Mathematical Explanation by Law

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 paper, a theory of extra-mathematical explanation is developed. The theory is modeled 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

Mathematical Explanation by Law

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 paper, a theory of extra-mathematical explanation is developed. The theory is modeled 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

Mathematical counterfactuals with number-theoretic antecedents and extra-mathematical explanation

A proposal by Baron, Colyvan, and Ripley to extend the counterfactual theory of explanation to include counterfactual reasoning about mathematical explanations of physical facts is discussed. Their suggestion is that the explanatory role of mathematics can best be captured counterfactually. This paper focuses on their example with a number-theoretic antecedent. Incorporating discussions on the structure and de re knowledge of numbers, it is argued that the approach leads to a change in the structure of numbers. As a result, the counterfactual is not about the natural numbers anymore. Linking the antecedent and consequent of the counterfactual also becomes problematic

Mathematical Explanation by Law

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 (DN) 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

Mathematical Explanation by Law

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 paper, a theory of extra-mathematical explanation is developed. The theory is modeled 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

Explaining Universality: Infinite Limit Systems in the Renormalization Group Method

I analyze the role of infinite idealizations used in the renormalization group (RG hereafter) method in explaining universality across microscopically different physical systems in critical phenomena. I argue that despite the reference to infinite limit systems such as systems with infinite correlation lengths during the RG process, the key to explaining universality in critical phenomena need not involve infinite limit systems. I develop my argument by introducing what I regard as the explanatorily relevant property in RG explanations: linearization* property; I then motivate and prove a proposition about the linearization* property in support of my view. As a result, infinite limit systems in RG explanations are dispensable

Mathematical Realism From Reflectance Physicalism

Do mirrors remain “reflective” objects in the dark, or does light shining onto a mirror instead give the mirror its reflective ability in the moment? More than an idle barstool question (like whether a tree falling in an abandoned forest makes a sound) the intrinsicality or light-independence of reflectance carries considerable philosophical import, since some philosophers reduce the human-visible colors to intrinsic surface reflectance. My dissertation, while remaining neutral on the best definition of color, argues that the received view of reflectance leaves it conceptually regressive and thus non-ascribable to surfaces. Rendering reflectance intrinsic to surfaces, I argue, requires a mathematized redefinition of reflectance, the literal interpretation of which implies a limited mathematical realism, itself a millennia-old philosophical bugbear. Without advocating mathematical realism per se, my thesis implicates a variety of current debates in scientific structural realism, metaphysical dispositional realism, mathematical nominalism, mathematical explanation, and even aesthetics, thanks to the philosophical precedent of reducing color to reflectance. Here is the argument whose implications I explore throughout my dissertation chapters. The received definition of reflectance is the per-wavelength efficiency of a surface to reflect “pulses” of light, pulses being finite-duration propagations. I object that according to a well-documented law of nature, all electromagnetic pulses exhibit an inverse relationship between their duration and bandwidth, and that this relationship generates a vicious regress of the purported reflectance value at any wavelength. I block the regress by redefining “pulses” as superpositions of Fourier harmonics, which are infinite-duration monochromes. If harmonics reflect from surfaces, however, then they must be real

The explanatory dispensability of idealizations

Enhanced indispensability arguments seek to establish realism about mathematics based on the explanatory role that mathematics plays in science. Idealizations pose a problem for such arguments. Idealizations, in a similar way to mathematics, boost the explanatory credentials of our best scientific theories. And yet, idealizations are not the sorts of things that are supposed to attract a realist attitude. I argue that the explanatory symmetry between idealizations and mathematics can potentially be broken as follows: although idealizations contribute to the explanatory power of our best theories, they do not carry the explanatory load. It is at least open however that mathematics is load-carrying. To give this idea substance, I offer an analysis of what it is to carry the explanatory load in terms of difference-making and counterfactuals