Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, how do (...) they relate to the sorts of explanation encountered in philosophy of science and metaphysics? (shrink)

Argues that there is no interpretation of the commonly-accepted idea that "explanation is that which produces understanding" on which it is of any use for finding what philosophers looking for a theory of explanation have been after. Contains a close examination of a couple of philosophers' attempts to use this idea for that purpose.

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...) of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities requested to the practitioners; (...) and (3) the specific forms of representation and notation shared and selected by the practitioners. Moreover, it is claimed that a broadening of the notion of ‘permissible action’ as introduced by Larvor (2012) with respect to mathematical arguments, allows for a consideration of all these three elements simultaneously. Second, a case from topology – the proof of Alexander’s theorem – is presented to illustrate a concrete analysis of a mathematical practice and to exemplify the proposed method. It is discussed that the attention to the three elements of the practice identified above brings to the emergence of philosophically relevant features in the practice of topology: the need for a revision in the definition of criteria of validity, the interest in tracking the operations that are performed on the notation, and the constant and fruitful back-and-forth from one representation to another in dealing with mathematical content. Finally, some suggestions for further research are given in the conclusions. (shrink)

This book argues against the view that mathematical knowledge is a priori,contending that mathematics is an empirical science and develops historically,just as ...

What does it mean to understand something? I approach this question by comparing understanding with knowledge. Like knowledge, understanding comes, at least prima facia, in three varieties: propositional, interrogative and objectual. I argue that explanatory understanding (this being the most important form of interrogative understanding) and objectual understanding are not reducible to one another and are neither identical with, nor even a form of, the corresponding type of knowledge (nor any other type of knowledge). My discussion suggests that definitions of (...) the concepts of explanatory and of objectual understanding must include a commitment condition, a grasping condition, an answering-the-facts condition, and an epistemically internal justification condition, but no further external anti-luck condition. On this basis I argue against reducing explanatory understanding to propositional understanding, and in favour of identifying propositional understanding with propositional knowledge. (shrink)

The official model of explanation proposed by the logical empiricists, the covering law model, is subject to familiar objections. The goal of the present paper is to explore an unofficial view of explanation which logical empiricists have sometimes suggested, the view of explanation as unification. I try to show that this view can be developed so as to provide insight into major episodes in the history of science, and that it can overcome some of the most serious difficulties besetting the (...) covering law model. (shrink)

Visual thinking in mathematics is widespread; it also has diverse kinds and uses. Which of these uses is legitimate? What epistemic roles, if any, can visualization play in mathematics? These are the central philosophical questions in this area. In this introduction I aim to show that visual thinking does have epistemically significant uses. The discussion focuses mainly on visual thinking in proof and discovery and touches lightly on its role in understanding.

In this paper, I argue that there are cases of explanatory induction in mathematics. To do so, I first introduce the notion of explanatory definition in the context of mathematical explanation. A large part of the paper is dedicated to introducing and analyzing this notion of explanatory definition and the role it plays in mathematics. After doing so, I discuss a particular inductive definition in advanced mathematics—CW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ CW}$$\end{document}-complexes—and argue that it is (...) explanatory. With this, we see that there are cases of explanatory induction. (shrink)

This paper investigates the following question: when can one reliably infer the existence of an intersection point from a diagram presenting crossing curves or lines? Two cases are considered, one from Euclid's geometry and the other from basic real analysis. I argue for the acceptability of such an inference in the geometric case but against in the analytic case. Though this question is somewhat specific, the investigation is intended to contribute to the more general question of the extent and limits (...) of reliable diagrammatic reasoning in mathematics. (shrink)

Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. In addition, mathematicians regard some proofs as explaining why the theorems being proved do in fact hold. This book proposes new philosophical accounts of many kinds of non-causal explanations in science and mathematics.

Mathematical explanations are poorly understood. Although mathematicians seem to regularly suggest that some proofs are explanatory whereas others are not, none of the philosophical accounts of what such claims mean has become widely accepted. In this paper we explore Wilkenfeld’s suggestion that explanations are those sorts of things that generate understanding. By considering a basic model of human cognitive architecture, we suggest that existing accounts of mathematical explanation are all derivable consequences of Wilkenfeld’s ‘functional explanation’ proposal. We therefore argue that (...) the explanatory criteria offered by earlier accounts can all be thought of as features that make it more likely that a mathematical proof will generate understanding. On the functional account, features such as characterising properties, unification, and salience correlate with explanatoriness, but they do not define explanatoriness. (shrink)

A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.

Understanding is a central aim of science and highly important in present-day society. But what precisely is scientific understanding and how can it be achieved? This book answers these questions, through philosophical analysis and historical case studies, and presents a philosophical theory of scientific understanding that highlights its contextual nature.

In this paper, I argue that explanations just ARE those sorts of things that, under the right circumstances and in the right sort of way, bring about understanding. This raises the question of why such a seemingly simple account of explanation, if correct, would not have been identified and agreed upon decades ago. The answer is that only recently has it been made possible to analyze explanation in terms of understanding without the risk of collapsing both to merely phenomenological states. (...) For the most part, theories of explanation were for 50 years held hostage to the historical accident that they far outstripped in sophistication corresponding accounts of understanding. (shrink)

In his long-awaited new edition of Philosophy of Mathematics, James Robert Brown tackles important new as well as enduring questions in the mathematical sciences. Can pictures go beyond being merely suggestive and actually prove anything? Are mathematical results certain? Are experiments of any real value?" "This clear and engaging book takes a unique approach, encompassing nonstandard topics such as the role of visual reasoning, the importance of notation, and the place of computers in mathematics, as well as traditional topics such (...) as formalism, Platonism, and constructivism. The combination of topics and clarity of presentation make it suitable for beginners and experts alike. The revised and updated second edition of Philosophy of Mathematics contains more examples, suggestions for further reading, and expanded material on several topics including a novel approach to the continuum hypothesis."--BOOK JACKET. (shrink)

Much recent work on mathematical explanation has presupposed that the phenomenon involves explanatory proofs in an essential way. I argue that this view, ‘proof chauvinism’, is false. I then look in some detail at the explanation of the solvability of polynomial equations provided by Galois theory, which has often been thought to revolve around an explanatory proof. The article concludes with some general worries about the effects of chauvinism on the theory of mathematical explanation. 1Introduction 2Why I Am Not a (...) Proof Chauvinist 2.1Proof chauvinism and mathematical practice 2.2Proof chauvinism and philosophy 3An Example: Galois Theory and Explanatory Proof 4Conclusion. (shrink)

Explanation in mathematics has recently attracted increased attention from philosophers. The central issue is taken to be how to distinguish between two types of mathematical proofs: those that explain why what they prove is true and those that merely prove theorems without explaining why they are true. This way of framing the issue neglects the possibility of mathematical explanations that are not proofs at all. This paper addresses what it would take for a non-proof to explain. The paper focuses on (...) a particular example of an explanatory non-proof: an argument that mathematicians regard as explaining why a given theorem holds regarding the derivative of an infinite sum of differentiable functions. The paper contrasts this explanatory non-proof with various non-explanatory proofs of the same theorem. The paper offers an account of what makes the given non-proof explanatory. This account is motivated by investigating the difficulties that arise when we try to extend Mark Steiner’s influential account of explanatory proofs to cover this explanatory non-proof. (shrink)

In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the (...) distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education. (shrink)