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.
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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.[1] 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.