Singularity (system theory)

The term singularity for an explanation of unstable systems was first, and in a most general meaning used in 1873 by James Clerk Maxwell. Maxwell does not differentiate between dynamical systems and social systems. Therefore, a singularity refers to a context in which a small change can cause a large effect. The existence of singularities is primarily an argument against determinism and absolute causality for Maxwell. Indeed, following the same initial conditions will always achieve the same results, but such a statement is of little value in a world in which the same initial conditions are never repeated.[1]

Characteristics

In summary, singularities are determined by the following characteristics which can vary in strength:

  1. Instability: Singularities are related to effect in which small causes produce great effects.
  2. System relatedness: Singularities represent a peculiarity based on a system and affect its identity.
  3. Uniqueness: Singularities do not stand out from quantitative singularity, but rather by qualitative uniqueness.
  4. Irreversibility: The caused changes of systems are largely irreversible.
  5. Subjectivity: The awareness is dependent on the human perception and experience.
  6. Randomness: Singularities are often considered as random because generally either the causes or their effects are not well known.
  7. Complexity: Their occurrence is often connected to the complexity of the system and its environment.
  8. Interaction: Singularities often arise when unexpected interactions occur between two systems.[2]

Singularities in dynamical systems

A further development of Maxwell's thoughts in relation to dynamic systems was carried out first by the French mathematician Henri Poincaré. Poincaré distinguished four different simple singularities (points singuliers) of differential equations. These are the node (les noeuds), the saddle (les cols), the focus (les foyers) and the center (les centers).[3] In recent times, the chaos theory found special attention. However, deterministic chaos is just a special case of a singularity, in which a small cause produces a large observable effect due to a nonlinear dynamic behavior. In contrast the singularities raised by Maxwell, such as a loose rock at a singular point on a slope, show a linear dynamic behavior as it was demonstrated by Poincaré. Singularities are the common staple of the chaos theory, catastrophe theory and bifurcation theory.[4]

Singularities in social systems

In social systems, a deterministic chaos is unlikely, because the elements of the system are some individuals that engage with awareness, will and foresight purposefully into the dynamic behavior of the system.[5] However, this does not exclude that approaches deterministic chaos in social systems are available. Rather, there is also an increase in the social development of nonlinear dynamics and instabilities .[6] Chaos in the colloquial sense of complete disorder or confusion, however, is to be found. It is often the basis for singularities, where cause-and-effect relationships are not clear. There are already numerous examples of singularities in social systems with Maxwell and Poincaré. Maxwell states that a word can start a war and all the great discoveries of man based on singular states. Poincaré gives the example of a roofer who drops a brick and randomly kills a passing man.

Singularities in natural history

The development of systems provides the science currently so before that by a singular Big Bang uniformly dispersed plasma spread after the creation of the universe in space, which is cooled with increasing expansion, so that formed atoms and finally for very small (singular) fluctuations in the uniform density inhomogeneities created self-reinforcing. They subsequently led to the formation of galaxies, stars and other systems in the universe, from which humans emerged at the end. Even if the singularity of the Big Bang can be avoided in the mathematical models, singularities remain an essential element of history. The evolutionary history shows that not only successful mutations can be perceived as positive singularities, but the humanization and the human becoming, the singular most important event in the evolution and represents a jump from the continuum of past evolutionary development of the planet Earth.[7][8] Recently, Ward and Kirschvink show that the history of life has been more influenced by disasters than by continuous evolution.[9] Disasters are here first destructive singularities that create space for new developments in the sense of innovations as productive singularities.[10]

Singularities and complexity

Closely related is the notion of singularity with the concept of complexity. J.C. Maxwell has already pointed out that a system has all the more singular points, the more complex it is. Complexity is also the basis of perceived chaos and singularities. Suppose a seemingly insignificant event that produces a great effect, even in a simple context, how difficult would it be to detect the reason in a complex situation with tremendously many elements and relationships. Complexity that is kind of a breeding ground for singularities, shows the downfall of ancient cultures. Causes such as intruders, internal conflicts or natural disasters are not sufficient alone to justify the destruction of a culture. Rather requirement is an increasing complexity and associated declining marginal returns.[11] The financial crisis of 2007-2008 shows how difficult decisions about a complex environment. Thus, the complexity of financial systems and financial products is a major challenge of the financial markets and institutions to look at.[12] One solution is to reduce complexity and increase the potential for adaptation and robustness. In a complex world with increasing singularities, it is therefore necessary to abandon optimization potential to gain adaptability to external shocks and disasters.[13]

Scientific literature

  • Maxwell, J.C.: “Does the Progress of Physical Science tend to give any Advantage to the Opinion of Necessity (or Determinism) over that of the Contingency of Events and the Freedom of the Will?”; in: L. Champbell/ W. Garnett: “The Life of James Clerk Maxwell”, London 1882, S. 440ff
  • Holzkämpfer, H.: Management von Singularitäten und Chaos, Wiesbaden 1996, ISBN 978-3-8244-0296-0
  • J.-H. Scharf (Hrsg.): Singularitäten, Nova Acta Leopoldina, Abhandlungen der Deutschen Akademie der Naturforscher Leopoldina, Vorträge anläßlich der Jahresversammlung vom 30. März bis 2. April 1985 zu Halle (Saale), Leipzig 1989

References

  1. ↑ Maxwell, J.C.: “Does the Progress of Physical Science tend to give any Advantage to the Opinion of Necessity (or Determinism) over that of the Contingency of Events and the Freedom of the Will?” in: L. Champbell/ W. Garnett: “The Life of James Clerk Maxwell”. London 1882, S. 440ff.
  2. ↑ Holzkämpfer, H.: Management von Singularitäten und Chaos, Wiesbaden 1996, S. 91
  3. ↑ Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle, in: Journal de mathématiques, 1881. S. 375-422.
  4. ↑ Tu, P.N.V.: Dynamical Systems: An Introduction with Applications in Economics and Biology, 2. überarb. und erw. Aufl., Berlin, Heidelberg u.a. 1994, S.195ff.
  5. ↑ Weizsäcker, C. C. von: Ordnung und Chaos in der Wirtschaft, in: W. Gerock/ H. Haken u.a. (Hrsg.): Ordnung und Chaos in der unbelebten und belebten Natur. Stuttgart 1989. S. 46.
  6. ↑ Bühl, W.L.: Sozialer Wandel im Ungleichgewicht: Zyklen, Fluktuationen, Katastrophen, Stuttgart 1990, S. 207.
  7. ↑ Hagemann, R.: Mutationen als produktive Singularitäten, in: J.-H. Scharf (Hrsg.): Singularitäten, Nova Acta Leopoldina, Abhandlungen der Deutschen Akademie der Naturforscher Leopoldina, Vorträge anläßlich der Jahresversammlung vom 30. März bis 2. April 1985 zu Halle (Saale), Leipzig 1989, S. 155-169.
  8. ↑ Vogel, C.: Die Hominisation, ein singulärer Sprung aus dem Kontinuum der Evolution?, in: J.-H. Scharf (Hrsg.): Singularitäten, Nova Acta Leopoldina, Abhandlungen der Deutschen Akademie der Naturforscher Leopoldina, Vorträge anläßlich der Jahresversammlung vom 30. März bis 2. April 1985 zu Halle (Saale). Leipzig 1989, S. 141–154.
  9. ↑ Ward, P., Kirschvink, J .: A New History of Life, Munich 2016, S.30.
  10. ↑ Holzkämpfer, H .: Management of Singularities and Chaos, Wiesbaden 1996, pp. 133ff and 139ff.
  11. ↑ Tainter, J.A.: The Collapse of Complex Societies, Cambridge, New York u.a. 1988, S. 42ff.
  12. ↑ Landau, J.-P.: Complexity and the financial crisis, Introductory remarks at the Conference on The Macroeconomy and Financial Systems in Normal Times and in Times of Stress, jointly organized by Banque de france and the Bundesbank, 8. June, 2009.
  13. ↑ Conrad, M.: Adaptability: The Significance of Variability from Molecule to Ecosystem, New York, London 1983.
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