The Joy of X: A Guided Tour of Mathematics, from One to Infinity

by Steven H. Strogatz

These are states of intractable conflict, with the network split into two hostile factions (of arbitrary sizes and compositions). All members of one faction are friendly with one another but antagonistic toward everybody in the other faction. (Sound familiar?) Perhaps even more surprisingly, these polarized states are the only states as stable as nirvana. In particular, no three-party split can have all its triangles balanced. Scholars have used these ideas to analyze the run-up to World War I. The diagram that follows shows the shifting alliances among Great Britain, France, Russia, Italy, Germany, and Austria-Hungary between 1872 and 1907.

As an undergraduate at Yale, he desperately wanted to stay in the Ivy League for graduate school. Unfortunately Princeton rejected his application. Harvard said yes but seemed to be dragging its feet about providing the financial support he needed. His best option, though he found it depressing, was MIT. In Gell-Mann’s eyes, MIT was a grubby technological institute, beneath his rarefied taste. Nevertheless, he accepted the offer. Years later he would explain that he had contemplated suicide at the time but decided against it once he realized that attending MIT and killing himself didn’t commute. He could always go to MIT and commit suicide later if he had to, but not the other way around.

Specifically, early in the development of quantum mechanics, Werner Heisenberg and Paul Dirac had discovered that nature follows a curious kind of logic in which p × q ≠ q × p‚ where p and q represent the momentum and position of a quantum particle. Without that breakdown of the commutative law, there would be no Heisenberg uncertainty principle, atoms would collapse, and nothing would exist. That’s why you’d better mind your p’s and q’s. And tell your kids to do the same.

Yet the allure of algebra is seductive, and in gullible hands it spawns such silliness as a formula for the socially acceptable age difference in a romance. According to some sites on the Internet, if your age is x, polite society will disapprove if you date someone younger than x/2 + 7. In other words, it would be creepy for anyone over eighty-two to eye my forty-eight-year-old wife, even if she were available. But eighty-one? No problem. Ick. Ick. Ick . . .

Electrical engineers love complex numbers for exactly this reason. Having such a compact way to represent 90-degree rotations is very useful when working with alternating currents and voltages, or with electric and magnetic fields, because these often involve oscillations or waves that are a quarter cycle (i.e., 90 degrees) out of phase.

Those of us who teach maths should try to turn this bug into a feature. We should be up front about the fact that word problems force us to make simplifying assumptions. That’s a valuable skill — it’s called mathematical modeling. Scientists do it all the time when they apply maths to the real world.