Ethan Walker

W.V.O Quine and Hilary Putnam constructed a well-known and popular argument for believing in the existence of mathematical objects. This argument is known as the “Indispensability Argument” (IA) for mathematical realism, and depends upon showing that mathematics is indispensably used and, from a logical standpoint, existentially quantified over in our best scientific theories. Being the types of naturalists that they were, Quine and Putnam took this to indicate that we should believe in the existence of mathematical objects. However, there has recently been some dissatisfaction with this approach to justifying mathematical realism, and has led to theorists proposing a new, ‘enhanced’ version of the IA (the EIA). This enhanced version argues that since mathematics is indispensably quantified over in our best explanations, we should believe in the existence of mathematical objects. In this paper, we will explore how the EIA emerged from the IA, and put forward some problems with it. We will argue that unclarity in both the meaning of ‘explanation’, the explanatory standards for when to classify an explanation as ‘best’, and the ontological relevance of those standards has plagued discussion of the EIA from the start and has created a confusing and muddled literature. As a remedy, we propose a new, enhanced version of the EIA, or the EEIA. Through this version, we hope to maintain clarity about what exactly should be shown in order for us to justify ontological inferences from explanatory considerations, regardless of what one takes ‘explanation’ to mean, the explanatory standards one uses to determine the best explanation, and the ontological import of those standards. We do this by invoking a higher order inference to the best explanation, where the best explanation of why mathematics is indispensably used in our best scientific theories is because it establishes dependency backing relations. One upshot from this discussion is that theorists should spend more time focusing on how mathematics could establish dependency backing relations, which may require developing in greater detail one’s favored ontology of mathematics and the world. Another upshot that will be touched on briefly at the end is that this will have ramifications for how one argues for scientific realism from inference to the best explanation.