In Hurshovki's construction of a new strongly minimal set, he uses a pre-dimension function to obtain a dimension function. I believe that it's a theorem that if you have a dimension function acting on the finite sets with given properties (see for example the comments at the beginning of page 3 : http://home.mathematik.uni-freiburg.de/ziegler/preprints/tutorial.pdf) then it has to come from a pregeometry. So it makes sense that you'd look for a function with these properties.
However, no intuition is offered as to why such a dimension function can be obtained from a pre-dimension function. The question is: what it the intuition behind using the $\delta$ (pre-dimension) function?