2.1 What is a mathematical statement?

As discussed in the previous chapter, mathematical language is precise, with words and symbols having exact meanings. In these notes, we use the term mathematical statement (or just statement) to mean a sentence that is either true or false, but not both.

At this stage, we have to balance strict mathematical formalism with practicality. There is a whole subject called ‘logic’ (often studied by philosophers) in which this working definition of a mathematical statement would not be considered formal enough. However, it is sufficient at this stage of your mathematical career.

Here are some examples of mathematical statements:

•

The integer 6 is even. [True.]
•

$0=1$ . [False.]
•

Every even integer greater than $2$ is the sum of two primes. [It is not known whether this is true or false, but it is definitely one or the other! This statement is known as Goldbach’s conjecture.]
•

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. [True; this is the Pythagorean theorem.]

The following are not mathematical statements:

•

$x^{2}-3x+2.$ [This is not a sentence as there is no verb, and it is neither true nor false. It is a mathematical expression.]
•

This statement is false. [A perfectly good sentence but it cannot be true or false, but not both. Think about what it would mean for it to be true.]
•

Let $x$ be a real number. [This does not claim anything that could be true or false.]

Mathematicians often use symbols to stand for objects, words or phrases. For example the symbol ‘ $=$ ’ is an abbreviation for the word ‘equals’ or the phrase ‘is equal to’. Mathematical statements when read aloud in English should always be well-formed sentences.

Mathematicians use common labels to refer to different types of statement. These labels help to communicate the status or role of a statement to other mathematicians. The labels do not have strict definitions, but the most common ones are listed below with a brief description of how they are typically used.

Conjecture

A statement which has not been disproved but also has no known proof.

A famous example of a conjecture made in 1742 is as follows.

Conjecture 2.1 (Goldbach’s conjecture).

Every even integer greater than $2$ is the sum of two prime numbers.

You might find it surprising that a statement as simple as this, and made so long ago, has no known proof. Wikipedia has a list of some well-known conjectures in mathematics. Some of these are still open problems, meaning that they remain genuine conjectures. Others have been proved or disproved since they were first made.

Another famous example of a conjecture is the Riemann hypothesis. Note that this is usually called a hypothesis rather than a conjecture. Fermat’s Last Theorem was another famous conjecture made in 1637 that remained an open problem until a proof was published in 1995. Despite only being a conjecture before that, it was still commonly known as Fermat’s Last Theorem. These two examples illustrate the point that the labels given to mathematical statements do not always obey strict rules.

Claim

In these notes, we use this name for a statement which we will attempt to prove. The statement may turn out to be true or false.

Proposition

A true statement with a known proof. Propositions are often results which are not particularly substantial in their own right.

The next three labels (lemma, theorem and corollary) can be thought of as a group.

Theorem

A substantial or important result. Theorems are sometimes named after mathematicians, though not always the ones who first proved them (see Wikipedia’s list of misnamed theorems). Theorems are also sometimes known by different names in different languages or countries.

Lemma

A result that is used in the proof of a theorem. Lemmas are often technical in nature and are written as separate results to help break down the proof of the theorem.

Corollary

A result that follows as a consequence of a theorem. Corollaries are often interesting results because they help us realise what a theorem means in particular contexts.