Advancements in Notation Enhance the Translation and Precision of Mathematics

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Advancements in Notation Enhance the Translation and Precision of Mathematics

Overview

During the Renaissance in Western Europe, a rediscovery and advancement of classical mathematics laid the foundation for the empiricism of the Scientific Revolution. One of the pillars of this intellectual reawakening in mathematics was the increased use of mathematical symbols that enabled scholars to communicate with each other more easily and accurately across geographical, national, and linguistic boarders.

Background

Although the use of symbols in mathematical operations dates to antiquity, scholars often differ in the awarding of credit for the first use of mathematical symbols. For example, according to some interpretations based on extant archaeological evidence, some scholars credit ancient Egyptian civilizations with the use of specific symbols for the operation of addition and with a symbol that denoted equality. Other scholars cite the first use of such symbols by Greek, Hindu, or Arabic cultures. Regardless, although symbolic use was certainly present in indigenous mathematical systems scattered throughout the world, there is no evidence of any concerted attempt at a universification of symbolism until after the advent of the printing press at the start of the European Renaissance, around 1450.

Beginning at this time, expansion in trade and foreign wars by European nations fueled the need for an easily translatable mathematics, especially arithmetic and geometry that were extensively used in commercial trades and in the development of weapons. In addition, the growing availability and capability of printing technology made it possible to quickly and accurately exchange scholarly ideas. Although these exchanges were not always altogether altruistic, with the goal of scholars often limited to the rapid incorporation of ideas that might benefit political or commercial interests, in general the introduction of mathematical symbols can be viewed as an attempt not only to streamline mathematical operations so that they could be more easily rendered by printed type but also to make such operations more precisely translatable.

Although the conventional view of historians for centuries held that Arab scholars merely preserved the classical mathematics of the Greeks until Europe emerged from its Medieval Age, recent reevaluations of the contributions of Arab mathematicians have cast light on many important mathematical innovations advanced by Arabic scholars. With regard to the development of mathematical symbolism, the writings of Arab (Moroccan) mathematician al-Marrakushi ibn Al-Banna (1256-1321) and Arab (Moorish and Tunisian) mathematician Abu'l Hasan ibn Ali al Qalasadi (1412-1486) exhibit a use of mathematical symbols to describe algebraic operations that scholars argue provides evidence of still earlier origins of algebraic notation in Arabic mathematics.

Regardless of intent or origin, an examination of mathematical works produced during the European Renaissance reveals an increasing reliance on mathematical symbols and the concurrent steady reduction in text to describe mathematical processes. Of particular benefit to those attempting to duplicate mathematical operations was a reduction in the extensive use of often confusing word abbreviations in the description of mathematical methods.

Impact

As a direct consequence of increased commercial trade during the Renaissance, there arose a form of commercial arithmetic made more easily understandable by the introduction of various symbols. Prior to the widespread use of symbols to denote numbers and operations, reading mathematics was a cumbersome and tedious exercise. Moreover, differences in language—and different usage of words within individual languages—increased the errors in translation that usually manifested themselves in errant mathematical calculations. In particular, prior to the introduction of algebraic symbols, algebraic expressions were laboriously scripted. As a result of the sensitivities of language, particularly when translating from Arabic, there was often great confusion and difficulty in reproducing and advancing algebraic concepts.

German mathematician Johann Widman's 1489 work, Behede und hubsche Rechnung was the first printed book to utilize and set out the plus (+) and minus (-) signs for the mathematical operations of addition and subtraction. Although the introduction of symbols for such elementary operations may seem a modest accomplishment by modern mathematical standards, it is important to remember the state of mathematics during the fifteenth and sixteenth centuries. As late as 1550 most learned scholars did not know how to multiply or divide. Accordingly, the advancement of mathematical symbols, even for elementary operations, contributed to the slow evolution of standardized notational systems that could be incorporated more easily into texts that, in turn, promoted a wider growth of scientific literacy upon which mathematics itself could flourish. In 1557 English mathematician Robert Recorde (1510-1558) introduced the equals sign (=) in a work titled Whetstone of Witte and made popular the plus and minus symbols published by Widman.

The rediscovery of the work of Pisan (later a part of Italy) mathematician Leonardo Pisano Fibonacci (1170-1250)—who borrowed extensively from other cultures during his travels—fueled the development of a notational system equipped to translate Fibonacci's mathematical puzzles and word problems.

The increased use of mathematical symbolism was given an important boost in French mathematician François Viète's (1540-1603) Canon Mathematicus, which covered many aspects of trigonometry and included the use of decimal fractions (printed in differential smaller type). Moreover, Viète made a systematic compilation of notation related to algebraic methods. In subsequent works, Viète used letter symbols to represent both known and unknown quantities (e.g., using vowels to represent unknowns and consonants to represent known quantities).

In his work De Thiende (Art of Tenths), Flemish-born Dutch mathematician Simon Stevin (1548-1620) demonstrated computation without the use of common fractions. Stevin formalized and synthesized prior concepts of decimal numbers into a notation system capable of representing decimal numbers, and he described the methods by which to add, subtract, multiply, and divide decimal numbers. Stevin's work was critical for the advancement of the empiricism and quantification of physics.

In his 1637 work Discours de la méthode (Discourse on Method) French philosopher René Descartes (1596-1650) championed the logic of mathematics as a paradigm for reasoning and, in so doing, elevated the use of mathematical symbolism into philosophical reflections on the nature and value of mathematics. Descartes incorporated the notation of English mathematician Thomas Harriot (1560-1621) with regard to the use of symbols for inequalities and made important modifications to the symbolism for exponential notation first advanced by Italian mathematician Rafael Bombelli (1526-1572).

In a later work titled La Géometrie, Descartes advanced the still-used convention wherein letters near the beginning of the alphabet are used to represent known quantities in equations and letters near the end of the alphabet are used to represent unknown quantities This innovation by Descartes allowed the rapid formulation and evaluation of equations with regard to the determination of unknown variables (e.g., in the equation ax = b it is apparent upon inspection that the quantity designated by x is an unknown variable).

The development of symbolism in coordinate geometry in the works of Descartes and French mathematician Pierre de Fermat (1601-1665) allowed the easier quantification of geometry and the application of algebraic methods to geometric problems in navigation and engineering. Moreover, the advancements by Descartes laid a solid foundation for English physicist and mathematician Sir Isaac Newton's (1642-1727) profoundly important Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy).

Curiously, not everyone readily accepted the increased use of symbolism in mathematics. The English philosopher Thomas Hobbes (1588-1679) argued that symbols only made the description of mathematics shorter—that symbols were nothing more than a convenience for the printer—and that they did nothing for the ultimate understandability of mathematical operations. Hobbs decried excessive use of symbolism as a "double labor" of the mind.

Other notable advances in seventeenth-century mathematical symbolism also include English mathematician William Oughtred's (1574-1660) advancement of a symbol for π as the ratio of the diameter of a circle to its circumference. In addition, in his work Clavis Mathematica, Oughtred advanced the use of "x" as a sign for mathematical multiplication. It was not, however, until the 1660s that the symbol ÷ was widely used to denote division. In his 1685 Treatise on Algebra English mathematician Rev. John Wallis (1616-1703) introduced a symbol used to represent numerical infinity in operations.

In the later decades of the seventeenth century German mathematician Gottfried Wilhelm von Leibniz (1646-1716) invented symbols of such operational utility that they quickly became the standard notation for the new calculus. The symbols still used for differentials (dx and dy) and the sign for the integral appear in papers that Leibniz published on the calculus.

This normalization of mathematics—occurring despite increased nationalism and provincialism in language—made possible the mathematical advances necessary to allow the quantification of accurate descriptions of uniform and accelerated motion (e.g. along parabolic curves) and the development of analytic geometry and calculus.

K. LEE LERNER