According to a widespread view in metaphysics and philosophy of science, all explanations involve relations of ontic dependence between the items appearing in the explanandum and the items appearing in the explanans. I argue that a family of mathematical cases, which I call “viewing-as explanations”, are incompatible with the Dependence Thesis. These cases, I claim, feature genuine explanations that aren’t supported by ontic dependence relations. Hence the thesis isn’t true in general. The first part of the paper defends this claim (...) and discusses its significance. The second part of the paper considers whether viewing-as explanations occur in the empirical sciences, focusing on the case of so-called fictional models. It’s sometimes suggested that fictional models can be explanatory even though they fail to represent actual worldly dependence relations. Whether or not such models explain, I suggest, depends on whether we think scientific explanations necessarily give information relevant to intervention and control. Finally, I argue that counterfactual approaches to explanation also have trouble accommodating viewing-as cases. (shrink)

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...) and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

If the laws of nature are deterministic, then it seems possible that a Laplacean intelligence that knows the initial conditions and the laws would be able to accurately predict everything that will ever happen. However, it would be easy to construct a counterpredictive device that falsifies any revealed prediction about its future behavior. What would then occur if a Laplacean intelligence encountered a counterpredictive device? This is the paradox of predictability. A number of philosophers have proposed solutions to it, though (...) part of my aim here is to argue that the paradox is more pernicious than has thus far been appreciated, and therefore that extant solutions are inadequate. My broader aim is to argue that the paradox motivates Humeanism about laws of nature. (shrink)

In this paper, I discuss the ways in which epistemic anxiety promotes well-being, specifically by examining the positive contributions that feelings of epistemic anxiety make toward intellectually virtuous inquiry. While the prospects for connecting the concept of epistemic anxiety to the two most prominent accounts of intellectual virtue, i.e., “virtue-reliabilism” and “virtue-responsibilism”, are promising, I primarily focus on whether the capacity for epistemic anxiety counts as an intellectual virtue in the reliabilist sense. As I argue, there is a close yet (...) unexplored connection between feelings of epistemic anxiety and the form of inference commonly known as “Inference to the Best Explanation” (IBE). Specifically, I argue that both the recognition that some fact requires an explanation—a necessary first step in applying IBE—and the subsequent motivation to employ IBE are typically facilitated by feelings of epistemic anxiety. So, provided IBE is truth-conducive the capacity for epistemic anxiety should count as an intellectual virtue in the reliabilist sense. After outlining my main argument, I address the challenge that the capacity for epistemic anxiety has the potential to be misleading. To respond to this challenge, I discuss how our recognition that a fact requires an explanation must in part be a species of practical knowledge, rather than theoretical knowledge. For the agent to skillfully distinguish between facts that require an explanation and facts that do not, she must develop the virtuous disposition to feel the appropriate amount of epistemic anxiety. Despite the many negative aspects associated with anxiety, as I conclude, being disposed to feel the appropriate amount of epistemic anxiety is ultimately good for us. (shrink)

Crucial to Hilary Putnam’s realism in the philosophy of mathematics is to maintain the objectivity of mathematics without the commitment to the existence of mathematical objects. Putnam’s indispensability argument was devised as part of this conception. In this paper, I reconstruct and reassess Putnam’s argument for the indispensability of mathematics, and distinguish it from the more familiar, Quinean version of the argument. Although I argue that Putnam’s approach ultimately fails, I develop an alternative way of implementing his form of realism (...) about mathematics that, by using different resources than those Putnam invokes, avoids the difficulties faced by his view. (shrink)

It is often assumed that the first stage of many classical arguments for theism depends upon some version of the Principle of Sufficient Reason being true. Unfortunately for classical theists, PSR is a controversial thesis that has come under rather severe criticism in the contemporary literature. In this article, I grant for the sake of argument that every version of PSR is false. Thus, I concede with the critics of PSR, that it is possible that there is, at least, one (...) fact that is ontologically brute. I then proffer an epistemological challenge to ontological bruteness. Specifically, I argue that whenever a knowledge seeker, S, perceives some unexplained fact, F, it is never reasonable for S to believe F is ontologically brute. I conclude that, even if PSR is false, it is more reasonable to believe that the key premise in the first stage of classical arguments for theism is true. (shrink)

It is often thought that metaphysical grounding underwrites a distinctive sort of metaphysical explanation. However, it would be a mistake to think that all metaphysical explanations are underwritten by metaphysical grounding. In service of this claim, I offer a novel kind of metaphysical explanation called metaphysical explanation by constraint, examples of which have been neglected in the literature. I argue that metaphysical explanations by constraint are not well understood as grounding explanations.

Surprises are important in our everyday lives as well as in our scientific and philosophical theorizing—in psychology, information theory, cognitive-neuroscience, philosophy of science, and confirmation theory. Nevertheless, there is no satisfactory theory of what makes something surprising. It has long been acknowledged that not everything unexpected is surprising. The reader had no reason to expect that there will be exactly 190 words in this abstract and yet there is nothing surprising about this fact. We offer a novel theory that explains (...) when and why an unexpected fact is surprising. We distinguish between descriptive and normative notions of what is surprising; clarify the sense in which surprising facts are unexpected; and, finally, develop and defend the significance account of surprise, according to which a fact is surprising to an agent if and to the extent that it is both unexpected and significant to the agent. Since a surprising fact can be significant to an agent in various ways—personal, moral, epistemic, and aesthetic—surprise is not merely or primarily epistemic. Fitting surprise reflects more than a person’s view of what is; it reflects a person’s view of what is significant. (shrink)

It has seemed, to many, that there is an important connection between the ways in which some theoretical posits explain our observations, and our reasons for being ontologically committed to those posits. One way to spell out this connection is in terms of what has become known as the explanatory criterion of ontological commitment. This is, roughly, the view that we ought to posit only those entities that are indispensable to our best explanations. Our primary aim is to argue that (...) the moral nonnaturalist places herself in an invidious position if she simply accepts that the nonnatural moral facts that she posits are not explanatory. Instead, we offer the nonnaturalist an alternative strategy for dealing with moral explanations. The strategy is to retain the explanatory criterion of ontological commitment and maintain that moral properties are, in fact, explanatory. The explanations they provide are not causal explanations; instead, moral properties make a non-causal difference to the physical facts. (shrink)

Mathematics appears to play a genuine explanatory role in science. But how do mathematical explanations work? Recently, a counterfactual approach to mathematical explanation has been suggested. I argue that such a view fails to differentiate the explanatory uses of mathematics within science from the non-explanatory uses. I go on to offer a solution to this problem by combining elements of the counterfactual theory of explanation with elements of a unification theory of explanation. The result is a theory according to which (...) a counterfactual is explanatory when it is an instance of a generalized counterfactual scheme. (shrink)

In this paper I present a case study of mathematical explanation in science that is new to the philosophical literature, and that arises in the context of estimating the energetic costs of running in bipedal animals. I refer to this as the Bipedal Gait Costs explanation. I argue that it is important for examples of applied mathematics to be driven not just by philosophical and mathematical concerns but also by scientific concerns. After a detailed presentation of the BGC case study, (...) I discuss ways in which it is different from standard examples of MES. One distinctive aspect of BGC is that substantive mathematics occurs at two different levels. I argue for a distinction to be drawn between the mathematical modeling that occurs at one level, which is non-explanatory, and the mathematical theorem which is applied to results gleaned from these models, which is explanatory. I conclude by drawing some connections with broader indispensability-based arguments for platonism in the philosophy of mathematics. (shrink)

Meta-laws, including conservation laws, are laws about the form of more specific, phenomenological, laws. Lange distinguishes between meta-laws as coincidences, where the meta-law happens to hold because the more specific laws hold, and meta-laws as constraints to which subsumed laws must conform. He defends this distinction as a genuine metaphysical possibility, such that metaphysics alone ought not to rule one way or another, leaving it an open question for physics. Lange’s distinction marks a genuine difference in how a given meta-law (...) can be used in explanations. Yet, I argue, it is not simply an empirical matter as to whether a given conservation law is a constraint or a coincidence. There is no set matter of fact about the world that determines this, and physics alone will not be able to return a determinate verdict on a law-by-law basis, even while there is a genuine difference between any given law as constraint and as coincidence. Rather, the difference marks different ways of treating the same law in a theoretical setting: by shifting the explanatory context, treating the same law as part of a different mathematical structure, it can be a genuine constraint and a genuine coincidence. The difference between constraint and coincidence relates to the way in which we use a law in specific theoretical and explanatory settings. Because the same law can appear in multiple contexts, it can be used in these genuinely different ways, without itself ‘‘really’’ being either one or the other as some atomistic empirical fact. Conservation laws as constraints and conservation laws as coincidences are both genuine theoretical roles that the same law can play. I conclude by considering how this pragmatist construal of constraints versus coincidences reveals how two parts of Lange’s work in this section of the book are unexpectedly independent of one another. -/- . (shrink)

Candidates for fundamental physical laws rarely, if ever, employ higher than second time derivatives. Easwaran sketches an enticing story that purports to explain away this puzzling fact and thereby provides indirect evidence for a particular set of metaphysical theses used in the explanation. I object to both the scope and coherence of Easwaran's account, before going on to defend an alternative, more metaphysically deflationary explanation: in interacting Lagrangian field theories, it is either impossible or very hard to incorporate higher than (...) second time derivatives without rendering the vacuum state unstable. The so-called Ostrogradski instability represents a powerful constraint on the construction of new field theories and supplies a novel, largely overlooked example of non-causal explanation in physics. (shrink)

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by appealing to what the (...) world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (shrink)

Thought experiments play a role in science and in some central parts of contemporary philosophy. They used to play a larger role in philosophy of science, but have been largely abandoned as part of the field’s “practice turn”. This chapter discusses possible roles for thought experimentation within a practice-oriented philosophy of science. Some of these roles are uncontroversial, such as exemplification and aiding discovery. A more controversial role is the reliance on thought experiments to justify philosophical claims. It is proposed (...) that if we adopt an underlying empiricist view of concepts, then thought experiments can be seen as affording us contact with scientific practice, despite their seemingly a priori character. The advantages and drawbacks of thought experiments are discussed via comparison with case studies, on the one, and simulations on the other hand. The chapter closes with some remarks on how to combine thought experiments with other methods. (shrink)

To a large extent, the evidential base of claims in the philosophy of science has switched from thought experiments to case studies. We argue that abandoning thought experiments was a wrong turn, since they can effectively complement case studies. We make our argument via an analogy with the relationship between experiments and observations within science. Just as experiments and ‘natural’ observations can together evidence claims in science, each mitigating the downsides of the other, so too can thought experiments and case (...) studies be mutually supporting. After presenting the main argument, we look at potential concerns about thought experiments, suggesting that a judiciously applied mixed-methods approach can overcome them. (shrink)

Far from being unwelcome or impossible in a mathematical setting, indeterminacy in various forms can be seen as playing an important role in driving mathematical research forward by providing “sources of newness” in the sense of Hutter and Farías :434–449, 2017). I argue here that mathematical coincidences, phenomena recently under discussion in the philosophy of mathematics, are usefully seen as inducers of indeterminacy and as put to work in guiding mathematical research. I suggest that to call a pair of mathematical (...) facts a coincidence is roughly to suggest that the investigation of connections between these facts isn’t worthwhile. To say of this pair, “That’s no coincidence!” is to suggest just the opposite. I further argue that this perspective on mathematical coincidence, which pays special attention to what mathematical coincidences do, may provide us with a better view of what mathematical coincidences are than extant accounts. I close by reflecting on how understanding mathematical coincidences as generating indeterminacy accords with a conception of mathematical research as ultimately aiming to reduce indeterminacy and complexity to triviality as proposed in Rota Indiscrete thoughts, Birkhäuser, 1997). (shrink)

Este capítulo es una guía para conocer los elementos básicos de la fundamentación metafísica. Ofrece una revisión accesible de sus características y usos, comparando este concepto con otras formas de dependencia que se pueden encontrar en la literatura. Enfatiza dos roles teóricos centrales que la fundamentación supuestamente juega en la teorización filosófica: (i) dar cuenta de una forma distintiva de determinación no causal y (ii) entender mejor la estructura jerárquica de la realidad. El capítulo apunta a persuadir a las lectoras (...) de la utilidad de la fundamentación discutiendo cómo pueden mitigarse algunas objeciones dirigidas a esos roles. (shrink)

Computer assisted theorem proving is an increasingly important part of mathematical methodology, as well as a long-standing topic in artificial intelligence (AI) research. However, the current generation of theorem proving software have limited functioning in terms of providing new proofs. Importantly, they are not able to discriminate interesting theorems and proofs from trivial ones. In order for computers to develop further in theorem proving, there would need to be a radical change in how the software functions. Recently, machine learning results (...) in solving mathematical tasks have shown early promise that deep artificial neural networks could learn symbolic mathematical processing. In this paper, I analyze the theoretical prospects of such neural networks in proving mathematical theorems. In particular, I focus on the question how such AI systems could be incorporated in practice to theorem proving and what consequences that could have. In the most optimistic scenario, this includes the possibility of autonomous automated theorem provers (AATP). Here I discuss whether such AI systems could, or should, become accepted as active agents in mathematical communities. (shrink)

In this paper we examine whether and how powers ontologies can back formal causation. We attempt to answer three questions: i) what is formal causation; ii) whether we need formal causation, and iii) whether formal causation need powers and whether it can be grounded in powers. We take formal causal explanations to be explanations in which something's essence features prominently in the explanans. Three kinds of essential explanations are distinguished: constitutive, consequential, and those singling out something's propria. This last kind (...) of explanation has been somewhat overlooked in the literature, but we argue that it features in much of the most relevant uses of formal explanations in philosophy and science. These are the explanations for why an object has certain propertis that are not, properly speaking, parts of its essence, but that belong to it more intimately than just being part of its consequential essence: they are the properties that ‘flow’ from something's essence. It is argued that a powers metaphysics is uniquely placed to make sense of this last phenomenon. We then distinguish three grades of involvement in which powers might be salient for formal causal explanations: i) powers might be the subject matter of the essence operator, ii) the essence of something might include (or be exhausted) by powers, or iii) powers can explain how propria can flow from something's constitutive existence. (shrink)

The idea that there is some fundamental “level” or “ground” where our description of the world bottoms out has acquired the status of ‘the received view’ in metaphysics ; for a more recent critical defense, see Cameron, 2008). Typically this view is cashed out in terms of some set of ‘basic building blocks’ populating this level, which sits at the bottom of a hierarchy ordered according to some set of compositional principles. These fundamental building blocks are thus taken to have (...) some form of “ultimate” ontological priority with regard to everything else in the hierarchy. In this chapter I shall consider two kinds of threats to this view: the first comes from arguments against the idea of such a level in general, whereas the second concerns the nature of these occupants. As we’ll see, both these threats become entwined in the context of modern physics but I’ll conclude with a suggestion as to how this “received view” may be maintained in this context. (shrink)

What is understanding? My goal in this paper is to lay out a new approach to this question and clarify how that approach deals with certain issues. The claim is that understanding is a matter of compressing information about the understood so that it can be mentally useful. On this account, understanding amounts to having a representational kernel and the ability to use it to generate the information one needs regarding the target phenomenon. I argue that this ambitious new account (...) can accommodate much of the data that has motivated theories of understanding in philosophy of science, and can also be generally applicable in epistemology and daily life as well. (shrink)

Husserl’s early picture of explanation in the sciences has never been completely provided. This lack represents an oversight, which we here redress. In contrast to currently accepted interpretations, we demonstrate that Husserl does not adhere to the much maligned deductive-nomological (DN) model of scientific explanation. Instead, via a close reading of early Husserlian texts, we reveal that he presents a unificationist account of scientific explanation. By doing so, we disclose that Husserl’s philosophy of scientific explanation is no mere anachronism. It (...) is, instead, tenable and relevant. We discuss how Husserl and other contemporary thinkers draw theoretical inspiration from the same source—namely, Bernard Bolzano. Husserl’s theory of scientific explanation shares a common language and discusses the same themes as, for example, Phillip Kitcher and Kit Fine. To advance our novel reading, we discuss Husserl’s investigations of grounding, inter-lawful explanation, intra-mathematical explanation, and scientific unification. (shrink)

Explanations are backed by many different relations: causation, grounding, and arguably others too. But why are these different relations capable of backing explanations? In virtue of what are they explanatory? In this paper, I propose and defend a monistic account of explanation-backing relations. On my account, there is a single relation which backs all cases of explanation, and which explains why those other relations are explanation-backing.

I argue that the best interpretation of the general theory of relativity has need of a causal entity, and causal structure that is not reducible to light cone structure. I suggest that this causal interpretation of GTR helps defeat a key premise in one of the most popular arguments for causal reductionism, viz., the argument from physics.

I show how Sir William Rowan Hamilton’s philosophical commitments led him to a causal interpretation of classical mechanics. I argue that Hamilton’s metaphysics of causation was injected into his dynamics by way of a causal interpretation of force. I then detail how forces are indispensable to both Hamilton’s formulation of classical mechanics and what we now call Hamiltonian mechanics (i.e., the modern formulation). On this point, my efforts primarily consist of showing that the contemporary orthodox interpretation of potential energy is (...) the interpretation found in Hamilton’s work. Hamilton called the potential energy function the “force-function” because he believed that it represents forces at work in the world. Various non-historical arguments for this orthodox interpretation of potential energy are provided, and matters are concluded by showing that in classical Hamiltonian mechanics, facts about the potential energies of systems are grounded in facts about forces. Thus, if one can tolerate the view that forces are causes of motion, then Hamilton provides one with a road map for transporting causation into one of the most mathematically sophisticated formulations of classical mechanics, viz., Hamiltonian mechanics. (shrink)

One can find in the literature two sets of views concerning the relationship between understanding and explanation: that one understands only if 1) one has knowledge of causes and 2) that knowledge is provided by an explanation. Taken together, these tenets characterize what I call the narrow knowledge account of understanding. While the first tenet has recently come under severe attack, the second has been more resistant to change. I argue that we have good reasons to reject it on the (...) basis of theoretical models that provide how-possibly explanations. These models, while they do not explain in the strict sense, afford understanding. In response, I propose an alternative epistemology of understanding, broad KAU, that takes cases of theoretical modelling into account. (shrink)

One puzzle concerning highly idealized models is whether they explain. Some suggest they provide so-called ‘how-possibly explanations’. However, this raises an important question about the nature of how-possibly explanations, namely what distinguishes them from ‘normal’, or how-actually, explanations? I provide an account of how-possibly explanations that clarifies their nature in the context of solving the puzzle of model-based explanation. I argue that the modal notions of actuality and possibility provide the relevant dividing lines between how-possibly and how-actually explanations. Whereas how-possibly (...) explanations establish claims of possible explanations, how-actually explanations establish claims of actual ones. Models, in turn, simply provide evidence for these claims. (shrink)

A lot of philosophical energy has been devoted recently in trying to determine if mathematics can contribute to our understanding of physical phenomena. Not many philosophers are interested, though, if the converse makes sense, i.e., if our cognitive interaction (scientific or otherwise) with the physical world can be helpful (in an explanatory or non-explanatory way) in our efforts to make sense of mathematical facts. My aim in this paper is to try to fill this important lacuna in the recent literature. (...) My answer to the question of this paper is negative. As I will argue, there are serious problems with the main reasons for believing in the first place that it is possible to have physical understanding of mathematical facts. (shrink)

Separatists about grounding take explanations to be separate from their corresponding grounding-facts. Grounding-facts are supposed to underlie, or back, such explanations. However, the backing relation hasn’t received much attention in the literature. The aim of this paper is to provide an informative definition of backing. First, I examine two prominent proposals: backing as explaining (Kovacs 2017; 2019a) and backing as grounding (see Sjölin Wirling 2020). Finally, I put forward my own proposal. I argue that under plausible assumptions about the role (...) of backing and the nature of explanation, backing should be understood as a form of truthmaking, minimally construed. (shrink)

Carl Hempel argued that probabilistic hypotheses are limited in what they can explain. He contended that a hypothesis cannot explain why E is true if the hypothesis says that E has a probability less than 0.5. Wesley Salmon and Richard Jeffrey argued to the contrary, contending that P can explain why E is true even when P says that E’s probability is very low. This debate concerned noncontrastive explananda. Here, a view of contrastive causal explanation is described and defended. It (...) provides a new limit on what probabilistic hypotheses can explain; the limitation is that P cannot explain why E is true rather than A if P assign E a probability that is less than or equal to the probability that P assigns to A. The view entails that a true deterministic theory and a true probabilistic theory that apply to the same explanandum partition are such that the deterministic theory explains all the true contrastive propositions constructable from that partition, whereas the probabilistic theory often fails to do so. (shrink)

Relationships of counterfactual dependence have played a major role in recent debates of explanation and understanding in the philosophy of science. Usually, counterfactual dependencies have been viewed as the explanantia of explanation, i.e., the things providing explanation and understanding. Sometimes, however, counterfactual dependencies are themselves the targets of explanations in science. These kinds of explanations are the focus of this paper. I argue that “micro-level model explanations” explain the particular form of the empirical regularity underlying a counterfactual dependency by representing (...) it as a physical necessity on the basis of postulated microscopic entities. By doing so, micro-level models rule out possible forms the regularity could have taken. Micro-model explanations, in other words, constrain empirical regularities and their associated counterfactual dependencies. I introduce and illustrate micro-level model explanations in detail, contrast them to other accounts of explanation, and consider potential problems. (shrink)

Social scientists appeal to various “structures” in their explanations including public policies, economic systems, and social hierarchies. Significant debate surrounds the explanatory relevance of these factors for various outcomes such as health, behavioral, and economic patterns. This paper provides a causal account of social structural explanation that is motivated by Haslanger (2016). This account suggests that social structure can be explanatory in virtue of operating as a causal constraint, which is a causal factor with unique characteristics. A novel causal framework (...) is provided for understanding these explanations–this framework addresses puzzles regarding the mysterious causal influence of social structure, how to understand its relation to individual choice, and what makes it the main explanatory (and causally responsible) factor for various outcomes. (shrink)

On a highly influential way to think of modality, that I call ‘relationalism’, the modality of a state is explained by its being composed of properties, and these properties being related by a higher-order and primitively modal relation. Examples of relationalism are the Dretske-Tooley-Armstrong account of natural necessity, many dispositional essentialist views, and Wang’s incompatibility primitivism. I argue that relationalism faces four difficulties: that the selection between modal relations is arbitrary, that the modal relation cannot belong to any logical order, (...) that to explain how the modal relation can relate properties of different adicities additional ideological complexity has to be introduced, and that not all modal constraints are relational. From the discussion, I will extract desiderata for a successor theory of modality. (shrink)

In this paper I study the activity of mathematical problem-solving in scientific practice, focussing on enquiries in mathematical social science. I identify three salient phases of mathematical problem-solving and adopt them as a reference frame to investigate aspects of applications that have not yet received extensive attention in the philosophical literature.

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)

Cartwright (Synthese 121(1/2):3–27, 1999a; The dappled world, Cambridge University Press, Cambridge, 1999b) attacked the view that causal relations conform to the Markov condition by providing a counterexample in which a common cause does not screen off its effects: the prominent chemical factory. In this paper we suggest a new way to handle counterexamples to Markov causation such as the chemical factory. We argue that Cartwright’s as well as similar scenarios feature a certain kind of non-causal dependence that kicks in once (...) the common cause occurs. We then develop a representation of this specific kind of non-causal dependence that allows for modeling the problematic scenarios in such a way that the Markov condition is not violated anymore. (shrink)

In “What Makes a Scientific Explanation Distinctively Mathematical?” (2013b), Lange uses several compelling examples to argue that certain explanations for natural phenomena appeal primarily to mathematical, rather than natural, facts. In such explanations, the core explanatory facts are modally stronger than facts about causation, regularity, and other natural relations. We show that Lange's account of distinctively mathematical explanation is flawed in that it fails to account for the implicit directionality in each of his examples. This inadequacy is remediable in each (...) case by appeal to ontic facts that account for why the explanation is acceptable in one direction and unacceptable in the other direction. The mathematics involved in these examples cannot play this crucial normative role. While Lange's examples fail to demonstrate the existence of distinctively mathematical explanations, they help to emphasize that many superficially natural scientific explanations rely for their explanatory force on relations of stronger-than-natural necessity. These are not opposing kinds of scientific explanations; they are different aspects of scientific explanation. (shrink)

Lange argues that some natural phenomena can be explained by appeal to mathematical, rather than natural, facts. In these “distinctively mathematical” explanations, the core explanatory facts are either modally stronger than facts about ordinary causal law or understood to be constitutive of the physical task or arrangement at issue. Craver and Povich argue that Lange’s account of DME fails to exclude certain “reversals”. Lange has replied that his account can avoid these directionality charges. Specifically, Lange argues that in legitimate DMEs, (...) but not in their “reversals,” the empirical fact appealed to in the explanation is “understood to be constitutive of the physical task or arrangement at issue” in the explanandum. I argue that Lange’s reply is unsatisfactory because it leaves the crucial notion of being “understood to be constitutive of the physical task or arrangement” obscure in ways that fail to block “reversals” except by an apparent ad hoc stipulation or by abandoning the reliance on understanding and instead accepting a strong realism about essence. (shrink)

According to the multiple realization argument, mental states or processes can be realized in diverse and heterogeneous physical systems; and that fact implies that mental state or process kinds cannot be identified with particular kinds of physical states or processes. More specifically, mental processes cannot be identified with brain processes. Moreover, the argument provides a general model for the autonomy of the special sciences. The multiple realization argument is widely influential, but over the last thirty years it has also faced (...) serious objections. Despite those objections, most philosophers regard the fact of multiple realization and the cogency of the multiple realization argument as plainly correct. Why is that? What is it about the multiple realization argument that makes it so resilient? One reason is that the multiple realization argument is deeply intertwined with a view that minds are, in some sense, computational. But we argue that the sense in which minds are computational does not support the conclusion that they are ipso facto multiply realized. We argue that the sense in which brains compute does not imply that brains implement multiply realizable computational processes, and it does not provide a general model for the autonomy of the special sciences. (shrink)

In this paper I present two cases, taken from the history of science, in which mathematics and physics successfully interplay. These cases provide, respectively, an example of the successful application of mathematics in astronomy and an example of the successful application of mechanics in mathematics. I claim that an illustration of these cases has a twofold value in the context of the applicability debate. First, it enriches the debate with an historical perspective which is largely omitted in the contemporary discussion. (...) Second, it reveals a component of the applicability problem that has received little attention. This component concerns the successful application of physical principles in mathematical practice. With the help of the two examples, in the final part of the paper I address the following question: are successful applications of mathematics to physics and successful applications of physics to mathematics two sides of the same problem? (shrink)

Many non-physicalists, including Chalmers, hold that the zombie argument succeeds in rejecting the physicalist view of consciousness. Some non-physicalists, including, again, Chalmers, hold that quantum collapse interactionism, i.e., the idea that non-physical consciousness causes collapse of the wave function in phenomena such as quantum measurement, is a viable interactionist solution for the problem of the relationship between the physical world and the non-physical consciousness. In this paper, I argue that if QCI is true, the zombie argument fails. In particular, I (...) show that if QCI is true, a zombie world physically identical to our world is impossible because there is at least one law of nature, a fundamental law of physics in particular, that exist only in the zombie world but not in our world. This shows that philosophers like Chalmers are committing an error in endorsing the zombie argument and QCI at the same time. (shrink)

Some explanations appeal to facts about the causal structure of a system in order to shed light on a particular phenomenon; these are explanations which do more than cite the causes X and Y of some state-of-affairs Z, but rather appeal to “macro-level” causal features—for example the fact that A causes B as well as C, or perhaps that D is a strong inhibitor of E—in order to explain Z. Appeals to these kinds of “macro-level” causal features appear in a (...) wide variety of social scientific and biological research; statements about features such as “patriarchy,” “healthcare infrastructure,” and “functioning DNA repair mechanism,” for instance, can be understood as claims about what would be different in a system with a different causal structure. I suggest interpreting counterfactual questions involving structural features as questions about alternative parameter settings of causal models, and propose an extension of the usual interventionist framework for causal explanation which enables scientists to explore the consequences of interventions on “macro-level” structure. (shrink)

Are there genuine mathematical explanations of physical phenomena, and if so, how can mathematical theories, which are typically thought to concern abstract mathematical objects, explain contingent empirical matters? The answer, I argue, is in seeing an important range of mathematical explanations as structural explanations, where structural explanations explain a phenomenon by showing it to have been an inevitable consequence of the structural features instantiated in the physical system under consideration. Such explanations are best cast as deductive arguments which, by virtue (...) of their form, establish that, given the mathematical structure instantiated in the physical system under consideration, the explanandum had to occur. Against the claims of platonists such as Alan Baker and Mark Colyvan, I argue that formulating mathematical explanations as structural explanations in this way shows that we can accept that mathematics can play an indispensable explanatory role in empirical science without committing to the existence of any abstract mathematical objects. (shrink)

The connection between understanding and explanation has recently been of interest to philosophers. Inglis and Mejía-Ramos (Synthese, 2019) propose that within mathematics, we should accept a functional account of explanation that characterizes explanations as those things that produce understanding. In this paper, I start with the assumption that this view of mathematical explanation is correct and consider what we can consequently learn about mathematical explanation. I argue that this view of explanation suggests that we should shift the question of explanation (...) away from why-questions and towards a “what’s going on here” question. Additionally, I argue that when we recognize the connection between understanding and explanation we naturally see how more than just proof can be explanatory. I expand this point by detailing how definitions and diagrams can be explanatory. In all, we see that when we take seriously the connection between understanding and explanation, we get a better sense of how explanation arises within mathematics. (shrink)

Wilson derives various broad philosophical morals from the scientific role played by the Principle of Virtual Work. He argues roughly that PVW conditionals cannot be understood in terms of things as large as possible worlds; that PVW conditionals are peculiar and so cannot be accommodated by general accounts of counterfactuals, thereby reflecting the piecemeal character of scientific practice and standing at odds with the one-size-fits-all approach of “analytic metaphysicians”; and that PVW counterfactuals are not made true partly by natural laws. (...) I distinguish, elaborate and critically examine various arguments for these morals suggested by the PVW and Wilson’s text, looking especially at what makes a displacement “virtual” and the operation of the conditionals that the PVW takes to express necessary and sufficient conditions for equilibrium. Ultimately, I do not find the PVW to be especially well suited to support Wilson’s morals; some of these arguments fail, whereas others arise from general considerations rather than having to appeal to anything like the PVW. (shrink)

According to Lange,?Really Statistical explanations? constitute an important kind of non-causalscientific explanation. However, Roski has argued that all alleged RS explanations are either causalexplanations or not explanations at all. In so arguing, Roski has invoked Kahneman?s interpretation of onealleged RS explanation. I employ Roski?s arguments as an opportunity to elaborate and defend RS explanations. Iargue that?RS explanations? genuinely explain rather than deny the presuppositions of why-questions. I argue thatthe RS model is not excessively permissive in allowing some explanations to work (...) purely statistically rather than bydescribing causal relations. I argue that Roski?s view that some?RS explanations? operate by describing causalrelations fails to capture the kind of explanatory insight that RS explanations provide. I elaborate the notion of a?characteristically statistical phenomenon? that figures in the RS model and thereby explain how an RS explanationreveals that its explanandum is independent of any causal facts or specific chances, but rather depends only on ageneric sort of arrangement of chances. Finally, I argue that Roski misinterprets Kahneman, who actually holds thatthe explanation he discusses is a non-causal explanation that nicely fits the RS model. RS explanations constitutean important kind of non-causal scientific explanation. (shrink)