Singularity (systems theory)

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The term singularity was first used in 1873 by James Clerk Maxwell in general to explain unstable systems. Maxwell does not differentiate between dynamic systems and social systems . In general, a singularity describes a context in which a small cause produces a large effect. For Maxwell, the existence of singularities is above all an argument against determinism and an absolute causality . Although the same initial conditions always follow the same events, such a statement is of little value in a world in which the same initial conditions are never repeated.

features

In summary, singularities are determined by the following features, which can be differently pronounced:

  1. Instability : They concern causal relationships in which small causes produce large effects.
  2. System-relatedness: They represent a special feature, each related to a system and influencing its identity.
  3. Uniqueness: You are not so much characterized by quantitative singularity, but rather by qualitative uniqueness.
  4. Irreversibility : The system changes caused are largely irreversible.
  5. Subjectivity : Your perception depends on human ideas and experiences.
  6. Randomness: They are often thought of as random because, in general, either the causes or their effects are not precisely known.
  7. Complexity : Their appearance is often related to the complexity of the respective system and its environment.
  8. Interaction : They often arise when unexpected interactions occur between two systems.

Singularities in dynamic systems

The French mathematician Henri Poincaré first developed Maxwell's ideas in relation to dynamic systems . Poincaré distinguished four different simple singularities ( points singuliers ) of differential equations . These are the nodes ( les noeuds ), the saddles ( les cols ), the foci / whirlpools ( les foyers ) and the centers ( les centers ).

In the further development of the theory of dynamic systems it becomes clear that even simple systems show a very complicated behavior that reflects the complexity of real phenomena. This includes:

  • Instabilities in the form of explosions or natural disasters,
  • Bifurcations (branches), for example in the case of developmental leaps in biological evolution
  • Turbulence with transition to chaos ( chaos theory )

Bifurcation points are of particular importance when the development of a system does not depend on clearly determinable parameters, but rather the smallest random fluctuations ( fluctuations ) at the branch determine which development the system takes. In recent times the chaos theory has received special attention. Ultimately, however, deterministic chaos is only a special case of a singularity, in which a small cause has a large effect from an observable non-linear dynamic behavior. The singularities mentioned by Maxwell, such as B. a loose boulder on a singular point on a slope, show a linear dynamic behavior, as it was investigated by Poincaré.

Singularities form the common bracket over the chaos, catastrophe and bifurcation theory ( fractal geometry ).

Singularities in social systems

In social systems, a deterministic chaos is rather unlikely, since the elements of the system are individual individuals who, with consciousness, will and foresight, intervene purposefully in the dynamic behavior of the system. However, this does not exclude the existence of deterministic chaos in social systems. Rather, there is also an increase in non-linear dynamics and instabilities in social development.

However, chaos in the colloquial sense of complete disorder or confusion is encountered. It is often the basis for singularities in which cause-effect relationships are not clearly recognizable. Numerous examples of singularities in social systems can already be found in Maxwell and Poincaré. Maxwell argues that a word can start a war and that all great human discoveries are based on singular states. As an example, Poincaré cites a roofer who drops a tile and kills a man who happens to be passing by. What Poincaré called the influence of two otherwise alien worlds can also be called the crossing of causal chains .

Singularities in natural history

Science currently envisions the development of systems in such a way that after the creation of the universe through a singular big bang, a uniformly distributed plasma spread in space, which cooled with increasing expansion, so that atoms were formed and finally due to the smallest (singular) fluctuations self-reinforcing inhomogeneities arose in the uniform density. They subsequently led to the formation of galaxies , stars and other systems in the universe, from which in the end humans also emerged. Even if the singularity of the Big Bang can be avoided in the mathematical models, singularities remain an essential element of the history of origin.

The history of evolution shows that not only successful mutations can be understood as positive singularities, but that hominization or becoming human is the most important singular event in evolution and represents a jump out of the continuum of the previous evolutionary development of planet earth.

More recently, Ward and Kirschvink show that the history of life has been more influenced by catastrophe than by continuous evolution. Catastrophes are initially destructive singularities that create space for new developments in the sense of innovations as productive singularities.

Singularities and complexity

The concept of singularity is closely related to the concept of complexity . JC Maxwell has already pointed out that the more complex a system, the more singular points it has. Complexity is also the basis for perceived chaos and singularities. If one does not perceive an apparently insignificant event that produces a great effect in a simple context, this is even less to be expected in a complicated situation with many elements and relationships. The decline of ancient cultures shows that complexity is the breeding ground for singularities. Causes such as intruders, internal conflicts or natural disasters are not enough on their own to justify the downfall of a culture. Rather, the prerequisite is increasing complexity and the associated decreasing marginal yields. Likewise, phenomena are increasingly found in the company-relevant environment that are quite comparable with phenomena of systems and chaos theory. The main signals for a chaotic environment are:

  • The high degree of networking of the environment with its numerous interdependencies , which increasingly produces singular events.
  • The low and only short-term predictability of market developments.
  • The compression of information in space and time that creates a feeling of chaos.
  • The increasingly lacking possibility of differentiating between right and wrong actions.

The 2007 financial crisis shows how difficult it is to make decisions in an overly complex environment. The complexity of financial systems and financial products is a major challenge that financial markets and institutions face. One solution is to reduce complexity and increase the adaptation potential or robustness. In an increasingly complex world with increasing singularities, it is therefore important to forego optimization potential in order to gain the ability to adapt to external shocks and catastrophes.

Web links

literature

  • JC Maxwell: 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, pp. 440ff.
  • H. Wood fighters: Management of singularities and chaos. Wiesbaden 1996, ISBN 3-8244-0296-3 .
  • Ch. Strub, K. Mainzer: Singular, Singularity. In: J. Ritter, K. Founder (Hrsg.): Historical dictionary of philosophy. Volume 9, Darmstadt 1992, Sp. 798ff.
  • J.-H. Scharf (Ed.): Singularitäten, Nova Acta Leopoldina, treatises of the German Academy of Natural Scientists Leopoldina, lectures on the occasion of the annual meeting from March 30th to April 2nd, 1985 in Halle (Saale). Leipzig 1989.

Individual evidence

  1. ^ JC Maxwell: 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, pp. 440ff.
  2. ↑ H. wood fighters: Management of singularities and chaos. Wiesbaden 1996, p. 91.
  3. ^ H. Poincaré: Mémoire sur les courbes définies par une equation différentielle. In: Journal de mathématiques. 1881, pp. 375-422.
  4. ^ IN Bronstein, KA Semendjajew: Taschenbuch der Mathematik. 25th edition. Stuttgart et al. 1991, p. 217.
  5. ^ I. Prigogine, I. Stengers: Dialogue with nature - new ways of scientific thinking. 6th edition. Munich 1990, p. 148ff.
  6. ^ PNV Tu: Dynamical Systems: An Introduction with Applications in Economics and Biology. 2., revised. and exp. Edition. Berlin u. a. 1994, p. 195ff.
  7. ^ CC von Weizsäcker: Order and chaos in the economy. In: W. Gerock, H. Haken u. a. (Ed.): Order and chaos in inanimate and animate nature. Stuttgart 1989, p. 46.
  8. ^ WL Bühl: Social change in imbalance: cycles, fluctuations, catastrophes. Stuttgart 1990, p. 207.
  9. ^ H. Staudinger: Singularity and Contingency. In: Report of the meeting of the Scientific Society at the Johann Wolfgang Goethe University in Frankfurt am Main. Volume 21, No. 3, Stuttgart 1985, p. 133.
  10. ^ R. Hagemann: Mutations as productive singularities. In: J.-H. Scharf (Ed.): Singularitäten, Nova Acta Leopoldina, treatises of the German Academy of Natural Scientists Leopoldina, lectures on the occasion of the annual meeting from March 30th to April 2nd, 1985 in Halle (Saale). Leipzig 1989, pp. 155-169.
  11. ↑ C. Vogel: Hominization, a singular leap out of the continuum of evolution? In: J.-H. Scharf (Ed.): Singularitäten, Nova Acta Leopoldina, treatises of the German Academy of Natural Scientists Leopoldina, lectures on the occasion of the annual meeting from March 30th to April 2nd, 1985 in Halle (Saale). Leipzig 1989, pp. 141-154.
  12. ↑ P. Ward, J. Kirschvink: A new story of life. Munich 2016, p. 30.
  13. ↑ H. wood fighters: Management of singularities and chaos. Wiesbaden 1996, pp. 133ff and 139ff
  14. ^ JA Tainter: The Collapse of Complex Societies. Cambridge, New York et al. a. 1988, p. 42ff.
  15. ↑ B. Heitger: Chaotic Organization - Organized Chaos? The contribution of management to the learning organization. In: T. Sattelberger (Ed.): The learning organization: Concepts for a new quality of corporate development. Wiesbaden 1991, p. 116ff.
  16. ↑ J.-P. Landau: 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. June 8, 2009.
  17. ^ M. Conrad: Adaptability: The Significance of Variability from Molecule to Ecosystem. New York / London 1983.