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RolfAndreassen comments on "What Is Wrong With Our Thoughts" - Less Wrong
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I'd unify your "So What" with "meaningless" into a single category "does not constrain observations". Math passes the test inasmuch as it constrains observations about outcomes of proof checking.
But now some people will complain (are already complaining) that we reject the majority of humanity's thought.
Again, it does seem observable that nobody has explained why three is prime and four isn't. (I'm not sure you can actually use 'why' in an intelligible way here; possibly I'm being confused by non-mathematical language applied to math.) It's not an observation I would expect anyone to care about, and possibly it may be the equivalent of nobody having seen something invisible; but it does seem to make a statement that could in principle have gone the other way.
I agree that I'm not sure how you're intending to use 'why' here, and I'm pretty sure there's a good answer for any particular meaning.
To answer the question in a possibly unsatisfactory way, 3 is prime because it is a natural number which has exactly two distinct natural number factors, whereas 4 is not prime because it has more than two distinct natural number factors.