Advances in Experimental Philosophy of Logic and Mathematics

Andrew Aberdein, Matthew Inglis, Andrew Aberdein, Matthew Inglis

• Publisher: Bloomsbury Academic
• Year: 2019
• Print ISBN:
  9781350217959
• eBook ISBN:
  9781350039032
• Edition: 1
• Topic: Philosophy
• Subtopic: Logic in Philosophy

This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics.

Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, interviews, and data-mining. Distinct themes emerge that reflect recent developments in the field, such as issues concerning the logic of conditionals and the role played by visual elements in some mathematical proofs.

Featuring leading figures from experimental philosophy and the fields of philosophy of logic and mathematics, this collection reveals that empirical work in these disciplines has been quietly thriving for some time and stresses the importance of collaboration between philosophers and researchers in mathematics education and mathematical cognition.

1

Introduction

Andrew Aberdein and Matthew Inglis

There has been very little overt discussion on the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to logic or the philosophy of mathematics is impractical or unavailing. That this would be a mistake is exhibited by at least three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such as the psychology of reasoning and mathematics education. Before turning to the specific contribution that we hope this book will make, we will offer a snapshot of each trend and address how they intersect with some of the standard criticisms of experimental philosophy. First, although experimental philosophy is often thought of as a twenty-first-century phenomenon primarily focused on questions in ethics and epistemology, it has some important anticipations in earlier projects in the philosophy of logic. The most significant is the work of Arne Næss and the Oslo Group (Næss, 1938, 1959, 1982; Tönnessen, 1951). For instance, Ingemund Gullvåg argued that to understand the meaning of a word such as “truth,” it was “hardly sufficient that a single person registers his own reactions to this or that sentence, or makes pronouncements based on intuitions” (Gullvåg, 1955: 343). Instead, the Oslo group argued, systematic empirical investigations were required. The connections between the “empirical semantics” developed by the Oslo group and experimental philosophy have now begun to be made explicit by historians of philosophy and further developed by a new generation of researchers (Murphy, 2014; Barnard and Ulatowski, 2016; Chapman, 2018). This productive connection between the empirical methods of two different generations is continued in Barnard and Ulatowski’s chapter in the present volume, discussed in greater detail below.

Secondly, in recent decades there has been a “practice turn” in the philosophy of mathematics, focusing on how mathematical research is actually conducted, rather than on the search for foundations for mathematics (Van Kerkhove and Van Bendegem, 2007; Mancosu, 2008). This has naturally led to an interest in empirical data about mathematical practice, a program dubbed “Empirical Philosophy of Mathematics” by some of its practitioners (Buldt et al., 2008; Löwe et al., 2010; Pantsar, 2015). There are several distinct axes along which the connections between the philosophy of mathematical practice and empirical work have been drawn. A significant body of work applies cognitive science research on mathematical reasoning to philosophical questions (Pease et al., 2013). This includes work on the status of mathematical knowledge (Cappelletti and Giardino, 2007; Pantsar, 2014); on the symbol systems of mathematics (De Cruz and De Smedt, 2013; Dutilh Novaes, 2013; Marghetis and Núñez, 2013); and on the role of diagrams and visualization in mathematics (Giaquinto, 2007; Hamami and Mumma, 2013). Moreover, modern mathematicians increasingly employ online tools for collaboration. This produces a considerable amount of potential data for researchers interested in mathematical practice, giving rise to another strategy for the investigation of that practice (Martin and Pease, 2013; Martin, 2015; Pease et al., 2017). Although logical practice may have received less attention than its mathematical counterpart, some researchers in the philosophy of logic have pursued a practice turn of their own, modeled on that in the philosophy of mathematics (Dutilh Novaes, 2012). As with its sister program in philosophy of mathematics, advocates of the philosophy of logical practice stress that too much attention has been paid to foundational issues at the expense of philosophical questions that arise elsewhere, such as in the application of logic to artificial intelligence, game theory, linguistics, and other disciplines. For example, the burgeoning research program of “argumentation mining,” which applies corpus-based techniques to extract and analyze arguments across large bodies of text, may be seen as a logical counterpart to the use of big data techniques in analysis of mathematical practice (Moens, 2018).

Thirdly, there is a growing awareness of how much research in adjacent disciplines has anticipated the research questions of experimental philosophy of logic and mathematics. Philosophers of logic have an extensive body of research on the psychology of reasoning to draw upon (Johnson-Laird, 2006). Lately, some work in the intersection of philosophy of logic and psychology of reasoning has made the relationship to experimental philosophy explicit (Pfeifer, 2012; Pfeifer and Douven, 2014; Ripley, 2016). There is also a substantial research tradition in mathematics education that addresses questions of immediate relevance to the philosophy of mathematical practice (Heinze, 2010; Weber et al., 2014; Weber and Mejía-Ramos, 2015; Alcock et al., 2016). Understanding mathematical practice is important for education researchers for at least two reasons. First, understanding the behavior of expert mathematicians helps to decide what the purpose of a mathematics curriculum should be. If a particular activity is highly valued in expert mathematical practice then this perhaps provides a reason for mathematics students to be exposed to some appropriate version of it (see, for example, Ball and Bass, 2000; Harel and Sowder, 2007; Lampert, 1990; Weber et al., 2014). Second, studying the in-the-moment strategies adopted by expert practitioners (in any domain) might provide suggestions for how to develop interventions that assist learners to develop expertise. An example of this approach can be found in the work of Alcock, Hodds, Roy, and Inglis (2015). They studied the reading behavior of research mathematicians, and used these insights to develop training materials that encouraged undergraduates to adopt similar strategies. These training materials significantly increased the amount students learned from reading a mathematical text.

In addition, there is now an emerging tradition of interdisciplinary work, applying quantitative techniques to address traditionally philosophical questions, such as mathematical aesthetics (Inglis and Aberdein, 2015, 2016). Some of this work has been presented as an enquiry into “mathematical cultures” (Löwe, 2016; Larvor, 2016). Likewise, Reuben Hersh, one of the forerunners of the practice turn, has lately called for “a unified, distinct scholarly activity of mathematics studies: the study of mathematical activity and behavior” (Hersh, 2017: 335). We regard the present volume as, in part, a contribution to the integrative work required for this project.

The advent of experimental philosophy has not been without controversy and has provoked a salutary debate on the proper methods of philosophical enquiry. One of the most prominent critiques is the “expertise defense” of traditional philosophical practice (Nado, 2014; Mizrahi, 2015). This maintains that surveys of nonphilosophers have limited bearing on the arguments of philosophers since, as experts, philosophers can be expected to be immune from the errors and biases exhibited by nonexperts. This debate has given rise to a substantial literature. However, the experimental philosophies of mathematics and logic seem to have ready responses to the expertise defense. Many studies of mathematical practice focus on professional mathematicians, placing the expertise of the participants essentially beyond dispute. Nonetheless, this is not universally true; for instance, some philosophers (e.g., De Cruz, 2016) have used results from the numerical cognition literature to draw conclusions about the ontology of natural numbers. Participants in numerical cognition studies include nonmathematical adults, children, and even nonhuman animals.

An important difference between mainstream experimental philosophy and work focused on mathematics is that studies in the latter tradition typically ask their participants—be they mathematicians, children, or animals—about mathematics, not about philosophy. (This is just as well, for in Hersh’s famous formulation, “the typical working mathematician is a platonist on weekdays and a formalist on Sundays” (Hersh, 1979: 32). Such insouciance would not bode well for the resolution of philosophical dilemmas.) In this respect experimental philosophy of mathematics is similar to psychological work on reasoning relevant to debates in the philosophy of logic. Here too participants are typically drawn from a more general population. (Although there clearly is such a thing as logical expertise; for a start, people can be trained to be better at logical reasoning (Attridge et al., 2016).) Just as mathematicians/children/fish are asked about mathematics not philosophy, participants in reasoning studies are asked object-level questions about everyday reasoning, not specialized questions about logical hypotheses that might predict or explain such reasoning. On this basis David Ripley has argued that these studies are better placed to answer the expertise objection than studies relevant to debates in ethics or epistemology (Ripley, 2016).

Nonetheless, there is a substantial body of psychological research that reveals a divergence between best practice in reasoning (at least, as defined by logicians) and how lay people actually reas...

Table of contents

• Cover
• Half-Title
• Series
• Title
• Contents
• Notes on Contributors
• 1 Introduction
• 2 Methodological Triangulation in Empirical Philosophy (of Mathematics)
• 3 Animal Cognition, Species Invariantism, and Mathematical Realism
• 4 The Beauty (?) of Mathematical Proofs
• 5 Can a Picture Prove a Theorem? Using Empirical Methods to Investigate Visual Proofs by Induction
• 6 An Empirical Study on the Admissibility of Graphical Inferences in Mathematical Proofs
• 7 Does Anyone Really Think That ⌈φ⌉ Is True If and Only If φ?;
• 8 New Foundations for Fuzzy Set Theory
• 9 What Isn’t Obvious about “Obvious”: A Data-Driven Approach to Philosophy of Logic
• 10 Philosophy and the Psychology of Conditional Reasoning
• 11 Folk Judgments about Conditional Excluded Middle
• Index
• Copyright