I liked this article. One notable point that felt like it was missing in the article is that the Prime Number Theorem, that the count of primes grow like (n / l

I liked this article. One notable point that felt like it was missing in the article is that the Prime Number Theorem, that the count of primes grow like (n / ln n) was provided such a map by Riemann in the letter in which he put forward his infamous eponymous hypothesis. That letter introduced the idea of using analysis to the Prime Number Theorem, extending the groundbreaking work of Riemann's friend Dirichlet who introduced the world to analytic number theory in Dirichlet's Theorem on the infinitude of primes in arithmetic progressions. It would take nearly half a century for mathematicians to digest the application of Fourier Analysis put forward by Riemann, and the proof of the Prime Number Theorem came only in the early 1900's. By then the analytic machinery would have been more commonly taught -- probably largely due to the advent of electrical engineering. Erdos and Selberg eventually put out fully arithmetic proofs of the Prime Number Theorem. And generally the helicopter analogy from the article probably doesn't apply so well to mathematics because you can probably always reduce theories and encapsulate all the dependent proofs to arithmetic first principles, but of course you already have the map. Recently the proofs of the Sensitivity Conjecture by Hao Huang and of the Bounded Gaps Between Primes by Yitang Zhang surprised mathematicians in how little new machinery these seemingly intractable problems required -- in the case of Zhang application of "hard work" on top of GPY and Hao Huang, a single clever insight.