From the Greek ‘axiomein’ meaning ‘to demand’ or ‘to think worthy’, which creates the abstract noun ‘axioma’ (‘a thing which is demanded or worthy’), an axiom is an assumption that does not need to be expanded upon to be made true; it is true in and of itself.
In Euclidean geometry and philosophy, sets of axioms were drawn up meaning that it was universally common that one can draw a straight line from A to B and extend it infinitely to point C without deviation. A circle’s properties can be found with a centre, and a radial point extending from its centre to its edge, can say that right angles are equal to one another at ninety degrees, and that if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.
In mathematics, an axiom can be logical (‘A and B implies A’) or non-logical (a + b = b + a), and serves as a starting point from which other statements are logically derived. Unlike theorems, axioms are not demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems). Tautologies excluded, nothing can be deduced if nothing is assumed.
Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge; all other assertions (or, mathematically, theorems) must be proven with the aid of these basic assumptions. Sir Isaac Newton, for instance, built on Euclid and thought of an axiom on the non-relation of space-time and the physics taking place in it at any moment. Albert Einstein's work on Special Relativity, and later the General Relativity, replaced Newtonian axioms.
Examples of famous verbal axioms are the opening line of Jane Austen’s ‘Pride and Prejudice’ whereby marriage is a ‘truth universally acknowledged’, and the American Constitution which decrees that ‘all men are created equal’ is a truth which is ‘self-evident’.
In Euclidean geometry and philosophy, sets of axioms were drawn up meaning that it was universally common that one can draw a straight line from A to B and extend it infinitely to point C without deviation. A circle’s properties can be found with a centre, and a radial point extending from its centre to its edge, can say that right angles are equal to one another at ninety degrees, and that if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.
In mathematics, an axiom can be logical (‘A and B implies A’) or non-logical (a + b = b + a), and serves as a starting point from which other statements are logically derived. Unlike theorems, axioms are not demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems). Tautologies excluded, nothing can be deduced if nothing is assumed.
Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge; all other assertions (or, mathematically, theorems) must be proven with the aid of these basic assumptions. Sir Isaac Newton, for instance, built on Euclid and thought of an axiom on the non-relation of space-time and the physics taking place in it at any moment. Albert Einstein's work on Special Relativity, and later the General Relativity, replaced Newtonian axioms.
Examples of famous verbal axioms are the opening line of Jane Austen’s ‘Pride and Prejudice’ whereby marriage is a ‘truth universally acknowledged’, and the American Constitution which decrees that ‘all men are created equal’ is a truth which is ‘self-evident’.
