Cognitive Artifacts for Geometric Reasoning

Proceedings of the Logic and Cognition Workshop at ESSLLI 2012, 2012

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.

Proceedings of the 12th International Conference on the Evolution of Language (Evolang12), 2018

The aim of this paper is to employ the newly contextualised historiographical category of ''premodern algebra'' in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on ''geometrical algebra''. Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid’s geometrical proofs, the so-called ''semi-algebraic'' alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron’s practice, highlights the significance of contextualizing ''premodern algebra'', and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.

2007

Science in Context, 2016

ArgumentThe aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related toElem.II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic ...

Philosophia Mathematica, 2012

Not simply set out in accompaniment of the Greek geometrical text, the diagram also is coaxed into existence manually (using straightedge and compasses) by commands in the text. The marks that a diligent reader thus sequentially produces typically sum, however, to a figure more complex than the provided one and also not (as it is) artful for being synoptically instructive. To provide a figure artfully is to balance multiple desiderata, interlocking the timelessness of insight with the temporality of construction. Our account of the diagram complements those of Manders and Macbeth by more strongly emphasizing practical synthesis. † We especially thank Catherine Legg, Koji Tanaka and Zach Weber for early critical input, Max Cresswell for his questions, Danielle Macbeth for her comments on an advanced draft, and two referees, John Mumma and Jessica Carter, for diligence, insight, and patience. Thanks also to Stephen Sharp.

Why are Geometric Proofs (Usually) “Non-Visual”? We asked this question as a way to explore the similarities and differences between diagrams and text (visual thinking versus language thinking). Traditional text-based proofs are considered (by many to be) more rigorous than diagrams alone. In this paper we focus on human perceptual-cognitive characteristics that may encourage textual modes for proofs because of the ergonomic affordances of text relative to diagrams. We sug- gest that visual-spatial perception of physical objects, where an object is perceived with greater acuity through foveal vision rather than peripheral vision, is similar to attention navigating a conceptual visual-spatial structure. We suggest that at- tention has foveal-like and peripheral-like characteristics and that textual modes appeal to what we refer to here as foveal-focal attention, an extension of prior work in focused attention.

Aristotle’s way of conceiving the relationship between mathematics and other branches of scientific knowledge is completely different from the way a contemporary scientist conceives it. This is one of the causes of the fact that we look at the mathematical passage we find in Aristotle’s works with the wrong expectation. We expect to find more or less stringent proofs, while for the most part Aristotle employs mere analogies. Indeed, this is the primary function of mathematics when employed in a philosophical context: not a demonstrative tool, but a purely analogical model. In the case of the geometrical examples discussed in this paper, the diagrams are not conceived as part of a formalized proof, but as a work in progress. Aristotle is not interested in the final diagram but in the construction viewed in its process of development; namely in the figure a geometer draws, and gradually modifies, when he tries to solve a problem. The way in which the geometer makes use of the elements of his diagram, and the relation between these elements and his inner state of knowledge is the real feature which interests Aristotle. His goal is to use analogy in order to give the reader an idea of the states of mind involved in a more general process of knowing.

Bulletin of the British Society for the History of …, 2005