***** The abstract to my upcoming lecture at the 6th International Deleuze Studies Conference “The territory in-between” (Lisboa Portugal, July 8-10 2013). Info: http://deleuze2013.fc.ul.pt

Keywords: symbolic algebra, time, space, metaphysics, semiotics, articulation, glossematics, diagrammatics

“Le fondement c’est donc ce qui nous donnera ou ne nous ne donnera pas le droit.”

(Gilles Deleuze, Qu’est-ce que fonder?)

Departing from the central rôle played by what Deleuze calls ‘quantitability’ in Difference and Repetition (1968), this paper will explore the perspective on a symbolically algebraic understanding of logics. The guiding – yet abstract ! – analogy will be how we can conceive of logical ‘territoriality’ in a similar way as symbolic algebra has come to conceive of the ‘territoriality’ of the real numbers (the famously infinite and continuous “number line”). Such conceived symbolical territoriality, I will suggest to call ‘limitudinality’.

Since antiquity, limits have been considered as the form of magnitudes. This understanding is assailed by the symbolic procedure of determining limits that arose in the 19th century (the so-called Dedekind Cut), and which operates by encoding one infinity with another infinity. The abstractness of this procedure unfolds as an empirical ground, I will argue, once we conceive of the relationality of a multitude of such encodings among each other. Such an empirical ground constitutes a kind of logical territoriality where quantization as an ability (Deleuzes’ ‘quantitability’) precedes quantification. Limits, then, are no longer best conceived as the form of magnitudes, where forms are to be determined ideally while magnitudes are taken as givens (Aristotle’s Naturalism), nor in purely formalistic terms as an operation which cannot itself be considered (Hilbert), but as the symbolically indexable formality of magnitude’s negativity.  Such an infinitized negativity becomes analyzable (differentially) once we consider logical quantification as derived and dependent upon an ability to ‘quantize’ (rather than as axiomatically constituted). The paper will demonstrate how such a perspective leaves us with an inversion from where we started out: not the forms are to be determined ideally, and the magnitudes empirically, but the other way around. It is along these terms that the paper will present a reading of Deleuze’s transcendental empiricism which can be related to Hjelmslev’s ‘glossematics’ as well as to Peirce’s ‘diagrammatics’.