2B) However, in a system of co-ordinates, that distance can be reinterpreted in a representation in terms of the co-ordinates. There are two possibilities again. Either the distance can be defined as simply the formula dist[{x, y}, {a, b}]
((x - a)2 + (y - b)2 ) with {x, y} the origin - above {x, y} = {0, 0} - or it can be defined geometrically as the hypotenuse of the differences of the co-ordinates. If either definition is accepted, then one can use Pythagoras’s theorem to derive the other.
The essential difference between (2A) and (2B) is that (2A) is elementary and poor in concepts and results, while (2B) is complexer and rich in concepts and results. Viewpoint (2A) only allows us to use measuring rods between arbitrary points and little else. We are allowed to sweep the rod around the center, and thereby draw the circle, but then it somehow stops. Viewpoint (2B) allows us to do much more. A line between two points is interpreted in terms of a system of co-ordinates, and that opens the scope for new results.
We find that the opposition of (1) against (2) is rather messy, and (2) actually hides two suppositions. The ease of (2) depends directly upon the ease of (2A), while (1) actually compares with (2B) that is complexer. The phrase “In other words” in (2) above thus was misleading, and actually represents the introduction of another assumption.
With this clarified, we also note that (2) is stronger than (1), and that it was possible to seduce the human mind to accept (2) rather easily. There has been a progression in concepts, resulting in stronger definitions.
Note that behind all this there is a notion of empirical space. In (1) there is a hidden assumption of a flat space. In (2B) the assumption is made explicit, and then open to amendments (curved surfaces, or abstract spaces). The movement of (1) to (2) thus is, partly, (a) the advancement in concepts by means of the definition of distance (and the circle as a collection of equal distance points), (b) the introduction of the separate step of observation - with the difficulties: when does the definition apply to reality, or if there is some reality, how do I select the proper definition ?
The point that is relevant for this book then is: that the definition is so good, that it in practice substitutes for many everyday empirical problems. A criterion for a good definition is: that it can be such a substitute.