This essay aims to redress the contention that epistemic possibility cannot be a guide to the principles of modal metaphysics. I introduce a novel epistemic two-dimensional truthmaker semantics. I argue that the interaction between the two-dimensional framework and the mereological parthood relation, which is super-rigid, enables epistemic possibilities and truthmakers with regard to parthood to be a guide to its metaphysical profile. I specify, further, a two-dimensional formula encoding the relation between the epistemic possibility and verification of essential properties obtaining (...) and their metaphysical possibility or verification. I then generalize the approach to haecceitistic properties, and examine the Julius Caesar problem as a test case. (shrink)

This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. The goal of the paper is a rigorous analysis of these explanations. This will be done in two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof trees, together with a decomposition of their elements; secondly we will show that those mathematical proofs that are regarded as having (...) explanatory power all display an increase of conceptual complexity from the assumptions to the conclusion. (shrink)

Some mathematical proofs explain why the theorems they prove hold. This paper identifies several challenges for any counterfactual account of explanation in mathematics (that is, any account according to which an explanatory proof reveals how the explanandum would have been different, had facts in the explanans been different). The paper presumes that countermathematicals can be nontrivial. It argues that nevertheless, a counterfactual account portrays explanatory power as too easy to achieve, does not capture explanatory asymmetry, and fails to specify why (...) certain proofs are explanatory and others are not. Greater informativeness about counterfactual dependence can even yield less explanatory power. (shrink)

Recent years have seen increasing interest in interventionist analyses of metaphysical explanation. One area where interventionism traditionally shines, is in providing an account of explanatory depth; the sense in which explanation comes in degrees. However, the literature on metaphysical explanation has left the notion of depth almost entirely unexplored. In this paper I shall attempt to rectify this oversight by motivating an interventionist analysis of metaphysical explanatory depth (MED), in terms of the range of interventions under which a metaphysically explanatory (...) generalization remains invariant. After elucidating the notion through a toy-example, I demonstrate the important work which MED can perform in characterizing debate within contemporary metaphysics. Focusing upon rival approaches to explaining the identity and distinctness of concrete objects, I argue that the progress achieved in this debate can be characterized in terms of increasing explanatory depth. Having made an initial case for the utility of MED, I then turn this analysis to the metaphysics of explanation itself. By adopting an interventionist framework with respect to MED, I will show that we can assess the depth of competing theories of explanation. This application has two interesting results: first, it suggests that an interventionist analysis of explanation provides deeper explanations of the connection between explanans and explanandum than rival accounts; and second, it suggests that explanations provided by interventionism become deeper still, if one accepts that this methodology ranges over metaphysical, as well as causal, instances. (shrink)

This article presents a challenge that those philosophers who deny the causal interpretation of explanations provided by population genetics might have to address. Indeed, some philosophers, known as statisticalists, claim that the concept of natural selection is statistical in character and cannot be construed in causal terms. On the contrary, other philosophers, known as causalists, argue against the statistical view and support the causal interpretation of natural selection. The problem I am concerned with here arises for the statisticalists because the (...) debate on the nature of natural selection intersects the debate on whether mathematical explanations of empirical facts are genuine scientific explanations. I argue that if the explanations provided by population genetics are regarded by the statisticalists as non-causal explanations of that kind, then statisticalism risks being incompatible with a naturalist stance. The statisticalist faces a dilemma: either she maintains statisticalism but has to renounce naturalism; or she maintains naturalism but has to content herself with an account of the explanations provided by population genetics that she deems unsatisfactory. This challenge is relevant to the statisticalists because many of them see themselves as naturalists. (shrink)

Explanatory realism is the view that explanations work by providing information about relations of productive determination such as causation or grounding. The view has gained considerable popularity in the last decades, especially in the context of metaphysical debates about non-causal explanation. What makes the view particularly attractive is that it fits nicely with the idea that not all explanations are causal whilst avoiding an implausible pluralism about explanation. Another attractive feature of the view is that it allows explanation to be (...) a partially epistemic, context-dependent phenomenon. In spite of its attractiveness, explanatory realism has recently been subject to criticism. In particular, Taylor :197–219, 2018). has presented four types of explanation that the view allegedly cannot account for. This paper defends explanatory realism against Taylor’s challenges. We will show that Taylor’s counterexamples are either explanations that turn out to provide information about entities standing in productive determination relations or that they are not genuine explanations in the first place. (shrink)

We explore the prospects of a monist account of explanation for both non-causal explanations in science and pure mathematics. Our starting point is the counterfactual theory of explanation for explanations in science, as advocated in the recent literature on explanation. We argue that, despite the obvious differences between mathematical and scientific explanation, the CTE can be extended to cover both non-causal explanations in science and mathematical explanations. In particular, a successful application of the CTE to mathematical explanations requires us to (...) rely on counterpossibles. We conclude that the CTE is a promising candidate for a monist account of explanation in both science and mathematics. (shrink)

A counterpossible is a counterfactual with an impossible antecedent. Counterpossibles present a puzzle for standard theories of counterfactuals, which predict that all counterpossibles are semantically vacuous. Moreover, counterpossibles play an important role in many debates within metaphysics and epistemology, including debates over grounding, causation, modality, mathematics, science, and even God. In this article, we will explore various positions on counterpossibles as well as their potential philosophical consequences.

Recently, there have been several attempts to generalize the counterfactual theory of causal explanations to mathematical explanations. The central idea of these attempts is to use conditionals whose antecedents express a mathematical impossibility. Such countermathematical conditionals are plugged into the explanatory scheme of the counterfactual theory and—so is the hope—capture mathematical explanations. Here, I dash the hope that countermathematical explanations simply parallel counterfactual explanations. In particular, I show that explanations based on countermathematicals are susceptible to three problems counterfactual explanations do (...) not face. These problems seriously challenge the prospects for a counterfactual theory of explanation that is meant to cover mathematical explanations. (shrink)

In ordinary language, in the medical sciences, and in the overlap between them, we frequently make claims which imply that we might have had different gametic origins from the ones we actually have. Such statements seem intuitively true and coherent. But they counterfactually ascribe different DNA to their referents and therefore contradict material-origin essentialism, which Kripke and his followers argue is intuitively obvious. In this paper I argue, using examples from ordinary language and from philosophy of medicine and bioethics, that (...) statements which attribute alternative material origins to their referents are useful, common in political and medical reasoning, and in many cases best interpreted literally. So we must replace the doctrine of material-origin essentialism with one that can make sense of ordinary discourse and the language of the medical sciences. I propose an anti-essentialist account of such counterfactuals according to which individuals’ modal properties are relative to a given inquiry. (shrink)

Recent years have seen growing interest in modifying interventionist accounts of causal explanation in order to characterise noncausal explanation. However, one surprising element of such accounts is that they have typically jettisoned the core feature of interventionism: interventions. Indeed, the prevailing opinion within the philosophy of science literature suggests that interventions exclusively demarcate causal relationships. This position is so prevalent that, until now, no one has even thought to name it. We call it “intervention puritanism” (I-puritanism, for short). In this (...) paper, we mount the first sustained defence of the idea that there are distinctively noncausal explanations which can be characterized in terms of possible interventions; a position we call “intervention liberalism” (I-liberalism, for short). While many have followed James Woodward (2003) in committing to I-puritanism, we trace support for I-liberalism back to the work of Jaegwon Kim (1974). Furthermore, we analyse two recent sources of scepticism regarding I-liberalism: debate surrounding mechanistic constitution; and attempts to provide a monistic account of explanation. We show that neither literature provides compelling reasons for adopting I-puritanism. Finally, we present a novel taxonomy of available positions upon the role of possible interventions in explanation: weak causal imperialism; strong causal imperialism; monist intervention puritanism; pluralist intervention puritanism; monist intervention liberalism; and finally, the specific position defended in this paper, pluralist intervention liberalism. (shrink)

The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as (...) Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. (shrink)

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. (...) Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics. (shrink)

An overview of hyperintensionality is provided. Hyperintensional languages have expressions with meanings that are more fine-grained than necessary equivalence. That is, the expressions may necessarily co-apply and yet be distinct in meaning. Adequately accounting for theories cast in hyperintensional languages is important in the philosophy of language; the philosophy of mind; metaphysics; and elsewhere. This entry presents a number of areas in which hyperintensionality is important; a range of approaches to theorising about hyperintensional matters; and a range of debates that (...) attention to hyperintensional constructions has generated. (shrink)