One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory power of science by expressing conceptual rules, rules which allow the transformation of empirical descriptions. Mathematics should not be thought of as describing, in any substantive sense, (...) an abstract realm of eternal mathematical objects, as traditional platonists have thought. In Rules to Infinity, this view, which I call mathematical normativism, is updated with contemporary philosophical tools, and it is argued that normativism is compatible with mainstream semantic theory. This allows the normativist to accept that there are mathematical truths, while resisting the platonistic idea that there exist abstract mathematical objects that explain such truths. Furthermore, Rules to Infinity defends a particular account of the distinction between scientific explanations that are in some sense distinctively mathematical – those that explain natural phenomena in some uniquely mathematical way – and those that are only standardly mathematical, and it lays out desiderata for any account of this distinction. Normativism is compared with other prominent views in the philosophy of mathematics such as neo-Fregeanism, fictionalism, conventionalism, and structuralism. Rules to Infinity serves as an entry point into debates at the forefront of philosophy of science and mathematics, and it defends novel positions in these debates. (shrink)

So-called ‘distinctively mathematical explanations’ (DMEs) are said to explain physical phenomena, not in terms of contingent causal laws, but rather in terms of mathematical necessities that constrain the physical system in question. Lange argues that the existence of four or more equilibrium positions of any double pendulum has a DME. Here we refute both Lange’s claim itself and a strengthened and extended version of the claim that would pertain to any n-tuple pendulum system on the ground that such explanations are (...) actually causal explanations in disguise and their associated modal conditionals are not general enough to explain the said features of such dynamical systems. We argue and show that if circumscribing the antecedent for a necessarily true conditional in such explanations involves making a causal analysis of the problem, then the resulting explanation is not distinctively mathematical or non-causal. Our argument generalises to other dynamical systems that may have purported DMEs analogous to the one proposed by Lange, and even to some other counterfactual accounts of non-causal explanation given by Reutlinger and Rice. (shrink)

A central goal of the scientific endeavor is to explain phenomena. Scientists often attempt to explain a phenomenon by way of representing it in some manner—such as with mathematical equations, models, or theory—which allows for an explanation of the phenomenon under investigation. However, in developing scientific representations, scientists typically deploy simplifications and idealizations. As a result, scientific representations provide only partial, and often distorted, accounts of the phenomenon in question. Philosophers of science have analyzed the nature and function of how (...) scientists construct representations, deploy idealizations, and provide explanations. As such, our aim in this special issue is to bring these three pillars of research into closer contact with the contributions to it focusing on three main themes. -/- The first set of papers, Alan Baker (2021) and Marc Lange (2021), address mathematical explanations in science. Baker (2021), a proponent of mathematical Platonism, examines its capacity to evade the critique that the so-called Enhanced Indispensability Argument is circular. Lange (2021) examines distinctively mathematical explanations, arguing that neither Platonism nor representationalism are successful paths, and instead argues in favor of Aristotelian realism. A second theme emerging from the papers in this special issue is the impact that various conceptualizations of idealization have on our abilities to offer scientific explanations, to pro- duce an analysis of what explanations are or should be, and to understand scientific representation. Peter Tan (2021) suggests amending inferentialist accounts of scientific representation to account for inconsistent idealizations. Michael Strevens (2021) advocates a view wherein the introduction of idealizations into a model is legitimate so long as it pertains to non-difference-making factors, arguing for a logical reading of the notion of difference-making. Natalia Carrillo and Tarja Knuuttila (2022) offer an alternative account to the idealization-as-distortion view, emphasizing instead the holistic nature of idealization. -/- Finally, contributions by Carrillo and Knuuttila (2022), Terzian (2021), Valente (2021), and Rodriguez (2021) illustrate how issues regarding idealization, representation, and explanation are applied to specific contexts and across various sciences. Carrillo and Knuuttila (2022) examine conceptions of idealization in the context of models of the nerve impulse. Giulia Terzian (2021) extends the discussion of idealizations to the context of generative linguistics. Giovanni Valente (2021) examines how idealizations and evaluations of accurate representation impact capacities to explain phenomena in the context of statistical thermodynamics. Quentin Rodriguez (2021) examines the role of idealizations and analogies in various strategies to explain critical phenomena. -/- In what follows, we offer a brief overview of important philosophical issues connected to representation, idealization, and explanation in science. We then provide short summaries of the eight papers in this special issue. (shrink)