What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in 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)

We investigated whether mathematicians typically agree about the qualities of mathematical proofs. Between-mathematician consensus in proof appraisals is an implicit assumption of many arguments made by philosophers of mathematics, but to our knowledge the issue has not previously been empirically investigated. We asked a group of mathematicians to assess a specific proof on four dimensions, using the framework identified by Inglis and Aberdein (2015). We found widespread disagreement between our participants about the aesthetics, intricacy, precision and utility of the proof, (...) suggesting that a priori assumptions about the consistency of mathematical proof appraisals are unreasonable. (shrink)

In this chapter we use methods of corpus linguistics to investigate the ways in which mathematicians describe their work as explanatory in their research papers. We analyse use of the words explain/explanation (and various related words and expressions) in a large corpus of texts containing research papers in mathematics and in physical sciences, comparing this with their use in corpora of general, day-to-day English. We find that although mathematicians do use this family of words, such use is considerably less prevalent (...) in mathematics papers than in physics papers or in general English. Furthermore, we find that the proportion with which mathematicians use expressions related to ‘explaining why’ and ‘explaining how’ is significantly different to the equivalent proportion in physics and in general English. We discuss possible accounts for these differences. (shrink)

In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged it valid, (...) and (d) participants who judged the argument valid usually did not change their judgment when presented with a reason raised by other mathematicians for why the proof should be judged invalid. These findings suggest that, contrary to some claims in the literature, there is not a single standard of validity among contemporary mathematicians. (shrink)

Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects—those which visually resemble in some way the concepts they represent—offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative (...) relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd's hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance. (shrink)

Many of the methods commonly used to research mathematical practice, such as analyses of historical episodes or individual cases, are particularly well-suited to generating causal hypotheses, but less well-suited to testing causal hypotheses. In this paper we reflect on the contribution that the so-called hypothetico-deductive method, with a particular focus on experimental studies, can make to our understanding of mathematical practice. By way of illustration, we report an experiment that investigated how mathematicians attribute aesthetic properties to mathematical proofs. We demonstrate (...) that perceptions of the aesthetic properties of mathematical proofs are, in some cases at least, subject to social influence. Specifically, we show that mathematicians’ aesthetic judgements tend to conform to the judgements made by others. Pedagogical implications are discussed. (shrink)

Nonsymbolic comparison tasks are commonly used to index the acuity of an individual's Approximate Number System (ANS), a cognitive mechanism believed to be involved in the development of number skills. Here we asked whether the time that an individual spends observing numerical stimuli influences the precision of the resultant ANS representations. Contrary to standard computational models of the ANS, we found that the longer the stimulus was displayed, the more precise was the resultant representation. We propose an adaptation of the (...) standard model, and suggest that this finding has significant methodological implications for numerical cognition research. (shrink)

Mathematicians often conduct aesthetic judgements to evaluate mathematical objects such as equations or proofs. But is there a consensus about which mathematical objects are beautiful? We used a comparative judgement technique to measure aesthetic intuitions among British mathematicians, Chinese mathematicians, and British mathematics undergraduates, with the aim of assessing whether judgements of mathematical beauty are influenced by cultural differences or levels of expertise. We found aesthetic agreement both within and across these demographic groups. We conclude that judgements of mathematical beauty (...) are not strongly influenced by cultural difference, levels of expertise, and types of mathematical objects. Our findings contrast with recent studies that found mathematicians often disagree with each other about mathematical beauty. (shrink)

There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by 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. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book. (shrink)

The literature on mathematical explanation contains numerous examples of explanatory, and not so explanatory proofs. In this paper we report results of an empirical study aimed at investigating mathematicians’ notion of explanatoriness, and its relationship to accounts of mathematical explanation. Using a Comparative Judgement approach, we asked 38 mathematicians to assess the explanatory value of several proofs of the same proposition. We found an extremely high level of agreement among mathematicians, and some inconsistencies between their assessments and claims in the (...) literature regarding the explanatoriness of certain types of proofs. (shrink)

Two experiments are reported which investigate the factors that influence how persuaded mathematicians are by visual arguments. We demonstrate that if a visual argument is accompanied by a passage of text which describes the image, both research-active mathematicians and successful undergraduate mathematics students perceive it to be significantly more persuasive than if no text is given. We suggest that mathematicians’ epistemological concerns about supporting a claim using visual images are less prominent when the image is described in words. Finally we (...) suggest that empirical studies can make a useful contribution to our understanding of mathematical practice. (shrink)

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. (shrink)

"This book is interesting and well-written. The research methods were explained clearly and conclusions were summarized nicely. It is a relatively quick read at only 130 pages. Anyone who has been told, or who has told others, that mathematicians make better thinkers should read this book." MAA Reviews "The authors particularly attend to protecting positive correlations against the self-selection interpretation, merely that logical minds elect studying more mathematics. Here, one finds a stimulating survey of the systemic difficulties people have with (...) basic syllogisms and deductions." CHOICE connect "The authors particularly attend to protecting positive correlations against the self-selection interpretation, merely that logical minds elect studying more mathematics. Here, one finds a stimulating survey of the systemic difficulties people have with basic syllogisms and deductions." CHOICE Connect For centuries, educational policymakers have believed that studying mathematics is important, in part because it develops general thinking skills that are useful throughout life. This 'Theory of Formal Discipline' (TFD) has been used as a justification for mathematics education globally. Despite this, few empirical studies have directly investigated the issue, and those which have showed mixed results. Does Mathematical Study Develop Logical Thinking? describes a rigorous investigation of the TFD. It reviews the theory's history and prior research on the topic, followed by reports on a series of recent empirical studies. It argues that, contrary to the position held by sceptics, advanced mathematical study does develop certain general thinking skills, however these are much more restricted than those typically claimed by TFD proponents. Perfect for students, researchers and policymakers in education, further education and mathematics, this book provides much needed insight into the theory and practice of the foundations of modern educational policy. (shrink)

We examined a large set of conditional inference data compiled from several previous studies and asked three questions: How is normative performance related to intelligence? Does negative conclusion bias stem from Type 1 or Type 2 processing? Does implicit negation bias stem from Type 1 or Type 2 processing? Our analysis demonstrated that rejecting denial of the antecedent and affirmation of the consequent inferences was positively correlated with intelligence, while endorsing modus tollens inferences was not; that the occurrence of negative (...) conclusion bias was related to the extent of Type 2 processing; and that the occurrence of implicit negation bias was not related to the extent of Type 2 processing. We conclude that negative conclusion bias is, at least in part, a product of Type 2 processing, while implicit negation bias is not. (shrink)