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A Mathematical Science of Qualities: A Sequel

Abstract

Following a previous article published in Biological Theory, in this study we present a mathematical theory for a science of qualities as directly perceived by living organisms, and based on morphological patterns. We address a range of qualitative phenomena as observables of a psychological system seen as an impredicative system. The starting point of our study is the notion that perceptual phenomena are projections of underlying invariants, objects that remain unchanged when transformations of a certain class under consideration are applied. The study develops with the observables, the entailed total order and metric, whence the algebra and the geometry of such a science, presenting a formal phenomenological model for phenomena that are not rigidly Euclidean. We show how non-Euclidean perception can have many useful (non-rigid) Euclidean formalizations, as well as locally-homeomorphic-to-Euclidean-space models. The mathematical models we provide are tested on the basis of results from experimental psychology, in particular from the field of color, time, and space perception.

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References

  • Albers J (1975) Interaction of color. Yale University Press, New Haven

    Google Scholar 

  • Albertazzi L (1993) The time of presentness: a chapter in positivistic and descriptive psychology. Axiomathes 10(1–3):49–73. doi:10.1007/BF02681816

    Google Scholar 

  • Albertazzi L (2004) Stereokinetic shapes and their shadows. Perception 33:1437–1452

    Article  Google Scholar 

  • Albertazzi L (2012) Qualitative perceiving. J Conscious Stud 19(11–12):37–47

    Google Scholar 

  • Albertazzi L (2013a) Experimental phenomenology: an introduction. In: Albertazzi L (ed) The Wiley Blackwell handbook of experimental phenomenology. Visual perception of shape, space and appearance. Blackwell-Wiley, Chichester, pp 1–36

    Chapter  Google Scholar 

  • Albertazzi L (2013b) Appearances from a radical viewpoint. In: Albertazzi L (ed) The Wiley Blackwell handbook of experimental phenomenology. Visual perception of shape, space and appearance. Blackwell-Wiley, Chichester, pp 267–290

    Chapter  Google Scholar 

  • Albertazzi L (2015a) A science of qualities. Biol Theory. doi:10.1007/s13752-015-0213-3

    Google Scholar 

  • Albertazzi L (2015b) Spatial elements in visual awareness. Challenges for an intrinsic “geometry” of the visible. Philos Sci 19(3):95–125

    Google Scholar 

  • Albertazzi L (2015c) Introduction to the special issue on morphologies in nature and art. Art Percept 3(3):213–219

    Article  Google Scholar 

  • Albertazzi L, Da Pos O (2016) Colour names, stimulus colour, and their subjective links. Color Res Appl. doi:10.1002/col.22034

    Google Scholar 

  • Albertazzi L, Poli R (2014) Multi-leveled objects: colour as a case study. Front Psychol. doi:10.3389/fpsyg.2014.00592

    Google Scholar 

  • Albertazzi L, van Tonder G, Vishwanath D (2010) Information in perception. In: Albertazzi L, van Tonder G, Vishwanath D (eds) Perception beyond inference: the information content of perceptual processes. MIT Press, Cambridge, pp 1–26

    Google Scholar 

  • Albertazzi L, Canal L, Da Pos O et al (2012) The hue of shapes. J Exp Psychol Hum Percept Perform 39(1):37–47. doi:10.1037/a0028816

    Article  Google Scholar 

  • Albertazzi L, Malfatti M, Canal L, Micciolo R (2014a) The hue of angles: Was Kandinsky right? Art Percept 3(1):81–92. doi:10.1163/22134913-00002025

    Article  Google Scholar 

  • Albertazzi L, Canal L, Dadam J, Micciolo R (2014b) The semantics of biological forms. Perception 43:1365–1376. doi:10.1068/p7794

    Google Scholar 

  • Albertazzi L, Canal L, Micciolo R (2015a) Cross-modal associations between materic painting and Spanish classical music. Front Psychol. doi:10.3389/fpsyg.2015.00424

    Google Scholar 

  • Albertazzi L, Bacci F, Canal L, Micciolo R (2015b) The tactile dimensions of abstract paintings: a cross-modal study. Perception 45(7):805–822. doi:10.1177/0301006616643660

    Article  Google Scholar 

  • Astbury WT (1961) Molecular biology or ultrastructural biology? Nature 190(4781):1124. doi:10.1038/1901124a0

    Article  Google Scholar 

  • Benussi V (1913) Psychologie der Zeitauffassung. Winter, Heidelberg

    Google Scholar 

  • Burigana L, Da Pos O (2013) Qualitative inference rules for perceptual transparency. In: Albertazzi L (ed) The Wiley Blackwell handbook of experimental phenomenology. Visual perception of shape, space and appearance. Blackwell-Wiley, Chichester, pp 343–368

    Google Scholar 

  • Calabresi R (1930) La determinazione del presente psichico. Bemporad, Florence

    Google Scholar 

  • Canal L, Micciolo R (2013) The measurement of the unmeasurable. In: Albertazzi L (ed) The Wiley Blackwell handbook of experimental phenomenology. Visual perception of shape, space and appearance. Blackwell-Wiley, Chichester, pp 477–498

    Chapter  Google Scholar 

  • Chapanis A, Mankin DA (1967) The vertical–horizontal illusion in a visually-rich environment. Percept Psychophys 2(6):249–255. doi:10.3758/BF03212474

    Article  Google Scholar 

  • Coren S, Gyrus J (1978) Seeing is deceiving: the psychology of visual illusions. Lawrence Erlbaum, Hillsdale

    Google Scholar 

  • Da Pos O, Albertazzi L (2010) It is in the nature of colour…. Seeing and Perceiving 23(2010):39–73

    Article  Google Scholar 

  • Ebbinghaus H (1902) Grundzüge der Psychologie. Veit, Leipzig

    Google Scholar 

  • Ehm W, Wackermann J (2012) Modelling geometric-optical illusions: a variational approach. J Math Psychol 56:404–416

    Article  Google Scholar 

  • Ehm W, Wackermann J (2016) Geometric-optical illusions and Riemannian geometry. J Math Psychol 71:28–38

    Article  Google Scholar 

  • Gibson JJ (1979) The ecological approach to visual perception. Houghton Mifflin, Boston

    Google Scholar 

  • Gregory RL (1997) Visual illusions classified. Trends Cogn Sci 1:190–194

    Article  Google Scholar 

  • Hammond C (2012) Time warped: unlocking the mysteries of time perception. Canongate, London

    Google Scholar 

  • Hartmann N (1940) Der Aufbau der realen Welt. Grundiss der allgemeinen Kategorienlehre. De Gruyter, Berlin

    Book  Google Scholar 

  • Hartmann N (1950) Philosophie der Natur. Abriss der speziells en Kategorienlehre. De Gruyter, Berlin

    Book  Google Scholar 

  • Hering E (1861) Beiträge zur Physiologie. I. Zur Lehre vom Ortssinne der Netzhaut. Engelmann, Leipzig

    Google Scholar 

  • Hering E (1920/1964) Outlines of a theory of the light sense. Springer, Berlin

  • Hogeweg P (2011) The roots of bioinformatics in theoretical biology. PLoS Comput Biol 7(3):e1002021

    Article  Google Scholar 

  • Husserl E (1966/1991) Zur Phänomenologie des inneren Zeitbewusstseins. In: Boehm R, Husserliana X (eds) Martinus Nijhoff, Den Haag. Eng. tr. (1991) Brough JB (trans). Kluwer, Dordrecht

  • James W ([1950]1890) Principles of psychology. Holt, Boston. Repr. 1950, Dover, New York

  • Jammer M (1954) The history of theories of space in physics. Harvard University Press, Cambridge

    Google Scholar 

  • Kanizsa G (1979) Organization in vision: essays on gestalt perception. Praeger, New York

    Google Scholar 

  • Kanizsa G (1991) Vedere e pensare. Il Mulino, Bologna

    Google Scholar 

  • Katz D (1906) Experimentelle Beiträge zur Psychologie des Vergleichs im Gebiete des Zeitsinns. Z Psychol 42:302–340

    Google Scholar 

  • Kitano H (2002) Computational systems biology. Nature 420(6912):206–210. doi:10.1038/nature01254

    Article  Google Scholar 

  • Köhler W (1929) Gestalt psychology. Liveright, New York

    Google Scholar 

  • Libet B (1982) Brain stimulation in the study of neuronal functions for conscious sensory experience. Hum Neurobiol 1:235–242

    Google Scholar 

  • Libet B (1987) Consciousness: conscious, subjective experience. In: Adelman G (ed) Encyclopedia of neuroscience. Birkhäuser, Boston, pp 271–275

    Google Scholar 

  • Libet B (2004) Mind time: the temporal factors in consciousness. Harvard University Press, Cambridge

    Google Scholar 

  • Libet B, Wright EW Jr, Feinstein B, Peral DK (1992) Retroactive enhancement of a skin sensation by a delayed cortical stimulus in man: evidence for delay of a conscious sensory experience. Conscious Cogn 1:367–375

    Article  Google Scholar 

  • Lipps T (1897) Raumaesthetik und geometrisch-optische Täuschungen. Barth, Leipzig

    Google Scholar 

  • Löfgren L (1968) An axiomatic explanation of complete self-reproduction. Bull Math Biol 30:415–425

    Google Scholar 

  • Louie AH (2009) More than life itself: a synthetic continuation in relational biology. Ontos, Frankfurt

    Book  Google Scholar 

  • Louie AH (2013) The reflection of life: functional entailment and imminence in relational biology. Springer, New York

    Book  Google Scholar 

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

    Google Scholar 

  • McTaggart JME (1908) The unreality of time. Mind 17:457–474

    Article  Google Scholar 

  • Metzger W (1930) Optische Untersuchungen am Ganzfeld. Psychol Forsch 13:6–29

    Article  Google Scholar 

  • Metzger W (1936/2006) Laws of seeing. Spillmann L, Lehar S, Stromeyer M, Wertheimer M (trans). MIT Press, Cambridge

  • Metzger W (1941/1963) Psychologie: die Entwicklung ihrer Grundannahmen seit der Einführung des Experiments. Steinkopf, Dresden

  • Michotte A (1948/1991) L’énigme psychologique de la perspective dans le dessin linéaire, Bull l’Acad R Belg 5(34):268–288. English translation, Costal A (trans) (1991) The psychological enigma of perspective in outline pictures. In: Thinès G, Costall A, Butterworth G (eds) Michotte’s experimental phenomenology of perception. Lawrence Erlbaum, Hillsdale, p 174–187

  • Michotte A (1954) La perception de la causalité. Publications Universitaires de Louvain, Louvain

    Google Scholar 

  • Michotte A (1963) In: Miles TR, Miles E (eds) The perception of causality. Methuen, London

    Google Scholar 

  • Müller-Lyer FC (1889) Optische Urteilstäuschungen. Arch Anat Physiol Physiol Abt Suppl 2:263–270

    Google Scholar 

  • Musatti CL (1924) Sui fenomeni stereocinetici. Arch Ital Psicol 3:105–120

    Google Scholar 

  • Musatti CL (1955) La stereocinesi e la struttura dello spazio visibile. Riv Psicol 49:3–57

    Google Scholar 

  • Musatti CL (1975) Stereokinetic phenomena and their interpretation. In: D’Arcais GB (ed) Studies in perception, Festschrift for Fabio Metelli. Giunti, Milan, pp 166–189

    Google Scholar 

  • Oppel JJ (1854–1855) Über geometrisch-optische Täuschungen. Jahresbericht des physikalischen Vereins zu Frankfurt am Main, p 37–47

  • Osgood CE (1956) Method and theory in experimental psychology. Oxford University Press, Oxford

    Google Scholar 

  • Petitot J (2009) Neurogéométrie de la vision. Modèles mathématiques et physiques des architectures fonctionelles. Le Editions de L’Ecole Polytechnique, Paris

  • Pizlo Z (2001) Perception viewed as an inverse problem. Vis Res 41(24):3145–3161. doi:10.1016/S0042-6989(01)00173-0

    Article  Google Scholar 

  • Poli R (2001) The basic problem of the theory of levels of reality. Axiomathes 12(3–4):261–283

    Article  Google Scholar 

  • Poli R (2006a) First steps in experimental phenomenology. In: Loula A, Gudwin R, Queiroz J (eds) Artificial cognition systems. Idea Group Publishing, Hersey, pp 358–386

    Google Scholar 

  • Poli R (2006b) Levels of reality and the psychological stratum. Rev int philos 61(2):163–180

    Google Scholar 

  • Poli R (2007) Three obstructions: forms of causation, chronotopoids, and levels of reality. Axiomathes 17(1):1–18. doi:10.1007/s10516-007-9007-y

    Article  Google Scholar 

  • Poli R (2011) Ontology as categorial analysis. In: Slavicand A, Civallero E (eds) Classification and ontology: formal approaches and access to knowledge. ErgonVerlag, Würzburg, pp 145–157

    Google Scholar 

  • Poli R (2016) Pure and qualified time. In: Poli R, Peterson K (eds) New research on the philosophy of Nicolai Hartmann. De Gruyter, Berlin, pp 3–22

    Google Scholar 

  • Ponzo M (1912) Rapports de contraste angulaire et l’appréciation de grandeur des astres à l’horizon. Arch Ital Biol 58:327–329

    Google Scholar 

  • Pöppel E (1994) Temporal mechanism in perception. Int Rev Neurobiol 37:185–202

    Article  Google Scholar 

  • Pöppel E, Bao Y (2014) Temporal windows as a bridge from objective to subjective time. In: Arstila V, Lloyd D (eds) Subjective time. MIT Press, Cambridge, pp 241–262

    Google Scholar 

  • Rensink RA (2000) Seeing, sensing, scrutinizing. Vis Res 40:1469–1487

    Article  Google Scholar 

  • Rensink RA (2002) Change detection. Annu Rev Psychol 53:245–277

    Article  Google Scholar 

  • Richardson IW, Louie AH (1983) Projections as representations of phenomena. J Theor Biol 102:199–223

    Article  Google Scholar 

  • Rock I (1977) In defence of unconscious inferences. In: Epstein W (ed) Stability and constancy in visual perception: mechanism and processes. Wiley, New York, pp 321–373

    Google Scholar 

  • Rogers B (2014) Delusions about illusions. Perception 43:840–845

    Google Scholar 

  • Rosen R (1978) Fundamentals of measurement and representation of natural systems. North-Holland, New York

    Google Scholar 

  • Rosen R (1985/2012) Anticipatory systems: philosophical, mathematical, and methodological foundations. Pergamon Press, Oxford. (2012) 2nd edn, Springer, New York

  • Rosen R (1991) Life itself: a comprehensive inquiry into the nature, origin, and fabrication of life. Columbia University Press, New York

    Google Scholar 

  • Rosen R (2000) Essays on life itself. Columbia University Press, New York

    Google Scholar 

  • Rubin E (1949) Geräuschverschiebungsversuche. In: Experimenta psychologica. Munkgaard, Copenhagen

  • Sander F (1926) Optische Täuschungen und Psychologie. Neue Psychol Stud 1:159–166

    Google Scholar 

  • Sivik L (1974) Colour meaning and perceptual colour dimensions: a study of colour samples. Göteb Psychol Rep 4(1):1–17

    Google Scholar 

  • Smith DA (1978) A descriptive model for perception of optical illusions. J Math Psychol 17:64–85

    Article  Google Scholar 

  • Sovrano VA, Albertazzi L, Rosa Salva O (2013) The Ebbinghaus illusion in a fish (Xenotoca eiseni). Anim Cogn. doi:10.1007/s10071-014-0821-5

    Google Scholar 

  • Sovrano VA, Da Pos O, Albertazzi L (2015) The Mueller–Lyer illusion in the teleost fish Xenotoca eiseni. Anim Cogn. doi:10.1007/s10071-015-0917-6

    Google Scholar 

  • Stroyan KD, Luxemburg WAJ (1976) Introduction to the theory of infinitesimals. Academic, New York

    Google Scholar 

  • Thom R (1979) Towards a revival of natural philosophy. In: Guttinger W, Eikemeier H (eds) Structural stability in physics. Springer, Berlin, pp 5–11

    Chapter  Google Scholar 

  • Vicario GB (1998) Forms and events. In: Albertazzi L (ed) Shapes of forms: from gestalt psychology and phenomenology to ontology and mathematics. Kluwer, Dordrecht, pp 89–106

    Google Scholar 

  • Vishwanath D (2010) Visual information in surface and depth perception: reconciling pictures and reality. In: Albertazzi L, van Tonder G, Vishwanath D (eds) Perception beyond inference: the information content of perceptual processes. MIT Press, Cambridge, pp 201–240

    Google Scholar 

  • Vishwanath D (2014) Towards a new theory of stereopsis. Psychol Rev 121(2):151–178

    Article  Google Scholar 

  • Wagner M (1985) The metric of visual space. Percept Psychophys 38:483–495

    Article  Google Scholar 

  • Wertheimer M (1912/1967) Experimentelle Studien über das Sehen von Bewegungen. Z Psychol Physiol Sinnesorgane 61:161–265 (reprinted 1967 in Drei Abhandlungen zur Gestalttheorie. Wissenschaftliche Buchgesellschaft, Darmstadt, p 1–105)

  • Wertheimer M (1923) Untersuchungen zur Lehre von der Gestalt. Psychol Forsch 4:301–350

    Article  Google Scholar 

  • Wyszecki G, Stiles WS (1982) Colour science. Concepts and methods, quantitative data and formulae. Wiley, New York

    Google Scholar 

  • Zöllner JF (1860) Über eine neue Art von Pseudoskopie und ihre Beziehungen zu den von Plateau und Oppel beschriebenen Bewegungsphänomenen. Ann Phys Chem 110:500–523

    Article  Google Scholar 

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Acknowledgments

The authors thank the director and the scientific board of STIAS (Stellenbosch Institute for Advanced Study, Wallenberg Research Centre at Stellenbosch University, Stellenbosch, South Africa) for providing the perfect environment in which to write this paper during the authors’ visit in February to April 2016.

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Albertazzi, L., Louie, A.H. A Mathematical Science of Qualities: A Sequel. Biol Theory 11, 192–206 (2016). https://doi.org/10.1007/s13752-016-0248-0

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Keywords

  • Color
  • Geometric model
  • Qualitative
  • Manifold
  • Space
  • Time