Abstract

Hans Vaihinger tried to explain how mathematical theories can be useful without being true or even coherent, arguing that mathematicians employ a special kind of fictional or “as if” reasoning that reliably extracts truths from absurdities. Moritz Pasch insisted that Vaihinger was wrong about the incoherence of core mathematical theories, but right about the utility of fictional discourse in mathematics. This essay explores this area of agreement between Pasch and Vaihinger. Pasch’s position raises questions about structuralist interpretations of mathematics.