Mathematics - science about fictions...?

Abstract

According to mathematical realism, mathematics describes an abstract realm of mathematical entities, and mathematical theorems are true in the classical sense of this term. In particular, mathematical realism is claimed to be the best theoretical explanation of the applicability of mathematics in science. According to Quine's indispensability argument, applicability is the best argument available in favor of mathematical realism. However, Quine's point of view has been questioned several times by the adherents of antirealism. According to Field, it is possible to show, that - in principle - mathematics is dispensable, and that so called synthetic versions of empirical theories are available. In his 'Science Without Numbers' Field follows the 'geometric strategy' - his aim is to reconstruct standard mathematical techniques in a suitable language, acceptable from the point of view of the nominalist. In the first part of the article, the author briefly presents Field's strategy. The second part is devoted to Balaguer's fictionalism, according to which mathematics is indispensable in science, but nevertheless can be considered to be a merely useful fiction.