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A Mysterious Paradox

Time to post something on the philosophy of maths I guess. As far as I can see, the *mysterious* thing about the Banach-Tarski paradox (which recently cropped up in an irrelevant aside here) is how elusive are the intuitions that are offended by it, as Feferman observed in his 2000 'Mathematical Intuition Vs. Mathematical Monsters' (Synthese 125, 317-32).Very roughly, a unit sphere (within a standard 3-space) can be considered to be made up of five pieces, which can be rearranged to form *two* unit spheres. That appears to be particularly paradoxical in an imaginary 4-space. Think of an impenetrably rigid 3-dimensional sphere, made of some perfectly smooth material (unrealistic but classical), in a 4-dimensional space. According to the Banach-Tarski theorem, it can be broken up into 5 similarly rigid pieces (connected subsets of points) that can be moved rigidly (in the fourth dimension, since they cannot pass through each other) to form two new spheres, each identical to the original sphere (in intuitive contravention of some sort of conservation law).

The paradox seems to arise from some intuition that pieces of things should not behave like that, but I find it hard to pin that intuition down in such a way that it would still apply to classical (and therefore unrealistic) things. E.g. I recently thought that dropping one of the 5 pieces into a measuring flask would give us a more paradoxical result, whereas either the measuring fluid would be unable to fill up all the space around the piece, or the fluid would develop a weird surface (or else be useless for measuring anything). So I wonder how other people think of the paradoxicality of the Banach-Tarski...

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