*The Mathematics of Emergence*

__Coarse Change__

When you trade financial markets you quickly learn that 10% up and 10% down are not the same thing!

If we were to grow $100 in 1000 incremental steps of 0.0095315% we will arrive at $110. And if we subsequently unwind/decline from this $110 in 1000 incremental steps of 0.0095315% we will arrive back at $100. If however we were to grow the same $100 to $110 in one incremental step of 10%, and then decline from $110 in one incremental step of 10% we will not arrive back at $100 but at $99 instead.

So why is this? The reason is simple. Smooth incremental change will synchronize up and down movements, but coarse incremental change will not.

__Linearization__

In the simplest of mathematical terms an increase of x% followed by a decrease of x% is the same as multiplying $100 by the product of (1 + x) times (1 – x) which can be written mathematically as

$100 × (1 + x) × (1 – x) = $100 × (1 – x^{2}) = $100 – ($100 × (x^{2}))

Now if x is 10% then

$100 – ($100 × (0.10^{2})) = $100 – $1.00 = $99.00

However if x is 0.0095315% then

$100 – ($100 × (0.000095315^{2})) = $100 – $0.00000009 = $100.00

So when x is small we can ignore the (x^{2}) term, because this tiny error is negligible.

Now believe it or not, “*ignoring the (x ^{2}) term”* is the cornerstone of all mathematical science. Ignoring the (x

^{2}) term is referred to as

*“linear approximation”*or

*“linearization”*. Linearization is used is every area of science; from physics to engineering, from economics to ecology. By assuming that everything always changes in a smooth continuous fashion we can approximate the behaviour of a systems as being “linear” — but as we will see

*“linear behaviour”*is really just another way of saying that the systems behaviour is not affected (very much) by

*“feedback”*…

__Feedback__

We can see that, in the above equation that the (x^{2}) term is actually acting as a tiny amount of *“ extra”*

*“positive feedback”*from the previous move. Now when incremental change is virtually smooth and continuous, this extra positive feedback appears to be effectively negligible in each step, and so by ignoring it we effectively equalize the number of steps on the way up with the number on the way down and consequently mathematical synchronicity is achieved.

[Think about it: If this tiny amount of extra feedback was not ignored then the number of steps on the way up would be less than the number on the way back down to the original $100 amount]

However when the incremental change is coarse, then positive feedback is significant and ignoring it * is not an option* and so there is a lack of synchronization in the up and down steps. In other words coarse change leads to coarse positive feedback which causes desynchronization in self-stabilizing systems…

In the latter part of the 20^{th} century however, we were to be offered the opportunity to see how these coarse messy dynamics could in fact be our first insight into the *creative power of incompressible feedback…*

__Chaos Theory + Discrete Mathematics__

Nonlinear Dynamics was first discovered in the early part of the 20^{th} century, but they were effectively rediscovered with the birth of *“Chaos Theory”.*

By the early 1960’s the computer revolution had brought with it a computational microscope that allowed us to study coarse incremental change using discrete *“**Recursive* *Mathematics”*. Before this time we had been limited to the (linearized) mathematics of *“continuous change in continuous time”*. Computers gave us a tool to study the long-term effects of *“incremental step-by-step change in discrete incremental time”. *

Chaos occurs shows us that strange things can happen when positive feedback is no longer compressed by ignoring the (x^{2}) term. Coarse positive feedback within negative feedback systems causes ** coarse-negative-feedback**, and this

*“*

*coarse damping”*disrupts the natural mathematical pull of a finely-tuned, highly symmetric, but fundamentally bland and featureless equilibrium.

__Coarse Emergent Mathematics__

*“Coarse Emergent Mathematics” *is a coined phase that seems to best describe the study of the *“rich diversity that can spontaneously emerge from bland uniformity” *when discrete self-damping systems employ a *“Coarse Incremental Step-Size”*.

Here we will look at two examples of recursive mathematical models that can employ a coarse step-size…

- If you know about Chaos you probably know of the
*“logistic map”*. The logistic map – also known as the*“logistic difference equation” –*was made famous by Robert May in 1976 when he used it to model the behaviour of each generation of a biological species, and found to his amazement that from this simplest of mathematical models, complex dynamics can emerge…**The Logistic Map** - We think of oscillating bodies in terms of smooth continuous change, but what if oscillating bodies actually oscillated in a step-by-step fashion. Then bodies oscillating at very fast speeds would be akin to coarse jumpy motion which mathematically can produce some very interesting effects. Here once again coarse synchronicity leads to the emergence of structure…
**Coarse Harmonic Motion**