Lambda the Ultimate

I have been thinking a little about pointers, and what kind of mathematical structure they form. They seem to be like a one-dimensional vector (as opposed to a simple scalar). If we use the terms location for a point in n-dimensional space, and distance for a vector, then we get the following properties, adding a distance to a location results in a location, subtracting a distance from a location results in a location, subtracting a location from a location results in a distance.

Where is gets more tricky is that adding a location to a location would seem to be invalid (I don't know what type of thing this is), and yet to average two locations, to get the mid-point seems perfectly reasonable. Topologically what is going on here? What is the type of a location added to a location?

If we treat all locations as distances with an origin at zero, then everything makes sense again in terms of types, except for zero itself which seems to still be a location and not a distance (or it all gets horribly self-referential).

Any thoughts on how to make sense of this, what the type of the sum of two locations might be, etc?