On metaphors of mathematics: Between Blumenberg’s nonconceptuality and Grothendieck’s waves

Abstract How can metaphors account for the formation of mathematical concepts, for changes in mathematical practices, or for the handling of mathematical problems? Following Hans Blumenberg’s thought, this paper aims to unfold a possible answer to these questions by viewing the metaphorical frameworks accompanying these changes as essential for an understanding of how changes in mathematical practices have been accounted for. I will focus especially on cases in which these changes were caused by encounters with a mathematical object which did not yet have a well-defined concept, but also show that such indeterminacy remains with the mathematical concept even after it is considered ‘well-defined’. As the paper will show, this ‘forefield’ Vorfeld of the concept is addressed by Blumenberg’s account of metaphorology on the one hand, and accompanied by his later account of nonconceptuality Unbegrifflichkeit on the other hand. While Blumenberg himself did not develop a full-fledged philosophy of mathematics or of mathematical practices, I aim to show that one can nevertheless extract from his writings a unique position concerning the role metaphors play in mathematics. To this end, Blumenberg’s account of nautical and oceanic metaphors and Alexandre Grothendieck’s philosophy of mathematical practice provide fruitful starting points.