Fermat–Reyes method in the ring of Fermat reals

Abstract

To discover derivatives, Pierre de Fermat used to assume a non-zero increment h in the incremental ratio and, after some calculations, to set h = 0 in the final result. This method, which sounds as inconsistent, can be perfectly formalized with the Fermat–Reyes theorem about existence and uniqueness of a smooth incremental ratio. In the present work, we will introduce the cartesian closed category where to study and prove this theorem and describe in general the Fermat method. The framework is the theory of Fermat reals, an extension of the real field containing nilpotent infinitesimals which does not need any knowledge of mathematical logic. This key theorem will be essential in the development of differential and integral calculus for smooth functions defined on the ring of Fermat reals and also for infinite-dimensional operators like derivatives and integrals.