Outer Limits of Reason (32 page)

Along the same lines as Deep Blue is Watson. In 2011, IBM had one of their language-recognition computers named Watson play against humans in the television game show
Jeopardy
. The computer decidedly beat the humans. Rather than asking if computers can reach the level of human beings, perhaps we should ask if computers have already surpassed humans. After all, a typical personal computer can now beat 99.99 percent of humans at chess. As many companies have shown, computers are much more efficient at answering phone calls and handling other tasks once performed only by people. Rather than seeing this as a decline of the status of man, see this as a triumph of man's ingenuity. Man has programmed machines to transcend their limitations.

This section contains more questions than answers. For most of these questions, this humble scribe resists the urge to push any purported answers. The questions are enjoyable enough.

Further Reading

Sections 6.1–6.3

The undecidability of the Halting Problem and other problems mentioned in
section 6.3
can be found in detail in many theoretical computer science books—for example, Cutland 1980, Davis, Sigal, and Weyuker 1994, Sipser 2005, and Sudkamp 2006, to name a few. Rice's theorem can be found in his original paper, Rice 1953.

Davis 1980 and Harel 2003 provide popular accounts of undecidable results.

Section 6.4

Oracle computation can be found in the books listed above. Baker- Gill-Solovay 1975 and Hartmanis-Hopcroft 1976 are the original papers with the results described at the end of
section 6.4
.

Many of the ideas described in sections
6.2
through
6.4
emerged from the mind of Alan Turing. The life of this genius is wonderfully described in Hodges 1992. This is a very interesting scientific biography and is well worth the read.

Section 6.5

The literature on the questions posed in
section 6.5
is immense. Some of the many books and articles that discuss these issues are Rucker 1982, Hofstadter 1979 (which won the Pulitzer Prize), and a most intriguing new book, Hofstadter 2007. Chapter 6 of Wang 1996 contains an in-depth philosophical discussion of Gödel's beliefs. Penrose 1991 and 1994 present his arguments and are full of interesting topics. On the Web, there is also an immense amount of literature both supporting and opposing Penrose's arguments.

Shainberg 1989 is a funny novel written by a neurosurgeon about a neurosurgeon performing a neurological operation on his own brain. The self-reference can drive you to . . . neurosis.

7

Scientific Limitations

Reason rules the world.

—Anaxagoras (500–428 BC)

Twas brillig, and the slithy toves

Did gyre and gimble in the wabe

—Lewis Carroll (1832–1898)

But you can travel on for ten thousand

miles, and still stay where you are.

—Harry Chapin (1942–1981),
W*O*L*D

Science is the exact reasoning that we use to make sense of the physical world we inhabit. We use science to describe, understand, and sometimes predict physical phenomena. The limitations of the scientific endeavor are, in a sense, the most interesting.

I start with a short discussion of chaos theory and science's ability to predict the future. In
section 7.2
, I describe several different experiments in quantum mechanics that demonstrate the strangeness of our universe.
Section 7.3
gives some intuition about relativity theory and what it tells us about space, time, and causality.

7.1  Chaos and Cosmos

Henri Poincaré (1854–1912) tells a morbid tale of a man walking down a street and getting killed because a roofer accidently drops a tile.
¹
Had the man been there a few seconds later or earlier, he might have lived for many years. Had the roofer not dropped the tile or done so a fraction of a second earlier or later, the man would have continued his stroll through life.

What is the moral of the story? An obvious moral is that bad things happen in this world of happenstance. But that is not exactly a balanced judgment. Good things also happen. The vast majority of falling tiles do not hit anyone. The walking man could also have become a mass murderer. In that case, the falling tile is a good thing. Rather, the correct moral one should derive from the story is that small changes in events at one time can cause major changes at a later time. Had the man lingered home with his wife and kids for a few more seconds, he might have lived to play with his grandchildren. Had he walked a little quicker, he could have become a philanthropist who helped many people. Had the roofer's fingers been slightly less moist, the potential mass murderer below might have grown into a full-fledged mass murderer. One can imagine many variations of this shopworn tale for which the outcome would be totally different.

This obvious fact, that slight changes can cause major unpredictable changes, is known to everyone. If you had chosen a 42 instead of a 43 on that lottery ticket . . . If that death row pardon had just occurred two minutes earlier . . .
²
If only that bungee cord had been a little stronger . . . The reason why this fact is obvious to us is that we all live in a big, complicated world, and we know that there are so many things affecting every action that it is impossible to predict the future. But what about small systems in which we can perfectly describe how the different parts interact? Such small systems are described and investigated by scientists. One would believe that in such small systems, we would be able to predict the future. In this section, we will see that even in certain small describable systems, small changes can cause major changes.

Since the time of Newton, our vision of the universe has been that of a large, flawlessly functioning clock. We envision gears and springs that interact perfectly and with total predictability. It was the job of science to understand this functioning and to predict how this clock would continue over time. From Newton on, as the laws of physics were being formulated, there was optimism concerning our ability to know the entire universe. This optimism is expressed by one of the pioneers of mathematics and physics, Pierre-Simon Laplace (1749–1827), who wrote:

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 a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
³

Laplace and others believed that this progress would continue forever and that eventually every scientific problem would be solvable and the future would be exposed before everyone's eyes. One can simply sit down with the right physical laws and calculate anything. However, by the beginning of the twentieth century, this optimism seemed unjustifiable. Poincaré and others had discovered systems for which it is humanly impossible to predict the future. Such unpredictable systems are called
chaotic
.

In 1961, Edward Lorenz, who was both a mathematician and a meteorologist, was studying computer simulations of weather patterns. He found a few simple equations that could describe certain weather patterns. Lorenz plugged these equations into a computer and studied the outcomes, which were very similar to common weather patterns that can be found in the real world. One day he wanted to review a certain simulation that he had previously seen. Rather than starting the entire simulation from the beginning, he attempted to start the simulation somewhere in the middle. The computer was dealing with numbers with six decimal places. However, to save space, it was outputting numbers with only three decimal places. Instead of typing in 0.506127, he entered the number 0.506. Thinking that the difference is less than one part in a thousand, he expected to get the same weather pattern. To his shock and amazement, the weather pattern that emerged was totally different from the one he intended. Lorenz realized that for these simple equations, the way the different parts interacted with each other, and the way outcomes of some of the equations became inputs to other equations, caused major changes in weather patterns depending on starting positions. In other words, an ever-so-slight change in the initial conditions of the equations can radically alter the rest of the simulation. In the real world, this means that a slight change in a weather pattern now can cause a major change later.

After exploring this, Lorenz went on to write a paper on this phenomenon with the colorful title “Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?” The title implies that a small change to the weather caused by a butterfly flapping its wings might cause a severe weather pattern far, far away. Not that the butterfly really causes the tornado, but rather, the flapping of its wings means that there will be a totally different weather pattern. The flapping might send an impending tornado off course and away from Texas. People can never track all butterflies and hence will never be able to predict the weather.

This effect has come to be known as the “butterfly effect” and has entered popular consciousness. The more scientific way of stating this is that the system shows “sensitive dependence on initial conditions”—that is, small changes in the initial setup of the system can cause major changes in the outcome. Systems that have this property are said to be “chaotic.” The word
chaos
is from the Greek word for “gap,” “lacking order,” or “disorder.” In contrast, the word
cosmos
comes from the Greek word for “order.”

The opposite of chaotic systems—that is, systems that are not so sensitive to initial conditions—are called
stable systems
or
integrable systems
.
Figure 7.1
provides a nice way of looking at the difference between such systems. The left-hand diagram shows a stable system with four different points that start and end near each other. In contrast, the chaotic system on the right has four points that start near each other but end in wildly different places.

Once a system is known to be chaotic, we lose the ability to make any long-term predictions about it. There is no way in the world that anyone can keep track of all the flapping butterflies in Brazil. We cannot retain information about a system if it demands infinite precision. Yes, the system is deterministic and we can write equations and formulas that describe its actions, but we cannot use these equations and formulas to predict any long-term outcomes. Chaotic systems force us to make a distinction between determinism and predictability. Determinism is a fact about the existence of laws of nature, whereas predictability is about the ability of human beings to know the future.

Researchers have gone on to show that many systems besides the weather are also chaotic. For example,

• Economists have determined that prices and the stock markets are dependent on small fluctuations.

• Biologists who study population dynamics have found that the rise and fall of certain populations of species are very sensitive to minor effects.

• Epidemiologists have found that the spread of certain illnesses can be affected by small factors such as a particular individual contracting the disease.

• Physicists studying simple systems of fluids have identified chaotic processes.

It can be shown that in all of these different systems there are physical processes that are deterministic but are not predictable.

It's not hard to make one of the simplest chaotic systems in your own toolshed. A pendulum is a rigid rod with a weight on one end. If you permit the rod to swing back and forth, you get a stable system that follows the simple laws of physics. First-year undergraduates spend much time calculating everything there is to know about such a pendulum. However, if you couple one pendulum to the bottom of another and let both rods swing, you get a double pendulum. This very simple system is totally chaotic. Both pendulums will swing in strange and unpredictable ways. The swinging is totally deterministic—that is, a physicist can write equations describing the motion of a double pendulum. The equations will take into account the lengths of both rods, the two weights, and the starting angles. Nevertheless, the system will be chaotic and unpredictable. To appreciate the butterfly effect, start this pendulum at some position and watch the way it swings. Then, attempt to replicate this motion from almost the exact same starting position—you'll see it swing in a very different way. Infinite precision is necessary to place such a pendulum in the same exact position twice. If you're not good with tools or too lazy to make your own double pendulum, you can find lots of cool videos of such contraptions online.