Skip to main content
SpringerLink
Log in

Visualisation and proof: a brief survey of philosophical perspectives

  • Original article
  • Published:
ZDM Aims and scope Submit manuscript

Abstract

The contribution of visualisation to mathematics and to mathematics education raises a number of questions of an epistemological nature. This paper is a brief survey of the ways in which visualisation is discussed in the literature on the philosophy of mathematics. The survey is not exhaustive, but pays special attention to the ways in which visualisation is thought to be useful to some aspects of mathematical proof, in particular the ones connected with explanation and justification.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Barwise, J., & Etchemendy, J. (1991). Visual information and valid reasoning. In: W. Zimmerman, & S. Cunningham (Eds.), Visualisation in teaching and learning mathematics (pp. 9–24). Washington, DC: MAA.

    Google Scholar 

  • Barwise, J., & Etchemendy, J. (1996). Heterogeneous logic. In: G. Allwain, & J. Barwise (Eds.), Logical reasoning with diagrams (pp. 179–201). New York: Oxford University Press.

    Google Scholar 

  • Biehler, R., Scholz, R. W., Strässer, R., & Winkelmann, B. (Eds.). (1994). Didactics of mathematics as a scientific discipline. Dordrecht: Kluwer Academic Publishers. .

  • Borwein, P., & Jörgenson, L. (2001). Visible structures in number theory. American Mathematical Monthly, 108, 897–911

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Brown, J. R. (1999). Philosophy of mathematics: The world of proofs and pictures. New York: Routledge.

    Google Scholar 

  • Casselman, B. (2000). Pictures and proofs. Notices of the AMS, 47, 1257–1266.

    Google Scholar 

  • Davis, P. J. (1993). Visual theorems, Educational Studies in Mathematics, 24, 333–344.

    Article  Google Scholar 

  • Folina, J. (1999). Pictures, proofs and ‘mathematical practice’: Reply to James Robert Brown. British Journal for the Philosophy of Science, 50, 425–429.

    Article  Google Scholar 

  • Francis, G. (1996). Mathematical visualization: Standing at the crossroads. Retrieved July 20, 2006, from http://www.oldweb.cecm.sfu.ca/projects/PhilVisMath/vis96panel.html .

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

    Google Scholar 

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

    Article  Google Scholar 

  • Giaquinto, M. (2005). From symmetry perception to basic geometry. In: P. Mancosu, K. F. Jørgensen, & S. A. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 31–55). Synthese Library, Vol. 327. Dordrecht: Springer.

  • Hanna, G. (1983). Rigorous proof in mathematics education. Curriculum Series/48, Toronto: OISE Press.

  • Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1–2), 5–23.

    Article  Google Scholar 

  • Jones, K., Gutiérrez, Á., & Mariotti, M.A. (Guest Editors). (2000). Proof in dynamic geometry environments. Special issue of Educational Studies in Mathematics, 44(1–2).

  • Mancosu, P. (2005). Visualization in logic and mathematics. In: P. Mancosu, K.F. Jørgensen, & S.A. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 13–30). Synthese Library, Vol. 327. Dordrecht: Springer.

  • Nelson, R. (1993). Proofs without words. Washington, DC: MAA.

    Google Scholar 

  • Palais, R. S. (1999). The visualization of mathematics: Toward a mathematical exploratorium. Notices of the AMS, 46(6), 647–658.

    Google Scholar 

  • Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics: Emergence from psychology. In: Á. Gutiérrez, & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 205–235). Dordrecht: Sense Publishers.

    Google Scholar 

  • Tall, D. (2002). Differing modes of proof and belief in mathematics. In International conference on mathematics: Understanding proving and proving to understand (pp. 91–107). Taipei: National Taiwan Normal University. See also http://www.davidtall.com/.

  • Tufte, E. (1983). The visual display of quantitative information. Cheshire, CT: Graphic Press.

    Google Scholar 

  • Tufte, E. (1990). Envisioning information. Cheshire, CT: Graphic Press.

    Google Scholar 

  • Tufte, E. (1997). Visual explanations. Cheshire, CT: Graphic Press.

    Google Scholar 

Download references

Acknowledgments

Preparation of this paper was supported in part by the Social Sciences and Humanities Research Council of Canada. Parts of this paper appeared previously in Hanna (2000).

Author information

Authors and Affiliations

  1. Department of Curriculum, Teaching and Learning, Ontario Institute for Studies in Education of the University of Toronto, Toronto, ON, Canada, M5S 1V6

    Gila Hanna

  2. Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada, V5A 1S6

    Nathan Sidoli

Authors
  1. Gila Hanna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nathan Sidoli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gila Hanna.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hanna, G., Sidoli, N. Visualisation and proof: a brief survey of philosophical perspectives. ZDM Mathematics Education 39, 73–78 (2007). https://doi.org/10.1007/s11858-006-0005-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-006-0005-0

Keywords

Search

Navigation