Abstract
Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give a natural explanation of Saccheri’s proofs as well as standard geometric proofs and even number-theoretic proofs.
Similar content being viewed by others
References
Brown J. (1997) Proofs and pictures. British Journal for Philosophy of Science 48: 161–180
Brown J. (1999) Philosophy of mathematics: An introduction to the world of proofs and pictures. Routledge, London
Giaquinto M. (1992) Visualizing as a means of geometrical discovery. Mind and Language 7: 382–401
Giaquinto M. (1993) Diagrams: Socrates and Meno’s slave. International Journal of Philosophical Studies 1: 81–97
Giaquinto M. (1994) Epistemology of visual thinking in elementary real analysis. British Journal for Philosophy of Science 45: 789–813
Greaves M. (2002) The philosophical status of diagrams. CSLI Publications, Stanford
Heath T. (1956) The thirteen books of Euclid’s Elements. Dover, New York
Hilbert D. (1921) The foundations of geometry (2nd ed). Open Court, Chicago
Lomas D. (2002) What perception is doing, and what it is not doing, in mathematical reasoning. British Journal for Philosophy of Science 53: 205–223
Mueller I. (1969) Euclid’s elements and the axiomatic method. British Journal for Philosophy of Science 20: 289–309
Nagel, E. (1979). The formation of modern conceptions of formal logic in the development of geometry. In Teleology Revisited (pp. 195–259). New York: Columbia University Press.
Nelsen, R. (1993). Proofs without words. Washington, DC: Mathematical Association of America.
Plato (1961). Meno. In E. Hamilton & H. Cairns (Eds.) The collected dialogues of Plato (pp. 353–384). Princeton: Princeton University Press.
Saccheri G. (1920) Euclides vindicatus (Trans Halsted). Open Court Publishing, Chicago
Seidenberg A. (1975) Did Euclid’s Elements, Book I, develop geometry axiomatically?. Archive for History of Exact Science 14: 263–295
Seidenberg A. (1977) The origin of mathematics. Archive for History of Exact Science 18: 301–342
Sherry D. (1999) Construction and reductio proof. Kant-Studien 90: 23–39
Tennant N. (1986) The withering away of formal semantics?. Mind and Language 1: 302–318
Wittgenstein L. (1961) Tractatus logico-philosophicus. Routledge and Kegan Paul, London
Wittgenstein L. (1983) Remarks on the foundations of mathematics (3rd ed). MIT Press, Cambridge, MA
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sherry, D. The Role of Diagrams in Mathematical Arguments. Found Sci 14, 59–74 (2009). https://doi.org/10.1007/s10699-008-9147-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10699-008-9147-6