Monday, 12 October 2015


By Catarina Dutilh Novaes






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).[2] 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.[3] So this brings us to the concept of functional beauty.



Lange, Marc [2014]: Aspects of Mathematical Explanation: Symmetry, Unity, and Salience. Philosophical Review 123 (4):485-531.

Perelman, Chaïm and Lucie Olbrechts-Tyteca [1969]. The New Rhetoric: A Treatise on Argumentation. Notre Dame, [Ind.]University of Notre Dame Press.

[1] 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)
[2] See the distinction between the concepts of particular and universal audience in the New Rhetoric framework of Perelman and Olbrechts-Tyteca (1969).
[3] 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.


  1. I have a tiny quibble here -- tiny because it is partly just terminological and in any case doesn't affect your main points at all. You write, "in professional journals, proofs are more often than not no more than proof sketches, where the key ideas are presented." But actually what mathematicians write is almost always a lot more than just a description of the key ideas, though of course also a lot less than a fully formalized proof. And we use the word "sketch" in a way that clearly distinguishes a description of the main ideas from what we would count as a "full proof", which I would define roughly as a proof where the missing steps are sufficiently straightforward (for the intended expert audience) that to put them in would be unhelpful. Sometimes one decides not to give a full proof of this kind, and then one clearly signals that the argument one is giving is "just a sketch" and not a proper proof.

    1. Yes, I suppose this is mostly a terminological quibble, but certainly one worth paying attention too (especially as, in the next post, I discuss the matter of what counts as a proof in the first place, i.e. criteria of individuation). So it seems we should distinguish between a proof idea (or ideas), and a proof sketch, which is a mode of presentation of the proof -- and, according to you, it typically contains more than just the main ideas.
      Btw I saw you have a paper in this volume:
      The whole volume looks great!

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