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Evidence, proofs, and derivations

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Abstract

The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning.

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Notes

  1. This is sometimes referred to as Hilbert’s Thesis, although that name is more properly reserved for the narrower claim that every proof can be formalized as a derivation in first-order logic (Kahle 2019).

  2. This reflects what has been called Tait’s Maxim: ‘The notion of formal proof was invented to study the existence of proofs, not methods of proof’ (Baldwin 2013, 114).

  3. For example, for Saunders Mac Lane, ‘the test for the correctness of a proposed proof is by formal criteria and not by reference to the subject matter at issue’ (Mac Lane 1986, 378) and Thomas Hales characterizes formal proof as providing ‘a thorough verification of my own research that goes beyond what the traditional peer review process has been able to provide’ (Hales 2008, 1378).

  4. For a more protracted discussion of how these two models of reasoning are related, see (Pease and Aberdein 2011, 28 ff.). For an alternative account, see (Konstantinidou and Macagno 2013, 1070).

  5. The mathematical uses of Argument from Positive Consequences are also discussed, together with some other schemes not in Table 1, in work by Nikolaos Metaxas and colleagues (Metaxas 2015, 84; Metaxas et al. 2016, 387).

  6. For a more extensive discussion of mathematical uses of Scheme 2, see (Aberdein 2013a, 244).

  7. It may be objected that this results in a regress, since the software checking the derivation trace must itself be checked. However, it is what Hales has called ‘a rather manageable regress’ (Hales 2008, 1376). The kernel of such proof checking software is very carefully designed to be small enough and clear enough to be amenable to thorough human checking.

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Acknowledgements

I presented an earlier version of this paper at the interdisciplinary symposium on Mathematical Evidence and Argument held at the University of Bremen in 2017. I am grateful to the participants for their comments and particularly indebted to Christine Knipping and Eva Müller-Hill for their invitation and their hospitality in Bremen. I am also grateful to three anonymous referees for insightful and thorough comments.

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    Andrew Aberdein

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Aberdein, A. Evidence, proofs, and derivations. ZDM Mathematics Education 51, 825–834 (2019). https://doi.org/10.1007/s11858-019-01049-5

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