What is a fiction in mathematics? Leibniz and infinitesimals as fictions

Abstract

This paper aims to examine the Leibnizian concept of mathematical fiction, emphasizing Leibniz’s view on the fictional character of infinitary notions. Firstly, a set of five conditions that fiction has to fulfill to be mathematically admissible is proposed as a general framework for the investigation. Based on Leibniz’s conceptions of symbolic knowledge, mathematical fiction is proposed as the class of confused notions that lack denotation due to the impossibility of their object. Departing from the analysis of the impossibility in terms of inconsistency, it is shown that Leibniz admits other forms of impossibility, especially for the infinitary notions. Thus, we propose impossibility as geometric irrepresentability and impossibility on the grounds of incompatibility with architectonic principles. In this way, the output of our examination supports the admission of three types of mathematical fiction: fiction1, which corresponds to the inconsistent notions, fiction2 that includes geometrically unrepresentable notions, and fiction3, which applies to “architectonically” impossible notions. In conclusion, infinitary concepts, without being inconsistent, correspond to the type of fiction2 andfiction3. Finally, it is concluded that Leibniz’s concerns focus on the impossibility due to incompatibility with architectonic principles rather than on the issue of geometric irrepresentability. Also we propose  some general issues about the relationships between Mathematics and reality in Leibniz’s philosophy.