Lately, philosophers of mathematics have been exploring the notion of mathematical explanation within mathematics. This project is supposed to be analogous to the search for the correct analysis of scientific explanation. I argue here that given the way philosophers have been using “ explanation,” the term is not applicable to mathematics as it is in science.

I examine the relation between explanation and unification in both the original Kaluza–Klein theory, which originated in the works of Theodor Kaluza and Oskar Klein in the 1920s, and in the modern Kaluza–Klein theories which date back to the late 1970s and which are still considered by the majority of the physics community to be the best hope for a complete unified theory of all fundamental interactions. I use the conclusions of this case study to assess the merits of Philip (...) Kitcher's account of explanation as unification. I also draw lessons about physicists’ pursuit of the higher dimensional unification of the fundamental forces of nature. (shrink)

Whether simulation models provide the right kind of understanding comparable to that of analytic models has been and remains a contentious issue. The assessment of understanding provided by simulations is often hampered by a conflation between the sense of understanding and understanding proper. This paper presents a deflationist conception of understanding and argues for the need to replace appeals to the sense of understanding with explicit criteria of explanatory relevance and for rethinking the proper way of conceptualizing the role of (...) a single human mind in the collective scientific understanding. (shrink)

Kitcher's unification theory of explanation seems to suggest that only the most reductive accounts can legitimately be termed explanatory. This is not what we find in actual scientific practice. In this paper, I attempt to reconcile these ideas. I claim that Kitcher's theory picks out ideal explanations, but that our term explanation is used to cover other accounts that have a certain relationship with the ideal accounts. At times, versions and portions of ideal explanations can also be considered explanatory.

Comparisons of rival explanations or theories often involve vague appeals to explanatory power. In this paper, we dissect this metaphor by distinguishing between different dimensions of the goodness of an explanation: non-sensitivity, cognitive salience, precision, factual accuracy and degree of integration. These dimensions are partially independent and often come into conflict. Our main contribution is to go beyond simple stipulation or description by explicating why these factors are taken to be explanatory virtues. We accomplish this by using the contrastive-counterfactual approach (...) to explanation and the view of understanding as an inferential ability. By combining these perspectives, we show how the explanatory power of an explanation in a given dimension can be assessed by showing the range of answers it provides to what-if-things-had-been-different questions and the theoretical and pragmatic importance of these questions. Our account also explains intuitions linking explanation to unification or to exhibition of a mechanism. (shrink)

According to Philip Kitcher, scientific unification is achieved via the derivation of numerous scientific statements from economies of argument schemata. I demonstrate that the unification of selection phenomena across domains in which it is claimed to occur--evolutionary biology, immunology and, speculatively, neurobiology--is unattainable on Kitcher's view. I then introduce an alternative method for rendering the desired unification based on the concept of a mechanism schema. I conclude that the gain in unification provided by the alternative account suggests that Kitcher's view (...) is defective. (shrink)

We provide two programmatic frameworks for integrating philosophical research on understanding with complementary work in computer science, psychology, and neuroscience. First, philosophical theories of understanding have consequences about how agents should reason if they are to understand that can then be evaluated empirically by their concordance with findings in scientific studies of reasoning. Second, these studies use a multitude of explanations, and a philosophical theory of understanding is well suited to integrating these explanations in illuminating ways.

According to Michael Friedman’s theory of explanation, a law X explains laws Y1, Y2, …, Yn precisely when X unifies the Y’s, where unification is understood in terms of reducing the number of independently acceptable laws. Philip Kitcher criticized Friedman’s theory but did not analyze the concept of independent acceptability. Here we show that Kitcher’s objection can be met by modifying an element in Friedman’s account. In addition, we argue that there are serious objections to the use that Friedman makes (...) of the concept of independent acceptability. (shrink)

Political science and economic science . . . make use of the same language, the same mode of abstraction, the same instruments of thought and the same method of reasoning. (Black 1998, 354) Proponents as well as opponents of economics imperialism agree that imperialism is a matter of unification; providing a unified framework for social scientific analysis. Uskali Mäki distinguishes between derivational and ontological unification and argues that the latter should serve as a constraint for the former. We explore whether, (...) in the case of rational-choice political science, self-interested behavior can be seen as a common causal element and solution concepts as the common derivational element, and whether the former constraints the use of the latter. We find that this is not the case. Instead, what is common to economics and rational-choice political science is a set of research heuristics and a focus on institutions with similar structures and forms of organization. (shrink)

In this paper, we will explore one way in which understanding can—and, we will argue, should—be valuable. We will do this by drawing on what has been said about the different ways knowledge can be valuable. Our main contribution will be to identify one heretofore undiscussed way knowledge could be valuable, but isn’t—specifically, having value to someone other than the understander. We suggest that it is a desideratum on an account of understanding that understanding have the specified type of value; (...) our basis for this claim will come from recent work in cognitive psychology. This desideratum can then be used to measure the success of various accounts of understanding. We argue that accounts of understanding that have a particular structure will predict why understanding has that sort of value. For good measure, we then engage in a bit of a literature review, investigating which extant accounts of understanding satisfy this desideratum. (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)

The goal of this paper is to present and defend an empiricist, neo-Hempelian account of scientific explanation as ampliative, specialized embedding. The proposal aims to preserve what I take to be the core of Hempel’s empiricist account, by weakening it in some respects and strengthening it in others, introducing two new conditions that solve most of Hempel’s problems without abandoning his empiricist strictures. According to this proposal, to explain a phenomenon is to make it expectable by introducing new conceptual/ontological machinery (...) and using special, and non-ad hoc, non-accidental regularities. The new conditions are elaborated making essential use of two central structuralist ideas, namely T-theoreticity and specialization. I first introduce and qualify the project, then present the new account in detail and assess it vis-à-vis its rivals, and finally discuss some possible objections, concluding that the account fares better than its monistic rivals and well enough to qualify as a promising neo-Hempelian account. Even for those unpersuaded by its monistic goals, it has the merit of calling attention to two new necessary conditions not explicitly emphasized thus far and showing how they serve to answer many of the criticisms addressed against Hempel’s account. (shrink)

It's widely supposed that unification is an epistemic virtue: the degree to which a theory is unified contributes to its overall confirmation. However, this supposition has consequences which haven't been noted, and which undermine the leading accounts of unification. For, given Hempel's equivalence condition, any epistemic virtue must be such that logically equivalent theories must equally well unify any body of evidence, and logically equivalent bodies of evidence must be equally well unified by any theory. Yet the leading accounts of (...) unification in Bayesian terms, or those in terms of argument patterns, cannot satisfy these constraints conditions. The reason for this runs deep: these accounts of unification make unity depend on factors that vary between equivalent theories: the probabilistic relations of their components, or their relations to argument patterns. The solution is to abandon such accounts and instead adopt an account of unity based on worldly relations such as causation, rather than inferential relations. Such an account effortlessly satisfies the equivalence conditions, and so may describe the epistemic virtue of unity. (shrink)

The reshaping of much scientific research around computational methods is not just a technological curiosity. It results in a significant reshaping of conceptual and representational resources within science in ways with which many traditional philosophical positions are ill-equipped to cope. Some illustrations of this are provided and a consequence for the roles of science and the arts is noted.