Mathematics

D3

a

Gauss: All Fields

Carl Friedrich Gauss ranks as one of the greatest mathematicians who ever lived. Diaries from his childhood show that he had already made important discoveries in the theory of numbers. His book Disquisitiones Arithmeticae (Inquiries into Arithmetic), published in 1801, marks the beginning of the modern era in number theory.

Gauss also provided a geometric explanation of complex numbers as points on a plane, and the xy plane became known as the complex plane. (Complex numbers are of the form a + bi where a and b are real numbers and i is the imaginary number Á.) In his doctoral dissertation Gauss gave the first satisfactory proof of the fundamental theorem of algebra: that every polynomial equation has at least one root, or solution, in the complex plane. Complex numbers then formed a new field for analysis. Gauss called mathematics “the queen of the sciences” and number theory “the queen of mathematics.”

Often Gauss combined scientific and mathematical investigations. For example, his investigations of the orbit of Ceres, a newly discovered asteroid, led to his discovery of a statistical method called least squares. He used this method to predict the asteroid’s future appearances. His study of magnetism led to the creation of the potential theory in physics, which uses potential-energy functions to study gravity, electricity, and magnetism. His investigations in surveying produced important contributions to the geometry of curved surfaces.

D3

b

Analysis

Almost from the introduction of calculus, efforts had been made to supply a rigorous foundation for it. Calculus had introduced two new and complex concepts—the derivative and the definite integral. Newton and Leibniz were hazy on these matters, as were the mathematicians who extended calculus to analysis. Concepts such as limit, continuity, and differentiability were vague. Almost every mathematician of the 18th and early 19th centuries, including Euler and Lagrange, made some effort to produce a logical justification for calculus and failed. Although calculus clearly worked in solving problems, mathematicians lacked rigorous proof that explained why it worked.

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Finally, in 1821 French mathematician Augustin Louis Cauchy established a rigorous foundation for calculus with his theory of limits, a purely arithmetic theory. Later mathematicians found Cauchy’s formulation still too vague because it did not provide a logical definition of “real number.” The necessary precision for calculus and mathematical analysis was attained in the 1850s by German mathematician Karl T. W. Weierstrass and his followers.

Another important advance in analysis came from French mathematician Jean Baptiste Fourier, who studied infinite series in which the terms are trigonometric functions. Known today as Fourier series, they are still powerful tools in pure and applied mathematics. The investigation of which functions could be equal to Fourier series led German mathematician Georg Cantor to the study of infinite sets and to an arithmetic of infinite numbers.

D3

c

Cantor: Set Theory

Georg Cantor began his mathematical investigations in number theory and went on to create set theory. In the course of his early studies on Fourier series, he developed a theory of irrational numbers (real numbers that cannot be defined as a ratio of two integers—Ã, for example, or p). The character of irrational numbers had withstood mathematical explanation since ancient times. Although they cannot be written as finite numbers, they can easily be handled algebraically—in the case of Ã, by squaring it. Cantor and another German mathematician, Julius W. R. Dedekind, defined the irrational numbers and established their properties. These explanations hastened the abandonment of many 19th-century mathematical principles.

Cantor introduced the theory of sets in 1874 in a paper on different kinds of infinities. Cantor’s theory was considered abstract and attacked as a “disease from which mathematics will soon recover.” However, it now forms part of the foundations of mathematics. The application of set theory greatly advanced mathematics in the 20th century.

D3

d

Non-Euclidean Geometry

By the 19th century Euclidean geometry was the most solidly constructed branch of mathematics because many properties of the number system were still proved geometrically. Nevertheless, one of Euclid’s axioms known as the parallel postulate makes an assertion that experience cannot substantiate. This axiom states that through a point outside a given line it is possible to draw only one line parallel to the given line. Over the centuries mathematicians had tried to frame a substitute axiom, but each substitute proved to have a flaw. Nor could they deduce the questionable axiom from the other nine axioms.

One could start by assuming either that there was no parallel to a line through a point, or that there was more than one line through the point. The parallel postulate could be proved if, by adopting either of these assumptions in place of the Euclidean postulate, one could show that the new set of ten axioms led to a contradiction.

During the early 19th century Gauss concluded that it was possible to replace the parallel postulate and still produce a consistent, though non-Euclidean, geometry. The substitute postulate would state that an infinite number of parallel lines could be drawn through a given point. But Gauss did not publish his ideas because he feared they would expose him to ridicule.

Credit for the creation of non-Euclidean geometry is given to Russian mathematician Nikolay Lobachevsky and Hungarian mathematician János Bolyai. Each man published an organized presentation of a geometry that allows an infinite number of parallel lines through a given point. In the 1850s German mathematician Georg F. B. Riemann introduced the idea of a geometry in which there are no parallel lines. Riemann also introduced the idea of a curved line called a geodesic as the shortest distance between two points. Riemann’s geometrical interpretation of the theory of functions proved important for Einstein’s discoveries on relativity. Non-Euclidean geometry was not taken seriously until Einstein’s theory of special relativity in 1905 awakened the scientific world to the reality of Riemannian curved space.

Non-Euclidean geometry was the most impressive intellectual creation of the 19th century. It showed that mathematics could no longer be regarded as a body of unquestionable truths and that the observable world could not provide all the answers. Mathematicians were henceforth liberated to explore whatever ideas attracted them, and they turned more and more toward abstraction and theory. Individual mathematicians felt free to define their notions and to set up their axioms as they pleased, subject only to the limitation that the axioms do not give rise to theorems that contradict one another. The enormous expansion in mathematical activity in the 20th century was largely the consequence of this new freedom.

D3

e

Advances in Algebra

Algebra underwent a transformation during the 19th century, moving from the solution of polynomial equations to a study of the structure of algebraic systems. A first step in this direction was the publication of Treatise on Algebra (1830) by George Peacock, an English mathematician. Peacock attempted to provide algebra with the logical foundation Euclid had given geometry.

The creation of different systems of algebra began with Irish mathematician William Rowan Hamilton. In searching for general properties of complex numbers, Hamilton in 1843 discovered quaternions, a class of complex numbers that break the commutative law in algebra. This law states that a x b = b x a. Hamilton’s quaternions paved the way for the study of new algebraic systems.

Immediately after Hamilton’s discovery, German mathematician Hermann Grassmann and American mathematician and physicist J. Willard Gibbs began the analysis of three-dimensional vectors. From his investigations, Grassmann developed what is now called exterior algebra, which he applied to spaces of n (indefinitely many) dimensions. Gibbs used ideas of Grassmann to produce a system of vector analysis that could be applied to physics. He published his Elements of Vector Analysis in three parts from 1881 to 1884.

Another major step in algebra during the 19th century was the development of the theory of groups, which had its beginnings in the work of Lagrange. Norwegian mathematician Niels Henrik Abel demonstrated that it was impossible to solve by elementary algebra any equation of degree greater than four. Évariste Galois of France introduced the group concept to the solution of algebraic equations, showing that equations have associated groups of substitutions that govern their solubility. Galois’s work signaled a new direction in mathematics.

Just as Descartes had applied the algebra of his time to the study of geometry, so too did German mathematician Felix Klein and Norwegian mathematician Marius Sophus Lie apply the new algebra of the 19th century to geometry. Klein continued the group theory work of Galois, studying the properties that remained constant in a geometry when it underwent a group of transformations. Lie, too, worked in group theory, applying it not only to geometry but also to differential equations and other areas of mathematics.

Among the mathematicians who used the discoveries of Hamilton and Grassman was George Boole in England. Boole claimed mathematics could be investigated in terms of logic and provided symbolic notation for mathematical operations. Boole’s major contribution to mathematics is Boolean algebra, an algebra of sets that later formed the basis of symbolic logic and computer technology.

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