ABSTRACT
In Auguste Comte's classification of the sciences, mathematics took up a special place at the opening of the encyclopedic series, both as the first term and as the basis of his Cours de philosophie positive [Comte 1830–1842, vol. 1, 2nd lesson]. Indeed it constituted ‘the most powerful instrument that the human mind can use in its study of natural phenomena.’ Through its ‘rigorous universality,’ mathematics provided the best model for what a science should be. The other fundamental sciences in the series—astronomy, physics, chemistry, physiology, and sociology—were themselves classified according to the degree of abstraction of the phenomena they sought to describe and the degree of mathematization of their resulting laws. According to Comte, therefore, mathematical science ought to form ‘the true starting point of any scientific training, either general or specialized.’ Moreover, Comte divided mathematics internally between calculus (calcul) (its abstract part) and geometry and mechanics, which represented its concrete part. Under its various (arithmetic, algebraic, or analytic) guises, calculus was the science of relations par excellence, while geometry and mechanics, to which calculus could be applied, were natural sciences based on observations, just as astronomy and physics.
