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Abstract

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This paper sketches two classes of mathematical models. Both treat ambivalence and attentiveness as undefined terms. The first class, ambivalence-generated models, is finitistic and requires no ontological commitment to any mathematical construction as sophisticated as ‘real numbers’. The intended semantics suggests the axiom underlying topologists’ well-understood theory of finite simplicial complexes (FSCs). Semantically reinterpreted within psychology, this theory yields concrete, empirically testable hypotheses about human behaviour, which in turn suggest further axiomatic restrictions on the models. The second class of models treats attentiveness as a Morse function on some differential manifold, and uses its gradient flow to construct a lower-dimensional spine. Both classes of models have potential to capture much of the flexibility and concreteness that are attractive in qualitative methodologies, while retaining (by application of mathematical analysis) the formality of quantitative methodologies.

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References

Abbey

Valsiner

(2004). Emergence of meanings through ambivalence. FQS: Forum Qualitative Sozialforschung, 6 (1), Art. 23. http://www.qualitative-research.net/fqs-texte/1-05/05-1-23-e.htm

Abraham

Robbin

(1967). Transversal Mappings and Flows. New York: W. A. Benjamin, Inc.

Berlin

Kay

(1969). Basic Color Terms: Their Universality and Evolution. Berkeley, CA: University of California Press.

Dzhafarov

E. N.

Colonius

(2001). Multidimensional Fechnerian scaling: Basics. Journal of Mathematical Psychology, 45, 670–719.

Dzhafarov

E. N.

Colonius

(2005). Psychophysics without physics: a purely psychological theory of Fechnerian scaling in continuous stimulus spaces. Journal of Mathematical Psychology, 49 (1), 1–50.

Ekman

Friesen

W. V.

(1975). Unmasking the Face: A Guide to Recognizing Emotions from Facial Clues. Oxford, England: Prentice-Hall.

Eliot

T. S.

(1952). Burnt Norton. In T. S. Eliot, The Complete Poems and Plays (p. 118). New York: Harcourt, Brace and Company.

Empson

(1930/1947). Seven Types of Ambiguity. London: Chatto and Windus.

Grice

H. P.

(1965). The causal theory of perception. In Swartz

R. J.

(Ed.), Perceiving, Sensing, and Knowing (pp. 438–472). Garden City, NY: Doubleday-Anchor. Reprinted from Aristotelian Society, supp. vol. 35, 121–52, 1961.

10.

Kaiser

Wehrle

(1992). Automated coding of facial behavior in human-computer interactions with FACS. Journal of Nonverbal Behavior, 16 (2), 67–84.

11.

Kreisel

(1978). Review of Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939. Bulletin of the American Mathematical Society, 84, 79—90.

12.

Kuipers

Tecuci

Stankiewicz

(2003). The Skeleton in the Cognitive Map: A Computational and Empirical Exploration. Environment and Behavior, 35, 81—106.

13.

Lefschetz

(1942). Algebraic Topology (American Mathematical Society Colloquium Publications, vol. 27). New York: American Mathematical Society.

14.

Milnor

(1963). Morse theory. Annals of Mathematics Studies, No. 51. Princeton, NJ: Princeton University Press.

15.

Nahas

De Bonis

(2001). Image technology and facial expression of emotions. Proceedings of 10th IEEE International Workshop on Robot and Human Interactive Communication.

16.

Novikov

S. P.

(1981). Multivalued functions and functionals. An analogue of the Morse theory. Dokl. Akad. Nauk SSSR 260 (1), 31–35.

17.

Pajitnov

Rudolph

Weber

(2001). Morse-Novikov numbers of knots and links. Algebra i Analiz (St. Petersburg Mathematical Journal), 13, 105–118.

18.

Poddiakov

A. N.

Valsiner

(submitted). Intransitivity cycles and their transformations: how dynamically adapting systems function? Theory & Psychology.

19.

Russell

J. A.

(1980). A circumplex model of affect. Journal of Personality and Social Psychology, 39 (1, supp. 6), 1161–1178.

20.

Shepard

R. N.

(1978). The circumplex and related topological manifolds in the study of perception. In Shye

(Ed.), Theory construction and data analysis in the behavioral sciences (pp. 29–80). San Francisco: Jossey-Bass.

21.

Shepard

R. N.

(1992). The perceptual organization of colors: An adaptation to regularities of the terrestrial world? In Barkow

J. H.

Cosmides

Tooby

(Eds.), The Adapted Mind (pp. 495–532). New York: Oxford University Press.

22.

Tucker

A. W.

(1939). The algebraic structure of complexes. Proc. Nat. Acad. Sci. USA, 25, 643–647.

23.

Uexküll Von

(1920/1926). Theoretical Biology (D. I. MacKinnon, Trans.; German original, 1920, Frankfurt: Suhrkamp). New York: Harcourt, Brace & Company, Inc.

24.

Wigner

(1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13 (1), 1–14.

25.

Zeeman

(1962). The topology of the brain and visual perception. In Fort

M. K.

(Ed.), Topology of 3-manifolds and related topics (pp. 240–256). Englewood Cliffs, N.J.: Prentice-Hall.

Spaces of ambivalence: Qualitative mathematics in the modelling of complex fluid phenomena

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