Even if geometry can be recast as an exercise in manipulating meaningless strings of symbols, no human being can generate geometric ideas without drawing pictures, without imagining figures, without thinking of the objects of geometry as real things. My philosopher friends typically find this point of view, usually called Platonism, fairly disreputable; how can a fifteen-dimensional hypercube be a real thing? I can only reply that they seem as real to me as, say, mountains. After all, I can define a fifteen-dimensional hypercube. Can you do the same for the mountain?
