Overview: Mathematics 1950-present

Background: Mathematics Becomes the Language of Scientific, Philosophical, and Cultural Revolution

During the nineteenth century advances in mathematics pointed toward a universe not necessarily limited to three dimensions and not necessarily absolute in time and space. By developing new mathematical models and precise formulas with enormous predictive power, mathematicians profoundly shaped the understanding and application of twentieth-century relativity and quantum theories. In many cases, innovative mathematical models became the only means to describe profoundly revolutionary scientific and philosophical concepts regarding the structure and workings of nature.

Throughout the twentieth century there was a steady pace to the refinement and discovery of new applications for mathematical principles. In particular, advancements in differential equations (equations that relate the rates of change of physical quantities to the values of those quantities themselves) found continued application in astronomy and physics. Mathematicians and physicists labored to find mathematical formulas, expressions, and constants to that which, in essence, governed the cosmos. Along with the speed of light, Planck's constant, for example, was found to be a fundamental constant used in the mathematical expression of the Heisenberg uncertainty principle and, as a consequence, carried profound philosophical implications regarding limits on knowledge.

French mathematician Alexander Grothendieck (1928- ) once wrote that "mathematical activity involves essentially three things: studying numbers, studying shapes and measuring distances." Grothendieck contended that all mathematical reasoning and divisions of study (e.g., number theory, calculus, probability, topology, or algebraic geometry) branched from one or a combination of these methodologies. Indeed, just as modern physicists have sought grand unification theories to reconcile quantum and relativity theory, during the last half of the twentieth century mathematicians sought, with varying degrees of success, to interrelate mathematical theories. The use of statistics, for example—beyond being just a mathematical convenience useful in describing the average workings of large systems—became the only way to describe some of the finer, quantum level workings of nature.

It may be fairly argued that in 1931 German mathematician Kurt Gödel's (1906-1978) theorem regarding the limitations of mathematical proofs was the assertion of a mathematical "uncertainty principle." Regardless, it became one of the most powerful and philosophically influential mathematical discoveries of the century—especially with regard to postmodern, existential, and abstract expressionist movements. Advancements in later twentieth-century mathematics, however, often refocused on classical mathematical theory to advance man's understanding of non-linearity and of complex or chaotic phenomena. Without question, English physicist Sir Isaac Newton's (1642-1727) classical mechanics and French mathematician Jules Henri Poincaré's (1854-1912) studies of the chaotic behavior of systems provided a path for the development of twentieth-century chaos theory.

As concepts regarding the dualism of mind and body underwent philosophical revision in the twentieth century, advances in both pure and applied mathematics worked their way into new and exciting concepts of physical and social order. Just as there was an increasing emphasis on the duality between the need for diversity and the need for interdependence of world-wide cultures, in the later half of the twentieth century mathematicians, scientists, and philosophers freely crossed blurred intellectual boundaries in an effort to more accurately describe an increasingly complex non-Newtonian world in which no classical, linear, God'seye view of nature was possible.

Advances in Theory and Application

Especially for theorists, challenging mathematical terrain to scale was clearly mapped at the beginning the twentieth century with the posting of mathematician David Hilbert's (1862-1943) famous list of 23 problems. Throughout the century mathematicians wrestled with Hilbert's problems and, in some cases, only pinned down solutions or partial solutions in the last decades of the century. One of Hilbert's problems apparently finding resolution late in the 1990s, for example, included proof of Kepler's conjecture regarding the most efficient geometrical arrangement for stacked spheres. Although the "spherepacking" problem seemed proved by everyday experience, a mathematical proof eluded mathematicians for nearly four centuries. The utilization of computers in providing proofs, however, spurred philosophical discussion about the nature and future of mathematical proofs.

Highlights of twentieth-century mathematical advancements would be incomplete without mention of four-color mapping theory, advances in understanding of Georg Cantor's (1845-1918) continuum hypothesis, and René Thom's (1923- ) influential catastrophe theory. Moreover, during the later half of the twentieth century, mathematics often moved rapidly from theory to application. For example, American mathematician John Forbes Nash's (1918- ) work in noncooperative games became influential in economic and social science. Although still controversial in theoretical aspects, game theory found application in the development of strategy for war, politics, and business.

Mathematics and Science

After 1950 advances in science, especially in physics and cosmology, became increasingly dependent upon advances and application of mathematics. English mathematician Sir Roger Penrose (1931- ), for example, one of the leading mathematicians of the later half of the twentieth century, is perhaps best known for his collaborative work with fellow English physicist Stephen Hawking (1942- ) regarding the calculation and prediction of the fundamental properties associated with black holes.

At the other end of the cosmic scale, English mathematician Simon Donaldson's (1957- ) work in low-dimensional topological geometry has been used by particle physicists to describe short-lived subatomic particle-like wave packets called instantons. The development of chaos theory also fused scientific and mathematical efforts to seek order in complex and seemingly unpredictable systems. In the last decades of the twentieth century chaos theory became an important tool in the study of population trends, epidemiology (the study of the spread of disease), explosions, meteorology, and complex chemical reactions.

Mathematics and Emerging Technology

Almost all of the research and innovation in statistics during the last two decades of the twentieth century was a result of, or was deeply influenced by, the increasing availability and power of computers. Powerful computer-based techniques referred to by statisticians as "bootstrap statistics," for example, allow mathematicians, scientists, and scholars working with problems in statistics to determine with great accuracy the reliability of data. The techniques, invented in 1977 by Stanford University mathematician Bradley Efron, allow statisticians to analyze data and make predictions from small samples of data. Accordingly, bootstrap techniques have found wide use politics (e.g., polls), economics, biology, and astrophysics.

Using the emerging tools of computer graphics, Polish-born American mathematician Benoit Mandelbrot's (1924- ) work in fractal geometry created a mathematical school with broad scope and application. Fractals seemed to be everywhere—a universality in nature—and were used by astrophysicists, for example, to construct computer simulations depicting the dynamics involved in the highly complex formation of galaxies and planetary systems.

In addition to igniting a world-wide micro-electronics revolution, by the end of the twentieth century the invention of the hand-held pocket calculator and powerful computer software such as Mathematica placed at the fingertips of the average middle school student the most powerful and elegant of mathematical concepts.

Mathematics and Education

Although the tools of mathematics became cheaper and the mechanics of math more accessible, methods for teaching mathematics, especially in the United States, became mired in controversy. "New Math," for example, launched into American schools in the early 1960s, stressed conceptual understanding of the principles of mathematics and de-emphasized technical computing skills in an effort to teach children basic mathematical truths they could apply to more specific problems in a rapidly specializing scientific and technical world. New Math also, however, stirred controversy akin to a national strategic crisis and fostered sharply divided political opinions and passionate social debate regarding pedagogy (teaching methodologies) as schools sought to boost student's lagging mathematical skills.

Interest in bolstering mathematical skills was not, however, solely an American concern. Harvard professor Heisuke Hironaka (1931- ), one of a number of influential Japanese-born mathematicians and executive director for the Japan Association for Mathematical Sciences, is often credited with providing the inspiration for the International Math Olympics competition for schoolchildren in an effort to encourage mathematical accuracy and speed.

Mathematics and Popular Culture

Just as games of mathematical logic became popular in Victorian England a century earlier, in the last decade of the twentieth century mathematics once again provided a source of popular entertainment. During the 1990s a number of biographies of mathematicians and books on mathematical theory soared to the top tiers of many best-seller lists. Movies using the complexities and subtleties of mathematics and the culture of mathematicians became box-office hits and, as an increasingly technological society sought deeper meanings behind the science and mathematics enabling the information age, books and articles explaining often difficult and abstract mathematical concepts in simple terms gained popularity.

Books, for example, ranged in topics from biographies of the Greek mathematician Archimedes (c. 287-212 B.C.) to the brilliantly eccentric Paul Erdös (1913-1996). Other works treated specialized areas of mathematics such as the history of π as fresh, exciting, and readable history. Regarding these popular works, however, none captured more public attention than the controversy and scholarly drama surrounding the proof of Fermat's last theorem by Princeton mathematician Andrew Wiles (1953- ).

Mathematics and Twenty-First Century Society

From the darker—decidedly unpublic—worlds of political intrigue and espionage, advances in mathematics allowed cryptography to become a part of the everyday experience. Cryptography allows its users, whether governments, military, businesses, or individuals, to maintain privacy and confidentiality in their communications. Although the attempt to preserve the privacy of communications is an age-old quest, many cryptologists and communications specialists insist that a truly global electronic economy will be dependent on the development of cryptographic systems that allow transactions to carry the same legal weight as paper contracts. Development of such systems is highly dependent upon further advances in number theory.

Modern philosophers and social ethicists also assert that, in an expanding, electronically networked world, preservation of traditional notions of privacy may be dependent on cryptologic applications of higher mathematics. In a very real sense, mathematics developed during the later half of the twentieth century with the intent of helping us probe the innermost secrets of nature may, at the same time, provide the means to protect the sanctity of our innermost selves in the twenty-first century.

K. LEE LERNER