Friday, 9 October 2015

The beauty (?) of mathematical proofs -- explanatory persuasion as the function of proofs

By Catarina Dutilh Novaes

This is the fifth installment of my series of posts on the beauty, function, and explanation in mathematical proofs (Part I is herePart II is herePart III is here; Part IV is here). In this post I bring in my dialogical conception of proofs (did you really think you'd be spared of it this time, dear reader?) to spell out what I take to be one of the main functions of mathematical proofs: to produce explanatory persuasion.

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Framing the issue in these terms allows for the formulation of two different approaches to the matter: explanatoriness as an objective, absolute property of the proofs themselves; or as a property that is variously attributed to proofs first and foremost based on pragmatic reasons, which means that such judgments may by and large be context-dependent and agent-dependent. (A third approach may be described as ‘nihilist’: explanation is simply not a useful concept when it comes to understanding the mathematical notion of proof.) Some of those instantiating the first approach are Steiner (1978) and Colyvan (2010); some of those instantiating the second one are Heinzmann (2006) and Paseau (2011). (It is important to bear in mind that the 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.)

For reasons which will soon become apparent, the present analysis sides resolutely with so-called pragmatic approaches: the notion of explanation is in fact useful to explain the practices of mathematicians with respect to proofs, in particular the phenomenon of proof predilection, but it should not be conceived as an absolute, objective, human-independent property of proofs. One important prediction of this approach is that mathematicians will not converge in their judgments on the explanatoriness of a proof, given that these judgments will depend on contexts and agents (more on this in the final section of the paper).

Perhaps the conceptual core of pragmatic approaches to the explanatoriness of a mathematical proof is the idea that explanation is a triadic concept, involving the producer of the explanation, the explanation itself (the proof), and the receiver of the explanation. The idea is that explanation is always addressed at a potential audience; one explains something to someone else (or to oneself, in the limit).[1] And so, a functional perspective is called for: what is the function (or what are the functions) of a proof? What is it good for? Why do mathematicians bother producing proofs at all? While these questions are typically left aside by mathematicians and philosophers of mathematics, they have been raised and addressed by authors such as Hersh (1993), Rav (1999), and Dawson (2006).

One promising vantage point to address these questions is the historical development of deductive proof in ancient Greek mathematics,[2] and on this topic the most authoritative study remains (Netz 1999). Netz[3] emphasizes the importance of orality and dialogue for the emergence of classical, ‘Euclidean’ mathematics in ancient Greece:


Greek mathematics reflects the importance of persuasion. It reflects the role of orality, in the use of formulae, in the structure of proofs… But this orality is regimented into a written form, where vocabulary is limited, presentations follow a relatively rigid pattern… It is at once oral and written… (Netz 1999, 297/8)

Netz’s interpretation relies on earlier work by Lloyd (1996), who argues that the social, cultural and political context in ancient Greece, and in particular the role of practices of debating, was fundamental for the emergence of the technique of mathematical deductive proofs. So from this perspective, it seems that one of the main functions of deductive proofs (then as well as now) is to produce persuasion, in particular what one could call explanatory persuasion: to show not only that something is the case,[4] but also why it is the case.[5] As well put by Dawson (2006, 270):

[W]e shall take a proof to be an informal argument whose purpose is to convince those who endeavor to follow it that a certain mathematical statement is true (and, ideally, to explain why it is true).

What I add to Dawson’s description is an explicit multi-agent, dialogical component, which is only implicit in this description. On this conception, a deductive proof corresponds to a dialogue between the person wishing to establish the conclusion (given the presumed truth of the premises), and an interlocutor who will not be easily convinced and will bring up objections, counterexamples, and requests for further clarification. A good proof is one that convinces a fair but ‘tough’ opponent; as the mathematician Mark Kac allegedly said, “the beauty of a mathematical proof is that it convinces even a stubborn proponent” (Fisher 1989, 50). Now, if this is right, then mathematical proof is an inherently dialogical, multi-agent notion, given that it is essentially a piece of discourse aimed at a putative audience, typically composed of ‘stubborn’ interlocutors.

To be sure, there are different ways in which the claim that mathematical proofs are essentially dialogical can be understood. For example, the fact that a proof is only recognized as such by the mathematical community once it has been sufficiently scrutinized by trustworthy experts can also be viewed as a dialogical component, perhaps in a loose sense (the ‘dialogue’ between the mathematician who formulates a proof and the mathematical community who scrutinizes it).[6] But in what follows I present a more precise rational reconstruction of the (quite specialized) dialogues that would correspond to deductive proofs.

On this conception, proofs are semi-adversarial dialogues of a special kind involving two participants: Prover and Skeptic.[7] Prima facie, the (fictitious) participants have opposite goals, and this is why the adversarial component remains prominent: Prover wants to establish the truth of the conclusion, and Skeptic wants to block the establishment of the conclusion (though not ‘at all costs’).[8] The dialogue starts with Prover asking Skeptic to grant certain premises. Prover then puts forward further statements, which purportedly follow from what has been granted. (Prover may also ask Skeptic to grant additional auxiliary premises along the way.) Ultimately, it may seem that most of the work is done by Prover, but Skeptic has an important role to play, namely to ensure that the proof is persuasive, perspicuous, and valid.[9] Skeptic’s moves are: granting premises so as to get the game going; offering a counterexample when an inferential move by Prover is not really necessarily truth-preserving (or a global counterexample to the whole proof); asking for clarifications when a particular inferential step by Prover is not sufficiently compelling and perspicuous. These three moves correspond neatly to what are arguably the three main features of a mathematical proof: it starts off with certain premises, and it proceeds through necessarily truth-preserving inferential steps which should also be individually ‘evident’, i.e. compelling.

From this point of view, a mathematical proof is characterized by a complex interplay between adversariality and cooperation: the participants have opposite goals and ‘compete’ with one another at a lower level, but they are also engaged in a common project to investigate the truth or falsity of a given conclusion (given the truth of the premises) in a way that is not only persuasive but also (hopefully) explanatory. If both participants perform to the best of their abilities, then the common goal of producing novel mathematical knowledge will be optimally achieved.[10]



References

Bueno, Otávio (2009). Functional beauty: Some applications, some worries. Philosophical Books 50 (1):47-54.

Colyvan, Mark (2010). There is No Easy Road to Nominalism. Mind 119 (474):285 - 306.

Dawson Jr, John W. (2006). Why do mathematicians re-prove theorems? Philosophia Mathematica 14 (3):269-286.

Dutilh Novaes, Catarina (2015). Conceptual genealogy for analytic philosophy. In J. Bell, A. Cutrofello, P.M. Livingston (eds.), Beyond the Analytic-Continental Divide: Pluralist Philosophy in the Twenty-First Century (Routledge Studies in Contemporary Philosophy), 2015.

Ernest, Paul (1994). The dialogical nature of mathematics. In Ernest, P. (ed.) Mathematics, Education and Philosophy: An International Perspective, London, The Falmer Press, 1994, 33-48.

Fisher, Michael (1989). Phases and phase diagrams: Gibbs’ legacy today. In G.D. Mostow & D.G. Caldi (eds.), Proceedings of the Gibbs Symposium: Yale University, May 15-17, 1989, American Mathematical Society.

Van Fraassen, Bas C. (1980). The Scientific Image. Oxford University Press.

Heinzmann, Gerhard (2006). Naturalizing Dialogic Pragmatics. In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer: 285--297.

Hersh, Reuben (1993) 'Proving is convincing and explaining", Educational Studies in Mathematics 24(4), 389-399.

Lakatos, Imre (ed.) (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.

Lloyd, G.E.R.  (1996). ‘Science in Antiquity: the Greek and Chinese cases and their relevance to the problem of culture and cognition’. In D. Olson & N. Torrance (eds.), Modes of Thought: Explorations in Culture and Cognition. Cambridge, CUP, 15-33.

Netz, Reviel (1999). The Shaping of Deduction in Greek Mathematics: A study in cognitive history. Cambridge, CUP.

Netz, Reviel (2005). The aesthetics of mathematics: A study. In Paolo Mancosu, Klaus Frovin JØrgensen, and Stig Andur Pedersen, eds. Visualization, Explanation and Reasoning Stryles in Mathematics. Synthese Library, Vol. 327. Dordrecht: Springer,251–293.

Netz, Reviel (2009). Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic. Cambridge University Press.

Paseau, Alexander (2011). Proofs of the Compactness Theorem. History and Philosophy of Logic 31 (1):73-98.

Rav, Y. (1999). Why Do We Prove Theorems? Philosophia Mathematica 7 (1):5-41.

Sørensen, Morten Heine & Urzyczyn, Pawel (2006). Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics). New York, Elsevier.

Steiner, Mark (1978). Mathematical explanation. Philosophical Studies 34 (2):135 - 151. 




[1] Here there are similarities with van Fraassen’s (1980) pragmatic theory of explanation, according to which explanation is a “is a three-term relation, between theory, fact, and context” (Van Fraassen 1980, 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 (which again, may be one and the same agent in certain cases).
[2] When it comes to functionalist questions, it makes sense to inquire into what the first practitioners of a given practice thought they were doing, and why they were doing it, when the practice first came about. But this is not to exclude the possibility of shifts of function along the way (Dutilh Novaes 2015).
[3] Not coincidentally, later work by Netz (2005, 2009) focuses specifically on aesthetic issues in ancient mathematics.
[4] One might think that the primary, perhaps sole function of a mathematical proof is to establish the truth of a certain mathematical conjecture. But this does not sit well with the phenomenon of preferring certain proofs over others, which is part and parcel of mathematical practice. If establishing truth were the only function of a proof, then a mathematician would be equally satisfied with two correct proofs establishing the same theorem. But this is not what happens (as also noted by Bueno (2009, 52)), and in fact mathematicians often work on re-proving theorems, i.e. on finding more satisfying proofs for a given theorem (Dawson 2006).
[5] 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.
[6] Take for example the ongoing saga of Mochizuki’s purported proof of the ABC conjection, which is for now still impenetrable for the mathematical community at large, and so it remains in limbo. See http://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509
[7] This terminology comes from the computer science literature on proofs. The earliest occurrence that I am aware of is in (Sørensen & Urzyczyn 2006), who speak of prover-skeptic games. One may think of the interplay between proofs and refutations as described in Lakatos’ (1976) seminal text as an illustration of this general idea: Prover aims at proofs, Skeptic aims at refutations.
[8] The ‘semi’ qualification pertains to the equally strong cooperative component in a proof.
[9] Moreover, again on the Lakatosian picture, refutations and counterexamples brought up by Skeptic may play the fundamental role of refining the conjectures and their proofs.
[10] Compare to what happens in a court of law in adversarial justice systems: defense 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.

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