Abstract
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 a morphodynamics 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.
References
Bailly, F & G. Longo. 2008. The physical singularity of life. London: Imperial College Press.Search in Google Scholar
Bennett, J. 2010. Vibrant matter. Durham, NC: Duke University Press.Search in Google Scholar
Benveniste, E. 1971. Problems in general linguistics, M. E. Meek (trans.). Coral Gables, FL: University of Miami Press.Search in Google Scholar
Berardi, F. 2015. And: Phenomenology of the end. London: MIT Press.Search in Google Scholar
Bony, J.-M. 1969. Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Annales de l’Institut Fourier 19. 227–304.10.5802/aif.319Search in Google Scholar
Braidotti, R. 2002. Metamorphoses: Towards a materialist theory of becoming. Cambridge: Polity Press; Malden, MA: Blackwell.Search in Google Scholar
Capogna, L. & G. Citti. 2016. Regularity for subelliptic PDE through uniform estimates in multi-scale geometries. Bulletin of Mathematical Sciences 6(2). 173–230.10.1007/s13373-015-0076-8Search in Google Scholar
Citti, G. 1996. C regularity of solutions of a quasilinear equation related to the Levi operator. Annali della Scuola Normale Superiore di Pisa 23(3). 483–529.Search in Google Scholar
Citti, G. & A. Sarti. 2006. A cortical based model of perceptual completion in the roto-translation space. Journal of Mathematical Imaging and Vision 24(3). 307–326.10.1007/s10851-005-3630-2Search in Google Scholar
Citti, G. & A. Sarti. 2014. Neuromathematics of vision. Berlin: Springer.10.1007/978-3-642-34444-2Search in Google Scholar
Citti, G. & M. Manfredini. 2005. Blow-up in non homogeneous Lie groups and rectifiability. Houston Journal of Mathematics 31(2). 333–353.Search in Google Scholar
Citti, G., E. Lanconelli & A. Montanari. 2002. Smoothness of Lipchitz-continuous graphs with nonvanishing Levi curvature. Acta Mathematica 188(1). 87–128.10.1007/BF02392796Search in Google Scholar
Croft, W. & D. A. Cruse. 2004. Cognitive linguistics. Cambridge: Cambridge University Press.10.1017/CBO9780511803864Search in Google Scholar
Deledalle, G. 1979. Théorie et pratique du signe: Introduction à la sémiotique de Ch. S. Peirce. Paris: Payot.Search in Google Scholar
Deleuze, G. 1994. Difference and repetition. New York: Columbia University Press.Search in Google Scholar
Deleuze, G. 2001. Empiricism and subjectivity. New York: Columbia University Press.Search in Google Scholar
Deleuze, G. 2006. Foucault, Sean Hand (trans. & ed.). Minneapolis: University of Minnesota Press.Search in Google Scholar
Deleuze, G. 2003. The logic of sensation. London: Continuum.Search in Google Scholar
Deleuze, G. & F. Guattari. 1987. A thousand plateaus. Minneapolis: University of Minnesota Press.Search in Google Scholar
Deleuze, G. & F. Guattari. 1994. What is philosophy?. London: Verso.Search in Google Scholar
Duits, R. & E. M. Franken. 2010. Left invariant parabolic evolution equations on SE(2) and contour enhancement via invertible orientation scores, part I: Linear left-invariant diffusion equations on SE(2). Quarterly of Applied Mathematics 68. 255–292, 2010.10.1090/S0033-569X-10-01172-0Search in Google Scholar
Eco, U. 1980. Peirce et la sémantique contemporaine. Langages 58. 75–91.10.3406/lgge.1980.1848Search in Google Scholar
Eco, U. 1984. Semiotics and philosophy of language. London: MacMillan Press.10.1007/978-1-349-17338-9Search in Google Scholar
Eco, U. 1988. Sémiotique et philosophie du langage. Paris: PUF.Search in Google Scholar
Everaert, N. 1990. Le processus interprétatif: Introduction à la sémiotique de Ch. S. Peirce. Paris: Mardaga.Search in Google Scholar
Fabbri, P. 1998. L’oscuro principe spinozista: Deleuze, Hjelmslev, Bacon. Special issue, Discipline Filosofiche 1. 209–220.Search in Google Scholar
Fontanille, J. 1999. Modes du sensible et syntaxe figurative. Actes Sémiotiques 61–63. 1–69.Search in Google Scholar
Fontanille, J. 2004. Soma et séma: Figures du corps. Paris: Maisonneuve et Larose.Search in Google Scholar
Fontanille, J. 2006. Pratiques sémiotiques: Immanence et pertinence, efficience et optimisation. Nouveaux Actes Sémiotiques 104–106. 13–74.Search in Google Scholar
Friedman, A. 1964. Partial differential equations of parabolic type. Englewood Cliffs, NJ: Prentice-Hall.Search in Google Scholar
Gilbarg, D. & N. S. Trudinger. 1998. Elliptic partial differential equations of second order. Berlin: Springer-Verlag.Search in Google Scholar
Godel, R. 1969. Les sources manuscrites du Course de Linguistique Generale de F. de Saussure (Publications Romanes et Francaises 61). Geneve: Droz.Search in Google Scholar
Hardwick, C. (ed.). 1977. Semiotic and significs: The correspondence between Charles S. Peirce and Victoria Lady Welby. Bloomington: Indiana University Press.Search in Google Scholar
Hjelmslev, L. 1969. Prolegomena to a theory of language, F. J. Whitfield (trans.). Madison: University of Wisconsin Press.Search in Google Scholar
Hoffman, W. C. 1989. The visual cortex is a contact bundle. Applied Mathematics and Computation 32. 137–167.10.1016/0096-3003(89)90091-XSearch in Google Scholar
Hörmander, L. 1967. Hypoelliptic second order differential equations. Acta Mathematica 119. 147–171.10.1007/BF02392081Search in Google Scholar
Husserl, E. 2001. Logical investigations, 2 vols., J. N. Findlay (trans.). London New-York: Routledge.Search in Google Scholar
Husserl, E. 1995. Lecones sur la theorie de la signification. Paris: Vrin.Search in Google Scholar
Krtolica, I. 2015. “L’algèbre de la pensée pure”: Deleuze et le calcul des problèmes. Revista Trágica 8(2). 16–30.10.59488/tragica.v8i2.26814Search in Google Scholar
Laurent, A. 1994. L’individualisme méthodologique (Que sais-je? 2906). Paris: PUF.Search in Google Scholar
Langacker, R. W. 2008. Cognitive grammar: A basic introduction. New York: Oxford University Press.10.1093/acprof:oso/9780195331967.001.0001Search in Google Scholar
Longo, A. 2016. Le modele mathematique a la base de la philosophie de Deleuze permet-il d’acceder a une realite en soi? http://www.implications-philosophiques.org/actualite/une/deleuze-et-les-mathematiques/ accessed 18 June 2019).Search in Google Scholar
Longo, G. & M. Montevil. 2014. Perspective on organisms: Biological time, symmetries, and singularities. Berlin: Springer.10.1007/978-3-642-35938-5Search in Google Scholar
May, T. 2005. Gilles Deleuze, difference, and science. In G. Gutting (ed.), Continental philosophy of science, 239–257. Malden, MA: Blackwell.10.1002/9780470755501.ch18Search in Google Scholar
Merleau-Ponty, M. 2012. Phenomenology of perception, D. A. Landes (trans.). London: Routledge.10.4324/9780203720714Search in Google Scholar
Nagel, A., E. M. Stein & S. Wainger. 1985. Balls and metrics defined by vector fields. I. Basic properties. Acta Mathematica 155. 103–147.10.1007/BF02392539Search in Google Scholar
Peirce, Charles S. 1931–1966. The collected papers of Charles S. Peirce, 8 vols., C. Hartshorne, P. Weiss & A. W. Burks (eds.). Cambridge: Harvard University Press. [Reference to Peirce’s papers will be designated CP followed by volume and paragraph number.]Search in Google Scholar
Petitot, J. 2015. Complex methodological individualism. Cosmos + Taxis 3.Search in Google Scholar
Petitot, J. 2017. The formalisation of semiotic elementary structures. In Dario Compagno (ed.), Quantitative semiotic analysis, 33–54. Berlin: Springer.10.1007/978-3-319-61593-6_2Search in Google Scholar
Petitot, J. & Y. Tondut. 1999. Vers une neurogéométrie: Fibrations corticales, structures de contact et contours subjectifs modaux. Mathématiques et Sciences humaines 145. 5–101.10.4000/msh.2809Search in Google Scholar
Petitot, J. 2004. Morphogenesis of meaning, P-A. Brandt & W. Wildgen (eds.). Bern: Peter Lang.Search in Google Scholar
Piotrowski, D. 2017. Morphogenesis of the sign. Berlin: Springer.10.1007/978-3-319-55325-2Search in Google Scholar
Rothschild, L. & E. M. Stein. 1976. Hypoelliptic differential operators and nilpotent groups. Acta Mathematica 137. 247–320.10.1007/BF02392419Search in Google Scholar
Rosenthal, V. & Y.-M. Visetti. 2008. Modèles et pensées de l’expression: perspectives microgénétiques. Intellectica 50. 177–252.10.3406/intel.2008.1232Search in Google Scholar
Sarti, A. & D. Barbieri. 2017. Neuromorphology of meaning. In Dario Compagno (ed.), Quantitative semiotic analysis, 55–74. Berlin: Springer.10.1007/978-3-319-61593-6_3Search in Google Scholar
Sarti, A. & G. Citti. 2015. The constitution of perceptual units in the functional architecture of V1. Journal of Computational Neuroscience 38(2). 285–300.10.1007/s10827-014-0540-6Search in Google Scholar
Sarti, A. & D. Piotrowski. 2015. Individuation and semiogenesis: An interplay between geometric harmonics and structural morphodynamics. In A. Sarti, F. Montanari & F. Galofaro (eds.), Morphogenesis and individuation, 49–73. Berlin: Springer.10.1007/978-3-319-05101-7_3Search in Google Scholar
Sarti, A., G. Citti & J. Petitot. 2008. The symplectic structure of the primary visual cortex. Biological Cybernetics 98. 33–48.10.1007/s00422-007-0194-9Search in Google Scholar
Sarti, A., G. Citti & D. Piotrowski. in preparation. Differential heterogenesis: Post-structural dynamics and singular becoming.Search in Google Scholar
Sarti, A., F. Montanari & F. Galofaro (ed.). Morphogenesis and individuation. Berlin: Springer.Search in Google Scholar
Saussure, F. de. 1959. Course in general linguistics, W. Baskin (trans.). New York: Philosophical Library.Search in Google Scholar
Sauvagnargues, A. 2008. Deleuze, l’empirisme transcendantal. Paris: PUF.Search in Google Scholar
Simondon, G. 1995 [1964]. L’individu et sa genèse physico-biologique (l’individuation à la lumière des notions de forme et d’information). Paris: J. Millon.Search in Google Scholar
Stein, E. M. 1993. Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. Princeton, NJ: Princeton University Press.10.1515/9781400883929Search in Google Scholar
Thom, R. 1972. Stabilité structurelle et morphogenèse. New York: Benjamin.Search in Google Scholar
Thom, R. 1980. Modèles mathématiques de la morphogenèse. Paris: Christian Bourgois.Search in Google Scholar
Turing, A. 1952–1992. The chemical basis of morphogenesis. In Collected Works, vol. 4, 1–36. Amsterdam: North-Holland.10.2307/j.ctv2z0vtrq.20Search in Google Scholar
Zucker, S. W. 2006. Differential geometry from the Frenet point of view: Boundary detection, stereo, texture and color. In N. Paragios, Y. Chen & O. Faugeras (eds.), Handbook of mathematical models in computer vision, 357–373. Berlin: Springer, US 2006.10.1007/0-387-28831-7_22Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston