Fictionalism, indispensability, cicadas: a fictionalist approach to the (in)dispensability of mathematics and the ontological status of abstract objects

The paper deals with fictionalism in the philosophy of mathematics, namely, the claim that we can use mathematical theories and, at the same time, believe that they are false because mathematical objects do not exist. Mathematical objects (numbers, sets, and functions) are non-spatiotemporal, nor mental platonic entities that are causally isolated from us. This raises two questions. The first is known as Benacerraf’s problem: how can we think of the truth of mathematical propositions? The second is the problem of applicability of mathematics. It becomes a problematic field for nominalists who refuse to rationally believe in the existence of such entities. Namely, if one believes that mathematical objects do not exist, why does mathematics-based empirical science work? As a response to this question, the well-known and widely discussed «indispensability argument» emerges, postulating ontological commitments to mathematical objects on the basis that they are indispensable to our best scientific theories. According to this argument, realists about science must also accept Platonism about mathematical entities. Hartry Field disproves this argument and demonstrates the dispensability of mathematics by proposing his «science without numbers». Field replaces the criterion of truth with the criterion of conservatism and argues that the applicability of mathematics should be explained by whether a particular theory is conservative or not. We then consider the «enhanced» indispensability argument (Baker) based on the explanatory role of mathematics. In the final section, we describe the «new» fictionalist account (Balaguer). The new fictionalist strategies allow us to accept the ontological thesis of nominalism without asserting the indispensability of mathematics. We agree that the explanatory power of mathematics is an argument in favor of the indispensable role of mathematical objects in the natural sciences. Nevertheless, the appeal to indispensability is misguided. We do not have to rationally believe in the existence of those entities that are indispensable to science. We can rather consider these entities as useful (in explanation) heuristic fictions and, at the same time, believe that they do not exist and that mathematical propositions are false.