Mathematics and the Eighteenth-Century Physical World

Overview

During the eighteenth century mathematicians and physicists embraced mathematics in general, and the calculus in particular, as an increasing powerful set of analytic techniques useful in the description of the physical world. Advancements in mathematical methods fueled increasingly detailed descriptions and investigations of the physical world. Caught between the rise of empirical science and the demands of the fledgling Industrial Revolution, the application of mathematics often outpaced its theoretical footing. Although there were sporadic attempts throughout the century to reconcile mathematical theory with its practical applications, it was a mostly French school of mathematicians in the later half of the century that undertook the task of producing a rigorous, self-consistent, mathematical system that was closely correlated to natural phenomena. The end result of these efforts to reintroduce rigorous analysis provided the foundation for the construction and rise of pure mathematics during the nineteenth century.

Background

A set of mathematical techniques known as the calculus was developed during the last decades of the seventeenth century by English mathematician and physicist Sir Isaac Newton (1642-1727) and, independently, German mathematician Gottfried Wilhelm von Leibniz (1646-1716). Although not fully developed in terms of logic, the calculus quickly found wide application, and its use provided a number of key new insights into the fields of physics, astronomy, and mathematics. The calculus became an essential tool to describe a clockwork universe that supported Western theological concepts of an unchanging, immutable God who ruled the universe through mechanistic laws.

In this regard the calculus fulfilled the intents of Newton, who developed his techniques and nomenclature in his important and influential Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). In Principia Newton described the results of gravity (i.e. the effects of gravitational forces) in great detail. There was, however, no explanation of the underlying mechanisms of gravity. In a similar fashion, the calculus was shown to work in many situations, but the underlying mechanisms went unexplained. Although Leibniz's work was much more mathematically rigorous than Newton's, as the calculus developed throughout the eighteenth century it remained better used than it was understood. That calculus worked was often proved by its close correlation with natural phenomena (i.e. the predictions of calculus were in accord with experimental or observational evidence). Why calculus worked, however, remained a question that eluded mathematicians.

Despite great success reaped from the applications of the calculus, a philosophical void remained with regard to the logical underpinnings of calculus. Into this void stepped a number of critics, chief among them Anglican bishop George Berkeley (1685-1753), to cast doubt upon the logic of calculus. In his 1734 work titled The Analyst; Or, a Discourse Addressed to an Infidel Mathematician, Berkeley argued that the theorems of calculus were derived from logical fallacies and that its apparent harmony with the natural world resulted from the mutual cancellation of fundamental errors in reasoning. Through such arguments diminishing mathematics, critics hoped to reclaim some role for scripture as a basis of insight into natural phenomena. Mathematicians took these philosophical criticisms seriously and set out to support the logical foundations of calculus and other developing forms of analysis with well-reasoned proofs.

Impact

Eighteenth-century mathematics allowed mathematicians, scientists, and philosophers deep and precise insights into the workings of the natural world. Indeed, the Age of Enlightenment was chiefly characterized by an increasing reliance on scientific and mathematical descriptions of nature as opposed to purely philosophical or traditional theological descriptions. Among eighteenth-century scientists, mathematics triumphed over scripture as the preferred tool of reason.

Eighteenth-century mathematics emphasized a practical, engineering-like analysis of the material parts of physical systems. In Newtonian kinematics, for example, objects were often idealized as to shape, reduced to point masses, or treated only with regard to the motion of their center of mass. Instead of focusing on details, there was an emphasis on the elaboration of the behavior of systems as a whole.

Throughout the eighteenth century there was a continuing advancement of mathematical applications. Scottish mathematician John Napier's (1550-1617) seventeenth-century advancement of logarithms greatly simplified the mathematical descriptions of many phenomena, and during the eighteenth century Napier's work provided a mathematical basis for generations of important preelectronic calculating tools, including early versions of the slide rule. French mathematician Gaspard Monge (1746-1818) invented differential geometry. Swiss mathematicians Jakob Bernoulli (1654-1750) and Johann Bernoulli (1667-1748) both made substantial application of calculus to physical problems, and the Bernoulli brothers' work eventually allowed Swiss mathematician Leonhard Euler (1707-1783) to more fully develop variational calculus (calculus of variations) as an extension of calculus dealing with maxima and minima of definite integrals. Euler and French mathematician Jean Le Rond d'Alembert (1717-1783) applied theoretical, deductive thought to a variety of physical problems, in a process they termed "amixed mathematics," wherein pure mathematical analysis as embodied by such disciplines as algebra and geometry was separated from the "mixed" mathematics applied to astronomy, physics, mechanics, etc. In a very fundamental way, mathematics became increasingly linked with physical reality and, as such, separated from its logical and philosophical (latter to be specifically termed "logical") foundations.

Although the Bernoulli brothers, Euler, and other mathematicians used calculus to attack a variety of mathematical and physical problems ranging from thermodynamics to celestial mechanics, even Euler's texts devoted to the methods of calculus failed to offer formal mathematical proofs regarding the new techniques. This "looseness" with math was also reflected in its incorporation by the French philosopher Antoine Nicolas de Caritat, marquis de Condorcet (1743-1974) and by French economist Anne-Robert-Jacques, baron de L'auline Turgot (1727-1781) into sociological analysis. For the French mathematician and astronomer Pierre-Simon Laplace (1749-1827), mathematics was created as a tool to explain the universe.

For other mathematicians and scientists, however, the development of theory and proofs remained highly important. French mathematician Joseph-Louis Lagrange (1736-1813), in his 1788 work Mécanique analytique (Analytical Mechanics), applied differential equations to the field of mechanics. Mécanique analytique is notable for its attempt to provide a more rigorous logical framework for mathematical analysis. In fact, Lagrange's attempt to reintroduce analytical rigor in Mécanique analytique was not limited to calculus; Lagrange also made substantial effort toward the reintroduction of logical development in geometry. Lagrange subsequently attempted a theory of functions and the principles of differential calculus in his 1797 work titled Théorie des fonctions analytiques.

A modest disregard for formal mathematical proof was driven by the beginnings of the Industrial Revolution in the later half of the eighteenth century. The demands of navigation and especially the practical engineering embodied in Scottish inventor James Watt's (1736-1819) invention of the steam engine provided a rapid pace of innovation that placed emphasis on applications of mathematics as opposed to development of theory. The application of the calculus and other tools of mathematical analysis without a well-developed foundation of proof was, however, a departure from tradition and was indicative of a major philosophical shift. Driven by the practical need to measure and explore, eighteenth-century mathematicians and scientists essentially introduced a revolutionary epistemology (i.e. the study of human knowledge and the how truth is defined). Truth was increasingly defined by what worked—in other words, what results were in best accord with the natural world. More importantly, fully developed mathematical proofs were not required in order to apply tools of analysis to problems in order to obtain accurate, precise, and predictive data regarding the natural world. In essence, a working knowledge often supplanted philosophical reasoning.

With particular regard to the calculus, during the eighteenth century mathematicians and scientists failed to precisely define the derivative and the integral, nor were they able to provide proofs of the theorems underlying their use. Mathematicians were, however, able to determine and apply the correct derivatives of most functions. Although there was no satisfactory explanations as to why these derivatives were correct, calculations utilizing such derivatives proved useful and in accord with nature. Ironically, the tolerance for the working defense of calculus (i.e., it was mathematically and philosophically defensible simply because it worked) gained its ultimate expression in the early nineteenth-century work of French scientist Nicolas-Léonard-Sadi Carnot (1796-1832), who, despite his own emphasis on rigor and mathematical formalities, defended calculus as a "compensation of errors." Such defenses were, if not abhorrent, at least insufficient for many mathematicians.

During the eighteenth century explanations of the rational basis of mathematical analysis tended to be more descriptive than explanatory. For example, the explanation of derivatives as a quotient of infinitesimals (an infinitely small but nonzero quantity) was not supportable by formal proof. The theoretical proof, involving epsilon-delta notations, that a function tends to a limit at a given point under certain conditions was not fully developed until the nineteenth-century work of French mathematician Augustin Louis Cauchy (1789-1857) and German mathematician Karl Theodor Weierstrass (1815-1897).

By the early 1820s Cauchy developed rigorous definitions of an integral, and by 1830 in his work Leçons sur le Calcul Différential, Cauchy set forth a coherent definition of a complex function of a complex variable. In addition, through his work on limits and continuity, Cauchy introduced rigor into calculus (and to other branches of mathematics as well) that enabled the rise of pure mathematics and modern analysis. Cauchy's work provided a philosophically and mathematically logical approach to calculus based upon the concepts of finite quantities and the limit. Interestingly, in the later half of the twentieth century German-born mathematician Abraham Robinson (1918-1974) developed nonstandard analysis, which provides a rationale for calculus based on a concept of infinitesimals similar to the reasonings of Newton and Leibniz.

By the end of the eighteenth century the renewed emphasis on mathematical rigor began to silence the critics of analysis, and in intellectual circles there was a growing acceptance of an understanding of nature based upon mathematics and empirical science. In turn the development of such Enlightenment thinking sent sweeping changes across the scientific, political, and social landscape.

K. LEE LERNER