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The purpose of a PhD is to certify that you're able to do independent research. Many researchers retrain (or just add a research interest) in adjacent fields during their postdocs or later. At that point it's just research.

It's possible! From somewhat famous mathematicians, at least Bela Bollobas has 2 PhDs: one in discrete geometry and one in functional analysis.

Try doing that in the modern academic environment tho..

Beside the habilitation example of rando234789 (https://news.ycombinator.com/item?id=44498702), in Russia (and Ukraine) there indeed exist two "doctorate levels": кандидат наук [Candidate of Sciences] and доктор наук [Doctor of Science].

I feel like most sciences should have this, it would accelerate science a lot via the cross-pollination of ideas and techniques.

But I can imagine that drawing connections between different branches of maths would be especially powerful, yes

check out the idea of a habilitation: https://en.wikipedia.org/wiki/Habilitation at least in germany, it is pretty much what you describe