Abstract
The paper addresses the issue of human diagrammatic reasoning in the context of Euclidean geometry. It develops several philosophical categories which are useful for a description and an analysis of our experience while reasoning with diagrams. In particular, it draws the attention to the role of seeing-as; it analyzes its implications for proofs in Euclidean geometry and ventures the hypothesis that geometrical judgments are analytic and a priori, after all.
Similar content being viewed by others
References
Audi R.: Reasons/causes. In: Dancy, J., Sosa, E. (eds) A companion to epistemology, pp. 424–426. Blackwell, Oxford (1992)
Avigad, J., Dean, E., & Mumma J. (2008). A formal system for Euclid’s Elements. Review of Symbolic Logic, forthcoming.
Boghossian P.: Epistemic rules. Journal of Philosophy 105(9), 472–500 (2008)
Byrne O.: The first six books of the elements of Euclid. William Pickering, London (1847)
Coliva A.: Wright and McDowell on the content of experience and the justification of empirical beliefs. Lingua e Stile 36(1), 3–23 (2001)
Coliva A.: In difesa del contenuto non concettuale della percezione. In: Parrini, P. (eds) Conoscenza e cognizione, pp. 147–161. Guerini, Milano (2002)
Coliva A.: The finer-grained content of experience. A redefinition of its role within the debate between McDowell and nonconceptual theorists. Dialectica 57(1), 57–70 (2003)
Coliva, A. (2004/2006). I concetti. Roma: Carocci.
Euclid. (1959). Elements, published as Euclid’s Elements: All thirteen books complete in one volume (T. Heath, Trans., D. Densmore, Ed.). New York: Dover Books.
Giaquinto, M. (2007a). Visual thinking in mathematics. An epistemological study (esp. Chaps. 1–5). Oxford, OUP
Giaquinto M.: Visualizing in mathematics: An introduction. In: Mancosu, P. (eds) The philosophy of mathematical practice, pp. 32–58. Clarendon Press, Oxford (2007b)
Giardino, V. (ms.). Diagrams and manipulation practices: The margins for an empirical research.
Glasgow, J., Narayanan, N. H., Chandrasekaran, B. (eds): Diagrammatic reasoning: Cognitive and computational perspectives. MIT Press, Cambridge, MA (1995)
Macbeth D.: Diagrammatic reasoning in Euclid’s Elements”. In: Van Kerkhove, B., De Vuyst, J., Van Bendegem, J.-P. (eds) Philosophical perspectives on mathematical practice, pp. 235–267. College Publications, London (2010)
Macbeth, D. (2011). Diagrammatic reasoning in Frege’s Begriffßchrift. this issue, Vol. 2.
Malone M. E.: Kuhn reconstructed: Incommensurability without relativism. Studies in History and Philosophy of Science 24(1), 66–93 (1993)
Manders K.: Diagram-based geometric practice. In: Mancosu, P. (eds) The philosophy of mathematical practice, pp. 88–111. Clarendon Press, Oxford (2007a)
Manders K.: The Euclidean diagram. In: Mancosu, P. (eds) The philosophy of mathematical practice, pp. 112–183. Clarendon Press, Oxford (2007b)
McDowell J.: Mind and world. Harvard University Press, Cambridge, MA (1994)
Netz R.: The shaping of deduction in Greek mathematics. Cambridge University Press, Cambridge (1999)
Owens K., Outhred L.: The complexity of learning geometry and measurement. In: Gutiérrez, A., Boero, P. (eds) Handbook of research on the psychology of mathematics education, pp. 83–115. Sense Publishers, Rotterdam (2006)
Panza, M. (2011). The two-fold role of diagrams in Euclid’s plain geometry. this issue, Vol. 2.
Peacocke C.: A study of concepts. MIT Press, Cambridge, MA (1992)
Saito K.: A preliminary study in the critical assessment of diagrams in Greek mathematical works. Sciamus 7, 81–144 (2006)
Shin, S. J. (2008). Review of Giaquinto 2007a. Notre Dame Philosophical Reviews. Accessed July 29, 2008, from http://ndpr.nd.edu/review.cfm?id=13745.
Shin, S. J. (2011). The forgotten individual: Diagrammatic reasoning in mathematics. this issue, Vol. 2.
Wittgenstein L.: Philosophical investigations. Blackwell, Oxford (1953)
Wittgenstein L.: Remarks on the philosophy of psychology (Vols. 1, 2). Chicago University Press, Chicago (1980)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Coliva, A. Human diagrammatic reasoning and seeing-as. Synthese 186, 121–148 (2012). https://doi.org/10.1007/s11229-011-9982-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-011-9982-9