What seems obvious within a given space -- the shortest distance between two points, the way one travels from one point to another, the limits on how far one can travel at all -- is shown to be illusion when standing outside the space.
A priori propositions, they said, are (1) linguistic, (2) conventional, and (3) analytic: (1) They are statements primarily of how one proposes to use words; if one says that "a straight line is the shortest line between two points," this merely reports one's definition of "straight" and declares one's purpose to use it only of the shortest.
If you created a wormhole between two points in space, with one mouth moving relativistically relative to the other, observers at either traversible end would have aged by vastly different amounts.
Unlike the real number line, where there exists only one path and hence one distance between two points, there are multiple distinct paths between complex numbers.
The math lesson here is that a straight line does not always make for the fastest way between two points, that hills and curves speed one's ride.
Since a metric tree is a space in which there is only one path between two points x and y, this would imply that if z is a point between x and y, by which we mean if d x, z) + d z, y) = d x, y) then we know that z is actually on the path between x and y.
Roughly speaking, in a geodesic space the associated metric behaves in an orderly manner, i.e., making sure there is at least one shortest path between two points.
The principle of parsimony suggests that the most likely evolutionary path between two points in a sequence space is the one which requires the least number of events.
Calculating one-dimensional distance between two points when you know the value for each is a cinch.
