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slicko comments on Innate Mathematical Ability - LessWrong
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I'm reminded of Graham's number (g notation) as an example where new notation (kind of) was invented for the purposes of a math paper.
I read a riveting blog post a few months ago introducing several concepts and building up to graham's number in a very accessible read if anyone's interested:
http://waitbutwhy.com/2014/11/1000000-grahams-number.html