yume no ato
The summer grasses:
The high bravery of men-at-arms,
The vestiges of dream.
— Matsuo Basho, on visiting Hiraizumi, once home to the great Fujiwara clan whose splendid castles had been reduced to overgrown grass mounds.*
A good haiku not only arrests our attention, it also demands reflection and contemplation of deeper themes. In Basho’s Oku no Hosomichi, The Narrow Road to the Interior, the haiku often serve as a point of pause amidst the travelogue, asking the reader to slow down and take in all that is being said. The slow road to understanding is often the easiest way to get there. At the same time the travelogue itself provides context for the haiku. Without that context, both from the travelogue, and from our own experiences of the world upon which the haiku asks us to reflect, the poem becomes shallow: you can appreciate the sounds and the structure, but the deeper meaning — the real essence of the haiku — is lost.
Mathematics bears surprising similarities. A well crafted theorem or proof demands reflection and contemplation of its deep and wide ranging implications. As with the haiku, however, this depth is something that can only be provided by context. A traditional approach to advanced mathematics, and indeed the approach you will find in most textbooks, is the axiomatic approach: you lay down the rules you wish to play by, assuming the bare minimum of required knowledge, and rapidly build a path straight up the mountainside. This is certainly an efficient way to get to great heights, but the view from the top is often not rewarding unless you have spent time wandering through the landscape you now look out upon. Simply put, you lack the context to truly appreciate the elegant and deep insights that the theorems have to offer; like the haiku it becomes shallow.