In Section 7 of Châtelet's chapter on Grassmann, orientation is no longer a sticker, or "right-hand rule" we slap on afterwards; it lives inside the antisymmetric sign of the product, folded around the very body we imagine lying on one edge to see the next dimension appear. In this moment mathematics stops stacking inert lengths and starts wedging and multiplying the ‘living forces’ Naturphilosophie had only talked about, climbing an intensive/extensive ladder that addition alone could never build. And it does so by way of some strange virtual body at the heart of diagrammatic space...
Grassmann’s arrow—once just a mathematical mark—suddenly reveals itself as a philosophical force. From the “mobile point” to the zero point-vortex, we’re no longer stacking segments but generating space through gesture. What’s uncanny is how closely this aligns with Paul Klee’s cosmic egg, or even Kristeva’s rhythmic womb: all three are rethinking how form arises from force. Behind it all looms Naturphilosophie (Goethe, Schelling, the generative seed, polarity...) recast through mathematics, drawing, and language. And with Châtelet guiding us back to Grassmann’s 1844 Ausdehnungslehre, we begin to see these diagrams not as tools to represent space, but as pedagogical machines that create it.
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.
modern physics emerges from this projection of motion first into the viewable spectra
Mathematical abstraction is not, a la Aristotle, a stripping away of mobile content for stable form, but a kind of condensation of experimental gestures and problems.
Diagrams are in a degree the accomplices of poetic metaphor. But they are a little less impertinent––
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