European Journal for Philosophy of Science

, Volume 3, Issue 1, pp 33–67 | Cite as

What is a complex system?

  • James LadymanEmail author
  • James Lambert
  • Karoline Wiesner
Original Paper in Philosophy of Science


Complex systems research is becoming ever more important in both the natural and social sciences. It is commonly implied that there is such a thing as a complex system, different examples of which are studied across many disciplines. However, there is no concise definition of a complex system, let alone a definition on which all scientists agree. We review various attempts to characterize a complex system, and consider a core set of features that are widely associated with complex systems in the literature and by those in the field. We argue that some of these features are neither necessary nor sufficient for complexity, and that some of them are too vague or confused to be of any analytical use. In order to bring mathematical rigour to the issue we then review some standard measures of complexity from the scientific literature, and offer a taxonomy for them, before arguing that the one that best captures the qualitative notion of the order produced by complex systems is that of the Statistical Complexity. Finally, we offer our own list of necessary conditions as a characterization of complexity. These conditions are qualitative and may not be jointly sufficient for complexity. We close with some suggestions for future work.


Complexity Statistical complexity Information Complex system 



We are extremely grateful to several anonymous referees for this journal and to the editor for very helpful comments and criticisms, and also to the students of the Bristol Centre for the Complexity Sciences doctoral programme over several years for their comments on our ideas. James Ladyman acknowledges the support of the AHRC Foundations of Structuralism project. Karoline Wiesner acknowledges funding through EPSRC grant EP/E501214/1.


  1. Anderson, P. W. (1972). More is different: Broken symmerty and the nature of the hierarchical structure of science. Science, 177, 393–396.CrossRefGoogle Scholar
  2. Brian Arthur, W. (1999). Complexity and the economy. Science, 284, 107–109.CrossRefGoogle Scholar
  3. Badii, R., & Politi, A. (1999). Complexity: Hierarchical structures and scaling in physics. Cambridge University Press.Google Scholar
  4. Bennett, C. H. (1988). Logical depth and physical complexity. In R. Herken, (Ed.), The universal Turing machine, a half-century survey (pp. 227–257). Oxford: Oxford University Press.Google Scholar
  5. Cooper, J. M. (Ed.) (1997). Phaedrus in Plato complete works. Hackett.Google Scholar
  6. Cover, T. M., & Thomas, J. A. (2006). Elements of information theory (2nd edn.). Wiley-Blackwell, September.Google Scholar
  7. Crutchfield, J. P. (1994). The calculi of emergence: Computation, dynamics and induction. Physica D: Nonlinear Phenomena, 75(1–3), 11–54.CrossRefGoogle Scholar
  8. Crutchfield, J. P., & Shalizi, C. R. (1999). Thermodynamic depth of causal states: Objective complexity via minimal representations. Physical Review E, 59(1), 275–283.CrossRefGoogle Scholar
  9. Crutchfield, J. P., & Young, K. (1989). Inferring statistical complexity. Physical Review Letters, 63, 105.CrossRefGoogle Scholar
  10. Crutchfield, J. P., & Young, K. (1990). Computation at the onset of chaos. Entropy, Complexity and the Physics of Information, SFI Studies in the Sciences of Complexity, VIII, 223–269.Google Scholar
  11. Dennett, D. (1991). Real patterns. The Journal of Philosophy, 88(1), 27–51.CrossRefGoogle Scholar
  12. Editorial. (2009). No man is an island. Nature Physics, 5, 1.Google Scholar
  13. Feynman, R. (2000). Feynman lectures on computation. Westview Press.Google Scholar
  14. Foote, R. (2007). Mathematics and complex systems. Science, 318, 410–412.CrossRefGoogle Scholar
  15. Gell-Mann, M. (1995). What is complexity. Complexity, 1, 1.Google Scholar
  16. Gell-Mann, M., & Lloyd, S. (1996). Information measures, effective complexity, and total information. Complexity, 2(1), 44–52.CrossRefGoogle Scholar
  17. Gell-Mann, M., & Lloyd, S. (2004). Effective complexity. In M. Gell-Mann, & C. Tsallis, (Eds.), Nonextensive entropy – interdisciplinary applications. The Santa Fe Institute, OUP USA.Google Scholar
  18. Goldenfeld, N., & Kadanoff, L. P. (1999). Simple lessons from complexity. Science, 284, 87–89.CrossRefGoogle Scholar
  19. Grassberger, P. (1986). Toward a quantitative theory of self-generated complexity. International Journal of Theoretical Physics, 25(9), 907–938.CrossRefGoogle Scholar
  20. Grassberger, P. (1989). Problems in quantifying self-generated complexity. Helvetica Physica Acta, 62, 489–511.Google Scholar
  21. Holland, J. H. (1992). Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control and artificial intelligence (new edn.). MIT Press, July.Google Scholar
  22. Holland, J. H. (1992). Complex adaptive systems. Daedalus, 121(1), 17–30.Google Scholar
  23. Jaynes, E. T. (1957a). Information theory and statistical mechanics. The Physical Review, 106(4), 620–630.CrossRefGoogle Scholar
  24. Jaynes, E. T. (1957b). Information theory and statistical mechanics, ii. The Physical Review, 108(2), 171–190.CrossRefGoogle Scholar
  25. Kolmogorov, A. N. (1965). Three approaches to the quantitive definition of information. Problems of Information Transmission, 1, 1–17.Google Scholar
  26. Kolmogorov, A. N. (1983). Combinatorial foundations of information theory and the calculus of probabilities. Russian Mathematical Surveys, 38(4), 29–40.CrossRefGoogle Scholar
  27. Ladyman, J., Ross, D., Spurrett, D., & Collier, J. (2007). Everything must go: Metaphysics naturalized. Oxford University Press.Google Scholar
  28. Li, C.-B., Yang, H., & Komatsuzaki, T. (2008). Multiscale complex network of protein conformational fluctuations in single-molecule time series. Proceedings of the National Academy of Sciences, 105(2), 536–541.CrossRefGoogle Scholar
  29. Li, M., & Vitnyi, P. M. B. (2009). An introduction to Kolmogorov complexity and its applications (3rd ed.). Springer, March.Google Scholar
  30. Lloyd, S. (2001). Measures of complexity: A nonexhaustive list. Control Systems Magazine, IEEE, 21, 7–8.CrossRefGoogle Scholar
  31. Lloyd, S., & Pagels, H. (1988). Complexity as thermodynamic depth. Annals of Physics, 188, 186–213.CrossRefGoogle Scholar
  32. MacKay, R. S. (2008). Nonlinearity in complexity science. Nonlinearity, 21, T273.CrossRefGoogle Scholar
  33. Mainzer, K. (1994). Thinking in complexity: The complex dynamics of matter, mind and mankind. Springer.Google Scholar
  34. Merricks, T. (2001). Objects and persons. Oxford University Press.Google Scholar
  35. Mitchell, S. (2009). Unsimple truths: Science, complexity, and policy. University of Chicago Press.Google Scholar
  36. Morin, E., & Belanger, J. L. R. (1992). Method: Towards a study of humankind : The nature of nature: 001. Peter Lang Pub Inc, November.Google Scholar
  37. Paley, W. (2006). Natural theology. Oxford University Press.Google Scholar
  38. Palmer, A. J., Fairall, C. W., & Brewer, W. A. (2000). Complexity in the atmosphere. IEEE Transactions on Geoscience and Remote Sensing, 38(4), 2056–2063.CrossRefGoogle Scholar
  39. Parrish, J. K., & Edelstein-Keshet, L. (1999). Complexity, pattern, and evolutionary trade-offs in animal aggregation. Science, 284, 99–101.CrossRefGoogle Scholar
  40. Rind, D. (1999). Complexity and climate. Science, 284, 105–107.CrossRefGoogle Scholar
  41. Ross, D. (2000). Rainforrest realism: A Dennettian theory of existence. In D. Ross (Ed.), Dennett’s philosophy: A comprehensive assessment (Chapter 8, pp. 147–168). MIT Press.Google Scholar
  42. Shalizi, C. R., & Crutchfield, J. P. (2001). Computational mechanics: Pattern and prediction, structure and simplicity. Journal of Statistical Physics, 104(3), 817–879.CrossRefGoogle Scholar
  43. Shalizi, C. R., & Moore, C. (2003). What is a macrostate? Subjective observations and objective dynamics. cond-mat/0303625.Google Scholar
  44. Shalizi, C. R., Shalizi, K. L., & Haslinger, R. (2004). Quantifying self-organization with optimal predictors. Physical Review Letters, 93(11), 118701.CrossRefGoogle Scholar
  45. Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379–423; 623–656.Google Scholar
  46. Simon, H. A. (1962). The architecture of complexity. Proceedings of the American Philosophical Society, 106(6), 467–482.Google Scholar
  47. Timpson, C. G. (2006). The grammar of teleportation. British Journal of Philosophy of Science, 57(3), 587–621.CrossRefGoogle Scholar
  48. Wackerbauer, R., Witt, A., Atmanspacher, H., Kurths, J., & Scheingraber, H. (1994). A comparative classification of complexity measures. Chaos, Solitons & Fractals, 4(1), 133–173.CrossRefGoogle Scholar
  49. Wallace, D. (2003). Everett and structure. Studies In History and Philosophy of Modern Physics, 34(1), 87–105.CrossRefGoogle Scholar
  50. Weng, G., Bhalla, U. S., & Iyengar, R. (1999). Complexity in biological signaling systems. Science, 284, 92–96.CrossRefGoogle Scholar
  51. Werner, B. T. (1999). Complexity in natural landform patterns. Science, 284, 102–104.CrossRefGoogle Scholar
  52. Whitesides, G. M., & Ismagilov, R. F. (1999). Complexity in chemistry. Science, 284, 89–92.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2012

Authors and Affiliations

  • James Ladyman
    • 1
    Email author
  • James Lambert
    • 1
  • Karoline Wiesner
    • 2
  1. 1.Department of PhilosophyUniversity of BristolBristolUK
  2. 2.Department of Mathematics and Centre for Complexity SciencesUniversity of BristolBristolUK

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