publication . Article . 2011

Fermat–Reyes method in the ring of Fermat reals

Giordano, Paolo;
Open Access
  • Published: 01 Oct 2011 Journal: Advances in Mathematics, volume 228, pages 862-893 (issn: 0001-8708, Copyright policy)
  • Publisher: Elsevier BV
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 t...
free text keywords: General Mathematics, Topology, Fermat's theorem, symbols.namesake, symbols, Regular prime, Wieferich prime, Fermat's theorem on sums of two squares, Proofs of Fermat's theorem on sums of two squares, Mathematics, Mathematical analysis, Fermat's factorization method, Discrete mathematics, Fermat number, Fermat's little theorem, Nilpotent infinitesimals, Smooth infinitesimal analysis, Extension of the real field, Fermat–Reyes theorem, Mathematics(all)
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