Mathematical Explanation

by

Paolo Mancosu, Christopher Pincock

• LAST REVIEWED: 22 November 2022
• LAST MODIFIED: 29 August 2012
• DOI: 10.1093/obo/9780195396577-0029

Introduction

Mathematical explanations are explanations in which mathematics plays a crucial role. There is a rich historical tradition of debates about mathematical explanation, dating back to Aristotle. One central strand of this tradition considers the significance of explanatory goals for pure mathematics. It is common for mathematicians to prefer one proof of a theorem over another because one proof shows why a theorem is true, even though both proofs show that that theorem is true. There have been several attempts to make sense of explanations in pure mathematics. While broadly causal approaches to scientific explanation seem to have little hope of clarifying explanation in pure mathematics, Mark Steiner has proposed an account in terms of characterizing properties of mathematical entities that is a kind of generalization of a causal approach. An alternative strategy, pursued by Philip Kitcher, is to say that an explanatory proof in pure mathematics is, roughly, one that unifies disparate mathematical claims. This unification approach seems more likely to succeed for pure mathematics, but it also faces challenges tied to the particular way that Kitcher measures unifying power. The second central strand of discussions of mathematical explanation considers the contributions that mathematics makes to scientific explanations. Carl Hempel’s influential deductive-nomological account of scientific explanation left room for mathematical claims in scientific explanations, but this approach to scientific explanation faces serious challenges. Causal and unification approaches to scientific explanation reach different verdicts on the question of mathematics and scientific explanation. Causal accounts may allow mathematics in scientific explanations, but invariably maintain that the mathematics is valuable only because it reflects the genuine causes of what is being explained. Unification accounts can assign the mathematics some explanatory power in its own right, independently of any corresponding causes, because unification may be facilitated by mathematical concepts and theorems. The problem of understanding what mathematics brings to scientific explanation has recently received new impetus from debates about explanatory indispensability arguments for mathematical platonism. These new indispensability arguments build on the traditional indispensability arguments found in W. V. Quine and Hilary Putnam. However, advocates of an explanatory indispensability argument insist that explanatory benefits from pure mathematics are significant enough by themselves to warrant our belief in mathematical entities. Critics of these new arguments either complain that the account of explanation at work here remains unclear, or else they offer their own proposals for how to make sense of mathematical explanation in science. (Acknowledgments: The authors would like to thank Hein van der Berg and an anonymous referee for their valuable suggestions.)

General Overviews

There is as yet no major monograph on mathematical explanation that brings together the historical tradition of discussing explanation in mathematics and science, contemporary debates about explanation in pure mathematics, and the contemporary examination of the significance of mathematics in scientific explanation. Mancosu 2001 was the first thorough general overview to touch on each of these areas. He links an interest in mathematical explanation to the broader turn toward mathematical practice in the philosophy of mathematics. This movement emphasizes issues that arise outside of foundational areas like logic and set theory and tries to use these issues to illuminate traditional and novel philosophical questions about mathematics. Paolo Mancosu summarizes the rich history of debates about explanation in pure mathematics as well as several examples in which explanatory considerations seemed decisive for mathematical practice. In addition to the importance of the Aristotelian tradition, he also points out the significance of a “hypothetico-inductivist” approach to mathematics, as reflected in the writings of John Stuart Mill, Bertrand Russell, Imre Lakatos, and Kurt Gödel. These authors emphasize explanatory factors in theory acceptance besides traditional proofs from accepted axioms. For example, Gödel allowed that explanatory considerations might provide evidence for new axioms in set theory. Mancosu 2008 and Mancosu 2011 provide comprehensive summaries of the discussions of mathematical explanation. The former article emphasizes the importance of mathematics in scientific explanation. In particular, Mancosu discusses the explanatory indispensability argument for mathematical Platonism and the corresponding question for pure mathematics of whether explanatory considerations might justify the ontological commitment to entities mentioned in the explanans. Mancosu also provides a helpful critical summary of the proposal for explanations in pure mathematics found in Mark Steiner and Philip Kitcher’s work. Mancosu 2011 contains more information on the historical debates on explanation and on how debates on mathematical modeling and idealization are relevant to mathematical explanations in science.

• Mancosu, Paolo. “Mathematical Explanation: Problems and Prospects.” Topoi 20 (2001): 97–117.

  DOI: 10.1023/A:1010621314372

  A general survey of issues connected with mathematical explanation emphasizing significant cases and the hypothetico-inductivist tradition of Mill, Russell, Lakatos, and Gödel.

• Mancosu, Paolo. “Mathematical Explanation: Why It Matters.” In The Philosophy of Mathematical Practice. Edited by Paolo Mancosu, 134–150. Oxford: Oxford University Press, 2008.

  DOI: 10.1093/acprof:oso/9780199296453.003.0006

  A general discussion of the significance of mathematical explanation for the philosophy of mathematics and the philosophy of science. Summarizes the explanatory indispensability argument as well as the proposals of Steiner and Kitcher, with special emphasis on how generality and explanation are connected.

• Mancosu, Paolo. Explanation in mathematics. In The Stanford Encyclopedia of Philosophy. Edited by Eugene N. Zalta. 2011.

  This is the most balanced and up-to-date exposition of the topic of explanation in mathematics, covering both the historical and the systematic aspects of the problem. It includes discussions of mathematical explanation within mathematics and mathematical explanations in science.