Disjunction
Lecture given 10-11 March 2022 for Auerglass and Friends at SFMOMA
The title of Tauba Auerbach’s exhibition here at SFMOMA, which we might speak aloud as ess-vee-zee, hinges on a glyph (the vee) that logicians use to denote disjunction. In the intuitive context of natural language, we can understand disjunction to be a manner of joining clauses that works like the colloquial “and/or” — hence we could read ess-vee-zee to propose “ess and/or zee.” It seems clear from Auerbach’s work that they broadly adopt (from fiber art) the association of S and Z with opposite directions of twist, so we might now wonder what it really means to propose ess-vee-zee as a disjunction. If ess and zee are opposites, why the “and/or” rather than simply or? It’s like saying “turn left and/or right” rather than “turn left or right” — is the “and” really a viable option?
My aim in this talk will be to develop a response to this provocation — something about the disjunction of opposites — drawing upon resonant ideas from the humanities and sciences to build an argument that disjunction can in fact be an insightful lens through which to consider much of the artwork we see and hear in S v Z, and in the performances we’re experiencing here today. In parallel I’ll be making an ulterior case for generally associating disjunction with hidden higher dimensions, and the visuals of the lecture will accordingly progress from line to plane, to sphere. This will require some leaps of association, so I hope you’ll bear with me — we’re going to start by tossing coins, move on to deconstructing the modern Western worldview, and end up in the realm of quantum mechanics. Line — plane — sphere — disjunction. Ready?
First Dimension: The Odds of Outcomes. Consider two possible orientations of a coin lying flat on a surface: according to which side is facing up and visible, we consider its configuration to be either “heads” or “tails.” Assuming the coin is always made to lie flat and that we are concerned only with which side of the coin faces up, these two configurations are mutually exclusive and exhaustive — only one of them can be the case at any given moment, and there are no other possible configurations.
Now what happens if I toss this coin, let it fall flat on a tabletop, and cover it with my hand — all while you have your eyes closed? Let’s assume that it’s a fair coin and a fair toss, that you trust me not to manipulate the configuration of the coin as I cover it up, and that you don’t have any superpowers. How would you construe the state of the coin? A reasonable response would be to say that under my hand, the configuration of the coin is either heads or tails, but you don’t know which. To be more precise, we could incorporate our contextual information about the coin being evenly weighted and there being no artifice in the way it was tossed or covered up and say that there’s a 50% probability for the configuration of the coin to be heads, and the same 50% probability to be tails. This kind of succinct probabilistic summary of everything relevant that you know about how the coin has been prepared is fundamentally what we mean in modern physics when we use the term, state. If we were tossing a coin that was not evenly balanced, but rather a weighted coin that lands heads-up three times out of four, your characterization of its state after being tossed would assign a 75% probability to its configuration being heads, and the remaining 25% to tails. Generalizing, your assessment of any additional biases in the coin-tossing scenario would be factored into your description of the coin’s state by adjusting the shares of probability assigned to each configuration.
For any system of interest whose configuration space is binary, by which I mean that there are exactly two mutually exclusive alternatives like heads or tails, left or right, ess or zee, this kind of probability-state can be represented graphically by a point on a line segment. If one end of this line segment represents certainty that the configuration is heads and the opposite end of the line segment represents certainty that the configuration is tails, a point in-between represents an assignment of some probability (here, the fraction to the right of the point) to heads and the remainder (here, the fraction left of the point) to tails. In phone-age parlance, it’s like a slider bar. The probabilistic state space of a binary system is shaped like a line segment, bracketed by its exclusive alternatives on either end.
And so, we have our line — the one-dimensional starting point of our consideration of possible diagrams of disjunction, which in its simplicity can represent a range of scenarios in which we might have one configuration or its complement, but not yet anything beyond that. In this sense I would claim that it captures a wide swathe of conventional thinking about binaries. As we’ll discuss in the next few minutes, we can similarly represent a complementary wide swathe of conventional thinking about binaries with another bracketed line segment — one that represents the nonbinary as a hybrid mixture of “pure” (in quotes) essences drawn from complementary poles. To help guide us through and beyond this reductive slant we turn now to the work of philosopher, anthropologist, and sociologist Bruno Latour — more specifically, to a swirl of ideas Latour developed in a pivotal essay from the early 1990s titled We Have Never Been Modern. Latour will suggest a way of adding a second dimension in our consideration of possible diagrams of disjunction.
Second Dimension: Of Maps and Morphosis. In a focal chapter of We Have Never Been Modern, Latour considers a line segment bracketed by “nature” on one end and “culture” on the other. Latour uses this schematic to represent what he calls “the work of purification,” a kind of ideological pursuit that he characterizes as defining the modern worldview in the West. To practice the work of purification is to insist upon analyzing complex phenomena of lived experience by teasing them apart into distinct components drawn from nature, on one hand, and culture on the other. For Latour’s modern thinkers, not to recognize the fundamental opposition of nature and culture — for example by practicing a form of nature religion or basing political strategies on a reading of tea leaves — would be considered a superstitious deficiency of reason characteristic of “primitive” (in quotes) cosmologies.
According to Latour, the modern worldview understands phenomena such as ozone depletion, emerging infectious diseases, and global supply chains as “monstrous hybrids” resulting from the sinister interplay of human desires and societal pressures with the intricate inner workings of Earth’s otherwise-indifferent biosphere. The modern perspective sees every such phenomenon as a concoction of some part nature-essence with another part culture-essence, such that every phenomenon can be located at a point along a nature-culture line segment according to the ratio of its mix. If our probability line-segment (for the coin toss) was a diagram of just the “or” aspect of disjunction, this one represents the “and” through a trope of hybridity. What this construal of real-things-as-hybrids fails to account for, according to Latour, is that nature and culture are not in fact transcendent essences. Nature and culture are not timeless, absolute source materials like copper and aluminum that we combine in various ratios to make alloys with novel properties. Rather, the essences of nature and culture change continuously as history unfolds, and it is historical occurrences — that is, the contingent things and events that somehow come to exist — that drive this unpredictable evolution. The ways we understand nature and culture both change when we split the atom; they both change when we clone Dolly the sheep. Imagine the drastic shift in humanity’s understanding of nature and culture from before the industrial revolution to some future date when we may have drowned the Maldives and irreversibly overfished the oceans — from nature as an eternal backdrop for the play of culture and society, to human culture and society as a kind of parasite in danger of killing its host.
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To locate actual things and happenings on a diagram encompassing nature and culture, Latour says we must include a second dimension — the bracketed line segment is insufficient. If the original nature-culture line constitutes the horizontal upper edge of a new, planar diagram, the vertical dimension of the diagram should reflect what he calls an “ontological gradient” from essence to existence. Latour insists that we must not understand nature and culture as being timeless, transcendent references like zero and infinity, but rather as mutable, relative references like fashion or science fiction. So, to put things on Latour’s new two-dimensional map: as they first come to exist, actual occurrences generally appear near its lower edge, but as critical events exert broader and more profound influence on the way we grasp the essences of nature and culture, their imprints may drift upward along trajectories that ultimately merge into the nature-culture axis. When Dolly the sheep is born on July 5, 1996, the fact of her life appears as a point low on the diagram, but with the advances in stem cell research spurred by the successful demonstration of adult cloning; with the subsequent birth of cloned pigs, horses, and monkeys; and with the advent of attempts to patent living animals, Dolly’s point surges upward to harrow the nature-culture axis. Our essential understandings of “nature” and “culture” continually assimilate the complex particles of history, even if we choose to retain these terms (“nature” and “culture”) as fixed labels for the upper corners of our map.
Latour’s deconstruction of the modern worldview teaches us to question the notion that real-things-that-exist should be thought of as hybrid mixtures of “pure” opposing essences. We don’t need to think of the form of a guillemot egg, for example, as a material substantiation hybridizing the platonic ideals of sphere and tetrahedron; rather, we can understand that the existence of things like eggs, berries, nuts, and calcite crystals help us to recognize an axis of difference relating roundish-yet-triangular things. The perfect sphere and perfect pyramid correspond merely to imaginary limit points beyond the virtual horizons of some real axis of difference, while it is actual things in the world that fundamentally define these axes. The roundest things ever to exist on Earth have been the gyroscope rotors made for the Gravity Probe B experiment, but we know that even these are not mathematically exact spheres. The Gravity Probe B rotors may be imperfect substantiations of an ideal sphere, but the rotors are real while the ideal sphere is a figment. The interior points of our line segments imply their binary limits, not vice versa.
Getting back to S v Z and the occasion for this lecture, we may reflect on what this “inside-out” shift of perspective does for the way we think about conventional oppositions such as art versus science, symbol versus ornament, or poise versus flow. Perhaps one take-away could be that in considering a collection of creative works, it really is limiting to think that their principal virtue might just be something about their hybridity, as if the purified essences one might describe them as mixing are more fundamental points of reference than the actuality of the works themselves. Rather than speaking of “works that blend art and science” along an implied bracketed line segment, we could probe how new works trouble the constellation of historical things and events that have engendered our modern gloss of the art-science dichotomy; we can ask how new creative works regenerate the art-science axis of difference. What happens to the conventional opposition of art and science when we consider the materials research and instrument development embodied by Auerbach’s Extended Object series? How are our understandings of the “plastic space” of art and the “projective geometry” of mathematics similarly perturbed by their series of maps-after-Spilhaus?
In discussing his two-dimensional diagram of essence and existence, Latour refers to its longitude and latitude, and to its “poles” of nature and culture. But given its geometry, with these poles located in adjacent corners, I don’t think we can conceive of Latour’s map as a projection of the surface of a globe, in the way that a Cartesian map is a projection of the surface of the Earth. Latour gives us our two-dimensional plane, but to get all the way from our bracketed line segment to a three-dimensional sphere — as a diagram pertaining to the disjunction of opposites — let us finally turn to a brief consideration of quantum mechanics.
Third Dimension: Polar Wander. In quantum mechanics the state space of a binary system is shaped like a sphere, and our entire conception of what it means for a system to be “binary” gets upended. In a sense the bracketed line segment is still there, corresponding to something like a line through the center of the sphere connecting its north and south poles, which in the context of our coin example we would associate with the configurations, “heads” and “tails.” But the midpoint of this line segment, which is in fact the center of the entire sphere, now represents only one possible way for the system’s state to reflect a 50/50 balance of heads and tails. If we stick with globe terminology to reference parts of the spherical quantum state-space, any point in the equatorial plane represents such a 50/50 combination, and these all somehow represent distinct states. In quantum mechanics, probability does not exhaust possibility — there are manifold ways to be an even balance of heads and tails, or more generally, of any sort of configuration and its complement.
As an analogy to help us think about that, let’s consider the complementary configurations “left-handed” and “right-handed.” We can draw a spherical state space[i] with the north pole corresponding to a definite left-handed configuration and the south pole corresponding to right-handed. According to what we just said, the entire equatorial plane of this globe represents a space of states that, in some sense, are all an even balance of left- and right-handed and yet all distinct from one another. My aim here is to suggest that this expansion of possibilities somehow correlates with lifting our thinking into higher dimensions, so, let’s start by noting that if I have a flat left-handed object constrained to lie in a two-dimensional plane, there is no way to reorient it into a right-handed configuration. For example, if I have a paper cutout of a traced left hand lying on the surface of a table, there is no way I can shift or rotate it into a right hand without lifting it off the surface. If, however, I do allow transient excursions into a third dimension, there are many different rotations I can perform that will flip-over the left-hand cutout and thereby convert it into a right-handed configuration. To name just two, I can either flip it 180 degrees thumb-over-pinky, or I can flip it 180 degrees wrist-over-fingertips. Either way, the left-hand cutout has been converted into a right-hand cutout. (Mathematically speaking, there’s an analogous story for those ess and zee twists, which similarly can be smoothly rotated into one another if we allow rotations in four-dimensional space. For those who are interested, we can highly recommend Claude Bragdon’s illustrated monograph A Primer of Higher Space, which has of course been printed in a beautiful modern edition by Diagonal Press.) Getting back to our hand-cutout and all those states on the equatorial plane, note that if we rotate only 90 degrees — that is to say, only “half-way over” — in either of these ways (wrist-over-fingertips or thumb-over-pinky), we reach a state of the cutout that is in a sense neither more left-handed nor more right-handed in terms of its orientation with respect to the original tabletop plane. And these are two completely distinct states from our “lifted” three-dimensional perspective…
One way we might summarize all this would be to say that the handedness of our cutout is a binary dichotomy — either left-handed or right-handed — if we limit our perspective to lie strictly within the two-dimensional tabletop plane. But if we lift our perspective into three dimensions, we see that these are just two possible configurations among a continuum of others. Furthermore, the states that lie between them cannot be adequately characterized simply as a mix of some left-handedness with some right-handedness — there higher dimensions of diversity to account for. If we were to play a game of randomly tossing our flat cutout of a hand onto a tabletop — like the coin we started this lecture with — and again I did this without letting you see the result, we could get back to our bracketed line segment in which you might naturally construe the state of the cutout as being a 50/50 probabilistic mix of being either left- or right-handed, with an underlying belief that “really” it must be one or the other. If we consider rotations of the cutout in full three-dimensional space, however, we can appreciate that there are countless distinct orientations that are all equally real.
Circling back once again to S v Z, we may be inspired to imagine manifold ways of being a balanced superposition not only of simple geometric descriptors such as left- and right-handed, or ess and zee twist, but even of more complex categories such as symbol and ornament, poise and flow, or art and science. We may be inspired to contemplate what higher dimensions we transit in a semantic rotation from yes to no. As the quantum “binary” state space is shaped like a sphere, from a geometric perspective there’s nothing particularly special about the points we start out positioning at the north and south poles. To reflect on this a bit more concretely, let’s label the north and south poles “art” and “science” but then rotate the sphere by ninety degrees around a perpendicular axis. In this orientation “art” and “science” now appear as points in the equatorial plane, which we could therefore interpret as different sorts of interpolations of whatever is represented by the new north and south poles — neither perspective, neither orientation of the sphere is fundamentally “correct.” Perhaps we would be better served by thinking of art and science as sibling superpositions of tangibility and insight, of holism and particularity, of essence and existence. Approaching an exhibition such as Auerbach’s S v Z, we need not simply situate its works as points along the bracketed line segment of mixtures of “art” and “science” — rather, we can see them as works that spill out across a complementary space of possibilities and in so doing displace “art” and “science” as archetypes of disciplinary purity. Purity is a matter of descriptive convention; difference constitutes the richness of lived experience.
Line, plane, sphere; disjunction. To propose a disjunction of opposites is to impel higher-dimensional thinking, to derange the binary, to scramble the compass.
[i] Here for a combination of pedagogical and rhetorical reasons I’m conflating the Poincaré Sphere with the Bloch Sphere…
