Indefinite with Respect to Concepts and Proofs

A critical philosophy of mathematics considers mathematics as being indefinite. Mathematical constructions are not heading towards definite formats; they are tentative and always open to change. In this chapter I concentrate on showing mathematics as being indefinite with respect to concepts and proofs. I illustrate this with reference to the changing roles of the notion of infinitesimal. Before the nineteenth century the term infinitesimal was used liberally, but in the nineteenth century it became a concern to ensure rigour in mathematical reasoning, and to get the infinitesimals under control. The notion of function is indefinite; in fact, it has not always been part of the mathematical vocabulary. Neither Newton nor Leibniz operated with the notion of function. The very word function was first used by Bernoulli, and Euler made some further clarifications of the notion. The notion of infinity has, in its own way, brought about huge controversies in mathematics. The notion is indefinite, and this indefiniteness has a huge impact on what to consider as a valid mathematical proof.