Mathematics

G

Set Theory and Logic

Set theory is the branch of mathematics that seeks to establish statements that are true of sets, regardless of the kind of objects that make up the set. A set is a group of objects with a well-defined criterion for membership so that we can say definitely whether an object belongs to the set or not. The terminology and many of the results of set theory are used in symbolic logic, geometry, the theory of probability, and mathematical analysis.

Set theory and logic are closely related as we can see in diagrams in which circles represent sets and the relationships between them. The union of two sets A and B, written AÈB, is the set of all elements that belong to A or B. The intersection, AÇB, is the set of all elements that belong to A and B. The complement of a set A consists of elements not in A. The formal manipulation of expressions involving and, or, and not is called Boolean logic, and was developed in 1847 by British mathematician George Boole.

German mathematician Georg Cantor provided the first formulation of set theory. He extended the intuitive concept of a set to include the possibility of sets containing an infinite number of objects, and he showed that it is possible to conceive of infinities of different “sizes.”

Set theory can be used to provide an axiomatic, or logical, foundation for almost all of mathematics. The number of elements in a set, n(S), can be used to produce all the positive integers (whole numbers), 1, 2, 3, and so forth. Addition can be related to the operation of union. Subtraction, the inverse of addition, produces the negative integers, -1, -2, -3, and so on. Multiplication can be shown to be repeated addition, and introducing division, its inverse, produces rational numbers. Metamathematics examines the extent to which mathematics can proceed using set theory alone. The classic work in the field of metamathematics is the Principia Mathematica (1910-1913) by British mathematicians and philosophers Bertrand Russell and Albert North Whitehead.

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H

Number Theory

Number theory is the branch of mathematics that deals with the properties of numbers, primarily integers—whole numbers that may be positive, negative, or zero. One of the earliest problems studied in algebra was the division of integers: Is it possible to write an integer as the product of smaller integers? The integer 6, for example, can be written as 2 x 3. If an integer can be written in this way, it is called a composite number; if not, it is called a prime number. The first few prime numbers are 1, 2, 3, 5, 7, 11, 13, 17, 19, and 23. There are an infinite number of prime numbers.

Investigating prime numbers has generated many problems in pure mathematics. In the 19th century German mathematician Georg Friedrich Bernhard Riemann analyzed the distribution of prime numbers, and conjectured that they were related to the roots of a function now called the Riemann zeta function. The so-called Riemann conjecture has not yet been proved. Another famous problem in number theory is known as Fermat’s last theorem after 17th-century French mathematician Pierre de Fermat. Fermat stated that he had found a remarkable proof demonstrating that the equation aⁿ + bⁿ = cⁿ cannot be solved in integers when n is greater than 2. But he did not write down his proof, and it remained one of the most famous unsolved problems in mathematics until English mathematician Andrew Wiles proved the theorem in 1994.

The extreme difficulty in factoring large integers into prime numbers is the basis for modern cryptography. Breaking codes is essentially equivalent to finding the factors of large numbers—that is, the quantities that can be multiplied to yield the large number. Not surprisingly, the U.S. government ranks as one of the world’s largest employers of number theorists.

I

Systems Analysis

The mathematical study of systems is called systems analysis. It plays a vital role in the understanding of communications networks and computing networks. In most mathematical models a functional relationship exists between two quantities—that is, we can express one quantity in terms of the other by an equation y = f(x). Not all relationships are this simple, however. One approach to complex systems is to focus on the connectivity between quantities, rather than on the quantities themselves. In understanding the Internet, for example, the network is of interest rather than the individual computers attached to it. Similarly, in studying communications networks, the network itself is of interest rather than the hardware involved or actual information that is transmitted. The mathematical study of the connectivity between objects is sometimes called network analysis.

Ecosystems—interdependent organisms and their environments—that regulate life on this planet form a rather different network. Mathematical ecology and population dynamics are new research areas that use mathematics to study ecosystems and other biological systems and their behavior over long periods of time. Systems analysis has even played a role in the study of learning. One model for learning, called a neural network, simulates the interconnection of the neurons in the brain.

J

Chaos Theory

Despite advances made in systems analysis, many systems remain beyond the reach of current mathematics. Chaos theory, a relatively new area of mathematics, concerns the analysis of unpredictable systems that are extremely sensitive to initial conditions. One important example of a chaotic system is climate. Global climate modeling is an area of mathematical research that seeks to develop models for predicting the weather, given accurate data from weather satellites orbiting Earth. The problem in developing such models arises not from lack of data but from the difficulty of modeling such a complex system (Earth’s atmosphere) with a small number of equations. In such models even a thousand equations may be considered small. The solution of these equations is very sensitive to changes in the initial conditions. The term initial conditions refers to all the measurements at the starting time. A tiny inaccuracy in a single measurement of a chaotic system—such as a temperature variation of a fraction of a degree—can produce large errors in solutions to the model’s equations and predictions.

Meteorologist Edward Lorenz tried to model climate in a series of equations during the 1960s. In doing so, he produced a chaotic system of three related differential equations, now known as a Lorenz attractor, or strange attractor. Through his models he discovered the sensitivity of chaotic systems to initial conditions, which he phrased in the question “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”

The Lorenz attractor is an example of a fractal, a pattern produced by applying a function repeatedly, much like pushing a button on a calculator over and over. The sequence {x, f(x), f(f(x)), f(f(f(x))), ...}, when graphed in two dimensions, gives rise to beautiful, complex geometric images such as the Mandelbrot set pictured in this article. These fractal images are named after Benoit Mandelbrot, a Polish-born French mathematician who developed fractal geometry and coined the word fractal. The interesting relationship among fractals, chaos, and unstable phenomena such as turbulence is the subject of a field called nonlinear dynamics.

IV

History of Mathematics

Counting was the earliest mathematical activity. Early humans needed counts to keep track of herds and for trade. Primitive counting systems almost certainly used the fingers of one or both hands, as evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems today. The first advances in arithmetic were the conceptualization of numbers and the invention of the four fundamental operations: addition, subtraction, multiplication, and division. The earliest advances in geometry dealt with simple concepts such as the line and the circle. Further progress in mathematics had to await the Babylonians and Egyptians of about 2000 bc.

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