steppi

Like the article says, what they did was ahead-of-its-time, and a monumental influence on all subsequent work on formal systems, including Gödel's work, regardless of whether Russell and Whitehead achieved their initial aims.

Let's take the Fundamental Theorem of Calculus as an example[0]:

  f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h}

This isn't the Fundamental Theorem of Calculus, it's the usual definition of the derivative of a function of a single real variable. The Fundamental Theorem of Calculus establishes the inverse relationship between differentiation and integration [0].

Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall (or both!)

Ramanujan was no stranger to banging his head against the wall. My impression from Kanigel's The Man Who Knew Infinity is that his work ethic and mathematical fortitude were as astonishing as his creativity. For much of his career, he couldn't afford paper in quantity and did his hard work on stone slate, only recording the results. This could make it seem like his results were a product of pure inspiration because he left no trace of the furious activity and struggle that was involved.

From The Man Who Knew Infinity:

When he thought hard, his face scrunched up, his eyes narrowed into a squint. When he figured something out, he sometimes seemed to talk to himself, smile, shake his head with pleasure. When he made a mistake, too impatient to lay down his slate pencil, he twisted his forearm toward his body in a single fluid motion and used his elbow, now aimed at the slate, as an eraser. Ramanujan's was no cool, steady Intelligence, solemnly applied to the problem at hand; he was all energy, animation, force.