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 here; Part II is here; Part 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]

FULL SERIES:

**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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