Erkenntnis

, Volume 79, Issue 4, pp 943–968 | Cite as

A Representational Approach to Reduction in Dynamical Systems

Original Article

Abstract

According to the received view, reduction is a deductive relation between two formal theories. In this paper, I develop an alternative approach, according to which reduction is a representational relation between models, rather than a deductive relation between theories; more specifically, I maintain that this representational relation is the one of emulation. To support this thesis, I focus attention on mathematical dynamical systems and I argue that, as far as these systems are concerned, the emulation relation is sufficient for reduction. I then extend this representational model-based view of reduction to the case of empirically interpreted dynamical systems, as well as to a treatment of partial, approximate, and asymptotic reduction.

References

  1. Adams, E. W. (1955). Axiomatic foundations of rigid body mechanics. Ph.D. Dissertation. Dept. of Philosophy, Stanford University.Google Scholar
  2. Adams, E. W. (1959). The foundations of rigid body mechanics and the derivation of its laws from those of particle mechanics. In L. Henkin, P. Suppes, & A. Tarski (Eds.), The axiomatic method (pp. 250–265). Amsterdam: North-Holland.CrossRefGoogle Scholar
  3. Arnold, V. I. (1977). Ordinary differential equations. Cambridge, MA: The MIT Press.Google Scholar
  4. Balzer, W., Pearce, D. A., & Schmidt, H.-J. (Eds.). (1984). Reduction in science. Dordrecht: D. Reidel.Google Scholar
  5. Balzer, W., Moulines, C. U., & Sneed, J. D. (1987). An architectonic for science: The structuralist program. Dordrecht: D. Reidel.CrossRefGoogle Scholar
  6. Beckermann, A. (1992). Supervenience, emergence and reduction. In A. Beckermann, T. Toffoli, & J. Kim (Eds.), Emergence or reduction? Essays on the prospects of nonreductive physicalism (pp. 94–118). Berlin: Walter de Gruyter.Google Scholar
  7. Bickle, J. (1998). Psychoneural reduction: The new wave. Cambridge, MA: The MIT Press.Google Scholar
  8. Bickle, J. (2003). Philosophy and neuroscience: A ruthlessly reductive account. Dordrecht: Kluwer.CrossRefGoogle Scholar
  9. Bourbaki, N. (1968). Theory of sets. Vol. I of the elements of mathematics series. Reading, MA: Addison-Wesley.Google Scholar
  10. Churchland, P. M. (1979). Scientific realism and the plasticity of mind. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  11. Churchland, P. M. (1985). Reduction, qualia, and the direct introspection of brain states. Journal of Philosophy, 82(1), 8–28.CrossRefGoogle Scholar
  12. Day, M. A. (1985). Adams on theoretical reduction. Erkenntnis, 23(2), 161–184.CrossRefGoogle Scholar
  13. Dizadji-Bahmani, F., Frigg, R., & Hartmann, S. (2010). Who’s afraid of Nagelian reduction? Erkenntnis, 73, 393–412.CrossRefGoogle Scholar
  14. Endicott, R. P. (1998). Collapse of the new wave. The Journal of Philosophy, 95(2), 53–72.CrossRefGoogle Scholar
  15. Endicott, R. P. (2001). Post-structuralist angst-critical notice: John Bickle, psychoneural reduction: The new wave. Philosophy of Science, 68(3), 377–393.CrossRefGoogle Scholar
  16. Fodor, J. A. (1974). Special sciences (Or: the disunity of science as a working hypothesis). Synthese, 28(2), 97–115.CrossRefGoogle Scholar
  17. Giunti, M. (1995) Dynamical Models of cognition. In R. F. Port, & T. van Gelder (Eds.), Mind as motion: Explorations in the dynamics of cognition (pp. 549–571). Cambridge, MA: The MIT Press.Google Scholar
  18. Giunti, M. (1997). Computation, dynamics, and cognition. New York: Oxford University Press.Google Scholar
  19. Giunti, M. (2010). Reduction in dynamical systems. In M. D'Agostino, G. Giorello, F. Laudisa, T. Pievani, & C. Sinigaglia (Eds.), SILFS New essays in logic and philosophy of science (pp. 677–694). London: College Publications.Google Scholar
  20. Giunti, M., & Mazzola, C. (2012). Dynamical systems on monoids: Toward a general theory of deterministic systems and motion. In G. Minati, M. Abram, & E. Pessa (Eds.), Methods, models, simulations and approaches towards a general theory of change (pp. 173–185). Singapore: World Scientific.CrossRefGoogle Scholar
  21. Hooker, C. A. (1979). Critical notice: R. M. Yoshida’s reduction in the physical sciences. Dialogue, 18, 81–99.CrossRefGoogle Scholar
  22. Hooker, C. A. (1981). Towards a general theory of reduction. Dialogue, 20, 38–59, 201–236, 496–529.Google Scholar
  23. Hooker, C. A. (2004). Asymptotics, reduction and emergence. British Journal for the Philosophy of Science, 55, 435–479.CrossRefGoogle Scholar
  24. Hooker, C. A. (2005). Reduction as cognitive strategy. In B. L. Keeley (Ed.), Paul Churchland (pp. 154–174). New York: Cambridge University Press.CrossRefGoogle Scholar
  25. Kim, J. (1998). Mind in a physical world. Cambridge, MA: The MIT Press.Google Scholar
  26. Kulenovic, M. R. S., & Merino, O. (2002). Discrete dynamical systems and difference equations with Mathematica. Boca Raton: Chapman & Hall/CRC.CrossRefGoogle Scholar
  27. Marras, A. (2002). Kim on reduction. Erkenntnis, 57, 231–257.CrossRefGoogle Scholar
  28. Martelli, M. (1999). Introduction to discrete dynamical systems and chaos. New York: Wiley.CrossRefGoogle Scholar
  29. Mayr, D. (1976). Investigations of the concept of reduction, I. Erkenntnis, 10, 275–294.CrossRefGoogle Scholar
  30. Mayr, D. (1981). Investigations of the concept of reduction II. Erkenntnis, 16, 109–129.Google Scholar
  31. Nagel, E. (1961). The structure of science. New York: Harcourt, Brace & World.Google Scholar
  32. Pearce, D. (1982). Logical properties of the structuralist concept of reduction. Erkenntnis, 18(3), 307–333.CrossRefGoogle Scholar
  33. Sandefur, J. T. (1990). Discrete dynamical systems: Theory and applications. New York: Oxford University Press.Google Scholar
  34. Schaffner, K. F. (1967). Approaches to reduction. Philosophy of Science, 34(2), 137–147.CrossRefGoogle Scholar
  35. Schaffner, K. F. (1969). The Watson–Crick model and reductionism. The British Journal for the Philosophy of Science, 20(4), 325–348.CrossRefGoogle Scholar
  36. Schaffner, K. F. (1976). Reductionism in biology: Prospects and problems. In R. S. Cohen, et al. (Eds.), PSA 1974 (pp. 613–632). Dordrecht: D. Reidel.Google Scholar
  37. Schaffner, K. F. (1977). Reduction, reductionism, values, and progress in the biomedical sciences. In R. G. Colodny (Ed.), Logic, laws, and life (pp. 143–171). Pittsburgh: University of Pittsburgh Press.Google Scholar
  38. Schaffner, K. F. (1993). Discovery and explanation in biology and medicine. Chicago: Chicago University Press.Google Scholar
  39. Schaffner, K. F. (2006). Reduction: The Cheshire cat problem and a return to roots. Synthese, 151, 377–402.CrossRefGoogle Scholar
  40. Smith, A. R., III. (1971). Simple computation-universal cellular spaces. Journal of the Association for Computing Machinery, 18(3), 339–353.CrossRefGoogle Scholar
  41. Sneed, J. D. (1971). The logical structure of mathematical physics. Dordrecht: D. Reidel.CrossRefGoogle Scholar
  42. Stegmüller, W. (1976). The structure and dynamics of theories. New York: Springer.CrossRefGoogle Scholar
  43. Suppes, P. (1957). Introduction to logic. New York: D. Van Nostrand Company.Google Scholar
  44. Szlenk, W. (1984). An introduction to the theory of smooth dynamical systems. Chichister: Wiley.Google Scholar
  45. Turing, A. M. (1950). Computing machinery and intelligence. Mind, 59, 433–460.CrossRefGoogle Scholar
  46. van Fraassen, B. (1980). The scientific image. Oxford: Clarendon Press.CrossRefGoogle Scholar
  47. Wolfram, S. (1983a). Statistical mechanics of cellular automata. Reviews of Modern Physics, 55(3), 601–644.CrossRefGoogle Scholar
  48. Wolfram, S. (1983b). Cellular automata. Los Alamos Science, 9, 2–21.Google Scholar
  49. Wolfram, S. (1984a). Computer software in science and mathematics. Scientific American, 56, 188–203.CrossRefGoogle Scholar
  50. Wolfram, S. (1984b). Universality and complexity in cellular automata. In D. Farmer, T. Toffoli, & S. Wolfram (Eds.), Cellular automata (pp. 1–35). Amsterdam: North Holland.Google Scholar
  51. Wolfram, S. (2002). A new kind of science. Champaign, IL: Wolfram Media Inc.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Applied Logic, Language, Philosophy and History of Science (ALOPHIS)University of CagliariCagliariItaly

Personalised recommendations