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]]>Take the unit interval (the real numbers between 0 and 1, inclusive). Every continuous function on the unit interval is a limit of polynomial functions (that’s the Weierstrass approximation theorem). Likewise, any continuous function is a limit of sums of trigonometric functions (that’s Fourier series). But a trig function is not finitely expressible in terms of polynomial functions, and polynomial functions are not finitely expressible in terms of trig functions.

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]]>I think that this is arguable. If the incommensurability is a result of different property functions being singled out within those vector spaces, or even one property function intuitively or otherwise negating others in some way, then I can see this incommensurability as not only existing but also having significant importance as Kuhn did.

Whether it’s trying to reconcile the quantum realm with that of the macroscopic, there are certain ways of looking at the world (and any resultant property functions) that are incompatible with others (e.g. Copenhagen interpretation vs. the MWI for quantum mechanics). I do like the idea of “property functions being singled out for pragmatic reasons”, but I also realize that they may only look like functions when in reality there are either other hidden variables which cause them to not reconcile with other property functions, or they are just coincidental close approximations. It’s hard to say.

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