Thermodynamics is Easier Than I Thought

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 […]

Gravity is Stronger Than I Thought

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 […]

The Arithmetic Hierarchy Meets the Real World

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 […]

Two Constants: Khinchin and Chaitin

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) […]

A Good Definition of Randomness

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 […]