Lars Arthur Tump

Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to Baker :223–238, 2005; Br J Philos Sci 60:611–633, 2009). Furthermore, the result is then also used to strengthen the platonist position :779–793, 2017a). We pick up the circularity problem brought up by Leng Mathematical reasoning, heuristics and the development of mathematics, King’s College Publications, London, pp 167–189, 2005) and Bangu :13–20, 2008). We will argue that Baker’s attempt to solve this problem fails, if Hume’s Principle is analytic. We will also provide the opponent of the Enhanced Indispensability Argument with the so-called ‘interpretability strategy’, which can be used to come up with alternative explanations in case Hume’s Principle is non-analytic.

A proposal by Baron, Colyvan, and Ripley to extend the counterfactual theory of explanation to include counterfactual reasoning about mathematical explanations of physical facts is discussed. Their suggestion is that the explanatory role of mathematics can best be captured counterfactually. This paper focuses on their example with a number-theoretic antecedent. Incorporating discussions on the structure and de re knowledge of numbers, it is argued that the approach leads to a change in the structure of numbers. As a result, the counterfactual is not about the natural numbers anymore. Linking the antecedent and consequent of the counterfactual also becomes problematic.