09 - Prouving - Prouver - Démontrer - 09

1: 3000 years of research

Does doubt exist in mathematics? Can one be satisfied with a collection of presumptions if they are 99% correct?
Demonstration is the basis of a mathematician's activity and is, in fact, its originality.
The first proofs were simple, written in a few lines and comprehensible to someone with a high school diploma.
Today we have proofs which represent hundreds of pages, which require the use of computers and which are verfiable by only a small number of specialists.
The complexity of the world poses mathematicians with more and more questions. To answer them they must work out models which must then be proved to be relevant.

2 From Pythagoras to Wiles

How can one demonstrate hypotheses that seem to be true? Do integers such as X2 + Y2 = Z2 exist? or such as Xn + Yn = Zn where n is more than 2?
The Greeks were the first to try to resolve these problems. Then Pythagoras gave his name to the theorem "The square of the hypotenuse...", while Euclid provided the oldest known proof.
Fermat later formulated that this result was not generalizable. Wiles demonstrated this conjecture in 1994! He used the most recent research in many areas of mathematics to produce his result.
Mathematicians regularly strive to make known the big problems which still have to be resolved.

• Pythagoras (6th cent. bc) • Euclid (3rd cent. bc)
• Pierre de Fermat (1601-1665)
• Andrew Wiles (Cambridge, 1953)

3 True but unprovable!

Can we always prove something that we know to be true?
In 1931, Kurt Gödel, in a genuine coup de théâtre, answered in the negative with his famous so-called "incompleteness" theorem.
He proved that the two notions of truth and provability do not coincide by discovering a formula about integers which is true but unprovable in elementary arithmetic.
Still more surprising, Gödel also showed, in the same spirit, that it is possible inside arithmetic neither to refute nor to prove that one will never arrive at a contradiction.
Elementary arithmetic is moreover undecidable. As a consequence, it is for instance impossible to write a computer programme that would check whether or not a given formula about integers is true.

• Kurt Gödel (1906-1978)

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