Euclidean Geometry - Methods of Proof

Methods of Proof

Euclidean Geometry is constructive. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge. In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory. Strictly speaking, the lines on paper are models of the objects defined within the formal system, rather than instances of those objects. For example a Euclidean straight line has no width, but any real drawn line will. Though nearly all modern mathematicians consider nonconstructive methods just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious nonconstructive ones—e.g., some of the Pythagoreans' proofs that involved irrational numbers, which usually required a statement such as "Find the greatest common measure of ..."

Euclid often used proof by contradiction. Euclidean geometry also allows the method of superposition, in which a figure is transferred to another point in space. For example, proposition I.4, side-angle-side congruence of triangles, is proved by moving one of the two triangles so that one of its sides coincides with the other triangle's equal side, and then proving that the other sides coincide as well. Some modern treatments add a sixth postulate, the rigidity of the triangle, which can be used as an alternative to superposition.

Read more about this topic:  Euclidean Geometry

Other articles related to "methods of proof, methods, proofs, proof":

Euclidean Plane Geometry - Methods of Proof
... not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge ... Though nearly all modern mathematicians consider nonconstructive methods just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious nonconstructive ones—e.g ... some of the Pythagoreans' proofs that involved irrational numbers, which usually required a statement such as "Find the greatest common measure of..." Euclid often used proof by ...

Famous quotes containing the words methods of, proof and/or methods:

    Methods of thought which claim to give the lead to our world in the name of revolution have become, in reality, ideologies of consent and not of rebellion.
    Albert Camus (1913–1960)

    If any proof were needed of the progress of the cause for which I have worked, it is here tonight. The presence on the stage of these college women, and in the audience of all those college girls who will some day be the nation’s greatest strength, will tell their own story to the world.
    Susan B. Anthony (1820–1906)

    I think it is a wise course for laborers to unite to defend their interests.... I think the employer who declines to deal with organized labor and to recognize it as a proper element in the settlement of wage controversies is behind the times.... Of course, when organized labor permits itself to sympathize with violent methods or undue duress, it is not entitled to our sympathy.
    William Howard Taft (1857–1930)