By Catarina Dutilh Novaes
This is the sixth installment (two more to come!) 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; Part V is here). After having introduced the dialogical conception of proofs in the previous post, in this post I explain why proofs do not appear to be dialogues, and what the prospects are for an absolute notion of the explanatoriness of proofs.
At this point, the reader may be wondering: this is all very well, but obviously deductive proofs are not really dialogues! They are typically presented in writing rather than produced orally (though of course they can also be presented orally, for example in the context of teaching), and if at all, there is only one ‘voice’ we hear, that of Prover. So at best, they must be viewed as monologues. My answer to this objection is that Skeptic may have been ‘silenced’, but he is still alive and well insofar as the deductive method has internalized the role of Skeptic by making it constitutive of the deductive method as such. Recall that the job of Skeptic is to look for counterexamples and to make sure the argumentation is perspicuous. This in turn corresponds to the requirement that each inferential step in a proof must be necessarily truth preserving (and so immune to counterexamples), and that a proof must have the right level of granularity, i.e. it must be sufficiently detailed for the intended audience, in order to achieve its explanatory purpose.
Let us discuss in more detail the phenomenon of different levels of granularity in mathematical proofs, as it is directly related to the issue of explanatoriness. It is well known that the level of detail with which the different steps in a proof are spelled out will vary according to the context: for example, in professional journals, proofs are more often than not no more than proof sketches, where the key ideas are presented. The presupposition is that the intended audience, namely professional mathematicians working on similar topics, would be able to reconstruct the details of the proof should they feel the need to do so (e.g. if they somehow doubt the results). In contrast, in the context of textbooks or in classroom situations, proofs tend to be presented in much more detail, precisely because the intended audience is not expected to have the level of expertise required to reconstruct the proof from a proof-sketch. What is more, the intended audience is in the process of learning the game of formulating and understanding mathematical proofs, and so proofs where each step is clearly spelled out is what is required. Furthermore, different areas within mathematics tend to have different standards of rigor for proofs, again in function of the intended audience.
What the phenomenon of different levels of granularity suggests when it comes to the explanatoriness of proofs is that, for a proof to be explanatory for its intended audience, the right level of granularity must be adopted. If a proof is to be explanatory in the sense of making “something that is initially puzzling less puzzling; an explanation reduces mystery” (Colyvan 2012, 76), the decrease of puzzlement is at least in first instance inherently tied to the agent to whom something should become less puzzling.
Another upshot of this conception is that explanatoriness is better seen as a matter of degrees, i.e. as a comparative notion: a proof may be more or less explanatory (for a given audience) than another proof (of the same or another theorem). For example, reductio ad absurdum proofs are typically viewed as less explanatory than direct proofs, and similarly proofs by cases and proofs by mathematical induction are generally thought to be less explanatory than other kinds of proofs (Lange 2014). But this does not mean that such proofs lack explanatoriness completely. On occasion, a reductio proof may even be more explanatory and convincing than a direct proof of the same theorem, for example if the direct proof is inordinately long whereas the reductio proof is short, perspicuous, and elegant. In fact, Colyvan (2012, 83) convincingly argues that discussing the explanatoriness or lack thereof of proofs in terms of structural families (reductio, by induction, by cases etc.) may not be the most promising approach to take, and that one should look at the details of individual proofs.
Does this mean that there is nothing left for a non-relative, objective notion of explanatory mathematical proofs? Not so, thanks to the notion of an internalized Skeptic. The internalized Skeptic is what could be described as the universal, arbitrary Skeptic: a mathematical proof should aim to be convincing and explanatory for the widest possible range of Skeptics (understood as those having the necessary credentials as mathematicians), and thus presuppose as little as possible that is specific to particular Skeptics (e.g. background knowledge). So we may say that an explanatory proof in an absolute sense is a proof that would be considered as explanatory by an arbitrary Skeptic.
But of course, the arbitrary Skeptic is an idealization, and in particular if the goal is to stay close to mathematical practices, it is not immediately obvious that it is a useful idealization. (As it so happens, actual mathematical proofs typically display features of context-dependence such as the desired level of granularity for the intended audience.) But this means that those seeking a non-relative, objective notion of explanatoriness need not reject the dialogical approach proposed here altogether. From this perspective, the non-relative notion emerges from maximally broadening the range of contexts/putative audiences considered and is thus a limit case of the relative notion.
At this point of the investigation we have the following elements in place:
- A functionalist account of proofs in dialogical terms.
- The idea that one of the main functions of a proof is to be persuasive in an explanatory way.
- Close conceptual connections between the notions of a proof being explanatory and a proof being beautiful, both in reductive (Rota’s) and, perhaps more surprisingly, non-reductive (Hardy’s) accounts of the beauty of mathematics.
These elements will now provide the background for the main thesis of the paper: a proof is (more) beautiful if it fulfills its function(s) well, in particular in the sense of being persuasive in an explanatory way. So this brings us to the concept of functional beauty.
Colyvan, Mark : Introduction to the Philosophy of Mathematics. Cambridge: CUP.
Lange, Marc :Aspects of Mathematical Explanation: Symmetry, Unity, and Salience. Philosophical Review 123 (4):485-531.
Perelman, Chaïm and Lucie Olbrechts-Tyteca . The New Rhetoric: A Treatise on Argumentation. Notre Dame, [Ind.]University of Notre Dame Press.
 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 and Olbrechts-Tyteca 1969, p. 19)
 See the distinction between the concepts of particular and universal audience in the New Rhetoric framework of Perelman and Olbrechts-Tyteca (1969).
 I’ve heard from practicing mathematicians that, to them, the beauty of a proof makes it more convincing, and so we may be dealing with a bidirectional relation: a proof may be beautiful because it is convincing, or it may be convincing because it is beautiful.