Mathematically speaking, what is an ‘Axiom’?
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In math, axioms are a set of statements which you simply assume to be true without proof. You then try to deduce what else is true, assuming that the axioms are. When you succeed in proving some interesting consequence of the axioms, you call it a theorem.
For example, there is an axiom in geometry called the triangle postulate. It states that the sum of the angles of any triangle is equal to 180 degrees. One interesting consequence of this axiom is the Pythagorean theorem – in a right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the two other sides. This theorem only holds true if you assume the triangle postulate first. If you don’t, then what you have is a non-Euclidean geometry, and the Pythagorean theorem does not work.
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