"Intentionality means that all the thoughts we’re having are directed towards something, an intentional object." A mug is for drinking out of; you can do this and that to it and you'll have such and such results. Same with a Lie group; it's defined by what you can do with it.
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This is what I was pointing at here: https://srconstantin.wordpress.com/2015/04/30/choice-of-ontology/comment-page-1/ … You can't have concepts without some kind of motivation or prioritization; your motives determine what counts as an "irrelevant" feature to abstract away, which in turn defines your concepts.
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