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“If it turns out P=NP, cryptography will break, and all kinds of practical problems will become easy” I can see why people like to speculate this sort of thing: it makes a compelling narrative. It’s just not accurate. If somebody proves P=NP, then maybe cryptography will break and computation power will jump by a few [...]

Zorn’s Lemma is one of the most beautiful and subtle axioms of mathematics. It is equivalent to its better-known sister, the Axiom of Choice, but in a certain sense, Zorn is the stronger statement– it’s certainly the more “mystical”. A mathematical joke goes: “The Axiom of Choice is obviously true, the Well-Ordering Theorem is obviously [...]

René-Louis Baire His “Baire space” and “Baire Category Theorem” help us understand topological spaces whose complements are very bare. George Cantor Taught us how to count infinite sets. Carl Friedrich Gauss Made a pretty good guess at how many primes there are below x. Also, some stuff about statistics. Kurt Gödel Played God by applying [...]

A subtle distinction in math is that between infinitely large things and arbitrarily large things. Suppose we have a set of elements, and each element has some “size”, where the sizes are allowed to range over the nonnegative numbers along with “infinity”. If one of the elements does happen to be infinite, then that implies [...]

Here’s the Cable Guy Anomaly: you’re waiting for a cable guy to come make an installation, and he has a 75% reliability, meaning a 75% chance he’ll show up at all. If he does show up, it’ll be sometime between noon and 2pm, with equal likelihood in each ten minute interval. As the minutes pass [...]

If the odds of an event are one in a billion, you shouldn’t bet the farm on it. But these events actually happen all the time; they are ubiquitous. Say you’re walking along the street, and you pass a dozen parked cars. What are the odds of those cars being parked in that exact order? [...]

Last quarter I taught business math, which was fun and new. Certain exercises from the book, however, made no sense. Equations of Value are interesting, but it seems some mathematicians who teach them (or who write the textbooks), don’t have a grasp of what’s really going on. The idea behind Equations of Value is that [...]

If mankind descends from an original matriarch and patriarch, then intuitively, there ought to be a giant family tree containing all of the human species. Granted, it might be a giant mess, but at the very least, it should intuitively have the property that you can’t find an infinite line of ancestors: if you start [...]

Imagine a scene in a fantasy novel. The wise old prophet declares there is a secret bloodline of heroes destined to overthrow the forces of evil. What’s more, this special bloodline has the following property: every twelfth and seventeenth descendant hero is a girl, and all the others are boys. In an effort to save [...]

Child psychologists once noticed an amazing pattern. Whenever six or more children got together, either three would become mutual friends, or three would become mutual non-friends. It was more than mere rule of thumb: it happened every single time, without fail, without even one single “exception that proves the rule”. The psychologists were baffled, and [...]