Grassmann's Chalk

The chalk in my hand is not romantic chalk, not the cinematic Cambridge chalk that arrives in little velvet-lined coffins like it’s a count who died in a duel; it’s the cheap, porous, slightly gritty Calcutta chalk that leaves a powdery corpse on your fingers and makes your dark T-shirt look like you’ve been hugged by a depressed ghost.

The room is doing that winter thing it does here—closed windows, trapped breath, a faint smell of damp cement, and the low-grade melancholy of sunlight that can’t quite commit to being sunlight. On the table: a battered ruler with centimeter marks half-erased, a notebook with squares like a city grid designed by someone who hates joy, and a pen that writes only when it feels spiritually supported.

I draw two arrows. Then a third. Then I stare at them as if they are going to confess something.

Because here’s the obscene miracle: with a small handful of arrows—directions, really, not quantities; a few stubborn “go this way” instructions—you can build an entire universe of movement. You can describe any point in a plane, any wobble of a bridge, any pixel shift in a camera, any portfolio of stocks that will betray you, any state of a quantum system that doesn’t even have the decency to be one thing at a time. You can take the chaos of “everything” and say: give me these few directions, and I’ll express the rest as a combination. That’s a basis. It’s a pedestal, a scaffold, a chosen set of bones.

And every time I write the word basis—that pompous little syllable that sounds like a brand of deodorant—I’m apparently doing something else too: I’m quietly lighting a candle for a dead German schoolteacher who was, for a while, treated like a man trying to sell umbrellas in a desert.

The Man Who Invented a Room Nobody Wanted to Enter

Hermann Grassmann. The name itself has that satisfyingly Teutonic heaviness, like it should be stamped into iron or used to label a type of stubborn bread. Grassmann was not, by the standards of mathematical folklore, a chosen one. No royal chair, no velvet-draped institute, no saintly circle of disciples who collected his chalk dust in little jars. He was a schoolteacher in the most literal sense—teaching adolescents, wrangling ordinary curricula, living among the quotidian irritations of timetables, examinations, and human mediocrity, which is to say: he lived among us.

In 1844 he published a book with a title that feels like it was designed to defeat the casual reader at the door: Die lineale Ausdehnungslehre—“The Theory of Linear Extension,” roughly, though the German has this delicious physicality to it, as if mathematics is not a cathedral of pure thought but a stretching of fabric, a pulling of taffy, a measuring-out of space like dokan-dar measuring rice. Ausdehnung is extension, yes, but it’s also expansion, the act of reaching out into more room.

And the book—this is the part that makes me want to laugh and swear and then go wash my face—essentially contains the engine of what we now call linear algebra. Vector spaces (without that later hygienic axiomatization), linear combinations, subspaces, basis, dimension, independence; an algebra of “directed magnitudes” that, if you’ve ever written a wedge product or heard the phrase “exterior algebra,” you’re already living inside. It’s the blueprint for a house people now cannot stop building, and he was standing there in 1844 shouting, I have invented a new kind of room, and the mathematical establishment, with the generosity of any establishment, responded with some version of: That’s nice, now please stop embarrassing yourself.

The tragedy is not that his ideas were wrong. The tragedy is that his ideas were too right in a way that made everyone else feel stupid, and people—Homo sapiens, that anxious ape with a little hat—do not tend to reward you for making them feel stupid.

If you want a metaphor (and I always do; my brain is basically a metaphor factory with inconsistent labor laws), imagine walking into a marketplace where everyone is selling spices, and you show up with a refrigerator. You’re not merely offering a better spice; you’re offering a fundamentally different way of keeping food from rotting. It’s not “competing.” It’s alien. It makes their entire mental furniture rearrange itself, and they don’t want to move the sofa.

So they ignore you. Or they mock you. Or, worst of all, they politely misread you. And then you go home and teach school again.

There’s something particularly cruel about being right in a way that cannot cash out immediately. Wrongness at least has drama: it can be refuted, corrected, turned into a moral tale. But being prematurely correct means you sit there in a long, quiet corridor of history, listening to your own footsteps echo, wondering if you’ve only produced very elegant nonsense.

Grassmann died without seeing the full vindication. The world took its time—half a century, give or take—before it realized that he’d built something like a universal grammar for space, for systems, for structure itself. His book became, in retrospect, a kind of mathematical palimpsest: the faint writing beneath the writing beneath the writing. Later people wrote over it—cleaner notation, better axioms, slicker pedagogy—but the old script was there all along, like a stubborn ghost refusing to vacate the building.

And yes: every time you define a basis, you are vindicating Grassmann. Which is a lovely sentence, and also a bit of a punch in the gluteus maximus, because it implies that vindication is something that can arrive decades late, strolling in with its shoes on, while the person who needed it is already compost.

“Basis” as a Word, as a Scam, as a Small Salvation

I’ve always been suspicious of words that sound too firm. Basis. It’s Greek, of course—básis, a step, a pedestal, a base. Something you stand on. The kind of word an institution loves because it suggests there is a floor and it is solid and nobody is going to fall through.

But in practice a basis is a weird kind of chosen fiction. You pick a set of vectors (or you’re given one by a textbook that pretends it fell from the sky), and then you agree—collectively, like a committee deciding what counts as “normal”—that these few directions will be the language in which everything else is spoken.

The thing itself is objective, but the representation is a choice. That’s the sly beauty: we pretend we’re doing pure truth, but we’re also doing convenience. We’re doing coordinates. We’re doing bureaucracy for space.

I like that, perversely. It’s honest in a back-alley way. It admits that understanding is not merely seeing; it’s also choosing how to describe what you see, which is why the same reality can be narrated as physics or poetry or propaganda. A basis is a coordinate system; a coordinate system is a story; and stories are how Homo sapiens sneaks meaning into indifferent matter and then acts surprised when the meaning leaks.

Grassmann, I suspect, saw this much earlier than people were emotionally prepared for. He wasn’t just computing; he was offering a way to think—a way to treat “space” not as a stage set with fixed props, but as an algebraic object you can manipulate, extend, factor, and generalize. He was, in effect, saying: stop treating geometry like an illustrated children’s book; it’s a grammar, and I’m going to give you new verbs.

And mathematicians, being mathematicians, eventually came around. But not because they are noble. Because the world—physics, engineering, electromagnetism, mechanics, later computing—started demanding a language that could handle systems, transformations, many variables moving at once like a crowded tram. Reality became too big and too entangled for cute Euclidean diagrams. The old tools started creaking. Then suddenly Grassmann’s weird room looked less like a prank and more like shelter.

A Bengali Man Meets German Abstraction (and Both Sigh)

There is also the delicious irony that Grassmann, this German abstraction machine, was not only a mathematician. He worked on language. He did philology. He wrestled with Sanskrit, of all things—Sanskrit, that ornate and ancient instrument that can name ten kinds of “being” while English is still trying to decide whether “is” is one thing or three.

This detail always makes me feel an odd, private tenderness. Because here I am, a Bengali man, compulsively bookish, raised in a culture where “respectable” means you can quote and conjugate, where parents dream of their sons as walking exam results, and here’s this 19th-century German man pouring his life into both algebra and the Rigveda, as if the universe is one continuous text and he’s trying to translate it.

It also makes the tragedy sharper. Because the kind of mind that builds a new algebra of space is often the kind of mind that can’t—or won’t—do the social algebra required to get applause on schedule. There is a social basis too: the right set of vectors (connections, institutions, patrons, fashionable topics) that can span the space of “recognition.” If you don’t have them, you can still be right, but you won’t be heard.

And this is where my left-leaning, anti-authoritarian reflex starts baring its teeth. Institutions love innovation the way landlords love tenants: only after the tenant has proven they can pay. The myth is that merit rises. The reality is that merit is often a poor, brown, stuttering thing without a microphone. The world is not a fair judge; it’s a distracted audience.

But here I have to check myself, because indignation is an addictive drug for people like me. It comes with a clean narrative: I am the victim; the world is the villain; the end. It’s delicious, and it’s also lazy.

Yes, institutions are cruel. Yes, they ignore the unfashionable. But also: I am complicit in my own invisibility. I hide. I procrastinate. I rewrite drafts obsessively until the original impulse is dead, like a frog dissected in the name of biology. I demand perfect conditions—silence, mood, clarity, a brain that behaves—and then act shocked when life refuses to provide a laboratory.

America, Then and Now, Through the Lens of a Whiteboard

When I first lived in America (1998 onward), the thing that struck me wasn’t the highways or the supermarkets—though, yes, those too, the aisles of cereal like a library of sugar—what struck me was the atmosphere of systems. Universities felt like machines that actually ran. You could walk into a classroom and find markers that worked, projectors that turned on, restrooms with an almost religious abundance of paper, and a kind of quiet assumption that the infrastructure would do its job.

In those rooms, linear algebra felt like part of the architecture. Whiteboards, clean grids, crisp notation; the idea that if you define your basis properly, everything else becomes manageable. Even life seemed to have a basis: degrees, internships, jobs, promotions, mortgages—vectors arranged in a socially recognized span. You could disagree with the system, but you couldn’t deny it had a shape.

Now, observing America more recently from afar, through the distorted funhouse mirror of the internet, I feel something has… frayed. The machine still runs, but it sounds louder. The discourse feels more feverish, more performative, like everyone is auditioning for a role in a tragedy that nobody rehearsed. The confidence in shared coordinates—shared facts, shared institutions, shared trust—seems weaker. It’s as if the basis vectors are wobbling, and people are pretending the wobble is freedom.

India, meanwhile, has its own version of this: we never had stable coordinates to begin with, so we’ve become virtuosos of improvisation. Our national superpower is doing linear combinations of nonsense and still arriving at a functioning day. We span the space of survival with whatever vectors are available: jugaad, family networks, bribery disguised as “speed money,” faith as anesthesia, cynicism as armor. It’s a kind of applied algebra of endurance, and it’s both impressive and disgusting.

The education system here—my god. It teaches mathematics the way some households teach manners: as a performance, not as understanding. You learn to recite. You learn to mimic. You learn to get the right answer without knowing what question you’ve answered. We produce students who can solve determinants by rote but cannot tell you what a determinant means—cannot see it as volume scaling, as orientation, as the shadow a transformation casts onto space.

The Bipolar Basis: How Mood Chooses Coordinates

There’s another layer here, and it’s not philosophical; it’s clinical.

My mood chooses my basis.

On some days—hypomanic, caffeinated by nothing, vibrating with connections—I can take a scrap of chalk and build an entire cathedral out of vectors. Everything relates. Everything rhymes. Grassmann becomes a symbol of all unrecognized labor, all ignored minds, all the quiet geniuses in government schools, all the women whose work got stapled to a man’s name, all the anonymous engineers keeping the world’s machines from exploding while some CEO receives an award for “vision.” I can hear, in my head, a choir of footnotes singing.

Then depression arrives, and it flattens everything into blunt geometry: a line segment of disappointment, a plane of futility, a dimensionless point of “who cares.” The same story—Grassmann ignored, later vindicated—stops being inspiring or tragic and becomes simply: people die without getting what they deserved. Full stop. The end. No poetry, no arc.

It’s humiliating to realize how much of my intellectual life is a function of neurotransmitters. I want to believe I am a sovereign mind, a little republic of reason. In reality I am often a weather system with opinions.

Vindication, Delayed: A Receipt Nobody Can Cash

What does it mean to be vindicated after you’re dead?

It means your work becomes part of the air, and you don’t get to breathe it.

It means students casually say “choose a basis” the way they say “open your laptop,” without any awareness that a man once had to carve that concept out of the stubborn wood of old geometry, splinter by splinter, with nothing but his mind and a schoolteacher’s stamina. It means your name becomes an adjective in a textbook—“Grassmann algebra”—and most people treat it as a label, not a life.

There’s also the uncomfortable thought that Grassmann’s work was eventually embraced not because people became more virtuous, but because the problems of the world forced them to adopt his framework. Physics needed it. Engineering needed it. Later, computers needed it. The universe, in its indifferent way, “paid” him by making his language necessary.

Which raises a question I don’t enjoy: what if some ideas never become necessary? What if they are beautiful but not demanded? What if they die quietly, not because they’re wrong, but because the world never hits the kind of wall that requires them?

A Small, Stupid Basis for Living

When I stop trying to make this grand, I come back to the chalk dust on my fingers.

A basis is, at heart, a small set of things you return to. Directions you trust enough to build from. In mathematics, it’s vectors. In life, it might be less dignified: making tea, taking a walk, reading a page, cleaning the desk, texting a friend, not being an asshole for one day. Tiny spanning sets of sanity.

My own basis is embarrassingly narrow. A few books. A few memories. A few habits that keep me from turning entirely into resentment. Sometimes even the act of drawing two arrows in a notebook feels like an assertion: space exists; structure exists; not everything is noise.

Grassmann’s story doesn’t redeem the world. It doesn’t even redeem mathematics, which can be as petty and status-obsessed as any other human tribe, just with cleaner handwriting. But it does offer a bleak kind of companionship: proof that a person can be ignored and still be building something real.

A little dignity, scribbled in chalk, in a room that smells faintly of damp winter and old paper, while the city outside conducts its usual opera of horns and exhaust and restless human hunger.

I look again at my three arrows. I think about the German schoolteacher and his unread book. I think about all the unseen labor holding up the world, like basis vectors holding up a space nobody notices until it collapses.

The chalk dust stays on my fingers.

I turn off the light anyway.