For Fano and his intellectual heirs, it doesn’t matter whether a line “looks like” a line, a circle, a mallard duck, or anything else—all that matters is that lines obey the laws of lines, set down by Euclid and his successors. If it walks like geometry, and it quacks like geometry, we call it geometry. To one way of thinking, this move constitutes a rupture between mathematics and reality, and is to be resisted. But that view is too conservative. The bold idea that we can think geometrically about systems that don’t look like Euclidean space,* and even call these systems “geometries” with
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