Abstract
This chapter explores applications of concepts from argumentation theory to mathematical proofs. Note is taken of the various contexts in which proofs occur and of the various objectives they may serve. Examples of strategic maneuvering are discussed when surveying, in proofs, the four stages of argumentation distinguished by pragma-dialectics. Derailments of strategies (fallacies) are seen to encompass more than logical fallacies and to occur both in alleged proofs that are completely out of bounds and in alleged proofs that are at least mathematical arguments. These considerations lead to a dialectical and rhetorical view of proofs.
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Notes
- 1.
Elsewhere I argued that this objective can be seen as a (secondary) dialectical objective (Krabbe, 2004).
- 2.
Proofs that are not arguments include: immediate proofs (where there is no reasoning), formal proofs, and proofs in a context where there is no difference of opinion (see Sect. 11.3).
- 3.
A philosophical argument that motion is impossible.
- 4.
On this point the grammar of the term “proof” differs from that of the term “argument,” for an invalid argument is still an argument.
- 5.
Other argumentative functions of reasoning mentioned by Krabbe and Van Laar are the polemic functions (in eristic dialogue) and the directive functions (in negotiation).
- 6.
Mathematicians are a kind of Frenchmen: If one speaks to them, they will translate it into their language and then it is in no time something totally different.
- 7.
It is a common experience of math students that authors of textbooks sometimes exasperate their readers by leaving not the routine parts but the more difficult parts of a proof to them.
- 8.
Often: Gerolamo.
- 9.
Clearly, for a given P and l, there are precisely three possible suppositions: no parallel, exactly one, or more than one. It should be remarked that when one of these suppositions holds, the same can be shown to hold for all other points Q and lines m (Q not on m) of the plane.
- 10.
Moreover, even as a nonmathematical argument it could be a fallacy (argumentum ad verecundiam).
- 11.
Fermat’s Last Theorem: There are no positive integers x, y, z, n (n > 2) such that \({x}^{n} + {y}^{n} = {z}^{n}\).
- 12.
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Acknowledgements
The chapter was first presented at the NWO-conference on “Strategic Manoeuvring in Institutionalised Contexts,” University of Amsterdam, 26 October 2007 and has been previously published in Argumentation (2008) 22:453–468.
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Krabbe, E.C.W. (2013). Strategic Maneuvering in Mathematical Proofs. In: Aberdein, A., Dove, I. (eds) The Argument of Mathematics. Logic, Epistemology, and the Unity of Science, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6534-4_11
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