From a children’s tale to Socrates and the Qur’an, the need for logic and sound reasoning is universal.

Nasraddin Hodja is a well-known folktale character in the Middle East. He is famous for his wit, which is apparent in the wisdom-filled stories that have been ascribed to him for eight centuries. Although Hodja was not a philosopher, many of his tales contain elements that demonstrate philosophical concepts and principles. As an example, the story of Hodja’s pot neatly exercises a fundamental philosophical method, i.e. the Socratic method or reductio ad absurdum. This story, though apparently simple and funny, has a deep philosophical dimension and teaches us the Socratic method in a very accessible way. The story goes as follows:

One day, Nasraddin Hodja borrowed a pot from his neighbor. After he had finished using it, he took it back to the neighbor with a smaller pot put inside it.
When the neighbor saw the smaller pot, he was surprised.


“What is that?” he asked.


“Well, said  Hodja, when I borrowed your pot it was pregnant and it gave birth.”


The man smiled and accepted both pots.

A few days later, Hodja borrowed the pot again but this time he did not return it.
The neighbor was rather cross. He went to Hodja and asked, “What about my pot?”


“I am very sorry,” said Hodja, “but it died.”


“Don’t make jokes with me,” replied the neighbor. “How can a pot die?”


“If you believe that it brought a child into the world,” said Hodja, “why can’t you believe that it died?”
 

In this story, Nasraddin Hodja suggests that his neighbor’s behavior of accepting the smaller pot is improper, by indicating an absurd consequence of his behavior, namely that a pot can die if it can give birth. This method is commonly used in philosophical argumentation since the Ancient Greeks. We can see similar examples in Plato’s dialogues. For example, in the Republic, Cephalus defines justice as speaking the truth and paying the debt. However, Socrates shows that this definition logically leads to some absurd consequences. Let us assume that somebody lends us arms when his mind is right, but wants them back later when he lost his mind. According to Cephalus’ definition of justice, we should give the arms to that insane and crazy person since they are not ours but his. Yet, no one would approve such an act. Once Cephalus encounters this counterexample, he understands that his initial definition has some problems; he approves Socrates’ point and tries to modify his idea of justice. Thus, the dialogue continues. Different definitions are introduced and checked by this method.

We can display the logical structure of the Socratic method in the following way. In a conversation, somebody holds an assumption, and makes a definition or claims something. The other person shows that this assumption, definition, or claim leads to an absurdity. For this reason, this method is also called reductio ad absurdum, i.e. reducing to absurdity. Since the initial assumption leads to something unacceptable, it is unacceptable as well by logical inference. Then the person who held that assumption is forced to abandon or modify it. See Table 1 for a comparison of these examples.

Table 1

The Socratic method is one of the basic methods that we use in reasoning. We use it in daily life, in sciences, and in any area where we engage in rational thinking. The strictest kind of absurdity is a contradiction, i.e. accepting and denying the exactly same thing under the same conditions. Because of this, logicians and mathematicians call the Socratic method, “the indirect proof,” or “proof by contradiction.” See the appendix for an example of the proof by contradiction in mathematics.

Interestingly enough, we also see this method in the holy books. For example, the Qur’an frequently suggests people use their rationality and carefully think about the universe. It is remarkable that, in the Qur’an, we find arguments that rely on the reductio ad absurdum method. For example, let us consider the following verse: “But the fact is that had there been in the heavens and the earth any deities other than God, both (of those realms) would certainly have fallen into ruin. All-Glorified God is, the Lord of the Supreme Throne, in that He is absolutely above all that they attribute to Him” (21:22). This verse proposes that one must have observed disorder in the universe if that person associated partners with God’s activity in the universe. Since this is not what we observe, the initial assumption is incorrect. This example shows that the Qur’an addresses the rationality of human beings, and suggests actively using that faculty.

As we have seen, deriving absurd consequences from an assumption and denying it on the basis of those absurdities is a common and fundamental way of reasoning that is exercised in many different areas and aspects of human life. It is called by different names over history, namely “the Socratic method,” “reductio ad absurdum,” or “proof by contradiction.” Yet, the basic idea behind these technical terms is the same. Most of us probably exercise this method without thinking about it.

Appendix

Theorem: The number that is equal to itself, when added to itself, is zero.

Prove that for any x, x is a number; if x +x = x, then x=0.

1. x+x=x (assumption for conditional/direct proof)
2. x is not equal to 0. (Assumption for indirect proof/reductio ad absurdum)
3. x+x=2x (by addition)
4. x=2x (since “x+x” is common for both the 1^st and 3^rd steps, by the principle of transitivity)
5. 1=2 (cancel x’s) –Contradiction
6. x=0. (The assumption that leads to the contradiction in the 5^th step must be false. Therefore, it is false that x is not equal to 0. Thus, x=0 is true.) Indirect proof is complete.
7. If x+x=x, then x=0. (Conditional proof is complete)

As a relatively new and exciting science, chaos science grew very slowly during its infancy. Yet in the last decade, due to active research in many areas, it has became one of the hottest topics in academia as well as the popular science press. Many books have been published, and millions of Internet pages have been designed full of fractal pictures.1 Given that chaos is associated with disorder or confusion in a system or condition, why does it continue to attract so many people?

History of chaos

To understand chaos, one first has to understand the Newtonian worldview. Sir Isaac Newton’s (1642-1727) development of the calculus and laws of classical mechanics began a scientific revolution in seventeenth-century Europe that caused all subsequent scientists to view nature from a profoundly different perspective. Now that they finally could determine the dynamics of bodies by simple equations, they believed that they had found the ultimate eternal rules that shape the universe.

French physicist Pierre-Simon Laplace (1749-1827), who based his work upon Newton’s work, is credited with the following famous quotation (often referred to as Laplace’s Demon): ‘We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.’2

Laplace’s Demon states the idea of determinism, that the past completely determines the future. One can clearly see why determinism was so attractive to scientists at that time. However, in Laplace’s word everything was predetermined: no chance, no choice, no uncertainty. A solid, inevitable destiny had frozen the events in every corner of the past and is continuously spreading out to the future to do same there. Determinism apparently invokes the idea that whole universe is like a clock. God set it in motion at the beginning of creation and then removed Himself, for everything had been planed before. The ideas that there was no place for free will and that God could not interfere killed the belief in a soul and, consequently, in spirituality. Philosophers and scientists have discussed this for many years. Determinism affected many philosophies and triggered the major ideological movements of during eighteenth century, especially in Europe.

Toward the end of the 1800s, mathematicians and scientists began encountering some very difficult equations, some of which we know today are unsolvable. The most troublesome are various nonlinear differential equations. Even though it looks like such a simple and totally deterministic system, the problem of three bodies attracting each other with purely gravitational forces (e.g., the sun, Earth, and moon triple) turns out to be missing an exact solution. At first, such problems were cast-off as special cases and largely ignored.

One reason for this also might come from the fascinating world of quantum mechanics, which dazzled even the great physicists, and the lack of fast computers at that time. When these equations finally were studied in detail, a fundamental change that would ultimately overthrow determinism began to occur in mathematics and science. An indication of the science that would be come to be known as ‘chaos’ began to appear.

Why does chaos interest people?

In contrast to its common usage, chaos does not actually mean disorder or confusion. By definition, it should occur in well-defined orderly systems. However, most natural physical systems often can exhibit an unpredictable or intractable behavior in the long run, even though the system is defined by clear-cut orderly mechanisms. In this sense, chaos can be defined as unpredictability rather than disorder.

For example, meteorologists use 12 sets of well-defined equations to forecast the weather. They relate such atmospheric parameters as pressure, temperature, flow speed, and time to each other. One can make a computer program that calculates the parameters’ final values by taking any initial conditions as the run’s starting point. In principle, therefore, if we know the initial temperature, pressure, and time values that describe today’s weather conditions, it is possible to derive tomorrow’s weather conditions by running a computer program, which is nothing more than a chain reaction of numerical iterations.

However, in practice, initial conditions cannot be measured exactly and so contain a degree of uncertainty. But since the system’s governing laws are known, one may estimate the effect of errors on future results. Hence, instead of giving the exact results, one can provide an approximate range of possibilities. This range can still be very useful, provided that the deviations do not stray too far from the actual values. In addition, knowing how the error grows in the system might help us understand and control the systems. But if we apply these error estimates to weather forecast equations, we will encounter a large problem, for errors grow exponentially in such systems. Even a tiny deviation at the beginning can create huge deviations from the actual values. It also can provide unrelated or nonsensical results.

An example of chaotic systems

This numerical behavior was first observed by the meteorologist Edward Lorenz, a pioneer in modern chaos work. Fascinated by the results he obtained, in the early 1960s he gave an interesting metaphor to explain the situation of high sensitivity to initial conditions: A butterfly’s slight wing movement (i.e., a little deviation from the initial conditions) can change the future in a way that causes some chain reaction that ultimately result in a hurricane.

Such systems that exhibit a very high sensitivity to initial conditions are called chaotic systems. Chaos comes from the mathematical properties hidden in the equations defining the system, and such unpredictability cannot be removed. Even if the measurements’ quality could be improved by minimizing errors, chaos never disappears from a chaotic system.

One may suppose that chaos occurs in complicated systems, such as weather forecast systems having 12 sets of equations. But even much simpler systems, such as billiards, can exhibit a very high degree chaos. A usual billiard system consists of many balls and a rectangular shaped table. I challenge master billiard players by requesting them to play the game in a stadium-shaped table. I am sure that they will find it difficult to do so, because such billiard tables would be chaotic systems.

If a system is defined as chaotic, this does not necessarily mean that its behavior is totally undefined all the time. As in stadium billiards, a ball has to be inside the billiards, so it should be somewhere on the table, even though sometimes we cannot foretell its exact position because of chaos. Besides, if you send the ball with a velocity perpendicular to a straight side, it will bounce back and forth between the two sides forever. Therefore, depending on which initial conditions are taken, chaotic systems also can show characteristics of regular motion.

A 3-body problem (in general n-body problems) such as the sun, Earth, and moon system, is a chaotic system. But since we can predict the motions of these celestial objects with great precision for many years in the future, why do we call this system chaotic? This triple system possesses a very special set of conditions: distance between bodies, their masses, and their velocities. These parameters cause it to exhibit near-regular behavior. It is analogous to the example of stadium billiards given above, for this triple system bounces back and forth between the table’s sides.

Conditions for chaos

Chaos also can be caused by other factors than just uncertainties measured in the initial conditions. Scientists generally define hypothetical systems by isolating them from the outside world in order to simplify them as much as possible. However, as even objects in the real world that are far apart interact with each other, no system in the real world can be isolated. Given this, a closed (isolated) system might be defined as a fluctuating approximation to its real counterpart, which is changing in an unpredictable manner all the time. In short, even though we would know the exact initial conditions, the actual system could be chaotic due to changes in the approximate system. In this sense, many physical systems have an inclination toward being chaotic.

Due to its maximum complexity, the universe is the largest chaotic system. Observable regular patterns in special parts of that system repeat themselves in time. While the rest of the flows are unpredictable, they are not totally irregular, abrupt, or disordered. Just like whirls in a flowing river, they are in a kind of free motion searching for convenient conditions in which to give birth to organized structures.

The future of chaos

Chaos gives today’s scientist a new worldview, for Newton’s concrete, cold, and deterministic one has been shaken by the uncertainty principle of quantum mechanics. No one ever thought that the Newtonian worldview could be replaced. However, now scientists are more likely to be open to chance and choice than their predecessors. The question is whether chaos theory will cause large revolutions in how we understand the universe. However, the existing excitement, expanding research and growing number of articles, and its numerous applications from economy to biology, seem to indicate that a surprise improvement might not be so far off. ‘

Footnotes

1. Fractal: A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
2. ‘Chaos and Fractals: Laplace’s Demon.’ Online at: www.pha.jhu.edu/ ldb/seminar/laplace.html.