A World of Points
To deal with points in a plane the mathematician draws two perpendicular lines, and locates any point, as P, by measuring its distances, X and Y, from these “coordinate axes.” The directions of his axes acquire for him a peculiar significance, standing out above other directions; he is apt to measure the distances
and
between the points P and Q in these directions, instead of measuring the single distance PQ. We do the same thing when we say that the railroad station is five blocks north and two east.
The mathematician visualizes himself as an observer, located on his coordinate framework. For another observer on another framework, the horizontal and vertical distances
and
between P and Q are different. But for both, the distance from P direct to Q is the same. In each case the right triangle tells us that:
Imagine an observer so dominated by his coordinate system that he knows no way of relating P with Q save by their horizontal and vertical separation. His whole scheme of things would be shattered by the suggestion that other observers on other reference frames find different horizontal and vertical components. We have to show him the line PQ. We have to convince him that this length is the absolute property enjoyed by his pair of points; that horizontals and verticals are merely relations between the points and the observer, result of the observer’s having analyzed the distance PQ into two components; that different observers effect this decomposition differently; that this seems not to make sense to him only because of his erroneous concept of a fundamental difference between verticals and horizontals.