Foundations of Physics

, Volume 41, Issue 6, pp 1065–1135 | Cite as

Less is Different: Emergence and Reduction Reconciled

Article

Abstract

This is a companion to another paper. Together they rebut two widespread philosophical doctrines about emergence. The first, and main, doctrine is that emergence is incompatible with reduction. The second is that emergence is supervenience; or more exactly, supervenience without reduction.

In the other paper, I develop these rebuttals in general terms, emphasising the second rebuttal. Here I discuss the situation in physics, emphasising the first rebuttal. I focus on limiting relations between theories and illustrate my claims with four examples, each of them a model or a framework for modelling, from well-established mathematics or physics.

I take emergence as behaviour that is novel and robust relative to some comparison class. I take reduction as, essentially, deduction. The main idea of my first rebuttal will be to perform the deduction after taking a limit of some parameter. Thus my first main claim will be that in my four examples (and many others), we can deduce a novel and robust behaviour, by taking the limit N→∞ of a parameter N.

But on the other hand, this does not show that the N=∞ limit is “physically real”, as some authors have alleged. For my second main claim is that in these same examples, there is a weaker, yet still vivid, novel and robust behaviour that occurs before we get to the limit, i.e. for finite N. And it is this weaker behaviour which is physically real.

My examples are: the method of arbitrary functions (in probability theory); fractals (in geometry); superselection for infinite systems (in quantum theory); and phase transitions for infinite systems (in statistical mechanics).

Keywords

Emergence Reduction Asymptotics Method of arbitrary functions Fractals Superselection Phase transitions Thermodynamic limit 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avnir, D., Biham, O., et al.: Is the geometry of nature fractal? Science 279, 39–40 (1998) ADSCrossRefMATHGoogle Scholar
  2. 2.
    Anderson, P.: More is different. Science 177, 393–396 (1972); reprinted in Bedau and Humphreys (eds.): Emergence: Contemporary Readings in Philosophy and Science. MIT Press/Bradford Books, Cambridge (2008) ADSCrossRefGoogle Scholar
  3. 3.
    Bangu, S.: Understanding thermodynamic singularities: phase transitions, data and phenomena. Philos. Sci. 76, 488–505 (2009) CrossRefGoogle Scholar
  4. 4.
    Barnsley, M.: Fractals Everywhere. Academic Press, Boston (1988) MATHGoogle Scholar
  5. 5.
    Barrenblatt, G.: Scaling, Self-similarity and Intermediate Asymptotics. Cambridge University Press, Cambridge (1996) Google Scholar
  6. 6.
    Batterman, R.: The Devil in the Details. Oxford University Press, London (2002) MATHGoogle Scholar
  7. 7.
    Batterman, R.: Critical phenomena and breaking drops: infinite idealizations in physics. Stud. Hist. Philos. Mod. Phys. 36B, 225–244 (2005) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Batterman, R.: Hydrodynamic vs. molecular dynamics: intertheory relations in condensed matters physics. Philos. Sci. 73, 888–904 (2006) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Batterman, R.: Emergence, singularities, and symmetry breaking. Pittsburgh arXive (2009) Google Scholar
  10. 10.
    Batterman, R.: On the explanatory role of mathematics in empirical science. Br. J. Philos. Sci. 61, 1–25 (2010) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Belot, G.: Whose Devil? Which Details? Philos. Sci. 72, 128–153 (2005); a fuller version is available at: http://philsci-archive.pitt.edu/archive/00001515/ MathSciNetCrossRefGoogle Scholar
  12. 12.
    Berry, M.: Asymptotics, singularities and the reduction of theories. In: Prawitz, D., Skyrms, B., Westerdahl, D. (eds.) Logic, Methodology and Philosophy of Science IX: Proceedings of the Ninth International Congress of Logic, Methodology and Philosophy of Science, Uppsala, Sweden 1991, pp. 597–607. Elsevier, Amsterdam (1994) Google Scholar
  13. 13.
    Bogen, J., Woodward, J.: Saving the phenomena. Philos. Rev. 97, 303–352 (1988) CrossRefGoogle Scholar
  14. 14.
    Bouatta, N., Butterfield, J.: Emergence and reduction combined in phase transitions. In: Kouneiher, J. (ed.) Frontiers of Fundamental Physics, vol. 11, Proceedings of the Conference. American Institute of Physics (2011, forthcoming) Google Scholar
  15. 15.
    Brady, R., Ball, R.: Fractal growth of copper electrodeposits. Nature 309(5965), 225–229 (1984) ADSCrossRefGoogle Scholar
  16. 16.
    Butterfield, J.: Emergence, reduction and supervenience: a varied landscape. Found. Phys. (2010, this issue) Google Scholar
  17. 17.
    Callender, C.: Taking thermodynamics too seriously. Stud. Hist. Philos. Mod. Phys. 32B, 539–554 (2001) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cardy, J.: Scaling and Renormalization in Statistical Physics. Cambridge Lecture Notes in Physics, vol. 5. Cambridge University Press, Cambridge (1997) MATHGoogle Scholar
  19. 19.
    Castaing, B., Gunaratne, G., et al.: Scaling of hard thermal turbulence in Rayleigh-Benard convection. J. Fluid Mech. 204, 1–30 (1989) ADSCrossRefGoogle Scholar
  20. 20.
    Cat, J.: The physicists’ debates on unification in physics at the end of the twentieth century. Hist. Stud. Phys. Biol. Sci. 28, 253–300 (1998) Google Scholar
  21. 21.
    Chaikin, P., Lubensky, T.: Principles of Condensed Matter Physics. Cambridge University Press, Cambridge (2000) Google Scholar
  22. 22.
    Colyvan, M.: Probability and ecological complexity’: a review of Strevens (2003). Biol. Philos. 20, 869–879 (2005) CrossRefGoogle Scholar
  23. 23.
    Diaconis, P.: Finite forms of de Finetti’s theorem on exchangeability. Synthese 36, 271–281 (1977) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Diaconis, P., Freedman, D.: De Finetti’s generalizations of exchangeability. In: Jeffrey, R. (ed.) Studies in Inductive Logic and Probability, vol. 2, pp. 233–249. University of California Press, Berkeley (1980) Google Scholar
  25. 25.
    Diaconis, P., Holmes, S., Montgomery, R.: Dynamical bias in the coin toss. SIAM Rev. 49, 211–235 (2007) MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Dixon, P., Wu, L., et al.: Scaling in the relaxation of supercooled liquids. Phys. Rev. Lett. 65, 1108–1111 (1990) ADSCrossRefGoogle Scholar
  27. 27.
    Emch, G.: Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley, New York (1972) MATHGoogle Scholar
  28. 28.
    Emch, G.: Quantum statistical physics. In: Butterfield, J., Earman, J. (eds.) Philosophy of Physics, Part B. The Handbook of the Philosophy of Science, pp. 1075–1182. North Holland, Amsterdam (2006) Google Scholar
  29. 29.
    Emch, G., Liu, C.: The Logic of Thermo-statistical Physics. Springer, Berlin (2002) Google Scholar
  30. 30.
    Engel, E.: A Road to Randomness in Physical Systems. Springer, Berlin (1992) MATHCrossRefGoogle Scholar
  31. 31.
    Falconer, K.: Fractal Geometry. Wiley, New York (2003) MATHCrossRefGoogle Scholar
  32. 32.
    Frigg, R., Hoefer, C.: Determinism and chance from a Humean perspective. In: Dieks, D., Gonzalez, W., Hartmann, S., Stadler, F., Uebel, T., Weber, M. (eds.) The Present Situation in the Philosophy of Science. Springer, Berlin (2010, forthcoming) Google Scholar
  33. 33.
    Goldenfeld, N., Martin, O., Oono, Y.: Intermediate asymptotics and renormalization group theory. J. Sci. Comput. 4, 355–372 (1989) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Gross, D.: Microcanonical Thermodynamics; Phase Transitions in Small Systems. World Scientific, Singapore (2001) CrossRefGoogle Scholar
  35. 35.
    Hadzibabic, Z., et al.: Berezinskii-Kosterlitz-Thouless crossover in a trapped atomic gas. Nature 441, 1118–1121 (2006) ADSCrossRefGoogle Scholar
  36. 36.
    Hastings, H., Sugihara, G.: Fractals: A User’s Guide for the Natural Sciences. Oxford University Press, London (1993) Google Scholar
  37. 37.
    Hooker, C.: Asymptotics, reduction and emergence. Br. J. Philos. Sci. 55, 435–479 (2004) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Jeffrey, R.: Conditioning, kinematics and exchangeability. In: Skyrms, B., Harper, W. (eds.) Causation, Chance and Credence, vol. 1, pp. 221–255. Kluwer, Dordrecht (1988) Google Scholar
  39. 39.
    Kadanoff, L.: More is the same: phase transitions and mean field theories. J. Stat. Phys. 137, 777–797 (2009); available at http://arxiv.org/abs/0906.0653 MathSciNetADSMATHCrossRefGoogle Scholar
  40. 40.
    Kadanoff, L.: Theories of matter: infinities and renormalization. In: Batterman, R. (ed.) The Oxford Handbook of the Philosophy of Physics. Oxford University Press (2010, forthcoming); available at http://arxiv.org/abs/1002.2985; and at http://jfi.uchicago.edu/~leop/AboutPapers/Trans2.pdf
  41. 41.
    Kadanoff, L.: Relating theories via renormalization (2010a); available at http://jfi.uchicago.edu/~leop/AboutPapers/RenormalizationV4.0.pdf
  42. 42.
    Keller, J.: The probability of heads. Am. Math. Mon. 93, 191–197 (1986) MATHCrossRefGoogle Scholar
  43. 43.
    Koenig, R., Renner, R.: A de Finetti representation for finite symmetric quantum states. J. Math. Phys. 46, 012105 (2005); available at arXiv:quant-ph/0410229v1 MathSciNetCrossRefGoogle Scholar
  44. 44.
    Koenig, R., Mitchison, G.: A most compendious and facile quantum de Finetti theorem. J. Math. Phys. (2007) 50, 012105; available at arXiv:quant-ph/0703210 ADSCrossRefGoogle Scholar
  45. 45.
    Kritzer, P.: Sensitivity and Randomness: The Development of the Theory of Arbitrary Functions. Diplom thesis, University of Salzburg (2003) Google Scholar
  46. 46.
    Landsman, N.: Between classical and quantum. In: Butterfield, J., Earman, J. (eds.) Philosophy of Physics, Part A. The Handbook of the Philosophy of Science, pp. 417–554. North-Holland, Amsterdam (2006); available at arxiv:quant-ph/0506082 and at http://philsci-archive.pitt.edu/archive/00002328 Google Scholar
  47. 47.
    Lavis, D., Bell, G.: Statistical Mechanics of Lattice Systems 1; Closed Forms and Exact Solutions, 2nd enlarged edn. Springer, Berlin (1999) Google Scholar
  48. 48.
    Lavis, D., Bell, G.: Statistical Mechanics of Lattice Systems. 2. Exact, Series and Renormalization Group Methods. Springer, Berlin (1999a) Google Scholar
  49. 49.
    Le Bellac, M.: Quantum and Statistical Field Theory. Oxford University Press, London (1991) (translated by G. Barton) Google Scholar
  50. 50.
    Liu, C.: Infinite systems in SM explanation: thermodynamic limit, renormalization (semi)-groups and irreversibility. Proc. Philos. Sci. 68, S325–S344 (2001) Google Scholar
  51. 51.
    Liu, C., Emch, G.: Explaining quantum spontaneous symmetry breaking. Stud. Hist. Philos. Mod. Phys. 36, 137–164 (2005) MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Mainwood, P.: Is More Different? Emergent Properties in Physics. D. Phil. dissertation, Oxford University (2006). At: http://philsci-archive.pitt.edu/8339/
  53. 53.
    Mainwood, P.: Phase transitions in finite systems, unpublished MS (corresponds to Chapter 4 of Mainwood (2006)) (2006a). At: http://philsci-archive.pitt.edu/8340/
  54. 54.
    Mandelbrot, B.: The Fractal Geometry of Nature. Freeman, San Francisco (1982) MATHGoogle Scholar
  55. 55.
    Menon, T., Callender, C.: Going through a phase: philosophical questions raised by phase transitions. In: Batterman, R. (ed.) The Oxford Handbook of Philosophy of Physics. Oxford University Press (2011, forthcoming) Google Scholar
  56. 56.
    Myrvold, W.: Deterministic laws and epistemic chances. Unpublished manuscript (2011) Google Scholar
  57. 57.
    Nagel, E.: The Structure of Science: Problems in the Logic of Scientific Explanation. Harcourt, New York (1961) Google Scholar
  58. 58.
    Nelson, E.: Radically Elementary Probability Theory. Annals of Mathematics Studies, vol. 117. Princeton University Press, Princeton (1987) MATHGoogle Scholar
  59. 59.
    Peitgen, H.-O., Richter, P.: The Beauty of Fractals. Springer, Heidelberg (1986) MATHCrossRefGoogle Scholar
  60. 60.
    Poincaré, H.: Calcul de Probabilities. Gauthier-Villars, Paris (1896); page reference to 1912 edition Google Scholar
  61. 61.
    Richardson, L.: The problem of contiguity: an appendix of statistics of deadly quarrels. Gen. Syst. Yearbook 6, 139–187 (1961) Google Scholar
  62. 62.
    Renner, R.: Symmetry implies independence. Nat. Phys. 3, 645–649 (2007) CrossRefGoogle Scholar
  63. 63.
    Rueger, A.: Physical emergence, diachronic and synchronic. Synthese 124, 297–322 (2000) MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Rueger, A.: Functional reduction and emergence in the physical sciences. Synthese 151, 335–346 (2006) MathSciNetCrossRefGoogle Scholar
  65. 65.
    Ruelle, D.: Statistical Mechanics: Rigorous Results. Benjamin, Elmsford (1969) MATHGoogle Scholar
  66. 66.
    Sewell, G.: Quantum Theory of Collective Phenomena. Oxford University Press, London (1986) Google Scholar
  67. 67.
    Sewell, G.: Quantum Mechanics and its Emergent Microphysics. Princeton University Press, Princeton (2002) Google Scholar
  68. 68.
    Shenker, O.: Fractal geometry is not the geometry of nature. Stud. Hist. Philos. Sci. 25, 967–982 (1994) MathSciNetCrossRefGoogle Scholar
  69. 69.
    Simon, H.: Alternative views of complexity. In: The Sciences of the Artificial, 3rd edn. MIT Press, Cambridge (1996); the Chapter is reprinted in Bedau and Humphreys (2008); page reference to the reprint Google Scholar
  70. 70.
    Smith, P.: Explaining Chaos. Cambridge University Press, Cambridge (1998) MATHCrossRefGoogle Scholar
  71. 71.
    Sober, E.: Evolutionary theory and the reality of macro-probabilities. In: Eells, E., Fetzer, J. (eds.) The Place of Probability in Science. Boston Studies in the Philosophy of Science, vol. 284, pp. 133–161. Springer, Berlin (2010) CrossRefGoogle Scholar
  72. 72.
    Stoppard, T.: Arcadia. Faber and Faber, London (1993) Google Scholar
  73. 73.
    Strevens, M.: Bigger than Chaos: Understanding Complexity through Probability. Harvard University Press, Cambridge (2003) Google Scholar
  74. 74.
    Thompson, C.: Mathematical Statistical Mechanics. Princeton University Press, Princeton (1972) MATHGoogle Scholar
  75. 75.
    von Plato, J.: The method of arbitrary functions. Br. J. Philos. Sci. 34, 37–47 (1983) MATHCrossRefGoogle Scholar
  76. 76.
    von Plato, J.: Creating Modern Probability. Cambridge University Press, Cambridge (1994) CrossRefGoogle Scholar
  77. 77.
    Wayne, A.: Emergence and singular limits. Synthese (2009, forthcoming) available at http://philsci-archive.pitt.edu/archive/00004962/
  78. 78.
    Weinberg, S.: Newtonianism, reductionism and the art of congressional testimony. Nature 330, 433–437 (1987); reprinted in Bedau and Humphreys (2008); page reference to the reprint; also reprinted in Weinberg, S. Facing Up: Science and its Cultural Adversaries, Harvard University Press, pp. 8–25 ADSCrossRefGoogle Scholar
  79. 79.
    Werndl, C.: Review of Strevens (2003). Br. J. Philos. Sci. (2010, forthcoming) Google Scholar
  80. 80.
    Wimsatt, W.: Aggregativity: reductive heuristics for finding emergence. Philos. Sci. 64, S372–S384 (1997); reprinted in Bedau and Humphreys (2008); page reference to the reprint CrossRefGoogle Scholar
  81. 81.
    Yeomans, J.: Statistical Mechanics of Phase Transitions. Oxford University Press, London (1992) Google Scholar
  82. 82.
    Feynman, R.: The Feynman Lectures on Physics, vol. 2. Addison-Wesley, Reading (1964) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Trinity CollegeCambridge UniversityCambridgeUK

Personalised recommendations