Sobre la continuidad y lo problemático en la matemática arquimidea

“The notion of continuity has roots not only in the history of philosophy but also in the history of mathematics. One might view Greek mathematics, in particular geometry, as having two poles. One pole, represented by Euclid, deals with the properties of static geometrical figures. The other pole, represented by Archimedes, concerns the construction of actual geometric figures. To apply the terms from the discussion of the history of philosophy, Euclid pursues a geometry predicated on discontinuity, while Archimedes pursues a geometry predicated on continuity. That is, Euclid is concerned about the universal, intelligible components of geometric figures, while Archimedes is concerned about the processes by which actual, different geometric figures might be generated. More important, however, is the way in which the Euclidean geometry of stable objects came not only to dominate geometry but also to be seen as the model for mathematics. Ultimately, this privileging of discontinuity expresses itself in mathematics as the preference for the discrete over the continuous. From the perspective of mathematics, the discrete has the trernendous advantage of making things countable and thereby subject to algebraic analysis. The continuous, in contrast to the discrete, is messy, slippery, and unstable. One doesn’t extract timeless truths from the continuous; one intervenes strategically. The discrete is axiomatic; the continuous is problematic. The difficulty that Deleuze sees with the triumph of the discrete in mathernatics is that life is not discrete. Life presents constant and continuous variability. It is for precisely this reason that Deleuze deploys the resources provided by calculus. Calculus thinks the continuous and variable without making it discrete. Calculus allows one to think without recourse to discontinuity. There is no immutable, intelligible component that the object of analysis more or less conforms to. There is only constant variability that tends toward infinity (…)”

The same battle between continuity and discontinuity is “played out in the history of mathematics, first in the distinction between axiomatic and problematic geometry and more recently in the attempt to ground calculus in set theory. If we recast this battle as a response to the question, What is a thing?, we discoyer that the same tension between stability and change is at work. The preferences for axiomatic geometry and for grounding calculus in set theory both reflect the idea that mathematics properly deals
with the discrete, the quantifiable, that for which we can produce an axiom. Thus, the “thing” in mathematics is the countable. Deleuze’s use of mathematics, particularly his use of calculus in Difference and Repetition, shows that he remains interested in that “lost” object of mathematics, the intensive, continuous, abstract “thing.” The reasons behind the choice of calculus in that book are still operative in A Thousand Plateaus. These reasons become explicit in Deleuze and Guattari’s distinction between royal science and minor or nomad science. “Royal science is inseparable from a ‘hylomorphic’
model implying both a form that organizes matter and a matter pre-pared for the form” (TP 369). Royal science is designed to work in concert with the state and deals in discrete and ideal essences, such as circles. In contrast to this, nomad science is inseparable from a “hylozoic” model in which matter is self-organizing and generates its own form. Nomad science is not interested in circles as an ideal type, but it is interested in “roundness,” the continuous curve, which sometimes appears as a circle. But, for nomad science the circle is not what roundness necessarily or even ideally tends toward.”

Atkins, Brent, Deleuze and Guattari’s A Thousand Plateaus, Edinburgh: Edinburgh University Press, 2015.