Paycha, in Encyclopedia of Mathematical Physics, 2006 Introductionįunctional analysis is concerned with the study of functions and function spaces, combining techniques borrowed from classical analysis with algebraic techniques. The record leaves little doubt that Leibniz considered the latter the foremost foundational problem in mathematics. If we read Leibniz on his own terms, and if we read what he actually published, then it soon becomes obvious that he wrote next to nothing about the nature of infinitesimals but paper after paper after paper about the problem of transcendental curves. 45 But in reality this skewed coverage says more about the interests of modern philosopher-historians than it does about Leibniz. Judging by the bulk of recent scholarship on Leibniz’s mathematics, for example, one could easily get the impression that he must have spent most of his time philosophising about infinitesimals. The same present-day values also lead to vastly disproportionate attention being paid to the few places where issues pertaining to foundations of infinitesimal methods are discussed in 17th-century sources. On the contrary, one finds them one after the other asserting again and again that their infinitesimal methods are perfectly well established on sound foundations.
Rarely, if ever, does one find any statements recording such recognition of foundational inadequacy in writings of leading practitioners of the early calculus. It is a projection onto the past of present-day values.
It is commonly said that “mathematicians of the seventeenth and eighteenth centuries … recognized that their methods were unsatisfactory, but were willing to tolerate them because they yielded correct results.” 44 In my view, this is not accurate. Nieuwentijt is either obstinate or ignorant,” 42 and “by no means worthy of a response.” 43 were perfectly happy to proceed with intuitive and flexible notions such as “ dx means to me the speed with which the x grow, or the difference between two subsequent x,” 39 and casual justifications such as that the calculus is “nothing but the Ancients Method of Exhaustions, a little disguised.” 40 When Nieuwentijt (1694) criticised the foundations of the infinitesimal calculus, the response, from Johann Bernoulli at any rate, was: “could anyone refrain from laughing?” 41 “I say again: Mr.
The historical record is quite straightforward: the matter was simply not seen as a very serious issue. In Transcendental Curves in the Leibnizian Calculus, 2017 2.3.4 Foundations of infinitesimal methodsĪnother point on which the preconceived opinions of modern authors lead to a distorted view of history is that of the foundations of infinitesimal methods. Although Euler had already noticed that the coordinates of a point on a surface could be expressed as functions of two independent variables, it was Gauss who first made a systematic use of such a parametric representation, thereby initiating the concept of “local chart” which underlies differential geometry. Until Gauss’ fundamental article Disquisitiones generales circa superficies curvas (General investigations of curved surfaces) published in Latin in 1827 (of which one can find a partial translation to English in Spivak (1979)), surfaces embedded in R 3 were either described by an equation, W( x, y, z) = 0, or by expressing one variable in terms of the others. The study of differential properties of curves and surfaces resulted from a combination of the coordinate method (or analytic geometry) developed by Descartes and Fermat during the first half of the seventeenth century and infinitesimal calculus developed by Leibniz and Newton during the second half of the seventeenth and beginning of the eighteenth century.ĭifferential geometry appeared later in the eighteenth century with the works of Euler Recherches sur la courbure des surfaces (1760) (Investigations on the curvature of surfaces) and Monge Une application de l’analyse à la géométrie (1795) (An application of analysis to geometry). Paycha, in Encyclopedia of Mathematical Physics, 2006 Curves and Surfaces