Mathematics, Definition of

views updated

Mathematics, Definition of

Over the centuries, people have thought of mathematics, and have defined it, in many different ways. Mathematics is constantly developing, and yet the mathematics of 2,000 years ago in Greece and of 4,000 years ago in Babylonia would look familiar to a student of the twenty-first century.

Mathematics, says the mathematician Asgar Aaboe, is characterized by its permanence and its universality and by its independence of time and cultural setting. Try to think, for a moment, of another field of knowledge that is thus characterized. "In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to the old structure," noted Hermann Henkel in 1884.

From Truth to Application

The mathematician and philosopher Bertrand Russell said that math is "the subject in which we never know what we are talking about nor whether what we are saying is true." Mathematics, in its purest form, is a system that is complete in itself, without worrying about whether it is useful or true.

Mathematical truth is not based on experience but on inner consistency within the system. Yet, at the same time, mathematics has many important practical applications in every facet of life, including computers, space exploration, engineering, physics, and economics and commerce.

In fact, mathematics and its applications have, throughout history, been inextricably intertwined. For example, mathematicians knew about binary arithmetic , using only the digits 0 and 1, for years before this knowledge became practical in computers to describe switches that are either off (0) or on (1). Gamblers playing games of chance led to the development of the laws of probability . This knowledge in turn led to our ability to predict behaviors of large populations by sampling .

The desire to explain the patterns in 100 years of weather data led, in part, to the development of mathematical chaos theory . Therefore, mathematics develops as it is needed as a language to describe the real world, and the language of mathematics in turn leads to practical developments in the real world.

The Rules of the Game

Another way to think of mathematics is as a game. When players decide to join in a game—say a game of cards, a board game, or a baseball game—they agree to play by the rules. It may not be "fair" or "true" in the real world that a player is "out" if someone touches the player with a ball before the player's foot touches the base, but within the game of baseball, that is the rule, and everyone agrees to abide by it.

One of the rules of the game of mathematics is that a particular problem must have the same answer every time. So, if Bill says that 3 divided by 2 is 1½, and Maria says that 3 divided by 2 is 1.5, then mathematics asks if these two different-looking answers really represent the same number (as they do). The form of the answers may differ, but the value of the two answers must be identical if both answers are correct. Another rule of the game of mathematics is consistency. If a new rule is introduced, it must not contradict or lead to different results from any of the rules that went before.

These rules of the game explain why division by 0 must be undefined. For example, when checking division by multiplication it is clear that 10 divided by 2 is 5 because 2 × 5 is 10. Suppose 10/0 is defined as 0. Then 0 × 0 must be 10, and that contradicts the rule that 0 times anything is 0. One may believe that 0 divided by 0 is 5 because 0 × 5 is 0, but then 0 divided by 0 is 4, because 0 × 4 is also 0. There is another rule in the game of mathematics that says if 0 divided by 0 is 5 and 0 divided by 0 is 4, then 5 must be equal to 4—and that is a contradiction that no mathematician or student will accept.

Mathematics depends on its own internal rules to test whether something is valid. This means that validity in mathematics does not depend on authority or opinion. A third-grade student and a college professor can disagree about an answer, and they can appeal to the rules of the game to decide who is correct. Whoever can prove the point, using the rules of the game, must be correct, regardless of age, experience, or authority.

Learning the Language

Mathematics is often called a language. Numbers and symbols are understood without the barrier of translation, and mathematics can be used to describe many aspects of today's world, from airline reservation systems to theories about the shape of space.

Yet learning the vocabulary of mathematics is often a challenge and can be confusing. For example, mathematicians speak of the "bottom" of a fraction as the "denominator," which is a pretty frightening word to a beginner. But, like any language, mathematics vocabulary can be learned, just as Spanish speakers learn to say anaranjado, and English speakers learn to say "orange" for the same color.

In Islands of Truth (1990), the mathematician Ivars Peterson says that "the understanding of mathematics requires hard, concentrated work. It combines the learning of a new language and the rigor of logical thinking, with little room for error." He goes on to say "I've also learned that mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve."