Skip to main content
SpringerLink
Log in

Theorems that admit exceptions, including a remark on Toulmin

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

Abstract

It is a plausible assumption that proof-novices try to make sense of the meaning of mathematical proof out of the perspective of every day thinking. In every day thinking, however, the domain of objects to which a general statement refers is not completely and definitely determined. Thus the very notion of a “universally valid statement” is not as obvious as it might seem. The phenomenon of a statement with an indefinite domain of reference can also be found in the history of mathematics when authors spoke of “theorems that admit exceptions”. Without having understood and accepted the theoretical nature of the idea of a universally valid statement the logical distinctions between, for example an implication and its converse loose their meaning for the learner. This might explain some disappointing findings of empirical research. Following a proposal by Inglis, Mejia–Ramos and Simpson it is suggested that in modelling mathematical thinking in proof situations the full scheme of Toulmin should be used including qualifications and rebuttals rather than a reduced version as is frequently done.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–238). London: Hodder & Stoughton.

    Google Scholar 

  • Durand-Guerrier, V. (2008). Truth versus validity in proving in mathematics. ZDM—The International Journal on Mathematics Education (this issue)

  • Fraser, C. (1987). Joseph Louis Lagrange’s algebraic vision of the calculus. Historia Mathematica, 14, 38–53.

    Article  Google Scholar 

  • Heath, T. L. (ed. and transl.) (1956). The thirteen books of Euclid’s elements: 2nd edn revised with additions. New York: Dover

  • Hoyles, C., & Küchemann, D. (2002). Students’ understanding of logical implication. EducationalStudies in Mathematics, 51(3), 193–223.

    Google Scholar 

  • Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: the importance of qualification. Educational Studies in Mathematics, 66, 3–21.

    Article  Google Scholar 

  • Jahnke, H. N. (1987). Motive und Probleme der Arithmetisierung der Mathematik in der ersten Hälfte des 19. Jahrhunderts—Cauchys Analysis in der Sicht des Mathematikers Martin Ohm. Archive for History of Exact Sciences, 37, 101–182.

    Google Scholar 

  • Jahnke, H. N. (2003). Algebraic analysis in the 18th Century. In H. N. Jahnke (Ed.), A History of Analysis. American Mathematical Society & London Mathematical Society, 105–153

  • Jahnke, H. N.: (2005). A genetic approach to proof. In: Bosch, M. (ed.), Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education, Sant Feliu de Guíxols, 428–437

  • Jahnke, H. N. (2007). Proofs and Hypotheses. ZDM—The International Journal on Mathematics Education, 39(1–2), 79–86.

    Article  Google Scholar 

  • Knorr, W. R. (1986). The ancient tradition of geometric problems. New York: Dover.

    Google Scholar 

  • Küchemann, D. & Hoyles, C. (2005). Pupils’ awareness of structure on two number/algebra questions. In M. Bosch (Ed.), Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education, Sant Feliu de Guíxols (pp. 438–447)

  • Lagrange, J. L. (1797). Théorie des fonctions analytiques. Paris: Imprimerie de la République.

    Google Scholar 

  • Newton, I. (1669). De analysi per aequationes numero terminorum infinitas. Quoted according to the English translation in D. T. Whiteside (Ed.), The mathematical papers of Isaac Newton, Cambridge, Vol. II (pp. 206–247)

  • Ohm, M. (1823) Aufsätze aus dem Gebiete der höhern Mathematik. Berlin

  • Pedemonte, B. (2008). Argumentation and Algebraic Proof. ZDM—The International Journal on Mathematics Education (this issue)

  • Sørensen, H. K. (2005). Exceptions and counterexamples: Understanding Abel’s comment on Cauchy’s Theorem. Historia Mathematica, 32, 453–480.

    Article  Google Scholar 

  • Toulmin, S. (1974). The uses of argument. UK: Cambridge University Press, Paperback edition.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Duisburg-Essen, Universitätsstr. 12, 45117, Essen, Germany

    Hans Niels Jahnke

Authors
  1. Hans Niels Jahnke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hans Niels Jahnke.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jahnke, H.N. Theorems that admit exceptions, including a remark on Toulmin. ZDM Mathematics Education 40, 363–371 (2008). https://doi.org/10.1007/s11858-008-0097-9

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-008-0097-9

Keywords

Search

Navigation