Alessandro Sarti

What is the dynamics underlying the constitution of a new form? Reconsidering the original Deleuzian-Guattarian idea of singular becoming and using contemporary instruments of geometric analysis, we introduce the concept of “differential heterogenesis” as a mathematical framework to envisage the emergence of singular forms from the assemblages of heterogeneous operators. In opposition to the kind of differential calculus that is usually adopted in mathematical-physics, which tends to assume a homogeneous differential equation applied to an entire homogeneous region, heterogenesis allows differential constraints of qualitatively different kinds in different points of space and time. We will discuss the impact of this approach to the individuation of living, perceptual, semiotic and ecological forms.

This contributed volume aims to reconsider the concept of individuation, clarifying its articulation with respect to contemporary problems in perceptual, neural, developmental, semiotic and social morphogenesis. The authors approach the ontogenetical issue by taking into account the morphogenetical process, involving the concept of individuation proposed by Gilbert Simondon and Gilles Deleuze. The target audience primarily comprises experts in the field but the book may also be beneficial for graduate students. The challenge of the genesis and constitution of "units" has always been at the center of philosophical and scientific research. This ontogenetical issue is common to every discipline but it is articulated in different ways: in phenomenology of perception the constitution of perceptual units is at the base of gestalt field theories, in theoretical neuroscience synchronized neural assemblies are considered as correlates of conscious processes, in developmental embryogenesis the constitution of organs is the principle outcome of morphodynamic evolution while in social morphogenesis the constitution of coherent units is common to segmentary, gerarchic and functional differentiation.

This contribution continues our work on the constitution of visual forms; more specifically, of semiotic plastic forms. We examine emergence from the perspective described by Paolo Fabbri: “unlike logic, the point is not to construct preliminary logical systems and then see how they work in language, but to assemble some very simple units and then observe emergent properties.”. The aim is to “see how, at a certain point, some emergent properties organize themselves and acquire meaning, which is exactly the opposite of old semiotics and old logics.”.

In this study, we analyse the notion of “differential heterogenesis” proposed by Deleuze and Guattari on a morphogenetic perspective. We propose a mathematical framework to envisage the emergence of singular forms from the assemblages of heterogeneous operators. In opposition to the kind of differential calculus that is usually adopted in mathematical-physical modelling, which tends to assume a homogeneous differential equation applied to an entire homogeneous region, heterogenesis allows differential constraints of qualitatively different kinds in different points of space and time. These constraints can then change in time, opening the possibility for new kinds of differential dynamics and the emergence of distinct entities and forms. Formally, we show that operators with different phase spaces can be assembled on the basis of a result of Rothschild & Stein (1976. Hypoelliptic differential operators and nilpotent groups. Acta Mathematica 137. 247–320). Furthermore, operators with different dynamics can be assembled by means of a partition of the unit.After stating the concept of differential heterogenesis in terms of contemporary mathematics, we show that this construction sheds light on the constitution of the semiotic function. In fact, both the Merleau-Pontian and the Deleuzian approaches share a common conceptualisation of the semiotic function and its emergence in terms of amorphodynamics of heterogeneous assemblages with a divergent actualisation. This divergent actualisation allows the co-constitution of various expression and content planes. Finally, we show that the divergent actualisation can be interpreted as the directions of principal eigenvectors of the actualized flow.