Outer Limits of Reason (12 page)

This question was posed to Marilyn vos Savant, who wrote a special puzzle column in
Parade Magazine
. Vos Savant recommended that you switch doors. She said it was more likely for the car to be behind the other unopened door than the one you originally picked. If you thought that there was no reason to switch, do not feel bad: you are in good company. Her answer generated over 10,000 letters from readers telling her she was wrong. Included were over 1,000 letters from people who identified themselves as PhDs. The article and the letters made such a big impression that the story made it onto the front page of the
New York Times
.

To see why you should change doors, let us look at all the possibilities, as in
figure 3.7
.

Assume Monty placed the car behind the third door. You have three possible doors to choose from. The three choices are depicted with the three rows. The left column shows what would happen if you stay with your original choice and the right column shows what would happen if you switch. Using the staying strategy gets you the car one out of three times, while the switching strategy has you winning two out of three times. You should indeed switch.

What's going on here? Why does switching help? The answer is that when Monty Hall opens the other door, he is giving you more information. Monty knows where the car is and is not going to open the door with the car. By avoiding the other door he is giving information that the other door was avoided. When he gives you information, the probabilities of what is behind each door change.

The way to see this more clearly is by imagining that Monty presents twenty-five doors to you and tells you that the car is behind one of the doors and there are goats behind the other twenty-four doors. You choose one of the doors and then Monty proceeds to open twenty-three other doors. Each door he opens reveals a goat, as in
figure 3.8
.

Now there are two doors that Monty did not open: the one you chose and one other that he avoided. It could very well be that (a) the one you chose is the one with the car (a 1-out-of-25 chance) and that Monty is simply bluffing and hoping you change. Or it could be that (b) you picked a door with a goat behind it (a 24-out-of-25 chance) and since Monty knows where the car is, he is not going to open that door. It is obvious that you should switch. Here Monty is subtly giving you information about the whereabouts of the car
by not telling you where the car is
.

Here is an interesting scenario to think about. Imagine that Monty himself does not know where the car is. Then he will be randomly opening doors. He might accidentally open the door with the car in it and the game is over. But if he does not accidentally open the door with the car, then should you switch? Answer: nope! There is nothing to gain. You should switch only when you know that Monty knows, and he is subtly giving you the information.

This is just one of the strange aspects of knowledge and information that we explore in this section.

Probably the simplest paradox about knowledge is the cousin of the famous liar paradox that we met in the last chapter. Simply hold the following idea in your head:

This idea is false.

As with the liar paradox, this idea is false if and only if it is true. This self-referential paradox also has many variants. For example, on Tuesday you can have the idea that while today you cannot think straight,

Tomorrow, all my ideas will be clear and true.

Then, on Wednesday, you can realize that

All my thoughts of yesterday were false.

Question: Was Tuesday's thought true or false? A short argument following the implications will show that Tuesday's idea was true if and only if it was false.

One possible solution to this is that the human mind is full of contradictions. As I mentioned in
chapter 1
, the human mind is not a perfect machine and has conflicting ideas. A little introspection will show that we all believe ideas that contradict each other.

One of the more interesting paradoxes about knowledge is called the
surprise-test paradox
. A teacher announces that there will be a surprise test in the forthcoming week. The last day of class is Friday of that week. What day can the surprise test happen on? If the test is going to be on Friday, then after school on Thursday night the students will already know that the test is on Friday and it will not be a surprise test. So the test cannot happen on Friday. Since this was purely logical reasoning, everyone knows this. Can the test be on Thursday? After class on Wednesday night the students can deduce that since the test has not happened already and it cannot be on Friday, it must be on Thursday. But again, since they know that it must be on Thursday, it will no longer be a surprise test. So the test cannot occur on Thursday or Friday. We can continue reasoning in the same way and conclude that the test cannot happen on Wednesday, Tuesday, or Monday. When exactly will this surprise test occur?

Logic has shown us that a teacher cannot give a surprise test within a given time interval. This is a paradox because it goes against the obvious fact that teachers have been torturing students with surprise tests for millennia.

It is interesting to note that the paradox would not arise if the teacher just remained silent. The problems only arise because of the teacher announcing to the students that there will be a surprise test. The instant the students are told of the surprise test, they must hold the two contradictory thoughts simultaneously: there will be a surprise test and there cannot be a surprise test.

 

In 2006, Adam Brandenburger and Jerome Keisler published a groundbreaking paper about the very nature of reason and beliefs. When playing a game of chess, you must play rationally and take into account the position of the pieces on the board. You also must take into account that your opponent is rational. Realize that just as you are going to make a rational move, so too will your opponent see what move you make and similarly make a rational move. Your opponent also takes into account that you are rational and she knows that you know she is rational. This goes back and forth and happens anytime there are strategies involved (as in
figure 3.9
). There are, however, problems with such scenarios. The ability of beliefs to deal with themselves will cause a self-referential paradox and hence a type of limitation.

A simple example has come to be known as the
Brandenburger-Keisler paradox
. It is a type of two-person liar paradox. Imagine Ann and Bob thinking about each other's thoughts. Now consider the situation described by these two lines:

Ann believes that Bob assumes that

Ann believes that Bob's assumption is wrong.

Pose the following question:

Does Ann believe that Bob's assumption is wrong?

If you answer yes, then you are agreeing with the second line. The first line says that Ann believes that this assumption is
correct
and not
wrong
. Hence the answer is no. Let us try the other way: the answer to the question is no. It is not the case that Ann believes that Bob's assumption is wrong. Therefore Ann believes Bob's assumption is correct. That is, the second line, which says
Ann believes that Bob's assumption is wrong
, is true. So the answer must have been yes. This is a contradiction.

Brandenburger and Keisler take such ideas and go much farther. Their revolutionary work proceeds to show that there will be limitations or “holes” in any type of game where two players reason about each other. That is, there will be situations where contradictions can happen.

Further Reading

Section 3.1

The section on the ship of Theseus, the problem of identity, and the problem of personal identity was mostly motivated by very passionate classroom discussions with my students at Brooklyn College and by reading too much David Hume. Unger 1979 comes to similar conclusions from a slightly different perspective.

Section 3.2

Zeno's motion paradoxes can be found in book VI of Aristotle's
Physics
. My discussion benefited greatly from the following publications: Grünbaum 1955, Huggett 2010, Makim 1998, Vlastos 1972, and especially Glazebrook 2001. There are also many wonderful papers in Salmon 1972. Chapter 1 of Sainbury 2007 has a nice exposition as well. Mazur 2007 is a popular history book on Zeno's paradoxes.

The discussion of Gödel's take on time travel can be found in Rucker 1982. In Yanofsky 2003, I show that the time-traveler paradoxes can be put into the same scheme as all other self-referential paradoxes.

Section 3.3

Sorensen 2001 is an important work on the general concept of vagueness. Chapter 2 of Sainsbury 2007 covers some of the same material. More on dialetheism and paraconsistent logic can be found in the works of Graham Priest, such as Priest 2003. Parikh 1994 is an interesting discussion of vague terms that is worth studying.

Section 3.4

The magazine article that made the Monty Hall problem famous was in
Parade Magazine
, September 9, 1990, 16. The front-page
New York Times
article (July 21, 1991) was by John Tierney: “Behind Monty Hall's Doors: Puzzle, Debate and Answer?”

You can read about the surprise-test paradox and many other epistemic paradoxes in Sorensen 2006. The Brandenburger-Keisler paradox and much more can be found in Brandenburger and Keisler 2006.

4

Infinity Puzzles

The last function of reason is to recognize that there is an infinity of things which are beyond it. It is but feeble if it does not see so far as to know this.
¹

—Blaise Pascal (1623–1662)

To infinity and beyond!

—Buzz Lightyear,
Toy Story
(1995)

There's an infinite number of monkeys outside who want to talk to us about this script for “Hamlet” they've worked out.

—Douglas Adams (1952–2001),
The Hitchhiker's Guide to the Galaxy

Since ancient times, people have contemplated the infinite and its properties. For most of that time, our thoughts on the infinite were mired in strange ideas that could not withstand the rigor of exact reasoning. With such confusion, the medievals would endlessly discuss inane questions, like “How many angels can dance on the head of a pin?” In the late nineteenth century, Georg Cantor (1845–1918) and several associates were finally able to grab ahold of this slippery topic and make some progress. However, the new science of infinity has many counterintuitive concepts that are challenging to our intuition.

It is important to realize that ideas about infinity are not abstract scholastic thoughts that plague absentminded professors in the ivy-covered towers of academia. Rather, all of calculus is based on the modern notions of infinity mentioned in this chapter. Calculus, in turn, is the basis of all of the modern mathematics, physics, and engineering that make our advanced technological civilization possible. The reason the counterintuitive ideas of infinity are central to modern science is that they work. We cannot simply ignore them.