# Mel Bochner

## Sonnabend Gallery

Mel Bochner’s drawings, which can be written about in connection to his current show at Sonnabend, operate as materialized idea pointing toward signification rather than attempting to define it. Much of his work is drawing—the setting forth of boundaries for the purpose of location and placement. His pieces have often involved the nonentities of relational connectives between substantive states, represented in language by conjunctions and prepositions—“between,” “on,” “in,” “and,” “or.” Bochner finds models for postulates, Euclidean axioms about space and objects in space, so that each postulate is converted into a true statement about the model. Bochner’s formal system may be considered in relation to that of Hilbert:

*We consider three different systems of things; the things of the first system we call points . . . those of the second, straight lines . . . those of the third, planes . . . We think of the points, straight lines, planes in certain relations to each other and denote these relations by words as “lying,” “between,” “parallel,” “congruent,” “continuous.” The precise and, for mathematical purposes, complete description of these relations is contained in the axioms of geometry*.

These axioms are those of connection (and), order (between) of congruence (equality), and of the parallel, always consistent within the formalized system. Bochner’s work is not mathematics; the mathematician is concerned with conclusions as necessary logical consequences of initial assumptions while Bochner is interested in discovering position and physical property as well as inducing and deducing logical necessity. In mathematics, relational connectives are implicitly defined by the axioms, while Bochner investigates them for their palpable value, his use of mathematics close to Russell’s definition of the discipline as “The subject in which we do not know what we are talking about, nor whether what we are saying is true.” Bochner’s use of prepositions can be seen in the drawing for *Theory of Boundaries*, 1969, which examines the properties “at,” “in,” and “out” by means of shaded areas on the wall, and in the drawing *Over and Under*, 1972, which uses masking tape and numbers in order to demonstrate “on,” “on/over,” “on/under,” “on/over/under,” and “over.”

Another aspect of Bochner’s work has been that of measurement and number. Numerals are characterized by order—the arrangement or sequence that makes measurement significant. Bochner has used measurement both as it is deduced from existing spaces and objects, and as ordinal and cardinal systems by themselves, metamathematical signs in a visual calculus. A drawing in Ginsburg’s exhibition, *Counting Stones*, 1972, is closely related to the piece in his show at Sonnabend, *To Count*. The drawing is subtitled *Rules of Inference; Rules for manipulation of the terms so that each new configuration can be derived from the initial ones*. The stones (or notations for stones) are ordered into prime and nonprime numbers, separated into prime factors (one, two, or three), and noted for the properties of closure of configuration and the right angles derived from the arrangement or connections of the points. This quality of inference is also operative in the piece *To Count: Transitive*. As well as indicating the non-substantial states between substantives and the process through which permutations occur, “transitive” also means the relation of entities such that if the first is related to the second and the second to the third, then the first is so related to the third. This is the logical proposition: “If p implies q and q implies r, then p implies r.” However, Bochner also uses orders which are not continuous in counting, properties drawn from the physical configurations such as the number of right angles in each. The piece operates on the level of sensuous manipulation, as if counting on an abacus or on one’s fingers, giving substance to the abstract idea of number. The choice of matchsticks is particularly opportune because they provide both point (head) and line (stick) which form actual shapes or diagrams; the rocks in earlier works had remained as points.

The title of the other piece in the show at Sonnabend, *The Axiom of Indifference*, seems to be a play on the rule of detachment (modus ponens) by which anything implied by a true elementary proposition is true. The work involves two sets of two squares each, executed in masking tape and labeled with black magic marker, three pennies to each square. They are placed precisely in the center of the wall dividing the large room of the gallery, lining up exactly from one side of the wall to the other. The squares on the left of each set are placed on a diagonal, while those on the right are on a straight axis. On the left side of the wall as one walks into the gallery, the diagonal boxes contain the following information: 1) The left box: “Some are not in” (two tails outside of the box, one heads inside); 2) The right box: “Some are in” (two heads in, one tails out). In the vertical boxes: 1) The lower box: “All are not in” (three tails out); 2) The upper box: “All are in” (three heads in). On the opposite side of the wall, in the diagonal boxes: 1) The left box: “Some are not out” (two tails in, one heads out); 2) The right box: “Some are out” (one tails in, two heads out). In the vertical boxes: 1) The lower box: “All are not out” (three tails in); 2). The upper box: “All are out” (three heads out).

Through slight shifts in physical conditions, the piece points toward the properties of “out,” “in,” “some,” and “all.” These qualities are made tangibly different through the information available in the coins, with two sides as markers. For example, on the left side of the gallery, “some” and “all” are “in” and “not in,” while on the right side of the gallery, “some” and “all” are “out” and “not out,”—marked by the readings of the pennies. On the left, the heads are all in while the tails are out. On the right side of the gallery, the tails are in and the heads are out. This palpable difference between “in” and “out” is counterpointed by the placement of the coins in the diagonal boxes on both sides of the wall, and those in the vertical boxes on both sides. They are in identical geometric configurations, but in visual crossover so that the left box in the left of the gallery is like the right box in the right and vice versa, and the upper box in the left becomes the lower on the right side of the gallery, etc. “Some are not in,” for example, lies in the same triangular configuration as “some are out.” This appears to demonstrate the visual identification of “in” and “not out,” and “not in” and “out” even though these functions have been differentiated by the information on the coins. In this way, Bochner deals with both the tangible and the abstract qualities of propositions. Because the pieces point toward these realizations without illustrating them, they can be viewed as drawings or maps which concretize properties they do not contain. Bochner’s interest in location and placement indicates a physical meta-criticism, not theoretical but as practical as G.E. Moore's statement; “Here is one hand and here is the other.” The truths of the propositions are self-evident—the information does not contradict our testings of logical truths—and Bochner uses them as scaffoldings for perceptions which would normally be implicit and consequently invisible in perception.