Paul Erdős imagined a heavenly book holding the most beautiful proof of every theorem
Paul Erdős liked to say that an especially elegant proof came straight from The Book, an imaginary volume containing the perfect argument for each theorem. In 2003 a real volume, Proofs from THE BOOK, gathered 32 favourites. What separates such an argument from mere persuasion is that it must cover every possible case.
A mathematical proof is a deductive argument showing that certain assumptions force a conclusion. It may lean on earlier theorems, but in principle everything traces back to axioms and accepted rules of inference. Piling up examples never counts, however many there are; a claim believed but unproved is a conjecture. The word descends from Latin probare, to test, a relative of probe, probation and probability, and of Spanish probar, to taste.
Demonstration probably began with geometry, itself born from practical land measurement, after a long era of pictures and analogies. Thales and Hippocrates of Chios produced some of the first known geometric proofs, while Eudoxus and Theaetetus stated theorems without proving them. Euclid, around 300 BCE, transformed the subject with the axiomatic method: start from undefined terms and self-evident axioms, then deduce. His Elements, which also proves that the square root of two is irrational and that primes never run out, was standard reading for educated Westerners until the mid-20th century. In the 10th century the Iraqi mathematician Al-Hashimi used numbers treated as lines to prove algebraic results, including that irrational numbers exist, and Al-Karaji introduced an inductive proof around 1000.
In practice proofs are written in careful natural language for a particular audience, and the required rigour has shifted over time; a vague argument can simply be rejected by the community. Mathematical logic formalises the idea as a sequence of formulas in a formal language, each following from those before. Published proofs are assumed convertible into this form, though that is rarely done outside automated proof assistants. Proof theory treats proofs as data structures, which allows rival axiom systems such as non-Euclidean geometry, and has shown that almost every axiomatic system contains statements it cannot decide.
Techniques vary. A direct proof chains definitions and known facts: writing two even numbers as 2a and 2b shows their sum is 2 times a plus b, hence even. Induction, despite its name, is deductive, proving a base case and a rule carrying each case to the next.
Source: Mathematical proof
