1 October 2019
Traditionally Chaos Theory is associated with Sensitive Dependence on Initial Conditions (SDIC) aka “The Butterfly Effect” – the idea that a small change can have a large and disproportional effect at some point in the future. But this is a naive understanding of Chaos.
Chaos is much more than the butterfly effect, but the very concept of the butterfly effect is demonstrative of how intellectual progress is often hindered by lockedin thinking.
To describe something as being SDIC is to be locked into linear thinking, more specifically locked into the ubiquitousness of linear mathematics in physics.
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The Mathematics of Change
About 350 years ago Isaac Newton and Gottfried Leibniz, independently of each other and around about the same time, introduced the world to “Calculus”.
This was a big event in human history, because up until that point in time mathematics was primarily a subject about static things like, geometry, trigonometry, and algebra (which was just beginning to emerge); consequently the introduction of calculus was effectively an injection of “dynamics” into mathematics for the very first time…
The introduction of calculus changed everything. Suddenly mathematics could be used to describe behaviour, but more importantly, this same mathematics could now also be used to predict future behaviour…
Over the following decades and centuries Differential and Integral Calculus found their way into virtually every field of science. However, something that has been generally overlooked about these powerful mathematical tools is that they rely on a fundamental assumption that is build into calculus; that assumption is that all change fundamentally occurs in a smooth continuous fashion.
The Mathematics of Chaos
The cosy mathematical world of smooth linear dynamics was upset somewhat in the latter part of the 20^{th} Century when, seemingly out of nowhere, Chaos Theory came along…
Chaos Theory shows us that not all change occurs in a smooth and continuous way, some types of change are distinctly nonlinear.
Moreover, Chaos Theory shows us that nonlinear dynamics are essentially unpredictable, but given the widespread allegiance to the belief in a predictable clockworklike universe, physicists have described this chaotic lack of prediction in the language of linear dynamics, and hence equated chaos with the concept of SDIC (aka The Butterfly Effect)…
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So where do we stand today?
Well, all the excitement about Chaos Theory more or less came and went and the locked in thinking about real dynamics remains in place. Science in general, and physics in particular, is still pretty much weddedto linear dynamics describing and predicting change that occurs in a smooth and continuous fashion…
Unfortunately such lockedin thinking can blind one to the truth. And the truth, in this case, is that much of our mathematics, that describes real physical things, is devoid of any form of abrupt noncontinuous change.
With continuous change things are easier to predict, but with abrupt change, prediction becomes a tad more difficult; and the more abrupt the change, the more unpredictable the behaviour becomes…
Such abrupt change is, of course, ubiquitous in all types of systems. And so Chaos Theory seems to imply that there is little that we can ever know about the future behaviour of these systems…
But we have, to date, been looking at Chaos Theory the wrong way around. Too often is the focus on the cocalled Butterfly Effect. The Butterfly Effect focuses on a lack of order – a lack of equilibrium – but what chaos mathematics is really trying to tell us is that, in reality, there is an invisible hand at play in all things in the universe (even if it looks like there isn’t)…
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What Chaos Theory Reveals
Some 300 years ago Adam Smith used the idea of “an invisible hand” to illustrate how economies selforganize themselves to an optimal economic equilibrium. Chaos Theory suggests that Smith’s invisible hand is not limited to economics, by showing us that there is the same type of mathematical hand at play in all things in our universe.
Chaos Theory is about the Mathematics of SelfGenerating Systems.
It is about how mathematics act like an invisible hand to generate emergent complexity, not just in economic systems but in every conceivable type of system.
So, forget about the socalled “Butterfly Effect” that distracting notion is virtually irrelevant; Chaos Theory is so much more than the butterfly effect!
Chaos Theory explains The Mathematics of “Dynamic Stability”. It reveals four things about all forms of dynamics:

 An Invisible Mathematical Hand is at play in all system behaviour…
 The existence of mathematical “Attractors” (both orderly and chaotic) means that some things are more likely than others…
 The type of attractor we see is determined by the amount of energy in the system…
 These attractors exist at all levels of scale (including at the scale of Quantum Mechanics)…