II. THE MATHEMATICAL SCIENCES:
THEIR STRUCTURE AND CONTRIBUTIONS
The mathematical sciences are the most abstract of the sciences, as suggested
in Table 1.
Table 1: The Intellectual Foci of the Sciences
Field The Study of
Mathematical sciences Patterns, structures, the modeling of reality Physics Energy, matter, time Chemistry Molecules Biology Life Materials science Materials, structures Earth sciences The earth: continents, oceans, the atmosphere Astronomy Origin and evolution of planets, stars, and the universe
The mathematical sciences have two major aspects. The first and more
abstract aspect can be described as the study of structures, patterns, and the
structural harmony of patterns. The search for symmetries and regularities in
the structure of abstract patterns lies at the core of pure mathematics. These
searches usually have the objective of understanding abstract concepts, but
frequently they have significant practical and theoretical impact on other
fields as well. For example, integral geometry underlies the development of
x-ray tomography (the CAT scan), the arithmetic over prime numbers leads to
generation of perfect codes for secure transmission of data on the Internet, and
infinite dimensional representations of groups enable the design of large,
economically efficient networks of high connectivity in telecommunications.
The second aspect of mathematical science is motivated by the desire to model
events or systems which occur in the world — usually the physical,
biological, and business worlds. This aspect involves three steps:
Examples of mathematical modeling include the quantum computer project,
DNA-based molecular design, pattern formation in biology, and the fast Fourier
transform and multiple algorithms used daily by engineers for numerical
computation.
The mathematical sciences are disciplines in themselves, with their own
internal vitality and need for nourishment. But they also serve as the
fundamental tools and language for science, engineering, industry, management,
and finance. They are inextricably linked to these "user" fields and
they frequently draw inspiration from them. The mathematical sciences represent
a mode of thought based on abstraction that sustains precision and permits
careful analysis and explicit calculation. Thus mathematics has a dual nature:
it is both an independent discipline valued for precision and intrinsic beauty,
and it is a rich source of tools for the world of applications. Mathematics
might be described as having abstractness internally and effectiveness
externally.
The two parts of this duality are intimately connected. The search for
order, symmetries, and regularities in patterns is the heart of research in pure
mathematics. Results of this research are very durable, sometimes finding
important application in unexpected ways decades after their discovery. A major
reason for this is that results in mathematics, once proven, are never disproved
-- even though they may be superceded by more powerful results. Other sciences,
by contrast, move towards truth by a process of successive approximations.
In the United States, mathematics research, which is carried out principally
at universities, may be segmented (somewhat arbitrarily) into nine sub-fields,
as described in Table 2.:
Table 2: Major Subfields of Mathematical Sciences
Subfield The Study of
Foundations Logical underpinnings of mathematics Algebra and Combinatorics Structures, discreteness Number Theory and Algebraic Properties of numbers and polynomials Geometry Topology and Geometry Spatial structures, patterns,shapes Analysis Extensions and generalizations of the calculus Probability Randomness and indeterminate phenomena Applied Mathematics Problems arising in nature Computational Mathematics Problems whose solution uses the computer Statistics Analysis of data
The boundaries between these subfields are neither fixed not solid, and some
of the most interesting and fruitful developments in mathematics come at the
interfaces of subfields. Some areas of research appear in more than one of
these categories; e.g., for example, Theoretical/Mathematical Physics appears in
Topology/Geometry, Analysis, and Applied Mathematics.
The Mathematical Sciences Research Community
The mathematical sciences research community differs from other research
communities in several ways. Mathematical research is the epitome of "small"
science; that is, much research is done by individuals working alone, with
modest equipment needs such as workstations. (Increasingly, however, some
mathematicians need access to supercomputers and visualization labs.) Also,
mathematical research is long-lasting, and rich in references to older
literature, so that mathematicians are more dependent than other scientists on
good libraries. Finally, mathematicians are more closely associated with
teaching and with educational institutions than other scientists. Most research
mathematicians are university based, so that their culture has an academic
orientation.
In 1995(see Endnote 5), approximately 16,000
(over 65%) of the doctoral mathematical scientists
in the United States were located at institutions of higher education. Of
these, 6,427 worked at doctorate-granting institutions and represent the heart
of the U.S. academic research community. Less than 25% of doctoral
mathematicians were employed in private industry, and 4.2% were employed in
government. Of the 1994-95 cohort of U.S. doctorate recipients, more
than 50% anticipated faculty positions at educational institutions, with an
additional 25% planning U.S. postdoctoral appointments, presumably as a
precursor to academic careers.
Mathematical scientists in industry seldom carry the title "mathematician;"
they are usually known as "engineers," "systems analysts," or by other
titles, (see SIAM report)(see Endnote 6). Thus
they lack the mathematical identity and
consciousness of their academic counterparts and in contrast to chemists and
engineers, tend to be poorly connected to the university community.
Mathematical Sciences as an International Discipline
Both by its abstract nature and by convention, mathematics knows neither
linguistic nor political boundaries. Its language is usually decipherable from
equations and relations alone; when words are needed, mathematicians around the
world use English by common agreement -- just as scholars once used Latin. In
the same spirit, mathematicians have managed to transcend political differences
and borders, even during the Cold War. And because mathematicians do not
require specialized laboratories to conduct their research, they travel freely
between universities and between countries. The result of these customs and
agreements is that mathematics is an extraordinarily open and international
activity.
The number of highly active research mathematical scientists worldwide is
small —probably well under 10,000 — so that a given subarea may be
populated by only a tiny number of highly specialized individuals. They know
each other well regardless of their country of residence; share a common,
specialized vocabulary; and collaborate extensively even over long distances.
Mathematical science conferences typically host participants from many
countries; meeting one’s peers is essential for the exchange of ideas which
may not appear in published work.
Because of this international culture, mathematicians frequently take up
sequential residencies in different countries or alternate between countries.
The United States, with its commitment to freedom, a high standard of living,
and excellent universities, has benefited enormously from flows of foreign-born
mathematicians; in the same spirit, Americans serve on mathematical science
faculties in almost every country in Europe. For these reasons, local changes
in the support of the mathematical sciences in any country can result in the
rapid migration of mathematicians, such as the great emigrations from Europe
before World War II and the former Soviet Union at the end of the Cold War.
Mathematicians also collaborate internationally on research, a trend that has
been growing consistently for nearly two decades. The number of papers
co-authored by mathematicians in the five major mathematical nations (see Endnote 7) with researchers in other countries rose
about 50% between 1981 and 1993, and this tendency continues.
The growth in co-authored papers by researchers in the United Kingdom,
France, and Germany reflects the growing unification of the countries in the
European Union.
Mathematics students tend to gather in the strongest research centers, a
tradition that began over a century ago. Before 1940, it was common for the
best U.S. students to study in Europe; after World War II, the U.S. reputation
in mathematics grew rapidly, and for the past 15 years, a majority of Ph.D.
graduates of U.S. institutions have been non-U.S. citizens. In 1996, non-U.S.
citizens earned 55% of total doctoral degrees in mathematical and computer
sciences(see Endnote 8). Other strong international
research centers are also attracting foreign students. In France, international
students now earn one out of three doctoral degrees awarded in all fields of
science; in Japan, that proportion is 40%; and in England, 27%, with many
students from commonwealth countries and the United States(see Endnote 9). Germany supports foreign graduate
students and postdoctorates on Humboldt Fellowships.
The Role of Mathematics in Society
Although most of the mathematical research community is
university-based, the impact of mathematics on society is pervasive.
Mathematics underpins most current scientific and technological activities.
Whole new areas of mathematics are evolving in response to problems in
experimental science (biology, chemistry, geophysics, medical science), in
government (defense, security), and in business (industry, technology,
manufacturing, services, finance). All of these areas now require the analysis
and management of huge amounts of loosely structured data, and all need
mathematical models to simulate phenomena and make predictions. Modeling and
simulation are essential to fields where observable data are scarce or involve a
great deal of uncertainty, such as astronomy, climatology, and public policy
analysis. Addressing such complex problems calls for openness to all of
mathematics and to the emergence of new mathematics. Progress requires radical
theoretical ideas as well as significantly greater collaboration between pure
mathematicians, statisticians, computer scientists, and experimental
scientists.
The applications of mathematics in the future will require closer
partnerships between mathematical scientists and the broader universe of
scientists and engineers. Meeting the complexity of tomorrow’s challenges
will demand insights across the full spectrum of the mathematical sciences.
Both the theoretical and the industrial impact of this development will be
enormous. Table 3 illustrates some of the present and potential contributions
of mathematics to society.
Table 3: Illustrations of Some Uses of Mathematics in Society
Problem/Application |
Contribution from Mathematics |
MRI and CAT Imaging |
Integral geometry |
Air traffic control |
Control theory |
Options valuation |
Black-Scholes options model and Monte Carlo simulation |
Global reconnaissance |
Signal processing, image processing, data mining |
Stockpile stewardship |
Operations research, optimization theory |
Stability of complex networks |
Logic, computer science, combinatorics |
Confidentiality and integrity |
Number theory,cryptology/combinatorics |
Modeling of atmospheres and oceans |
Wavelets, statistics, numerical analysis |
Agile, automated manufacturing |
Geometry, visualization, robotics, control theory, in process quality control |
Design and training |
Simulation, modeling, discrete mathematics |
Analysis of the human genome |
Data mining, pattern recognition, algorithms |
Rational drug design |
Data mining, combinatorics, statistics |
Seiberg-Witten questions (string theory) |
Geometry |
Interpreting data on the universe |
Data mining, modeling, singularity theory |
Design systems for composite materials |
Control theory, computation, partial differential equations |
Earthquake analysis and prediction |
Statistics, dynamical systems/turbulence, modeling, in process control |
The sciences have always used mathematics to formulate theory and underpin
simulation and statistics to design productive experiments. Wherever numbers or
symbols are manipulated, the manipulations rest on mathematical relationships.
With the advent of high-speed computers and sensors, some experimental sciences
can now generate enormous volumes of data --- the human genome project is an
example -- and the new tools needed to organize this data and extract
significant information from it will depend on the mathematical sciences. Hence
the mathematical sciences are now essential to all three aspects of science:
observation, theory, and simulation.
The following examples illustrate ways in which mathematics contribute
to areas of broad concern to our nation:
National security. The security of complex communications
systems--voice, data, and electronic--rests on mathematically sophisticated
tools. Stockpile stewardship--the maintenance of the nuclear arsenal without
testing--will be based on mathematical modeling and advanced computation. The
operation of national surveillance systems requires extensive use of mathematics
for collection and analysis of data. Military systems are being transformed by
the application of mathematical-based systems for intelligence, logistics, and
warfighting.
Technology. Mathematics is ubiquitous in the design, manufacturing,
and use of technology. No complex system--from microprocessors to aircraft
engines, from satellite communications networks to home marketing systems, and
from the air traffic control system to the laptop computer--could exist without
the application of mathematics.
Education. The role of mathematics in educating the work force is
crucial for the well-being of the nation. Scientists and engineers depend on the
mathematical sciences and need a sound foundation in that discipline to succeed.
For the average citizen, a grounding in mathematics, at least through the
secondary level, is essential to modern citizenship. Innumeracy is as crippling
as illiteracy.
Medicine. The operation of modern medical imaging systems--CAT
scanners, nuclear imagers, Magnetic Resonance Imaging (MRI)--depends on the
mathematical processing of signals. The success of the human genome project
will require the use of mathematics to search for information that correlates
genetic sequences to human disease. Elucidating the complex geometry of protein
folding is the key to understanding protein functions. The management of
hospital patient records will increasingly require the application of
mathematics to construct efficient databases.
Finance. Mathematics has become indispensable in measuring risk and
modeling the behavior of financial instruments, financial institutions, and
financial systems (individual countries, trading blocs, and global systems, such
as international settlements). The combination of probability theory and
advanced financial models with increased data, capacity, algorithm efficiency,
and computational speed facilitates the sophisticated modeling of interest
rates, currencies, commodities, equities, and other financial instruments.
Better understanding of areas such as value at risk, portfolio theory for credit
exposures, and non-linear instruments depends on the application of
mathematics.
Environmental monitoring. Building useful models of oceans and
atmospheres to predict the impact of human activities on the environment is
essential to the formulation of sound public and regulatory policies. Climate
models require the manipulation of massive quantities of data and the study of
complex simultaneous interactions (for example, among the many trace chemicals
in the atmosphere). All such models are based on uncertainty; to judge their
validity will require heavy use of mathematics.
In broad terms, both the techniques of the sciences and the needs of
society are dramatically more complex than those of the past. The ability to
understand new needs and systems, and to predict and control their behavior,
will require two elements: i) new mathematical ideas and methods, and ii) more
effective collaboration between all groups actively concerned with them --
disciplinary scientists, engineers, computer scientists, and members of relevant
professions, from medicine to public policy.