In what way does a mathematical proof depend on the notation used in its presentation? This paper examines this question by analysing the computational differences, in the sense of Larkin and Simon's 'Why a diagram is (sometimes) worth 10,000 words', between diagrammatic and sentential notations as a means for presenting geometric proofs. Wittgenstein takes up the question of mathematical notation and proof in Section III of Remarks on the Foundations of Mathematics. After discussing his observations on a proof's 'characteristic visual shape' in Section III with respect to arithmetical proofs, the paper shows how the notion of a characteristic visual shape illuminates the special effectiveness of diagrammatic notation in geometry.

Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object-wise, could underlie diagram-based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions.

The cognitive processing of spatial relations in Euclidean diagrams is central to the diagram-based geometric practice of Euclid's Elements. In this study, we investigate this processing through two dichotomies among spatial relations—metric vs topological and exact vs co-exact—introduced by Manders in his seminal epistemological analysis of Euclid's geometric practice. To this end, we carried out a two-part experiment where participants were asked to judge spatial relations in Euclidean diagrams in a visual half field task design. In the first part, we tested whether the processing of metric vs topological relations yielded the same hemispheric specialization as the processing of coordinate vs categorical relations. In the second part, we investigated the specific performance patterns for the processing of five pairs of exact/co-exact relations, where stimuli for the co-exact relations were divided into three categories depending on their distance from the exact case. Regarding the processing of metric vs topological relations, hemispheric differences were found for only a few of the stimuli used, which may indicate that other processing mechanisms might be at play. Regarding the processing of exact vs co-exact relations, results show that the level of agreement among participants in judging co-exact relations decreases with the distance from the exact case, and this for the five pairs of exact/co-exact relations tested. The philosophical implications of these empirical findings for the epistemological analysis of Euclid's diagram-based geometric practice are spelled out and discussed.

The infinite regress of Carroll's 'What the Tortoise said to Achilles' is interpreted as a problem in the epistemology of mathematical proof. An approach to the problem that is both diagrammatic and non-logical is presented with respect to a specific inference of elementary geometry.

In line with Ken Manders's seminal account of Euclid's diagrammatic method in the "The Euclidean Diagram," two proof systems with a diagrammatic syntax have been advanced as formalizations of the method FG and Eu. In a paper examining Eu, Nathaniel Miller, the creator of FG, has identified a variety of technical problems with the formal details of Eu. This response shows how the problems are remedied.

A key piece of data in understanding mathematics from the perspective of model-based reasoning is the use of diagrams to discover and to convey mathematical concepts and proofs. A paradigmatic example of such use is found in the classical demonstrations of elementary Euclidean geometry. These are invariably presented with accompanying geometric diagrams. Great progress has been made recently with respect to the precise role the diagrams plays in the demonstrations, so much so that diagrammatic formalizations of elementary Euclidean geometry have been developed. The purpose of this chapter is to introduce these formalizations to those who seek to understand mathematics from the perspective of model-based reasoning. The formalizations are named FG and Eu. Both are based on insights articulated in Ken Manders' seminal analysis of Euclid's diagrammatic proofs. The chapter presents these insights, the challenges involved in realizing them in a formalization, and the way FG and Eu each meet these challenges. The chapter closes with a discussion of how the formalizations can each be thought to prespecify a species of model-based reasoning.

Spatial relations are central to geometrical thinking. With respect to the classical elementary geometry of Euclid's Elements, a distinction between co-exact, or qualitative, and exact, or metric, spatial relations has recently been advanced as fundamental. We tested the universality of intuitions of these relations in a group of Senegalese and Dutch participants. Participants performed an odd-one-out task with stimuli that in all but one case display a particular spatial relation between geometric objects. As the exact/co-exact distinction is closely related to Kosslyn's categorical/ coordinate distinction, a set of stimuli for testing all four types was used. Results suggest that intuitions of all spatial relations tested are universal. Yet, culture has an important effect on performance: Dutch participants outperformed Senegalese participants and stimulus layouts affect the categorical and coordinate processing in different ways for the two groups. Differences in level of education within the Senegalese participants did not affect performance.

Euclidean diagrammatic reasoning refers to the diagrammatic inferential practice that originated in the geometrical proofs of Euclid's Elements . A seminal philosophical analysis of this practice by Manders (`The Euclidean diagram', 2008 ) has revealed that a systematic method of reasoning underlies the use of diagrams in Euclid's proofs, leading in turn to a logical analysis aiming to capture this method formally via proof systems. The central premise of this paper is that our understanding of Euclidean diagrammatic reasoning can be fruitfully advanced by confronting these logical and philosophical analyses with the field of cognitive science. Surprisingly, central aspects of the philosophical and logical analyses resonate in very natural ways with research topics in mathematical cognition, spatial cognition and the psychology of reasoning. The paper develops these connections, concentrating on four issues: (1) the cognitive origins of Euclidean diagrammatic reasoning, (2) the cognitive representations of spatial relations in Euclidean diagrams, (3) the nature of the cognitive processes and cognitive representations involved in Euclidean diagrammatic reasoning seen as a form of visuospatial relational reasoning and (4) the complexity of Euclidean diagrammatic reasoning for the human cognitive system. For each of these issues, our analysis generates concrete experiment proposals, opening thereby the way for further empirical investigations. The paper is thus a prolegomenon to a research program on Euclidean diagrammatic reasoning at the crossroads of logic, philosophy and cognitive science.

Modern formal accounts of the constructive nature of elementary geometry do not aim to capture the intuitive or concrete character of geometrical construction. In line with the general abstract approach of modern axiomatics, nothing is presumed of the objects that a geometric construction produces. This study explores the possibility of a formal account of geometric construction where the basic geometric objects are understood from the outset to possess certain spatial properties. The discussion is centered around Eu , a recently developed formal system of proof (presented in Mumma (Synthese 175:255–287, 2010 )) within which Euclid’s diagrammatic proofs can be represented.

This is the introduction to a Special issue of Synthese, Diagrams in Mathematics: History and Philosophy, edited by the authors together with Gabriel Sandu.

For more than two millennia, Euclid’s Elements set the standard for rigorous mathematical reasoning. The reasoning practice the text embodied is essentially diagrammatic, and this aspect of it has been captured formally in a logical system termed Eu [2, 3]. In this paper, we review empirical and theoretical works in mathematical cognition and the psychology of reasoning in the light of Eu. We argue that cognitive intuitions of Euclidean geometry might play a role in the interpretation of diagrams, and we show that neither the mental rules nor the mental models approaches to reasoning constitutes by itself a good candidate for investigating geometrical reasoning. We conclude that a cognitive framework for investigating geometrical reasoning empirically will have to account for both the interpretation of diagrams and the reasoning with diagrammatic information. The framework developed by Stenning and van Lambalgen [1] is a good candidate for this purpose.

Résumé Rav (1999 et 2007) et Leitgeb (2009) défendent la thèse de l’autonomie des preuves informelles par rapport aux systèmes formels de preuve. Azzouni (2004, 2005, et 2009), au contraire développe une explication qui réduit les preuves informelles à un réseau de systèmes formels sous-jacents. L’objectif principal de cet article est de démontrer la possibilité d’une position tierce médiane mettant en avant une explication quasi formelle de la méthode de preuve dans les Éléments . L’explication est quasi formelle, plutôt que formelle, en ce qu’elle donne au contenu géométrique un rôle irréductible dans les preuves d’Euclide en ce que ce rôle est sujet à des contraintes formelles. Les inférences qui sont basées sur concepts géométriques ont une occurrence à l’intérieur d’un cadre formel précisément défini.

Ray (1999, 2007) and Leitgeb (2009) argue for the autonomy of informal proofs from formal systems of proof. In contrast, Azzouni (2004, 2005 and 2009) develops an account which reduces informal proofs to a network of underlying formal systems. The general aim of this paper is to demonstrate the possibility of a third, middle position with a quasi-formal account of Euclid's proof method in the Elements. The account is quasi-formal, rather than simply formal, in that it gives geometric content an irreducible role in Euclid's proofs. It is quasi-formal, rather than simply informal, in that this role is subject to formal constraints. Inferences which are based on geometric concepts occur within a precisely defined formal framework.

The paper presents an account of the diagrammatic arguments of Euclid’s Elements as an example of a proof checking process that is guided by specific mathematical concepts rather than universal, abstract logical rules.

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