Abstract
In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.
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Notes
- 1.
Page numbers are to the anthologized version (Finocchiaro, 2005).
- 2.
What Finocchiaro calls ‘interpretation’ is often called ‘analysis’. I prefer the latter term and generally use it instead.
- 3.
I unceremoniously identify informal logic and argumentation theory and argumentation studies as a single entity—really a single kind of response to practical difficulties in theorizing about and teaching principles of argument appraisal/assessment. That the phrase informal logic is more often used by philosophers or that the phrase argumentation theory is more often used by researchers in communication studies is beside the point. The goal is the same: to produce a workable theory of argument appraisal.
- 4.
Where ‘formal logic’ is understood as a theory of entailment which may include both formal semantics and formal proof theory.
- 5.
- 6.
- 7.
Johnson nowhere, as far as I can tell, dismisses the use of informal logic in mathematics. Instead, his concern is to distinguish proof from argument-proper.
- 8.
If this assertion is too strong or too hasty, qualify it to read: The methods for assessing mathematical proofs resemble the methods for assessing non-mathematical arguments.
- 9.
The angles in the original triangle, ABC, are labeled simply A, B or C respectively. The other angles are labeled more fully, e.g. ∠ADC, to avoid ambiguity.
- 10.
One way to understand the mistake is that the argument treats the ‘f’ of f(x) to be capable of manipulation. Suppose f(x) = x 2. Then for any n, \(f(n) = f(-n)\). Then, dividing both sides by f, one gets \(n = -n\). Since sin is a function symbol, it isn’t itself manipulable. Instead, the function is manipulable only when it is given with an attending argument.
- 11.
James Franklin (1987, revised version reprinted in this collection) gives a brief sample of some of the acceptable non-deductive methods.
- 12.
In what follows I’ve left out the qualifier ‘gastronomic’ because, for the purposes of providing an example, it is unnecessary. Gastronomic Cannibalism is distinguished by physical anthropologists from Ritualistic Cannibalism solely in terms of purpose (Cáceres et al., 2007, 899).
- 13.
For a textbook treatment of this scheme, see (Walton, 2006, 112–4).
- 14.
The use of square brackets, ‘[’ and ‘]’, indicates that this material is either paraphrased from the original material or added to the material.
- 15.
The conclusion is clearly indicated by the phrase ‘suggests a clear case of gastronomic cannibalism’.
- 16.
For an illuminating philosophical discussion of the use of computers in the proof of the Four-Color Theorem see (Detlefsen and Luker, 1980).
- 17.
Whereas DNA proofs will remain defeasible, computer proofs can have formal verifications.
- 18.
For a different assessment using the tools of Toulmin diagrams, see (Aberdein, 2007). There, Aberdein reconstructs (a part of) the proof of the four-color theorem as an explicit Toulmin diagram. The use of Argumentation Schemes vs. Toulmin Diagrams shouldn’t be thought of as necessarily opposed. One may be able to capture all of the elements of a scheme in a diagram and vice versa.
- 19.
Below I suggest that proofs by Mathematical Induction have much in common with Argumentation Schemes. Perhaps, Mathematical Induction is, simply put, a mathematical argumentation scheme. Again, Poincaré’s views on mathematical induction could importantly prefigure this idea, see (Detlefsen, 1992, 1993).
- 20.
Heath’s footnote at (Euclid, 1956, 284–5).
- 21.
A distressing caveat: Euclid abbreviates his proofs, especially in the last step where the reasoning from a particular figure is generalized. This means that, strictly speaking, each of Euclid’s proofs is a proof sketch. This is different from the usual complaint about the incompleteness of Euclidean proofs. The usual complaint is that the diagram fills in tacit assumptions, like continuity, that should be made explicit, as in Pasch’s Axiom. I discuss proof sketches in more detail below.
- 22.
There is nothing in the conception of informal logic I accept that would preclude the informal logician from partaking in the fruits of the formal logician’s labor. The informal logician simply denies that the method by which mathematical (or really any) reasoning gets appraised is translation into formal language and comparison with accepted formal results.
- 23.
See Michael Malone’s argument that this is a theoretical as well as practical problem (Malone, 2003).
- 24.
Other possible mathematical argumentation schemes include: mathematical symmetry arguments and mathematical analogies, e.g., (Steiner, 1998, 48ff.).
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Acknowledgements
A prior version of this paper was published in Foundations of Science (2009), 14(1–2):137–152. It received careful and helpful criticism from Andrew Aberdein and David Sherry. I thank them both. Previous versions of this paper were presented in Las Vegas and Amsterdam. I thank the audiences for their helpful questions and comments.
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Dove, I.J. (2013). Towards a Theory of Mathematical Argument. 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_15
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