Mathematical proof

Proof is the foundation of all mathematics. Some proofs are just a few lines long while others run to thousands of pages, but all proofs share a common structure. Beginning with a set of reasonable assumptions, a proof follows logical steps that demonstrate a result that must be true. Without this logical process, mathematicians could not build on the work of others and the whole of maths would come crumbling down.

The ancient Greek mathematician Euclid was one of the first to realise this. He wrote a textbook called Elements that, with just a few assumptions, set out to prove the entire system of plane and solid geometry. The proof of Elements is so elegant that it was still widely read until the 20th century, when Euclid’s teachings were incorporated into the school curriculum.

Even the greatest mathematicians can make mistakes though, and Euclid was no exception. In the 19th century, a number of mathematicians found that one of Euclid’s assumptions wasn’t universally true.

Euclid had stated: for any straight line and a point not on the line, there is exactly one other line that passes through the point and is parallel to the first line. In other words, parallel lines don’t cross. That sounds reasonable – but it turns out to only be true in flat geometry. For example, on the curved surface of the Earth longitude lines are parallel but all meet at the North and South poles.

This excerpt from history teaches an important lesson – be careful about making assumptions! Most people who use maths in their careers don’t bother to sit down and prove every theorem; they leave that to academic mathematicians. Even so, being able to set out a logical argument to solve a complex problem is a valuable skill for anyone working with maths. This logical pedantry and scepticism of any unknown assumption gives mathematicians the edge over others lacking the skills of proof. For more information about proof, check out these links.

The Origins of Proof is a four-part series from Plus magazine. Read parts one, two, three and four.

Related Links

Euclid of Alexandria
A biography of Euclid.

Pythagoras of Samos
A biography of Pythagoras.

Mathematical mysteries: Strange Geometries
An article from Plus Magazine

Mathematical mysteries: Transcendental meditation
An article from Plus Magazine