Order of Nature and Orders of Science: On the Mathematical Philosophy of Nature and Its Changing Concepts of Science from Newton and Euler to Lagrange and Kant

Abstract

It is common knowledge that, next to experimentation, mathematics is the most important pillar of modern natural science. Less well known is how strongly the mathematical ideal of knowledge shaped modern science, especially so-called ‘rational mechanics’, which was regarded by most scientists and philosophers as the foundation and backbone of all natural sciences. The following chapter examines how this ideal of a ‘mechanical Euclideanism’, as I call it, shaped different programmes of mathematical natural philosophy in the late seventeenth and eighteenth centuries. It shows that this ideal was in opposition to the modern, hypothetico-decuctive understanding of science, and reveals how it was supported by epistemological and methodological arguments from both traditional rationalism and empiricism. The analysis of these processes is directed against an understanding of the development in question as one that was shaped primarily by Newton’s mechanics (as Ernst Mach, Thomas S. Kuhn and others claimed). Rather, it attempts to reconstruct this development as a dispute and competition between different programmes, guided by different scientific metaphysics and striving for different conceptual foundations of rational mechanics. It also tries to reveal how the attempts to integrate the achievements of these programmes into a unified formal framework of analytical mechanics alter and ultimately undermine the ideal of mechanical Euclideanism. The first edition of ‘Between Leibniz, Newton and Kant’ already contained a slightly different version of this paper as a chapter. As the former one was quite widely received and positively evaluated by the readership, this new version could make do with some updates and a few corrections. Readers interested in a more detailed account than can be given in this outline, perhaps also in continuing the story beyond Lagrange and Kant into the nineteenth century up to Neumann and Einstein, are referred to my book Axiomatik und Empirie.