Actually, thermodynamics is hard and I don’t understand it.  But even without totally understanding thermodynamics, it turns out its possible to do a surprising number of useful calculations with just a couple of simple rules about entropy. The setup is as follows: Imagine that there is some set of states of the world, called the […]

I’m not a physicist, and I’d always supposed that, while the Earth has a significant gravitational pull because it’s so massive, the gravitational pull between everyday objects must be completely undetectable, or maybe only detectable with modern laboratory equipment. But I only thought that because I never bothered to actually plug in any numbers.  Using […]

Mathematical logic has a categorization of sentences in terms of increasing complexity called the Arithmetic Hierarchy.  This hierarchy defines sets of sentences and for all nonnegative integers .  The definition is as follows: and are both equal to the set of sentences such that a computer can determine the truth or falsity of in finite […]

Popular explanations of physical phenomena like gyroscopes or magnetic fields often end up having to explain the right-hand rule to explain how rotational quantities add (say, by using the right-hand rule to convert angular momenta into vectors, then adding the vectors). This is bad, not just because the right-hand rule is confusing, but because it […]

The Strong Law of Small Numbers (see also Wikipedia) says that “There aren’t enough small numbers to meet the many demands made of them.” It means that when you look at small numbers, it’s easy to see compelling patterns that turn out to be false when you look at larger numbers. Using complexity theory, we […]

Take a real number, .  Write out its continued fraction: It’s an intriguing fact that if you look at the sequence of geometric means this approaches a single constant, called Khinchin’s constant, which is approximately , for almost every .  This means that if you were to pick (for convenience, say it’s between 0 and 1) […]

Most mathy people have a pretty good mental model of what a random process is (for example, generating a sequence of 20 independent bits). I think most mathy people also have the intuition that there’s a sense in which an individual string like 10101001110000100101 is more “random” than 0000000000000000000 even though both strings are equally […]