"The tacit assumption on which analytic geometry operated was that it was possible to represent the points on a line... by means of numbers. This assumption is... equivalent to the assertion that a perfect correspondence can be established... The great success of analytic geometry... gave this assumption an irresistible pragmatic force. It was essential to include this principle... But how? Under such circumstances mathematics proceeds by fiat. It bridges the chasm between intuition and reason by a convenient postulate. ...The very vagueness of all intuition renders such a substitution... highly acceptable. ... On the one hand there was the logically consistent concept of a real number and its aggregate, the arithmetic continuum; on the other hand, the vague notions of the point and its aggregate, the linear continuum. All that was necessary was to declare the identity of the two... to assert that: It is possible to assign to any point on a line a unique real number, and, conversely, any real number can be represented in a unique manner by a point on a line. This is the famous Dedekind-Cantor axiom."