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In my senior year of undergrad, sitting in a graduate level course in real analysis, I was initially a little startled by this fact: it’s possible to find a subset of the real line of “measure” (length) smaller than 1, which nevertheless contains every rational number. In fact, you can even find such a “tiny [...]

In the endless pursuit of real-world applications of abstract logic, I turn today to a concept invented by Eliezer Yudkowsky: Semantic Stopsigns. I won’t use Yudkowsky’s concept precisely, but will modify it in a certain way. Eliezer opens his article with the following intriguing dialog: Q (child): Where did this rock come from? A (adult): [...]

History has been full of arbitrary kings, temporary monarchies built in the sand, artificial and forced. But there is an intrinsic and natural monarchy, a hidden and subtle monarchy with no explicit decrees or crowns or throne rooms, but in the long run this monarchy has absolute authority over all mankind. This is Mathematical Royalty, [...]

Did you know that deep inside your computer, there’s a giant spool of tape? Actually, I’m lying. I was thinking of Turing Machines, the most popular idealized version of a theoretical computer. Alan Turing deserves his immortal seat at the table of computability theorists, he was among the first pioneers and his Turing Machines were [...]

The best math is the kind that makes you do a double-take, when you read a theorem and can’t believe it’s true. That’s how Goodstein Sequences struck me when I was introduced to them. These number-sequences have the property that they grow for awhile but eventually stop growing and shrink down to zero. The way [...]

I don’t just do math, I break math. In this article, among other things, I’ll show how to get a more general version of the quadratic equation. You see, mathematical formulas often hinge on certain conditions being true. For example, the quadratic equation says that if ax²+bx+c=0 then x=(-b±√(b²-4ac))/2a. But this is implicitly assuming a≠0. [...]

“Too good to be true.” That’s what Tom Foremski initially wrote about Algebraix Data, a brand new startup in the lucrative database business. The company claims to use something called “Extended Set Algebra” to revolutionize the way database queries are performed. I was immediately skeptical, having never heard of this in all my years studying [...]

One of the side benefits of moving to a stronger server was that I was able to install LaTeX (Bluehost, believe it or not, doesn’t have LaTeX). If you didn’t already know, LaTeX is the program which is used to typeset all manner of mathematical equations. All your math books are written in LaTeX. And [...]

Writers of math books should realize: no matter how many irrelevant pictures you plaster on every page, people are still going to think your math textbook is an arcane tome of black magic. You might as well make it look like a tome of spells, at least that way it’s cool. No, I don’t need [...]

Here’s a theoretical game, played by two players– you and me and, for example. Hidden from your view, I choose an infinite sequence of numbers, like “1,2,3,4,…” or “1,4,9,16,…” or whatever. It can be absolutely any sequence, but I have to nail it down, I’m not allowed to change it later. And you don’t know [...]