Nagging Question about Islam on What May Possibly be the Eve of Eid-el-Hajj, but We Won't Actually Know Yet if it Really is the Eve, or the Day Before the Eve, or Some Unrelated Day Until it Actually Happens

There are many things I don't understand in this world, matrix multiplication for example.  The non-commutative nature of it deeply disturbs me.  Another thing, how is it that Muslims never know when their holidays are going to be?  I know it depends on the moon, but what the moon is going to look like on any given day is a well-solved problem, and Muslim astronomers did most of the solving.  They were, in fact, renowned for using empirical astronomical observations.  So the fact that I never know when a holiday is going to be until I show up is annoying and baffling.  I asked a Muslim volunteer about this, and her unsatisfactory response was that it was like a period, you know about when it is going to happen but you never quite know exactly.  This is not a valid analogy, as the phases of the moon are affected by constant forces that do not involve anything so unpredictable as hormones, environment, nutrition, general physical condition, stress, age, etc.  So I still don't understand why Muslims aren't drawing on their own heritage of excellence in astronomy to figure out beforehand when holidays happen.  I am all for people getting time off from work for their religious occasions, but when religious holidays are also national holidays it would help a lot to know actually when they happen.  At least a day or two in advance would be nice.