# Demonstration

In Mathematical, a demonstration (one crosses sometimes, in particular in the scientific community, the Anglicisme “  proof   ”, which is to be proscribed) is a reasoning which makes it possible to establish Assertion S starting from properties previously established or allowed. The shown assertion can then be itself used in evidence. A demonstration cannot be disputed, only the assumptions can be discussed. The demonstrations are articulated on the Logique.

A Conjecture is an assertion, confirmed by examples, for which no proof exists. The refutation of a conjecture passes not by a demonstration but by the search for Contre-exemple S.

## Methods of demonstration

• the direct Démonstration is the simplest method of demonstration being based on the logical deduction.
One opposes sometimes deductive demonstration (used for example showing the existence of an object starting from theorem ensuring its existence without to have built this object explicitly) and constructive Démonstration (which consists in building a concrete example having a certain property, to show that there exists at least an object having this property).
• the reductio ad absurdum is a method which consists in establishing the nonsense of the contrary proposal.
• the inductive demonstration

In the context of the Théorie of the demonstration, in which purely formal demonstrations are considered, of the demonstrations which are not entirely formal “social demonstrations are called”. In fact demonstrations are based on assertions considered as exact because they are allowed by a whole of people. The idea is accepted like exact when it makes the consensus.

The result which is shown calls a Théorème.

Once the shown theorem, it can be used as bases to show other assertions.

An assertion which is supposed to be true but which was not shown yet is called a Conjecture.

Sometimes it is possible to show that a certain assertion cannot be shown starting from a given whole of axioms; it is the case of the Axiome of the choice and of the Hypothèse of continuous the with respect to the axiomatic one of the set theory of Zermelo-Fraenkel, one says whereas this assertion is independent of this system of axioms. In much of the systems of common axioms in mathematics, there exist assertions which can be neither shown nor refuted; to see the Theorem of incomplétude of Gödel.

The formal demonstrations require a very great rigor and an special attention; we must specify the rules of logic the type of reasoning which we use, to define if required of new mathematical objects for which we require, to point out the axioms or the theorems to which we refer, to check that we are goods under the conditions for application of a theorem before using it, etc

In addition, it is possible to write all the demonstrations in formal language, but that is seldom done. Too much formalism would make almost impossible the comprehension of a demonstration and would dissimulate the general idea in an excessive symbolism. However as that by using a software of assistance to the demonstration that is possible starts to be done more and more.

There is thus a compromise between a very thorough rigor in the field of logic and a surface rigor using the natural language more.

A demonstration, within the academic framework (course, delivers, exposed…) is not in general detailed at the point to be “right” within the meaning of logic; in general one is satisfied to give sufficiently precise elements so that the auditoire/le assistantship concerned is convinced. Indeed, to define $1$ properly, one would need already pages and pages! This is why for example a demonstration given to the level license will not be appropriate for a pupil of sup: for this last, the demonstration will not be detailed enough. In this direction, a demonstration is something of very relative.

## See too

• Theory of the automatic demonstration
• Demonstration of theorems

### External bonds

• has what serves the demonstration. The example of the theorem of Pythagore (Flash)
• Can one all show?
• To make a demonstration in mathematics

Zh-classical: 證明 Zh-min-nan: Chèng-bêng

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