Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

Detailed examinations of scientific practice have revealed that the use of idealized models in the sciences is pervasive. These models play a central role in not only the investigation and prediction of phenomena, but in their received scientific explanations as well. This has led philosophers of science to begin revising the traditional philosophical accounts of scientific explanation in order to make sense of this practice. These new model-based accounts of scientific explanation, however, raise a number of key questions: Can the (...) fictions and falsehoods inherent in the modeling practice do real explanatory work? Do some highly abstract and mathematical models exhibit a noncausal form of scientific explanation? How can one distinguish an exploratory "how-possibly" model explanation from a genuine "how-actually" model explanation? Do modelers face tradeoffs such that a model that is optimized for yielding explanatory insight, for example, might fail to be the most predictively accurate, and vice versa? This chapter explores the various answers that have been given to these questions. (shrink)

This book presents the concept of “complementary science” which contributes to scientific knowledge through historical and philosophical investigations. It emphasizes the fact that many simple items of knowledge that we take for granted were actually spectacular achievements obtained only after a great deal of innovative thinking, painstaking experiments, bold conjectures, and serious controversies. Each chapter in the book consists of two parts: a narrative part that states the philosophical puzzle and gives a problem-centred narrative on the historical attempts to solve (...) the puzzle; and the analysis part which provides in-depth analyses of certain scientific, historical, and philosophical aspects of the story. (shrink)

Woodward's long awaited book is an attempt to construct a comprehensive account of causation explanation that applies to a wide variety of causal and explanatory claims in different areas of science and everyday life. The book engages some of the relevant literature from other disciplines, as Woodward weaves together examples, counterexamples, criticisms, defenses, objections, and replies into a convincing defense of the core of his theory, which is that we can analyze causation by appeal to the notion of manipulation.

The following article treats the 'applicational turn' of modern fluid dynamics as it set in at the beginning of the 20th century with Ludwig Prandtl's concept of the boundary layer. It seeks to show that there is much more to applying a theory in a highly mathematical field like fluid dynamics than deriving a special case from a general explanatory theory under particular antecedent conditions. In Prandtl's case, the decisive move was to introduce a model that provided a physical/causal conception (...) of viscous flow at high Reynolds numbers. It facilitated an approximate solution to the Navier-Stokes equations, which in turn gave rise to many special applications. After a detailed account of Prandtl's achievement, the article discusses the role of the physical model and its experimental and mathematical significance. It is shown that the mathematical simplification provided by the physical model greatly expanded the explanatory capacity of the theory which the Navier-Stokes equations alone could not provide. (shrink)

Science is the study of our world, as it is in its messy reality. Nonetheless, science requires idealization to function—if we are to attempt to understand the world, we have to find ways to reduce its complexity. Idealization and the Aims of Science shows just how crucial idealization is to science and why it matters. Beginning with the acknowledgment of our status as limited human agents trying to make sense of an exceedingly complex world, Angela Potochnik moves on to explain (...) how science aims to depict and make use of causal patterns—a project that makes essential use of idealization. She offers case studies from a number of branches of science to demonstrate the ubiquity of idealization, shows how causal patterns are used to develop scientific explanations, and describes how the necessarily imperfect connection between science and truth leads to researchers’ values influencing their findings. The resulting book is a tour de force, a synthesis of the study of idealization that also offers countless new insights and avenues for future exploration. (shrink)

The first of its kind, this book is an in-depth history of hydrodynamics from its eighteenth-century foundations to its first major successes in twentieth-century hydraulics and aeronautics. It documents the foundational role of fluid mechanics in developing a new mathematical physics. It gives full and clear accounts of the conceptual breakthroughs of physicists and engineers who tried to meet challenges in the practical worlds of hydraulics, navigation, blood circulation, meteorology, and aeronautics, and it shows how hydrodynamics at last began to (...) fulfill its early promise to unify the different worlds of flow. Richly illustrated, technically thorough, and sensitive to cross-cultural effects, this history should attract a broad range of historians, scientists, engineers, and philosophers and be a standard reference for anyone interested in fluid mechanics. (shrink)

In many applications of physics, boundary conditions have an essential role. The purpose of this paper is to examine from both a historical and philosophical perspective one such boundary condition, namely, the no-slip condition of fluid dynamics. The historical perspective is based on the works of George Stokes and serves as the foundation for the philosophical perspective. It is seen that historically the acceptance of the no-slip condition was problematic. Philosophically, the no-slip condition is interesting since the use of the (...) no-slip condition illustrates nicely the use of scientific models. But more importantly, both the use and justification of the no-slip condition illustrate clearly how theories can holistically approach the world through model construction. Further, since much of the debate over scientific realism occurs in the realm of models, a case is made that an understanding of the role of the no-slip condition has something to offer to this debate. (shrink)

Can there be mathematical explanations of physical phenomena? In this paper, I suggest an affirmative answer to this question. I outline a strategy to reconstruct several typical examples of such explanations, and I show that they fit a common model. The model reveals that the role of mathematics is explicatory. Isolating this role may help to re-focus the current debate on the more specific question as to whether this explicatory role is, as proposed here, also an explanatory one.