“The logical space of reasons” is a figure of speech, and a telling one. C20th philosophy is full of phase spaces and logical spaces, rhetorical spaces and epistemic spaces, from Wittgenstein’s Logische Raum and Heidegger’s Spielraum through to the smooth and striated spaces of Deleuze & Guattari. Spatialisation is one way to “go transcendental”: the movement from point to space is always a virtualising movement, a movement in the direction of overarching (rather than “underlying”) logic, higher-order organisation.
Henri Lefebvre’s argument about spatial metaphors is, more or less, a reworking of Merleau-Ponty’s phenomenological centreing of spatial intuition on the body: for “the body” as that of an individual phenomenal subject, Lefebvre substitutes the social body. For Lefebvre, the contouring of “mental space”, with all its various metaphorical deployments in philosophy and mathematics, is consequent upon the structuring and restructuring of social and geographical space, in particular the space of the city. One could always see Lefebvre’s own project of tracing shifts in the structure of social space as outlining a kind of space of spaces within which such transformations could be plotted; but that would seem somewhat against the spirit of it.
The critical question here concerns the status of the transcendental, relative to the social (and historical, geographical, physical etc) matrix of which it is the transcendental. A “logical space of reasons” might be one sort of thing in a city where argumentative cliques congregated in coffee shops and bars to debate the latest pamphlets and manifestos, and another in a remote village with a primarily oral culture, where collective decision-making and tribal identity were mediated by the same stock of stories and story-tellers. Or – and the use of “space” as metaphor predisposes us towards this alternative – we might be trying to talk about differently-situated instantiations of the same thing, transcendentally speaking.
The key notion that I draw from Zalamea’s metaphorical deployment of sheaf theory is that there is not one “space of reasons”; that the transcendental is only available on condition that one navigates from space to space, constructing “spaces of spaces” through transcendentalising operations. That is why it makes sense to me to argue that the space of reasons is not a “full body”, and is in fact incompletable. There is not an independent space of types, governing a subordinate domain of values: type and value are inextricably entwined.