Lars Arthur Tump

The enhanced indispensability argument is at risk of being circular. In this chapter it is shown that paraphrasing the explanandum into first-order logic with identity only works when a bridge principle, i.e. Hume’s Principle, is used to bridge the difference in context between the mathematical explanans and the translated explanandum. However, Hume’s Principle must not be analytic, something which is not a given. An alternative ‘interpretability strategy’ is provided, which makes use of Robinson Arithmetic. Keeping in mind the broadly empiricist context of the argument, the difference in mathematical strength is discussed.

This chapter introduces the enhanced indispensability argument. Given the criticism of confirmation holism in the original Quine-Putnam indispensability argument, in conjunction with the idea that indispensability is not considered to be sufficient for drawing ontological conclusions, the enhanced indispensability argument offers an explanatory version of the indispensability argument and states that the entities involved should be indispensable in the right way, namely indispensable for explanations.

The second part shifts focus towards embedding mathematical explanations within a framework of counterfactual reasoning. This chapter sets up the context. It discusses a proposal for a non-interventionist counterfactual account for explanation. If the counterfactual information can indeed be separated from the causal information, then mathematical explanations could in principle be incorporated into a counterfactual theory of explanation. Because this inevitably leads to impossible antecedents, as mathematical properties are necessary, the topic of counterpossibles is introduced, specifically with regard to the question whether they are trivially true.

This chapter discusses a concrete example of counterfactual reasoning about a mathematical explanation of physical facts, which serves a starting point for number-theoretic counterpossibles. Special attention will be given to contradictions that arise from varying the necessary properties in the antecedent. Even though a contradiction tolerant logic would be able to deal with those, this is not the only problem. Since the numbers are not closed under succession anymore, the resulting model of arithmetic will not be isomorphic to the standard interpretation.

Based on insights from discussions on computation, the structure of natural numbers, de re knowledge of numbers and the standard interpretation of arithmetic, it is questioned whether a counterfactual account of explanation can work for number-theoretic counterpossibles. Making use of Peano arithmetic, emphasis is put on the role of the standard induction scheme, as it contains the symbol for the successor function, as such suggesting an intimate connection between Peano numerals and de re knowledge of numbers. Because of the resulting shift in context in the antecedent, the way a contexts stops an explosion of ramifications in the case of ordinary counterfactuals might not be available in the case of number-theoretic counterpossibles.

In this introductory chapter, the subject of mathematical explanation of empirical phenomena is positioned in relation to the topics of scientific explanation and mathematical explanation. This serves as the stepping stone towards the main matter of the book. Part one, The Indispensability of Mathematics, pertains to the role of mathematical entities in mathematical explanations of empirical phenomena. Part two, Mathematical Counterfactual Dependence, pertains to the topic of mathematical explanation within the framework of a counterfactual theory of explanation.

This book addresses contemporary issues in the philosophy of mathematics that deal with the role of mathematics in explanations of empirical phenomena. It brings together various debates, such as on indispensability, number theory, abstraction principles, and counterpossibles, which turn out to be highly relevant for evaluating the role of the mathematics in question. The book consists of two parts and has a general introduction of the broader context in which the discussions take place. The first part focuses on the possibility of extracting an argument for mathematical realism in relation to the explanatory indispensability argument, and shows that circularity looms unless a controversial abstraction principle is assumed. It also offers an alternative non-mathematical explanation that makes use of relative interpretation. The second part focuses on the possibility of bringing out the explanatory role of mathematics counterfactually, and shows that, due to the necessary nature of mathematics, any proposal should take into account discussions on the knowledge and the structure of numbers. As a whole, this book is of great use to academic research in the field of philosophy of mathematics.

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.

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.