May 142000

Chaos is not a mere disorder. What is chaos, then? Let’s consider why chaos does not result in skepticism, though it is neither non-linear nor indeterminate.

A plot of Lorenz attractor for values r = 28, σ = 10, b = 8/3. “Projection of trajectory of Lorenz system” by Wikimol is licensed under CC-BY-SA.

1. Why were natural sciences determinate?

One night a man was looking for something under the streetlight.

Passerby: What are you looking for?

Man: I have dropped my wallet and I am looking for it.

Passerby: Did you drop it under the streetlight?

Man: No. I am searching under the streetlight, just because here I can easily find my wallet.

This is a caricature of natural sciences. Natural sciences are often said to be stricter than the humanities or social sciences, but the fact is that they have studied only what can be mathematically and strictly studied. It is, however, worth attention that in recent years the science of complexity exceeds such a strait limitation and tries to elucidate chaos. This column treats of what influence the study in chaos has on the traditional philosophical controversy between determinism and anti-determinism.

2. Chaos is nonlinear

First, I must explain what chaos is. The word "chaos" usually means mere disorder but it is used among the scientists of complexity as a technical term indicating the systems that behave in an unpredictable manner according to deterministic nonlinear rules.

Here the nonlinear rule means such a function as is


This function is quadratic and not linear, drawing a parabola. It is called a logistic map because it approximates the evolution of an animal population over time. In this equation x is population density of a certain animal in a certain generation within a limited territory and f(x) that of the next generation (1 means saturation; 0 extinction). Each individual bears four young but not all of them come of age. As the exponential growth of their population is hindered by the physical limitations of their surrounding or internal disturbances, 4x is multiplied by (1-X).

When this simple map is applied over numbers of generations, the result is as follows:

Chaotic behavior of the logistic map
Generation Population 1 Population 2
0 0.40000 0.40001
1 0.96000 0.96001
2 0.15360 0.15357
3 0.52003 0.51995
4 0.99840 0.99841
5 0.00641 0.00636
6 0.02547 0.02526
7 0.09928 0.09850
8 0.35768 0.35518
9 0.91898 0.91610
10 0.29782 0.30743
11 0.83650 0.85167
12 0.54707 0.50531
13 0.99114 0.99989
14 0.03514 0.00045
15 0.13561 0.00180

Population does not converge on a fixed value, nor does it vary periodically. If the number of generations is enlarged infinitely, even a supercomputer cannot predict the population. Such a system is called chaos.

3. Chaos is indeterminate

Another feature of chaos is a sensitive dependence on initial conditions. The initial value of Population 1 differs from that of Population 2 by only 0.00001. Such a difference is out of significant figures and usually disregarded but this virtually immeasurable difference in initial conditions can lead to the wildly differing result at the 15th generation.

The sensitive dependence on initial conditions is also called "butterfly effect"; a flap of a butterfly’s wings might set off an otherwise avoidable Tornado. This goes too far but it is certain that weather phenomena are typical chaos and it is because of this butterfly effect that a weather forecaster can scarcely predict weather two weeks ahead, though the probability of him predicting tomorrow’s weather is high.

The paradigm of the modern natural science set by Galileo and Newton is mechanistic determinism. Laplace, a mathematician in France in the 19th century, regarded nature as consisting of particles bound in causal relations and moving according to Newton dynamics, and he thought a demon with the ability to recognize all of present initial conditions and restraint conditions could predict the state of all the particles at any time of future. According to him, men cannot predict the future just because they don’t have such a demonic ability and, as everything is determined beforehand, freedom of will is but an illusion. Although Newton dynamics is still valid in limited cases, universal application of it brings about philosophical gibberish like Laplace’s demon.

Mechanical determinism had big influence also on the humanities and social sciences. The idealistic philosophers made all efforts to protect free will from natural necessity. Social scientists, feeling inferior because their theories lacked strictness, are eager to find an inevitable "law" of history. Marx, who overcame mechanism and criticized the ideology of modern bourgeois sciences, was also within the deterministic paradigm.

Mechanical determinism collapses gradually during the 20th century. The uncertainty principle in quantum mechanics denied the determinism at a microscopic level, and the butterfly effect in the science of complexity denied the determinism at a macroscopic level.

According to the Copenhagen interpretation of quantum mechanics, a microscopic system is originally an indeterminate wave motion, which contracts and gets determinate only when a recognizing subject interferes with the object rather than the object is originally determinate, seeming indeterminate and unpredictable just for lack of our full recognition capability.

Against this anti-determinism, Marxists insisted that accidentalness at the microscopic level was sublated up (aufgehoben) to necessity at the macroscopic level; like a car running on a highway that had freedom to change lanes or adjust speed but was destined for a certain destination, our human society, though it might go through some unpredictable accidents, was destined for the ultimate goal, the proletarian dictatorship through the communist revolution.

But if the indeterminacy at the microscopic level is enlarged at the macroscopic level as the butterfly effect, you can no longer believe in the philosophy of Aufheben. The deterministic paradigm collapsed concurrently with the collapse of communism and the science of complexity and the indeterminate global market economy took their place.

4. Chaos does not result in skepticism

Chaos behaves in an unpredictable manner according to deterministic rules. This is equivalent to the more ambitious proposition; unpredictable and seemingly lawless disorder can be ascribed to rather simple deterministic rules. Some scientists criticize this ascription as an old reductionism and propose conceiving the complex whole as it is. But isn’t their holism just a revival of far older mystical agnosticism?

The novelty of chaos consists in a new position;

the natural law is determinate but the future is unpredictable

while the old determinism insists

the natural law is determinate and the future is predictable

and anti-determinism insists

the natural law is indeterminate and the future is unpredictable.

If you still cannot make sense of chaos, I will illustrate chaos with a metaphor of games. In a game of sports, the players follow determinate rules but players behave chaotically and nobody can predict which team will win. Even in the game where it is evident that a certain team will win, a little mistake made by the team might be amplified to result in their surprising defeat as a butterfly effect. It is because games are chaos that people are charmed with sports.

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