Mathematics

I

Introduction

Mathematics, a way of describing relationships between numbers and other measurable quantities. Mathematics can express simple equations as well as interactions among the smallest particles and the farthest objects in the known universe. Mathematics allows scientists to communicate ideas using universally accepted terminology. It is truly the language of science.

We benefit from the results of mathematical research every day. The fiber-optic network carrying our telephone conversations was designed with the help of mathematics. Our computers are the result of millions of hours of mathematical analysis. Weather prediction, the design of fuel-efficient automobiles and airplanes, traffic control, and medical imaging all depend upon mathematical analysis.

For the most part, mathematics remains behind the scenes. We use the end results without really thinking about the complexity underlying the technology in our lives. But the phenomenal advances in technology over the last 100 years parallel the rise of mathematics as an independent scientific discipline.

Until the 17th century, arithmetic, algebra, and geometry were the only mathematical disciplines, and mathematics was virtually indistinguishable from science and philosophy. Developed by the ancient Greeks, these systems for investigating the world were preserved by Islamic scholars and passed on by Christian monks during the Middle Ages. Mathematics finally became a field in its own right with the development of calculus by English mathematician Isaac Newton and German philosopher and mathematician Gottfried Wilhelm Leibniz during the 17th century and the creation of rigorous mathematical analysis during the 18th century by French mathematician Augustin Louis Cauchy and his contemporaries. Until the late 19th century, however, mathematics was used mainly by physicists, chemists, and engineers.

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At the end of the 1800s, scientific researchers began probing the limits of observation, investigating the parts of the atom and the nature of light. Scientists discovered the electron in 1897. They had learned that light consisted of electromagnetic waves in the 1860s, but physicist Albert Einstein showed in 1905 that light could also behave as particles. These discoveries, along with inquiries into the wavelike nature of matter, led in turn to the rise of theoretical physics and to the creation of complex mathematical models that demonstrated physical laws. Einstein mathematically demonstrated the equivalence of mass and energy, summarized by the famous equation E=mc², in his special theory of relativity in 1905. Later, Einstein’s general theory of relativity (1915) extended special relativity to accelerated systems and showed gravity to be an effect of acceleration. These mathematical models marked the creation of modern physics. Their success in predicting new physical phenomena, such as black holes and antimatter, led to an explosion of mathematical analysis. Areas in pure mathematics—that is, theory as opposed to applied, or practical, mathematics—became particularly active.

A similar explosion of activity began in applied mathematics after the invention of the electronic computer, the ENIAC (Electronic Numerical Integrator and Calculator), in 1946. Initially built to calculate the trajectory of artillery shells, ENIAC was later used for nuclear weapons research, weather prediction, and wind-tunnel design. Computers aided the development of efficient numerical methods for solving complex mathematical systems.

Without mathematics to describe physical phenomena, we might be living in a world with beautiful art, literature, and philosophy, but no technology. Even the medical advances of the last 50 years might not have occurred. Science and technology, in their turn, have provided many of the problems that motivated progress in mathematics. Such problems include the behavior of weather systems, the motion of subatomic particles, and the creation of speedier and smaller computers that can perform multiple tasks simultaneously.

II

Mathematics: The Language of Science

Experimental scientists observe phenomena and conduct experiments to obtain data about the way the universe behaves. Theoretical scientists generalize and draw conclusions from these results to form models of how the universe works. Mathematical scientists then study these models to understand their underlying principles and try to deduce what the models predict about unknown behavior or phenomena. Computational scientists use numerical simulations to study these models on computers. The cycle repeats as experimental scientists try to verify the predictions of mathematical and computational scientists through experiments. Social scientists also use mathematical techniques, primarily probability and statistics, to help resolve uncertainty about questions such as how various factors affect human behavior, how these variable factors are related, and how groups differ in their responses.

Mathematics attempts to capture the complexity of a problem using mathematical notation (signs and symbols) and concepts (theorems and proofs). Mathematical notation is a powerful tool, especially for representing entities, processes, or relationships that are impossible to visualize. For example, in modern geometry, mathematicians may work with more than three dimensions of space, even with infinite dimensions. Although these spaces are difficult to imagine, objects in these spaces can be studied through mathematics. Einstein’s discovery of relativity depended on studying objects in four dimensions, with time as the fourth dimension. Mathematicians develop simple corresponding models in two or three dimensions, then use the symbols and logic of mathematics to extend their intuition to infinite dimensions.

A

Symbols, Equations, and Theories

Mathematics studies relationships using symbols (numbers or letters), logic, and formal proof. Equality is one of the most fundamental relationships that two objects can have. If two things are equal, and we know something about one object, we can then deduce the same thing about the other object. Expressions of equality, called equations, are one of the main subjects of mathematical analysis. We often express equality in terms of variable quantities, such as x and y. A main tool of mathematics involves transforming one form of equality to another by changing variables. In a very simple example, if we know that x = y and y = z, then we also know that x = z.

Mathematicians strive for simplicity and generality, which lie at the core of what they call elegance. Simplicity means the use of a minimal number of assumptions or hypotheses in a proof or theory. Generality is the ability to apply the mathematical theory to different situations. A 14th-century Franciscan friar, William of Ockham, expressed this principle when he said, “Entities should not be multiplied unnecessarily.” The principle of economy in logic is sometimes known as Ockham’s razor. To put it another way: If you have two competing theories that both explain the observed results, choose the one that is the simplest, until additional evidence comes along.

Scientists continue to search for one of the ultimate expressions of mathematical elegance: a unified field theory. Such a theory would describe the behavior of all things in the universe in a consistent set of equations and unify the four known interactions—the strong, weak, electromagnetic, and gravitational forces. Einstein hoped that a unified field theory could be found, and he worked on this project from 1928 until his death in 1955.

How do we reconcile these grand mathematical ideas that seek a fixed order with what we know about real life, where things are unpredictable, random events occur, and order and structure often disappear and are replaced by chaos? Mathematicians study real-world change, such as the behavior of weather systems, by means of chaos theory. They have determined that any nonlinear system (system that cannot be predicted on the basis of past behavior) that has sufficient variables (unknown quantities) can behave in a chaotic manner. Systems besides weather known to be chaotic include heart rhythms, the rise and fall of animal populations, and chemical reactions. In some cases chaotic behavior may barely be observable. Scientists long thought that if they could eliminate randomness from chaotic systems, the systems would then follow predictable rules. They know now that this is not the case.

B

Pure Mathematics and Applied Mathematics

Mathematics, the language of science, has two dialects: pure mathematics and applied mathematics. Both kinds of mathematics are used to solve problems. Pure mathematics is the study of abstract relationships, whereas applied mathematics applies mathematical analysis to real-world problems, such as the rate of global warming. The relationship between pure and applied mathematics is a complex one, and the boundary between the two is constantly shifting.

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