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> Ask anyone who hasn't actually done pure math

Here I am.

I think pure math is simply not applied math - so math that is not being done in direct service of a particular goal, such as working out insurance pricing or perhaps ML/AI etc.

So it's essentially about who pays you. Seems fairly clear to me.

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What I meant was what it is as a practice, not how it is used. "Applied vs pure" anything could be defined as such and it's not particularly helpful, beyond showing you have a basic grasp of English.

> anything could be defined as such

Not really, because many disciplines are inherently applied. There is, for example, no such thing as "pure teaching", since teaching is always practical.

To me the lack of a practical purpose at the moment is the defining characteristic.

I guess the real question is what do you think "do math" means.

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