Applied mathematics and society

Summary: Virtually all human pursuits depend on or were made possible by some application of mathematics, and historically applied mathematics often preceded the study of pure mathematics.

In the 1920s, the German military adapted, from a then recently developed business device, and began using an encryption code device known as the “Enigma” machine. Believing it to be an unbreakable encryption device, they continued to employ it into World War II. A significant effort to crack the Enigma code was undertaken, first by Polish mathematicians prior to the German invasion of Poland and then by British mathematicians at Bletchley Park, culminating in the breaking of the code in 1940. This success was instrumental in the ultimate Allied victory and in shortening the war significantly.

To this day, a key role of many applied mathematicians still involves encryption and cryptography, not just for military and defense purposes, but for a wide range of life activities, including computer and ATM security. In fact, more generally, there is almost no area of science, technology, and culture that is not heavily dependent upon the application of mathematical concepts and techniques. Applied mathematics therefore represents, in many ways, the ultimate multidisciplinary subject.

Historical Context

Although archaeological evidence is spotty and incomplete, it appears that the first mathematical efforts of civilized society involved either commerce, including accounting for transactions and inventories, or the measurement of land holdings for agricultural purposes. For these purposes, ancient Egyptians and Babylonians developed and applied basic concepts and techniques in arithmetic and geometry. Both peoples also used geometry in support of their building efforts and in the placement of monuments.

A large part of what is known about Egyptian mathematics comes from examination of the Rhind Papyrus. This document includes practical mathematical examples and exercises. It is apparent that the early development and purposes of mathematics were in response to, and in support of, practical, real-world problems, often of an engineering nature.

As with so many aspects of human culture and cognition, ancient Greece represented a shift—or, at least, the beginnings of a shift—in its approach to and philosophy regarding mathematics. There were certainly still applied mathematics problems, for example, involving navigation and astronomy. However, apropos of the birth in Greece of philosophical thought and reasoning, there was also some movement toward a reasoned approach to advancing mathematical knowledge. Thus, a divergence between pure and applied mathematics began to emerge.

There were several areas of mathematics in which inroads were made by the Greeks, for example, in geometry, trigonometry, logic and proof, and algebra (although work in algebra began with later Greeks). The Greeks also noted and struggled with irrational numbers (numbers that cannot be expressed as a ratio of integers). One can imagine, in an age where immediate physical needs and practical problems were paramount, that the inability to precisely measure a length (for example, not being able to precisely express the length of the diagonal of a square in terms of the known length of the sides) would have provided a conundrum.

Because of the Greek willingness to consider the theoretical, they were able to deal with, or accept, such situations to a degree. These situations led, however, to certain philosophical problems or paradoxes, such as Zeno’s paradox, named for Zeno of Elea, that continue to challenge mathematicians. It can still be difficult for modern people, who initially in life are cognitively dependent upon experience and observation, to make the jump from the world of physically demonstrable, practical, applied mathematics to that of abstract and representative theory.

The Roman Empire had a largely practical, engineering-oriented approach to life, and this was manifested in their approach to mathematics. They were not particularly interested in expanding the horizons of mathematical theory. Instead, they used mathematics for applied engineering purposes from which emerged remarkable achievements that have survived through history.

A key application of mathematics beginning in the seventeenth century involved trying to understand and mathematically model the natural world. Certainly, there had been earlier efforts in that direction going back thousands of years—perhaps most notably by Ptolemy of Alexandria, with his extensive system of cycles and epicycles geared toward explaining, and ultimately predicting, the movements of heavenly bodies. But in the seventeenth century, with mathematicians and physicists such as Galileo Galilei and Isaac Newton, the modern effort to explain the world began in earnest.

Into and through the nineteenth century, a mathematician, like a scientist, was largely capable of understanding and keeping up with mathematical developments. With the explosion of mathematical activity in the twentieth century, it became impossible to do so, leading to a splitting of different specializations and mathematical disciplines and also a split between pure and applied mathematics, particularly in academic institutions. Interestingly, toward the end of the twentieth century and beginning of the twenty-first century, that separation seems to have lessened as each area began to appreciate more the usefulness of the other.

Substance of Applied Mathematics

It is difficult to comprehensively identify the substance of applied mathematics. In part, the difficulty is because of the overlap, which can take several forms, between pure and applied mathematics. First, a mathematical discovery or technique that initially seems without a practical application can, over time, become adopted and embraced by science and technology for practical application. Thus, to complain that an area of mathematics has no current usefulness can be potentially shortsighted; no one knows what future advances in society might be welcoming of—or possibly even made possible by—those pure mathematical excursions.

Another way in which pure and applied mathematics can overlap is simply in how such things are labeled. It is impossible to draw a clear line of demarcation between pure and applied mathematics. A new proof or technique made in a pure mathematics context may have very real practical applications, either now or later. Similarly, a practical, real-world problem may result in the development of a new approach with conceptual implications for theoretical mathematics. Furthermore, while many jobs require mathematical skills and techniques, such as architecture and engineering, they may not be technically classified as “applied mathematics” careers.

For example, a mathematical subject area such as number theory would generally not be considered an area of applied mathematics; and yet, it has significant implications and relevance for certain types of industrial applications, such as encoding. Similarly, abstract algebra would not, on the surface, seem to be applied; nevertheless, physicists now use group theory to better understand the world of elementary particles and quantum physics.

An important organization for applied mathematics is the Society for Industrial and Applied Mathematics (SIAM). According to its Web site, SIAM was organized in 1952 “to convey useful mathematical knowledge to other professionals who could implement mathematical theory for practical, industrial, or scientific use,” and its membership in 2011 consisted of some 13,000 individuals and nearly 500 institutions.

A listing of some of the activity groups within SIAM serves to indicate the wide range of mathematics with important applications:

• Computational science and engineering
• Control and systems theory
• Dynamical systems
• Financial mathematics and engineering
• Geosciences
• Imaging science
• Life sciences
• Mathematical aspects of materials science
• Nonlinear waves and coherent structures
• Optimization
• Supercomputing

Example of an Applied Mathematics Discipline: Actuarial Science

Mathematics can be—and is—applicable to most any discipline. An example of an important and well-respected applied mathematics profession (which is generally ranked in the top five and sometimes at the very top of job-ratings surveys) is actuarial science. This applied mathematics career is representative of others and gives a sense of the general nature of applied mathematics work as well as its impact on society.

The ability to manage risk—not necessarily to eradicate or even reduce it, but at least to “manage” its potential impact—is critical in a complex socioeconomic environment. Without a way to manage risk, for example via an effective insurance industry, many activities that humans rely on might never happen (bridges might not be built and surgical procedures might not be undertaken) without the protection to society, organizations, and individuals that insurance provides. The ability to offer protection against the impact of risks is based on some key statistical ideas: the Law of Large Numbers and its related concepts. Only with a sophisticated understanding and application of probability and statistics can an effective risk management industry be sustained.

Actuarial science developed as the mathematical discipline underlying the analysis of risk contingencies. There are basically three types of actuaries: (1) life actuaries, who deal primarily with human mortality issues and life insurance; (2) pension actuaries, who focus on pension and retirement systems; and (3) property-casualty actuaries, who deal with other areas of risk and insurance, such as auto, homeowners, workers compensation, and medical malpractice insurance. Actuarial science is, in some ways, the ultimate interdisciplinary field.

Since risk applies to any type of endeavor or situation, an actuary attempting to quantify risk should potentially understand at least the fundamentals associated with almost all topics. One cannot adequately comprehend or evaluate a set of data without understanding where it came from and under what specific conditions it emerged. Thus, being an actuary or a risk analyst involves not only the relevant mathematics but also asking questions and learning about the context of the situation and using the findings to tailor mathematical methods appropriately. Furthermore, as with any quantitative discipline that uses sophisticated techniques, an effective actuary must be a very good communicator—able to translate mathematical concepts and techniques into understandable descriptions for nonmathematicians.

Becoming an actuary is a significant accomplishment. After earning an undergraduate degree (most often in either actuarial science or mathematics), actuaries spend several of their first careers both working at a job and studying for an extensive series of professional exams in an attempt to earn a designation or certification. These exams cover a variety of relevant areas, including specific actuarial techniques, finance and economics, and business processes.

On the job, actuaries use mathematics in an attempt to model real-world stochastic processes, such as the frequency and size of insurance losses, as well as economic and financial variables, such as interest rates, inflation, and investment performance. For example, based largely on historical data, an actuary might estimate that the frequency, or number, of claims that will occur in a given year is well-represented by a certain statistical distribution, such as a Poisson, named for Siméon-Denis Poisson, or a Negative Binomial. Similarly, given that a claim has occurred, historical loss information might suggest that the dollar size of a particular claim probabilistically follows another type of distribution, such as a Normal, Gamma, Lognormal, or Pareto, named for Vilfredo Pareto. Such decisions are largely based upon a thorough analysis of historical data, but other factors are also taken into account, including a qualitative understanding of the nature of the risks and hazards that the insurer is indemnifying and the entire socioeconomic context of the insurance activity. Once a model is developed, it provides a basis for not only prediction and analysis of appropriate future insurance policy rates but also testing the potential impact of making a variety of possible strategic or operational decisions, such as changes to the types of policyholders targeted and changes in policy provisions.

In the last few decades of the twentieth century, actuarial science and risk management became more technically sophisticated and more enterprise-wide in perspective. Part of the actuary’s job is to understand the behavior of economic and financial variables and how they may impact the insurance and risk management process. For example, Brownian motion equations and concepts, named for Robert Brown, are frequently used to model the movements of interest rates and equity prices over time. Because insurance companies take in premiums but may not pay out corresponding losses for months or years, it is important to model how the insurer’s investments may perform in the future. Insurers may even decide to sell some of their policies at an underwriting loss because they know that they have the opportunity to earn an adequate return on equity from the potential investment earnings on the premiums they take in as well as on their equity. By considering all aspects of an insurer’s operations, including the effect of economic and financial conditions, the actuary’s job has become much more holistic, or multidisciplinary.

Overall, an actuary’s or risk analyst’s job is one that is completely predicated upon mathematical techniques and quantitative skills, but it is also a business position. Skills involving communications, problem solving, project management, and teamwork are also essential for success in this environment.

Other Applied Mathematics Fields and Careers

The above description of actuarial science is representative of a variety of areas of applied mathematics. There are several other areas, including the following:

• Biomathematics and biostatistics. Applications of mathematics to biology have the potential to advance society and the human condition in substantial ways. Some of that advancement will come from mathematical modeling and analysis of genome and DNA mapping and sequencing. Another important area involves applying network analysis and dynamical systems techniques to the potential spread of infectious disease. Other areas include using geometry, topology, and other mathematical tools to examine and image brain activity; using differential equations and geometry to locate and attack tumors; and modeling human organs to allow testing of new surgical or other medical techniques.
• Operations research. Anything involving sequential processes can potentially be made more efficient and effective with the application of mathematical techniques and modeling. A few examples of such processes to be modeled include queue lines to limited resources, such as ATMs or grocery checkout machines; automobile traffic patterns; and Internet traffic. Like much of applied mathematics, the ultimate goal of operations research is to improve operational and strategic decision making.
• Natural hazard modeling. Hurricane and earthquake modeling are examples of interdisciplinary applied mathematics areas. Modelers need not only appropriate quantitative skills, such as geometry and systems of differential equations, but also an understanding of the appropriate science or technology, such as atmospheric sciences or geosciences.
• Software, computers, and data. Applied mathematics disciplines make use of computers, and some are very heavily dependent upon computational techniques and resources. In addition, numerous areas of mathematics play a role in careers in software engineering, data analysis, digitization, and compression.

Looking Forward

Ian Stewart, in the 2002 book The Next 50 Years: Science in the First Half of the Twenty-First Century, offers an essay titled “The Mathematics of 2050.” In that chapter, he opines that several areas of mathematical exploration will undergo, and indeed are already undergoing, upswings or even revolutions. Among those he mentions are several areas of applied mathematics, including biomathematics and financial mathematics. Biomathematics certainly seems to be coming of age, and people’s lives, and those of their immediate descendants, are being overwhelmingly affected by developments in this area.

One might argue, after the financial and economic crises of the first decade of the twenty-first century, that financial mathematics sustained a “black eye” that will suppress its credibility and potential. However, these same crises certainly made clear the importance of understanding the nature and potential impact of “risk” in the world, perhaps especially economic and financial risks. Being able to identify, quantify, and manage risk is critical to the smooth operation and advancement of society. This ability is simply impossible without a good understanding of the mathematical underpinnings of economics and finance and their attendant risks, as well as the ability to model different approaches and solutions to managing those risks.

It is, of course, difficult to hazard any guesses about long-term societal developments. However, one prospective application from the realm of science fiction is interesting to note. Isaac Asimov, in his Foundation series of stories and books, posited a mathematics-based “psychohistory.” The stories focus on the legacy of Hari Seldon, a mathematician who built psychohistory into a statistical basis for modeling and predicting how human society will likely respond to various factors and stimuli. In the early twenty-first century, applied mathematicians are far from exhausting the potential of mathematics to change and advance society.

Bibliography

Holmes, Mark H. Introduction to the Foundations of Applied Mathematics. New York: Springer, 2009.

The Operational Research Society. http://www.orsoc.org.uk.

Society for Industrial and Applied Mathematics. “Thinking of a Career in Applied Mathematics?” http://www.siam.org/careers/thinking.php.

Tan, Soo Tang. Applied Mathematics for the Managerial, Life, and Social Sciences. Belmont, CA: Brooks Cole/Cengage Learning, 2008.

U. S. Department of Labor. Bureau of Labor Statistics. “Occupational Outlook Handbook 2010–11.” http://www.bls.gov/oco/ocos043.htm.