It’s used to signal when you have a pair of the same letters but the second has a different pronunciation.

So “deer” has two e’s but no umlaut because they make the same sound but “coördinate” has the umlaut since the second o is a new sound.

I personally like them and wish they would come back if only as a small push back against the horrific ambiguity that is the English language

> It’s used to signal when you have a pair of the same letters but the second has a different pronunciation.

That's a somewhat naïve take.

As far as I understand, it's actually used to indicate a hiatus instead of a diphtong. E.g. naïvety is meant to indicate that aï are part of different syllables and shouldn't be pronounced in the same syllable, just like the oö in coördinate.

The catch is that the diaeresis in Türkiye does something completely different that the diaeresis you mentioned, so actually it would be less confusing if English didn't have it at all.

...then again, even some Germans are using the diaeresis for this purpose, such as this guy https://en.m.wikipedia.org/wiki/Bernhard_Ho%C3%ABcker - and you can't blame him, because usually "oe" is a transliteration of "ö", and then his name would mean "hump" (as in camel's hump).

Where did you go to school, if you don't mind me asking? I didn't realize that was ever something that was taught -- I'm curious if it's a geographic thing?

The only place I've ever seen the diaeresis used like that is in the New Yorker.

Ontario, Canada, in the '90s. We weren't required to use it but there was a brief lesson on co-operate, cooperate, coöperate, etc. being valid variants. Maybe Mr. B. aspired to have students who would one day read The New Yorker.