"Like novelists, mathematicians are creative authors. With diagrams, symbolism, metaphor, double entendre and elements of surprise, a good proof reads like a good story. (...) [Reviel] Netz reveals the stunning stylistic similarities between Hellenistic poetry and mathematical texts from the same era. (...) In the very layout, in the use of a particular formulaic language, in the structuring of the text (...) 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. (...) 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." - Amira

"The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." -- G. H. Hardy, A Mathematician’s Apology (1941) - Amira

the difference between communicating a proof orally in person, and formally in writing, is so vast -- the former does rely on storytelling of sorts, then there's the twist in the plot where the innovation or the key insight is made... and there's also a lot of "hand waving" (an official mathematical term :-) - Adriano

Sometimes also some sort of poetry or storytelling seems to be 'hand waving' in itself. :-) - Amira

Indeed, Amira, linguists have mostly ignored the grammar and syntax of hand gestures which is more primitive than oral or written communication. We implicitly accept its deep structure and understand its rich semantics, conveying more than words sometimes :-) - Adriano