Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement. It shows that specific assumptions logically guarantee a conclusion. This process creates logical certainty, which is different from empirical arguments. Empirical arguments rely on observation to create a reasonable expectation. In contrast, a proof must demonstrate that a statement is true in all possible cases. Showing many examples where a statement holds is not enough. If a proposition is believed to be true but lacks a proof, it is called a conjecture. If it is used as an assumption for more work, it is called a hypothesis.

To build a proof, mathematicians use logic expressed through symbols and natural language. Most mathematical literature uses rigorous informal logic. This means it uses words to guide the reader through the reasoning. Purely formal proofs use only symbolic language without natural language. This specialized study is known as proof theory. A formal proof is a sequence of formulas. Each formula must be a logical consequence of the ones before it. This structure allows the concept of proof to be studied as a mathematical object itself.

There are several distinct methods used to establish mathematical truth. A direct proof establishes a conclusion by combining axioms, definitions, and previously proven theorems. For example, one can prove the sum of two even integers is even by using the definition of even numbers. Another method is proof by contraposition. This method proves the statement "if p, then q" by proving its logically equivalent version: "if not q, then not p." Proof by construction involves building a concrete example to show that something with a specific property exists. For instance, Joseph Liouville proved the existence of transcendental numbers by constructing an explicit example.

Other methods are used to handle complex or infinite scenarios. Proof by contradiction, or reductio ad absurdum, assumes a statement is false to show this leads to an impossible result. A famous use of this is proving the square root of two is irrational. Proof by mathematical induction is a deductive method used for sequences. It requires proving a single "base case" and an "induction rule." The rule must show that if one case is true, the next case is also true. This allows mathematicians to prove properties for infinitely many cases without checking each one individually.

Finally, proof by exhaustion divides a problem into a finite number of cases. Each case must be proven separately to reach the conclusion. The first proof of the four color theorem used this method. It required checking 1,936 different cases. This specific proof was controversial because a computer program checked most of them. The development of mathematical proof is tied to the history of ancient Greece. It likely began with geometry and practical land measurement. Thales and Hippocrates of Chios provided some of the earliest known proofs of geometric theorems.

Euclid revolutionized the field around 300 BCE with the axiomatic method. This method starts with axioms, which are propositions assumed to be self-evidently true. From these axioms, mathematicians prove theorems using deductive logic. Euclid's Elements was a foundational text for centuries. It covered geometry and number theory, including proofs about prime numbers. Later, in the 10th century, the Iraqi mathematician Al-Hashimi worked with numbers as "lines." He used them to prove algebraic propositions about multiplication and division. In the year 1000, Al-Karaji introduced an inductive proof for arithmetic progressions. This helped prove the binomial theorem and properties of Pascal's triangle.

Modern mathematics continues to explore the nature of these logical structures. Proof theory has discovered that almost all axiomatic systems can generate undecidable statements. These are statements that cannot be proved within that specific system. Philosophers also debate whether proofs are analytic or synthetic. The mathematician Paul Erdős famously admired elegant proofs. He spoke of "The Book," a hypothetical collection of the most beautiful proofs. In 2003, a book titled "Proofs from THE BOOK" was published. It contains 32 proofs that editors found particularly pleasing.