Outer Limits of Reason (2 page)

A word must be said about Professor Heller's unique method of mentoring. After learning much from him during two years of coursework, I was disinvited from attending any more of his classes. He said that I had gained enough from him. From then until his passing, even though we talked about mathematics constantly, he never
taught
me a scintilla of mathematics. My job was to present my work or what I was studying. His job was to find flaws in my presentation or my understanding. He would harangue me about developing a correct definition, indicate where my proofs had failed, and point out when I was not exact. Although a gentleman—always with a kind word—his method of mentoring was intimidating and disheartening, to say the least. He never articulated this sink-or-swim philosophy. Nevertheless, he had a valid point: it was my job to learn mathematics and I had to struggle with it on my own. I am forever grateful for his confidence in me and for the independence he insisted I develop. He was the greatest of teachers.

 

I had the benefit of having a world-class mathematician as a neighbor. Professor Leon Ehrenpreis lived a few blocks from me and I took advantage of it by visiting him on a regular basis. My Friday-night visits were always greeted with a warm, welcoming smile. Besides being a first-rate mathematician, he was also a rabbinic scholar, marathon runner, handball player, classical pianist, and father of eight. This Renaissance man's breadth of knowledge was truly astounding. I have many fond memories of sitting at his kitchen table chatting about the subtleties of Hebrew grammar, the edge-of-the-wedge theorem, raising children, the consequences of the Kochen-Specker theorem, the role of cows in the Book of Genesis, hypergeometric functions, and many other topics. Professor Ehrenpreis was blessed with the most pleasant disposition and always had a kind, encouraging word. I learned much from him. He passed away in August of 2010.

All three are painfully missed.

 

This book is dedicated to my wife, Shayna Leah, whose warmth and loving support made this work possible, and to my daughters, Hadassah and Rivka, who fill our home and hearts with laughter and joy. My love and gratitude toward them are limitless.

1

Introduction

Human reason, in one sphere of its cognition, is called upon to consider questions, which it cannot decline, as they are presented by its own nature, but which it cannot answer, as they transcend every faculty of the mind.
¹

—Immanuel Kant (1724–1804)

As the circle of light increases, so does the circumference of darkness.
²

—Attributed to Albert Einstein

Zorba:  Why do the young die? Why does anybody die?

Basil:  I don't know.

Zorba:  What's the use of all your damn books if they can't answer that?

Basil:  They tell me about the agony of men who can't answer questions like yours.

Zorba:  I spit on this agony!

—
Zorba the Greek
(1964)

A civilization can be measured by how much progress its science and technology have made. The more advanced their science and technology are, the more advanced their civilization is. Our civilization is deemed more advanced than what we call primitive societies because of all the technological progress we have made. In contrast, if an alien civilization visited Earth, we would be considered primitive, almost by definition, since they have mastered interstellar space travel while we have not. The reason for using science and technology as a measuring stick is that these activities are the only aspect of culture that builds on itself. What was done by one generation is used by the next generation. This was expressed nicely by one of the greatest scientists of all time, Isaac Newton (1643–1727), who is quoted as saying, “If I have seen further it is only by standing on the shoulders of giants.” This constant accumulated progress makes science a good measuring stick to compare civilizations. In contrast to science and technology, other areas of culture, such as the arts, human relations, literature, politics, morality, and so on, do not build on themselves.
³

Another way to measure a civilization is by the extent to which it has banished unscientific and irrational ideas. We are more advanced today because we have cast alchemy into the wastebasket of silly dreams and study only chemistry. Centuries of treatises on astrology have been deemed nonsense while we retain our study of astronomy. As a civilization progresses, it subjects its beliefs and mythologies to logical analysis and disregards what is not within the bounds of reason.

The tool a civilization uses to make this progress is reason. Rationality and reason are the methodologies used by a society to advance. When a culture acts reasonably it will progress. When it deviates from reason, or steps beyond the limits of reason, it stagnates or regresses.

Reason comes in many forms. In broad (and perhaps inexact) terms, science is the language that we use to describe and predict the physical and measurable universe. The more abstract mathematics can be split into two areas: applied mathematics is the language of science, and pure mathematics is the language of reason. Logic is also a language of reason. Since science, technology, reason, rationality, logic, and mathematics are all intimately connected to each other, much of what I say about one will usually be true about all. At times I will just use the word
reason
to describe them all.

Philosophers have reflected and argued for centuries about what humans can and cannot know. The branch of philosophy that deals with human knowledge and its limitations is called
epistemology
. While the ideas of such philosophers are fascinating, their work will not be our central focus. Instead, we will be interested in what scientists, mathematicians, and current researchers have to tell us about the limits of human knowledge and reason.

One of the most amazing aspects of modern science, mathematics, and rationality is that they have matured to the level where they are able to see their own limits. As of late, scientists and mathematicians have joined philosophers in discussing the limitations of man's ability to know the world. These scientific limitations of reason are the central subject of this book.

The following is a cute little puzzle that gives a taste of what it means for reason to describe a limitation.
⁴
The puzzle is loads of fun, is worth pondering, and is also strongly recommended as a challenge at any cocktail party. Take a normal 8-by-8 chessboard and some dominoes that are of size 2-by-1. Try to cover the chessboard with the dominoes. There are sixty-four squares on the chessboard and each domino covers two squares, so thirty-two dominoes will be needed. There are millions of ways to perform this task.
Figure 1.1
shows how we might start the process.

That was pretty easy. Now let's try something a little more challenging. Put two queens on the opposite corners of the chessboard. Try to cover all the squares except the ones with queens, as in
figure 1.2
. There are sixty-two squares that need to be covered, which means thirty-one dominoes will be required. Try it!

After trying this problem for a while and not being able to cover every square, you might consider showing it to others—in particular, puzzle fans. They will have a similar experience. You might want to get a computer to work on the problem since a machine can quickly try many possibilities. There are millions, if not billions, of possible ways to try to start placing the dominoes on the board. Nevertheless, there is no way anyone or any computer will ever finish this task.

The reason why this simple problem of placing thirty-one dominos on a chessboard seems so hard is because
it cannot be done
. It is not a hard problem; it is an
impossible
problem. It is actually easy to explain why. Every domino is 2-by-1 and hence must cover a black-and-white square on the chessboard. The original board in
figure 1.1
had thirty-two black squares and thirty-two white squares that needed to be covered. There was total symmetry on the board. By contrast, the board in
figure 1.2
has thirty black squares and thirty-two white squares that need to be covered. The symmetry has been broken. There is no way anyone is going to be able to cover these sixty-two squares with dominoes where each covers one black square and one white square. Move the queens so that one is on a black square and one is on a white square. Now try it.

This small puzzle has a lot of nice features. It is easy to explain, easy to attempt a solution, and a computer can be used to try to solve the problem. Nevertheless it cannot be solved. It is not that we are not smart enough to solve the problem or that the problem is beyond the capabilities of present-day technology, but that it cannot be solved at all. It is not someone's opinion that this problem cannot be solved. Rather, it is a fact about the world. Reason dictates that there is a limitation in our ability to solve this problem. The best part of this problem is that the explanation of why it is unsolvable is easy to explain. Once it is stated, you are totally convinced and find it trivial.

This book will demonstrate a myriad of such unsolvable problems and limitations.

 

Rather than giving an orderly synopsis of each chapter, I'll provide a classification of the types of limitations the book covers. For every type of limitation, I will look at examples found in the different chapters. This will give the book a more unified structure.

Examples of limitations abound. Computer scientists have shown that there are many tasks that computers cannot perform in a reasonable amount of time (
chapter 5
). They have also shown that there are certain tasks that computers cannot perform in any amount of time (
chapter 6
). Physicists discuss how complex the world is and how some phenomena are just too complicated for science and mathematics to predict (
section 7.1
). Mathematicians have identified certain types of equations that cannot be solved by normal means (
section 9.2
). Logicians have proved that there are limitations to the power of proofs. They have described logical statements that are true but cannot be proved (
section 9.4
). Philosophers of language have shown that our ability to describe the world we live in is limited (
chapter 2
).

There are other types of limitations that, in a sense, are deeper. These are limitations that show that our naive intuition about the world we live in and our relationship to that world is faulty. The way we think about the universe and its properties has to be updated. Our very assumption that there is an objective definition of a particular physical object needs to be reevaluated (
section 3.1
). The classical philosopher Zeno has shown that our usual notions of space, time, and motion demand a deeper analysis (
section 3.2
). Quantum mechanics has taught us that the relationship between the knower and the known is not simple. This branch of physics has also shown us that the world is more interconnected than previously thought (
section 7.2
). Researchers have shown that our simple intuitions about infinity are faulty and need to be adjusted (
chapter 4
). Relativity theory has demonstrated that our notions of space, time, and causality are wrong and need to be corrected. Physicists have shown that there is no objective measure of length or duration (
section 7.3
). The very relationship between us, our world, and the science and mathematics that we use to describe the world is not simple (
chapter 8
). All these topics, and many more, are explored in depth in the pages that follow.

The limitations just mentioned are demonstrated in myriad ways. One of the more interesting ways is with paradoxes. The word comes from the Greek
para-
“contrary to” and
doxa
“opinion.” The
Oxford English Dictionary
gives many overlapping definitions, including