Everyone has experienced the non-predictability of the weather system. It is the goal of meteorologists to predict the weather precisely. However, they find that no matter how sophisticated their model is and how much computer power they have, they can only predict the weather precisely within a few days. It turns out that an exceedingly slight variation, even as slight as a butterfly flapping its wing in Beijing, can cause a storm in America. This effect is often referred to as the “butterfly effect.” We should note that a butterfly can cause a storm in America, but not all butterflies in Beijing will cause a storm in America. Chaos is a new science branch where people study complex systems’ chaotic behavior. The weather system is very complex; however, scientists found that even simple mathematical systems can show chaotic behavior. For example, in the famous Mandelbrot set, the equation is very simple, zn+1 = zn2 + c. where Z and C are complex numbers. Simply put, the next point on the point equals the square of the current value plus some constant. If you plot the location of all the points in the series, you will get a very interesting graph like this
Zooming into the graph will show a similar pattern, but the pattern doesn’t repeat itself exactly. The shape of the curve highly depends on the initial Z and C values. This kind of never-ending pattern is called fractal. Fractals are infinitely complex patterns that are self-similar across different scales. Nature is full of examples of fractals; snowflakes and fern leaves are classic examples.
When you change the initial conditions of a chaotic system by even a tiny amount, the values of the first few iterations may look similar. However, after a few hundred iterations, the values will start to deviate, and after a few thousand iterations, they will be nothing like the original. The smaller the change in initial conditions, the more iterations it takes to deviate, but eventually, the system will deviate. This is why weather prediction can only be accurate for a few days. It’s impossible to account for all the factors that contribute to weather patterns, such as the flapping of butterfly wings, so the precision of weather prediction diminishes over time.
Biologists have been studying the variability in populations of various species and found a straightforward equation that predicts animal populations. This model fits reasonably well with real-life data. This equation is called Logistic Difference Equation.
Xn+1 = r Xn(1-Xn)
Xn is the normalized population at a given year n, and Xn+1 is the population in the next year.
R is called the driving factor, which describes the growth rate.
When r is less than 3, the population converges to a stable value. But once r is larger than 3, the population fluctuates between two stable values. Increase the value of r further; it fluctuates between four values. As we gradually increase the value of r, it fluctuates between 8 values, then 16, and eventually becomes chaotic. This kind of fluctuation is called bifurcations.
In chaos theory, a phenomenon called a strange attractor describes a system that exhibits unpredictable behavior. The system moves back and forth within the strange attractor, transitioning between different regions in a non-linear and complex way. Although the specific paths it takes are always unique, they display similar patterns. An example of such a system is a magnetic pendulum swinging over fixed magnets. While its behavior may not be precisely predictable, its general patterns are identifiable. You don’t know whether it will be cold next week, but you know that, in general, it will be cold in winter. You don’t know whether your unborn baby will look like you, but you surely know they will not be a monkey. The word “Chaos” can add confusion to the theory; some think it should be called “orderly disorder” instead.
One can argue that, in principle, things are still deterministic; it is just that we don’t know all the variables precisely enough to predict the outcome. It is slightly more complicated than Newton’s clockwork universe, but the world is still predetermined! There is no free will; our choices are just an illusion; we are just characters in cosmic drama.
I’m in the field of experimental physics. In my Ph.D. research, I made scientific instruments. My area of research was in the field of Scanning Tunnelling Microscopy (STM). With this microscope, we were able to see individual atoms. The principle of this microscope is very simple, we use a very sharp tip (ideally a single atom at the end) and scan this tip across a metal surface. When two metals are incredibly close together of the order of 1nm (25000 times thinner than a human hair), a current will flow between them even in a vacuum. The current is exponentially dependent on the separation between them. You can think of atoms as balls 0.3nm in diameter. We can map out individual atoms by measuring the current as we scan the sample. This microscope revolutionized surface science, and the inventors Bining and Rohror got their Nobel prize within five years of their invention. I was very fortunate to work with Dr. Rohror as a summer student at IBM Zurich.
When the tip is so close to the surface, atomic forces between the atoms become significant. It will distort the atoms you want to image and create all sorts of artifacts; one of my research areas is understanding these artifacts. Simply put, the act of measurement affects the subject. Therefore, even in principle, we cannot know the variables precisely enough to predict the outcome. Modern physics tells us that some events, called “Quantum events, ” happen for no reason. These effects happen on an atomic scale; therefore, one cannot observe them daily. Furthermore, the uncertainty principle tells us that we cannot simultaneously have all information available. For example, one can tell where an electron is, but one cannot tell what it is doing; if you know what it’s doing, you don’t know where it is. Simply put, nature has the means to prevent you from knowing exactly everything; uncertainty is part of nature! Returning to the chaos theory, since we cannot know and there is no way to tell where and what all the atoms in the universe are doing simultaneously, there is no way to determine the future. The universe is an open system, not a giant clockwork system, as viewed by scientists in the nineteenth century.
I believe that God can interact with the world through what is known as bottom-up causation by using the smallest quantum events to influence larger systems. Chaotic systems are capable of amplifying the effects of microscopic events. The quantum world is a realm where the logic of everyday life does not apply and in which multiple states can coexist in the absence of an observer. However, when an observer intervenes, the wave function collapses, and potentiality becomes a reality. In the famous thought experiment by Edwin Schrödinger (the great pioneer of Quantum Physics), a cat is placed inside a closed chamber with radioactive material and some poison gas. Radioactive reactions in the nuclear material will emit alpha particles in a non-deterministic way. In the chamber, there is also a trigger mechanism. When it detects enough alpha particles, it releases the poison gas and kills the cat. The cat is both alive and dead until the chamber is opened and observed.
While there are many interpretations of this experiment and many philosophical issues related to the interpretation of quantum events, the limitations of this model must be acknowledged. The link between microscopic and macroscopic events is still not fully understood, and we must be cautious in asserting that God uses quantum events to influence the world. Nevertheless, the world remains open, even though physical laws are deterministic. God may use other means to influence the world, but the fact that the universe is open remains unchanged.
Very often, people associate bottom-up causation with reductionism. I don’t think it is necessary to be the case; we are associating events at the lower hierarchical level as trigger events in the higher hierarchical level, not as the explanation of higher-level activities. For example, electron flow can trigger a transistor to turn on an electrical circuit, but it will not and cannot explain the behavior of a transistor.
The theory of chaos provides a strong argument for the existence of free will; even though equations governing physical laws are already known and well-established, the future is not predetermined. Although physical laws are rigid and deterministic, an openness in the world still defies predictability. Chaos is not random or out of control, and underlying patterns may exist. God may have provided a pathway within the laws of physics for free will to emerge.
As we have discussed in the section on AI, the emergence of consciousness and the workings of the human brain are still not fully understood. While chaos theory and AI cannot provide us with all the answers, they provide us with insights into the existence of free will. At the very least, we can be confident that our future is not predetermined and that we have the freedom to choose.