Outer Limits of Reason (3 page)

• A statement or tenet contrary to received opinion or belief. (For example, “Second-hand smoke is not so bad for you.” “Democracy is not always the best form of government.”)

• An apparently absurd or self-contradictory statement or proposition, or a strongly counter-intuitive one, which investigation, analysis or explanation may nevertheless prove to be well-founded or true. (For example, “In the long-run, the stock market is a bad place to invest.” “Standing is more strenuous than walking.”)

To us, the most important definition will be

• An argument, based on (apparently) acceptable premises and using (apparently) valid reasoning, which leads to a conclusion that is against sense, logically unacceptable, or self-contradictory.

Such paradoxes will be our main concern. Here one has a premise or makes an assumption and using valid logic derives a falsehood. We might envision this paradox or derivation as

assumption ⇒ falsehood.

Since falsehoods cannot occur and since our derivation followed valid logic, the only conclusion is that our assumption was not true. In a way, the paradox is a test to see if an assumption is a legitimate addition to reason. If one can use valid reason and the assumption to derive a falsehood, then the assumption is wrong. The paradox shows that we have stepped beyond the boundaries of reason. A paradox in this sense is a pointer to an incorrect view. It points to the fact that the assumption is wrong. Since the assumption is wrong, it cannot be added to reason. This is a limitation of reason.

The type of falsehood that we will mostly encounter is a contradiction. By a contradiction I mean a fact that is shown to be both true and false. This is written as

assumption ⇒ contradiction.

Since the universe does not have contradictions, there must be something wrong with the assumption. For example, in
chapter 6
, we will see that if we assume that a computer can perform a certain task, then we can derive a contradiction about certain computers. Since there are no contradictions about physical objects like computers, there must be something wrong with our assumption.

Such paradoxes work the same way as a commonly found mathematical proof. A “proof by contradiction” or in Latin,
reductio ad absurdum
(“reduction to the absurd”), is as follows. If you want to show that some statement is true, simply assume that the statement is false and derive a contradiction:

statement is false ⇒ contradiction.

Since contradictions are not permitted in the exact world of mathematical reasoning, it must be that the assumption was incorrect, and the statement is, in fact, true. A simple example is the mathematical proof that the square root of 2 is not a rational number (
section 9.1
). If we assume that the square root of 2
is
a rational number, then we derive a contradiction. From this we conclude that the square root of 2 is not a rational number. In
section 4.3
I show that if we assume two particular sets are the same size, we can derive a contradiction. From this we conclude that one of the sets is larger than the other. Proofs by contradiction are ubiquitous.

One need not derive a full-fledged contradiction for a paradox. All that is needed is to derive a fact that is different from observation or simply false:

assumption ⇒ false fact.

Once again, because we derived something false, our assumption must be in error. Zeno's paradoxes are examples of this type (
section 3.2
). Zeno assumes something and then proceeds to show that movement is impossible. Anyone who has ever walked down the street knows that movement occurs all the time and hence the assumption is false. The difficulty with Zeno's paradoxes is to identify the bad assumptions.

Many times paradoxes arise and highlight previously hidden assumptions. It could be that these assumptions are so deep within us that we do not even consider them (for example, that space is continuous and not discrete, or that physical objects have exact definitions). Such paradoxes will be a challenge to our intuitions about the universe we live in. By showing that our intuitions are false, we can disregard them and be propelled forward. The American philosopher Willard Van Orman Quine (1908–2000) eloquently wrote:

The argument that sustains a paradox may expose the absurdity of a buried premise or of some preconception previously reckoned as central to physical theory, to mathematics or to the thinking process. Catastrophe may lurk, therefore, in the most innocent-seeming paradox. More than once in history the discovery of paradox has been the occasion for major reconstruction at the foundation of thought.
⁵

This method of exploring paradoxes and looking for their assumptions will be one of our focuses throughout the book.

Particular types of paradoxes play a major role in the tale we tell. Self-referential paradoxes are paradoxical situations that come from a system where the objects of the system can deal with / handle / manipulate themselves. The classic example of a self-referential paradox is the so-called
liar paradox
. Consider the English sentence:

“This sentence is false.”

If the sentence is true, then the sentence is, in fact, false because it says so. If the sentence is false, then since the sentence expresses its own falsehood, the sentence is true. This is a genuine contradiction. The problem arises from the fact that English sentences have the ability to describe true and false statements about themselves. For example, “This sentence has five words” is a legitimate English sentence that expresses something true about itself. In contrast, “This sentence has six words” is a false statement about itself. We will see that whenever a system can discuss properties about itself, a paradoxical situation can occur. We will find that language, thought, sets, logic, math, and computers are all systems with the ability to deal with themselves. Within each of these areas, the potential for self-reference will lead to paradoxes and hence some type of limitation. The amazing fact is that although these areas are very different, the form of the paradoxes are the same.

Another method of describing a limitation is by piggybacking on an already established limitation. Before I explain what this is all about, let's discuss some mountain climbing. Mount Everest is 29,000 feet high and Mount McKinley is “only” 20,000 feet high. The following fact seems obvious: if you can climb Mount Everest, then you can most definitely (
a fortiori
) climb Mount McKinley. We write this as

climbing Everest ⇒ climbing McKinley.

If you are able to climb Mount McKinley, you would feel great pride. We write this as

climbing McKinley ⇒ pride.

Putting the two implications together, we get

climbing Everest ⇒ climbing McKinley ⇒ pride,

which leads to the obvious conclusion that if you are able to climb Mount Everest, you would feel great pride. Now let us look at the dark side of mountain climbing. Suppose your doctor told you that bad things might happen to you if you try to climb Mount McKinley. We write this as

climbing McKinley ⇒ bad.

This is expressing a limitation of your abilities: you should not climb Mount McKinley. Combining this implication with the first one, gives us

climbing Everest ⇒ climbing McKinley ⇒ bad.

This states the obvious fact that if you should refrain from climbing Mount McKinley, then you most definitely should refrain from climbing Mount Everest. In other words, the obvious implication that

climbing Everest ⇒ climbing McKinley

can be used to transfer or piggyback a known limitation about climbing Mount McKinley into a limitation about climbing Mount Everest. I use these simple ideas in the following pages.

Now let us use this intuition about mountain climbing to understand the general concept of one limitation piggybacking on another limitation. Imagine that a limitation was established by a contradiction as follows:

assumption A ⇒ contradiction.

That is, assumption A cannot be correct because we can derive a contradiction from it. Now consider assumption B. If we can show that from assumption B, we can derive assumption A, that is,

assumption B ⇒ assumption A,

then we have

assumption B ⇒ assumption A ⇒ contradiction.

To elaborate, if assumption B is correct, then assumption A is correct and since we already established that assumption A is not correct, we conclude that assumption B also cannot be correct. This is called a
reduction
: one assumption was reduced to another. With a reduction there is a transfer of already-known limitations to other areas.

Examples of reductions are found throughout the book:

• I show that if a certain problem takes a long time for a computer to solve, then other harder problems will take a longer time for a computer to solve (
section 5.3
).

• I show that if a certain problem cannot be solved by a computer, than harder problems also cannot be solved by a computer (
section 6.3
).

• I use similar methods to show that certain simply stated math problems are unsolvable (
section 9.3
).

• Other comparable reductions are found in our discussion of logic (
section 9.5
).

A few words about contradictions. The physical universe does not permit any contradictions:

• A certain molecule cannot both be hydrochloric acid and not hydrochloric acid.

• It cannot both be Monday and not Monday simultaneously in the same place.

• The diagonal of a square cannot be the same length as its side.

Similarly science, which is a description of the physical universe, also cannot express contradictions:

• It cannot be that the formulas
E
=
mc
²
and
E
≠
mc
²
are both true.

• A calculation about a chemical process cannot be both true and not true.

• A prediction cannot predict two incompatible events.

If there were a contradiction in science, it would not be an exact description of the contradiction-free universe. Similarly for mathematics and logic: to the extent that they are used in describing the universe and science, they cannot contain any contradictions.

There is, however, a place where contradictions do occur: inside the human mind. We are all fraught with contradictions; we desire contradictory things; we believe contradictory ideas; and we predict contradictory events. Anyone who has ever been in a relationship knows the feeling of simultaneously being in love with a person and hating them. We desire to have our cake and to be thin. As the Queen says to Alice in
Through the Looking Glass
, “Why, sometimes I've believed as many as six impossible things before breakfast.” The human mind is not a perfect machine. We are conflicted and confused. Similarly, human language, which expresses states of mind, must also have contradictions. There is nothing strange when we state “I love her and I hate her.” It is not unusual for a person to express a desire to be thin while having another piece of cake.
⁶

When we meet a paradox in the physical world and derive a contradiction, we know that there must be something wrong with the assumption of the paradox. However, when we meet a contradiction in the realm of human thought or in human language, then we need not abandon the assumption. More subtlety is possible. Why not permit the contradiction? Consider the liar paradox discussed earlier. Why not simply say that the sentence

This sentence is false.

is both true and false or perhaps meaningless? It is only an English sentence and many English sentences express contradictions. Similarly, the belief

This belief is false.

is both true and false. Why not permit such contradictory beliefs in our already-confused minds?

The relationship between the contradiction-free universe and our feeble human minds and languages raises many more interesting questions. How is it that the human mind can understand any part of the universe? How can a language formulated by human beings describe the universe? Why does science work? Why is mathematics so good at describing science and the universe? Do the laws of science have an external existence or are they only in our mind? Can there be a final description of the universe—that is, will science ever complete its mission and end? Are the truths of science and mathematics time dependent or culturally dependent? How can human beings tell when a scientific theory is true? As Albert Einstein wrote, “The eternal mystery of the world is its comprehensibility.
⁷
These and a host of other questions from the philosophy of science and mathematics are addressed in
chapter 8
.

Between the contradiction-free universe and the contradiction-laden human mind, a landscape full of vagueness exists:

• A person who stands in the doorway of a room is both in the room and not in the room.

• How many hairs does a man have to lose in order to be considered bald? Depending on which way the wind blows, he is sometimes considered bald and sometimes considered not bald.

• Is 42 a small or large number?

Human beings use vague ideas all the time. Our mindset and our concomitant human language are full of vague statements:

• Sometimes we say people in a doorway are in the room and sometimes we say they are not in the room.

• We call certain people with a few hairs bald and others not bald.

• If our bank account contains only $42, we say that 42 is a small number, but if we are talking about the number of diseases a person has, 42 is a large number.

Because vague ideas are outside the pristine world of science and mathematics, we cannot rely on some of the usual tools in addressing these ideas. Vagueness plays a major role in our discussions in
chapter 3
.

 

As a slight aside, special types of jokes are of interest for our discussion. We have seen that paradoxes are ways of showing that one has gone too far with reason. Violating a paradox means you stepped beyond the boundaries of reason and entered the land of the absurd. There are jokes that also play on the fact that we are taking reason too far. Such jokes take logic and reason to places where they were not intended. They start off with concepts that are well understood and then go farther or beyond their usual meaning. Consider the following: