Plerosis and Atomic Gestalts

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Franz Brentano, 1838–1917, introduced the intriguing concept of “plerosis” in order to account for aspects of the continuum that were “explained” by formal mathematics in ways that he considered absurd from the perspective of intuition, especially visual awareness and imagery. In doing this, he pointed in directions later developed by the Dutch mathematician Luitzen Brouwer. Brentano’s notion of plerosis involves distinct though coincident points, which one might call “atomic entities with parts”. This notion fits the modern concepts of “receptive field” in neurophysiology, “perceptive field” in psychology and “differential operator” in the formal theory of scale space. We identify Brentano’s boundary points as the primordial atomic Gestalts of visual imagery. The concept deserves to play a key role in Gestalt theory.

Albertazzi, L. (2002). Towards a neo-Aristotelian theory of continua: Elements of an empirical geometry. In L. Albertazzi (Ed.), Unfolding perceptual continua (pp. 29–80; esp. pp. 44–52). Amsterdam, Netherlands: Benjamins.

Albertazzi, L. (2006). Immanent realism: An introduction to Brentano. (pp. 233–268; esp. pp. 248–267). Dordrecht, Netherlands: Springer.

Albertazzi, L. (2015). Spatial elements in visual awareness. Challenges for an intrinsic “Geometry” of the visible. Philosophia Scientiæ, 19(3), 95–125.

Attneave, F. (1954). Some informational aspects of visual perception. Psychological Review, 61, 184–193.

Bakalis, N. (2005). Handbook of Greek philosophy: From Thales to the Stoics analysis and fragments. Victoria, BC, Canada: Trafford Publishing.

Bell, J. (2006). The continuous and the infinitesimal in mathematics and philosophy. Monza, Italy: Polimetrica.

Bell, J. (2009). Cohesiveness. Intellectica, 9(1), 51.

Bijl, P., Koenderink, J. J., & Toet, A. (1989). Visibility of blobs with a Gaussian luminance profile. Vision Research, 29(4), 447–456.

Blum, M. (1967). A transformation for extracting new descriptors of shape. In W. Wathen-Dunn (Ed.), Models for the perception of speech and visual form (pp. 362–380). Cambridge, MA: MIT Press.

Boime, A. (1993). The art of the Macchia and the Risorgimento. Chicago and London: The University of Chicago Press.

Bouma, H. (1970). Interaction effects in parafoveal letter recognition. Nature, 226(241), 177–178.

Brentano, F. (1974, [1874]). Psychologie vom empirischen Standpunkt. Leipzig, Germany: Duncker und Humblot. (English quotations from: “Psychology from an Empirical Standpoint”. New York: Humanities Press).

Brentano, F. (1988). Philosophical investigations on space, time and the continuum (Smith, Trans.). London, England: Croom Helm.

Broude, N. (1987). The Macchiaioli: Italian painters of the nineteenth century. New Haven and London: Yale University Press.

Brouwer, L. E. J. (1918). Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Erster Teil, Allgemeine Mengenlehre. KNAW Verhandlungen, 5, 1–43.

Brown, J. W. (1977). Mind, brain and consciousness. New York, NY: Academic Press.

Cartan, É (1923). Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie). Annales Scientifiques de l’École Normale Supérieure, 40, 325–412.

Cateura, L. (1995). Oil painting secrets from a master. New York, NY: Watson & Guptill.

Chadwick, H. (1992). St. Augustine, Confessions. [orig. 397-400CE]. Oxford, England: Oxford University Press.

Dedekind, R. (1872). Stetigkeit und irrationale Zahlen. Braunschweig, Germany: Vieweg, p. 18.

Dunn, C. (1995). Conversations in paint, a notebook of fundamentals. New York, NY: Workman Publishing.

Ehrlich, P. (2006). The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Archive for History of Exact Sciences, 60(1), 1–121.

Elkind, D. (1964). Studies in perceptual development II, part-whole. Child Development, 35(1), 81–90.

Euclid. (1956, [fl. 300BCE]). The thirteen books of Euclid’s elements, (Translation and Commentaries by Heath, T. L.), Three volumes. New York, NY: Dover Publications.

Feynman, R. (1966). The character of physical law. Cambridge, MA: MIT Press.

Florack, L. (1997). Image structure. Dordrecht, Netherlands: Kluwer Academic.

Geem, Z. W., Kim, J. H., & Loganathan, G. V. (2001). A new heuristic optimization algorithm: Harmony search. Simulation, 76(2), 60–68.

Giordano, P. (2001). Nilpotent infinitesimals and synthetic differential geometry in classical logic. In P. Schuster, U. Berger, & H. Osswald (Eds.), Reuniting the antipodes–constructive and nonstandard views of the continuum (pp. 75–92). Dordrecht, Netherlands: Kluwer.

Gödel, K. (1960, [1947]). What is Cantor’s continuum problem? The American Mathematical Monthly, 54, 515–525. (originally published in 1947, Page references are to the reprinting of the expanded 1960 version in Benacerraf and Putnam 1983, pp. 420–485).

Hartline, H. K. (1938). The response of single optic nerve fibers of the vertebrate eye to illumination of the retina. The American Journal of Physiology, 121, 400–415.

Hilbert, D. (1980, [1899]). The foundations of geometry (2nd ed.). Chicago, IL: Open Court.

Hoffman, D. D. (2009). The interface theory of perception. In S. Dickinson, M. Tarr, A. Leonardis, & B. Schiele (Eds.), Object categorization: Computer and human vision perspectives (pp. 148–165). New York, NY: Cambridge University Press.

Hubel, D. H., & Wiesel, T. N. (1968). Receptive fields and functional architecture of monkey striate cortex. Journal of Neuroscience, 195, 215–243.

Husserl, E. (1991, [1893-1917]). On the phenomenology of the consciousness of internal time (1893-1917) (J. B. Brough, Trans.). Dordrecht, Netherlands: Kluwer.

Imbriano, V. (1868). La quinta promotrice 1878-1868: Appendici di Vittorio Imbriani. Napoli, Italy: Tipografia Napolitana.

Jacobs, T. S. (1986). Light for the Artist. NewYork, NY: Watson-Guptill.

Kandinsky, W. (1926). In W. Gropius & L. Moholy-Nagy (Eds.), Punkt und Linie zu Fläche. Bauhaus Bücher, Schriftleitung München, Germany: Verlag Albert Langen.

Kanizsa, G. (1980). Grammatica del vedere. Saggi su percezione e Gestalt. Bologna, Italy: IL Mulino.

Koenderink, J. J. (1984). The structure of images. Biological Cybernetics, 50, 363–370.

Koenderink, J. J. (1988). Operational significance of receptive field assemblies. Biological Cybernetics, 58(3), 163–171.

Koenderink, J. J. (1990). The brain a geometry engine. Psychological Research, 52(2–3), 122–127.

Koenderink, J. J. (1993). What is a “feature”? Journal of Intelligent Systems, 3(1), 49–82.

Koenderink, J. J. (2002). Continua in vision. In L. Albertazzi (Ed.), Unfolding perceptual continua, advances in consciousness research 41 (pp. 101–118). Amsterdam and Philadelphia: John Benjamins.

Koenderink, J. J. (2011). Vision as a user interface. In B. E. Rogowitz & T. N. Pappas (Eds.), Proceedings of SPIE-IS&T Electronic Imaging, SPIE, Vol. 7865: Human Vision and Electronic Imaging XVI (pp. 1–13). San Francisco: SPIE-IS&T.

Koenderink, J. J. (2014). The all seeing eye. Perception, 43, 1–6.

Koenderink, J. J. (2015). Ontology of the mirror world. Gestalt Theory, 37(2), 119–140.

Koenderink, J. J., Albertazzi, L., van Doorn, A. J., van Ee, R., van de Grind, W. A., Kappers, A. M. L., … de Vries, S. (2010). Does monocular visual space contain planes? Acta Psychologica (Amst), 134(1), 40–47.

Koenderink, J. J., & Richards, W. A. (1988). Two-dimensional curvature operators. Journal of the Optical Society of America A, 5(7), 1136–1141.

Koenderink, J. J., & Van Doorn, A. J. (1990). Receptive field families. Biological Cybernetics, 63(4), 291–297.

Koenderink, J. J., & Van Doorn, A. J. (1992a). Generic neighbourhood operators. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14, 597–605.

Koenderink, J. J., & Van Doorn, A. J. (1992b). Receptive field assembly pattern specificity. Journal of Visual Communication and Image Representation, 3(1), 1–12.

Koenderink, J. J., & van Doorn, A. J. (1997). Local image operators and iconic structure. In G. Sommer & J. J. Koenderink (Eds.), Proceedings of the AFPAC’97. Algebraic Frames for the Perception-Action Cycle (pp. 66–93). Berlin, Heidelberg: Springer.

Koenderink, J. J., van Doorn, A. J., & Pinna, B. (2015). Psychogenesis of Gestalt. Gestalt Theory, 37(3), 287–304.

Koenderink, J. J., van Doorn, A. J., Albertazzi, L., & Wagemans, J. (2015). Relief articulation techniques. Art & Perception, 3(2), 151–171.

Koenderink, J. J., van Doorn, A. J., Pinna, B., & Wagemans, J. (2016). Boundaries, transitions and passages. Art & Perception, 4(3), 185–204.

Marr, D. (1982). Vision: A computational investigation into the human representation and processing of visual information. New York, NY: Freeman.

Marr, D., & Hildreth, E. (1980). Theory of edge detection. Proceedings of Royal Society of London B, 207, 187–217.

Metzger, W. (1936). Gesetze des Sehens. Frankfurt am Main, Germany: W. Kramer & Co.

Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. San Francisco, CA: W. H. Freeman.

Panconi, T. (1999). Antologia dei Macchiaioli, La trasformazione sociale e artica nella Toscana di metà 800. Pisa, Italy: Pacini Editore.

Pasch, M. (1912). [first edition 1882], Vorlesungen uber neuere Geometrie (2nd ed.). Leipzig, Germany: B.G. Teubner, p. 21.

Photoshop. (2014). Adobe, Retrieved from

Pinna, B. (2008). Watercolor illusion. Scholarpedia, 3(1), 5352.

Pinna, B. (2010). New Gestalt principles of perceptual organization: An extension from grouping to shape and meaning. Gestalt Theory, 32(1), 11–78.

Pinna, B. (2012). Perceptual organization of shape, color, shade and lighting in visual and pictorial objects. i-Perception, 3, 257–281.

Pinna, B., & Albertazzi, L. (2011). From grouping to visual meanings: A new theory of perceptual organization. In L. Albertazzi, G. van Tonder, & D. Vishwanath (Eds.), Information in perception (pp. 288–344). Cambridge, MA: MIT Press.

Pinna, B., Brelstaff, G., & Spillmann, L. (2001). Surface color from boundaries: A new ‘watercolor’ illusion. Vision Research, 41, 2669–2676.

Pinna, B., & Deiana, K. (2015). The syntax organization of shape and color and the laws of coloration in vision, art and biology. Art & Perception, 3(3), 319–345.

Pinna, B., & Grossberg, S. (2005). The watercolor illusion and neon color spreading: A unified analysis of new cases and neural mechanisms. Journal of the Optical Society of America A, 22, 2207–2221.

Pinna, B., & Reeves, A. (2006). Lighting, backlighting and watercolor illusions and the laws of figurality. Spatial Vision, 19, 341–373.

Pinna, B., Spillmann, L., & Werner, J. (2003). The watercolor effect: A new principle of grouping and figure-ground organization. Vision Research, 43, 43–52.

Richards, W. R., & Bobick, A. (1988). Playing twenty questions with nature. In Z. W. Pylyshyn (Ed.), Computational processes in human vision: An interdisciplinary perspective (pp. 3–26). The Canadian Institute for Advanced Research Series in Artificial Intelligence. Norwood, NJ: Ablex Publishing.

Richards, W.R. (1982, December): How to play twenty questions with nature and win (M.I.T. A.I. Memo 660).

Riedl, R. (1975). Die Ordnung des Lebendigen. Systembedingungen der Evolution. Hamburg, Germany: Paul Parey.

Riemann, B. (2013). [Habilitationsvortrag 10. Juni 1854]. Über die Hypothesen, Welche der Geometrie zu Grunde Liegen. Berlin, Germany: Springer.

Roeper, P. (2006). The Aristotelian continuum. A formal characterization. Notre Dame Journal of Formal Logic, 47(2), 211–232.

Schrödinger, E. (1944). What is life?. Cambridge, UK: Cambridge University Press.

Spillmann, L. (1971). Foveal perceptive fields in the human visual system measured with simultaneous contrast in grids and bars. Pflügers Archiv, 326(4), 281–299.

Stevin, S. (1585). In A. J. E. M. Smeur (Ed.), De Thiende (p. 1965). Nieuwkoop, Netherlands: De Graaf.

Stoltz, O. (1883). Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes. Mathematische Annalen, 22, 504–519.

ter Haar Romeny, B. M. (2003). Front-end vision and multi-scale image analysis. Dordrecht, Netherlands: Kluwer Academic.

Tinbergen, N. (1952). The curious behavior of the stickleback. Scientific American, 187(6), 22–26.

van Dalen, D. (1997). How connected is the intuitionistic continuum? The Journal of Symbolic Logic, 62(4), 1147–1150.

Veronese, G. (1894). Grundzüge der Geometrie von mehreren Dimensionen und mehreren Arten gradliniger Einheiten in elementarer Form entwickelt (Uebersetzt von A. Schepp). Leipzig, Germany: B.G. Teubner.

von Uexküll, J. (1909). Umwelt und Innenwelt der Tiere. Berlin, Germany: Springer.

von Uexküll, J. (1920). Theoretische Biologie. Berlin, Germany: Springer.

Weber, E. H. (1846). Tastsinn und Gemeingefühl. In R. Wagner (Ed.), Handwörterbuch der Physiologie mit Rücksicht auf physiologische Pathologie (Band 3 ed., pp. 481–588). Braunschweig, Germany: Vieweg.

Wertheimer, M. (1922): Untersuchungen zur Lehre von der Gestalt, I: Prinzipielle Bemerkungen. Psychologische Forschung, 1, pp. 47–58. [Translated extract reprinted as “The general theoretical situation.” In W. D. Ellis (Ed.), (1938). A source book of Gestalt psychology (pp. 12–16). London, UK: Routledge & Kegan Paul Ltd.].

Wertheimer, M. (1923): Untersuchungen zur Lehre von der Gestalt, II. Psychologische Forschung, 4, pp. 301–350. [Translated as “Investigations on Gestalt principles, II. In L. Spillmann (Ed.), (2012). On motion and figure-ground organization (pp. 127–182). Cambridge, MA: M.I.T. Press].

Weyl, H. (1925). On the current epistemological situation in mathematics [English translation of Die Heutige Erkenntnislage in der Mathematik, Symposion, 1, 1925–1927, 1–32.] In P. Mancosu (Ed.) (1998), pp. 123–142.

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