### The beauty (?) of mathematical proofs -- beauty and explanatoriness

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This is the third installment of my series of posts on the beauty, function, and explanation in mathematical proofs (Part I is here; Part II is here). In this post I start drawing connections (later to be discussed in more detail) between beauty and explanatoriness.

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While brief, these considerations should in any
case be sufficient to offer initial motivation for the pertinence of the
concept of explanatoriness in mathematical proofs for the present purposes (i.e. a discussion of the presumed beauty of mathematical proofs). With
this in mind, in the next post we will take a brief look at what has been said on explanation
and mathematical proofs in the philosophical literature.

Colyvan, Mark [2012]: Introduction to the Philosophy of Mathematics. Cambridge: CUP.

Hafner, Johannes, and Paolo Mancosu [2005]: The Varieties of Mathematical Explanation. In Visualization, Explanation and Reasoning Styles in Mathematics. Edited by Paolo Mancosu, et al., 215–250. Berlin: Springer.

Hardy, G.H. [1940]: A Mathematician’s Apology. Cambridge: Cambridge University Press.

Rota, G.C. [1997]: ‘The phenomenology of mathematical beauty’, Synthese 111, 171–182.

**Catarina Dutilh Novaes**This is the third installment of my series of posts on the beauty, function, and explanation in mathematical proofs (Part I is here; Part II is here). In this post I start drawing connections (later to be discussed in more detail) between beauty and explanatoriness.

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A hypothesis to be investigated in more detail in what
follows is that there seems to be an intimate connection between attributions
of beauty to mathematical proofs and the idea that mathematical proofs should
be

*explanatory*. Indeed, the reductive account of Rota in terms of enlightenment immediately brings to mind the ideal of explanatoriness:We acknowledge a theorem's beauty when we see how the theorem "fits" in its place, how it sheds light around itself, like aLichtung, a clearing in the woods. We say that a proof is beautiful when such a proof finally gives away the secret of the theorem, when it leads us to perceive the actual, not the logical inevitability of the statement that is being proved. (Rota 1997, 182).

It is not surprising that there should be such a connection
for non-literal, reductive accounts such as Rota’s; explanatoriness is a very
plausible candidate as the epistemic property that is actually being tracked by
these apparently aesthetic judgments. However, the connection is arguably present
both in reductive and in non-reductive accounts. Indeed, it is striking to
notice that many of the properties that Hardy (1940) attributes to beautiful
proofs are in fact properties typically associated with explanatoriness in the
literature (to be discussed in an upcoming post). According to Hardy, a beautiful mathematical proof is:

- · Serious: connected to other mathematical ideas

- · General: idea used in proofs of different kinds

- · Deep: pertaining to deeper ‘strata’ of mathematical ideas

- · Unexpected: argument takes a surprising form

- · Inevitable: there is no escape from the conclusion

(Notice that the depth criterion is closely related to his
realist, Platonist conception of mathematics: some concepts are inherently more
fundamental, i.e. ‘deeper’, than others according to a ‘strata’ organization.)
As we will see shortly, (what Hardy describes as) seriousness, generality,
depth, inevitability, and economy are properties associated with explanatory
proofs.

Do judgments of explanatoriness play a role in the practices
of mathematicians similar to the role of aesthetic judgments? Prima facie, it
seems that aesthetic vocabulary is more often present in how mathematicians
talk about proofs than explanation-related vocabulary.[1] So
one may wonder whether mathematicians themselves do (or do not) view the
categories of ‘explanatory’ and ‘non-explanatory’ as relevant for their
practices. Colyvan (2012, 79) recognizes that judgments of explanatoriness are
typically left out of the published work of mathematicians; he also notes that
“it is difficult to find a great deal of agreement in the mathematical
literature on which proofs are explanatory and which are not”.[2]
Nevertheless, he maintains that these categories are in fact relevant for the
practices of mathematicians, and thus that explanatoriness is an important
topic for the philosopher of mathematics.[3] Hafner
and Mancosu (2005) reach a similar conclusion on the relevance of explanation
for mathematical practice on the basis of some case studies. (It
is to the credit of these authors that they raise the question at all; most
philosophers writing on explanation in mathematics seem to take the relevance
of the topic as a given.)

FULL SERIES:

**References**Colyvan, Mark [2012]: Introduction to the Philosophy of Mathematics. Cambridge: CUP.

Hafner, Johannes, and Paolo Mancosu [2005]: The Varieties of Mathematical Explanation. In Visualization, Explanation and Reasoning Styles in Mathematics. Edited by Paolo Mancosu, et al., 215–250. Berlin: Springer.

Hardy, G.H. [1940]: A Mathematician’s Apology. Cambridge: Cambridge University Press.

Rota, G.C. [1997]: ‘The phenomenology of mathematical beauty’, Synthese 111, 171–182.

[1] If Rota is right, then this is not
surprising: for some reason, mathematicians would eschew vocabulary related to
explanation and enlightenment and use instead aesthetic vocabulary to talk
about these epistemic properties instead.

[2] Again, whether mathematicians
converge in their attributions of beauty and of explanatoriness to proofs is
still essentially an open question. Much of the literature on both topics seems
to presuppose that they do, but this is not to be taken for granted (Inglis and
Aberdein have been conducting some pilot studies which seem to suggest
significant variations in proof appraisal). The philosophical significance of
consensus or lack thereof on the beauty and/or explanatoriness of proofs will
be discussed in more detail shortly.

[3] Some philosophers may want to
maintain that whether mathematicians themselves view explanation as an
important concept is irrelevant to determine the

*philosophical*significance of the topic. But it seems to me to be a mistaken attitude to disregard mathematical practice in this way.
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