Essays in abstraction

If my Readers have followed me with any attention up to this point, they will not be surprised to hear that life is somewhat dull in Flatland. I do not, of course, mean that there are not battles, conspiracies, tumults, factions, and all those other phenomena which are supposed to make History interesting; nor would I deny that the strange mixture of the problems of life and the problems of Mathematics, continually inducing conjecture and giving the opportunity of immediate verification, imparts to our existence a zest which you in Spaceland can hardly comprehend. I speak now from the æsthetic and artistic point of view when I say that life with us is dull; æsthetically and artistically, very dull indeed.

A square, Flatland, 1884

Shot on D-4. Nothing. Water. Now on the other side. Shot at B-3. Also, water. Again, the other player: J-7, right on the edge of the board. Water, again. And this alternate shooting continues, to explore the squares of a grid whose coordinates – arranged vertically in alphabetical order, horizontally (abscissa axis) in numerical order – represent any place in the middle of a grid space, which is like a fictional portion of the sea, with the aim of hitting one of the five ships hidden in the opponent’s grid. It’s a game of strategy and guesswork, where each player tries their luck, firing speculative torpedoes, knowing only that somewhere, horizontally or vertically, a battleship, a submarine, an aircraft carrier, a destroyer and a cruiser occupy small successive series of two to five squares. The shooting is so speculative that each player has two square grids, one horizontal and one vertical, which relate respectively to the distribution of their shots and ships and the supposed distribution of the opponent’s ships.

The shots themselves aren’t represented until they’ve erupted on the square they hit, which isn’t surprising given the fact that the grids only represent two-dimensional spaces. Shots occur in higher dimensions, but their effects are expressed in orthogonal planes. This means that two-dimensional entities cannot see entities of other dimensions unless they intersect their planes.

Something similar was described at the end of the 19th century in a satirical mathematical fiction novel written by an English schoolmaster, Edwin Abbott Abbott, who signed it simply as “A Square,” his native narrator. In Flatland, a two-dimensional world, only polygons lived—flat figures composed of edges, vertices, or closed curved lines and the areas they enclose—but also, in an inferior position (because Flatland is a hierarchical and inflexible world, much like the stratified Victorian society in which it emerged), line segments, that is, entities with only one dimension, and, occasionally, points. Points, obviously, have no dimensions; they have no length or width, but are located somewhere in a Cartesian space that can be defined by coordinates, like shots in the game Battleship. On a New Year’s Eve, the square dreams of a world with only one dimension, Lineland, where men are lines and women are bright points. Neither men nor women can see the square, except as an occasional one-dimensional manifestation—that is, a series of points on a line segment. In the dream, the square tries to show the king of Lineland that there is a second dimension, but fails, and wakes up under threat of death.

After this dream, the square is visited by a sphere, but just as lines and points couldn’t see him in his dream, now it is the square who can only see the sphere as a circle—one that grows or shrinks in area as it intersects the plane (Flatland), and sometimes disappears completely when the volume of the sphere lies outside the plane. He cannot see it nor understand the existence of a third dimension, no matter how hard the sphere tries to make him see it or uses analogies with lower-dimensional objects, to which one would simply need to add a dimension: a point becomes a line, a line a polygon, and a polygon a polyhedron or another three-dimensional shape. Only through a journey to Spaceland, a three-dimensional world, does the square, led by the sphere, finally understand the existence of higher-dimensional entities. Inspired by this transformative revelation, the square—just as in the dream, but now in a speculative projection—tries to convince the sphere of the theoretical possibility of a fourth dimension, which appears absurd to the three-dimensional solid. The rigidity and intolerance toward different realities was similar in Spaceland to that of Flatland, or even Lineland.

The square has another dream, in which he is again visited by the sphere, who this time introduces him to a dimensionless world: Pointland. There, a single inhabitant—the point—is simultaneously the subject, the monarch, and the entire universe. In its solipsistic autism, the point does not even comprehend the square’s attempt to communicate as anything other than its own thought. Back in his own world, the square is also never able to convince his peers of the existence of a three-dimensional world. In fact, his attempts to show it brought him disgrace and persecution, ending in prison, where he wrote this memoir.

This parable—or fable, though the status of the characters is ambiguous here—like all allegorical narratives, oscillates between movements of abstraction and concreteness. Although in this case, the scale clearly tips toward abstraction, especially since the characters are mathematical objects—abstract entities. Not even the attempt to individualize the narrator and main character—a square—escapes the indeterminacy and vagueness of the indefinite article. If a square is that square, he is also just any square.

The “a” in “a square” is less a cardinal numeral quantifier than an existential quantifier. That is, although “a” is one, one is not the one—that is to say, it is not the number “one,” the first of the integers, but one among many. It’s an any one. And yet, it is a square—not a triangle, nor a circle, nor any other polygon. But let’s take it step by step, even though one has no parts, no whole parts, only rational parts: a half, a quarter, an eighth, a twelfth, or n other parts. And n here can also be one, or two, three, four, and so on. n is a variable. But n could also be x. It doesn’t really matter in this case. That much is clear.

What might not yet be clear is this: if the “a” in “a square” is not the one, but any one, does it even really exist? And the funny thing is—even if I don’t want to confuse my reader—is that whether it is any one or the one, the “one” [i.e, the “a”] in “a square” perhaps does not exist, at least not in the way these letters printed on paper—or backlit on the screen you are probably reading this text on, since this is as we now read in the 21st century—exist. Because this screen—yes, this one—and these letters, the ones that follow one another to form these sentences, are very concrete. Even though the meaning we attribute to them is a bit more abstract. Especially because the meaning attributed to them could be this one—the one the patient and attentive reader (if still reading) is giving to them—or another meaning given by someone else, or yet another given in a future reading.

But what makes these letters, this screen, this paper that the reader holds, exist, while letters in general, a screen, a piece of paper, the idea of those letters, that screen, that paper—their concept—do not exist in the same way? Just like the concept of a square, the fact that it has four sides, four edges, and its four angles add up to 360°, do not really exist? That’s because the former—those letters, that screen, that paper—are concrete objects, and the latter—the general idea of letters, the concept of a screen, the concept of a square—are abstract objects. Those letters—these letters—are here and now, even if on a backlit screen they are volatile and can disappear at any moment. Letters in general, a screen, the concept of a square—in fact, the square itself, because there are no squares in nature, and , to be sure, here nature means the physical world— [they] are nowhere, not even in our heads, in our consciousness, some would say.

This doesn’t mean that they are not real; it just means that they are not actual, here and now. They are virtual, a potentiality, a power of becoming or allowing something to be.Because, if we think about it, in the end, when all is said and done, what is theatre? What do artists do? What happens on a stage? Where is the art? Or, as Nelson Goodman would rather say, when is it art? And these questions aren’t as simple as they might seem. But let’s even start with the easiest: abstract art (abstract here is not italicized, because it’s just an adjective, a property—which, ironically, already makes it an abstract object, but let’s not go down that road now). From the outset, in a certain sense, art—as we’ve come to understand it since at least the 18th century—is all abstract. In a way, even a painting by Poussin, like one of those titled Et in Arcadia ego, contains many elements of abstract art. It’s not simply because it obeys mimetic principles, because it’s figurative or representational, that it lacks abstract elements. An allegory is an abstraction. It conveys meanings, concepts, common notions—often moral, but also aesthetic and political—derived from countless concrete experiences and situations and ultimately represented by a figure that cannot be reduced to a literal sense.

There are three shepherds bent over a tomb trying to decipher an inscription, and a woman—somewhat more enigmatic—who seems to be watching over and calming one of the shepherds. Around them, a bucolic landscape. Ah, and also the shadow of one of the shepherds—too overt not to assign its relevance—in the second version, from 1637/8, which can be seen at the Louvre. The title helps, especially in a hermeneutic context where one assumes the power of the pictorial nominalism of an artist. (This isn’t Duchamp, and we’re not facing a Fountain, let alone Craig-Martin’s An Oak Tree, but one could almost speculate on the retroactive or retrotracted resonances of a ready-made, or the apocatastic power of a conceptual transubstantiation.) Bellori said the subject of the phrase—ego—was Death. “Even in Arcadia, I, Death, exist.” Death, the ultimate abstraction. Félibien, on the other hand, interpreted it as the deceased—the concrete dead one—spectrally underlined by the shepherd’s shadow. “I too, the one who lies here, have once lived [and was happy, presumably] in Arcadia.” Panofsky agreed with him. As we can see, Poussin painted more with his mind than with his hands (though he painted very well, of course). Each line, each brushstroke, each shadow and figure, the relationship between background and figures, the perspective—as is evident—operate within a semantic and rhetorical structure, also intertextual (Virgil and Pliny the Elder are here implicit interlocutors), composing a text, narrating a story readable to the viewer. In a letter to Paul Fréart de Chantelou, written in 1639, though about another painting depicting a biblical episode with several figures expressing different emotions, Poussin asked the collector: “read the story and the painting, so that you may judge whether everything is appropriate to the subject matter.”

Caravaggio, who came into the world to “destroy painting,” was, single handedly, more abstract and more concrete than Poussin. His figures—biblical, historical, or everyday—seemed to exist right then and there, in that moment, in that place, right there with the viewer, in their carnality, trompe l’oeil, trompe le monde, due to those patches of colour, that captivating black, those textured brushstrokes, which were painted more with Caravaggio’s hands and cremaster muscle than with his eyes, obfuscating narrative and mimesis in favour of sensory experience.

Unlike traditional narrative, the history of abstract painting—abstractionism—could be seen as the history of the decomposition of the mimetic apparatus, not so much in order to free the artist from the burden of imitation, but to situate the viewer, to bring them into the present moment, to make them live and vibrate with the colours, the textures, the iconic three-dimensionality of the painting, its haecceity or paradoxical concreteness. From Cézanne to Rothko, from Fauvism to Minimalism, the artwork increasingly became a deictic device that calls the viewer to itself and its context. “Ce sont les regardeurs qui font les tableaux,” said Duchamp. But one might add: because “les tableaux [les œuvres d’art] font [aussi] les regardeurs.” The experience of art is an energetic, sensory, and semantic feedback loop, acting between the work and the viewer—the experiencer. The abstract objects that artworks also are increasingly gave way to the experiences enabled by their installations and devices. “Les immatériaux” (1985) were not so much a symptom of the dematerialization of the artwork as of its sublimation. There is nothing more concrete than Klein’s International Blue—but only at the moment it is instantiated on a gallery wall, on a suspended canvas, or in the fabric of a jacket. The metal cubes of Donald Judd, Carl Andre’s bricks, Dan Flavin’s fluorescent lamps, or Richard Serra’s corten steel plates are as much experiences of abstraction in form and material as they are processes of concretizing sensory and environmental experience. They surely help to set the climate for a gaseous state of contemporary art, but also for an aesthetic experience of urban daily life.

Still on the subject of misunderstandings in the time-based arts, one might say, for example, that abstract music is not necessarily abstract, simply because it is pure, instrumental music without words or extra-musical references, as it began to be characterized from the late 18th century. To understand what makes abstract music abstract, consider, by contrast, concrete music. Concrete music—musique concrète—“invented” by Pierre Schaeffer in the mid-20th century through his radio experiments with locked-groove records, is not concrete because it uses recordings of sounds produced by everyday objects, but because it is made from concrete sound material—a kind of sculpture in sound, of sound in its very materiality. It is concrete because it is the opposite of abstract music: that is, music fixed and inscribed in scores, using musical notes. What is an A, a D, a quaver, a treble clef? They are abstract objects, codified and inscribed in scores that can be read—like Poussin’s painting—and interpreted with instruments that give them life and instantiate them. A musical composition does not exist in time and space, but is instantiated every time it is interpreted, performed, and heard. In this sense, abstract music should be understood as such. It only becomes concrete when it transmutes into an event, in time and space—an unrepeatable event. The sounds heard in one performance will never be the same as those in the next, nor even in a repeated playback of a record, or each time you click a Spotify link (for those who practice that kind of listening).

Ah, but we still haven’t said anything about theatre… even though by now, I think the direction of all this is becoming pretty clear. Let me say it very quickly: Theatre, like music, has its score, which perhaps belongs more to a literary genre—drama—than to a performative art, a score that includes dramatis personae, didascaliae, and other instructions… It is also a building, a scenic device, and a liturgy, with distinct places for actors, directors, spectators, and so on… At first, it has been a religious festival—goats were sacrificed, wine was drunk—but then masks began to be used for actors to speak through, along with a chorus, and an orchestra… There were liturgical dramas or spiritual plays in the churchyard—still partly religious, but already playing at make-believe… And then came a bourgeois, mimetic, illusionistic theatre, which people enjoyed immensely. But as other arts began to deconstruct the apparatuses of mimesis and representation, theatre could not remain untouched. It became necessary to interrogate the spaces, the temporal frames, the forces at play, the sense of liturgy, the power relations, the materiality of bodies, to experiment with Elizabethan collars, checkered patterns, Hitchcockian cuts, cards to shuffle and deal again, to rehearse difference and repetition, life on stage… Yes, it’s all live, but all of this still feels very abstract. So maybe, after all, not all’s well that ends well?

C-5. Water. On the other side: D-2. Also, water. Another rehearsal, another shot: F-7. Aircraft carrier! Oh no, sorry, my mistake. Water again. Your turn.

I. But my Lord has shown me the intestines of all my countrymen in the Land of Two Dimensions by taking me with him into the Land of Three. What therefore more easy than now to take his servant on a second journey into the blessed region of the Fourth Dimension, where I shall look down with him once more upon this land of Three Dimensions, and see the inside of every three-dimensioned house, the secrets of the solid earth, the treasures of the mines in Spaceland, and the intestines of every solid living creature, even of the noble and adorable Spheres.

Ibidem

Nuno Fonseca