I am afraid you mix up Greek and Latin. Indeed, 'museum' is rooted in Greek 'muse'. And 'a' in Greek means 'absense of'.
However, amusement is a French word, via Latin. The 'a' here means 'to', as in the Fench word 'à'. 'Muser', while possibly distantly related to Greek muses, here has a different meaning: 'a trifle', 'a diversion'. (Note the a's here are English 'a', and do not mean 'not' or 'to')
Amusement then, is 'to divert oneself with entertaining matter'. The very wod preserves
the notion that it is trivial, and that hence there are graver matters. A man's mind should not be exclusively given to the pursuit of only either one.
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