Scary Alignment: Grassmann, Klee, Kristeva

Perhaps the real shock of this week’s discussion, of sections 5-6 of Chapter 4 of Figuring Space, was that we should have seen it coming. Grassmann’s emphasis on the arrow as a gestural force is “scarily aligned” with Paul Klee’s project. (And—via the “womb”—Kristeva’s pulsational rhythmic opening of language and the symbolic up to flux, could also be woven into this strange alignment.) 

So before we follow the arrow, I did a small bit of contextual digging. Grassmann died two years after Klee was born, and wasn’t widely known even in math circles at the time. And yet, we could say that what ties them together is a fundamental commitment to unfolding the effects of gesture as it leaves a kind of trace: for Klee through drawing and painting, and for Grassmann through math as a formal space of gestures referring back to themselves. And this project, for is deeply informed by the larger context of Naturphilosophie. Châtelet, going back to the philosophical language of the first, 1844, edition, Die lineale Ausdehnungslehre (“Linear Extension Theory”), shows how fundamental this tradition is to Grassmann’s mathematical innovation. While Klee was explicitly drawing from Goethe’s Theory of Colours and the Metamorphosis of Plants (the latter, by the way, is essentially a way to think the genesis of form through observational drawing!), and less obviously Schelling. (See this article, which attempts to make sense of the “scientific diversion” Klee mentions in his Diaries, via the influence of “a visual model of kinetic energy derived from Jean Perrin’s diagram of Brownian trajectories.”) In any case, we can locate in the broader context of Naturphilosophie a common set of ideas: complementarity, polarity, seeds, arrows, a dynamic generative field animated by gestures and forces, and the inseparability of pedagogy from the world. 

Beyond the stump

Ok, back to the arrow. 

One of the things we returned to was the weird gestural shift in perspective that allows Grassmann to oscillate between continuity and the linking of discreet gestures. Figure 2 is something like the meta-perspective that maps out the way in which these shifts bring together four different perspectival modes of math. And as always with Grassmann, it is this fourfold loop that affords both a proliferation of gestures and their coherence. Section 5, “The Intensive/Extensive Dialectic,” shows how this establishment of a point-force establishes an intensive pole, while the extensive pole scatters a horizon. But now Grassmann, in a very Klee-like move, will propel the horizon into the next higher steps. Peeling the gestural space of discrete and continuous lines of sections 1–4 into a generative space. In Section 6, “The Additive Generation of Vectorial Systems,” Châtelet insists that we “denounce” the rigid segment as a mere stump of a motion, a paralysed unit. Now space must be regenerated from the “mobile point,” Grassmann’s generative element which converts length into trajectory, a stump into an arrow. Here, echoing the “indifference point” zero is not simply a neutral nothingness but a kind of point-vortex that splits into two opposed motions. If sections 1–4 lay out the perspectiveal cuts that generate different mathematical fields, Grassmann is now beginning to supply a (gestural-perspectival) mechanism for moving between them. 

And here we are hovering between addition and multiplication, that will “verticalise” space not through addition’s ladder but through a leap into another dimension through the point-vortex. For Châtelet a genuine, oriented multiplication will allow a “reciprocal penetration of factors” that intensifies space and mirrors living synthesis. Klee and aesthetics? Could we at least say that this, in the end, collapses the distinction between the real and formal sciences, via the pedagogic force of gestural diagrams?

Next up: Gilles Châtelet, Figuring Space, Chapter 4 § 7 "Grassmann's Products", §§ a-c (122-134). Look, we're all in the weeds here, so you might as well join the Diagrams reading group...