Outer Limits of Reason (34 page)

The three-body problem and its generalization, the
n
-body problem, is not an abstract problem for absentminded physicists. Rather it occurs all the time. You are held to the Earth because you and the Earth are two bodies that follow Newton's beautiful formula. What happens if there is a pen near you? The pen is being pulled toward the Earth (release it from your grasp and watch it fall to the Earth), and there is a very subtle pull between you and the pen. Is there a simple formula that tells us how the three bodies will interact?

In the late nineteenth and early twentieth centuries, Ernst Heinrich Bruns (1848–1919) and Henri Poincaré showed that there is no simple formula that solves the three-body problem. By “simple” I mean a formula consisting of the usual operations as opposed to infinite sums and integrals. In short, they showed that the three-body problem is essentially unsolvable. That means that the complex relationship between you, the pen, and the Earth is beyond the limits of science.

It can also be shown that the three-body problem is chaotic—that is, it is extremely sensitive to initial conditions. However, such a chaotic system has a somewhat different character than the systems discussed at the beginning of this section. Yes, the three-body problem is deterministic in the sense that the components of the system follow set rules of how to act. Unlike Lorenz's weather formation, for which we can jot down some nice short formulas that describe the motion, no such nice formulas exist for the three-body problem. The system does follow set rules but we cannot easily describe these rules. This is yet a further step away from Newton's dream. Such systems are even further beyond the boundaries of predictability and reason.

Let us take a look at an example to show that the inability to solve the three-body problem is relevant to our world. Consider the Earth and its two largest, most influential neighbors: the sun and the moon. Using observations and Newton's formulas for two bodies, physicists have calculated that the Earth makes a full rotation around the sun in 365.2421897 days. One might argue that it is inappropriate to use Newton's formula for two bodies because the moon is also a player in this drama. In fact, the moon does have an effect on the Earth: the tides are influenced by the moon. However, since the sun is so much larger than the moon, and its influence on the Earth is so great in comparison to the moon's, we might as well ignore the moon when calculating the length of the year.

Contrast that with the calculation of a lunar month. If you investigate how long it takes the moon to go around the Earth you will find statements like “the approximate average length of the lunar month is 29.53 days.” What does “approximate average length” mean? Can't someone tell us exactly how long it takes for the moon to go around the Earth? The answer is no! There are two bodies pulling at the moon: the Earth and the sun. Although the Earth is smaller than the sun, since the Earth is closer to the moon than the sun is, its influence cannot be ignored. This makes the problem of determining the moon's path a three-body problem. Such problems are unsolvable and there is nothing we can do about it. In fact, the lunar month can be up to 15 hours longer or shorter than 29.53 days. Since it cannot be calculated exactly, the average is approximated by keeping track of how many days passed in many months and then calculating the average length of the month. Hindu priests have kept records of this for over three thousand years and have a very exact average. The point is that the simple question of the length of the lunar month is unsolvable because the three-body problem is unsolvable.

It should be noted that modern mathematicians have partially solved the three-body problem and even its generalization, the
n
-body problem. Donald G. Saari, Quidong (Don) Wang, and Zhihong (Jeff) Xia, building on the earlier work of Finnish mathematician Karl Sundman (1873–1949), have produced formulas describing these systems. However, these formulas are immensely complex and not made of a finite number of simple operations. They demand an unreasonably large amount of computation to achieve even partial solutions. So, there are equations, but they are essentially useless for long-term predictions.

Statistical mechanics is a related area of physics that describes systems that are deterministic but not predictable. This branch of physics deals with phenomena such as heat, energy, water flows, and other systems where there are a huge number of components. Each component of these systems follows deterministic laws, but the systems are not predictable as with chaotic systems. Within statistical mechanics the nonpredictability occurs not only from sensitive dependence on initial conditions but from the huge numbers of components of the system. There is no way we can keep track of all the water molecules in a glass of tea. Whether we are dealing with hot air atoms in a combustion engine or water molecules hitting some piece of pollen in a flask, the number of items that we would have to track in order to make exact predictions is simply beyond our capability. In order to deal with such large ensembles, physicists must state their laws for such systems in a probabilistic language. With such laws they have become amazingly adept at predicting the large-scale phenomena of these ensembles of elements. The statistical laws coincide with experimental observations. It must be stressed that each component of the system obeys deterministic laws. Every atom bounces in ways that it is supposed to, like billiard balls on a pool table. Every water molecule hits the pollen in a deterministic manner. However, since there are so many such atoms and molecules, and since we can never know where each is, the laws must be given as probabilities. As previously stated, unpredictability or seeming randomness is simply a subjective result of lack of information. It is an expression of the limitations on our knowledge.

In this section I have concentrated on
practical
barriers to predicting the future of certain systems: there is simply no way we can know (with enough precision) the initial conditions of all the components of certain systems to make a valid calculation of the future. Too much information exists to be knowable. There is, however, a curious little puzzle that shows the inherent
logical
impossibility of perfectly predicting the future. This puzzle is a version of our old familiar friend: it is a self-referential paradox.

Imagine for a moment that we are capable of perfectly predicting the future. The simplest formulation of this puzzle is that we program two computers with this ability. Call one computer Mimic and the other Contrary. Both will simply print out the word
true
or
false
. The Mimic computer will predict what Contrary will print at some specific time in the future and print the same thing. In contrast, the Contrary computer will predict what Mimic will do at that specific time and print the opposite.
⁶
If Mimic will print
true
, then Contrary will print
false
. If Mimic will print
false
, then Contrary will print
true
. This is a paradox. The astute reader will notice that this is nothing more than a simple formulation of the liar paradox that we met in
chapter 2
:

L
₂
: L
₃
is false.

L
₃
: L
₂
is true.

In a similar vein, we can formulate the same puzzle with one computer. Program a single computer to predict what it will do at some specific time in the future and have it do the opposite. In other words, the computer will negate its own prediction. Such a computer would cause a contradiction and hence cannot exist.

This is nothing more than the
crocodile's dilemma
paradox of classical Greek philosophy.
⁷
A crocodile steals a child and the mother of the child begs for the return of her beloved baby. The crocodile responds, “I will return the child if and only if you correctly guess whether or not I will return your child.” The mother cleverly responds that he will keep the child. What is an honest crocodile to do?

In 
section 3.2
we saw that reason does not permit us to change the past. Here we see that reason also restricts us from knowing the future.

Let us summarize the types of physical systems that we have discussed. To us, the most interesting aspect of a system is the amount of human knowledge we can acquire about it.
Figure 7.5
presents a crude hierarchy of such systems and examples that we have mentioned.

Begin with the innermost circle. These are deterministic systems that we know the most about and can predict with ease. They are stable systems. As we have discussed, regular single pendulums and the two-body problem that can be solved with Newton's small formula are examples of such systems. The next level is the central focus of this section, namely chaotic systems. These are also deterministic and we can write simple formulas to describe their short-term behavior, but because they are extremely sensitive to initial conditions, long-term predictions cannot be made. The weather and the double pendulum are just a few of the many examples that we have discussed. Further away from predictability are chaotic systems that do not possess even simple formulas describing the short-term behavior of the system. These systems are still deterministic but because of their complexity and/or the fact that they have a huge number of components, they cannot be described by simple formulas. Quintessential examples of such systems are three-body problems and systems described by statistical mechanics. Finally, stepping outside of deterministic systems, we find random systems. Here, no formula exists that determines the future of a system nor can there be a formula to predict the short-term behavior of the system. The only
⁸
known example of such random behavior is quantum mechanics, covered in the next section.

I conclude with a little meditation on the number of physical phenomena that can and cannot be explained/predicated by science.
⁹
In a sense, language, be it spoken or written, be it natural language or exact formulas, is countably infinite. There is no longest word or longest novel, because there is no limit to the longest formula, and so on. This makes language infinite. However, it can be alphabetized or counted, which makes language countably infinite. In contrast to language, which can be used to describe or predict phenomena, let us examine what is really “out there.” It is plausible to say that there is an uncountably infinite number of phenomena that can occur.
¹⁰
This is stated without proof because I cannot quantify all phenomena. To quantify them, I would have to describe them and I cannot do that without language. So there might be an uncountably infinite number of phenomena and only a small, countably infinite subset describable by science. This is the ultimate, nonscientific (science must stay within the bounds of language) limitation on science's ability. At this point we must take Wittgenstein's dictum to heart: “What we cannot speak about we must pass over in silence.”
¹¹

7.2  Quantum Mechanics

Probably the greatest development in all of physics is quantum mechanics. With the exception of gravity, all physical phenomena are described by this theory. Phenomena ranging from the interactions in an atom to the workings of the sun follow the laws of quantum mechanics. However, quantum mechanics has also taught us that we have a severe limitation when it comes to understanding how the particles of our universe behave. They are extremely mysterious and defy our attempts to make sense of them.

In this section, I discuss some highlights of quantum mechanics and show that our universe is a very strange place indeed. There are ideas and concepts here that are counterintuitive and will blow your mind! Nevertheless, they are all true. It is important to realize that quantum mechanics is not an approximation to a theory. It is the most exact science we know. The weirdness I describe cannot be brushed away. As strange as the results sound, they must be accepted as science and not as science fiction.

Although there are many different, strange, and counterintuitive parts of quantum mechanics, I will show that most of its bizarre features can be understood as consequences of the following intuitive idea:

The Wholeness Postulate:
  The outcome of an experiment depends on the
whole
setup of the experiment.

This makes sense. After all, you would expect that different experiments would yield different outcomes. What is unexpected is the dependency on the entire experiment as opposed to just part of it. I emphasized the word
whole
because, as we will see, most of the strange aspects of quantum mechanics can be understood as simple consequences of what we mean by that word. I will return to this postulate over and over throughout this section.

Rather than getting into the nitty-gritty details behind quantum theory, I will go through several experiments and explain what they tell us about our world. The physical experiments are stressed because we want to emphasize that this is not just some strange theorizing. Rather, we are talking about the real world.