Abstract
In this chapter, I argue that the issue of explanatoriness in mathematical proofs can be fruitfully addressed within the dialogical conceptualization of proofs that I have been developing in recent years. The key idea is to emphasize the observation that a proof is a piece of discourse aimed at an intended audience, with the intent to produce explanatory persuasion. This approach explains both why explanatory proofs are to be preferred over non- or less explanatory ones, and why explanatoriness is an audience-relative property of a proof. This account is also able to clarify a number of features of mathematical practice.
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Notes
- 1.
See Baracco (2017) for a critique of this idea, which is described as ‘someism’. Instead, Baracco defends the view that all proofs are explanatory, at least in some sense.
- 2.
It is important to bear in mind that the whole discussion here pertains to so-called ‘informal’ deductive proofs (such as proofs presented in mathematical journals or textbooks), not to proofs within specific formal systems.
- 3.
An interesting recent development is Gijsbers’ quasi-interventionist theory of mathematical explanation, which draws on Woodward’s interventionist theory of causation (Gijsbers, 2017). However, the resulting conception of explanatoriness is according to the author himself “subjective in the sense that it does not depend on the objective structure of mathematics itself” (p. 47).
- 4.
- 5.
It is revealing that in van Fraassen’s account of scientific explanation based on the idea of why-questions, explanation “is a three-term relation, between theory, fact, and context” (van Fraassen, 1980, p. 153). What is conspicuously missing in van Fraassen’s account from the present perspective are the agents consuming and producing these explanations, i.e. the agents asking the why-questions and the agents answering them. (More on van Fraassen shortly.)
- 6.
Recently, in the literature on scientific explanation, epistemic approaches focusing on understanding have also gained some traction (Grimm, Baumberger, & Ammon, 2016; de Regt 2009; Khalifa, 2012). To my knowledge, this trend has not yet reached discussions on mathematical explanation (with the exception of Dufour (2013a, b), who discusses explanation and understanding with respect to mathematical proofs), though the question of understanding with respect to proofs has been discussed by some philosophers (Avigad, 2008).
- 7.
Naturally, I am not the first one to emphasize the dialogical nature of mathematical proofs; this idea is at the heart of Lakatos’ Proofs and Refutations, and has been developed in detail by Ernest (1994).
- 8.
- 9.
Whether mathematicians converge in their attributions of explanatoriness to proofs is essentially an open question. Much of the philosophical literature seems to presuppose that they do, but this is of course ultimately an empirical question. The philosophical significance of consensus or lack thereof on the explanatoriness of proofs, as well as preliminary empirical results, will be discussed in more detail in the final section of this chapter.
- 10.
Some philosophers may wish to maintain that whether mathematicians themselves view explanation as an important concept is irrelevant to determine the philosophical significance of the topic. But to disregard mathematical practice in this way seems to me to be a highly problematic move.
- 11.
Pincock (2015) adopts the comparative perspective, thus asking the question ‘what makes a proof more explanatory than other proofs?’ rather than ‘what makes a proof explanatory simpliciter?’ (More on this point shortly.)
- 12.
“Like Hempel, Kitcher remains half-pragmatic and says nothing about dialectical or interactive features in the use of explanation and argument. His very formulation is symptomatic: “scientific explanation advances our understanding”. Whose understanding? Yours, mine?” (Dufour, 2013b, p. 11).
- 13.
Some authors take the two notions of argument and explanation to be disjoint, but Walton, Dufour, and Aberdein have convincingly disputed this idea.
- 14.
For Hersh (1993), proof is also about convincing and explaining, but on his account these two aspects come apart. According to him, convincing is aimed at one’s mathematical peers, while explaining is relevant in particular in the context of teaching. Similarly, in argumentation theory, persuasion and understanding are often thought to be orthogonal phenomena: persuasion would be related to a mild form of coercion via argumentation—one cannot but assent to the conclusion—whereas understanding via explanation presupposes a certain amount of cognitive freedom. (See Wright, 1990 on the two phenomena.) On my story however, persuasion and explanation go hand in hand in mathematical proofs; a proof will be more persuasive precisely if it is viewed as (more) explanatory. Smale’s proof of the eversion of the sphere, for example, was viewed as paradoxical precisely for its lack of explanatoriness.
- 15.
Take for example the ongoing saga of Mochizuki’s purported proof of the ABC conjecture, which is for now still impenetrable for the mathematical community at large, and so it remains in a limbo. See http://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509. See also (Auslander, 2008) for proof as certification.
- 16.
This terminology comes from the computer science literature on proofs. The earliest occurrence that I am aware of is in Sørensen and Urzyczyn (2006) who speak of prover-skeptic games. One may think of the interplay between proofs and refutations as described in Lakatos’ (1976) seminal work as an illustration of this general idea: Prover aims at proofs, Skeptic aims at refutations. The ‘semi’ qualification pertains to the equally strong cooperative component in a proof, to be discussed shortly. See also Pease, Lawrence, Budzynska, Corneli, and Reed (2017) for a formalized version of Lakatosian games of proofs and refutations, again very much in the spirit of the Prover-Skeptic dialogues here described.
- 17.
Moreover, again on a Lakatosian picture, refutations and counterexamples brought up by Skeptic may play the fundamental role of refining the conjectures and their proofs (see quote in footnote below).
- 18.
Lakatos (1976) distinguishes between global and local counter examples.
- 19.
As well put by one of the characters in Proofs and Refutations: “Then not only do refutations act as fermenting agents for proof-analysis, but proof-analysis may act as a fermenting agent for refutations! What an unholy alliance between seeming enemies!” (Lakatos, 1976, p. 48). See also recent discussions on ‘adversarial collaboration’ in the social sciences (Tetlock & Mitchell, 2009).
- 20.
Compare to what happens in a court of law in adversarial justice systems: defence and prosecution are defending different viewpoints, and thus in some sense competing with one another, but the ultimate common goal is to achieve justice. The presupposition is that justice will be best served if all parties perform to the best of their abilities.
- 21.
The cooperative component becomes immediately apparent if one considers that a one-line ‘proof’ from premises to conclusion—say, from the axioms of number theory straight to Fermat’s Last Theorem—will be necessarily truth-preserving, and yet will not count as an adequate proof. Of course, Skeptic may also make misuse of why-questions and refuse to be convinced even when a particular inferential step is as clear as it can get (such as the tortoise in L. Carroll’s famous story of Achilles and the Tortoise).
- 22.
Though of course they can also be presented orally, for example in the context of teaching, but even then writing also typically occurs.
- 23.
The work of L. Andersen interviewing mathematicians on their refereeing practices shows that, when refereeing a paper, mathematicians behave very much like the fictive Skeptic (Andersen, 2017). This suggests that, in the broader social context of mathematical practices, Skeptic does remain active insofar as this role is played by members of the community who scrutinize proofs (in particular, but not exclusively, in their capacity of referees).
- 24.
“[T]he mature mathematician understands the entire proof from a brief outline.” (Lakatos, 1976, p. 51).
- 25.
The reader versed in argumentation theory may notice some similarities with the ‘New Rhetoric’ framework introduced by Perelman and Olbrechts-Tyteca: “since argumentation aims at securing the adherence of those to whom it is addressed, it is, in its entirety, relative to the audience to be influenced.” (Perelman & Olbrechts-Tyteca, 1969, p. 19) (More on connections with the New Rhetoric shortly.)
- 26.
See also Raman-Sundström (2012) on fit.
- 27.
See the distinction between the concepts of a particular vs. a universal audience in the New Rhetoric framework of Perelman and Olbrechts-Tyteca (1969). See Dufour (2013a, b) for an application of the New Rhetoric framework to mathematical argumentation. See also Malink (2015) on the emergence of formal logic in the Prior Analytics as related to the concern of making premises fully explicit, which can be understood in terms of addressing a universal audience.
- 28.
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Thanks to Jonathan Schaffer for helpful comments on an earlier draft.
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Dutilh Novaes, C. (2018). A Dialogical Conception of Explanation in Mathematical Proofs. In: Ernest, P. (eds) The Philosophy of Mathematics Education Today. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-77760-3_5
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