the Newtonian theory makes predictions about the orbits of planets; if one puts into the equations a slightly wrong position, the theory’s prediction of the orbit will be wrong, but only slightly. Similarly, Euclid’s results on perfect spheres apply to real spheres, which are imperfect, not because of some magical idealization but because of the purely geometrical fact that the volume of, for example, an approximate sphere is close to the volume of a perfect sphere that it approximates. Physical systems that are not like this are unpredictable, even though deterministic; they are now much
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