Towards a history of mathematical consistency: tacit knowledge and conceptual change in early modern mathematics

In the crucial early modern period, fundamental notions like number, geometrical magnitude, and ratio, essentially defined à la Euclid in 1500, had been abandoned by 1700. By then infinitesimals, a notion inconsistent with traditional mathematics, were widely used in mathematics and rational mechanics. No satisfactory account exists for such a revolution, although traditional histories of mathematics stressed the "lack of rigour" of early modern mathematicians. In fact, the new notions of number and magnitude and the legitimation of infinitesimal thinking provide striking evidence that in critical episodes the evolution of mathematics is not guided by deductive rules nor by mathematical consistency. Motivations and rationale must be social in nature.

This project studies the emergence of implicit novel mathematical notions in mathematical practice, as opposed to mathematical texts. In particular, we focus on practical geometry and the practitioners of the art of measuring as sources for tacitly introducing some form of arithmetical continuum. We want to reevaluate the criticism of "lack of rigour" in early modern calculus by situating the foundational discussions within the wider social context in which the sceptical crisis of the early enlightenment discredited traditional axiomatico-deductive structures of knowledge. In particular we think we will be able to show how fruitful turns out to be a Wittgensteinian account of 17th century philosophy of mathematics.