Mathematical Proof

Mathematical Proof, an argument that is used to show the truth of a mathematical assertion. In modern mathematics, a proof begins with one or more statements called premises and demonstrates, using the rules of logic, that if the premises are true then a particular conclusion must also be true.

The accepted methods and strategies used to construct a convincing mathematical argument have evolved since ancient times and continue to change. Consider the Pythagorean theorem, named after the 5th century bc Greek mathematician and philosopher Pythagoras, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Many early civilizations considered this theorem true because it agreed with their observations in practical situations. But the early Greeks, among others, realized that observation and commonly held opinion do not guarantee mathematical truth. For example, before the 5th century bc it was widely believed that all lengths could be expressed as the ratio of two whole numbers. But an unknown Greek mathematician proved that this was not true by showing that the length of the diagonal of a square with an area of 1 is the irrational number à (see Number).

An example of a mathematical proof is the following argument, which proves that the Pythagorean theorem is true. Figure 1 and figure 2 demonstrate that the relationship A² + B² = C² holds in a right-angled triangle with sides A and B and hypotenuse C. Figure 1 shows that a square of side A + B can be divided into four of the right-angled triangles, a square of side A, and a square of side B. Figure 2 shows that a square of side A + B can also be dissected into four of the right-angled triangles and a square of side C. Since the two squares of side A + B have the same area, they must still have the same area once the four triangles are removed from each of them. The total area of the squares that remain on the left side is A ² + B², and the area of the square remaining on the right side is C². Thus A² + B² = C².

The Greek mathematician Euclid laid down some of the conventions central to modern mathematical proofs. His book The Elements, written about 300 bc, contains many proofs in the fields of geometry and algebra. This book illustrates the Greek practice of writing mathematical proofs by first clearly identifying the initial assumptions and then reasoning from them in a logical way in order to obtain a desired conclusion. As part of such an argument, Euclid used results that had already been shown to be true, called theorems, or statements that were explicitly acknowledged to be self-evident, called axioms; this practice continues today.

In the 20th century, proofs have been written that are so complex that no one person understands every argument used in them. In 1976 a computer was used to complete the proof of the four-color theorem. This theorem states that four colors are sufficient to color any map in such a way that regions with a common boundary line have different colors. The use of a computer in this proof inspired considerable debate in the mathematical community. At issue was whether a theorem can be considered proven if human beings have not actually checked every detail of the proof.