# aha! Gotcha

## Preface

These are old fond paradoxes to make fools laugh i' the alehouse.

Desdemona, Othello, Act 1, Scene 1

If we alter Desdemona's remark to "These are old and new paradoxes to make us laugh during lunch time," then it is not a bad Description of this book. The word paradox has many meanings, but I use it here in a broad sense to include any result so contrary to common sense and intuition that it invokes an immediate emotion of surprise. Such paradoxes are of four main types:

1. An assertion that seems false but actually is true.
2. An assertion that seems true but actually is false.
3. A line of reasoning that seems impeccable but which leads to a logical contradiction. (This type of paradox is more commonly called a fallacy.)
4. An assertion whose truth or falsity is undecidable.

Paradoxes in mathematics, like those in science, can be much more than jokes. They can lead to deep insights. For early Greek thinkers it was a bothersome paradox that the diagonal of a unit square could not be measured accurately, no matter how finely graduated the ruler. This disturbing fact opened up the vast realm of the theory of irrational numbers. To nineteenth century mathematicians it was enormously paradoxical that all the members of an infinite set could be put in one-to-one correspondence with the members of one of its subsets, and that two infinite sets could exist whose members could not be put into one-to-one correspondence. These paradoxes led to the development of modern set theory, which in turn had a strong influence on the philosophy of science.

We can learn much from paradoxes. Like good magic tricks they are so astonishing that we instantly want to know how they are done. Magicians never reveal how they do what they do, but mathematicians have no need to keep secrets. Throughout, I have done my best to explain in nontechnical language, and as briefly as possible, why each paradox is paradoxical. If this stimulates you to go on to other books and articles where you can learn more, you will not only absorb a great deal of significant mathematics but also enjoy your-self in the process. Some easily accessible readings are starred in the References and Suggested Readings at the end of the book.

November 1981              Martin Gardner