I was reading the wikipedia page on intersection number: http://en.wikipedia.org/wiki/Intersection_number#Intersection_multiplicities_for_plane_curves
and I was confused by property number 6 they want there. I cannot justify why this is a natural property to have.
I have seen the definition of intersection number via dimension of the complete local ring of the scheme theoretic intersection over the base field. From this definition property 6 is natural, but I would like to see a low-brow (i.e. in classical language), intuitive reason for expecting 6.