Expecting the unexpected.

How do we know that the creations of worlds are not determined by falling grains of sand?

Victor Hugo, Les Misérables

In a whirlwind of dust, raised by elemental force; in the most frightful tempest excited by contrary winds, when the waves roll as high as mountains; there is not a single particle of dust or drop of water that has been placed by chance which has not a cause for occupying the place where it is found. In the most rigorous sense of the word, things act after the manner in which they ought to act; that is, according to their own peculiar essence and that of the energy from whom they receive this communicated force. 1 Until you introduce chaos.

Chaos is regarded as the smallest of changes in a system that can result in very large ripples in its behavior. The so-called butterfly effect has become one of the most popular images of chaos. It is the concept that the flapping of a butterfly’s wings in Argentina could cause a tornado in Texas three weeks later.

An extraordinary amount of time has been spent attempting to calculate, predict, and understand the “order” of chaos. In overview, my opinion considers this time to be spent frivolously; the very essence of chaos is to be unpredictable. Chaos is a “science of surprises,” to attempt to make sense of it is an exercise in expert level expecting the unexpected. The curious senses within me twinge when considering the depth at which humanity has sought to understand the very definition of disorder; however, I am little surprised. Driven by fear, we spent most of our lives avoiding feelings or states of powerlessness and loss of control; the concept of chaos undoubtedly can be too much to bear.

“A very small cause, which eludes us, determines a considerable effect that we cannot fail to see, and so we say that this effect is due to chance. If we knew exactly the laws of nature and the state of the universe at the initial moment, we could accurately predict the state of the same universe at a subsequent moment. But even if the natural laws no longer held any secrets for us, we could still only know the state approximately. If this enables us to predict the succeeding state to the same approximation, that is all we require – the phenomenon has been predicted, that it is governed by laws. But this is not always so, and small differences in the initial conditions may generate very large differences in the final phenomena. A small error in the former will lead to an enormous error in the latter. The prediction then becomes impossible, and we have a random phenomenon. This was the birth of chaos theory.” 1

Photo by Daniele Pelusi


Consider, if a geometrician knew the precise energies acting in each case, the properties of the particles moved, and could demonstrate that after causes given, each particle had precisely acted as expected and could not have acted otherwise than it did. Chaos explanations do not rely on gnomic considerations at all; rather, they rely on patterns of behavior and various properties characterizing this behavior. In summary, chaos studies search for patterns rather than laws.

Chaos has been invoked as an explanation for or as substantially contributing to explanations of actual-world behaviors. Links are thought to exist between “causal mechanisms” and behaviors. In the causal-mechanical model, behaviors are supposed to be reliable links along the lines of a cause and effect system: If mechanism CC is present, behavior BB typically follows. In this sense, chaos explanations understood on the causal-mechanical model are envisioned as providing reliable connections between mechanisms and the chaotic behavior exhibited by systems containing such mechanisms. 1

The basic idea of attempting to unify broad ideas is that science provides an understanding of diverse facts and events by showing how these may be unified by much smaller sets of factors (e.g., laws or causes). It stands to reason that chaos is (or could be) a domain or set of a limited number of patterns and tools for explaining/understanding a set of characteristic behaviors found in diverse phenomena spread across physics, chemistry, biology, economics, social psychology, etc. In this sense, the set patterns or structures might make up the explanatory story unifying our understanding of all diverse phenomena behaving chaotically. 1

The Principles of Chaos Theory:

  • Causality Principle: Every effect has a preceding cause. Every real event is the result of a prior occurrence.
    (Feeling nerdy? Research Causality Theory… “One version of the causal theory claims that a perceiver sees an object only if the object is a cause of the perceiver’s seeing it.” 3)

  • Determinism: A philosophical proposition that every event is determined by an unbroken chain of prior occurrences.
    Four sorts of determinism have at various times been put forward and have been felt to threaten the freedom of the will and human responsibility. They are: logical determinism, theological determinism, psychological determinism, and physical determinism.
    • Logical determinism maintains that the future is already fixed as unalterable as the past.
    • Theological determinism argues that since God is omniscient, He knows everything, the future included.
    • Psychological determinism maintains that there are certain psychological laws that we are beginning to discover, enabling us to predict, usually on the basis of experiences in early infancy, how a person will respond to different situations throughout their later life.
    • Physical determinism is based on there being physical laws of nature, and of whose truth we can reasonably hope to be quite certain, together with the claim that all other features of the world are dependent on physical factors. 4
  • Predictability: This refers to the degree that a correct forecast of a system’s state can be made either qualitatively or quantitatively – i.e., data in the form of descriptive words that can be examined for patterns, meaning, qualities, or characteristics; versus, data that can be counted, measured, and expressed using numbers.
  • Unpredictability: Because we can never know every intimate detail, we cannot hope to predict the ultimate fate of a complex system. Even slight errors in measuring the state of a system will be amplified dramatically, rendering any prediction useless.
  • Transition: Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed.
    • Example: Two neighboring water molecules may end up in different parts of the ocean or even in different oceans.
  • Model: A pattern, plan, representation, or description designed to show the structure or workings of an object, system, or concept.
  • Dynamical System: A system that changes over time in both a causal and a deterministic manner; i.e., its future depends only on phenomena from its past and its present (causality) and each given initial condition will lead to only one given later state of the system (determinism). Systems that are noisy or stochastic, in the sense of showing randomness, are not dynamical systems, and the probability theory is the one to apply to their analysis.
  • Integrable system: In mathematics, this refers to a system of differential equations for which solutions can be found.
  • Linear System: A system is said to be linear when the whole is exactly equal to the sum of its components.
    • [Nonlinear example – fluids cannot be unmixed].
    • Mixing/Nonlinear patterns: nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on.
  • Feedback: A response to stimuli or information; the process whereby a change to the system triggers a chain of events to either increase the change (positive feedback) or reduce it to bring the system back to normal (negative feedback) — or induce a cyclic phenomenon. 5

    Systems often become chaotic when there is positive feedback present. A good example of feedback is the behavior of the stock market. As the value of a stock rises or falls, people are inclined to buy or sell that stock. This in turn further affects the price of the stock, causing it to rise or fall chaotically.

  • Self-similarity: This means that an object is composed of subunits and sub subunits on multiple levels that (statistically) resemble the structure of the whole object. In everyday life, there are lower and upper boundaries over which such self-similar behaviors apply.
    • The Koch Snowflake is an example of a figure that is self-similar, meaning it looks the same on any scale. 6
    • The Hilbert Curve is a self-similar object, but the way we construct it is an iterative process. 6
      • A Hilbert Curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891. A space-filling curve (SFC) is a way of mapping the multi-dimensional space into one-dimensional space. It acts like a thread that passes through every cell element (or pixel) in the multi-dimensional space so that every cell is visited exactly once. 7
    • Iteration: repeating a set of rules or steps over and over; one step is called an iterate.
    • Recursion: given some starting information and a rule for how to use it to get new information, the rule is then repeated using the new information.
  • Why self-similarity is different from iteration and recursion? — Self-similarity is a property of the object, not of the steps used to build the object.
  • Fractal: Is a geometrical object satisfying two criteria: self-similarity and fractional dimensionality. A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. 8
    • Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar since nature is full of fractals.
    • This Fern consists of many small leaves that branch off a larger one.  – –  This Romanesco broccoli consists of smaller cones spiraling around a larger one.
  • Initially appearing like highly complex shapes; however, you might notice that they both follow a relatively simple pattern: all the of the plants look exactly the same, just smaller. The same pattern is repeated over and over again, at smaller scales.
    • In mathematics, this is self-similarity, and shapes that have it are called fractals. 8
  • A fractal is a geometric shape that has a fractional dimension. Fractals contain patterns at every level of magnification, and they can be created by repeating a procedure or iterating an equation infinitely. 8

“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”

-Albert Einstein

I feel no calling to reason with chaos and I hold no space of fear for those elements outside my realm of control. There are energies at work I may never understand and I do not suffer from the unquenchable thirst to explain the inner workings at the core of all matter. However, with firm resolve I reason, just as the light balances the dark, so does order remain balanced by chaos. According to the laws of thermodynamics, energy cannot be created or destroyed but only transformed. Likewise, the entropy of an isolated system always increases; the act of putting one system in order requires that the other system become disordered.

Ilya Prigogine, author of “a theory of dissipative structures in thermodynamics,” considers that the universe is neither totally deterministic nor totally stochastic (randomly determined; having a random probability distribution or pattern that may be analyzed statistically but may not be predicted precisely).1

René Thorn is the author of catastrophe theory. This theory is akin to chaos theory, but it was constructed from the study of singularities, (i.e., continuous actions that produce discontinuous results). Catastrophe theory is interesting in that it places much emphasis on explanation rather than measurement. Thorn’s work has been received with much debate and was at the origin of a renewed debate on the issue of determinism (a philosophical proposition that every event is determined by an unbroken chain of prior occurrences).

In 1980, Thorn wrote an article “stop change, silence the noise” expressing his ideas on the issues stating, “I’d like to say straight away that this fascination with randomness above all bears witness to an unscientific attitude. It is also to a large degree the result of a certain mental confusion, which is forgivable in authors with literary training, but hard to excuse in scientists experienced in the rigors of rational enquiry. What in fact is randomness? Only a purely negative definition can be given: a random process cannot be simulated by any mechanism or described by any formalism. Asserting that ‘chance exists’ is tantamount to the ontological position that there are natural phenomena that we will never be able to describe, nor therefore to understand.” 1

A superficial analysis might conclude that the successful study of chaos theory can only lead to semantic truths, my final opinion falls somewhere in the very middle of the road. Some researchers in the field of social sciences propose that chaos theory offers a revolutionary new paradigm, away from the materialistic Utopia, coinciding with the paradox of progress. A social system maintained at the edge of chaos, between too much and too little authoritarian control; this concept concerning politics rather than physics. I see more value in the philosophical consideration of chaos theory than the analytical. Randomness is the heart of chaos; the fact that chance exists is the reason we seek after our curiosity even if we often fail. Unfortunately, in fear of things beyond our control we wish for rational explanations when it might be more rational to accept, there are none.

“Our survival depends not on eliminating the dangers of life but on relearning to embrace and acknowledge that which terrifies us – including altered states of consciousness.”

Christopher Ryan, Civilized to Death.

The French philosopher d’Holbach, coauthor of the Encyclopédie de Diderot et d’Alembert was the first to include, his 1770 book Le système de la nature, a deterministic statement about the feasibility of calculating the effects of a given cause. However, it was the mathematician and astronomer Pierre-Simon Laplace who most clearly stated the concept of universal determinism shortly after d’Holbach, in 1778:

We ought then to regard the present state of the universe as the effect of Its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose It — an intelligence sufficiently vast to submit these data to analysis– It would embrace in the same formula the motions of the greatest bodies of the universe and those of the lightest atom; for It nothing would be uncertain and the future, as the past, would be present to Its eyes.

Laplace is also famous for an exchange with Napoleon asking about his work, “You have written this huge book on the system of the world without once mentioning the author of the universe.” To this Laplace responded, “Sire, I had no need of that hypothesis. These words attest to the self-confidence of this man.” 1

Man is an animal suspended in a web of significance he himself has spun.

clifford geertz

To be continued….


1 – Oestreicher, Christian PhD* A history of chaos theory. National Center for Biotechnology Information, 2007 [ncbi.nlm.nih.gov]

2 – Bishop, Robert. Chaos. Stanford Encyclopedia of Philosophy, 2008. [https://plato.stanford.edu/entries/chaos/]

3 – Gold. Perception: Philosophical Aspects. International Encyclopedia of the Social & Behavioral Sciences, 2001. [https://www.sciencedirect.com/topics/computer-science/causal-theory#:~:text=The%20causal%20theory%20holds%20that,of%20the%20perceiver’s%20seeing%20it.]

4 – Lucas, J.R. The Freedom of the Will. Oxford University Press Scholarship Online, 2011. [https://oxford.universitypressscholarship.com/view/10.1093/acprof:oso/9780198243434.001.0001/acprof-9780198243434-chapter-12]

5 – The Albert Team. Positive and Negative Feedback Loops in Biology. Learn By Doing, Inc., 2021. [https://www.albert.io/blog/positive-negative-feedback-loops-biology/]

6 – Bradley, Larry. Chaos & Fractals. Department of Physics and Astronomy at the Johns Hopkins University, 2010 [https://www.stsci.edu/~lbradley/seminar/index.html]

7 – Mokbel, Mohamed. Performance of Multi-Dimensional Space- Filling Curves. Purdue University, Department of Computer Science, 2002. [https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=2545&context=cstech]

8 – Legner, Phillip. Fractals, introduction. Mathigon,2021. [https://mathigon.org/course/fractals/introduction]

29. Thorn R. Modèles mathématiques de la morphogenèse. English translation: Mathematical Models of Morphogenesis. Chichester, UK: Wiley,1983 [Google Scholar]

30. Thorn R. Halte au hasard, silence au bruit. Le Débat n°3. Paris, France: Gallimard,1980 [Google Scholar]

31. Prigogine I., Stenger I. La Fin des Certitudes. English translation: The End of Certainty: Time, Chaos and the New Laws of Nature. New York, NY; London, UK; Toronto, Canada; Sydney, Australia; Singapore: The Free Press;1997 [Google Scholar]

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