The directionality of distinctively mathematical explanations

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.     

Historicity and explanation

Some scientific explanations are distinctively historical. The aim of this paper is to say what gives such explanations their historical character. A secondary aim is to describe what makes an explanation a stronger or weaker historical explanation. We begin with a critical discussion of John Beatty's and Eric Desjardins' work on historicity and historical narrative. We then offer an alternative account of historical explanation that draws on the work of earlier philosophers (Gallie, Danto, Mink, and Hull). In that alternative account, we highlight four features of narrative explanation that Beatty and Desjardins underemphasize: central subjects; historical trajectories; the idea that historical narratives are known retrospectively; and criteria for determining what is a stronger or weaker historical narrative.     

Realizability and the varieties of explanation

What realization is has been convincingly presented in relation to the way we determine what counts as the realizers of realized properties. The way we explain a fact of realization includes a reference to what realization should be; therefore it informs in turn our understanding of the nature of realization. Conceptions of explanation are thereby included in the views of realization as a metaphysical property.Recently, several major views of realization such as Polger and Shapiro's or Gillett and Aizawa's, however competing, have relied on the neo-mechanicist theory of explanations (e.g,. Darden and Caver 2013), currently popular among philosophers of science. However, it has also been increasingly argued that some explanations are not mechanistic (e.g., Batterman 2009).Using an account given in Huneman (2017), I argue that within those explanations the fact that some mathematical properties are instantiated is explanatory, and that this defines a specific explanatory type called “structural explanation”, whose subtypes could be: optimality explanations (usually found in economics), topological explanations, etc. This paper thereby argues that all subtypes of structural explanation define several kinds of realizability, which are not equivalent to the usual notion of realization tied to mechanistic explanations, onto which many of the philosophical investigations are focused. Then it draws some consequences concerning the notion of multiple realizability.     

Mechanistic artefact explanation

One thing about technical artefacts that needs to be explained is how their physical make-up, or structure, enables them to fulfil the behaviour associated with their function, or, more colloquially, how they work. In this paper I develop an account of such explanations based on the familiar notion of mechanistic explanation. To accomplish this, I (1) outline two explanatory strategies that provide two different types of insight into an artefact’s functioning, and (2) show how human action inevitably plays a role in artefact explanation. I then use my own account to criticize other recent work on mechanistic explanation and conclude with some general implications for the philosophy of explanation.     

How could models possibly provide how-possibly explanations?

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.     

Collaborative explanation and biological mechanisms

This paper motivates and outlines a new account of scientific explanation, which I term ‘collaborative explanation.’ My approach is pluralist: I do not claim that all scientific explanations are collaborative, but only that some important scientific explanations are—notably those of complex organic processes like development. Collaborative explanation is closely related to what philosophers of biology term ‘mechanistic explanation’ (e.g., Machamer et al., Craver, 2007). I begin with minimal conditions for mechanisms: complexity, causality, and multilevel structure. Different accounts of mechanistic explanation interpret and prioritize these conditions in different ways. This framework reveals two distinct varieties of mechanistic explanation: causal and constitutive. The two have heretofore been conflated, with philosophical discussion focusing on the former. This paper addresses the imbalance, using a case study of modeling practices in Systems Biology to reveals key features of constitutive mechanistic explanation. I then propose an analysis of this variety of mechanistic explanation, in terms of collaborative concepts, and sketch the outlines of a general theory of collaborative explanation. I conclude with some reflections on the connection between this variety of explanation and social aspects of scientific practice.     

Essentially narrative explanations

This essay argues that narrative explanations prove uniquely suited to answering certain explanatory questions, and offers reasons why recognizing a type of statement that requires narrative explanations crucially informs on their assessment. My explication of narrative explanation begins by identifying two interrelated sources of philosophical unhappiness with them. The first I term the problem of logical formlessness and the second the problem of evaluative intractability. With regard to the first, narratives simply do not appear to instantiate any logical form recognized as inference licensing. But absent a means of identifying inferential links, what justifies connecting explanans and explanandum? Evaluative intractability, the second problem, thus seems a direct consequence. This essay shows exactly why these complaints prove unfounded by explicating narrative explanations in the process of answering three interrelated questions. First, what determines that an explanation has in some critical or essential respect a narrative form? Second, how does a narrative in such cases come to constitute a plausible explanation? Third, how do the first two considerations yield a basis for evaluating an explanation offered as a narrative? Answers to each of these questions include illustrations of actual narrative explanations and also function to underline attendant dimensions of evaluation.     

A pragmatic account of mechanistic artifact explanation

A mechanistic artifact explanation is an explanation that accounts for an artifact behavior by describing the underlying mechanism. The article shows that there are different kinds of mechanistic artifact explanation: top-down and bottom-up explanation, and I also distinguish between less and more inclusive top-down explanations. To illustrate these different kinds of explanation, the behavior of a simple, fictional artifact is explained in different ways. I defend that which explanation is ideal, depends on pragmatic factors (e.g., the background knowledge of the explainee and the specific goal for which the explanation will be used). For each kind of explanation, the situations, goals and interests for which it is most appropriate are specified, resulting in a pragmatic theory of mechanistic artifact explanation. This theory is compared to Jeroen de Ridder’s account of the pragmatics of mechanistic artifact explanation.     

Hobbes on natural philosophy as “True Physics” and mixed mathematics

In this paper, I offer an alternative account of the relationship of Hobbesian geometry to natural philosophy by arguing that mixed mathematics provided Hobbes with a model for thinking about it. In mixed mathematics, one may borrow causal principles from one science and use them in another science without there being a deductive relationship between those two sciences. Natural philosophy for Hobbes is mixed because an explanation may combine observations from experience (the ‘that’) with causal principles from geometry (the ‘why’). My argument shows that Hobbesian natural philosophy relies upon suppositions that bodies plausibly behave according to these borrowed causal principles from geometry, acknowledging that bodies in the world may not actually behave this way. First, I consider Hobbes's relation to Aristotelian mixed mathematics and to Isaac Barrow's broadening of mixed mathematics in Mathematical Lectures (1683). I show that for Hobbes maker's knowledge from geometry provides the ‘why’ in mixed-mathematical explanations. Next, I examine two explanations from De corpore Part IV: (1) the explanation of sense in De corpore 25.1-2; and (2) the explanation of the swelling of parts of the body when they become warm in De corpore 27.3. In both explanations, I show Hobbes borrowing and citing geometrical principles and mixing these principles with appeals to experience.     

Ernst Cassirer's transcendental account of mathematical reasoning

Cassirer's philosophical agenda revolved around what appears to be a paradoxical goal, that is, to reconcile the Kantian explanation of the possibility of knowledge with the conceptual changes of nineteenth and early twentieth-century science. This paper offers a new discussion of one way in which this paradox manifests itself in Cassirer's philosophy of mathematics. Cassirer articulated a unitary perspective on mathematics as an investigation of structures independently of the nature of individual objects making up those structures. However, this posed the problem of how to account for the applicability of abstract mathematical concepts to empirical reality. My suggestion is that Cassirer was able to address this problem by giving a transcendental account of mathematical reasoning, according to which the very formation of mathematical concepts provides an explanation of the extensibility of mathematical knowledge. In order to spell out what this argument entails, the first part of the paper considers how Cassirer positioned himself within the Marburg neo-Kantian debate over intellectual and sensible conditions of knowledge in 1902–1910. The second part compares what Cassirer says about mathematics in 1910 with some relevant examples of how structural procedures developed in nineteenth-century mathematics.     

Re-orienting discussions of scientific explanation: A functional perspective

Philosophy of science offers a rich lineage of analysis concerning the nature of scientific explanation, but the vast majority of this work, aiming to provide an analysis of the relation that binds a given explanans to its corresponding explanandum, presumes the proper analytic focus rests at the level of individual explanations. There are, however, other questions we could ask about explanation in science, such as: What role(s) does explanatory practice play in science? Shifting focus away from explanations, as achievements, toward explaining, as a coordinated activity of communities, the functional perspective aims to reveal how the practice of explanatory discourse functions within scientific communities given their more comprehensive aims and practices. In this paper, I outline the functional perspective, argue that taking the functional perspective can reveal important methodological roles for explanation in science, and consequently, that beginning here provides resources for developing more adequate responses to traditional concerns. In particular, through an examination of the ideal gas law, I emphasize the normative status of explanations within scientific communities and discuss how such status underwrites a compelling rationale for explanatory power as a theoretical virtue.     

Determinism and Chance

It is generally thought that objective chances for particular events different from 1 and 0 and determinism are incompatible. However, there are important scientific theories whose laws are deterministic but which also assign non-trivial probabilities to events. The most important of these is statistical mechanics whose probabilities are essential to the explanations of thermodynamic phenomena. These probabilities are often construed as ‘ignorance’ probabilities representing our lack of knowledge concerning the microstate. I argue that this construal is incompatible with the role of probability in explanation and laws. This is the ‘paradox of deterministic probabilities’. After surveying the usual list of accounts of objective chance and finding them inadequate I argue that an account of chance sketched by David Lewis can be modified to solve the paradox of deterministic probabilities and provide an adequate account of the probabilities in deterministic theories like statistical mechanics.     

Why the ultimate argument for scientific realism ultimately fails

In this paper, I argue that the ultimate argument for Scientific Realism, also known as the No-Miracles Argument (NMA), ultimately fails as an abductive defence of Epistemic Scientific Realism (ESR), where (ESR) is the thesis that successful theories of mature sciences are approximately true. The NMA is supposed to be an Inference to the Best Explanation (IBE) that purports to explain the success of science. However, the explanation offered as the best explanation for success, namely (ESR), fails to yield independently testable predictions that alternative explanations for success do not yield. If this is correct, then there seems to be no good reason to prefer (ESR) over alternative explanations for success.     

Can the behavioral sciences self-correct? A social epistemic study

Advocates of the self-corrective thesis argue that scientific method will refute false theories and find closer approximations to the truth in the long run. I discuss a contemporary interpretation of this thesis in terms of frequentist statistics in the context of the behavioral sciences. First, I identify experimental replications and systematic aggregation of evidence (meta-analysis) as the self-corrective mechanism. Then, I present a computer simulation study of scientific communities that implement this mechanism to argue that frequentist statistics may converge upon a correct estimate or not depending on the social structure of the community that uses it. Based on this study, I argue that methodological explanations of the “replicability crisis” in psychology are limited and propose an alternative explanation in terms of biases. Finally, I conclude suggesting that scientific self-correction should be understood as an interaction effect between inference methods and social structures.     

Circularity,indispensability, and mathematical explanation in science

In this paper I consider the objection that the Enhanced Indispensability Argument (EIA) is circular and hence fails to support mathematical platonism. The objection is that the explanandum in any mathematical explanation of a physical phenomenon is itself identified using mathematical concepts. Hence the explanandum is only genuine if the truth of some mathematical theory is already presupposed. I argue that this objection deserves to be taken seriously, that it does sometimes undermine support for EIA, but that there is no reason to think that circularity is an unavoidable feature of mathematical explanation in science.     

Symmetries and the explanation of conservation laws in the light of the inverse problem in Lagrangian mechanics

Many have thought that symmetries of a Lagrangian explain the standard laws of energy, momentum, and angular momentum conservation in a rather straightforward way. In this paper, I argue that the explanation of conservation laws via symmetries of Lagrangians involves complications that have not been adequately noted in the philosophical literature and some of the physics literature on the subject. In fact, such complications show that the principles that are commonly appealed to to drive explanations of conservation laws are not generally correct without caveats. I hope here to give a clearer picture of the relationship between symmetries and conservation laws in Lagrangian mechanics via an examination of the bearing that results in the inverse problem in the calculus of variations have on this topic.     

Narrative possibility and narrative explanation

Narratives are about not only what actually happened, but also what might have. And narrative explanations make productive use of these unrealized possibilities. I discuss narrative explanation as a form of counterfactual, difference-making explanation, with a demanding qualification: the counterfactual conditions are historically or narratively (not merely logically or physically) possible. I consider these issues in connection with literary, historical and scientific narratives.     

Detail and generality in mechanistic explanation

This article is about the role of abstraction in mechanistic explanations. Abstraction is widely recognised as a necessary concession to the practicalities of scientific work, but some mechanist philosophers argue that it is also a positive explanatory feature in its own right. I claim that in as much as these arguments are based on the idea that mechanistic explanation exhibits a trade-off between fine-grained detail and generality, they are unsuccessful. Detail and generality both appear to be important sources of explanatory power, but investigators do not need to make a choice between these desiderata, at least when an explanation incorporates further detail through the decomposition of the mechanism's parts.     

Maimon's criticism of Kant's doctrine of mathematical cognition and the possibility of metaphysics as a science

The aim of this paper is to discuss Maimon's criticism of Kant's doctrine of mathematical cognition. In particular, we will focus on the consequences of this criticism for the problem of the possibility of metaphysics as a science. Maimon criticizes Kant's explanation of the synthetic a priori character of mathematics and develops a philosophical interpretation of differential calculus according to which mathematics and metaphysics become deeply interwoven. Maimon establishes a parallelism between two relationships: on the one hand, the mathematical relationship between the integral and the differential and on the other, the metaphysical relationship between the sensible and the supersensible. Such a parallelism will be the clue to the Maimonian solution to the Kantian problem of the possibility of metaphysics as a science.