Mathematical Explanation (philpapers)

Studies in History and Philosophy of Science Part A, 2021

Philosophical Review, 2014

No��s, 2012

Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader issues in the philosophy of mathematics; in particular it may help platonists respond to a recent challenge by Joseph Melia concerning the force of the Indispensability Argument.

Quantum, Probability, Logic, 2020

The mathematical nature of modern physics suggests that mathematics is bound to play some role in explaining physical reality. Yet, there is an ongoing controversy about the prospects of mathematical explanations of physical facts and their nature. A common view has it that mathematics provides a rich and indispensable language for representing physical reality but that, ontologically, physical facts are not mathematical and, accordingly, mathematical facts cannot really explain physical facts. In what follows, I challenge this common view. I argue that, in addition to its representational role, in modern physics mathematics is constitutive of the physical. Granted the mathematical constitution of the physical, I propose an account of explanation in which mathematical frameworks, structures, and facts explain physical facts. In this account, mathematical explanations of physical facts are either species of physical explanations of physical facts in which the mathematical constitution of some physical facts in the explanans are highlighted, or simply explanations in which the mathematical constitution of physical facts are highlighted. In highlighting the mathematical constitution of physical facts, mathematical explanations of physical facts deepen and increase the scope of the understanding of the explained physical facts. I argue that, unlike other accounts of mathematical explanations of physical facts, the proposed account is not subject to the objection that mathematics only represents the physical facts that actually do the explanation. I conclude by briefly considering the implications that the mathematical constitution of the physical has for the question of the unreasonable effectiveness of the use of mathematics in physics. Outline 1 The orthodoxy 2 An alternative perspective 3 On the relationship between mathematics and physics 4 On conceptions of mathematical constitution of the physical 5 On the common view of how mathematical models represent physical reality 6 On the notion of the physical 7 On the scope of the mathematical constitution of the physical 8 A sketch of a new account of mathematical explanation of physical facts 9 On mathematical explanations of physical facts 9.1 On a D-N explanation of the life cycle of 'periodical' cicadas 9.2 On structural explanation of the uncertainty relations 9.3 On abstract explanation of the impossibility of a minimal tour across the bridges of Königsberg 9.4 On explanations by constraints that are more necessary than laws of nature 10 Is the effectiveness of mathematics in physics unreasonable?

THEORIA. An International Journal for Theory, History and Foundations of Science

This paper discusses Baker’s Enhanced Indispensability Argument (EIA) for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such use still does not support realism about mathematical objects.

European Journal for Philosophy of Science, 2020

Lange (2013; 2016) argues that some natural phenomena can be explained by appeal to mathematical, rather than natural, facts. In these "distinctively mathematical" explanations (DMEs), the core explanatory facts are either modally stronger than facts about ordinary causal law or understood to be constitutive of the physical task or arrangement at issue. Craver and Povich (2017) argue that Lange's account of DME fails to exclude certain "reversals". Lange (forthcoming) has replied that his account can avoid these directionality charges. Specifically, Lange argues that in legitimate DMEs, but not in their "reversals," the empirical fact appealed to in the explanation is "understood to be constitutive of the physical task or arrangement at issue" in the explanandum. I argue that Lange's reply is unsatisfactory because it leaves the crucial notion of being "understood to be constitutive of the physical task or arrangement" obscure in ways that fail to block "reversals" except by an apparent ad hoc stipulation or by abandoning the reliance on understanding and instead accepting a strong realism about essence.

The Philosophical Quarterly, 2009

Synthese, 2008

Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.

The British Journal for the Philosophy of Science, 2013

Certain scientific explanations of physical facts have recently been characterized as distinctively mathematical-that is, as mathematical in a different way from ordinary explanations that employ mathematics. This article identifies what it is that makes some scientific explanations distinctively mathematical and how such explanations work. These explanations are non-causal, but this does not mean that they fail to cite the explanandum's causes, that they abstract away from detailed causal histories, or that they cite no natural laws. Rather, in these explanations, the facts doing the explaining are modally stronger than ordinary causal laws (since they concern the framework of any possible causal relation) or are understood in the why question's context to be constitutive of the physical arrangement at issue. A distinctively mathematical explanation works by showing the explanandum to be more necessary (given the physical arrangement in question) than ordinary causal laws could render it. Distinctively mathematical explanations thus supply a kind of understanding that causal explanations cannot.

Metascience, 2013

Christopher Pincock's Mathematics and Scientific Representation is an interesting and important new book on the applications of mathematics to science. The book is loaded with fascinating case studies that bring out the varied and interesting ways that mathematics is used in science. Pincock provides a much more nuanced picture of the roles that mathematics plays in empirical science than we usually get from philosophers of mathematics writing on the topic of applications. He describes various dimensions along which different uses of mathematics can differ; for instance, to pick out just one of the dimensions he discusses, we sometimes use mathematics to help us represent the causal structure of a series of events, whereas in other cases, we use it to provide an acausal representation of a system in a static, fixed state. The overall picture we get from Pincock is a sort of taxonomy of different kinds of uses of mathematics. He also draws interesting connections