Chapter 4 Mathematical statements

4.1 Definition

Quite simply, if a statement is good (clear and non-paradoxical) and has a clearly-defined truth value (even if you don’t know what that truth value is), it is described as a mathematical statement.

Statements which fail to qualify as mathematical statements are called non-mathematical statements.

4.2 Mathematical statements don’t need to be about mathematics!

Before we move on, we need to address a really important point. When we say something is a mathematical statement, it does not have to be about mathematics!!

Let’s look at some examples of mathematical and non-mathematical statements:

Example 4.1 Bishkek is the capital of the Kyrgyz Republic.

This is a mathematical statement, even though it’s not talking about mathematics! It is mathematical because it is clear, all terms are defined, and it will have a clearly-defined truth value.

Example 4.2 Malta is the largest country in the world

This is also a mathematical statement, despite not being about mathematics. It is clear, all terms are defined, and it will have a clearly-defined truth value.

Example 4.3 \(356\) and \(989\) have no common divisors.

This is a mathematical statement - it will have a clearly-defined truth value.

Example 4.4 There is a smallest elementary particle, one that can’t be divided into smaller particles.

This is a mathematical statement. All terms are easy to Google, and the statement will have a clearly-defined truth value (though it’s currently unknown⁵).

Example 4.5 Pears are the best fruit.

This is not a mathematical statement because it is an opinion.

Example 4.6 \(8\) is better than \(14\).

This statement is not mathematical because it is an opinion, and because “better” is not defined, and therefore will not have a clearly-defined truth value.

Example 4.7 The Earth is big.

This is also not mathematical because it is vague and therefore will not have a clearly-defined truth value.

Example 4.8 Poetry is difficult to understand.

‘Understand’ is not clearly defined (do we mean to get a personal understanding; to understand other people’s interpretation of the poem; to grasp the poet’s meaning; or just to understand what the words mean?). ‘Difficult’ will have a subjectivity to it which means different people might come to different conclusions about the truth value of this statement. It’s also not clear which type of poetry, or which poet, the statement is referring to. Therefore the truth value of this statement will not be clearly defined, and so the statement is not mathematical.

4.3 Statements containing mathematics aren’t necessarily mathematical!

We should also be clear on another point: Just because a statement contains mathematics, that doesn’t mean it’s mathematical!

The statement “8 is a better number than 14” might be about numbers, but it’s not mathematical because it doesn’t have a clearly-defined truth value.

Equally, the statement “\(ab = cd\)” is definitely mathematical in its content, but it is not classified a mathematical statement because it doesn’t contain enough information to decide on its truth value.

Careful!

Mathematical statements don’t have to be about mathematics. They just have to be good, clear statements with a clearly-defined (though possibly unknown) truth value. We use the word mathematical here to describe the nature of the statement, rather than its content.

Exercise 4.1 For this exercise, you should write:

1. A mathematical statement that is not about mathematics.
2. A mathematical statement that is about mathematics.
3. A non-mathematical statement that is about mathematics.
4. A non-mathematical statement that is not about mathematics.

Exercise 4.2 Do you know the truth values of the two mathematical statements you wrote in the previous question? If not, do you think you could find out?

Exercise 4.3 Look back at the twenty-six statements in Exercise 3.8.

1. How many of them are mathematical statements?
2. How many are non-mathematical statements?

Exercise 4.4 Of those statements in Exercise 3.8 which are mathematical statements:

1. Which of them do you know are true?
2. Which do you know are false?
3. Which don’t you know the truth value of but you think someone in the world does?
4. Do you think there are any to which no one in the world knows the answer to?

Exercise 4.5 From Exercise 4.4, part d, we know that there are mathematical statements to which no one can ever know the answer? Can you think of any others? Remember, they don’t have to be about mathematics!

Exercise 4.6 Below are four more statements. Repeat Exercises 4.3 and 4.4 for these statements.

1. There are infinitely many square numbers.
2. There are infinitely many prime numbers.⁶
3. There are infinitely many perfect numbers.
4. There is no set⁷ whose cardinality⁸ is strictly between that of the integers⁹ and the real numbers¹⁰.
   

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5. See https://www.youtube.com/watch?v=ZorRPAD32i4 for what might be smaller than leptons and quarks, and https://www.youtube.com/watch?v=ehHoOYqAT\_U for a nice introduction to elementary particles.↩︎

6. Watch from 12:30 in this video: https://www.youtube.com/watch?v=OihQPf4mJH4.↩︎

7. A set is a collection of items.↩︎

8. Cardinality tells you the number of items in a set.↩︎

9. Integers are the whole numbers: …-3, -2, -1, 0, 1, 2, 3…↩︎

10. Real number include both whole and non-whole numbers.↩︎