Inductive reasoning is sometimes compared to thinking that something will always be a certain way just because it always has been that way: When you predict that the sun will rise in the East tomorrow morning, you are confident in your forecast because the sun has never failed to so in all of written history. Sometimes, this kind of argument can carry great weight, but generally, the fact that something has been going on for a very long time does not guarantee that it will do so indefinitely. After all, the Roman Empire lasted quite a while in its day and was thought permanent and invincible, until it was overrun by Germanic tribes.

Mathematical induction is more precisely defined, and consequently, a more sound form of reasoning. Its most common use is in number theory, although it is also essential in other branches of...