Outer Limits of Reason (7 page)

the real number between 0 and 1 defined by its
n
th digit being 9 minus the
n
th digit of the
n
th Richard phrase.

That is, if the digit is a 5, this phrase will describe a 4. If the digit is an 8, this phrase will describe a 1. And if the digit is a 9, this phrase will describe a 0. This phrase is a legitimate English phrase that precisely describes a number between 0 and 1, yet it is different from every single Richard phrase. The phrase does describe a number if and only if it does not describe a number. What to do?
⁹

These last two paradoxes can be seen as self-referential paradoxes. In a sense, they can be summarized by the following two descriptions:

• “the Berry phrase that is different from all Berry phrases”

• “the Richard phrase that is different from all Richard phrases”

From this point of view, they are simple extensions of the liar paradox. Self-reference is very common and we must be careful with it.

Further Reading

Many of the paradoxes can be found in places such as Quine 1966, Hofstadter 1979, 2007, Barrow 1999, and Poundstone 1989. Sorenson 2003 is a clear and well-written introduction to paradoxes. Chapter 5 of Sainsbury 2007 covers the liar paradox and other forms of self-reference. Chapter 3 of Paulos 1980 provides a humorous look at all self-referential paradoxes. Yablo's paradox is found in Yablo 1993.

A formal version of self-referential paradoxes can be found in Yanofsky 2003, which is derived from Lawvere 1969.

3

Philosophical Conundrums

Moreover, although these opinions appear to follow logically in a dialectical discussion, yet to believe them seems next door to madness when one considers the facts. For indeed no lunatic seems to be so far out of his senses.

—Aristotle (384–322 BC),
On Generation and Corruption
, 325a15

All are lunatics, but he who can analyze his delusion is called a philosopher.

—Ambrose Bierce,
The Collected Works of Ambrose Bierce

It depends on what the meaning of the word “is” is.

—William Jefferson Clinton

Long before modern scientists took up the task of investigating the limits of reason, philosophers were analyzing the complexities of our world and our knowledge of it. In this chapter I explore some of the ancient and contemporary philosophical aspects of reason's limitations.

In
section 3.1
, I begin by discussing some very fundamental questions about concrete and abstract objects and the way we define them. In
section 3.2
, the very nature of space, time, and motion are analyzed using some of Zeno's paradoxes. The section ends with a short discussion of time-travel paradoxes.
Section 3.3
is concerned with vagueness.
Section 3.4
is centered on the very notion of knowing and having information. These sections are independent of each other and of the rest of the chapters. They can be read in any order.

3.1  Ships, People, and Other Objects

In ancient Greece, there was a legendary king named Theseus who supposedly founded the city of Athens. Since he fought many naval battles, the people of Athens dedicated a memorial in his honor by preserving his ship in the port.
¹
This “ship of Theseus” stayed there for hundreds of years. As time went on, some of the wooden planks of Theseus' ship started rotting away. To keep the ship nice and complete, the rotting planks were replaced with new planks made of the same material. Here is the key question: If you replace one of the planks, is it still the same ship of Theseus? This question about a mythical ship is the poster child for one of the most interesting problems in all of philosophy, namely the
problem of identity.
What is a physical object? How do things stay the same even after they change? At what point does an object become different? When we talk about a certain object and say that “it changed,” what exactly is “it”?

What happens if you change two of the ship's planks? Would that make it somehow less of the original ship than after one plank is changed? What if the ship consists of a hundred planks and forty-nine of the planks are changed? How about fifty-one changed planks? What about changing ninety-nine of the hundred planks? Is the single plank at the bottom of the ship enough to maintain the original lofty status of the ship? And what if all of the planks are changed? If the change is gradual, does the ship still maintain its status as the ship of Theseus? How gradual must the change be?

We are not answering these questions simply because there are no objective correct answers. Some maintain that changing one plank changes the ship and makes it no longer the ship of Theseus. Others claim that as long as there is at least one plank from the original, it is still the original. There are also those who maintain that the changed ship is always the same as the original ship because it has the form of the original. None of these different positions are wrong. However, there is no reason to say that any of them are correct either.

Let us continue asking more questions about our beleaguered boat. What happens if we switch the old wooden planks for more modern plastic planks? Then, as we change more and more of the planks, the ship will be made of a different material than the original. What happens if the people who replace the planks make mistakes in putting in the new planks and the ship has a slightly different form? Another question: Does it matter who is making all these changes to the ship—that is, whether one group of workers does it or another? If the ship is to be preserved for hundreds of years, then surely many different people will have to be making the changes. What if we make so many changes to the boat that it can no longer float out to sea? Can we still call it the ship of mighty Theseus if it cannot perform the same function as the original?
²

Such questions go on indefinitely. I will restrain myself and discuss just one more scenario. Imagine that every time a plank is changed, rather than consigning the old planks to the scrap heap, we store them in a warehouse. After some time, all the old planks are assembled into a ship. This new construction is made to look exactly like the old ship with the planks in their original position. Question: Which ship has the right to call itself the ship of Theseus, the ship with the replaced planks or the ship constructed out of the old planks?

A common answer to some of these questions is that the ship remains the same because the changes are gradual. However, it is not clear why that should make a difference. How gradual must the changes be in order for the original ship to maintain its status? Is there a minimum speed limit for changes? To put the question of what is “gradual” in perspective, consider the case of Washington's ax. A certain museum wanted to preserve the ax of the founding father of the United States. The ax consists of two parts: a handle and a head. As time went on, the wooden handle would rot and the metal head would rust. When needed, each of these two parts was replaced. As the years passed, the head was changed four times and the handle was replaced three times. Is it still Washington's ax? Notice that here there is no question of the change being gradual. Every time a change is made, half the parts of the ax are replaced.

Our discussion is not limited to ships and axes. A tree is lush and green in the summer and bare and brown during the winter. Mountains rise and fall. Cars and computers get refurbished. Any physical object changes over time. This is the content of Heraclitus' famous dictum that you cannot step into the same river twice. For Heraclitus, the river changes at every instant.

Physical objects are not the only things that change. Businesses, institutions, and organizations are also dynamic entities that constantly change and evolve. Barings Bank was in existence from 1762 through 1995. In that time, the owners, workers, and customers all changed. The Brooklyn Dodgers have been around since 1883. Their players, managers, owners, and fans have definitely changed. What remains the same about a baseball team? After heartlessly betraying their city of birth, the Dodgers cannot even claim that they play in the same city as they originally did. In colleges, the students change every four years. Even the professors change over the years. The only real heart and soul of a college are the beloved secretaries. But, alas, even they change. Political parties are also not immune to change. The Democratic Party was founded in the 1790s to support states' rights over federal rights, the opposite of their current platform. Everything changes!

We are not only talking about change. Rather, we are discussing what it means for an object to be that object. What does it mean for a certain institution to be that institution? When we say that a certain object changes, we mean that it had a certain property beforehand and after the change it does not. In the beginning, the ship of Theseus had planks that Theseus himself touched. At the end, there were planks that he did not touch. That is a change in the properties of the ship. Our fundamental question is: What are the core properties of the ship of Theseus? We have shown that there are no clear answers to this question.

 

This discussion becomes far more interesting when we stop talking about ancient ships and start talking about human beings. Every person changes over time. We grow from infants to old people. What properties does a three-year-old have in common with their eighty-three-year-old self? These philosophical questions are called the
problems of personal identity
. What are the properties that make up a particular human being? We are not the same person we were several years ago. Nevertheless, we are still considered the same person.

Philosophers usually fall into one of several camps on this question. Some thinkers push the notion that a person is essentially their body. We each have different bodies and can say that every person is identified with their body. By postulating that a human being is their body, we are subject to the same insoluble questions that we faced with the ship of Theseus and other physical objects. Our bodies are in constant flux. Old cells die and new cells are constantly being born. In fact, most of the cells in our body are replaced every seven years. This leads to hundreds of questions that philosophers have posed over the centuries. Why should a person stay in jail after seven years? After all, “he” did not perform the crime. It was someone else. Should a person own anything after seven years? The old person bought it. In what sense is a person the same after having a limb amputated? Science fiction writers are adept at discussing challenging questions like cloning, mind transfers, identical twins, conjoined twins, and other interesting topics related to the notion that a person is the same as their body. When an ameba splits, which is the original and which is the daughter? When your body loses cells it loses atoms. These atoms can go on to belong to others. Similarly, other peoples' atoms can become part of your body. What about death? We usually think in terms of the end of a person's existence when they are dead even though the body is still there. Sometimes we use sentences like “She is buried there” as if “she” were still a person. And sometimes we use sentences like “His body is buried there” as if there is a difference between “him” and his body. In short, it is problematic to say that a human being is identified with their body.

Other thinkers favor the notion that a person is really their mental state or psyche. After all, human beings are not simply their bodies. A person is more than a physical object because there is thought. To such philosophers, a person is a continuous stream of consciousness—they are memories, intentions, thoughts, and desires. This leads us to ask other insoluble questions: What if a person has amnesia? Are they the same person? Doesn't a person's personality change over time? Who is the real you: the one who is madly in love with someone or the one who is bored with the same person two months later? Literally hundreds of questions can be posed about change in a person's thoughts, memories, and desires. Again, philosophers and science fiction writers have become quite adept at describing interesting scenarios that challenge our notion of a human being as a continuous stream of mental states. These scenarios are concerned with Alzheimer's disease, amnesia, personality changes, split-brain experiments, multiple personality disorders, computers as minds, and so on. There are also many questions along the lines of the mind-body problem. How much is the mind—that characterizes a human being—independent of the brain, which is a part of the body?

One of the more interesting challenges to the position that continuity of mental states characterizes a human being is the question of
transitivity of identity
. My mental states are essentially the same as they were ten years ago. That means I am the same person I was ten years ago. Furthermore, ten years ago, my mental states were essentially the same as they were ten years earlier. Hence the person I was ten years ago is the same as the person I was twenty years ago. However, at present, I do not have similar mental states to those I had twenty years ago. So how can it be that I am the same person I was ten years ago, and that person is the same as I was twenty years ago, but I am
not
the same as I was twenty years ago?

Yet another option is that everyone has a unique soul that determines who they are. Avoiding the questions of the definition or existence of a soul, let us concentrate instead on how this answers our question of the essential nature of a human being. Assuming the existence of a soul, what is the relationship between the soul and the body? What is the relationship between a soul and a person's actions, psyche, and personality? If there is no connection, then in what sense is one soul different from another soul? How can you differentiate between souls if they have no influence over any part of you? What would the purpose of a soul be? If, on the other hand, a connection exists, then does the soul change when the body, actions, psyche, or personality changes? Is the soul in flux? If the soul does change, we are back to the same questions we had previously asked: Who is the real you? Are you the one with the soul prior to the change or are you the one with the changed soul?