Abstract
Contemporary argumentation theory tends to steer away from traditional formal logic. In the case of argumentation theory applied to mathematics, though, it is proper for argumentation theory to revisit formal logic owing to the in-principle formalizability of mathematical arguments. Completely formal proofs of substantial mathematical arguments suffer from well-known problems. But practical formalizations of substantial mathematical results are now available, thanks to the help provided by modern automated reasoning systems. In-principle formalizability has become in-practice formalizability. Such efforts are a resource for argumentation theory applied to mathematics because topics that might be thought to be essentially informal reappear in the computer-assisted, formal setting, prompting a fresh appraisal.
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Notes
- 1.
It is not always the case that modus ponens is the only rule of inference available in a Hilbert-style system. In certain systems of modal logic, for example, one typically finds a rule of necessitation as part of a Hilbert-style formalism. But for classical propositional and predicate logic, as well as for others, it is standard to assume that in a Hilbert-style calculus modus ponens is the only rule of inference. The main feature of Hilbert-style calculi is that they have very few rules, placing the deductive burden of the formalism on its axioms, which are formulas, rather than rules of inference.
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- 9.
Avigad actually proposes methods—which correspond, for instance, to tactics in the Isabelle theorem prover—as alternatives (cf. Avigad, 2006).
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“I have described the Classification as a theorem, and at this time I believe that to be true. Twenty years ago I would also have described the Classification as a theorem. On the other hand, 10 years ago, while I often referred to the Classification as a theorem, I knew formally that that was not the case, since experts had by then become aware that a significant part of the proof had not been completely worked out and written down” (Aschbacher, 2004, 737 f.).
- 11.
This is not to say that such phenomena are not worth studying. One way of coming to grasp the meaning of a statement is by arguing with it; we may find, for example, that if we have reached an unacceptable conclusion through sound reasoning from premises that we accept, we find ourselves having reached a better understanding of the conclusion. Thus the B we reach at the end is, in some sense, different from the B (in A → B) from which the argument commenced. Such a phenomenon might be understood as argument-based discovery of meaning. Such argumentation—which might be seen as fallacious—is present in mathematics, but we shall not consider it here.
- 12.
“Classical” means that the law of excluded middle is assumed to be valid: the disjunction ϕ ∨ ¬ϕ is assumed to hold for any formula ϕ.
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This example appears in Rudnicki (1987).
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There are many counterexamples to this general outlook. The solution, by an automated theorem prover, of the long-outstanding Robbins problem required 8 days of continuous computation (McCune, 1997).
- 16.
If one is not satisfied with this example, we could replace the fundamental theorem of calculus by some other significant mathematical fact, perhaps even one that has not yet been discovered.
- 17.
We thank Artur Korniłowicz for this example.
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Acknowledgements
Both authors were partially supported by the ESF research project Dialogical Foundations of Semantics within the ESF Eurocores programme ‘LogICCC’, LogICCC/0001/2007, and the project ‘The Notion of Mathematical Proof’, PTDC/MHC-FIL/5363/2012, both funded by the Portuguese Science Foundation FCT. Alama’s research was conducted in part as a visiting fellow at the Isaac Newton Institute for the Mathematical Sciences, Cambridge, in the programme ‘Semantics & Syntax’. Kahle was partially supported by the FCT project ‘Hilbert’s Legacy in the Philosophy of Mathematics’, PTDC/FIL-FCI/109991/2009.
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Alama, J., Kahle, R. (2013). Checking 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_9
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