Because it has only conceptual existence - - - it's a concept, not a thing.
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Mathematics was not so much discovered as developed over time by many different sources and cultures.
Physics
pure mathematics is mathematics studied and used reasons other than application. So some algebra certainly might be, but not all algebra. Modern or abstract algebra would fit better in that category.
Mathematics is the language of physics, and all the sciences are branches of physics. So if we are to explain, describe, or understand anything in our universe, we must use mathematics in one form or another to come to that understanding.
It tells you the answer so you don't waste paper