With lateral thinking one goes on generating as many approaches as one can even after one has found a promising one.
Vertical thinking is selection by exclusion. One works within a frame of reference and throws out what is not relevant. With lateral thinking one realizes that a pattern cannot be restructured from within itself but only as the result of some outside influence.
Design is really a special case of problem solving.
Philosophers may look out at the world from a stained-glass window, but after a while they stop looking at the world and start looking at the stained glass.