Outer Limits of Reason (45 page)

There is an interesting way to actually measure the curvature of space. Your weight is different depending on whether you measure it on Mount Everest or at sea level. All other things being equal, an increase in altitude from sea level to the top of Mount Everest (29,035 feet) causes a weight decrease of about 0.28 percent. Someone weighing 190 pounds at sea level will weigh 190 × 0.0028 = 0.532 pounds less on top of Mount Everest. Though this might not be enough to notice, it is true nevertheless.
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This can be explained from the Newtonian point of view: since the distance between you and the center of the Earth is greater at the top of Mount Everest, Newton's equation predicts that the force tugging at you will be less. However, one can also see this from the perspective of relativity theory. If you think of the Earth as making an indentation in the very fabric of spacetime, then the top of Mount Everest is farther away from the indentation and closer to the flat part of spacetime. Therefore, your weight on Mount Everest—that is, the tug on you toward Earth—is less than when you are at sea level.

General relativity has been experimentally proved in many other ways since Einstein formulated it. This idea that space, time, length, duration, mass, energy, gravity and so on are all relative notions is a surprising fact about our ability to know our world.

Unifying Quantum Mechanics and Relativity Theory

In the last two sections, we have discussed the two greatest revolutions in our understanding of the universe: quantum mechanics and relativity theory. These two theories are extremely successful, but they contradict each other in several ways:

• For the most part, the two theories deal with different domains. Quantum mechanics deals with objects that are very small, while relativity theory deals with very large objects. However, there are areas where their domains overlap: places called singularities or black holes. In these overlapping areas these two theories give conflicting predictions.

• These theories also reflect different conceptions of the fundamental nature of space and time. For example, the entanglement phenomenon in quantum mechanics seems to show that space is more intertwined with itself than is the notion of space in relativity theory. Also, general relativity uses the fact that space is continuous, while quantum theory considers space and time discrete.

• The laws of classical physics and general relativity are deterministic, whereas the laws of quantum mechanics are nondeterministic.

In short, it can be shown that

quantum mechanics and relativity theory ⇒ contradiction.

As with many paradoxes, this contradiction shows that we are in need of a new paradigm. A new theory is required. This new theory should unify both quantum mechanics and relativity theory. It should make the same predictions as each of the two theories it is replacing. Furthermore, in the domains in which they overlap, this theory should make a single prediction that conforms to observation. It is expected that this theory will provide new conceptions of space, time, matter, and causality.

Although, at present, no such theory is agreed on by everyone, it already has a name:
quantum gravity
. Since this theory will describe both gravity and the quantum mechanical forces, it will be a
Theory of Everything
or a
Grand Unified Theory
. Many different theories are vying for that lofty position. These theories have esoteric names such as
string theory
,
loop quantum gravity
, and
noncommutative geometry
. Each of these different theories has its own counterintuitive properties. At the moment, string theory seems to be the leading contender. However, it is too soon to tell. Any one of them could be the true Theory of Everything. Or, the true theory may not have been developed yet. Perhaps there will never be a Theory of Everything. One thing seems certain about quantum gravity: it will show us that our naive conception of the universe is wrong and that the universe is a far more interesting place than we believed.

Further Reading

Section 7.1

The story of Lorenz, the basics of chaos theory, the butterfly effect, and many other topics are wonderfully covered in Gleick 1987. Chaotic systems are also discussed in chapter 11 of Tavel 2002. Statistical mechanics is treated in chapter 9 of Tavel 2002. Complex systems that have feedback and self-organization are discussed in Waldrop 1992. Turing's role in morphogenesis can be found in Hodges 1983. The three-body problem and the many attempts to find solutions to it are covered in Diacu and Holmes 1996 and Diacu 1996.

Section 7.2

Contrary to popular belief, it is possible to learn the mysteries of quantum theory . . . you just have to find the right book, because some are simpler or more challenging than others. Here is a list of references from easiest to hardest:

• History: Gamow 1966; Gilder 2009 (an interesting, slightly fictionalized history of the notion of entanglement); Guillemin 1968; and Pickering 1984 (a fascinating book on the relationship between group theory and quantum theory)

• Popular expositions: Greene 2004, 2011; Gribbin 1984, 1995; Pagels 1982; Peat 1991; Penrose 1991, 1994, 2005; and Tavel 2002, chap. 10

• Philosophy: Bohr 1935; Bub 1997; Casti 1989, chap. 7; d'Espagnat 1983; Heisenberg 2007; Herbert 1985; Malin 2001; and Wick 1995

• Simple textbooks: Gillespie 1970; Jordan 1986; Scarani 2006; White 1966; and Yanofsky and Mannucci 2008 (chapter 4 has a short, easy exposition of the basic ideas of quantum mechanics)

• “Real” textbooks: Dirac 1986 (an amazingly readable and deep book, but interestingly, there is nothing about entanglement even in the fourth edition); Hannabuss 1997 (a more algebraic treatment); Sakurai 1994 (used in graduate-level physics courses); and Sudbery 1986 (for people familiar with some mathematics)

Our version of Bell's theorem was taken from d'Espagnat 1979. It can also be found in a delightfully clear article by Bell himself, Bell 1981. A more exact formulation can be found in Sakurai 1994, section 3.9.

This is, as far as I know, the first popular exposition of the Kochen-Specker theorem. However, a readable, philosophical, and slightly technical exposition is given in Held 2006. It is formally proved in section II.12 of Manin 2010.

Section 7.3

There are some very popular and readable expositions of relativity theory. Here are some resources from easier to more difficult: Gardner 1997; Greene 2004; Rindler 1969; Schwartz 1989; and Tavel 2002, chaps. 5–8.

The lightning-and-train thought experiment comes from chapters 8 and 9 of Albert Einstein's readable 1920 exposition,
Relativity: The Special and General Theory.

There is a fascinating BBC Horizon documentary titled
How Long Is a Piece of String?
Clips from the documentary are available at
http://www.bbc.co.uk/programmes/b00p1fpc
.

8

Metascientific Perplexities

All logical arguments can be defeated by the simple refusal to reason logically.

—Steven Weinberg
¹

Not only is the universe stranger than we imagine, it is stranger than we can imagine.

—Arthur Stanley Eddington

It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.

—Henri Poincaré

Scientists are not the only ones who deal with the physical world. Philosophers and other researchers are interested in how the universe works and how we learn about it. They are interested not only in what the structure is, but also in why there is structure and how it is described.

Philosophical questions about the relationship between science, the universe, and our mind are dealt with in
section 8.1
.
Section 8.2
discusses the relationship between science and mathematics.
Section 8.3
takes up the question of why the universe seems so perfectly suited for life and rationality.

8.1  Philosophical Limitations of Science

In this section I explore different aspects of the philosophy of science. This large and fascinating branch of philosophy covers the nature of science and the way it progresses. Rather than attempting a detailed survey of the philosophy of science, I simply cherry-pick several topics at the core of the field and examine how they pertain to the limits of science.

The Problem of Induction

One of the major issues in the philosophy of science is the problem of induction. Simply stated, why should one believe that if every swan ever seen is white, then all swans are white? The problem of induction asks what right we have to generalize from our few observations to a universal law. If we observe a phenomenon over and over, why does that mean it is always true? There is no logical reason to come to such a conclusion. It could very well be that pink swans exist and we simply did not see them. There is no logical reason why swans are white and not pink.
²

We use induction every moment of our lives. We turn the light switch on with the expectation that the light will go on. We turn on the shower with the expectation that water and not mud will come out of the faucet. We plan our schedule with the assumption that the sun will rise tomorrow simply because it has happened every morning until now.

In all these different cases, we are making conclusions from a limited set of observations. We have only seen some swans in our lifetime. We have not seen all of them. The sun has risen every morning until now, but we do not know about the future and yet we make predictions about the future. Why do our past experiences give us any reason to think that the future will be the same?

This is not a new problem. More than two hundred years ago, David Hume showed that there is no logical reason why induction should work. One might counter and say that the reason why the light goes on when the light switch is turned on is because the switch causes the circuit to be completed and the bulb must light up when electricity passes through it. Hume would counter that this long chain of reasoning is simply a series of cause-and-effect operations that work only because we assume they work by induction. Each action has in the past caused a particular effect that we assume is going to happen in the future. Hume says that a person who is using induction is making an assumption that the universe is somehow uniform over time. There is no reason to believe this assumption.

Induction is going from observing many single instances to a general rule. Going in the opposite direction—from a general rule to a conclusion about a particular instance—is called
deduction
. If there is a general rule that says that all swans are white, then we can safely conclude that a particular swan is white. In contrast to induction, deduction is a reasonable process. From the statements “All men are mortal” and “Socrates is a man,” it is a logical deduction that “Socrates is mortal.” There is no way one can deny this reasoning. The major problem with deduction is that the general rules usually come from induction.

The problem of induction is at the very core of science. Scientific laws are formulated by looking at phenomena and generalizing them to universal rules we call laws of nature. There are, however, no real reasons why we have the right to come up with such generalizations. The law that Newton gave us that describes the motion of pairs of bodies was not formulated because Newton examined all pairs of bodies in the universe. Rather he formulated the law by using understanding and induction on what he saw. In fact, that law is simply false when applied to
all
pairs of objects. Quantum mechanics has shown us that subatomic particles do not follow Newton's simple law. General relativity has also shown that Newton's laws are not the whole story. We conclude that Newton's laws were formulated with induction and they turned out to be false. They did not work for very small or very large objects, as revealed to us by the physics revolutions of the twentieth century.

These abstract epistemological topics are at the core of the contemporary battle concerning global warming. While most scientists look at the data available and come to the conclusion that human beings are causing the Earth to get hotter, some scientists are not convinced. They say that there is not enough data to arrive at that conclusion. They see other times in history where there were ice ages and thaws. They do not see the current global warming as different from those other ages. Such scientists feel that much more data, perhaps even data that we will never be able to acquire, must be examined before we can come to such a conclusion.

Not only science, but our entire worldview is built from induction. We observe phenomena and formulate theories about the true nature of the world. Every time we close the refrigerator door, we are sure the light goes off, even though we do not see it go off. As Wheeler wrote, “What we call ‘reality' . . . consists of an elaborate papier-mâché construction of imagination and theory fitted in between a few iron posts of observation.”
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Philosophers have formulated different responses to the problem of induction. The most popular response is to agree that induction does not always give absolute truths, but it does give probabilistic truths. If all the swans that have been seen so far are white, it is highly probable that all the swans that exist are white. Furthermore, the more white swans that you see, the more sure we will be that all swans are white. As for the sun rising tomorrow, there are no logical proofs to prove it. However, because the sun has risen every morning until now, the probability that it will also rise tomorrow is very good and you can “bet your bottom dollar that tomorrow there'll be sun!”

Another possible answer to the problem of induction is that while making inductive inferences might not be logical, it is definitely a human activity. In other words, humans have learned over time how to go from particulars to general rules. Not all inductive laws that human beings make are perfectly true, but many are true. While it might not be strictly reason, it is nevertheless a justified human activity.