The Stakes of the Mobile (4:1–4): Grassmann’s Gestures of Space Itself

Chapter Four, “Grassmann’s Capture of the Extension: Geometry and Dialectic,” of Châtelet’s Figuring Space, is less a capture and more a kind of folding and unfolding of gesture so that space itself recovers its mobility. Grassmann’s insight is to show how not only do gestures not take place within a predetermined space, but through their cascading effects generate different spaces themselves. Let’s just stay with the first few pages (sections 1–4) to see this unfolding drama.

To situate this within the longer arc of Châtelet’s alternative genealogy of math, attempting to recover space from the axiomatic gestures of the Surveyor, we can trace two distinct lineages. The first activates the gestural force of the virtual in math: from Oresme’s intensive diagramatic cuts, to the thought experiments that throw phenomenon onto a virtual screen/horizon to see what happens, to the hinge of the indifference point that embraces local ambiguity as a generator of individuation, Grassmann retains and encodes these operations into an indefinite extension of space itself. All of this, to keep fighting off the mummification of figures by the stultifying monstrosity of axiomatic space. Grassmann proposes a kind of internalised process of this gestural trajectory that allows it to scale and cascade indefinitely.

At the same time, we can trace another, philosophical arc that is evident in Grassmann’s original 1844 version of the Theory of Extension, but stripped out for various reasons in the 1862 standard second edition. There Châtelet recovers another genealogy fighting the same fight on different terrain. Against the foreclosed space of Descartes, Châtelet evokes an early alternative precursor: 

However, Plato saw that the receptacle is a part of the intelligible, but in a very obscure way…. If he wishes to cry with the poet: ‘Space was splendid!’ the philosopher must remain at the outposts of the obscure, accept full-on the verticality of Being, the springing forth of dimensions; he must let himself be haunted by space… [103]

Let us not dwell on Plato’s limitations. His insight, for Châtelet, was that reality is not just divided between the eternal forms and the sensible copies that we perceive, but that there must be a third thing: a receptacle or χώρα (khôra) for what appears to us.

Space is a kind of imperceptible universal womb of becoming. Jumping forward to Kant, the uncertainty of space shifts from a cosmic womb to a form of sensible intuition in the subject ourselves. This brings us to the upheavals of German Idealism that Grassmann will invoke in 1844.

Attempting to solve for the gap between intuition and the external world itself, Hegel’s dialectic approach is to suggest that space emerges from a process of self-othering. In other words, space emerges out of the tension between being and nothingness that generates a differential movement in which space spreads out. It is Grassman, steeped in German Idealism, who will supply the missing calculus for this extensible space.

Which is to say, that he will locate the gestural sequences and cascades that not only allow for extension but open it up as an active mode of writing and rewriting space itself.

Gesture

What Châtelet is bringing to our attention is the way in which Grassmann both opens up a gestural space and at the same time shows how it folds back on itself in generative ways. Let’s follow the gestures. First, Grassmann borrows the simple cut from early German pedagogic taxonomy (“in the manner approved since Aristotle”) between Real and Formal Science. The gestures of real science are oriented to the bodies and forces of the sensible world that face us. We can manipulate and measure the things themselves. The gestures of the formal sciences however takes place reflexively within thought itself: 

…it is posited by thought which itself now faces, in its turn as a being, a second act of thought. […] Thus in the formal sciences, demonstration does not leave the sphere of thought and remains purely in the combination of various acts of thought. (107)

There are thus two distinct modes of gestures of hand-or-mind, those that engage with the world they face and those that turn back on the previous gestures themselves. (It is almost as if “the mind” is simply this recursive mode, and not that it is fundamentally distinct from the hand!) AND THEN, within formal science we can differentiate two other types of gestures: continuous one-stroke gestures that unfurl all at once and discrete two-stroke gesture that link one to the next. (These obviously draw on our understanding of physical gestures for their sensibility, but for Grassmann they are formal, symbolic gestures of thought itself.)

Continuous magnitude and counting. But here is where the gestural simultaneously gets wild and “captured.” just as these distinct formal gestures oddly evoke the gestures we might make in and with the material world, each formal gesture in turn starts to emulate the other. Enough tiny discrete gestures strung together start to look continuous, and if one slows down a sweeping gesture the line starts to look like a row of discreet, stitching gestures. In other words, we might say that there are at least two “perspectival gestures” that allow these distinct gestures to shift into each other. (But what is perspective then?! And does it also follow the same logic? In other words, does perspective toggle, like a discrete form, or does it flow from one orientation to another?!)

Diagram

In any case, what Grassmann is articulating then, is something like a continuous gestural sequence that will form the famous Quadrilateral that articulates the moves between arithmetic and geometry out of only the two gestures, smooth and discrete. In four discrete moves you return to the beginning with no displacement, forming a kind of pocket of continuity. (Note: this is also reflects the strange going out and return of our hand gestures.) In other words, this both shows how counted gestures can self-organise into smooth balance. And at the same time, since the quadrilateral is not organized in a pre-existing axial space (despite the impression of figure 2!) this opens the door, in both the Plato-Kant-Hegel lineage and the virtual screen-horizon-hinge lineage, to the constructing of space from the inside out, as it were. We are now on the cusp of Grassmann’s algebra in which these emergent spaces can be generated through symbolic rules as a kind or writable and re-writable gestural possibility. But it is this diagram, as a kind of minimal gesture lab, that will afford this by bringing together the hand and eye, before any coordinate or formula is introduced. 

Pedagogy

Which brings us, finally, to the educational inflection that Châtelet highlights in Grassmann. First we see Grassmann’s insistence that it is not enough to proceed axiomatically, either formally or pedagogically. You have to engage with gestures, and it is through these that we are able to move between two different gestural registers which mirror the discreet and the continuous at the level of both proof and pedagogy. You need a chain of steps, each following from the last, but you also need an overview that directs the process that moves by way of a guiding hunch or intuition. This is what the gestural diagram does. Châtelet insists that Grassmann’s greatness hinges on (in the spirit of the German philosophies of nature) not separating the generative functions of math from pedagogy. 

Space thus appears as a visible understanding and the understanding as an invisible space. The point is to clarify the link between the process of individuation of the knowing subject and the process of articulation of the forms of the grasping of space (101).

Next up: Gilles Châtelet, Figuring Space, Chapter 4 § 5-6, (111-122).