Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-05T13:58:30.463Z Has data issue: false hasContentIssue false
  • English
  • Français

Thinking through drawing

Diagram constructions as epistemic mediators in geometrical discovery

Published online by Cambridge University Press:  30 July 2013

Lorenzo Magnani
Lorenzo Magnani*
Affiliation:
Department of Humanities, Philosophy Section, Computational Philosophy Laboratory, University of Pavia, Pavia, Italy; e-mail: lmagnani@unipv.it
Get access Rights & Permissions [Opens in a new window]

Abstract

The concept of manipulative abduction is devoted to capture the role of action in many interesting cognitive situations: action provides otherwise unavailable information that enables the agent to solve problems by starting and performing a suitable abductive process of generation or selection of hypotheses. We observe that many external things, usually inert from an epistemological point of view, can be transformed into epistemic mediators. I will present some details derived from the history of the discovery of the non-Euclidean geometries that illustrate the relationships between strategies for anomaly resolution and visual thinking. Geometrical diagrams are external representations that play both a mirror role (to externalize rough mental models) and an unveiling role (as gateways to imaginary entities). I describe them as epistemic mediators able to perform various explanatory, non-explanatory, and instrumental abductive tasks (discovery of new properties or new propositions/hypotheses, provision of suitable sequences of models as able to convincingly verifying theorems, etc.). I am also convinced that they can be exploited and studied in everyday non-mathematical applications also to the aim of promoting new trends in artificial intelligence modeling of various aspects of hypothetical reasoning: finding routes, road signs, buildings maps, for example, in connection with various zooming effects of spatial reasoning. I also think that the cognitive activities of optical, mirror, and unveiling diagrams can be studied in other areas of manipulative and model-based reasoning, such as the ones involving creative, analogical, and spatial inferences, both in science and everyday situations so that this can extend the epistemological, computational, and the psychological theory.

Type
Articles
Information
The Knowledge Engineering Review , Volume 28 , Special Issue 3: Visualization, Visual Representation and Reasoning , September 2013 , pp. 303 - 326
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allwein, G., Barwise, J. (eds) 1996. Logical Reasoning with Diagrams. Oxford University Press.CrossRefGoogle Scholar
Anderson, M., Cheng, P., Haarslev, V. (eds) 2000. Theory and Application of Diagrams: First International Conference, Diagrams 2000. Springer.CrossRefGoogle Scholar
Brown, J. R. 1997. Proofs and pictures. Philosophy of Science 48, 161–180.Google Scholar
Brown, J. R. 1999. Philosophy of Mathematics. An Introduction to the World of Proofs and Pictures. Routledge.Google Scholar
Darden, L. 1991. Theory Change in Science: Strategies from Mendelian Genetics. Oxford University Press.CrossRefGoogle Scholar
Dossena, R., Magnani, L. 2007. Mathematics through diagrams: microscopes in non-standard and smooth analysis. In Model-Based Reasoning in Science, Technology, and Medicine, Magnani, L. & Li, P. (eds). Springer, 193–213.CrossRefGoogle Scholar
Gelertner, H. 1963. Realization of a geometry-theorem proving machine. In Computers and Thought, Feigenbaum E. A. & Feldman J. (eds). MacGraw Hill, 134–152.Google Scholar
Giaquinto, M. 1992. Visualizing as a means of geometrical discovery. Mind and Language 7, 381–401.CrossRefGoogle Scholar
Giaquinto, M. 1994. Epistemology of visual thinking in elementary real analysis. British Journal for the Philosophy of Science 45, 789–813.CrossRefGoogle Scholar
Giaquinto, M. 2007. Visual Thinking in Mathematics: An Epistemological Study. Oxford University Press.CrossRefGoogle Scholar
Glasgow, J., Narayanan, N. H., Chandrasekaran, B. (eds) 1995. Diagrammatic Reasoning: Cognitive and Computational Perspectives. AAAI Press.Google Scholar
Glasgow, J. I., Papadias, D. 1992. Computational imagery. Cognitive Science 16, 255–394.CrossRefGoogle Scholar
Greenberg, M. J. 1974. Euclidean and Non-Euclidean Geometries. Freeman and Company.Google Scholar
Hutchins, E. 1995. Cognition in the Wild. The MIT Press.Google Scholar
Hutchins, E. 2005. Material anchors for conceptual blends. Journal of Pragmatics 37, 1555–1577.CrossRefGoogle Scholar
Kant, I. 1929. Critique of Pure Reason. MacMillan, translated by N. Kemp Smith, originally published 1787, reprint 1998.Google Scholar
Kant, I. 1968. Inaugural dissertation on the forms and principles of the sensible and intelligible world (1770). In Kant. Selected Pre-Critical Writings, Kerferd, G. & Walford, D. (eds). Manchester University Press, 45–92. translated by G. B. Kerferd and D. E. Walford. Also translated by J. Handyside, in I. Kant, Kant's Inaugural Dissertation and Early Writings on Space. Open Court, Chicago, IL (1929) (pp. 35–85).Google Scholar
Kosslyn, S. M., Koenig, O. 1992. Wet Mind, the New Cognitive Neuroscience. Free Press.Google Scholar
Lambert, J. H. 1786. Theorie der Parallellinien. Magazin für die reine und angewandte Mathematik 2-3, 137–164, 325–358, written about 1766; posthumously published by J. Bernoulli.Google Scholar
Lindsay, R. K. 1994. Understanding diagrammatic demonstrations. In Proceedings of the 16th Annual Conference of the Cognitive Science Society, Ram, A. & Eiselt, K. (eds). Erlbaum, 572–576.Google Scholar
Lindsay, R. K. 1998. Using diagrams to understand geometry. Computational Intelligence 9(4), 343–345.CrossRefGoogle Scholar
Lindsay, R. K. 2000a. Playing with diagrams. In Diagrams 2000, Anderson, M., Cheng, P. & Haarslev, V. (eds). Springer, 300–313.Google Scholar
Lindsay, R. K. 2000b. Using spatial semantics to discover and verify diagrammatic demonstrations of geometric propositions. In Spatial Cognition. Proceedings of the 16th Annual Conference of the Cognitive Science Society, O'Nuallian, S. (ed.). John Benjamins, 199–212.Google Scholar
Lobachevsky, N. I. 1829–1830, 1835–1838. Zwei geometrische Abhandlungen, aus dem Russischen bersetzt, mit Anmerkungen und mit einer Biographie des Verfassers von Friedrich Engel. B. G. Teubner.Google Scholar
Lobachevsky, N. I. [1840] 1891. Geometrical Researches on the Theory of Parallels. University of Texas.Google Scholar
Lobachevsky, N. I. 1897. The “Introduction’’ to Lobachevsky's New Elements of Geometry. Transactions of Texas Academy, 2: 1–17, translated by G. B. Halsted. Originally published in N. I. Lobachevsky, Novye nachala geometrii, Uchonia sapiski Kasanskava Universiteta 3, 1835: 3–48.Google Scholar
Lobachevsky, N. I. 1929. Pangeometry or a summary of geometry founded upon a general and rigorous theory of parallels. In A Source Book in Mathematics, Smith, D. E. (ed.). McGraw Hill, 360–374.Google Scholar
Magnani, L. 2001a. Abduction, Reason, and Science. Processes of Discovery and Explanation. Kluwer Academic/Plenum Publishers.CrossRefGoogle Scholar
Magnani, L. 2001b. Philosophy and Geometry. Theoretical and Historical Issues. Kluwer Academic Publisher.CrossRefGoogle Scholar
Magnani, L. 2009a. Multimodal abduction in knowledge development. In International Workshop on Abductive and Inductive Knowledge Development. iJCAI2009, Preworkshop Proceedings, 21–26.Google Scholar
Magnani, L. 2009b. Abductive Cognition. The Epistemological and Eco-Cognitive Dimensions of Hypothetical Reasoning. Springer.CrossRefGoogle Scholar
Magnani, L. 2013. Is abduction ignorance-preserving? Conventions, models, and fictions in science. The Logic Journal of the IGPL, doi: 10.1093/jigpal/jzt012. First published online: April 4, 2013.CrossRefGoogle Scholar
Magnani, L., Bardone, E. 2008. Sharing representations and creating chances through cognitive niche construction. The role of affordances and abduction. In Communications and Discoveries from Multidisciplinary Data, Iwata, S., Oshawa, Y., Tsumoto, S., Zhong, N., Shi, Y. & Magnani, L. (eds). Springer, 3–40.CrossRefGoogle Scholar
Magnani, L., Dossena, R. 2005. Perceiving the infinite and the infinitesimal world: unveiling and optical diagrams and the construction of mathematical concepts. Foundations of Science 10, 7–23.CrossRefGoogle Scholar
Nersessian, N. J. 1992. How do scientists think? Capturing the dynamics of conceptual change in science. In Cognitive Models of Science. Minnesota Studies in the Philosophy of Science., Giere, R. N. (ed.). University of Minnesota Press, 3–44.Google Scholar
Nersessian, N. J. 1995. Should physicists preach what they practice? Constructive modeling in doing and learning physics. Science and Education 4, 203–226.CrossRefGoogle Scholar
Peirce, C. S. 1931–1958. Collected Papers of Charles Sanders Peirce. Harvard University Press, vols. 1–6, Hartshorne, C. and Weiss, P. (eds); vols. 7–8, Burks, A. W. (ed.).Google Scholar
Rosenfeld, B. A. 1988. A History of Non-Euclidean Geometry. Evolution of the Concept of Geometric Space. Springer.CrossRefGoogle Scholar
Saccheri, G. 1920. Euclides Vindicatus. Euclid Freed of Every Fleck. Open Court, translated by G. B. Halsted. Originally published as Euclides ab omni naevo vindicatus, Ex Typographia Pauli Antonii Montani, Mediolani (Milan), 1733.Google Scholar
Smith, D. E. [1925] 1958. History of Mathematics. Dover Publications, 2 vols.Google Scholar
Stroyan, K. D. 2005. Uniform continuity and rates of growth of meromorphic functions. In Contributions to Non-Standard Analysis, Luxemburg, W. J. & Robinson, A. (eds). North-Holland, 47–64.Google Scholar
Tall, D. 2001. Natural and formal infinities. Educational Studies in Mathematics 48, 199–238.CrossRefGoogle Scholar
Torretti, R. 1978. Philosophy of Geometry from Riemann to Poincaré. Reidel.CrossRefGoogle Scholar
Torretti, R. 2003. Review of L. Magnani, Philosophy and Geometry: Theoretical and Historical Issues, Dordrecht: Kluwer Academic Publishers (2001). Studies in History and Philosophy of Modern Physics 34b(1), 158–160.CrossRefGoogle Scholar
Trafton, J. G., Trickett, S. B., Mintz, F. E. 2005. Connecting internal and external representations: spatial transformations of scientific visualizations. Foundations of Science 10, 89–106.CrossRefGoogle Scholar
Zhang, J. 1997. The nature of external representations in problem solving. Cognitive Science 21(2), 179–217.CrossRefGoogle Scholar