***** 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’.