calculus a central branch of mathematics, originally conceived in connection with the determination of the tangent (or normal) to a curve and of the area between it and some fixed axis; but it also embraced the calculation of volumes and of areas of curved surfaces, the lengths of curved lines, and so on. Mathematical analysis is a still broader branch that subsumed the calculus under its rubric (see below), together with the theories of functions and of infinite series. Still more general and/or abstract versions of analysis have been developed during the twentieth century, with applications to other branches of mathematics, such as probability theory.
The origins of the calculus go back to Greek mathematics, usually in problems of determining the slope of a tangent to a curve and the area enclosed underneath it by some fixed axes or by a closed curve; sometimes related questions such as the length of an arc of a curve, or the area of a curved surface, were considered. The subject flourished in the seventeenth century when the analytical geometry of Descartes gave algebraic means to extend the procedures. It developed further when the problems of slope and area were seen to require the finding of new functions, and that the pertaining processes were seen to be inverse. Newton and Leibniz had these insights in the late seventeenth century, independently and in different forms.
In the Leibnizian differential calculus the differential dx was proposed as an infinitesimal increment on x, and of the same dimension as x; the slope of the tangent to a curve with y as a function of x was the ratio dy/dx. The integral, ex, was infinitely large and of the dimension of x; thus for linear variables x and y the area ey dx was the sum of the areas of rectangles y high and dx wide. All these quantities were variable, and so could admit higher-order differentials and integrals (ddx, eex, and so on). This theory was extended during the eighteenth century, especially by Euler, to functions of several independent variables, and with the creation of the calculus of variations. The chief motivation was to solve differential equations: they were motivated largely by problems in mechanics, which was then the single largest branch of mathematics.
Newton’s less successful fluxional calculus used limits in its basic definitions, thereby changing dimensions for the defined terms. The fluxion was the rate of change of a variable quantity relative to ‘time’; conversely, that variable was the ‘fluent’ of its fluxion. These quantities were also variable; fluxions and fluents of higher orders could be defined from them.
A third tradition was developed during the late eighteenth century by J. L. Lagrange. For him the ‘derived functions’ of a function f(x) were definable by purely algebraic means from its Taylorian power-series expansion about any value of x. By these means it was hoped to avoid the use of both infinitesimals and limits, which exhibited conceptual difficulties, the former due to their unclear ontology as values greater than zero but smaller than any orthodox quantity, the latter because of the naive theories of their deployment. In the early nineteenth century the Newtonian tradition died away, and Lagrange’s did not gain general conviction; however, the Leibniz- Euler line kept some of its health, for its utility in physical applications. But all these theories gradually became eclipsed by the mathematical analysis of A. L. Cauchy. As with Newton’s calculus, the theory of limits was central, but they were handled in a much more sophisticated way. He replaced the usual practice of defining the integral as (more or less) automatically the inverse of the differential (or fluxion or whatever) by giving independent definitions of the derivative and the integral; thus for the first time the fundamental ‘theorem’ of the calculus, stating their inverse relationship, became a genuine theorem, requiring sufficient conditions upon the function to ensure its truth. Indeed, Cauchy pioneered the routine specification of necessary and/or sufficient conditions for truth of theorems in analysis. His discipline also incorporated the theory of (dis)continuous functions and the convergence or divergence of infinite series. Again, general definitions were proffered and conditions sought for properties to hold. Cauchy’s discipline was refined and extended in the second half of the nineteenth century by K. Weierstrass and his followers at Berlin. The study of existence theorems (as for irrational numbers), and also technical questions largely concerned with trigonometric series, led to the emergence of set topology. In addition, special attention was given to processes involving several variables changing in value together, and as a result the importance of quantifiers was recognized – for example, reversing their order from ‘there is a y such that for all . . .’ to ‘for all x, there is a . . .’. This developed later into general set theory, and then to mathematical logic: Cantor was the major figure in the first aspect, while G. Peano pioneered much for the second. Under this regime of ‘rigor,’ infinitesimals such as dx became unacceptable as mathematical objects. However, they always kept an unofficial place because of their utility when applying the calculus, and since World War II theories have been put forward in which the established level of rigor and generality are preserved (and even improved) but in which infinitesimals are reinstated. The best-known of these theories, the non-standard analysis of A. Robinson, makes use of model theory by defining infinitesimals as arithmetical inverses of the transfinite integers generated by a ‘non-standard model’ of Peano’s postulates for the natural numbers. See also MATHEMATICAL ANALYSIS , PHIL — OSOPHY OF MATHEMATICS , SET THEOR.
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