Abstract
The role of audiences in mathematical proof has largely been neglected, in part due to misconceptions like those in Perelman and Olbrechts-Tyteca (The new rhetoric: A treatise on argumentation, University of Notre Dame Press, Notre Dame, 1969) which bar mathematical proofs from bearing reflections of audience consideration. In this paper, I argue that mathematical proof is typically argumentation and that a mathematician develops a proof with his universal audience in mind. In so doing, he creates a proof which reflects the standards of reasonableness embodied in his universal audience. Given this framework, we can better understand the introduction of proof methods based on the mathematician’s likely universal audience. I examine a case study from Alexander and Briggs’s work on knot invariants to show that we can fruitfully reconstruct mathematical methods in terms of audiences.
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Notes
See for example Larvor (2016a).
The most closely relevant example would be Dufour (2013) which, although it deals with similar argumentation literature, only looks at audience role in mathematical argumentation outside of proof.
For a collection on mathematics and argumentation theory, see Aberdein and Dove (2013). In that collection alone, very few papers discuss audiences and, when they do, it is related to presentational changes after the completion of a proof. After proof completion is one of the core parts of Dufour (2013). None of the papers examine the role of audiences in development of a proof.
There are various uses and definitions of “argumentation.” Some authors, like Corcoran (1989), define argumentation as a three part system comprised of a premise set, conclusion, and chain of reasoning. Others, like Tindale (2015), include the social processes of giving and taking reasons in their definition; they usually reserve argument for the name of the product devoid of process. In choosing my definition, I attempt to adhere more closely to Corcoran (1989) and view the argumentation as the product. To his definition, I add that an argumentation is a product which reflects the consideration of the audience. This makes it possible to encompass the argumentation that Perelman and Olbrechts-Tyteca (1969) are concerned with while examining their distinction between “argumentation” and “demonstration.” Under definitions without an audience-reflective component, both “demonstrations” and “argumentations” would be types of argumentation.
The “proof” discussed in this paper is not the object of what logicians call proof theory. Rather, I intend to encompass those proofs that mathematicians comfortably write and read. It is the proof typical of mathematical practice. They are found in textbooks, journals, and seminars. I intend it to be interpreted as a broad class.
For similar arguments, see Dove (2009), Aberdein (2019), and Larvor (2016b). All these have arguments to the effect that mathematical proof is not usually derivation. But the question usually involves accepting this fact and then asking whether or not mathematical proof is reducible to derivation which is clear in examples like Azzouni (2004), Azzouni (2009), Tanswell (2015). It is also assumed in discussions like Antonutti Marfori (2010), Larvor (2016b), Carrascal (2015) which are all interested in determining the best approach to rigor in proof, given that they’re not typically derivations.
The “follows from” here is broadly construed. In many normal argumentations, we would be happy with the truth following with a certain probability. In addition, conclusions could follow from emotional appeal. In mathematical proof the truth of the theorem should follow with certainty. This is not to say that they only do so by deduction, as many proof methods fall short of deductive entailment, see Dove (2009) for examples.
The barring of audiences from derivation is continued in modern rhetorical argumentation theory and follows in Perelman and Olbrechts-Tyteca (1969)’s tradition. See, for example, (Tindale 2015, p. 63) which claims that a “[derivation] aims to derive conclusions from strict premises. In these terms it is an altogether different enterprise to argumentation, which involves, essentially, a meeting of minds.”
See (Perelman and Olbrechts-Tyteca 1969, p. 13) for more about “demonstrations.”
For more on acceptable gaps in mathematical proof, see Fallis (2003).
Some argue that proofs are derivations with the “obvious” steps left out. This is compatible with the argument above. I make no claim about whether or not all mathematical proofs could be made derivations. There may be a system which makes all mathematical proofs into derivations without changing the mathematical proof itself. But it suffices for my argument that mathematicians do not need that system. They convince each other with component arguments in their mathematical proof that succeed because they reflect a consideration for that audience. This is true even of positions like Azzouni (2004)’s where a proof is taken to indicate a derivation. These proofs are still reflective of their audiences since they are argumentations. It is a problem outside the scope of this paper as to whether or not the derivations that they indicate is also audience-reflective.
The universal audience was first introduced in Perelman and Olbrechts-Tyteca (1969). The version I discuss here mainly follows Tindale (2015). For Perelman and Olbrechts-Tyteca (1969), it is never clear if the universal audience actually physically exists or if one could access it. Tindale (2015) argues that the universal audience must be a representation defined by the standards of reasonableness abstracted from particular audiences.
I’m grateful to an anonymous referee for pointing out this alternative interpretation.
How many mathematicians agree with this is an empirical question. Given the interpretive issues and the evidence that quotes provide, an empirical study into this identified thread is worthwhile. The arguments here can provide the basis for such an investigation.
In Perelman and Olbrechts-Tyteca (1969), persuading is associated with particular audiences while convincing is associated with the universal audience. I follow their usage here. But nothing vital to my argument hangs on the technical differences between persuading and convincing.
The potential particular audiences I explore here by no means enumerate all potential particular audiences. But they are representative of a number of suggestions and objections raised against the view. These few examples should suffice to highlight the underlying problem in trying to model proof development on particular audiences.
The one outright exception I’ve encountered is in Davis and Hersh (1986). They explicitly state that the goal is convincing people like the mathematician. So accepting this particular audience is not impossible or contradictory but it falls short of the reasons discussed above.
Some evidence for this can be found in (Andersen et al. 2019) which describes the process of turning a PhD student’s proof into a publishable research paper. The PhD student who is experienced in mathematics but inexperienced in publishing, produces a proof which has little concern for what the journal reviewers will be convinced by. The experienced mathematician helps to improve the proof’s presentation so as to increase chances of publication.
I stress again, that this is not a full answer to the issue regarding journal reviewers. This is because I have differentiated development and presentation in a preliminary manner. A full account, which I have no intention of giving here, should be able to make sense of the interplay between development and presentation. I hope to explore this in future work.
‘Incarnations’ is the term employed by Perelman and Olbrechts-Tyteca (1969). An incarnation of the universal audience should be understood as the real audiences whose primary reasons for being convinced are in line with the universal audience’s.
Reasonable person and reasonableness are loaded terms. I would like to remain as neutral as possible on discussions of what is and is not reasonable. Perelman and Olbrechts-Tyteca (1969) are coming from a background in law and so the terms carry, to some extent, a legal notion. Perelman and Olbrechts-Tyteca (1969) and Tindale (2015) both seem to take reasonableness as a learned feature. There are no theoretical definitions of reasonable by which arguers judge audiences or people. Given a certain set of experiences, one may come to bar unreasonable people, see for example discussions in 12.3 of Tindale (2015). A full account of reasonableness is outside the scope of this paper. For present purposes it should suffice that mathematicians do make judgments of reasonableness. Moreover, this discussion deserves extensive discussion in the future, as it seems to me that the reflective arguer should be allowed to place certain meta-theoretic conditions on reasonableness to help further target his audience.
This is a psychological claim which I have neither the room nor the evidence to defend in the specifically mathematical case. This is a limitation I address in the concluding remarks.
I’m grateful to an anonymous reviewer for making this point which forced me to get clearer about what the case study provides to this paper. This point entails that looking at proofs themselves will not generate any evidence for the main thesis. But the argument above stands without attempting to draw evidence from these easily generated claims.
My case study draws on Epple (2004). Epple’s argument looks at how local research becomes more universal mathematical knowledge. I focus on the methods in the proof, in particular Alexander and Briggs’ dotting notation, and look at standards of reasonableness abstracted from local particular audiences.
I refer to Alexander’s universal audience here and afterward. The 1927 paper is co-authored with his student G. B. Briggs. But the 1928 paper is written solely by Alexander. Briggs makes vital contributions, but I take these facts—that Briggs was his student and Alexander published a later paper on it alone—as evidence that Alexander is mostly the one determining standards. Epple (2004) also relies on Alexander’s background but not Briggs’s.
For more theoretical work on the importance of knot diagrams as evidence, see De Toffoli and Giardino (2014).
The dotting notation is only mentioned once by Epple as a “seemingly marginal difference” between Reidemeister and Alexander and Briggs Epple (2004).
One excellent objection, worth briefly sketching, to this involves the concept of knots. The objection provides an explanation for the use of “better” by Alexander. It is that the dotting notation introduces direction into the concept of knots. And that Alexander and Briggs were claiming superiority because adding directedness made for a better understanding of knots. Plausible as that may be, even the diagrams with the broken line notation indicate directedness in Alexander and Briggs’ paper. So it’s not likely that they were adding directedness.
The biographical and historical picture gestured at in this section follows Epple (2004)’s account.
Recall that the view expressed by Hadamard is most likely too extreme. But the less extreme claim involving generality of reasoning that it expresses is really what’s at play in arguments for the universal audience.
References
Aberdein, A. (2019). Evidence, proofs, and derivations. ZDM Mathematics Education, 51(4).
Aberdein, A., & Dove, I. J. (Eds.). (2013). The argument of mathematics. Dordrecht: Springer.
Alexander, J. W. (1928). Topological invariants of knots and links. Transactions of the American Mathematical Society, 30(2), 275–306.
Alexander, J. W., & Briggs, G. B. (1927). On types of knotted curves. Annals of Mathematics, 28(1), 562–586.
Andersen, L. E., Johansen, M. W., & Sørensen, H. K. (2019). Mathematicians writing for mathematicians. Synthese. https://doi.org/10.1007/s11229-019-02145-5.
Antonutti Marfori, M. (2010). Informal proofs and mathematical rigour. Studia Logica, 96, 261–272.
Azzouni, J. (2004). The derivation-indicator view of mathematical practice. Philosophia Mathematica, 12(2), 81–105.
Azzouni, J. (2009). Why do informal proofs conform to formal norms? Foundations of Science, 14(1–2), 9–26.
Carrascal, B. (2015). Proofs, mathematical practice and argumentation. Argumentation, 29(3), 305–324.
Corcoran, J. (1989). Argumentations and logic. Argumentation, 3, 17–43.
Davis, P. J., & Hersh, R. (1986). Mathematics and rhetoric. In P. J. Davis & R. Hersh (Eds.), Descartes’ dream: The world according to mathematics (pp. 57–73). London: Penguin.
De Toffoli, S., & Giardino, V. (2014). Forms and roles of diagrams in knot theory. Erkenntnis, 79(4), 829–842.
Dove, I. J. (2009). Towards a theory of mathematical argument. Foundations of Science, 14(1–2), 137–152.
Dufour, M. (2013). Arguing around mathematical proofs. In A. Aberdein & I. J. Dove (Eds.), The argument of mathematics (pp. 61–76). Dordrecht: Springer.
Epple, M. (2004). Knot invariants in Vienna and princeton during the 1920s: Epistemic configurations of mathematical research. Science in Context, 17(1), 131–164.
Fallis, D. (2003). Intentional gaps in mathematical proofs. Synthese, 134, 45–69.
Hadamard, J. (1945). An essay on the psychology of invention in the mathematical field. Princeton, NJ: Princeton University Press.
Hadamard, J. (2004). Thoughts on the heuristic method. In R. G. Ayoub (Ed.), Musings of the masters. The athematical Association of America.
Hardy, G. H. (1929). Mathematical proof. Mind, 38, 1–25.
Hume, D. (1739). A treatise of human nature. Penguin Books, London, 1969 Reprint
Krantz, S. G. (2011). The proof is in the pudding: The changing nature of mathematical proof. New York, NY: Springer.
Larvor, B. (Ed.). (2016a). Mathematical cultures: The London meetings 2012–2014. Basel: Birkhäuser.
Larvor, B. (2016b). Why the naïve derivation recipe model cannot explain how mathematicians’ proofs secure mathematical knowledge. Philosophia Mathematica, 24(3), 401–404.
Perelman, C., & Olbrechts-Tyteca, L. (1969). The new rhetoric: A treatise on argumentation. Notre Dame, IN: University of Notre Dame Press.
Sigler, J. (2015). The new Rhetoric’s concept of universal audience, misconceived. Argumentation, 29(3), 325–349.
Tanswell, F. (2015). A problem with the dependence of informal proofs on formal proofs. Philosophia Mathematica, 23, 295–310.
Tindale, C. W. (2015). The philosophy of argument and audience reception. Cambridge: Cambridge University Press.
Wiedijk, F. (2004). Formal proof sketches. In Berardi, S., Coppo, M., Damiani, F. (Eds.) Types for proofs and programs: International workshop, TYPES 2003, Torino, Italy, April 30–May 4, 2003, Revised Selected Papers, Springer, Berlin, LNCS, vol 3085, pp. 378–393.
Acknowledgements
I am incredibly grateful for the comments from (and conversations with) Andrew Aberdein, Nic Fillion, Tom Archibald, Silvia de Toffoli, two anonymous referees, and members of the audience at the CSHPS 2017, CSHPM 2018, the 2019 Masterclass on Philosophy of Mathematical Practice, and the 2020 APMP meetings. Although they will not agree with all of the arguments and conclusions in the paper, their comments greatly improved the manuscript.
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Ashton, Z. Audience role in mathematical proof development. Synthese 198 (Suppl 26), 6251–6275 (2021). https://doi.org/10.1007/s11229-020-02619-x
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DOI: https://doi.org/10.1007/s11229-020-02619-x