Outer Limits of Reason (10 page)

This paradox can also be solved if we introduce discrete ideas into the mix. But rather than saying that space is discrete, here we say that time is discrete. At each separate point in time there is no motion. But time leaps from one separated point to another and motion happens at that instant leap. In other words, say that time is discrete and not continuous. We do not see these magical leaps for the same reason we think we see continuous motion when we are watching a movie. In fact, a movie is made out of numerous discrete frames and there is no motion between them. Because the separate time points are so close to each other and there are so many, there is an illusion of continuity.

This paradox basically describes the following derivation:

Time is continuous ⇒ movement is impossible.

Once more, since it is an obvious fact that movement is possible, we conclude that time is not continuous but is discrete.

There is also a mathematical analogy to this paradox. Consider the real-number line. Think of the real line as time. Each point of the real line corresponds to a “now.” And yet each “now” has no thickness. In ninth grade you learned that the real line is made of an infinite number of points. Each point has zero length. So how can a finite line be made of points that do not have any thickness? Zero times anything is zero. Was your ninth-grade teacher lying to you? Does the real-number line make sense? Should we abandon it?

Again, there is a problem abandoning the notion of continuous time for discrete time. Modern physics and engineering are based on the fact that time is continuous. All the equations have a continuous-time variable usually denoted by
t
. And yet, as Zeno has shown us, the notion of continuous time is illogical.

The fourth and final paradox against motion is the
stadium paradox
. Zeno wants us to imagine three marching bands as in
figure 3.5
.

The A's are standing still, and behind them, the B's and C's are moving in opposite directions at the same speed. After some time, the marching bands will look as they do in
figure 3.6
.

Notice that in the same time period, the leading B passed two A's and four C's. Since the A's and C's are the same size, how can the B's pass different numbers of each? The obvious answer, and the reason that Aristotle dismisses this paradox out of hand, is that the A's are standing still while the C's are moving. There is a difference between velocity and relative velocity. We are used to such distinctions when driving or riding in a car and seeing how fast the houses pass as opposed to how fast the cars moving in the opposite direction pass.

Perhaps we should not be so dismissive of Parmenides' faithful student. There is really no way of ascertaining what Zeno's original intention was since we only have a brief discussion from Aristotle. Modern thinkers have postulated a scenario that is a bit more sophisticated. In the previous three paradoxes, we saw that our problems would evaporate if we thought of space and time as discrete or quantized. In this final paradox, let us assume that both space and time are discrete. Think of the members of the marching bands as having the smallest possible discrete size. At the same time, imagine that the B's are moving at the fastest possible speed. Two time clicks are needed for the B's to get from
figure 3.5
to
figure 3.6
. At this speed, each of the B's crosses one box per time click. This is the smallest possible discrete time period for the B's to pass in front of both A's. How can it be that in this smallest time period, the B's passed twice as many C's? That means that the B's are passing the C's at an even faster rate. What would this look like to a member of the B marching band? It would appear to the B's that the C's are skipping boxes or are going faster than permitted. This reading of the stadium paradox demonstrates that the assumption that space and time are discrete is also problematic.

Space and time are discrete ⇒ false fact.

We must conclude that space and time are not discrete.

Which is it? Are space and time continuous or discrete? On the one hand, the stadium paradox is pointing toward space and time being continuous. On the other hand, the first three paradoxes would be solved if we thought of space and time as discrete. The answer is that we simply do not know the nature of space and time.

This conflict is a microcosm of a battle in contemporary physics. The two great achievements of twentieth-century physics are relativity theory and quantum theory.
¹¹
These two revolutionary sciences essentially describe most of the phenomena in the physical universe. Relativity theory deals with gravity and large objects, while quantum theory deals with the other forces and small objects. However, these two theories are in conflict with each other. One of the main reasons for their conflict is that relativity theory considers space and time to be continuous while quantum theory believes space and time to be discrete. For the most part, since the theories deal with different realms, the conflict does not bother us. Nonetheless, the conflict is apparent with certain phenomena such as black holes, which are termed the “edge of space.” Since we cannot have conflicting physical theories, it must be that we do not know the final story. The jury is still out regarding the structure of space and time.

The most amazing aspect of Zeno's paradoxes is that they are 2,500 years old and they deal with such simple topics. What is the nature of space, time, and motion? It is doubtful that we have heard the last of our Elean friend.

 

Since we are discussing the relationship of space, time, and logic, let us talk about time-travel paradoxes. We first have to ask ourselves what it means to travel back
¹²
through time. What would it mean for me to go back to the Continental Congress held in Philadelphia in 1776, in order to witness the signing of the Declaration of Independence? If I am miraculously transported back there and see the signing, then the very fact that I am in the room on that hot day in July means that it is not the original Continental Congress. After all, I was not there during the original. In other words, if there were 150 people present at the original Continental Congress, when I go back there will be 151 people present. That is not the original. It is a major difference between what I was transported to and the original. What exactly am I being transported to? One thing is certain: not to the Continental Congress of 1776.
¹³
This conundrum shows how hard it is to understand the very basic concepts of time travel.

Be that as it may, let us imagine for a moment that we understood what traveling through time actually means, and furthermore, let us imagine that such a process was, in fact, possible. If time travel was possible, a time traveler might go back in time and shoot his bachelor grandfather, ensuring that the time traveler was never born. If he was never born, then he could not have shot his grandfather. Homicidal behavior is not necessary to achieve such paradoxical results. The time traveler might just ensure that his parents never have children,
¹⁴
or he might simply go back in time and make sure that he does not enter the time machine. These actions would entail a contradiction and hence cannot happen. The time traveler should not shoot his own grandfather (moral reasons notwithstanding) because if he shoots his own grandfather, he will not exist and will not be able to travel back in time to shoot his own grandfather. So by performing an action he is ensuring that the action cannot be performed. The event is self-referential. Usually, one event affects other events, but here an event affects itself. In the language of
chapter 1
, we are showing that

Time travel ⇒ contradiction.

Since the universe does not permit contradictions, we must somehow avoid this paradox. Either time travel is impossible, or even if it was possible, one would still not be able to cause a contradiction by killing an earlier version of oneself. Which impossibility should we prefer?

Albert Einstein's theory of relativity tells us that the usual way that we conceive the universe makes time travel impossible. In 1949, Einstein's friend and Princeton neighbor, Kurt Gödel, did some moonlighting as a physicist and wrote a paper on relativity theory. Gödel constructed a mathematical way of looking at the universe in which time travel would be possible. In this “Gödel universe,” it would be very hard, but not impossible, to travel back in time. Gödel, the greatest logician since Aristotle, was well aware of the logical problems of time travel. The mathematician and writer Rudy Rucker tells of an interview with Gödel in which Rucker asks about the time-travel paradoxes. The relevant passage is worth quoting: “Time-travel is possible, but no person will ever manage to kill his past self.' Gödel laughed his laugh then, and concluded, ‘The
a priori
is greatly neglected. Logic is very powerful.”
¹⁵
Gödel replies that the universe simply will not allow you to kill your past self. Just as the barber paradox shows that certain villages with strict rules cannot exist, so too the physical universe will not allow you to perform an action that will cause a contradiction.

This leads us to even more mind-blowing questions. What would happen if someone took a gun back in time to shoot an earlier version of himself? How will the universe stop him? Will he not have the free will to perform the dastardly deed? Will the gun fail to shoot? If the bullet fires and is properly aimed, will the bullet stop short of his body? It is indeed bewildering to live in a world that does not permit contradictions.

3.3  Bald Men, Heaps, and Vagueness

At what point does a man lose enough hair that he is considered bald? Do we have to be able to see his scalp? What if his hair is long but thin? Does that make a difference? When is someone considered tall? Is there a difference between a “pile” of toys and a “heap” of toys? Is that color red or maroon? All these questions are based on concepts that are somewhat vague. There does not seem to be universal agreement on when someone is bald and when someone is not bald. Nor is there a generally agreed on use of the terms
tall
and
short
. Even your interior decorator might have a hard time distinguishing dark red from maroon. In this section I explore the pervasive element of vagueness in our language and thought.

One of our core ways of describing limitations of reason is by finding contradictions. As I stressed in
chapter 1
, there are no contradictions in the physical universe. In contrast to the physical universe, in human language and thought there can be contradictions. Humans are not perfect beings. Our language and thought are rife with contradictory statements and beliefs. When we want to reason and talk about the physical world we must ensure that our language and thought do not have contradictions. There are, however, times when we are ostensibly thinking about or discussing the physical world and our meaning is not clear. This happens when there is vagueness. In contrast to contradictions where a statement is both true and false, a vague statement can be thought of as neither true nor false.

Vagueness is applied to terms that are not always perfectly defined. For example, a five-year-old is clearly a child. In contrast, a twenty-five-year-old is definitely not a child. At what point is a person no longer considered a child? There are
borderline cases
where someone is neither a child nor older than a child. Such terms with borderline cases are vague. Other terms with borderline cases are
tall
,
smart
, and
red
. Where does red end and maroon begin? How about scarlet, cardinal, crimson, cherry, puce, pink, ruby, and fuchsia?
¹⁶

One must make a distinction between vague statements and
ambiguous statements
. An ambiguous statement is one in which the subject of the statement is unclear. For example, “Jack is above six feet” is ambiguous since you do not know which Jack is being discussed. Jack Baxter is above six feet, but Jack Miller is below six feet. However, this statement is not vague since six feet is an exact amount. Of course, we can make a statement that is both vague and ambiguous: “Jack is tall.”