I don't know whether it is genetic or not, but can at least attest to the diversity. I have a friend who's thought is so multi-sensory and vivid, that considering an action is often very difficult to disassociate from performing that action. For someone else I know, thought is visual alchemy, with strange image barrages combining with each other to inexplicably repersent the world - consequently, articulating his emotions or ideas is challenging. For another friend, everything has an example in memory, and his thoughts are like the old-fashioned detective's bulletin board with all the lines drawn between experiences and events (except without that image - he just cross-references directly). Myself, I rarely, rarely ever see a single image. When I do, even the act of visualising it is abstract, unconscious. I simply know it is in memory currently, and capable of being analyzed for whatever content I desire. Everything is verbal, or at least has a verbal texture to it, if it isn't explicitly articulated.
So maybe somasimple is right in saying that the average person is visual; I don't really know, but I'm willing to believe since most people at least have a visual aspect. Mine usually don't. Yours seems very interesting - what were you thoughts like before you were introduced to mathematics?
lates,
cotarded