There’s a beautiful novel by Italian writer Italo Calvino called Invisible Cities, which narrates a fictional encounter between Mongol emperor Kublai Khan and a certain young Venetian explorer. Nearing the end of his life, the aging emperor begins to realize that the boundless reaches of his empire encompass cities unseen and unknown to him — cities pulsating with unfamiliar patterns and memories. The Khan’s own footprints of conquest are among these patterns and memories; yet so, too, are the cycles of history and empire intertwined with the objects and signs of human dreams and remembrances. The young explorer, a man by the name of Marco Polo, recounts stories of his travels to the emperor. Though the emperor is skeptical of the veracity of Marco Polo’s stories, he listens attentively to the descriptions of these invisible cities. Soon their names and rhythms become entangled in his own imagination.
“Invisible Cities follows a beautiful mathematical structure that repeats chapter titles to form a cascading pattern”
Invisible Cities follows a beautiful mathematical structure that repeats chapter titles to form a cascading pattern (see here for a visual). Marco Polo speaks of 55 unique cities, each in its own short chapter titled with one of 11 possible titles, including those such as “Cities and memory,” “Hidden cities,” “Cities and names,” or “Cities and eyes.” Calvino’s structure was intentional — in a posthumously published interview in The Paris Review, Calvino said of Invisible Cities that “the design … became the plot of a book that had no plot” and that “the architecture is the book itself.”
Calvino was a part of an experimental literature group known as “OuLiPo” (Ouvroir de Littérature Potentielle) which was formed by writers and mathematicians in France in the 1960s. Its members explored the possibilities for verse and prose written under predetermined structural constraints.
OuLiPo’s focus was not new but rather a continuation of the centuries-old practice of studying mathematics and poetry together. The10th-century Indian mathematician Halayudha, for example, wrote extensively about combinatorial mathematics in his studies of Sanskrit prosody. In his writing he presented the construction of what’s now known in the West as Pascal’s triangle as a way of generalizing possibility for poetic metres. Even earlier than Halayudha in around the 2nd or 3rd century BCE, another Indian mathematician Pingala wrote the Chandaḥśāstra, a treatise on Sanskrit poetry. The Chandaḥśāstra contains an algorithm for calculating powers of 2, discussions of the Fibonacci numbers, one of the earliest uses of the number 0 and descriptions of binary numeral systems.
Outside of rigorous studies of meter and form, mathematicians have for a long time dabbled in poetry in other ways. Archimedes’ famous cattle problem was presented as a poem in a letter to Eratosthenes, and the 16th-century mathematician Tartaglia revealed his discovery of how to solve certain kinds of cubic equations in a poem to another mathematician.
Aside from its structure, I also find Calvino’s novel to be beautiful for the ways its abstractions and images extend themselves in my mind to my own invisible cities. For me, studying math often feels like being led through unknown parts of unseen cities, or like trying to untangle architectural blueprints of the buildings within familiar cities. Sometimes studying math feels like no more (or no less) than standing in the shadow of one of these buildings, unable to see the top and unsure of where the entrance is or even what lies inside.
“For me, studying math often feels like being led through unknown parts of unseen cities, or like trying to untangle architectural blueprints of the buildings within familiar cities.”
When describing the fictional city of Zaira, Marco Polo says to the Khan, “I could tell you how many steps make up the streets…and the degree of the arcades’ curves … but I already know this would be the same as telling you nothing. The city does not consist of this, but of relationships between the measurements of its space and the events of its past …” I wrote about metric spaces last semester as an example of a structure in math that is defined by how distance is measured within the space. This semester I’m taking a class in measure theory where most of the stories begin with “Let (X, A, μ) be a measure space …” A measure space is defined by a measure. It is not something imposed on the space afterwards the space is created; it is an essential characteristic of the space. Just as the measure defines the measure space, measurements between objects in the city of Zaira define Zaira.
Later, while describing the city of Tamara, Marco Polo says, “Your gaze scans the streets as if they were written pages: the city says everything you must think, makes you repeat her discourse, and while you believe you are visiting Tamara you are only recording the names with which she defines herself and all her parts.” The predominant feeling I’ve had this semester in my measure theory class has been bafflement. As I struggle through writing proofs for homework assignments I find myself reading and re-reading definitions and theorems in attempts to properly internalize ideas. While I may believe I am visiting the invisible city of measure theory, most of the time at my level I am only recording the names that define it and all its parts in the hopes of building a clearer image of the city in my mind.
“While I may believe I am visiting the invisible city of measure theory, most of the time at my level I am only recording the names that define it and all its parts in the hopes of building a clearer image of the city in my mind.”
Although measure theory often perplexes me, Calvino offers a reminder that there is immeasurable value in being lost in unfamiliarity. He writes, ”… the more one was lost in unfamiliar quarters of distant cities, the more one understood the other cities he had crossed to arrive there …” While there are many concepts in measure theory that I still barely understand, I now have a much deeper appreciation for concepts I’ve seen previously and which I now know were only specific cases of more generalizable, and perhaps therefore more sublime, results.
As an undergraduate, most of the invisible cities of mathematics in my mind are at best skeletal.
Even so, they’re there, sometimes on their own and sometimes as parts of others: cities of memory, hidden cities, cities of names, cities of eyes. Our invisible cities — the sciences we study, the languages we think in, the ideas we ponder, the art we devote ourselves to, the stories we learn and re-tell, the wisdom we seek — subliminally define our notions of potentiality. As a result, they characterize our creativity.