The clockwork Newtonian world recedes in the intellectual horizon. Its time at the frontiers of our science and intuition has come to an end, the gradual rise of a new science has already begun. The seemingly orderly universe which we encounter every second is more liberal than what we thought, instead of being completely predictable and deterministic nature is unpredictable and chaotic.
Does this mean Newton’s laws of motion and all the work derived from them are useless? Shall we hammer out our linear intuition? The answer is simply no. Simplifying data by formulating linear relationships is still useful, it still governs most of our science and economics. We cannot immediately discard what has granted us so much power and delve into a uncertain and new mathematical world. Linear science, that is, simple linear and exponential mathematical equations predicting natural phenomena is still a good investment. It is still reaping returns. Though like a new idea or business chaos deserves an investment; as it succeeds and evolves its share on the scientific market will grow.
You’re still asking the question… what is chaos? Okay I will briefly introduce it, though the concept and formulation of chaos itself is still growing.
Chaos can be defined as sensitive dependence on initial conditions. This means that a system which could be within nature e.g. organisms resulting from small changes in DNA or the weather is greatly dependent on how it starts. If you or nature changes how a system starts or tweak a factor in its initial conditions, you will observe a great difference in the behaviors of the system and as time continues the difference becomes greater. In a lecture about his paper on the chaotic behavior of the weather: ‘Non-periodic flow’ Edward Lorenz named this sensitive dependence as the butterfly effect. The effects of the butterfly on the surrounding air albeit minuscule may cause a tornado on the other side of the globe. This sensitive dependence plus the nonlinear laws that govern a system make it a chaotic one.
Here we have two images, one represents a butterfly the other a lorenz attractor (a 3 dimensional graph with lines representing variables accross time). The Lorenz attractor is determined by the other picture, that is, the variables represented on the graph is determined by the butterfly effect. The system is sensitive to its initial conditions. The graph is chaotic, it may look predictable but the variables oscillate un-predictably over time.
So why do we need to use chaos when it is unpredictable? Surely it is not very useful? It may seem that way when chaos is over-simplified. Chaos is merely a non-linear science which forms relationships and laws of nature, economics and other systems from non linear geometry and equations of a special kind. New medical research into predicting heart attacks contains chaotic methods. The pace-maker cells, blood flow and cholesterol all at first glance seem to be periodic and simple, however as small changes in the cells or small environmental or diet changes may change the conditions of this biological system. Over time chaos may perform it’s role… what at first seems minuscule may become a momentous heart attack. Like the butterfly causing the tornado.
Non-linear science arises when linear science fails to adequately explain natural phenomena. As classical physics failed to explain the wave-duality of particles and the characteristics of spin, quantum mechanics was born. The genre of phenomena that non-linear attempts to explain what linear science couldn’t is ‘self organisation’. Self organisation is where a system which is independent of external equipment or major forces behaves in patterns that are random. Linear science needs to know the forces acting on the system… but we don’t know it… the drive of the system is in the system itself. So we need to have a science that explains how certain systems can behave randomly in patterns by themselves.
Another important concept inherent in chaos is self similarity. This concept arose from one maverick genius… Benoit Mandelbrot. Self-similarity is where a pattern repeats itself at smaller and smaller stages and doing so infinitely. The equation that governed this fractal pattern as he coined was… Z <> Z^2 + C. The value of Z is continually used to create the pattern. A brilliant yet complex pattern has arose from a simple equation. Our linguistic and intuitive relationship between complexity and simplicity has been overturned. They are not distinct, they are interconnected.
A Mandelbrot set with beautiful geometry. Chaoticists argue that this fractal geometry is inherent in nature.
Chaos is an amazing mathematical discovery, it has applications in nearly every scientific discipline.
What wonders will it bring? Nobody knows. Scientific and mathematical discovery may itself be chaotic, dependent on its initial conditions. I do have intuition that the elegant mathematical butterfly of chaos may evoke an intellectual tornado, changing the landscape of knowledge forever.