Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 45
The Community of Mathematics
HISTORICAL PERSPECTIVES
Starting with the first expression of rational thought, mathematical
thinking influenced the development of our culture in general and
was decisive in the creation of science and technology. The inven-
tion and development of the infinitesimal calculus preceded the
eighteenth century age of enlightenment; these mathematical ideas
and techniques made possible the beginning of the industrial age.
The many mathematicians engaged in the development of analysis
extended the scope of mathematics itself and provided tools for
solving problems as diverse as those of the motions of dynamical
systems, the flow of liquids and gases, problems involving the
strength of materials, and, late in the nineteenth century, the theory
of electrical and magnetic phenomena. The definitive formulation
of this latter area by Maxwell required still more advanced mathe-
matics. Alongside these developments in analysis, general mathemati-
cal ideas in algebra and geometry found their important role in
foundations of physical theories. The non-Euclidean geometries and
more abstract spaces became the mathematical precursors of new
physical theories the theory of relativity, for example. Certain
abstract mathematical constructions constitute a prelude to formu-
lations of quantum theory. By the end of the nineteenth century
and at the beginning of the twentieth century a vast mathematical
apparatus was being used not only for dealing with problems in
astronomy and in physics and in many branches of engineering, but
also in sciences like chemistry, which are built upon theories in
45
OCR for page 46
46
The State of the Mathematical Sciences
physics and involve an increasing use of mathematics on all levels
from the elementary to the most abstract.
Because of the long, deep, and essential connection between the
growth of mathematics and that of physics, mathematics is often
classed with the physical sciences; its impact on civilization until very
recently was almost exclusively through its applications to astron-
omy, physics, chemistry, and engineering. Yet mathematics has
features that distinguish it from all experimental sciences and
perhaps from all other sciences.
One is the deep roots that mathematics has in the past. Some of
the problems on which mathematicians work today go back several
thousand years. Much of contemporary mathematics is a direct
continuation of the work begun by the ancient Greeks. A more basic
difference between mathematics and experimental sciences, includ-
ing even the most theoretical branches, lies in the extent to which
mathematical research originates within the body of mathematics
itself. Physicists, chemists, biologists, and psychologists are more or
less directly concerned with observable phenomena. Mathematicians
often receive the seed of their problems from the outside world,
sometimes through other sciences, but once a mathematical problem
or concept has been formulated, it acquires, as it were, a life of its
own. In a simplified way mathematics consists of abstractions of
real situations, abstractions of abstractions of real situations, and
so on. It is surprising but true that these abstractions of abstractions
often turn out to further our knowledge and control of the world
in which we live.
During and after the Second World War, mathematical thinking
penetrated the sciences and the development and organization of
technology at an ever-accelerating rate. Probability theory has
played an increasing role not only in the natural sciences but in-
creasingly in all work dealing with problems of organization, in
economics, and in sociology. In the last decades, problems of life
sciences in biology especially have begun to require mathemati-
cal formulations. Dramatic recent discoveries in molecular biology
open still newer prospects for the use of mathematical ideas and
models in the understanding of life processes. This increasing rate
of mathematization is further accelerated by the development and
use of automatic computers.
New fields of technology, made possible through the exploitation
of the discoveries in physics of this century, require sophisticated
use of mathematics as an absolutely necessary condition. The enor
OCR for page 47
The Community of Mathematics
47
mous progress in electronics that has changed the entire pattern of
communication including information theory, network synthesis,
feedback theor~would have been unthinkable without all the
mathematics that goes into the development of basic concepts and
into their application. The technology of nuclear energy and the
conquest of space are being made possible only through the advent
of the automatic computer, a development which in turn was predi-
cated upon all the work in foundations of mathematics and mathe-
matical logic, along with the technology of electronics.
Mathematics is involved in the problems of communication-
through the telephone, the radio, the transmission of pictures-
and, more recently, in the exponentially growing problems of collec-
tion, classification, and transmission of data in general. Physical
communication and transportation, on the ground and especially
in the air, require increasingly a study of organization and tra~c-
control problems of mathematical character. It is no exaggeration
to say, therefore, that the fundamental problems of national life
depend now, more than ever before, upon the existence and the
further growth of the mathematical sciences and upon the con-
tinuing activities of able people skilled in their use.
The developments sketched above amount to a mathematization
of our culture. The remarkable growth in this mathematization is
easily documented. For instance, an authoritative report on applied
mathematics submitted to President Roosevelt in 19408 estimated
that in the future industry might require as many as 10 mathe-
matics PhD's per year. By contrast, the figures of new mathematics
PhD's entering industry in 1966 has been estimated at 200. Several
new disciplines, like mathematical linguistics, hardly existed 25
years ago. Less dramatic but not less significant is the increase in
mathematical sophistication in all fields of engineering.
In all engineering journals, 50 percent or more of the papers in recent
issues would probably have been rejected by the editors as too mathematical
ten or fifteen years ago.
The preceding lines are from W. Prager's memorandum to
COSRIMS. In this same memorandum he mites that:
Twenty years ago, the typical mathematical preparation of the future engi-
neer consisted of a review course in trigonometry and analytic geometry and
a formal course in calculus aimed at proficiency in handling routine prob-
lems. Today, the review course has disappeared from the curriculum while
OCR for page 48
48
The State of the Mathematical Sciences
the emphasis in the second course has shifted from problem solving to the
understanding of basic concepts. A fairly rigorous course on ordinary differ-
ential equations has been added to the typical curriculum, and students are
encouraged to take as many mathematical electives as can be fitted into their
program. Foremost among these are vector and tensor analysis, complex
variables, transform methods, variational calculus, probability and statistics,
and numerical analysis.
Mathematics was, of course, always a traditional part of the edu-
cation of an engineer. But now it is becoming an essential part of
the education of biologists, psychologists, economists, and many
others. Therefore, the first obligation the mathematical community
has toward society is in the realm of education. We consider these
problems in Part III of this report.
Another novel situation is the greatly increased number of people
doing mathematical research. Mathematical sciences have shared
with all sciences the almost explosive increase in the number of
investigators. For instance, Mathematical Reviews, an international
abstracting journal, reviewed approximately 2,000 papers in 1940
and approximately 15,000 in 1966. The number of mathematical
journals published throughout the world has increased correspond-
ingly. These figures in themselves would be of little significance
had the increase in quantity been paid for by a drop in quality.
While objective studies of quality in mathematical or indeed any
research are difficult, it is the considered opinion of this Committee
that both the average quality of published research and the quality
of the best work today are comparable with what they were in the
1920's or in the decade preceding World War I.
THE NATURE OF MATHEMATICS
The essential features of mathematical search and discovery have
remained the same for centuries. So have its main sources of in
spiration: the external world and its own internal structure.
Mathematics done for its own sake is traditionally designated as
"pure mathematics" and mathematical investigations aimed at in-
creasing our understanding of the world are classified as "applied
mathematics." Such a division of mathematics into pure and ap-
plied, however, is difficult to maintain; the origin of many important
mathematical ideas can be ultimately traced to applications, and,
OCR for page 49
The Community of Mathematics
49
on the other hand, mathematics created for its own sake often turns
out to be important for applications.
The mode of origin of a mathematical concept or technique may
be only very loosely correlated with its ultimate applications.
Fourier considered mathematics as a tool for describing nature.
But the impact of "Fourier series," crucially important as these
series are in physics and engineering, has been particularly felt in
some of the "purest" branches of mathematics. Cayley, on the other
hand, believed that matrices, which he invented, would never be
applied to anything useful (and was happy about it). They are now
an everyday working tool of engineers, physicists, economists, and
.
statisticians.
Geometry originated in practical problems of land measurement.
It was developed by the Greeks into an axiomatic system and inten-
sively studied for its own sake. Among Euclid's axioms, there was
one (the parallel axiom) that was less intuitively evident than the
others. The realization that this axiom could not be logically de-
rived from the remaining ones came only after centuries of struggle
and led to the discovery of non-Euclidean geometries (see Coxeter's
essay in reference 7~. At the time of their birth, these geometries were
the answer to a purely logical problem, and even their creators did
not dare to attribute to them any counterparts in the real world.
But Riemann already suspected that the physical universe in the
large may be non-Euclidean, and his intuition was brilliantly con-
firmed by Einstein's theory of general relativity, a theory that still
provides mathematicians with challenging problems.
These examples show the futility of attempting to draw dividing
lines between pure and applied mathematics. The difference is not
so much in subject matter as in original motivation. Also the name
"pure mathematics" is unfortunate since it implies a monastic aloof-
ness from the world at large and an isolation from its scientific,
technological, and social concerns. Such an aloofness may be char-
acteristic of some mathematicians. It is certainly not characteristic
of mathematics as a collective intellectual endeavor. In fact, many
of the greatest mathematicians have attacked, with equal vigor,
enjoyment, and success problems posed by nature and problems
arising from mathematics itself. For example, Hermann Weyl con-
tributed in equal measure to the theory of groups as a pure mathe-
matical discipline and to the effective uses of this theory in the
theoretical constructions of atomic physics.
OCR for page 50
50
The State of the Mathematical Sciences
We, therefore, prefer not to use the term "pure mathematics,"
but replace it by the more descriptive term "core mathematics."
The core comprises those highly developed subdivisions of mathe-
matics that have been and are being investigated primarily for their
own sake. The term "core mathematics" also reminds us of the
central position of so-called "pure" mathematics with respect to all
mathematical sciences.
Intellectual curiosity and intellectual excitement are the main
motivating forces behind research in all mathematical sciences, as
they are in all sciences in general. The thrill of recognizing a pattern
in a seemingly chaotic situation and of reducing a large number of
apparently unrelated phenomena to a single simple principle are
again characteristic of all sciences; but in the mathematical sciences,
and especially in core mathematics, the part played by such con-
siderations is so predominant that some mathematicians consider
mathematics to be as much an art as a science.
Mathematics is typically done by individuals rather than by
groups; yet it is a collective effort. This is seen, for instance, in the
fact that important discoveries are often made by several people
working independently in different parts of the world. Personal
contact among mathematicians in the same field is often an essen-
tial condition for successful work. But the actual work is almost
always done by one man thinking about a problem. Very few
mathematical papers have three authors and practically none have
more. Since mathematics is one of the intellectual activities in which
one man unaided, unsupervised, and undirected can make im-
portant contributions, it continues to attract some of the brightest
young people.
One striking feature of mathematical research during the past
decade is the confluence of various mathematical disciplines. Dur-
ing the years around World War II the tendency toward speciali-
zation appeared serious. It was feared that the boundaries between
various fields of core mathematics would solidify so fast that mathe-
maticians of different specialties would be unable to talk to each
other. We see now that these fears were not justified. Recent devel-
opments have led to an interpenetration of fields, which will be
illustrated in Chapter 4.
Indeed many mathematicians believe that this is a golden age of
mathematics. They point out that many famous problems that have
baffled mathematicians of the past are being solved now, that mathe-
matics is being applied in an ever-increasing number of other dis
OCR for page 51
The Community of Mathematics
51
ciplines, and that young men entering upon research careers (and
their advisers) have no difficulty in identifying interesting problems.
CORE AND APPLICATIONS
"The similarities are essential, but the differences are vitall"
Progress in the central core of mathematics proper, where most of
the essential tools and new concepts are hammered out, has long
and in large measure depended upon abstraction, upon recogniz-
ing that seemingly very different things have enough in common
to be governed by common considerations, by an abstractly formu-
lated mathematical structure. History shows that the worker in the
core accomplishes more if he acts, and often feels, as if things ab-
stractly equivalent were really the same.
In the applied mathematical sciences' however, it is often neces-
sary to take almost the opposite attitude to that which is so helpful
ire the core. Here the important aspects are specific (sometimes
quite concrete, sometimes more abstract from the viewpoint of some
particular subject-matter field), rather than mathematically abstract.
Indeed, the mathematical abstraction is almost always approximate,
and it is usually vital to admit, at the proper time, the approximate
character of the answers.
For the applications of mathematics to grow at a rate com-
mensurate with the demands of society, they must continue to
receive an ever-growing supply of new results, new concepts, and
new approaches from the central core. Applied areas would ulti-
mately starve without what is made available by the core. Less
directly and obviously, but just as inevitably, the core would lose
much stimulation and implicit guidance, and indeed its develop-
ment would be badly distorted and stunted, if it were cut off from
applications. It is essential to recognize both the existence and com-
plexity of these relations between the core and applied areas and
the importance of maintaining communication between them. We
shall return to this question.
Those intellectual efforts coming half way between core mathe-
matics, on the one hand, and the sciences and technologies to which
mathematics is applied, on the other, are often referred to as
"applied mathematics." This is accurate only if it is understood that
these efforts include not only the traditional parts of mathematics
concerned with applications in physics, chemistry, and engineering
OCR for page 52
52
The State of the Mathematical Sciences
(which we shall refer to as physical mathematics or classical
applied mathematics), but also statistics, computer science, the gen-
eral field of optimized allocation, control, and decision-making, and
fields such as mathematical biology. Each of these fields has a char-
acter of its own, but they have in common a spirit that is well
described in reference 9:
. . . mathematics is "applied' if conceived in a spirit of ready cooperation
with sister sciences in the grand endeavor of comprehending our environ-
ment, making sense of ourselves, and bringing order into their interaction.
Like all good mathematics, good applied mathematics is original and imagi-
native in the invention and use of its concepts and in its tentative modeling
constructions. Its chief distinction from pure mathematics, which shows
self-motivated progress along dimly discernible natural paths of growth
towards intellectually satisfying goals, lies in adding to this conceptual
activity a deep concern for the world of outer experience and a ready in-
terest in problems beyond the confines of mathematics.
In common with the core mathematician, many applied mathe-
maticians are interested in the stimulation of the development of
new mathematics but with primary emphasis on those aspects
strongly and directly motivated by scientific, technological, or
sociological problems. In common with the theoretical scientist, the
applied mathematician seeks knowledge and understanding of facts
and phenomena through the use of mathematical methods.
The applied mathematician is most typically engaged in the
formulation, analysis, and interpretation of mathematical models:
in other words, (a) the formulation of scientific, technological, or
practical problems in terms of abstract mathematical models; (b)
the solution of the resultant mathematical problems, which may
consist of detailed answers or new techniques; and (c) the discus-
sion, interpretation, and evaluation of the results of this analysis,
especially in relation to the correspondence between the abstract
model and the real world. In different areas of the applied and
partly mathematical sciences, the character of these models may
be quite different. In computer science, for example, they some-
times take the form of a detailed computer program. The final goal
of these efforts is the creation of ideas, concepts, and methods that
are of basic significance to the subject in question. As mentioned
above, these efforts may lead to the creation of new mathematical
ideas and theories.
When thinking of "applications" and "mathematics" together,
the nonmathematician tends to think of an immediately applicable
OCR for page 53
The Community of Mathematics
53
solution to his own pressing problem. The applied mathematical
sciences do indeed contribute in such immediate ways, but they
also contribute throughout a range, of greater or less immediacy,
stretching to the very edge of core mathematics. All these varied
kinds of contribution are important, not least those that have the
same pure-science character as core mathematics.
CRITICISMS AND TENSIONS
The highly exciting achievements within core mathematics and
explosive penetration of mathematics into other fields are generally
recognized. At the same time, criticism of mathematicians by other
scientists as well as criticisms within the mathematical community
itself are far from uncommon. Certain of these criticisms may be
summarized as follows. It is said that during the recent past mathe-
maticians have alienated themselves from the mainstream of scien-
tific development.
One thing modern mathematicians tend to overlook is that the giants of
former days were all actively interested in physics as well as mathematics.
This is the first generation for which this is no longer true.
From a letter by a prominent physicist
It is also claimed that what contemporary pure mathematicians do
is of interest only to themselves and most, if not all, of it will never
be used in any other discipline. While all sciences require sophisti-
cated mathematical tools and present interesting and challenging
mathematical questions, mathematicians are said to be ignorant of
and not interested in these questions. This state of affairs is alleged
to be particularly and perhaps specifically characteristic of Ameri-
can mathematicians.
Has American mathematics become so remote that scientists will have to
turn to other schools of mathematics "Russia?] for help in the future?
From the same letter by the physicist
This criticism is not new. Mathematicians have long been criticized
for being too abstract and too remote from the needs of science.
Thus as knowledgeable and distinguished a mathematician as Felix
Klein considered the Hamilton-Jacobi theory of dynamics as a play-
thing of no use for physics (F. Klein, Vorlesungen uber die Entwick
OCR for page 54
54
The State of the Mathematical Sciences
lung der ~lathematik im 19. Jah~hundert, I, Springer, Berlin, 1926,
p. 207~. Yet the Hamilton-lacobi theory proved to be the basis of
quantum mechanics. Even the greatest of physicists may err in their
evaluation of mathematics. Einstein was at first reluctant to accept
Minkowski's four-dimensional geometry as the proper description of
special relativity, and several years later he was reluctant to express
his physical ideas in the language of tensors. The history of science
has shown time and time again that it is impossible to predict what
mathematical theories will turn out to be useful outside of pure
mathematics.
This point can hardly be stressed too much. Planning the over-all
emphasis of mathematics, either for the progress of mathematics as
an end in itself or for its application to other sciences and tech-
nologies, must fail lamentably. There is a process of natural selec-
tion in mathematics. For instance, abstraction for abstraction's sake
rots away rapidly if it leads to no new intellectual progress; abstrac-
tion that only appears to be for abstraction's sake often proves to
be of vital importance. Unconstrained opportunities to succeed or
fail; broad diversity, both of individual activities and organizational
patterns; ease of change, especially in giving up old patterns these
are the characteristics that have tended to allow the mathematical
sciences to make progress.
The claim that "the giants of former days were all actively inter-
ested in physics as well as mathematics" is illusory. To be sure,
Gauss, Riemann, and Poincare were actively interested in physics.
But one can easily think of first-rank mathematicians of the nine-
teenth century whose interests were exclusively mathematical.
Among them there were several whose work, unbeknown to them-
selves, turned out to be of vital importance for other sciences (for
instance, Frobenius, who founded the theory of group representa-
tions). There have been contemporary mathematicians with inter-
ests as catholic as those of the greatest men of the nineteenth cen-
tury for example, Hermann Weyl, Wiener, and Von Neumann
or Gelfand and Kolmogorov but it is unreasonable to expect more
than a few such men in a generation.
Still, the charge that mathematicians have separated themselves
from their sister disciplines like physics and astronomy reflects a
true state of affairs. Specialization is the inevitable price of the
increase of knowledge. One cannot expect many future scientists to
combine mathematical and physical achievements in the manner
of Newton or Gauss. Yet we must not allow ourselves or our stu
OCR for page 55
The Community of Mathematics
55
dents to become intimidated by the "explosion of knowledge." 'The
problem facing us is mainly that of communication and, by impli-
cation, that of education.
IMPROVING COMMUNICATIONS
In the mathematical sciences, as in most other sciences, the explo-
sive growth of activities has had an important side effect. The
problems of communication between scientists in various disciplines,
between the creators of new theories and the users of these theories,
between investigators and teachers, between teachers at various
levels, between working scientists and scientific administrators, and
last but not least between the scientific community and the citizenry
at large have become difficult and require a conscious effort at
solutions.
The volume of scientific literature has increased so much that
the average working mathematician cannot even scan all the review
journals. This results in the paradoxical situation that while many
more printed pages of mathematics appear than ever before, mathe-
maticians must depend to a large extent on personal contacts and
personal correspondence in order to keep abreast of the current
developments in their specialties. (This explains why conferences
and symposia are an indispensable part of contemporary scientific
research.) The ultimate solution may come from radical new
methods for storing, retrieving, and dispensing information. Per-
haps journals, books, and libraries as we know them will be replaced
by other techniques. This, however, will not happen at once.
A different problem is that of communication between various
branches of the mathematical community. Here the difficulties are
not technical but psychological and intellectual. In the central
area of core mathematics any threat to the exchange of ideas and
results between subdisciplines has been overcome through recent
unifying tendencies. On the other hand, this danger is acute as far
as the interfaces between core mathematics and the applied and
partly mathematical sciences are concerned and even more so be-
tween the mathematical sciences as a whole and the various users of
mathematics. What is needed is a conscious effort by highly qualified
people to overcome this "communication gap" by specially written
books and articles, by interdisciplinary conferences and courses, and
by other appropriate means. If this endeavor is to be successful,
OCR for page 56
56
The State of the Mathematical Sciences
leadership must come from the profession, specifically from indi-
viduals. Financial support, where necessary, will also have to be
found.
The problems of communication and publication have recently
been considered in more detail by the American Mathematical
Society's ad hoc Committee on Information Exchange and Publica-
tion in Mathematics. By special permission we have reproduced that
committee's final report in Appendix B. Its recommendations are
addressed to the American Mathematical Society but may provide
more general guidance.
During the past decade, the mathematical sciences have made
strenuous and successful efforts to establish communications be-
tween the investigators and teachers of mathematics at various
levels. The various programs for curriculum reform in high schools
and elementary schools involve joint work by university mathe-
maticians and teachers. The manifold activities of the Committee
on the Undergraduate Program in Mathematics (CUPM) establish
a link between graduate schools and colleges. We do not believe
that any of these groups has already found definitive solutions to
the problems of mathematics education. Valuable steps have
already been taken, however, and such work should be continued
and supported.
The problem of making mathematics understandable to the edu-
cated layman continues to be almost insurmountable. A Greek
tyrant who wanted a rapid explanation of mathematics was told
that there was no royal road to this science. This remains true. It
is equally true, however, that the mathematical community has a
duty to do all it can to make its work understandable to those who
want to understand it. The collection of essays7 to be published in
conjunction with our report is a modest contribution in this direc-
tion.
Representative terms from entire chapter:
core mathematics
