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Journal of Logic, Language and Information

, Volume 22, Issue 4, pp 421–448 | Cite as

Prolegomena to a Cognitive Investigation of Euclidean Diagrammatic Reasoning

  • Yacin HamamiEmail author
  • John Mumma
Article

Abstract

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.

Keywords

Diagrammatic reasoning Euclidean geometry Mathematical cognition Visuospatial relational reasoning Categorical and coordinate spatial relations Relational complexity 

Notes

Acknowledgments

Parts of this paper have been presented at the Logic and Cognition 2012 Conference in Poznań (Poland, May 2012), the International Conference on Thinking in London (UK, July 2012) and the Logic and Cognition Workshop at ESSLLI in Opole (Poland, August 2012). We would like to thank the audiences of these events for helpful feedback and comments, in particular Markus Knauff and Michiel van Lambalgen for stimulating discussions. We are thankful to Jean Paul van Bendegem for providing comments on the whole manuscript. Finally, we are very grateful to Jakub Szymanik, Rineke Verbrugge and three anonymous referees for providing extensive comments that led to substantial improvements of the paper. Yacin Hamami acknowledges support from a doctoral fellowship of the Research Foundation Flanders (FWO).

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Centre for Logic and Philosophy of ScienceVrije Universiteit BrusselBrusselsBelgium
  2. 2.Department of PhilosophyCalifornia State University, San BernardinoSan BernardinoUSA

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