Axioms and Postulates

Did you notice how it is difficult to precisely define the concepts of points and lines, and how we had to rely on our intuitive understanding to provide some definitions? Euclid faced the same problem.

For some geometrical concepts which are so fundamental as to be difficult to define, but which he thought are intuitively well-understood, Euclid assumed that no definitions or justifications were required. He took their truth to be granted. Such concepts and ideas can be thought of as obvious truths. Such obvious truths are referred to as axioms or postulates.

For example, one of Euclid’s postulates is that a unique straight line can be drawn from any one point to any other point. The truth of this statement seems to be obvious – if we were to plot two points A and B in the plane, we would be able to draw one (and only one) line passing through A and B. Since this seems to be such a fundamental idea, Euclid saw no reason to try and prove this somehow, and instead took its truth as granted. Thus, he talked about it as a postulate – a universal truth.

Axioms and postulates are almost the same thing, though historically, the descriptor “postulate” was used for a universal truth specific to geometry, whereas the descriptor “axiom” was used for a more general universal truth, which is applicable throughout Mathematics (nowadays, the two terms are used interchangeably; in fact, postulate is also a verb – to postulate something).

For example, one of Euclid’s axioms was that if A = B, and C = D, then

A + B = C + D

He stated this in words as follows: if equals are added to equals, the wholes are equal.