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The Role of Diagrams in Mathematical Arguments

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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.

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References

  • Brown J. (1997) Proofs and pictures. British Journal for Philosophy of Science 48: 161–180

    Article  Google Scholar 

  • Brown J. (1999) Philosophy of mathematics: An introduction to the world of proofs and pictures. Routledge, London

    Google Scholar 

  • Giaquinto M. (1992) Visualizing as a means of geometrical discovery. Mind and Language 7: 382–401

    Article  Google Scholar 

  • Giaquinto M. (1993) Diagrams: Socrates and Meno’s slave. International Journal of Philosophical Studies 1: 81–97

    Article  Google Scholar 

  • Giaquinto M. (1994) Epistemology of visual thinking in elementary real analysis. British Journal for Philosophy of Science 45: 789–813

    Article  Google Scholar 

  • Greaves M. (2002) The philosophical status of diagrams. CSLI Publications, Stanford

    Google Scholar 

  • Heath T. (1956) The thirteen books of Euclid’s Elements. Dover, New York

    Google Scholar 

  • Hilbert D. (1921) The foundations of geometry (2nd ed). Open Court, Chicago

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Mueller I. (1969) Euclid’s elements and the axiomatic method. British Journal for Philosophy of Science 20: 289–309

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Seidenberg A. (1975) Did Euclid’s Elements, Book I, develop geometry axiomatically?. Archive for History of Exact Science 14: 263–295

    Article  Google Scholar 

  • Seidenberg A. (1977) The origin of mathematics. Archive for History of Exact Science 18: 301–342

    Google Scholar 

  • Sherry D. (1999) Construction and reductio proof. Kant-Studien 90: 23–39

    Article  Google Scholar 

  • Tennant N. (1986) The withering away of formal semantics?. Mind and Language 1: 302–318

    Article  Google Scholar 

  • Wittgenstein L. (1961) Tractatus logico-philosophicus. Routledge and Kegan Paul, London

    Google Scholar 

  • Wittgenstein L. (1983) Remarks on the foundations of mathematics (3rd ed). MIT Press, Cambridge, MA

    Google Scholar 

Download references

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  1. Northern Arizona University, Flagstaff, AZ, USA

    David Sherry

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  1. David Sherry
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Correspondence to David Sherry.

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Sherry, D. The Role of Diagrams in Mathematical Arguments. Found Sci 14, 59–74 (2009). https://doi.org/10.1007/s10699-008-9147-6

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