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Truth versus validity in mathematical proof

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Abstract

In mathematics education, it is often said that mathematical statements are necessarily either true or false. It is also well known that this idea presents a great deal of difficulty for many students. Many authors as well as researchers in psychology and mathematics education emphasize the difference between common sense and mathematical logic. In this paper, we provide both epistemological and didactic arguments to reconsider this point of view, taking into account the distinction made in logic between truth and validity on one hand, and syntax and semantics on the other. In the first part, we provide epistemological arguments showing that a central concern for logicians working with a semantic approach has been finding an appropriate distance between common sense and their formal systems. In the second part, we turn from these epistemological considerations to a didactic analysis. Supported by empirical results, we argue for the relevance of the distinction and the relationship between truth and validity in mathematical proof for mathematics education.

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Notes

  1. In logic, a proposition is a linguistic entity that is either true or false.

  2. A syllogism has two premises and a conclusion; each premise is a proposition with a subject term and a predicate term (an attribute); the middle term occurs twice in the premises. It does not occur in the conclusion. Its position determines four figures. For example, in the first figure, the middle term is once the predicate, once the subject; in the second figure, it is twice the predicate.

  3. A very clear presentation of Aristotle’s Syllogistic can be found in Lukasiewicz (1951), who presents this text as an introduction to formal logic.

  4. English translation “On the scientific justification of a conceptual notation” in Frege 1972, pp. 83–90.

  5. The question of knowing which is the right notion of implication is discussed in (Durand-Guerrier 2003).

  6. There are obviously many others philosophical matters raised in this treatise. Here, I propose a reading based on a purely didactic perspective (Durand-Guerrier 2006).

  7. A contradiction take the truth-value “false” for every distribution.

  8. This is developed in (Durand-Guerrier 2006).

  9. In French, this expression means that you do something in order to solve a specific problem.

  10. see Sect. 2.1.

  11. http://www.mathkang.org/concours/kangsansf.html.

  12. The method of Copi (1954) is briefly described in the Appendix.

  13. This example is developed in (Durand-Guerrier 2005, pp. 128–139). Notice that we use “general statements,” where Jahnke (2008) use “closed general statements.”

  14. This example is developed in French in (Durand-Guerrier and Arsac 2003); in English in (Durand-Guerrier and Arsac 2005) and in (Durand-Guerrier 2004).

  15. For a polynomial with power one, the derivative is a constant; for a polynomial with power two, we have 2c = b + a.

  16. For functions x 2 and x 3 on the interval [0;1], the number c must be \( \frac{1}{2} \) and \( \sqrt {\frac{1}{3}}, \) respectively.

  17. Our translation.

  18. On this question, see also Epp (2003).

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Acknowledgments

We wish to thank the reviewers for their advice and Jonathan Simon for reading over our paper and improving our English. Of course, we keep the entire responsibility for what is written in this paper.

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  1. LEPS, Laboratoire d’étude du Phénomène Scientifique (Laboratory for Studying the Scientific Phenomenon), EA 4148, Université de Lyon; Université Lyon 1, 38 bd Niels Bohr, 69622, Villeurbanne Cedex, France

    Viviane Durand-Guerrier

  2. Institut Universitaire de Formation des maîtres (Teacher Training Institute), Université Lyon 1, 5 rue Anselme, 69004, Lyon, France

    Viviane Durand-Guerrier

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Appendix

Appendix

1.1 Natural deduction in predicate calculus

Natural deduction systems provide a theoretical framework that reflects quite well the way that mathematicians reason. These systems provide rules for the elimination and introduction of connectors and quantifiers. The first such system was due to Gentzen (1935, 1955), but it has since been modified by both Quine (1950) and Copi (1954). According to Prawitz (1965) “Because of their close correspondence to procedures common in informal reasoning, systems of natural deduction have often been used in textbooks for pedagogical purposes.” (ibid. p. 103). Besides classical rules for the introduction and elimination of propositional connectors, we find four rules for the elimination and introduction of quantifiers in one-place predicate formulae, accompanied by two restriction rules (Copi 1954, 2nd edition, 1965, pp. 79–83). In the way that logicians generally do, Copi uses a horizontal line between two statements to indicate a deduction (see Fig. 1). Besides these four rules and their two restrictions, in case of two-place (or more) predicate, it is necessary to introduce a third restriction rule: U.G. can be applied provided that fa contains no individual symbol introduced by E.I. (Copi 1954, 2nd edition, 1965, p. 112). As a consequence, if w has been introduced by applying E.I. after a was introduced by applying U.I., then U.G. cannot be applied to faw; it is necessary first to apply E.G. By combining the introduction and elimination of connectors and quantifiers, Copi’s system provides rules that on one hand allow local control of validity by analyzing deduction step by step, and on the other hand, indicate, by paying attention to change in logical status for letters, when global control of validity is required.

Fig. 1
figure 1

The four rules for the introduction and elimination of quantifiers (according to Copi 1954, 2nd edition, 1965, pp. 80–82)

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A more detailed presentation of this framework and an example of the way we use it are developed in (Durand-Guerrier 2005, pp. 163–168).

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Durand-Guerrier, V. Truth versus validity in mathematical proof. ZDM Mathematics Education 40, 373–384 (2008). https://doi.org/10.1007/s11858-008-0098-8

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