A tour of the main branches of mathematics, organised around the split between pure and applied mathematics, opens by noting that pure mathematics was first developed to satisfy curiosity and only later supplied methods for practical problems in engineering, finance, biochemistry and astronomy. Pure mathematics is then laid out by area: number systems from the natural numbers through integers, rationals, reals and complex numbers; sets and cardinal numbers; structure, covered by abstract algebra (groups, rings, fields) and linear algebra; space, ranging from Euclidean geometry, the Pythagorean theorem and trigonometry to non-Euclidean geometry, differential geometry, topology, fractal geometry and measure theory; and change, studied through calculus, real and complex analysis, differential equations, dynamical systems and chaos theory. Along the way it mentions the solved Poincaré conjecture, the still-open Hodge conjecture, and the computer-assisted proofs of the four colour theorem and the Kepler conjecture.
Applied mathematics is described through its overlap with statistics and probability, numerical analysis and scientific computing, game theory, mathematical physics and fluid dynamics. The piece ends by observing that mathematics touches every part of the real world and that, in the view of some philosophers, everything in the universe can be explained through it.
