What is a good definition?

Answer 1

A good definition gives an if-and-only-if condition for when an object satisfies the definition. For example: an integer greater than 1 is prime if and only if it has exactly two positive integer factors - 1 and itself. This tells us both what we have to prove, to show that an integer is prime, and what we know about an integer if we know it's prime. On the other hand, if I wrote that an integer is prime if it's 2, 3, 5, or 7, this is a true statement, but not a definition, because it doesn't completely describe prime numbers. True, if an integer is 2, 3, 5, or 7, then it's prime, but there are other prime integers. A good definition should also involve simpler terms than the term you're defining. This is true for the above example - the meaning of a factor is simpler than the meaning of a Prime number. On the other hand, if I were to write that an integer n is prime if and only if the ring Z/nZ is a field, this would be a bad definition - even though it's equally true, the terms it involves are much more complicated than the notion of a prime number.