The first known mathematical proof emerged in ancient Egypt and Babylon before the 6th century BCE, with Thales of Miletus introducing deductive reasoning between 624 and 546 BCE. Thales and his successor Hippocrates of Chios, who lived from approximately 470 to 410 BCE, gave some of the first known proofs of theorems in geometry. This marked a shift from empirical observation to logical necessity.
Euclid introduced the axiomatic method around 300 BCE, which remains in use today. The Elements was read by anyone considered educated in the West until the middle of the 20th century, serving as the primary textbook for teaching proof-writing techniques for over two millennia. It covered geometry, number theory, and proofs that the square root of two is irrational and that there are infinitely many prime numbers.
In the 10th century, the Iraqi mathematician Al-Hashimi worked with numbers as such to prove algebraic propositions concerning multiplication, division, and the existence of irrational numbers. An inductive proof for arithmetic progressions was introduced in the Al-Fakhri, written in 1000, by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. These developments allowed for the manipulation of abstract quantities independent of physical space.
Gödel's first incompleteness theorem showed that many axiom systems of mathematical interest will have undecidable statements. These are statements that are neither provable nor disprovable from a set of axioms, such as the parallel postulate in Euclidean geometry. This discovery fundamentally altered the understanding of what can be known within a mathematical system.
Q.E.D. stands for quod erat demonstrandum, which is Latin for that which was to be demonstrated. A more common alternative is to use a square or a rectangle, such as □ or , known as a tombstone or Halmos after its eponym Paul Halmos. The end of a proof signifies the completion of a logical journey where the conclusion has been established beyond any doubt.