Quotes from Steven H. Strogatz

First comes intuition. Rigor comes later.

we've come to realize that most systems of differential equations are unsolvable, in that same sense; it's impossible to find a formula for the answer. There is, however, one spectacular exception. Linear differential equations are solvable.

a big, messy linear problem can always be broken into smaller, more manageable parts. Then each part can be solved separately, and all the little answers can be recombined to solve the bigger problem. So it's literally true that in a linear problem, the whole is exactly equal to the sum of the parts.

Whenever the whole is different from the sum of the parts—whenever there's cooperation or competition going on—the governing equations must be nonlinear.

And because the PageRanks are defined as proportions, they have to add up to 1 when summed over the whole network. This conservation law suggests another, perhaps more palpable, way to visualize PageRank. Picture it as a fluid, a watery substance that flows through the network, draining away from bad pages and pooling at good ones. The algorithm seeks to determine how this fluid distributes itself across the network in the long run.

When a guitar string is plucked or when children jiggle a jump rope, the shape that appears is a sine wave. The ripples on a pond, the ridges of sand dunes, the stripes of a zebra—all are manifestations of nature's most basic mechanism of pattern formation: the emergence of sinusoidal structure from a background of bland uniformity.

Simulation is no substitute for math—it could never provide a proof—but if Peskin's conjecture was false, this approach would save me a lot of time by revealing a counterexample. This sort of evidence is extremely valuable in math. When you're trying to prove something, it helps to know it's true. That gives you the confidence you need to keep searching for a rigorous proof. Programming

Simulation is no substitute for math—it could never provide a proof—but if Peskin's conjecture was false, this approach would save me a lot of time by revealing a counterexample. This sort of evidence is extremely valuable in math. When you're trying to prove something, it helps to know it's true. That gives you the confidence you need to keep searching for a rigorous

That latter result, 3 + 10/70, reduces to 22/7, the famous approximation to ? that all students still learn today and that some unfortunately mistake for ? itself.

Nature—cue the theme from The Twilight Zone—somehow knows calculus.

If whole numbers and their ratios couldn't even measure something as basic as the diagonal of a perfect square, then all was not number. This deflating letdown may explain why later Greek mathematicians always elevated geometry over arithmetic. Numbers couldn't be trusted anymore. They were inadequate as a foundation for mathematics.