Department of Cognitive Science
The epistemology of geometry I: The problem of exactness
A G Newstead (firstname.lastname@example.org)
James Franklin (email@example.com)School of Mathematics and Statistics, Sydney
AbstractWe show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept.
Citation details for this article:Newstead, A., Franklin, J. (2010). The Epistemology of Geometry I: The Problem of Exactness. In W. Christensen, E. Schier, and J. Sutton (Eds.), ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science (pp. 254-260). Sydney: Macquarie Centre for Cognitive Science.
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- Armstrong, D.M. (1997). A World of States of Affairs. Cambridge University Press.
- Benacerraf, P. (1973). Mathematical Truth. In P. Benacerraf & H. Putnam (Eds.), The Philosophy of Mathematics. Harvard University Press.
- Cassam, Q. 2007. The Possibility of Knowledge. Oxford: Clarendon Press.
- Copleston, F. (2003). History of Western Philosophy I, Ancient Greece and Rome. New York: Continuum Press (reprinted from 1946).
- Cornford, F. (trans.), (1957). Plato's Theory of Knowledge: The Theateatus and Sophist of Plato. New York: Macmillan.
- Dehaene, S., V. Izard, P. Pica, and E. Spelke, (2006). Core Knowledge of Geometry in an Amazonian Indigene Group. Science, 311, 381-384.
- Field, H. (1989). Realism, Mathematics, and Modality. Oxford, England: Blackwell.
- Fine, G. (2003). Plato and Aristotle on Forms as Causes. In Plato on Knowledge and Forms. Oxford: Clarendon Press.
- Franklin, J. (2009). Aristotelian realism. In J. Wood & A. Irvine (Eds.), Handbook of Philosophy of Mathematics. North Holland Elsevier Press.
- Giaquinto, M. (2007). Visual Thinking in Mathematics. Oxford University Press.
- Kitcher, P. (1984). Mathematical Knowledge. Oxford: Oxford University Press.
- Mulligan, J., Prescott, A., & Mitchelmore, M. C. (2004). Children's development of structure in early mathematics. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 393-400). Bergen, Norway: Program Committee.
- Mulligan, J. Young Children's Difficulties in Mathematics Learning. Macquarie University Presentation, Macquarie University Centre for Research in Mathematics and Science Education, Retrieved from: http://www.cheri.com.au/PDF_Files/Assoc%20Prof%20Joanne%20Mulligan.pdf.
- Millar, S. (2002). Understanding and Representing Space. Oxford: Clarendon Press.
- Norman, J. (2006). After Euclid: Visual Reasoning and the Epistemology of Diagrams. Stanford: CSLI Publications.
- Pessoa, L., E. Thompson, and A. Noe,(1998). Finding Out about Filling In; A guide to Perceptual Completion for Visual Science and the Philosophy of Perception. Behavioral and Brain Sciences, 21, 723-802.
- Pettigrew, R. (2009). Aristotle and the subject matter of geometry. Phronesis, 54, 239-260.
- Putnam, H. (1971). Philosophy of Logic. New York: Harper.
- Rosen, G. (2009). Abstract Objects. In E. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), retrieved from: http://plato.stanford.edu/archives/fall2009/entries/abstract-objects/.
- Shapiro, S. (2000). Thinking About Mathematics. Oxford University Press.
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