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Algebras

ALGEBRAS.

The word algebra refers to a theory, usually mathematical, which is dominated by the use of words (often abbreviated), signs, and symbols to represent the objects under study (such as numbers), means of their combination (such as addition), and relationships between them (such as inequalities or equations). An algebra cannot be characterized solely as the determination of unknowns, for then most mathematics is algebra.

For a long time the only known algebra, which was and is widely taught at school, represented numbers and/or geometrical magnitudes, and was principally concerned with solving polynomial equations; this might be called "common algebra." But especially during the nineteenth century other algebras were developed.

The discussion below uses a distinction between three modes of algebraic mathematics that was made in 1837 by the great nineteenth-century Irish algebraist W. R. Hamilton (18051865): (1) The "practical" is an algebra of some kind, but it only provides a useful set of abbreviations or signs for quantities and operations; (2) In the "theological" mode the algebra furnishes the epistemological basis for the theory involved, which may belong to another branch of mathematics (for example, mechanics); (3) In the "philological" mode the algebra furnishes in some essential way the formal language of the theory.

Lack of space prevents much discussion of the motivations and applications of algebras. The most important were geometries, the differential and integral calculus, and algebraic number theory.

Not Distant Origins?

Several branches of mathematics must have primeval, unknown, origins: for example, arithmetic, geometry, trigonometry, and mechanics. But algebra is not one of them. While Mesopotamian and other ancient cultures show evidence of methods of determining numerical quantities, the means required need only arithmetical calculations; no symbolism is evident, or needed. Concerning the Greeks, the Elements of Euclid (fourth century b.c.e.), a discourse on plane and solid geometry with some arithmetic, was often regarded as "geometric algebra"; that is, the theories thought out in algebraic terms. While it can easily be so rendered, this reading has been discredited as historical. For one reason among many, in algebra one takes the square on length a to be a times a but Euclid worked with geometrical magnitudes such as lines, and never multiplied them together. The only extant Greek case of algebraization is the number theory of Diophantus of Alexandria (fl. c. 250 c.e.), who did use symbols for unknowns and means of their combination; however, others did not take up his system.

A similar judgment applies to ancient Chinese ways of solving systems of linear equations. While their brilliant collection of rules can be rendered in terms of the modern manipulation of matrices, they did not create matrix theory.

The Arabic Innovations

Common algebra is a theory of manipulating symbols representing constant and unknown numbers and geometrical magnitudes, and especially of expressing polynomial equations and finding roots by an algorithm that produces a formula. Its founders were the Arabs (that is, mathematicians usually writing in Arabic) from the ninth century, the main culture of the world outside the Far East. Some of the inspiration came from interpreting various Greek or Indian authors, including Euclid. The pioneer was Al-Khwarizmi (fl. c. 800847) with his work Al-jabr wa'l-muqabala, known in English as the Algebra, and over the next five centuries followers elaborated his theory.

The problems often came from elsewhere, such as commerce or geometry; solutions usually involved the roots of polynomial equations. Algebra was seen as an extension of arithmetic, working with unknowns in the same way as arithmetic works with knowns. The Arabic manner of expression was verbal: the word shay denoted the unknown, mal its square, ka'b its cube, mal mal for the fourth power, and so on. Arabs also adopted and adapted the Indian place-value system of numerals, including 0 for zero, that is called Hindu-Arabic. They were suspicious of negative numbers, as not being pukka quantities.

European Developments to the Seventeenth Century

Common algebra came into an awakening Europe during the thirteenth century. Among the various sources involved, Latin translations of some Arab authors were important. A significant homegrown source was the Italian Leonardo Fibonacci, who rendered the theory into Latin, with res, census, and cubus denoting the unknown and its powers. He and some translators of Arabic texts also adopted the Indian system of numerals. Communities developed, initially of Italian abbacists and later of German Rechenmeister, practicing arithmetic and common algebra with applicationssome for a living.

The title of al-Khwarizmi's book included the word al-jabr, which named the operation of adding terms to each side of an equation when necessary so that all of them were positive. Maybe following his successor Thabit ibn Qurra (836901), in the sixteenth century Europeans took this word to refer to the entire subject. Its theoretical side principally tackled properties of polynomial equations, especially finding their roots. An early authority was Girolano Cardano (15011576), with his Ars magna (1545); successors include François Viète (15401601) with In artem analyticem isagoge (1591), who applied algebra to both geometry and trigonometry. The Europeans gradually replaced the words for unknowns, their powers, means of combination (including taking roots), and relationships by symbols, either letters of the alphabet or special signs. Apart from the finally chosen symbols for the arithmetic numerals, no system became definitive.

In his book Algebra (1572), Rafael Bombelli (15261572) gave an extensive treatment on the theory of equations as then known, and puzzled over the mystery that the formula for the (positive) roots of a cubic equation with real coefficients could use complex numbers to determine them even if they were real; for example (one of his), given

The formula involved had been found early in the century by Scipione del Ferro (14651526), and controversially published later by Cardano. It could be adapted to solve the quartic equation, but no formula was found for the quintic.

Developments with Equations from Descartes to Abel

René Descartes's (15961650) Géométrie (1637) was an important publication in the history algebra. While its title shows his main concern, in it he introduced analytic geometry, representing constants and also variable geometric magnitudes by letters. He even found an algebraic means of determining the normal to a curve. Both this method and the representation of variables were to help in the creation of the calculus by Isaac Newton (16421727) in the 1660s and Gottfried Wilhelm Leibniz (16471716) a decade later.

During the seventeenth century algebra came to be a staple part of mathematics, with textbooks beginning to be published. The binomial theorem was studied, with Newton extending it to non-integral exponents; and functions were given algebraic expression, including as power series. Algebraic number theory developed further, especially with Pierre de Fermat (16011665). Negative and complex numbers found friends, including Newton and Leonhard Euler (17071783); but some anxiety continued, especially in Britain.

The theory of polynomial equations and their roots remained prominent. In particular, in Descartes's time "the fundamental theorem of algebra" (a later name) was recognized though not proved: that for any positive integer n a polynomial equation of degree n has n roots, real and/or complex. The Italian mathematician J. L. Lagrange (17361813) and others tried to prove it during the eighteenth century, but the real breakthrough came from 1799 by the (young) C. F. Gauss (17771855), who was to produce three more difficult and not always rigorous proofs in 1816 and 1850. He and others also interpreted complex numbers geometrically instead of algebraically, a reading that gradually became popular.

Another major question concerning equations was finding the roots of a quintic: Lagrange tried various procedures, some elaborated by his compatriot Italian Paolo Ruffini (17651822). The suspicion developed that there was no algebraic formula for the roots: the young Norwegian Niels Henrik Abel (18021829) showed its correctness in 1826 with a proof that was independent of Lagrange's procedures.

Lagrange was the leading algebraist of the time: from the 1770s he not only worked on problems in algebra but also tried philologically to algebraize other branches of mathematics. He based the calculus upon an infinite power series (the Taylor series); however, his assumption was to be refuted by Augustin-Louis Cauchy (17891857) and W. R. Hamilton (18051865). He also grounded mechanics upon principles such as that of "least action" because they could be formulated exclusively in algebraic terms: while much mechanics was encompassed, Newtonian and energy mechanics were more pliable in many contexts.

The Nineteenth Century: From Algebra to Algebras

Lagrange's algebraic ambitions inspired some new algebras from the late eighteenth century onward. The names used below are modern.

Firstly, in differential operators, the process of differentiating a function in the calculus was symbolized by D, with the converse operation of integration taken as 1/ D, with 1 denoting the identity operation; similarly, finite differencing was symbolized by, with summation taken as 1/. Much success followed, especially in solving differential and difference equations, though the workings of the method remained mysterious. One earnest practitioner from the 1840s was George Boole (18151864), who then imitated it to form another one, today called Boolean algebra, to found logic.

Secondly, in functional equations, the "object" was the function f itself ("sine of," say) rather than its values. In this context F.-J. Servois (17671847) individuated two properties in 1814: "commutative" (fg gf ) and "distributive" (f (gh ) fg fh ); they were to be important also in several other algebras.

As part of his effort to extend Lagrange's algebraization of applied mathematics, Hamilton introduced another new algebra in 1843. He enlarged complex numbers into quaternions q with four units 1, i, j and k :
q:= a + ib + jc + kd, where i 2 = j 2 = k 2 = ijk = 1; and ij = k and ji = k
and similar properties. He also individuated the property of associativity (his word), where i (jk ) (ij ) k.

At that time the German Hermann Grassmann (18091877) published Ausdehnungslehre (1844), a very general algebra for expressing relationships between geometrical magnitudes. It was capable of several other readings also; for example, later his brother Robert adapted it to rediscover parts of Boolean algebra. Reception of the Grassmanns was much slower than for Hamilton; but by the 1880s their theories were gaining much attention, with quaternions extended to, for example, the eight-unit "octaves," and boasting a supporting "International Association." However, the American J. W. Gibbs (18391903) was decomposing quaternions into separate theories of vector algebra and of vector analysis, and this revision came to prevail among mathematicians and physicists.

Another collection of algebras developed to refine means of handling systems of linear equations. The first step (1840s) was to introduce determinants, especially to express the formulae for the roots of systems of linear equations. The more profound move of inventing matrices as a manner of expressing and manipulating systems themselves dates from the 1860s. The Englishmen J. J. Sylvester (18141897) and Arthur Cayley (18211895) played important roles in developing matrices (Sylvester's word). An important inspiration was their study of quantics, homogeneous polynomials of some degree in any finite number of variables: the task was to find algebraic expressions that preserved their form under linear transformation of those variables. They and other figures also contributed to the important theory of the "latent roots and vectors" (Sylvester again) of matrices. Determinants and matrices together are known today as linear algebra; the analysis of quantics is part of invariant theory.

On polynomial equations, Lagrange's study of properties of functions of their roots led especially from the 1840s to a theory of substitution groups with Cauchy and others, where the operation of replacing one root by another one was treated as new algebra. Abel's even younger French contemporary Évariste Galois (18111832) found some remarkable properties of substitutions around 1830.

This theory of substitutions gradually generalized to group theory. In its abstract form, as pioneered by the German Richard Dedekind (18311916) in the 1850s, the theory was based upon a given collection of laws obeyed by objects that were not specified: substitutions provided one interpretation, but many others were found, such as their philological intrusion into projective and (non-)Euclidean geometries. The steady accumulation of these applications increased the importance of group theory.

Other algebras also appeared; for example, one to express the basic properties of probability theory. In analysis the Norwegian Sophus Lie (18421899) developed in the 1880s a theory of "infinitesimal transformations" as linear differential operators on functions, and formed it as an algebra that is now named after him, including a group version; it has become an important subject in its own right.

Consolidation and Extensions
in the Twentieth Century

At the end of the nineteenth century some major review works appeared. The German David Hilbert (18621943) published in 1897 a long report on algebraic number theory. The next year the Englishman Alfred North Whitehead (18611947) put out a detailed summary of several of them in his large book A Treatise on Universal Algebra, inspired by Grassmann but covering also Boole's logic, aspects of geometries, linear algebra, vectors, and parts of applied mathematics; an abandoned sequel was to have included quaternions. His title, taken from Sylvester, was not happy: no algebra is universal in the sense of embracing all others, and Whitehead did not offer one.

Elsewhere, group theory rose further in status, to be joined by other abstract algebras, such as rings, fields (already recognized by Abel and Galois in their studies of polynomial equations), ideals, integral domains, and lattices, each inspired by applications. German-speaking mathematicians were especially prominent, as was the rising new mathematical nationality, the Americans. Building upon the teaching of Emmy Noether (18821935) and Emil Artin (18981962), B. L. van der Waerden's (19031996) book Modern Algebra became a standard text for abstract algebras and several applications, from the first (19301931) of its many editions.

This abstract approach solved the mystery of the need for complex numbers when finding real roots of real polynomial equations. The key notion is closure: an algebra A is closed relative to an operation O on its objects a, or to a means of combining a and b, if Oa and a·b always belong to A. Now finding roots involved the operations of taking square, cube, roots and complex but not real numbers are closed relative to them.

One of the most striking features of mathematics in the twentieth century was the massive development of topology. Algebraic topology and topological groups are two of its parts, and algebras of various kinds have informed several others. Both (abstract) algebras and topology featured strongly in the formalization of pure mathematics expounded mainly after World War II by a team of French mathematicians writing under the collective name "Bourbaki."

Reflections

The proliferation of algebras has been nonstop: the classification of mathematics in the early twenty-first century devotes twelve of its sixty-three sections of mathematics to algebras, and they are also present in many other branches, including computer science and cryptography. The presence or absence in an algebra of properties such as commutativity, distributivity, and associativity is routinely emphasized, and (dis)analogies between algebras noted. Meta-properties such as duality (given a theorem about and ·, say, there is also one about · and) have long been exploited, and theologically imitated elsewhere in mathematics. A massive project, recently completed, is the complete classification of finite simple groups. Textbooks abound, especially on linear and abstract algebras.

Abstract algebras bring out the importance of structures in mathematics. A notable metamathematical elaboration, due among others to the American Saunders MacLane (b. 1909), is category theory: a category is a collection of mathematical objects (such as fields or sets) with mappings (such as ismorphisms) between them, and different kinds of category are studied and compared.

Yet this story of widespread success should be somewhat tempered. For example, linear algebra is one of the most widely taught branches of mathematics at undergraduate level; yet such teaching developed appreciably only from the 1930s, and textbooks date in quantity from twenty years later. Further, algebras have not always established their own theological foundations. In particular, operator algebras have been grounded elsewhere in mathematics: even Boole never fixed the foundations of the D -operator algebra, and a similar one proposed from the 1880s by the Englishman Oliver Heaviside (18501925) came to be based by others in the Laplace transform, which belongs to complex-variable analysis. However, a revised version of it was proposed in 1950 by the Polish theorist Jan Mikusinski (19131987), drawing upon ring theorythat is, one algebra helped another. Algebras have many fans.

See also Logic ; Logic and Philosophy of Mathematics, Modern ; Mathematics .

bibliography

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I. Grattan-Guinness

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