The Plan of a Square

Amidst many discussions on super-valuational algebras and their philosophical applications — on which I was writing my dissertation — Hans and I once paused to ponder the mystical experience of the square. I mean A Square, the hero of Flatland. I mean that perfectly two-dimensional being, with no depth whatsoever, citizen of an equally two-dimensional depthless world, who one day had the good fortune of receiving a visit from a Sphere. What's more, he had the fortune of being able to visit, albeit briefly, the undreamt-of three-dimensional world his guest came from — which is to say, our three-dimensional world. He visited and experienced our world before falling back for all eternity into the total flatness of his: Flatland, the plane world, the world with no aboves and no belows, the world in which cars and airplanes alike, so to speak, belong to the same category, and everything, literally everything, is reduced to fragile shadows on an enormous and eternally illuminated floor. (Which does not mean that Flatland was a perfectly democratic world. Power was in the hands of the caste of Circles, most certainly not in the hands of the infamous Irregular Polygons.)

I said that the Square's experience was a mystical one, and this was the point of my discussion with Hans. It is almost as if we were given the opportunity to visit a four-dimensional world, a world which we have no knowledge of, and whose shapes, beauties, and dangers we could not quite imagine. For us the three dimensions are everything: they constitute a habitat so natural that to us it seems impossible to envision different spaces, new and unknown dimensions, shapes that don't even fit into our imagination. And I am not thinking of the temporal dimension: that dimension we know far too well. I am thinking specifically of a fourth dimension of space, which I wouldn't even know what to call, just as the Square did not know what the Sphere's greatly extolled 'depth' was. Now, some will say this is just science fiction, a play of wit, late-Victorian satire at best. Perhaps. But I'm sure Hans would rather speak of mental horizons. And the capacity to extend our mental horizon is not a matter of science fiction, for it is there that we can measure our provincialism. It is there that we can see if we are truly able to think freely, to push ourselves beyond the obvious. And it is there that we can test our sense of possibility — that sense of possibility upon which Reverend Abbott invited us to reflect in such an unusual way but which is the driving force, I think, in many a writing by Hans himself, if not in his way of being a philosopher.

But I don't intend to dwell on these topics here. I simply wish to honour Hans by presenting him with a note that I happened to find while rummaging through some old papers in the attic and that speaks for itself. For I have reasons to believe that the author is no one else than the Square. Indeed I am certain about that. The date on the note is not readable, but it will be apparent that it is posterior to the Sphere's visit, even if Abbott doesn't mention it explicitly in his novel. So here it is, or rather here is the text, which I've strived to transcribe with the utmost accuracy.

The Sphere said that our world is flat, but can have different shapes. She said this more than once and for a long while I tried to understand what this could possibly mean. If the world is flat, is this not enough? What else could we add, if not to bemoan our limits and to recognize the scarcity of our horizons? Isn't it enough to say that the land in which we are forced to live is akin to a vast sheet of paper, with no width or depth? I will admit that some time ago I would have said 'the...