Fermat Reals

Abstract

The main aim of the present work is to start a new theory of actual infinitesimals, called theory of Fermat reals. After the work of A. Robinson on nonstandard analysis (NSA), several theories of infinitesimals have been developed: synthetic differential geometry, surreal numbers, Levi-Civita field, Weil functors, to cite only some of the most studied. We will discuss in details of these theories and their characteristics, first of all comparing them with our Fermat reals. One of the most important differences is the philosophical thread that guided us during all the development of the present work: we tried to construct a theory with a strong intuitive interpretation and with non trivial applications to the infinite-dimensional differential geometry of spaces of mappings. This driving thread tried to develop a good dialectic between formal properties, proved in the theory, and their informal interpretations. Almost all the present theories of actual infinitesimals are either based on formal approaches, or are not useful in differential geometry. As a meaningful example, we can say that the Fermat reals can be represented geometrically (i.e. they can be drawn) respecting the total order relation. The theory of Fermat reals takes a strong inspiration from synthetic differential geometry (SDG), a theory of infinitesimals grounded in Topos theory and incompatible with classical logic. SDG, also called smooth infinitesimal analysis, originates from the ideas of Lawvere and has been greatly developed by several categorists. The result is a powerful theory able to develop both finite and infinite dimensional differential geometry with a formalism that takes great advantage of the use of infinitesimals. This theory is however incompatible with classical logic and one is forced to work in intuitionistic logic and to construct models of SDG using very elaborated topoi. The theory of Fermat reals is sometimes formally very similar to SDG and indeed, several proofs are simply a reformulation in our theory of the corresponding proofs in SDG. However, our theory of Fermat reals is fully compatible with classical logic. We can thus describe our work as a way to bypass an impossibility theorem of SDG, i.e. a way considered as impossible by several researchers. The differences between the two theories are due to our constraint to have always a good intuitive interpretation, whereas SDG develops a more formal approach to infinitesimals. Generally speaking, we have constructed a theory of infinitesimals which does not need a background of logic to be understood.