Outer Limits of Reason (33 page)

How can a deterministic process produce an unpredictable or seemingly random event? After all, if it is deterministic and we can write a formula that describes its short-term future, then why is it unpredictable in the long-term? To understand this we must remind ourselves that whether a process is deterministic represents an objective fact about the universe. Does this process follow fixed deterministic law? In contrast, whether a process is predictable is a subjective question about the mind. Do you have enough information about this system and its initial conditions to predict its long-term future or will it seem random? What is random to one person might be predicable to another person. My desk may look like a chaotic mess to you, but I know where everything is.

Flipping a coin is the classic example of an unpredictable process. However, if you had a laboratory with an exact machine that flipped a perfectly fair coin with no air interference, then you could flip a coin and know what the result would be beforehand.
⁴
The lack of precise information about the speed at which you are flipping a coin, lack of information about the exact weight of the coin, and lack of exact information about air interference while the coin is flying cause unpredictability. In a chaotic system, the imprecision of the initial conditions is so strong that the system becomes objectively unpredictable. No human being in the world possessing all the desired computing power can ever determine the the system's future. In conclusion, a deterministic process can cause an event that cannot be predicted.

Chaotic systems have shown that the Newtonian dream of calculating the future of every system is over. Laplace's optimism was for naught: the universe is far more complicated than he thought.

The truth is that science was never really about predicting. Geologists do not really have to predict earthquakes; they have to understand the process of earthquakes. Meteorologists don't have to predict when lightning will strike. Biologists do not have to predict future species. What is important in science and what makes science significant is explanation and understanding.

I once humorously pointed out to my thesis advisor, Alex Heller, that there are subatomic particles in nature that follow equations of motion that human beings cannot solve. And even though humans do not know where the particles will go, the particles seem to know exactly where to go. Professor Heller responded by saying that this shows that science has nothing to do with calculating or predicting. Calculations can be done by computers. Predictions can be performed by subatomic particles. Science is about
understanding
—an ability only human beings possess.

 

One might try to disregard the butterfly effect by saying that if two initial conditions were close enough, the results would be the same. We saw in the case of the weather that being the same to the third decimal number is not close enough. Perhaps initial conditions have to be the same to the fifth or the tenth decimal number? Although this sounds reasonable, it is, in fact, wrong. The best way to demonstrate that this is erroneous is by looking at something called the
Mandelbrot set
, which was formulated by mathematicians in the late 1970s and can be characterized as one of the prettiest parts of mathematics. The Mandelbrot set is an easily describable set of complex numbers. Start with a complex number
c
, square it, and add
c
to the result. This gives you another complex number. Square this number and add
c
to it again. Iteratively continue this procedure over and over again—that is, take the complex number
z
, calculate
z
²
+
c
, and repeat.
⁵
Either of two things can happen to the numbers in this iteration:

• They can get bigger and bigger until they go off to infinity.

• The complex numbers can remain small.

The Mandelbrot set consists of complex numbers that remain small after this iteration process. The set is the central black part shown in
figure 7.2
.

The fascinating part of the Mandelbrot set is the boundary between those numbers whose iterations go off to infinity and those that stay small. The boundary does not have any straight lines. It twists and turns and makes more and more shapes. The shapes are self-similar in the sense that within the boundary of the Mandelbrot set you will find shapes that are similar to the Mandelbrot set. This twisting and turning goes on forever and ever. Such shapes are called
fractals
. Since this is a mathematical shape and not a physical object, one can continuously magnify the image, as in
figure 7.3
.

Two 2-dimensional pictures simply cannot do justice to the true splendor of this shape. With modern computers one can magnify the boundary with ease and in real time. It is worth looking online for some videos of the Mandelbrot set.

What is the area of a two-dimensional figure? Since the shape of the Mandelbrot set can be shown in a picture, its area is fixed and finite. It might not be possible to know exactly what the area is, but we can approximate it pretty well. In contrast, since the boundary is forever twisting and turning and getting more and more complicated as you look deeper and deeper, the length of the boundary can be shown to be infinite. This is a seemingly paradoxical situation where there is a finite area bounded by an infinite border.

The complexity of the boundary of the Mandelbrot set shows that the question of whether an iteration of a complex number goes off to infinity is not so simple. The answer depends on many digits of the decimal expansion of the complex number. In fact, it depends on all of its infinite digits. Similarly, when we have any chaotic system, we would need to know the initial conditions with infinite precision in order to make any reasonable predictions. Human beings cannot deal with infinite precision.

 

This sensitive dependence on initial conditions is related to other interesting phenomena that researchers are currently investigating. Scientists deal with complex chaotic systems that have characteristics that are called self-organizing, emergent, feedback, and so on. These properties make the world a very interesting place.

As an interesting sideline that brings to light some of these properties, let us consider a complex process called
morphogenesis
(from the Greek for “form” and “creation,” i.e., the “creation of form”). This is a field of developmental biology that deals with how organisms take shape. One of the first people to study this subject in a serious manner was the computer scientist Alan Turing, who we met in the last chapter. Consider a zygote, a single-cell fertilized egg. It has the DNA of an organism. Through the process of mitosis, the zygote divides into two cells. Each of the cells has an exact copy of the same DNA. This process continues to four, eight, sixteen, etc., cells. As this process continues, something amazing happens. Certain cells become skin cells and some become bone cells. Some cells go on to form the nervous system, while others become the stomach muscles. The question arises, how does each cell know what to become? Why should one cell become part of a nail, while another becomes part of the brain? After all, they all have the exact same DNA. This is similar to handing complete blueprints of a building to construction workers as they enter a construction site without telling any worker where to go or what part of the building they should work on. And yet the building gets built! The zygote becomes a complete organism with many different parts. How do the cells self-organize? If all the cells are the same, why does the heart come out on the left? Turing was able to make some progress on these questions by coming up with certain equations that show how parts of the process work. (He did this before Watson and Crick actually described DNA. He was truly a genius!) The cells differentiate themselves by their relative position within the multicellular organism. They also differentiate themselves by where they are in relation to the outside of the organism. As they continue to differentiate themselves, they affect other cells. This is a type of feedback mechanism. There is extreme sensitivity to the position of each cell within the multicellular mass. The position determines the type of cell it will become. From a single-cell zygote, multicellular life emerges.

Certain systems are deterministic and unpredictable in a special way. Their determinism is a little less clear than the systems discussed above. An example of such a system is the three-body problem.

First some history. Newton taught us how two physical bodies interact. He provided a small, fantastic formula that determines most of the movement that we see in the world. This formula,

applies to rotations of the planets around the sun, as well as to an apple falling to the ground. In detail, the force (
F
) between any two bodies depends on the mass of one body (
m
₁
), the mass of the second body (
m
₂
), and the distance (
r
) between the two masses squared. The
G
is simply a constant number that is needed to put all the measurements in order.

Obviously, there is a desire to generalize this formula to three bodies. In other words, we would like a formula for three different bodies and the forces between them, as shown in
figure 7.4
. One would imagine that such a formula would have variables
m
₁
,m
₂
,
and
m
₃
representing the three masses. There would also be variables for the distance between the first and second bodies (
r
₁₂
), as well as the other distances (
r
₁₃
and
r
₂₃
). This problem of determining how three bodies interact is called the
three-body problem.