The distinction between "pure" and "applied" math is, when you get down to it, a little nonsensical. Areas of math that originally seemed entirely abstract and "pure" have often had surprising applications in many of the sciences - should we define pure math as "all math for which we haven't found a practical use yet"? I'm not sure this is a useful way of thinking about it.
I think this also leads to another false dichotomy between abstraction and practicality. We use abstractions all the time in more practical fields: consider the idea of evolution, and all its applications in modern science. It is certainly a broad abstraction of many much more complicated interactions, and it does not "exist" in any physical sense, and yet we do not dismiss it as "too abstract".
In the end, abstractions are just ideas of generalization, and generalization is a useful tool. While the the ancient mathematicians lacked some of the more powerful tools of generalization (like symbolic algebra), they evidently saw the value of these ideas and put them into practice.
Good! So many of these binaries or dichotomies are pushed beyond their usefulness (as a provisional way of sorting things out) and are taken as absolute. I like the example of evolution as an abstraction that doesn't appear to be 'too abstract'!
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