Definitions, Propositions and Theorems

Three kinds of sentence

A mathematical argument is built from sentences of three kinds, and each does a different job. A definition fixes what a word means. A proposition claims something, and the claim is either true or false. A theorem is a proposition that has been proved.

Telling them apart matters because they are handled differently. A definition is accepted as it stands, a proposition is tested, and a theorem can be used, as a step in the proof of something else.

A definition fixes a meaning

“A whole number n is even when n = 2k for some whole number k.” This sentence says what the word even will mean from now on. It claims nothing about the world, so there is nothing in it to agree or disagree with, and it is neither true nor false. 8 is even because 8 = 2 × 4; 7 is not, because no whole number k gives 2k = 7.

Definitions are choices, made to be useful. The definition of a prime, a whole number with exactly two whole-number factors, leaves out 1, which has only one. That is a choice, not a discovery: with 1 left out, every whole number above 1 has exactly one factorization into primes, apart from the order of the factors. Arguing about whether 1 is really prime goes nowhere, because the answer is fixed by the definition.

8 counters in pairs: 4 pairs, none left over, so 8 = 2 × 4 and 8 is even by the definition.

A proposition claims something

“Every even number above 2 is a sum of two primes.” This sentence makes a claim about every even number, and the claim is either true or false. Small cases fit: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7, 12 = 5 + 7, 14 = 7 + 7, 16 = 3 + 13, 18 = 5 + 13 and 20 = 7 + 13.

But there are infinitely many even numbers, and no list of cases, however long, covers them all. Nobody has proved this claim, and nobody has found an even number that breaks it. A proposition that seems true but has no proof yet is called a conjecture.

Some propositions are false. “Every prime is odd” is a proposition, and 2 breaks it: 2 is prime and even. One such counterexample is enough to make a claim about every case false.

A theorem is a claim with a proof

“In a right triangle with legs a and b and hypotenuse c, $a^2+b^2=c^2$a² + b² = c².” This is a proposition, and there are proofs of it, so it is a theorem. A proof is an argument that starts from definitions and statements already proved, and shows that the claim holds in every case it covers.

One proof cuts up a square of side a + b in two ways. Four copies of the triangle in its corners leave a square hole of area $c^2$c²; the same four copies paired into two rectangles leave two holes of areas $a^2$a² and $b^2$b². The square and the triangles are the same both times, so the holes have the same total area: $a^2+b^2=c^2$a² + b² = c². Nothing in the argument depends on particular lengths.

Four copies of a right triangle in a square of side a + b. The shape left in the middle is the square on the hypotenuse, $c^2$c².

The same four triangles, paired into two a by b rectangles. The two squares left over are $a^2$a² and $b^2$b².

Lemmas and corollaries

A lemma is a smaller theorem proved on the way to a bigger one. The proof above needs the hole in the first drawing to have right-angled corners, and that rests on a lemma: the two acute angles of a right triangle add to 90°.

A corollary follows from a theorem in a step or two. Define a multiple of 3 to be a number 3k for a whole number k. The theorem “the sum of two multiples of 3 is a multiple of 3” is proved by 3a + 3b = 3(a + b). Its corollary, that the sum of three multiples of 3 is a multiple of 3, follows by using the theorem twice: add two of them, then add the third.

An example is not a proof. 6 + 9 = 15 illustrates the theorem, since 15 = 3 × 5, but it shows only that one pair works; the line 3a + 3b = 3(a + b) covers every pair at once.

The usual mistakes

Treating a definition as a claim to be proved. “A square number is a whole number times itself” names a kind of number; it cannot be false.

Calling a well-tested claim a theorem. The claim about even numbers has no known counterexample, and it is still a conjecture, because it has no proof.

Taking examples as a proof. However many cases fit, a claim about every case needs an argument that covers them all.

Using a statement that has not been proved. A proof may rest only on definitions and on statements already proved.

The calendar’s leap years

In the application below, the calendar’s definition of a leap year is applied clause by clause to four years, and a proposition about every year from 2001 to 2100 is tested against that definition. One year breaks it.