> As far as the written, formal language is concerned, English really has only three dialects

This is true, but I don't see why the "formal" qualifier is needed here :) There are much more than 3 dialects of English, both written & spoken.

Especially there's a fair few extremely common notable differences in (casual, written) Irish English: the word "amn't" (among other less common contractions), the alternative present tense of the verb "to be" (i.e. "do be"), various regional plurals of "you", and - perhaps the most common - prepositional pronouns, etc. etc.