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I argue that fictionalism about grounding is unmotivated, focusing on Naomi Thompson's (2022) recent proposal on which the utility of the grounding fiction lies in its facilitating communication about what metaphysically explains what. I show that, despite its apparent dialectic kinship with other metaphysical debates in which fictionalism has a healthy tradition, the grounding debate is different in two key respects. Firstly, grounding talk is not indispensable, nor even particularly convenient as a means of communicating about metaphysical explanation. This undermines the revolutionary proposal. Secondly, talk of grounding primarily occurs within metaphysics, which means the usual options for motivating a non-literal interpretation are ineffective. This undermines the hermeneutic proposal.

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This paper provides a sorely-needed evaluation of the view that mathematical explanations in science explain by unifying. Illustrating with some novel examples, I argue that the view is misguided. For believers in mathematical explanations in science, my discussion rules out one way of spelling out how they work, bringing us one step closer to the right way. For non-believers, it contributes to a divide-and-conquer strategy for showing that there are no such explanations in science. My discussion also undermines the appeal to unifying power in support of the enhanced indispensability argument.

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Some scientific explanations appear to turn on pure mathematical claims. The enhanced indispensability argument appeals to these ‘mathematical explanations’ in support of mathematical platonism. I argue that the success of this argument rests on the claim that mathematical explanations locate pure mathematical facts on which their physical explananda depend, and that any account of mathematical explanation that supports this claim fails to provide an adequate understanding of mathematical explanation.

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We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to key intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned.

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I show how mathematical platonism combined with belief in the God of classical theism can respond to Field's epistemological objection. I defend an account of divine mathematical knowledge by showing that it falls out of an independently motivated general account of divine knowledge. I use this to explain the accuracy of God's mathematical beliefs, which in turn explains the accuracy of our own. My arguments provide good news for theistic platonists, while also shedding new light on Field's influential objection.

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The standard semantic analysis of sentences such as ‘The number of planets in the solar system is eight’ is that they are identity statements that identify certain mathematical objects, namely numbers. The analysis thereby facilitates arguments for a controversial philosophical position, namely realism about mathematical objects. Accordingly, whether or not this analysis is accurate should concern philosophers greatly. Recently, several authors have offered rival analyses of sentences such as these. In this paper, I will consider a wide range of linguistic evidence and show that all of these analyses, including the standard analysis, suffer significant drawbacks. I will then outline and present further evidence in favour of my own analysis, developed elsewhere, according to which such sentences are identity statements that identify certain kinds of facts. I also defend a novel and plausible approach to the semantics of interrogative clauses that corroborates my analysis. Finally, I discuss how realists about mathematical objects should proceed in light of the arguments presented in this paper.

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The ‘indispensability argument’ for the existence of mathematical objects appeals to the role mathematics plays in science. In a series of publications, Joseph Melia has offered a distinctive reply to the indispensability argument. The purpose of this paper is to clarify Melia’s response to the indispensability argument and to advise Melia and his critics on how best to carry forward the debate. We will begin by presenting Melia’s response and diagnosing some recent misunderstandings of it. Then we will discuss four avenues for replying to Melia. We will argue that the three replies pursued in the literature so far are unpromising. We will then propose one new reply that is much more powerful, and—in the light of this—advise participants in the debate where to focus their energies.

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‘The following sentence is true only if numbers exist: (1) The number of planets is eight. It is true; hence, numbers exist.’ So runs a familiar argument for realism about mathematical objects. But this argument relies on a controversial semantic thesis: that ‘The number of planets’ and ‘eight’ are singular terms standing for the number eight, and the copula expresses identity. This is the ‘Fregean analysis’. I show that the Fregean analysis is false by providing an analysis of sentences such as (1) that best explains the available linguistic data, and according to which no terms in (1) purport to stand for numbers.

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Heavy duty platonism (HDP) is of great dialectical importance in the philosophy of mathematics. It is the view that physical magnitudes, such as mass and temperature, are cases of physical objects being related to numbers. Many theorists have assumed HDP’s falsity in order to reach their own conclusions, but they are only justified in doing so if there are good arguments against HDP. In this paper, I present all five arguments against HDP alluded to in the literature and show that they all fail. In doing so, I establish two related truths: HDP has been unfairly ignored, and the arguments which take its falsity as a key premise should be re-assessed.