13 Facts About Mathematical proof

Mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.

The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of its greatest achievements.

Mathematical proof was revolutionized by Euclid, who introduced the axiomatic method still in use today.

For example, direct Mathematical proof can be used to prove that the sum of two even integers is always even:.

Probabilistic Mathematical proof is one in which an example is shown to exist, with certainty, by using methods of probability theory.

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Probabilistic Mathematical proof is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'.

Combinatorial Mathematical proof establishes the equivalence of different expressions by showing that they count the same object in different ways.

Nonconstructive proof establishes that a mathematical object with a certain property exists—without explaining how such an object can be found.

In contrast, a constructive Mathematical proof establishes that a particular object exists by providing a method of finding it.

However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first Mathematical proof of the four color theorem is an example of a computer-assisted Mathematical proof.

In practice, the chances of an error invalidating a computer-assisted Mathematical proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs.

An elementary Mathematical proof is a Mathematical proof which only uses basic techniques.

Particular way of organising a proof using two parallel columns is often used as a mathematical exercise in elementary geometry classes in the United States.