You ever read a passage that comes out of nowhere and just sticks, and you find yourself repeating its general idea long after reading it? That happened--and is happening--to me with the following passage from Concepts of Modern Mathematics. (The first paragraph sets up the flow.)
"A point lies on a line, geometrically, if it is a set-theoretic member of the line. So a point lies on two lines L and M if it is a member of L and a member of M, in other words, if it is a member of the intersection L & M. Set-theoretic intersection corresponds to geometrical intersection.
"Proceeding in this way, using coordinate geometry as inspiration, you can set up the whole of Euclidean geometry as part of set theory. From the way that you want geometry to behave, you can construct a purely mathematical theory. But now, instead of indulging in deep metaphysical arguments about the 'real' geometry, you can say: here is a mathematical theory. It deals with things which I call 'points' and 'lines.' I suspect that in the real world very small dots and very thin lines will behave in approximately the same way. And then people can go away and do experiments, to see if you are right. And even if it turns out that with very exact measurements you are wrong, you will still have a nice theory."
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