What counts as proof?

In everyday life “proof” means good evidence. In mathematics it means something much stronger: an argument that shows a claim must be true, in every case, without measuring anything. This explainer follows the lesson “Can you prove it?”.

1. A4 paper has a secret

A sheet of A4 is 210 mm × 297 mm. Fold it in half and you get A5, which is exactly the same shape, only smaller. Fold A5 and you get A6, the same shape again.

Only one shape does this. Call the ratio long side ÷ short side r. Folding in half halves the long side, and the old short side becomes the new long side, so the new ratio is short ÷ (long ÷ 2) = 2 ÷ r. If the shape stays the same, r = 2 ÷ r, so r × r = 2. The ratio must be √2 (about 1.414).

2. Can measuring settle it?

Official A4 is 297 ÷ 210 = 1.41428571… but √2 = 1.41421356… They agree to three decimal places. A ruler marked in millimetres can’t tell them apart: a sheet exactly √2 times 210 mm long would be 296.985 mm, just 0.015 mm shorter, much thinner than a human hair.

However carefully you measure, a measurement is always “this close, as far as I can tell”. Measuring can never tell you that a number is exactly √2.

3. Why √2 can’t be a fraction

1. Imagine √2 is a fraction, cancelled down as far as it will go.
2. Squaring it shows that the top number must be even.
3. Follow that through, and the bottom number must be even too.
4. But then we could cancel by 2 again, and we said we couldn’t. Impossible! So √2 is not a fraction.

The trick is called proof by contradiction: assume it is possible, show that this leads to something impossible, so it can’t be possible. Every fraction like 297/210 is a fraction, so official A4 can never be exactly √2. We know that without measuring anything.

4. The broken chessboard

Two opposite corners have been cut off a chessboard, leaving 62 squares. A domino covers two squares side by side, across or down. Can you cover all 62 squares with 31 dominoes? Try it before you read on.

• Every domino covers one dark and one light square.
• So 31 dominoes always cover 31 dark and 31 light squares.
• But both cut-off corners were dark: the board has 30 dark and 32 light squares.

So it is impossible, for every attempt, even the ones nobody has tried yet. There are millions of ways to lay the dominoes, and nobody needs to try them all. That is what a proof gives you.

5. What a proof gives you, and what it doesn’t

• Certainty for every case, including cases nobody will ever check, as long as you accept the starting points (axioms and definitions).
• Knowledge that doesn’t wear out: Euclid’s proof that there are infinitely many prime numbers is as true now as it was 2,300 years ago.
• But proofs are written and checked by people. Andrew Wiles announced a proof of Fermat’s Last Theorem in 1993; reviewers found a gap, and it took him and Richard Taylor another year to repair it. Mathematicians check each other too.
• A proof tells you about the mathematics, not the world. Whether real A4 paper, cut by a machine, is close enough to √2 is a question of measurement again.

6. “Proof” in other areas of knowledge

Outside mathematics, people still say “proof”, but they usually mean strong evidence. Use these questions to compare, and add your answers to your journal and the grid on your handout:

• Natural sciences: can any number of experiments prove a theory, or can one result overturn it?
• Human sciences: your Dunning-Kruger designs turned people into numbers. What does that do to how sure we can be?
• History: historians weigh sources. When would a historian say something has been “proved”?
• Law: a court needs proof “beyond reasonable doubt”. How is that different from a mathematical proof?

Key words

Answer the lesson’s questions in your journal →