By Catarina Dutilh Novaes
This is the seventh 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; Part V is here; Part VI is here). I now turn to beauty properly speaking, and discuss ways in which mathematical proofs are beautiful both in a functional and in a non-functional way.
Prima facie, the concept of functional beauty is strikingly simple: a thing is beautiful insofar as it performs its function(s) well. It seems clear that, generally speaking, for something to fulfill its function is a good thing: normally, it will be useful and advantageous (e.g. it typically enhances fitness for organisms). So it is not surprising that function and beauty should become closely associated. As detailed in (Parsons & Carlson 2009), to date the most comprehensive study of this concept, functional beauty has a venerable pedigree, dating back to classical Greek philosophy (Aristotle in particular, which is not surprising given his interest in function and teleology), and having been particularly popular in the 18th century. As famously noted by Hume:
This observation extends to tables, chairs, scritoires, chimneys, coaches, saddles, ploughs, and indeed to every work of art; it being a universal rule, that their beauty is chiefly deriv’d from their utility, and from their fitness for that purpose, to which they are destin’d. (Hume 1960, 364)
But of course, much complexity lies behind the concept of function itself, which is what is doing all the work. What determines the function(s) of an object, artifact or organism? The concept of function occupies a prominent role in biology, in fact since Aristotle but with renewed strength since the advent of evolutionary biology. (Indeed, Parsons and Carlson (2009) rely extensively on work on function within philosophy of biology, e.g. Godfrey-Smith’s work.) Here however we should focus on artifacts, given that the goal is to increase our understanding of the (putative) beauty of mathematical proofs, which, despite a potentially problematic ontological status (more on which shortly), come closer to artifacts than to organisms or natural objects such as e.g. planets or rocks. Parsons and Carlson (2009, 75) offer the following definition of the (proper) function of an artifact:
An artifact has proper function if and only if it currently exists because, in recent past, its ancestors were successful in meeting some need or want in the marketplace because they performed that function, leading to the manufacture and distribution of that artifact.
As argued above and detailed in (Netz 1999), mathematical proofs emerged in ancient Greece against the background of needs for persuasion in argumentative contexts. This aspect of their origins is arguably still present in many of the key features that we attribute to proofs, such as the requirement that each inferential step be necessarily truth-preserving (which makes the proof as a whole correspond to a winning strategy for Prover), and that each step be individually perspicuous, so that the proof as a whole is persuasive. And precisely because Euclid-style proof is such a successful argumentative strategy in mathematical contexts (and perhaps elsewhere, as suggested by the many uses of the more geometrico in the history of philosophy), it became the dominant approach to argumentation in mathematics for centuries. (Indeed, the concept of a mathematical proof has remained surprisingly stable over millennia, so much so that modern readers still recognize the proofs in Euclid’s Elements as real proofs, even if the standards of rigor are different.) Even today, while there is more variation in what counts as a successful proof (as well as a number of different functions attributed to proofs), the classical, Euclidean conception remains influential. In other words, the history of mathematics seems to conform to the description of mathematical proofs having a (proper) function as described above, i.e. as having established themselves in the ‘marketplace’ precisely because they fulfill that function well.
But what kinds of artifacts are proofs? Paterson (2013) defends the view that mathematical proofs are conceptual artifacts, which presumably means that a proof is something other than the particular presentations thereof. So for instance, presentations with different levels of granularity of the same proof idea would, on this conception, count as one and the same proof, given that they share the same conceptual core. The present analysis, however, takes the audience-relativity of proofs to be a fundamental component, and thus for the present purposes we need to distinguish between different presentations of the same proof idea as being different proofs altogether. Naturally, we need not go to extreme levels of token fine-grainedness (e.g. presentations of the proof in two different languages may still be considered as ‘the same proof’ for present purposes, despite the different intended audiences), but even typographical, perceptual differences in the notations used may have a real cognitive impact on how the proof is processed and understood. (See (Dutilh Novaes 2012, chap. 5) for the relevance of perceptual features of mathematical notation.)
This does not mean that proof-presentation is, in an absolute sense, the one and only suitable level of analysis to spell out the ontological status of proofs and their conditions of individuation. Rather, the claim is simply that, given the goals and premises of the present investigation (a focus on proofs as produced to persuade a given audience), it is the right level of analysis for this investigation. For other projects, it may well be that a more coarse-grained criterion such as conceptual cohesion – the identity of a proof thus being defined in terms of the proof ideas, not of particular presentations – is what is required.
With these considerations in place, we may then conclude that, for present purposes, mathematical proofs are best considered as artifacts of the textual kind. (Which, again, should not make us forget oral-mixed presentations of proofs such as in classroom situations.) This is also the perspective adopted by Netz (2005). And so, literary genres such as poetry, rhetoric, narrative etc. seem to offer a suitable starting point for a detailed analysis of the aesthetic dimension of mathematical proofs. Indeed, mathematical proofs display interesting similarities with poems (to be discussed below); similarly, if the chief goal of a proof is to produce explanatory persuasion, then elements from theories of rhetoric will also come in handy (see references to the ‘New Rhetoric’ in a previous post). Moreover, there has been quite some interest in the connections between proofs and narratives recently, e.g. the volume edited by Doxiadis and Mazur (2012).
Of course, and as repeatedly emphasized by Netz (1999) and many others, visual elements such as diagrams and special notations are also widely present in mathematical proofs, and may very well be a crucial component for the aesthetic appreciation of such proofs. But above all, mathematical proofs as currently produced and consumed belong primarily to the category of texts. (Not coincidentally, even in classroom oral presentations, writing (on a blackboard or what have you) typically plays an essential role.)
It may seem that, by comparing mathematical proofs to literary texts, I am veering away from the idea of functional beauty specifically. Indeed, while it is natural to speak of functional beauty when it comes to artifacts such as tables and chairs, as in Hume’s quote above, it is not in any way obvious what the function(s) of, say, a poem would be, other than to give rise to artistic beauty. I will return to the idea of functional beauty and explanatory persuasion in mathematical proofs in the next section, but at this point let me clarify that, while the main thesis of this paper is that much of what is perceived as aesthetic features in proofs comes down to explanatory persuasion (and is thus essentially functional and epistemic), it seems that there should also be room for a purely aesthetic (non-functional), non-epistemic residue in any account of aesthetic assessments of proofs. And it seems to me that the comparison with poetry, for example, can bring out at least one way in which proofs can be beautiful in a non-epistemic, non-functional way.
In a number of poetic traditions (Greek, Chinese, Arabic, French) what is usually referred to as ‘classical poetry’ is characterizes by a fairly strict set of rules, which poets should abide to. Take the poetic form sonnet, for example, which according to historians originated in the court of Emperor Frederick II in Sicily (first half of 13th century), and was later made famous by Petrach and exported to France, England, and even India centuries later. (In 19th century French poetry, the sonnet was still very popular, e.g. with Baudelaire.) As is well known, a sonnet is a poem of fourteen lines that follows a strict rhyme scheme and specific structure, though conventions associated with the format have evolved over its history. These conventions are not entirely arbitrary though, and indeed often serve as ‘recipes’ for constructing interesting, unusual sounds.
In classical poetry thus understood (i.e. as abiding to fairly strict rules for rhyme, number of syllables, focus etc.), one of the ways in which poetic beauty emerges is precisely through the phenomenon of creativity within constraints, that is the ‘tension’ between the constraints imposed by the rules and how the poet still manages to come up with something innovative and unexpected. It is the difference between, say, a dull sonnet which perfectly complies with the rules for the format and yet fails to ignite a spark, and an inspired sonnet, say Baudelaire’s ‘Correspondances’, which goes much beyond simply complying with the rules by introducing unusual combinations of sounds and words.
Now back to the beauty of mathematical proofs, we now encounter Hardy’s remaining feature of beautiful proofs, namely unexpectedness. Recall that the other five features could be traced back to epistemic ideals linked to explanatoriness, and can (under various forms) be found in the literature on mathematical explanation. But unexpectedness could not be reduced to anything primarily epistemic, and is thus potentially one aspect of the non-functional aesthetic residue in proofs. A beautiful proof is one that displays ingenuity and creativity while also complying with the fairly strict rules of the deductive method; indeed, mathematicians often express a dislike for what are known as ‘brute force’ proofs (e.g. proofs by cases, combinatorial proofs), which typically lack ingenuity. I submit that the aesthetic pleasure emerging from creative proofs is analogous to the aesthetic pleasure emerging from excellent (classical) poetry; ‘bending the rules’ in creative ways, displaying innovation rather than simply mechanically applying the rules. (I do not wish to maintain though that this is the only way in which beauty emerges both in classical poetry and for mathematical theorems; that is, this is merely an existential claim.)
Nevertheless, I still maintain that when mathematicians attribute beauty to certain proofs, it is still by and large something along the lines of functional beauty that guides their judgment, function here being predominantly understood in epistemic and dialogical terms (explanation and persuasion). A beautiful proof is one that is explanatory and illuminating, one that ‘at a glance’ reveals robust patterns among the concepts in question. And so, while functional beauty may not be sufficient to account for the beauty (or lack thereof) of artworks in general (as argued for example by (Bueno 2009)), it seems to be particularly suited for the case of mathematical proofs (even if, also in this case, a non-functional residue of beauty seems to remain).
Bueno, Otavio (2009). Functional beauty: some applications, some worries. Philosophical Books 50, 47–54.
Doxiadis, Apostolos and Barry Mazur (2012). Circles Disturbed: The Interplay of Mathematics and Narrative. Princeton University Press.
Dutilh Novaes, Catarina (2012). Formal Languages in Logic: A philosophical and cognitive analysis. Cambridge University Press.
Hume, David (1960). A Treatise of Human Nature. Oxford University Press.
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.
Parsons, Glenn and Allen Carlson (2008). Functional Beauty. Clarendon Press.
Paterson, Grace (2013). The Aesthetics of Mathematical Proofs. MA thesis, University of Alberta.
 As pointed out by Bueno (2009, 49), “in the case of artworks, in order to identify their proper function, it is important to be able to determine which functional category the works under consideration belong to. For example, religious, historical, or mythological paintings exhibit very distinctive proper functions, given their ties to the communities that produced them and kept them in existence.” In this vein, it may well be that ‘mathematical proof’ is best understood as a heterogeneous class comprising different functional categories. Hersh (1993), for example, suggests that proofs in the context of teaching and in the context of research have different functions. But it seems to me that there is sufficient commonality in that they all aim to produce explanatory persuasion.
 See (Paterson, section 2.5.1) for a discussion on whether the ‘marketplace’ analogy carries over to the case of proofs.
 This is indeed what Netz (2005) does, by taking the concepts of the narrative structure of prose works and the prosodic structure of poems as the guiding threads for his investigation of the aesthetic dimensions of geometric proofs in ancient Greek mathematics.
 The similarities between poetry and (my conception of) mathematical proofs was first pointed out to me by Rachael Briggs (personal communication), who is not only a mathematically-trained philosopher but also an accomplished poet.
 However, in a competitive context of argumentation, unexpectedness may represent a strategic advantage for Prover, who can thus compel the audience to concede certain statements without realizing where they will lead.