central role in
support of basic research and higher education in the mathematical
sciences.
DISCUSSION The National Science Foundation is the agency of the
federal government whose direct mission is the promotion and
support of basic research and education in the sciences. In carrying
out this mission it has evolved a versatile and highly effective array
of programs. A vital ingredient in the success of these programs has
been the system of peer-evaluation for ensuring high quality in the
research and educational activities supported.
The National Science Foundation has been a very important
source of support for the mathematical sciences, furnishing in
recent years close to one third of the total federal support of mathe-
matical activities in basic research and approximately half of the
federal support specifically allotted to mathematical activities in
higher education. As the sum total of our discussion in various parts
# In studies, for example, of the dynamics of the ocean, the structure of galaxies,
the physics of low-density gases, and the optimal use of water-supply systems.

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20
Summary
of the present report indicates, we feel that the whole range of
National Science Foundation programs in the mathematical sci-
ences has been valuable and well conceived. We urge a natural rate
of growth in most of these programs and a more rapid growth in
several. We also suggest a few new programs.
AL ~
Several mission-oriented agencies of the government rely on ad-
vanced mathematical techniques in accomplishing their tasks. For
such agencies it is important to maintain close contact with con-
temporary activity and competence in the mathematical sciences.
6. We recommend that mission-oriented agencies that expect to
derive significant benefits from the use of mathematical sciences
continue and expand their partnership with the community of
mathematicians by:
(:aJ participating in the sponsorship, not only of research that
promises predictable returns in applications, but also of basic inves-
tigations that enlarge the intellectual foundations of the field, and
(bJ evolving organized plans for bringing their unsolved scientific
problems to the attention of the mathematical-sciences community
and for provid ing the opportunity to qualified research mathe-
maticians to further, at times and in the depth of their choosing,,
the mathematization of major realms of scientific and technical
eff ort of national concern.
DISCUSSION The past record of sponsorship of mathematics research
by the mission-oriented agencies shows its effectiveness. The pos-
sibility of rapid adaptation and £ollow-through in connection with
newly developed techniques of mathematical analysis has greatly
assisted the mission-oriented agencies; and familiarity with some of
the difficult scientific and technological obstacles that must be
overcome has been instrumental in stimulating fruitful funda-
mental research endeavors.
Time may be lost and effort wasted in the achievement of tech-
nology-dependent objectives, and delays may occur in the progress
of mathematical techniques for the advancement of other sciences,
if we fail to develop and to maintain continuing channels of com-
munication between the mathematicians and the heavy users of
mathematical sciences. Thus we believe that it is vital to continue,

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Recommendations
21
and to strengthen where it is already established this pattern of
cooperation between mission-oriented agencies and the mathemati-
cal community, and to extend it further as our national commit-
ments venture into areas in which the role played by science and
technology becomes ever more intricate. Areas of possible expansion
include housing and urban development, transportation, manage-
ment of this country's natural resources, and the guidance of its
educational efforts.
Of the federally sponsored research in the mathematical sciences,
reported as basic by the supporting agencies, 60 percent is con-
ducted in academic institutions. Of this fraction, a little less than
one half is supported by the National Science Foundation. Other
agencies, mainly the Department of Defense, the Atomic Energy
Commission, and the National Aeronautics and Space Administra-
tion, account for the remainder of the university research and for the
bulk of the basic work conducted under government sponsorship at
nonacademic establishments, amounting to just under three fourths
of the total federal commitment to basic research in the mathe
mat~cal sciences.
7. We recommend that the Department of Defense, the Atomic
Energy Commission, the National Aeronautics and Space Admin-
istration, and the National Institutes of Health continue programs
for the sponsorship of basic research in the mathematical sciences,
and especially in physical and engineering mathematics, statistics,
computer science, and operations research and management science.
This support should increase at rates that will enable these agencies
to share responsibly in maintaining at least the natural growth
rate and that provide for higher rates of expansion in areas with
long-term relevance to the agency's mission.
DISCUSSION At issue here for the most part is basic research, con-
ducted at universities, government laboratories, and industrial
establishments in the areas of applied mathematics and statistics,
computer science, and operations research and management science.
The Department of Defense has the longest history of cooperation
with the community of mathematics, and there can be no question
that over the years defense technology has benefited in many and

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22
Summary
vital ways therefrom (e.g., in computer and communications tech-
nology, quality control arid life testing, and programming of mas-
sive supply operations).
The agencies in question bear an important share in the steward-
ship of one of this country's vital resources its research potential
in the mathematical sciences. Moreover, the multiple character of
the support has itself contributed greatly to the vigorous state of
American mathematics.
Forms of Supports
Since World War II, the overwhelming bulk of federal support of
research in mathematical sciences has beers support of individual
or group projects. Two decades of experience have demonstrated
the effectiveness of the project system.
S. We recommend that federal agencies sponsoring basic academic
research in the mathematical sciences continue to use the project
system as the primary mechanism for support.
DISCUSSION The project system has proved compatible with almost
every pattern of departmental university organization; it has also
proved flexible in adjustment to the tasks required and eRective in
linking the problems of sponsoring agencies with relevant contem-
porary mathematical research. A more detailed discussion and
evaluation of the project system is given in Chapter 10.
~ik ~
We believe, however, that project grants and contracts are not
always best suited for fulfilling several necessary tasks: assisting
departments of quality and promise to become truly outstanding,
developing new centers of leadership in the applied mathematical
sciences, and providing centers of research and graduate education
in geographical regions so far deprived of them.
# This committee is aware that authoritative voices have proposed very radical
revisions of the whole federal system for supporting academic research and uni-
versity education, abandoning the present forms of support in favor of direct
federal subsidy to universities. We feel that a discussion of this problem lies out-
side our competence. The fact that we do not mention these possibilities in our
report, however, should not be taken as evidence that we oppose them. It is self-
evident that in any thorough discussion of such radical changes the special prob-
lems of the mathematical sciences would have to be taken into account.

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Recommendations
23
9. We recommend increased exploitation of departmental grants
to supplement the traditional project grants and the recently estab-
lished university science development awards.
DISCUSSION The departmental Cant provides a mechanism for
support of the mathematical sciences with desirable flexibility in
meeting the diverse needs and capacities for academic mathematical
research. The university science development grants have proved
useful for the support of research and graduate educational activ-
ities that correlate well with those of other departments. In the
mathematical sciences these have usually tended to be better suited
to the needs of computing, statistics, and traditional applied mathe-
matics than to those of core mathematics. The departmental grants
should extend such opportunities to core mathematics, as well as
permitting other areas of the mathematical sciences to develop
their identities and programs.
Peer Evaluation
Federal support of basic academic research has relied on judgment
by qualified investigators in the evaluation of proposals for research
grants as well as in processing applications for fellowships. This
practice is doubtless one reason why such support has worked as
well as it has.
10. We recommend that the principle of peer judgment continue
to be used with respect to all forms of support for basic academic
research and research education. An essential part of peer judgment
should be representation of specialized areas, especially applied
mathematical sciences, on evaluation groups that are likely to deal
with applications from these areas.
DISCUSSION The purpose of such representation is to ensure that
proposals from an area of specialization receive a fair evaluation by
people familiar with the aims and standards of the field. (The
range of judgment required in peer evaluation is indicated by the
# We are aware of the twin dangers of "gimmickitis" and "hit and run" financing
so eloquently described by George Pake t"Basic Research and Financial Crisis in
the Universities," Science, 157, 517-520 (Aug. 4, 1967) i. \Ve hope that pro-
grams of departmental grants (and indeed all grants) will be so administered as
to minimize these dangers.

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32
Summary
for the U.S. Government. As its needs for mathematical techniques
have grown, however, the government has come increasingly to rely
on outside contractors rather than on the expansion of mathemati-
cal resources within its own research and development establish-
ments.
21. We recommend that a broadly qualified ad hoc group be con-
vened to study the desirability and feasibility of creating research
units within one or a few of the government's key research and
development establishments whose mission would be the develop-
ment and imaginative application of mathematical-science results
and techniques in contexts pertinent to federal efforts.
New National Goals
Growing up alongside the national programs that call for physics
and heavy engineering, there are now programs that, with increasing
frequency, receive at least equal priority and that are designed to
ameliorate the lives of individuals or to develop beneficial social
organizations, and hence involve the problems of environment and
people. The use of mathematical techniques in these contexts can
be very substantial and must be expected to depend largely on the
applications that are made of electronic data-processing facilities as
well as the constructions of systems analysis, operations research,
and the management sciences.
22. We recommend that recently established agencies of the fed-
eral government, whose missions strongly depend on science and
technology, cooperate with the National Science Foundation in a
thorough review of those activities in the mathematical sciences
that deserve attention in the context of their missions.
IMPLICATIONS FOR PROGRAM AND
RESOURCES PLANNING
Most of the recommendations of the previous sections make budget-
ing and policy demands on the conduct of the federally sponsored
research and advanced education programs in the mathematical
sciences. Relevant information and data have been developed

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Recommendations
33
throughout the report. For the convenience of program planners
and managers, the key elements are summarized here in direct
juxtaposition with our key recommendations.
The baseline for our projections is constituted by the allocations
of fiscal year 1966, the last complete year for which data became
available while the present survey was in progress. For that year,
the agencies of the federal government reported research and devel-
opment obligations totaling $125 million for research in the mathe-
matical sciences. Of this amount a conservative estimate identifies
at least $45 million as having been spent on basic research. The
remainder of $80 million has served to fund applied research
directly supporting the missions of the sponsoring agencies, as well
as a few major projects principally under the aegis of the Ad-
vanced Research Projects Agency of the Department of Defense-
in which it proved impossible to separate basic from applied com-
ponents. No long-term rates of growth have been projected for this
remainder item; its future level will be established by the needs
and opportunities as they are identified by the individual agencies.
Returning to the allocation of $45 million to basic research in
1966, we have estimated that about $35 million of this went for the
support of academic research, leaving a remainder of roughly $10
million for the conduct of basic mathematical-sciences research
under programs administered at the local level by the major fed-
erally sponsored research and development centers. Again, no
attempt has been made to project the growth of this fraction over
the next few years. Within the $35 million for academic research,
we have identified slightly over $90 million as allocated to project
support, the remainder accounting for other than project-type sup-
port, such as portions of interdisciplinary efforts, departmental and
institutional grants, and conference activities.
In addition to the $125 million in research and development
obligations in base year 1966, approximately $10 million of other
federal funds were allocated in that year to the support of graduate
study in the mathematical sciences and approximately $5 million
to programs of further faculty training and various activities in
undergraduate educational improvement in these fields. Our recom-
mendations dealing with levels and forms of support are addressed
principally to the budget for academic research ($35 million in
1966) and for these closely related items in higher education ($15
million in 1966~.

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34
The Staging of the Colleges; Research and
Research Education
Summary
The current level and rate of growth of the demand for profes-
sional competence in the mathematical sciences to conduct research
and research education has been estimated in connection with:
(a) Projected requirements for the teaching of mathematical
sciences at the college level;
(b) The funding of applied research in the mathematical sciences
by the mission-oriented agencies of the federal government;
(c) Manpower demands in certain of the applied mathematical
sciences, especially computer science and those intervening in the
field of operations research.
Conservative estimates anticipate a need of some 8,000 additional
full-time college faculty members by the academic year 1970-1971
over the 10,750 in service in 1965-1966. According to these estimates
only about 41 percent of these new faculty members will have
doctorates, even on the optimistic assumption that in the interven-
ing five years no less than 70 percent of the new PhD's will be teach-
ing mathematical sciences at universities and four-year colleges.
This would represent a lowering of quality in the sense that cur-
rently 46 percent of those teaching the mathematical sciences in
universities and four-year colleges have PhD degrees in the mathe-
matical sciences. (Another 6 percent have PhD degrees in other
fields, primarily education.)
Government support of applied research in the mathematical
sciences has grown at an average rate of 51 percent per year during
the period 1960 to 1966, largely because of rapidly increasing
commitments In computer research and development and in oper-
ations research. Having now reached a level of about $80 million
per year, this support shows a slackening growth rate, which is,
however, still running well ahead of the annual growth rate for
support of basic research. Correspondingly, growing manpower de-
mands in the applied mathematical sciences indicate that current
shortages will become even more severe in the next few years.
Thus, shortages of mathematical manpower, for both teaching
and research, are increasing. These are occurring in spite of a rela-
tively high rate of growth in PhD production during recent years
averaging 18 percent per year over the period 1960-1965 and in
.. . .

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Recommendations
35
spite of the fact that not many capable graduate students appear
to have been lost because of lack of support during that period.
As a consequence, we have had to conclude that even optimal
planning and management of sponsored programs in research and
professional education will not build up the mathematical-sciences
community fast enough to meet national needs. It is therefore im-
portant that economic deterrents not retard the replenishment of
the group on which the responsibility for innovation, research
training, and college education devolves. Hence, our recommenda-
tions (Recommendations 1, 8, 9, and 11) call for an expansion in
the support of basic academic research and research apprenticeship
in the mathematical sciences at least at a rate that will not inter-
pose economic barriers to the achievement of competence in re-
search and research education.
In the absence of much of the necessary information, only rela-
tively crude planning factors can be established. Taking Recom-
mendations 1 and 11 together because they involve common ele
. ~
meets, we estimate that:
(a) In the $20 million worth of project research, a total of
approximately 920 tenure research investigators (TRI) participated,
so that the average expenditure of such funds (investigators'
salaries, visitors, research associates and assistants, secretarial sup-
port, publication, overhead) amounted to around $22,000 per TRI.
(b) One out of six PhD's in the mathematical sciences ends up
doing research found worthy of support. Allowing for the elapse
of approximately five years between receipt of the doctorate and
the acquisition of tenure, the figures on earned doctorates provide
an estimate of approximately 1,400 for the group of Trues by 1971.
(c) The planning factor of $22,000 per TRY will MOW, because of
(i) the increasing cost of research, which is certainly no less than
4 percent per year, (ii) the cost of growing requirements for machine
computing, which cannot be reliably estimated at present but, in
particular instances, reach magnitudes that dwarf all other costs,
and, finally, (iii) the increase in the number of research assistants
per TR! called for by Recommendation 11. We arrive, conservatively,
at a minimum of $29,000 per All by 1971.~
(d) Comparable growth should be provided in the nonproject
forms of academic research support, under the assumption that no
~ In computer science itself the corresponding average annual cost is estimated
in the section on Computer Science (page 205) to be approximately $60,000 per
TRI.

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36
Summary
radical change is made in the balance with project support (Recom-
mendation 9~.
(e) Graduate-student enrollment will grow from its 1966 level
of 9,400 to approximately 18,400 by 1971. With 1,300 covered by re-
search assistantships See (b) above], a total of 4,800 will have to be
accommodated by fellowship and traineeship programs if one third
of them are to be given research apprenticeship support, in accord-
ance with Recommendation 11. The corresponding figure for 1966
is 1,834, and the at least 4 percent per year increase in cost of re-
search also affects the rate per research apprentice in these programs.
Total costs can now be projected for 1971 and converted into an
annual percent rate of growth for the period 1966-1971. Specifically,
there would be 566 million for academic research, of which at least
$38 million would be in the form of project research, and another
$30 million in fellowship and traineeship support. The equivalent
annual growth rates turn out to be 14 percent for research, 24
percent for research apprenticeship' and 16 percent overall. Of
special significance and thus to be emphasized is the relatively
greater increase in support for research apprenticeship than for
research, in order to prepare for meeting national needs in the
mid-1970's.
# There is another, simpler, kind of calculation that also leads to this over-all
annual growth rate for the period 1966-1971. The Westheimer reports on chem-
istry (page 166) tied PhD production to total federal obligations for basic re-
search in the field, leading to a figure for "Federal support cost per PhD
produced." For 1962, the year reported on in the Westheimer report, the mathe-
matical sciences had the lowest such cost, namely approximately
$22.6 million _ $55 OOO/PhD
410 PhD's '
For 1966, we find that it was approximately
$50 million $65 OOO/PhD
indicating that from 1962 to 1966 the cost per PhD had increased by approxi-
mately 4 percent per year. Supposing it to continue to increase at this rate, this
cost will be approximately $80,000/PhD by 1971. With about 1,300 PhD's pro-
jected to be produced in 1971, this gives, for federal support of basic research in
1971,
$80,000/PhD X 1,300 PhD's = $104 million.
This is slightly more than double the 1966 figure and may be computed to call
for growth at an average annual rate of just about 16 percent.

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Recommendations
37
The final sense of our recommendations, however, does not lie
in these particular figures but in identifying the factors that must
serve to determine them. As our knowledge regarding the latter
improves, both budget projections and growth rates can be adjusted
accordingly. Hence, our recommendations (especially Recommenda-
tions 3 and 20) call for continuing efforts of investigation and
analysis to develop suitable planning factors and information about
research and education in the mathematical sciences so that evolving
needs and trends can be appraised more reliably.
As the demand for mathematical-science instructors with PhD
education will continue to outrun supply in the shorter time frame
in any case, their number must be increased by using opportunities
that have so far been neglected for one reason or another. Par-
ticular programs toward such an end are the subject of Recom-
mendations 13 and 14. For the 50 postdoctoral teaching fellowships
of Recommendation 12, the cost of supplementary stipends (about
58-9 thousand each) and administration should not exceed $600
thousand per year. The special part-time graduate fellowships for
women under Recommendation 13 would constitute about 10 per-
cent of all available full-time graduate fellowships in the mathemati-
cal sciences, i.e., about 100 initially and perhaps 200 five years
hence. Cost, including administration, would range correspondingly
from $300 thousand to $600 thousand.
In addition to the need for a basic policy that maintains the
present momentum of research and research apprenticeship in the
mathematical sciences across the board, our recommendations recog-
nize certain critical areas in which more than ordinary efforts are
needed if the mathematical-sciences community is to render the
required services in today's social fabric. These are Recommen-
dations 2, 3, and 4 relating to the support of research and research
education in computer science and in the applied mathematical
sciences as such.
Planning factors to gauge the development of research and re-
search education in computer science are provided by the size, the
cost, and the growth rate of this country's computer establishment.
It is clear that it will be some time before the schools and univer-
sities will have caught up even approximately with the require-
ments that this is generating. Not the least among these is the grow-
ing use of computers in the educational process itself, the full-scale
expansion of which has been recommended by the President's Sci-
ence Advisory Committee in the Pierce report.3 At the same time,

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38
Summary
the availability of support for computer science as a field of research
in its own right has been minuscule in comparison to its economic
and intellectual importance. Under the conditions, there is no
choice except to stretch the capacity of high-quality departments as
far as this is possible with resources in faculty, space, and computer
facilities, potentially available to them, in order to engage in
original research and, especially, to create opportunities for research
· . .
apprenticeship.
The appropriate level of support for such programs is notoriously
difficult to project, one of the more recent proposals suggesting that
it be made a flat percentage (e.g., ~ percent) of the $415 million esti-
mated by the Pierce report as being required by 1971 to cover the
educational use of computers in universities. There are at present
a dozen or more departments of computer science that would qual-
ify for support under Recommendation 2; five years from now, their
number may well have tripled. This suggests a program, starting at
$6 million and stabilizing at $15-20 million no later than five years
hence. If, however, the doubling time of the number of eligible
departments should be two years, rather than the three years esti-
mated above, these projections would represent gross underestimates
for the latter years.
In contrast, the support of a few research and research training
programs of exceptional quality in the applied mathematical sci-
ences for their own sake will be of low cost. There are today prob-
ably no more than half a dozen universities that would qualify for
such a program. Program cost would therefore amount to an initial
$500,000, growing to $1.~-2 million per year in the course of the
next three or four years and stabilizing at that point.
Undergraduate anct Early Graduate Education
Of the recommendations in these critical areas, three relate to {acuity
improvement in undergraduate colleges. Two of these (Recom-
mendations 16 and 17) call for appropriately directed expansion of
existing programs. Doubling in the course of five years the number
of available National Science Foundation Science Faculty Fellow-
ships in the mathematical sciences alone would increase the program
by only about $1.5 million a year. Planning in this connection, how-
ever, will have to take into consideration the Science Faculty Fel-
lowship program as a whole in the establishment of proper balances.
With respect to summer institutes, as proposed in Recommendation

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Recommendations
39
17, the underlying current estimates are that, of the roughly 10,000
college teachers in the mathematical sciences, approximately 10
percent should have the opportunity each year of participating in
summer institutes. Effective training groups run about 30 students
each, which would lead to some 35 institutes per summer, tripling
the currently supported number. Costs per institute will range from
$70,000 to $100,000, so that initial program totals would lie at
around $3 million, rising in future years.
Recommendations 14 and 15 propose certain forms of student
assistance. Neither of these programs is likely to be very expensive.
The support of graduate students in colleges and universities, offer-
ing no PhD degree but a high-quality master's degree in the mathe-
matical sciences, is meant to be experimental and therefore limited
to perhaps 15 to 20 typical such schools. The program of special
fellowships or forgivable loans to promising students, emerging
from colleges with inadequate departments in the mathematical
sciences, would be gauged to make 200 awards per year at a total
program cost of $1 million.
The development, finally, of undergraduate programs in the
applied mathematical sciences is largely an internal decision of
university administrations, which might be expedited only periph-
erally by the possibility of federal support. If the primary resources
exist to offer such programs be they in comprehensive applied
mathematics, statistics, computer science, or operations research-
the difference of providing the necessary management, housekeep-
ing, once and classroom space, and other facilities is more a matter
of support of university infrastructure as a whole than of particular
fields. The moral pressure of concerned interest, backed up as neces-
sary by an occasional subsidy of the right sort, is all that is needed
to implement Recommendation 19.
Applied Research
Applied research in the mathematical sciences, supported by mis-
sion-oriented agencies in the context of and for immediate utiliza-
tion in specific applications of significance to the sponsoring agency,
is of very recent origin. As late as 1960, it amounted to no more
than $6.6 million out of total research obligation of $23.6 million-
all of 28 percent. By 1966, it had increased to certainly not less than
$62.4 million and probably more nearly S78.4 million, amounting
to between 53 percent and 63 percent of the total for mathematical

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40
Summary
sciences research. Recommendations 6 and 7 essentially call for
agencies that traditionally sponsored applied research in the
mathematical sciences to continue to do so at levels and in direc-
tions in which it has been found useful, and for newly established
agencies to contemplate comparable participation in the develop-
ment, adaptation, and use of mathematical techniques relevant to
their problems. No long-term rates of growth are projected, and
levels are expected to be set by needs and opportunities as they
are identified. The yearly rate of growth of this portion of the
research budget has been decreasing, but the latest figures place it
still above the growth rate for academic research in comparable
periods.
Sources and Forms
Continued participation in the support of academic research by
the National Science Foundation and by other agencies is called
for by Recommendations 5 and 6. No quantitative apportionment
of relative shares is proposed, provided the levels of Recommenda-
tions 11 and 12 are met.
Recommendations 8 and 9 identify the relative functions of
project funds and other forms of support for academic research.
Again trends rather than absolute quantities are stressed. It is pro-
posed that project support remain pre-eminent and, among the
various forms of broader support, areas as well as departmental
grants be given increased utilization as against interdepartmental
and institution-wide grants for the development of quality in the
mathematical sciences.
Cautionary Remarks
The mathematical sciences, perhaps more than any other major
discipline of modern science, play a pivotal role in a wide variety
of contexts, both in opportunities for application and in require-
ments of education. The programs recommended in our report
reflect this diversity. Each of these programs is designed to meet
needs that in some contexts have emerged as urgent, but no single
priority scale applies across the board; hence their adoption as a
whole or in part, down to some given level of priority, cannot be
made the subject of a single action at the national level. Instead,
we expect our recommendations to be implemented, either indi

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Recommendations
41
vidually or jointly, as permitted by the internal priorities of the
various agencies involved and by the national emphasis given to
the goals of which our objectives are a part. Many of our recom-
mended programs may, of course, be completely merged into other
similar but broader programs.
The same considerations apply to our crude projections of costs,
with their wide differences in reliability, amounts involved, and
periods spanned. To combine them all into one grand balance
sheet would not be very useful. Each of the contexts for implemen-
tation of these programs has its own scale of benefits relative to
which the programs must be weighed.
Since a number of these programs will be funded by agencies
that share in their implementation as parts of other, broader activ-
ities, often of a more applied nature, a good deal of support calf the
mathematical sciences may be termed "implicit." In particular, this
implies that the federal government must be alert to the impact on
the mathematical sciences of abrupt shifts in criteria for support
· · . . . .
Wit tin mlsslon-orlentec agencies.

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