Outer Limits of Reason (64 page)

Physical Limitations

The simplest type of limitation is one that shows reason does not permit a certain physical object or physical process to exist. The very first limitation we met (in
chapter 1
) was with the chessboard and dominoes. This is an example of a physical process that cannot exist. There is no way to place the dominoes on the chessboard with the two black corners removed. The barber paradox of
section 2.2
also demonstrates that a certain isolated village with a particular rule cannot exist. The same section discussed a reference book that cannot exist. At the end of
section 3.2
we saw that the time-traveler paradox shows either that time travel is impossible, or that even if it were possible, some actions by the time traveler will not be permitted. The universe simply will not permit a contradiction-causing process. All of
chapter 6
and section 9.3 show that certain physical computers or algorithmic processes cannot exist. There are tasks that simply cannot be performed in this world. And finally in
section 7.2
, we talked about the possibility that quantum mechanics is inherently nondeterministic. In that case, no physical process can anticipate a quantum mechanical outcome. In all these examples we see that the physical universe is constrained by the dictates of reason.

Mental-Construct Limitations

A second, more subtle type of limitation states that a certain idea or mental construct cannot exist. I consider language a mental construct that is used to describe a mental state or a part of the universe. When discussing the liar paradox in
section 2.1
, I showed that certain sentences are neither true nor false. If a sentence is true then it is false, and if it is false then it is true. The mind cannot give such sentences any meaning. Similarly for other linguistic paradoxes—like the heterological paradox of
section 2.2
as well as the interesting-number paradox, Berry's paradox, and Richard's paradox of
section 3.2
—Zeno's paradoxes of
section 3.2
deserve some thought. They are not physical limitations since the slacker will get to the door and Achilles will win the race. Rather, they demonstrate a problem with the descriptions of certain actions. The descriptions are faulty because they seen to demand an infinite process. In contrast, the actions are perfectly legitimate. Zeno's paradoxes show that there are problems with certain mental and linguistic descriptions of basic movements. Similarly, our discussion of vagueness in
section 3.3
showed a limitation of this type. Deciding whether a certain collection is a heap, and whether someone is considered bald, is a mental and/or a linguistic problem. We showed there were certain problems with such vague predicates. The inability to prove or disprove statements like the continuum hypothesis and the axiom of choice (
section 4.4
) demonstrates limitations of our logical ability. Similarly, most of the mathematical limitations discussed in
chapter 9
are limitations of mental constructs. Human beings like their mathematics free of contradictions. Systems with contradictions do not have the right to be called mathematics and are ignored by researchers. Mathematicians avoid axioms and definitions that will bring about contradictions.

Several times throughout the book, I have had to restrain myself from taking an obvious step so as not to cross the limits of reason. A limitation of a mental construct was needed so as to avoid contradictions.

• In
section 2.1
we saw that it is wrong to demand that every declarative sentence is either true or false. If we were to make such a demand, then we would have to say a liar sentence is true or false and get into contradictions.

• In
section 2.2
we saw that there are phrases (e.g., “not true about itself”) or even words (e.g.,
heterological
) that—regardless of their obvious meaning—cannot be given meaning.

• In
section 3.3
, we saw that we had to restrict the use of the logical law of modus ponens when there are vague terms involved for fear of proving false statements.

• In
section 4.4
, we saw that we are constrained from making the obvious assumption that for every property there is a set of objects that satisfy that property. We simply cannot say that for fear of tripping on a predicament like Russell's paradox.

• In
section 9.4
, we saw that we must restrict our use of the logical fixed-point machine for fear of Löb's paradox.

In all these cases, we have an obvious logical step to take. But we realize that we are at the edge of the precipice and we restrict ourselves so as not to fall into the world of contradictions. Notice that all these restrictions are of mental constructs, not physical objects.

There is a slight exit strategy for natural language and for some mental constructs. As I stressed in
chapter 1
, we as humans are not bothered by having some contradictions in our everyday language, nor are we perturbed by contradictions in our mind. So when we meet a limitation of natural language and of certain mental constructs, we might simply ignore the limitation and have a contradiction. However, when our language is used to describe the physical universe (science) or made to describe mathematics, we do not have the luxury of having contradictions. We must keep such mental constructs and language contradiction-free so as to describe the contradiction-free physical universe.

Practical Limitations

Another type of limitation that we have met is of a slightly less fundamental character. They are not limitations that show it is impossible for something (physical or mental) to exist. Rather, they show that it is extremely impractical for something to exist. That is, it is impossible to make some prediction or find some solution in a normal amount of time or with a normal amount of resources.
Chapter 5
is concerned with certain solvable computer problems that demand trillions of centuries to solve. While computers can theoretically solve such problems, for all practical purposes it is beyond human ability.
Section 7.1
discussed chaos theory, which shows that systems with extreme sensitivity to initial conditions are essentially impractical to solve. In terms of the butterfly effect, while theoretically we might be able to keep track of all butterflies in Brazil to predict tornadoes in Texas, it is simply impractical to do so. These and several other practical limitations were discussed in this book.

Limitations of Intuition

We have seen that our naive intuition is somewhat flawed, reflecting more an error than a limitation. Our basic intuitions about the universe around us were challenged many times and the usual view was shown to be wrong. In
section 3.1
our discussion of the ship of Theseus showed that objects do not really have an existence
as that object
(outside of a mind). We saw that this is not only a problem with physical objects and people but also with institutions and concepts.
Chapter 4
showed that our naive intuitions about infinities are somewhat problematic.
Section 3.4
, where we met the Monty Hall problem, indicated that our concept of knowing needs adjusting.

Another point we have seen, especially in quantum mechanics (
section 7.2
) and in relativity theory (
section 7.3
), is that the observer plays a major role in the observed universe. There is a naive belief that the world is objective and external to the observer. This naive belief says that we can learn about this external world without changing it. The belief needs to be updated. The truth is that our view of the external world depends on how it is observed. The results of our experiments depend on what types of experiments are employed. The answers to our questions depend on what questions are asked and how the questions are posed. This places the conscious mind of the observer in a more central position in the study of the universe. Scientists are not outside the universe looking in. Rather, they are part of the universe and trying to make sense of it. They are part of the phenomena they are studying. It is hard to separate the experimenter from the experiment. That is, the universe is the ultimate self-referential system: the universe uses scientists to study itself.
⁴
We have seen several other counterintuitive ideas in both quantum mechanics and relativity theory.

There are also many questions about the sciences and the physical world for which our intuitions are in need of adjustments (
chapter 8
). The very nature of science and its relationship with mathematics, the universe, and the mind is open. The big questions of how and why we perceive structure and order in the universe are far from being resolved.

 

The various limitations were found or proved with different methodologies. However, there are some similar patterns that are worth highlighting.

Many of the limitations that we found were simply byproducts of self-reference. Once a system has the ability to talk about itself and deal with its own properties, there will be limitations of the system.
Table 10.1
provides a list of some diverse areas where we found self-reference.

These diverse areas all have self-reference and have paradoxical situations and limitations.
⁵
We can also classify these self-referentail limitations into the four types of limitations listed in the beginning of this section. The situations in sections
2.2
,
3.2
,
5.2
, and
7.1
are about physical limitations. The situations in sections
4.3
,
4.4
, and
9.4
are about limitations of mental constructs (math, set theory, and logic), and these limitations show that such exact mental constructs are not permissible. Sections
2.1
and
3.4
deal with human language and beliefs where contradictions are commonplace. What is remarkable is the common scheme of all the self-referential paradoxes: they all negate some basic aspect of themselves. This scheme seems to be a fundamental facet of reason.
⁶

Once there is a limitation (from a self-referential paradox or in any other way), it is not hard to find other limitations by looking at reductions. One limitation can piggyback on another. In fact, a careful reading of
chapter 6
and of
section 9.3
will demonstrate that only one problem was shown to be computationally unsolvable: the Halting Problem. All the other problems were simply shown to be reductions of the Halting Problem. From a single limitation one can go on to build an entire edifice of limitations. We wonder if perhaps every type of limitation somehow comes from some form of self-referential limitation or a reduction from such a limitation.

Another common theme we have seen is the distinction between what is describable and what is indescribable.
⁷
By the very nature of language, what can be described is countably infinite. In contrast, what actually exists “out there” is uncountably infinite.
Table 10.2
reminds us of some of the differences we have encountered.

The immense difference between countably and uncountably infinite exemplifies that what can be captured by language and formal reasoning is minuscule compared to all that exists. As Isaac Newton is quoted as saying, “What we know is a drop, what we don't know is an ocean.”
⁸

10.2  Defining Reason

At the end of
chapter 1
, I asked for a definition of reason. An entire book on what is the limits of reason begs for a definition of what is within the boundaries of reason. Why is it that some processes are reasonable and some are not? Why is it reasonable to check your blood pressure and not reasonable to check your horoscope? Why is it reasonable to agree with the chemists of today and to ignore the alchemists of centuries ago?

Over the centuries, philosophers have proposed many different definitions of reason and discussed their properties. Philosophers have also spilled much ink distinguishing between reason and related terms such as
intelligence
,
intellect
,
rationality, understanding
, and
wisdom
. There have been many attempts as well to distinguish human reason from animal reasoning and computer reasoning.
⁹
A summary of all the various theories and opinions on the topic is beyond the scope of this book.