The Geometry of Straight Line Figures

Abstract

We have seen that the step from arithmetic to algebra is merely the transference of the rules of numbers to letters, which, in a sense, makes algebra more abstract than arithmetic. But this difference is more apparent than real because numbers are abstractions also unless we associate them with measurable entities. Since a measurement is never precise, however, the number assigned to such a measurement is something of an abstraction and must be taken on faith. We accepted this abstraction when we introduced the ordinal aspect of numbers by assigning integers or fractions to points on a line because the points themselves are abstractions. Numbers become abstract even when we assign them to collections of discrete entities such as people or atoms. We can certainly speak of 5 people or the population of a town but things are not so clear when we are dealing with atomic entities. To say that we have one atom or one electron in a box is an abstraction since we can never isolate an atom or an electron with a certainty that permits us to speak of only one.

One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.

—HEINRICH HERTZ