1.4 What is proof in mathematics?

You will be familiar with the concept of ‘proof’ in everyday language. A proof is a convincing argument that something is true. But whether an argument is convincing enough depends on the context and on precisely what is being claimed.

For instance, imagine tossing a coin. If you wished to determine whether the coin was fair (meaning that it is equally likely to come up heads as it is tails) then you might carry out an experiment. You could toss the coin $10\>000$ times and count how many times it comes up heads. Even if it came up heads exactly $5\>000$ times, that would not be a mathematical proof that the coin is fair. Maybe you just got lucky.

Note that we can make some mathematical deductions about this situation. For example, we can work out the probability that the coin is fair given our experimental results (see the Year 2 course Elementary Probability and Statistics), but this is not the same as giving a mathematical proof that the coin is fair.

A mathematical proof is a checkable record of reasoning and has many purposes [2].

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  A proof should convince a knowledgeable mathematician that a claim is true.

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  A proof should be a logically-sound deductive argument, meaning that it does not leave room to draw incorrect conclusions.

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  A proof should be transparent, meaning that a knowledgeable mathematician can fill any gaps (given enough time and motivation).

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  A proof should be presented in a way that conforms to norms of the mathematical community.

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  Some mathematicians believe a proof should explain why a claim is true, rather than only establishing that it is true.

Deductive reasoning is central to mathematics. Deductive reasoning means starting with some assumptions and making a logical argument that leads to a logically certain conclusion.

For example, suppose we want to prove the following statement about integers¹¹ 1 Do not worry if you have not seen the term ‘integer’ before. You can safely think of this as being a positive or negative ‘whole’ number, or zero..

‘The square of any even integer is even.’

We might argue by checking some examples:

‘$2^{2}=4$ is even, $4^{2}=16$ is even, $100^{2}=10\>000$ is even. We have checked three even numbers, their squares are all even, and so the square of any even number must be even.’

This is not a proof (even though it is true that the square of every even number is even). All this argument has done is check that a statement is true for three examples; there are infinitely many examples we have not checked! While it will help your understanding of a statement to try it out with specific examples, this is not the same as proving that the statement must always be true.

We will discuss mathematical proofs further in Chapter 4 4   Proving mathematical statements.