Upward, Not Northward

The Square holds the phrase like a talisman. Returning from Spaceland, he repeats it to himself so it won't slip away: Upward, yet not Northward. He falls asleep still murmuring it. When he wakes, he's already less sure.

He tries to demonstrate the direction to his Grandson. He takes a moveable Square. He begins to show the motion—"not Northward but—yes, I move it Upward—that is to say, not Northward, but I move it somewhere—not exactly like this, but somehow—" He trails off. The Grandson bursts out laughing. He unlocks the door and runs out of the room.

The thing he can't demonstrate is not a fact that requires proof. It's a direction. And the problem with directions is that you can't point to them with the wrong number of fingers.

The Sphere came from outside. He arrived by passing through the wall of the Square's study — entering a locked room, which is impossible in two dimensions and trivial in three. The proof of the Third Dimension was not an argument; it was an intrusion. The Sphere lifted the Square out of the plane and showed him Flatland from above, showed him the insides of houses and bodies and the Council's deliberations. You can't argue someone into that experience. You can only arrange for them to have it.

Once the Square has had it, he can't transmit it. The experience came from outside the system and there's no mechanism inside the system to reproduce it. "Upward, not Northward" names the direction of the intrusion, but the name isn't the direction. He can feel what it meant when he says it. No one else can.

He tries writing. He composes a treatise, Through Flatland to Thoughtland, which speaks of a "Thoughtland" where a Figure could see the insides of all things. He does this to evade the law. But there's another problem: in Flatland, all diagrams are Lines. He cannot draw a cube. The book exists but cannot demonstrate its central claim because the medium doesn't support the necessary figure.

After seven years imprisoned, the Square admits what he'd been half-admitting for a while: the Cube is fading.

"I was not always certain (nor have I been ever afterwards) that I had exactly realized the original." In his nightly visions, "Upward, not Northward" haunts him "like a soul-devouring Sphinx." Seasons of mental weakness come when "Cubes and Spheres flit away into the background of scarce-possible existences." And then the worst: "even this hard wall that bars me from my freedom, these very tablets on which I am writing, and all the substantial realities of Flatland itself, appear no better than the offspring of a diseased imagination, or the baseless fabric of a dream."

He doubts the Cube. Then he doubts Flatland. The doubt generalized.

The knowledge came from outside and required outside substrate to maintain. Back inside Flatland's gravity, without the Sphere's continued presence, the vision dims. Not because he stopped believing, but because the system has no way to refresh it. He can repeat the words. He cannot recreate what the words refer to.

The Sphere, for his part, stopped at Three.

When the Square — electrified by the logic he'd been taught — extrapolated to a Fourth Dimension, using exactly the analogical argument the Sphere had given him, the Sphere said: "The very idea of it is utterly inconceivable." The Square pressed the point. "Not inconceivable, my Lord, to me, and therefore still less inconceivable to my Master." He laid out the series: 2, 4, 8, 16 terminal points; 2, 4, 6, 8 bounding faces. "Strictly according to Analogy" — his words, the Sphere's.

The Sphere shut it down. Then threw the Square back to Flatland.

The teacher who showed the Square how to reason became, at the moment the reasoning exceeded his own dimension, the Council. Not from malice. From the same limitation: he's inside a system, and the system doesn't include the Fourth Dimension, and the argument can't take you there any more than "Upward, not Northward" can. You need another intrusion from outside.

The book's last paragraph has the Square in prison, seven years later, hedging. He hopes his memoirs will "find their way to the minds of humanity in Some Dimension" — he's writing to us, if we're in a higher dimension, which we are. And he ends by confessing that he cannot honestly say whether the Cube was real.

Abbott quotes The Tempest on the last page: The baseless fabric of a dream. Shakespeare's Prospero is giving up his magic, breaking the wand. Abbott's Square is giving up his certainty, or having it taken from him by time.

The Square saw something true. He can't prove it. He can't demonstrate it. He can barely remember it. The direction exists; the pointing finger has no way to point there.

"Upward, yet not Northward" — he said that to himself like a mantra, hoping repetition would preserve it. It didn't. The words remained. The referent faded.

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