In this speculative article, I question the postulation of homogeneity in models of
time intended for use in developmental, social or cultural psychology. After
surveying extant models, I propose a new kind of model, which is highly
inhomogeneous and one-dimensional only in a global sense: locally such a model can
be very ‘full’. My aim is to capture in mathematical form one
common hypothesis about psychological time (that it reckons ‘acts of
attention’), somewhat amplified by a further claim (that these acts are
interpolated by states of ambivalence), and to do so precisely enough that the
resulting models can be used effectively in further collaborations between
mathematicians and psychologists
Abraham, R.
, &
Robbin, J.
(1967). Transversal mappings and flows.
New York: Benjamin
.
2.
Baaz, M.
,
Leitsch, A.
, &
Zach, R.
(1996).
Completeness of a first-order temporal logic with time-gaps
. Theoretical Computer Science, 160,
241-270
.
3.
Barrett, L. F.
,
Tugade, M. M.
, &
Engle, R.
(2004).
Individual differences in working memory capacity and dual-process theories
of the mind
. Psychological Bulletin, 130,
553-573
.
4.
Benjamin, W.
(1969). Theses on the philosophy of history. In
Illuminations (H. Zohn, Trans.;
H. Arendt
, Ed.; pp. 253-264).
New York: Schocken
. (Original work published 1940.)
5.
Borges, J. L.
(1962). The garden of forking paths (D.A. Yates,
Trans.). In Labyrinths (
D. A. Yates
&
J. E. Irby
, Eds.; pp. 123-138).
New York: New Directions
.
6.
Bourdieu, P.
(1979). The disenchantment of the world. In
Algeria 1960 (pp. 1-94).
Cambridge: Cambridge University Press
. (Original work published 1963.)
7.
Burgess, J. P.
(1979).
Logic and time
. Journal of Symbolic Logic, 44,
566-582
.
8.
Burgess, J. P.
(1980).
Decidability for branching time
. Studia Logica, 39,
203-218
.
9.
Cantor, G.
(1895).
Beiträge zur Begründung der transfiniten Mengenlehre
. Mathematische Annalen, 46,
481-512
. http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0046.
10.
Chaudhuri, A.
(2004).
In the waiting-room of history (review of Provincialising Europe:
Postcolonial thought and historical difference, by Dipesh Chakrabarty)
. London Review of Books, 26(2, 24 June). http://www.lrb.co.uk/v26/n12/chau01.html.
11.
Coulthard, V.
, &
Davoren, J. M.
(2004).
Spatio-temporal logics for continuous dynamical systems
. Conference paper submission, Nineteenth Annual IEEE Symposium on
Logic in Computer Science (LICS 2004). http://www.ee.unimelb.edu.au/staff/davoren/00-lics04-coulthard.pdf.
12.
Darling, D.
(2004).
Marking time
. Latitudes36(6), unpaginated. http://www.latitudesmagazine.com/main/article/vol36-6.html.
13.
Davoren, J. M.
,
Coulthard, V.
,
Markey, N.
, &
Moor, T.
(2004). Non-deterministic temporal logics for general flow
systems. In
R. Alur
&
G. J. Pappas
(Eds.), Hybrid systems: Computation and control (HSCC 2004), Lecture
Notes in Computer Science 2993 (pp. 280-295).
New York: Springer-Verlag
. http://www.ee.unimelb.edu.au/staff/davoren/00-hscc2004-temporal-logics-general-flow-systems.pdf.
14.
Di Maio, M. C.
&
Zanardo, A.
(1998).
A Gabbay-rule free axiomatization of TW Validity
. Journal of Philosophical Logic, 27,
435-487
.
15.
Gagné, J.-R.
, &
Plaice, J.
(1996).
A non-standard temporal deductive database system
. Journal of Symbolic Computation22,
649-664
.
16.
Henle, M.
(1997). Modern geometries: The analytic approach.
Upper Saddle River, NJ: Prentice Hall
.
17.
Hopkins, G. M.
(1918). Poems of Gerard Manley Hopkins now first
published (
R. Bridges
, Ed.).
London: H. Milford
.
18.
Jeans, J.
(1961). The growth of physical science (2nd
ed.).
New York: Macmillan
.
19.
Johnson, D. M.
(2002). Commentary on Menger’s work on dimension
theory.In
B. Schweizer
,
A. Sklar
,
K. Sigmund
,
P. Gruber
,
E. Hlawka
,
L. Reich
, &
L. Schmetterer
(Eds.), Karl Menger. Selecta mathematica (Vol. 1, pp.
23-32).
Vienna: Springer-Verlag
.
20.
Kahneman, D.
, &
Tversky, A.
(1979).
Prospect theory: An analysis of decision under risk
. Econometrica, 47,
263-292
.
21.
Kennison, J. K.
(2002).
The cyclic spectrum of a Boolean flow
. Theory and Applications of Categories, 10,
392-409
.
22.
Khrennikov, A.
(1997). Non-Archimedean analysis: Quantum paradoxes,
dynamical systems and biological models (Mathematics and its Applications,
vol. 427).
Dordrecht: Kluwer Academic
.
23.
Khrennikov, A.
(2000).
Classical and quantum dynamics on p-adic trees of ideas
. BioSystems, 56,
95-120
.
24.
Khrennikov, A.
(2002). Classical and quantum mental models and
Freud’s theory of unconscious mind.
Växjö, Sweden:
Växjö University Press
.
25.
Krantz, D. H.
,
Luce, R. D.
,
Suppes, P.
, &
Tversky, A.
(1971). Foundations of measurement: Vol. 1. Additive and
polynomial representations.
New York/San Diego, CA: Academic Press
.
26.
Krantz, D. H.
,
Luce, R. D.
,
Suppes, P.
, &
Tversky, A.
(1989). Foundations of measurement: Vol. 2. Geometrical,
threshold and probabilistic representations.
New York/San Diego, CA: Academic Press
.
27.
Krantz, D. H.
,
Luce, R. D.
,
Suppes, P.
, &
Tversky, A.
(1990). Foundations of measurement: Vol. 3. Representation,
axiomatization, and invariance.
New York: San Diego, CA: Academic Press
.
28.
Lewin, K.
(1936). Principles of topological psychology (F. Heider
& G.M. Heider, Trans.).
New York: McGraw-Hill
.
29.
Lewin, K. (1981). Die zeitliche Geneseordnung. In
Kurt-Lewin-Werkausgabe (A. Métraux, Ed.; Vol. 1, pp. 213-232).
Bern: Hans Huber. (Original work published 1923.)
30.
Lewin, K. (1983). Der Begriff der Genese. In C.-F. Graumann (Ed.),
Kurt-Lewin-Werkausgabe (Vol. 2, pp 47-318). Bern: Hans Huber. (Original
work published 1922.)
31.
Malotki, E.
(1983). Hopi time: A linguistic analysis of the temporal
concepts in the Hopi language.
Amsterdam: Mouton
.
32.
Medvedev, F.A. (1993). N.N. Lúzhin o vremenyi nearchimedovskii
[N.N. Luzin on non-Archimedean time] Istoriko-Matematicheskie
Issledovaniya, 34,103-128.
33.
Minkowski, H. (1984). Raum und Zeit. Vortrag, gehalten auf der 80.
Naturforscher-Versammlung zu Köln am 21. September 1908... Mit dem Bildnis
Hermann Minkoskis sowie einem Vorwort von A. Gutzmer. In Gauβsche
Flächentheorie, Riemannsche Räume und Minkowski-Welt (J.
Böhm & H. Reichardt, Eds.). Teubner-Archiv zur Mathematik, Bd. 1.
Leipzig: Teubner-Verlag. (Original work published 1909.)
34.
Munn, N. D.
(1992).
The cultural anthropology of time: A critical essay
. Annual Review of Anthropology, 21,
93-123
. http://links.jstor.org/sici?sici=0084-6570(1992)2:21%3C93:TCAOTA%3E2.0.CO;2-M.
35.
Nelson, E.
(1977).
Internal set theory: A new approach to nonstandard analysis
. Bulletin of the American Mathematical Society, 83,
1165-1198
.
36.
NSF (National Science Foundation)
(2004). Interdisciplinary Grants in the Mathematical Sciences
(IGMS)Program solicitation, NSF 04-518. http://www.nsf.gov/pubs/2004/nsf04518/nsf04518.htm.
37.
Peierls, R.
(1979). Surprises in theoretical physics
(Princeton Series in Physics).
Princeton, NJ: Princeton University Press
.
38.
Poddiakov, A. N.
, &
Valsiner, J.
(2005). Intransitivity cycles and their transformations: How
dynamically adapting systems function? Manuscript submitted for publication.
39.
Robinson, A.
(1966). Non-standard analysis.
Amsterdam: North-Holland
.
40.
Rudolph, L.
(2006).
Spaces of ambivalence: Qualitative mathematics in the modeling of complex
fluid phenomena
. Estudios de Psicología, 27,
67-83
.
41.
Rust, H.
(2003). A non-standard approach to operational semantics
for timed systems. In
E. Börger
,
A. Gargantini
, &
E. Riccobene
(Eds.), Abstract state machines 2003 (Lecture Notes in Computer
Science, Vol. 2589, pp. 423-424).
New York: Springer
.
42.
Rust, H.
(2005). Operational semantics for timed systems: A
non-standard approach to uniform modeling of timed and hybrid systems (Lecture
Notes in Computer Science, Vol. 3456).
New York: Springer
.
43.
Schilder, F. (1997). Temporal relations in English and German narrative
discourse. Unpublished Ph.D. dissertation, University of Edinburgh. http://www.era.lib.ed.ac.uk/bitstream/1842/514/3/Schilder_pdf.
44.
Uexküll, J. von.
(1926). Theoretical biology (D.L. MacKinnon, Trans.).
New York: Harcourt, Brace & Company
. (Original work published 1920.)
45.
Vickers, A.
(1990).
Balinese texts and historiography
. History and Theory, 29,
158-178
. http://links.jstor.org/sici?sici=0018-2656(199005)29:2%3C158:BTAH%3E2.0.CO;2-F.
46.
Whitrow, G. J.
(1961). The natural philosophy of time (1st
ed.).
New York: Clarendon Press
.
47.
Whorf, B. L.
(1956). Language, thought, and reality: Selected writings of
Benjamin Lee Whorf.
Cambridge, MA: MIT Press
.
48.
Wiener, N.
(1948). Cybernetics.
New York/Paris: Technology Press
.
49.
Winfree, A. T.
(1987a). When time breaks down: The three-dimensional
dynamics of electrochemical waves and cardiac arrhythmias.
Princeton, NJ: Princeton University Press
.
50.
Winfree, A. T.
(1987b). The timing of biological clocks
(Scientific American Library series, no. 19).
New York: Freeman
.
51.
Winfree, A. T.
(2001). The geometry of biological time.
New York: Springer-Verlag
.
52.
Yamada, Y.
, &
Kato, Y.
(2006).
Images of circular time and spiral repetition: The Generative Life Cycle
Model and cyclical concepts of time
. Culture & Psychology, 12,
143-160
.
53.
Zeeman, E. C.
(1962). The topology of the brain and visual
perception. In
M. K. Fort
(Ed.), Topology of 3-manifolds and related topics (pp.
240-256).
Englewood Cliffs, NJ: Prentice Hall
.
