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jsteinhardt comments on Are Interesting Problems Useful? - Less Wrong Discussion
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At least in math, one common indication of of interestingness is how much something is connected to other things or shows up in different forms. Thus for example groups are very interesting in the abstract because they show up in many different mathematical contexts. This is a good metric for also determining usefulness since the more things something is connected to in the abstract the more likely it is that one of those connections will be practical.
I am advocating testing assertions like this, as well as possible. Like you say, groups are interesting in the abstract because of their many connections to other branches of mathematics. I am not convinced that research in abstract group theory is useful. I suspect be that basic notions and (very basic) results of group theory are useful, but that further work motivated only by its connection to group theory (as opposed to some other domain where the language of groups can be applied) has little value.