Like novelists, mathematicians are creative authors. With diagrams, symbolism, metaphor, double entendre and elements of surprise, a good proof reads like a good story.
So starts Corrie Goldman’s May 8, 2012 article about Stanford University (California, US) professor Reviel Netz and his new book, Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic. (Goldman’s article was republished from the May 7, 2012 article in the Stanford Report.) The article was written in a Question and Answer format and here is an excerpt from the Stanford website,
You have said that a math proof is more focused on the properties of text than any other human endeavor, short of poetry.
Mathematics is structured around texts – proofs – that have very rich protocols in terms of their textual arrangement, whether in the use of extra-verbal elements – diagrams – in the very layout, in the use of a particular formulaic language, in the structuring of the text. And its success or failure depends entirely on features residing in the text itself. It is really an activity very powerfully concentrated around the manipulation of written documents, more perhaps than anywhere else in science, and comparable, then, to modern poetry.
How do you define or identify literary-like elements like metaphors in a mathematical proof?
Metaphor is fairly standard in mathematics. Mathematics can only become truly interesting and original when it involves the operation of seeing something as something else – a pair of similarly looking triangles, say, as a site for an abstract proportion; a diagonal crossing through the set of all real numbers.
You have said that a proof can be seen as having a complex narrative and even elements of surprise much like how a story unfolds. Can you give me an example?
You tell me, “I’m going to find the volume of a sphere.” And then you do nothing of the kind, going instead through an array of unrelated results – a cone here, a funny polygon there, various proportion results and general problems; then you make a thought experiment that shows how a sphere is like a series of cones produced from a certain funny polygon and, lo and behold, all the results do allow one a very quick determination of the volume of the sphere. Here is surprise and narrative. That’s Archimedes’ “Sphere and Cylinder” proof; it’s a typical mechanism in his works. Other authors are often much more sedate and progress in a more stately manner; this is Euclid’s approach.
These questions are answers derived from an April 13, 2012 workshop (Mathematics as Literature / Mathematics as Text Workshop) held at Stanford University.
The description for Netz’s book, Ludic Proof, provides more insight into his work (excerpted from the description),
This book represents a new departure in science studies: an analysis of a scientific style of writing, situating it within the context of the contemporary style of literature. Its philosophical significance is that it provides a novel way of making sense of the notion of a scientific style. For the first time, the Hellenistic mathematical corpus – one of the most substantial extant for the period – is placed centre-stage in the discussion of Hellenistic culture as a whole. Professor Netz argues that Hellenistic mathematical writings adopt a narrative strategy based on surprise, a compositional form based on a mosaic of apparently unrelated elements, and a carnivalesque profusion of detail. He further investigates how such stylistic preferences derive from, and throw light on, the style of Hellenistic poetry.
The word ‘carnivalesque’ made me think of literary theorist Mikhail Mikhailovich Bakhtin, from the Wikipedia essay (footnotes and links removed),
Bakhtin had a difficult life and career, and few of his works were published in an authoritative form during his lifetime.As a result, there is substantial disagreement over matters that are normally taken for granted: in which discipline he worked (was he a philosopher or literary critic?), how to periodize his work, and even which texts he wrote (see below). He is known for a series of concepts that have been used and adapted in a number of disciplines: dialogism, the carnivalesque, the chronotope, heteroglossia and “outsidedness” (the English translation of a Russian term vnenakhodimost, sometimes rendered into English—from French rather than from Russian—as “exotopy”). [emphasis mine] Together these concepts outline a distinctive philosophy of language and culture that has at its center the claims that all discourse is in essence a dialogical exchange and that this endows all language with a particular ethical or ethico-political force.
I didn’t find that description as helpful as I hoped and so clicked to Carnivalesque and here I found a liaison between this term and Netz’s response about mathematical proofs unfolding as complex narratives with surprises,
Carnivalesque is a term coined by the Russian critic Mikhail Bakhtin, which refers to a literary mode that subverts and liberates the assumptions of the dominant style or atmosphere through humor and chaos.
It’s not the first time I’ve across a reference to Bakhtin’s theories, specifically ‘carnivalesque’, in the context of scientific and/or technical writing. Somehow one doesn’t usually associate chaos, humour, and surprise with those writing forms and yet, ‘carnivalesque’ keeps popping up.