Development of Higher-Dimensional Algebraic Concepts

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Development of Higher-Dimensional Algebraic Concepts

Overview

During the nineteenth century several attempts were made to generalize the algebra of complex numbers, which provided an adequate description of displacements and rotations in the plane, to describe objects in three dimensions. A number of generalized or hypercomplex systems were found, but none that preserved the properties of multiplication that were common to the real and complex numbers. Over time the form of vector algebra developed and popularized by the American J. Willard Gibbs came to be accepted as standard in scientific use, although some of the issues raised by the earlier work have left a lasting impact on pure and applied mathematics.

Background

At the start of the eighteenth century, mathematicians were generally familiar with the appearance of the square root of negative integers in the solution of algebraic equations, but were distrustful of such results, considering them spurious or fictitious. By the middle of that century the great Swiss mathematician Leonhard Euler 1707-1783) had shown by manipulating infinite series that one could make the identification: where is an angle measured in radians, e = 2.718... is the basis of natural logarithms, and i denotes the square root of -1. A more complete interpretation was provided in 1797 by Caspar Wessel (1745-1818), a Norwegian surveyor whose paper was published in a journal little read by mathematicians, so that credit is generally given to the French mathematician J. R. Argand (1768-1822) who independently obtained the same result in 1806. Wessel and Argand showed that every point in the plane could be represented by a combination of the form x+iy, or equivalently, by a line connecting the origin of coordinates with the point (x,y). Euler's result then represented a line of unit length. The addition of complex numbers corresponded to what we now call vector addition in two dimensions. The multiplication of the two complex numbers (a+ib) and (c+id) yields (acbd) + i(ad+bc) or a line of length numerically equal to products of the lengths corresponding to the two numbers multiplied and making an angle equal to the sum of the angles made by the two lines with the x-axis. With Argand's interpretation it was easy to interpret the subtraction and division of two complex numbers so that complex numbers possessed all the arithmetic properties of real numbers.

Given the success of mathematics in defining complex numbers that obeyed ordinary arithmetic rules, it seemed plausible that it would be possible to find a similar interpretation of points in three-dimensional space. The Irish astronomer Sir William Rowan Hamilton (1788-1856) reported the first significant success in this direction in 1843. Hamilton was, however, unable to find any procedure involving triples of numbers that would obey the right arithmetic rules. Instead, he introduced a system of quaternions, a set of numbers of the form w+ix+jy+kz, where the new units i, j, and k obey the multiplication rules Multiplication of quaternions was unlike the multiplication of real or complex numbers in that it was not commutative, that is, the product of two quaternions was different when the order is interchanged, so that ab was not necessarily equal to ba.

The quaternion system clearly included more information than was needed to describe displacements in three-dimensional space. Hamilton suggested calling the first term the scalar term and the remainder the vector term. When two quaternions with zero scalar terms are multiplied, one obtains a product with a scalar term equal to the modern scalar product of the two vectors and a vector term equal to the modern vector product.

Hamilton considered the discovery of quaternions to be his greatest achievement, devoting most of his attention to them for the remainder of his life. One of his corespondents, Peter Guthrie Tait, assumed leadership of the field on Hamilton's death in 1865, producing eight books on the subject with various collaborators Tait had assumed the chair of Natural Philosophy at Edinburgh University in 1960, becoming one of the collaborators of William Thomson (1824-1907), who would later become Lord Kelvin and a leader of British physics. Thomson considered quaternions unnecessarily elaborate, and placed Tait in the curious position of co-authoring several books making no mention of quaternions while promoting their use in his other works. Some of Tait's methods were adopted, however, by the Scottish mathematical physicist James Clerk Maxwell (1831-1879), who had been his childhood friend, in his analysis of electromagnetic phenomena.

The modern system of vector analysis, which has come to supplant quaternions, is generally attributed to the American physicist Josiah Willard Gibbs (1839-1903), a professor of mathematical physics, and the Englishman Oliver Heaviside (1850-1925), a self-taught scientist interested in the new electrical technology, who worked independently. Both individuals traced their interest in vector quantities to Maxwell's Treatise on Electricity and Magnetism, published in 1873, which had employed quaternion methods. Gibbs and Heaviside found the combination of vector and scalar products implicit in quaternions to be artificial and produced a streamlined version with the scalar and vector products considered as separate entities. This resulted in a simpler formalism that met the needs of electromagnetic theory and physics and engineering in general. Gibbs wrote the first textbook of vector analysis and taught the first university courses in the subject. In twentieth-century physics texts, quaternions are mentioned as a historical footnote, if at all.

Impact

Over the course of the history, the concept of number has had to be broadened numerous times. To the natural or counting numbers, 1,2,3, ..., had to be added a zero, then negative numbers, then rational numbers, then irrational and complex numbers. In each case a guiding principle had been that the basic properties of the fundamental arithmetic operations, addition and multiplication, would still apply to the broadened set of numbers. This expectation was even formalized by the English mathematician George Peacock in 1830 as a "principle of permanence of formal operations."

The fact that a consistent algebra in which numbers do not commute under multiplication could exist and be useful in the description of nature came as something of a shock to mathematicians, who tended to regard mathematics as a set of self-evident truths about the universe. This shock followed on the discovery that alternative geometries exist, geometries in which the sum of the angles of a triangle could exceed or be less than the sum of two right angles. This loss of certainty about what were once considered self-apparent rules of arithmetic and geometry stimulated an increased attention to the foundations of mathematics, an investigation which was to occupy many of the major figures in mathematics and mathematical logic in the twentieth century.

Among the sets of noncommuting mathematical objects to be introduced into mathematics are the matrices introduced by the English mathematician Arthur Cayley (1821-1895) in 1858. These are arrays of numbers, real or complex, which obey a multiplication rule:

If A and B are matrices then in general AB is a different matrix from BA so the multiplication of matrices, like that of quaternions does not obey the commutative law. In fact sets of 3×3 matrices of real numbers exist which obey the same multiplication rules as Hamilton's quaternions, provided the number "1" is interpreted as the matrix with each diagonal element equal to l and 0s in all the other positions.

Matrices are used in many branches of higher mathematics and in all areas of physics. One very important use is the representation or rotations of the xyz coordinate system by 3×3 matrices of real elements. If the components of a vector are listed in a column, then placing the vector on the right-hand side of such a matrix and performing the matrix multiplication operations yields a column with the vectors x-, y-, and zcomponents in the rotated coordinate system. The effect of two subsequent rotations about two different axes is then represented by the multiplication of the two rotation matrices. Since the result of two rotations in three-dimensional spaces depends on the order in which they are performed, it is not surprising that matrix multiplication is not commutative. The difference in the products of two matrices is called the commutator or commutation relation, denoted [A,B]=AB-BA.

In modern physics, the Heisenberg uncertainty principle states that it is generally not possible to measure two properties of a system at the same time, since measurements generally affect the states of the systems. Thus the outcome of two measurements performed in sequence will depend on the order in which they are performed. Heisenberg used this line of reasoning as a basis for developing a matrix mechanics to describe subatomic states. In this matrix approach, which is still used in quantum mechanics and elementary particle physics, it is the commutation relations which are used to calculate all the measurable properties of the systems.

It is noteworthy that almost all the individuals involved in the development of the theory of vectors were employed as physical scientists. At the start of the nineteenth century almost all mathematics was done by people concerned with the applications of mathematics to physical problems or engineering. By the end of the century a new group of professional mathematicians, with their own journals and academic posts and societies, had come into being, in part because of the demand for teachers of mathematics for students of the new technical fields. The pragmatic vector analysis of Gibbs met the needs of science for a time, leaving the new mathematicians to contemplate the significance and potential of noncommuting algebras, until they too were needed by the physicists about a quarter century later.

DONALD R. FRANCESCHETTI