Moscow’s elaborate subway system has stations dotting the entire city. Riding for five minutes in the fluorescent glow of a train car, only to emerge forth through the station’s doors, I find myself transported to a new, fantastic, and completely arbitrary world. Thus exploring, I “collect universes”. On foot, I travel through them; underground, I travel between them.
In my mind, another world is explored: the abstract landscapes of mathematics itself. These universes are expanses of terrain; they’re built and connected, however, out of pure ideas. They’re abysses of blackness, filled only by the shifting machinery of logical structures. They’re universes of mechanical ideas. I explore them too.
There’s Taganskaya, a station on the far eastern edge of the city’s “circle line”, where the sun’s slanting orange rays, in the late afternoon, illuminate crumbling, rustic European-style stucco walls. Walking away from the pulsing metro station as the roar of traffic subsided, I entered Taganskaya’s quieter back-alleys; I walked past many a small, nice restaurant, with the owner calmly smoking a cigarette out front.
There’s Krasnopresnenskaya, a humble, worn-down collection of buildings in the city’s northwest corner. Early in a cold morning, under a light blue sky, I passed through the quiet, enclosed courtyards of Russian-style living complexes. Moving from the brisk air of the main station, I pushed deeper into the maze of enclosed open-air squares. I walked past playgrounds and basketball courts – either deserted, or full of shouting children – as adults strolling past, bundled up against the cold, slowly walked their dogs.
Meanwhile, as I walk, my mind is working. Passing through difficult thoughts of life and the future, I also struggle to assemble mathematical ideas.
In the universe of algebra, we consider sets with abstract operations defined on them. These operations do more, however: they endow these sets with structure. This structure takes on a universe of intricacy, folding and shifting like an unimaginable machine. Subsets of these sets fold and permute under their own smaller group structure; these subsets decompose the group itself, generating a new group altogether. Fields, the largest structures of them all, are systematically augmented to produce even larger ones, special ones, massive fields which support their own mechanical ecosystem. Massive machines contain smaller ones. Mapping one machine to another, we convert it, breaking it apart and assembling it into yet a new whole… I stepped beneath the street into the train station.
There’s Paveletskaya – shining metropolis of the night – where a bustling food market in the station’s local courtyard is overshadowed by the towering lights of the distant buildings. Walking alone through the market’s crowded stores, past shouting shwarma vendors and colorful flower shops, I ducked into a fruit stand and bought a delicious fig. I sat on a bench in the night air and ate it in silence, looking out at the city lights, as the group of drunkards near me smoked and joked amongst themselves.
My mind was light and open. I continued to think. In the topological universe, we systematically – relentlessly systematically – explore the massive, uncharted topological space. This space is perhaps, though not necessarily, a manifold, a generalization of space itself, conceivably as complex as the human body and as large as the sun. We deploy an infinite army of minuscule loops, each crawling over the surface of this sun, shifting and scanning for subtleties and snags. These loops spontaneously organize, coagulating into groups laden with structural meaning. Next, we deploy an infinite army of two-dimensional sheets, similarly stretching, wrapping, and identifying as they explore the surface of the sun. Next comes the three-dimensional army. Generalizing to arbitrary dimension, we generating a huge array of structural information. The topological space becomes known.
Kievskaya, western and austere, is home of the “Moscow City” business district, a small clump of skyscrapers that’s as startlingly modern as it is entirely out-of-place. Rising above the backdrop of a small, underdeveloped park, the handful of sparkling buildings give one the impression of a city trying to be something it’s not.
I was the only one at the park. Is math a set of truths or a set of tools? Calculus, a prodigious collection of mathematical tools, allows us to make mathematical sense of properties – such as, for example, curvature and volume – which only emerge upon contact with the “infinitely tiny”. These tools extend, with stunning complexity, into arbitrary dimension – imagine calculus in twenty-dimensional space! Much more profound, however, is the extension of calculus to manifolds, the most abstract spaces conceivably possible, and the theoretical apparatus designed to facilitate this extension. Now, we can mathematically define and investigate – given a map between manifolds – whether this map is smooth, or differentiable, or integrable, or, truly, anything we can hope to imagine.
Prospekt Mira, in the city’s northeastern end, presents the run-down and inexplicably sad sight of a grey-green park, wide and flat, with the towers of old soviet apartment blocks rising in the distance. Walking deeper through the park, one finds a lone, Orthodox Church – as isolated physically as it is architecturally – where, in spite of it all, one might still find a Saturday night service packed with women in headscarves and bowed, grey-haired men.
Moscow is an eternal city, ripe for exploration. I strap on my backpack once in a while – when I have the time – and walk to the nearby station; the small blue metro card in my wallet serves as a portal to new universes.
The deepest exploration, however, occurs in my own mind, as I push through, and try to make sense of, the mathematical worlds unfolding before me. These worlds are without time and without space; they’re inexplicably fascinating and inextricably lonely. Here – and only here – lies the eternal city.