The notion of science in the Enlightenment has been strongly influenced by mathematics. Mathematics was understood as a science of reason that made new knowledge possible as well as a useful means that enabled a deeper understanding of the physical world and a better human life within this world. The new infinitesimal calculus, discovered independently by Newton and Leibniz, became the decisive mathematical instrument of science under the rule of mechanism as it was—in quite different forms. Lagrange and others understood a ‘higher’ form of the calculus, the calculus of variations, as universal key for a proper understanding of the whole of nature. This process was accompanied by the development of further mathematical tools such as potential theory and the theory of differential equations. Towards the end of the century, analytical mechanics dominated the understanding of mathematical physics. Analysis, since Pappus first and foremost a mathematical method, was closely related to the success of the new calculus and became the key concept of a broader reflection on the methodology of science and even of philosophy. The talk will focus on this development, for which Newton played a decisive role. It will end with a short glimpse on the new understanding of pure mathematics in the German-speaking lands at the end of the century, which was internally influenced by the development of abstract analysis and algebra.

Helmut Pulte is Professor of Philosophy at the University of Bochum. He studied mathematics, physics and philosophy and earned his PhD with a thesis on the foundation of mechanics in the works of Euler, Maupertius and Lagrange. After a research period as an Alexander von Humboldt fellow at the University of Cambridge, he received his venia legendi in 2002. His research focuses on the philosophy and history of physics and mathematics, as well as on modern epistemology and the changing notions of science since the Enlightenment. He was Co-Editor of the 12th volume of the Historisches Wörterbuch der Philosophie; since 2006 he is Co-Editor in Chief of the Journal for General Philosophy of Science. He is, inter alia, author of Axiomatik und Empirie. Eine wissenschaftsgeschichtliche Untersuchung zur Mathematischen Naturphilosophie von Newton bis Neumann (2005). His recent publications include The Reception of Isaac Newton in Europe (3 vols., 2019, co-ed. with S. Mandelbrote) as well as a series of studies on Isaac Newton, Leonhard Euler, Jakob Friedrich Fries, Hermann von Helmholtz, Johann von Kries, Moritz Schlick and other philosopher-scientists of modern times.