People have been storing information since the stone ages, ever since they’ve been writing or putting art on tablets and walls. With the invention of paper and ink, the “density of information” increased significantly, packing a lot more information into a tighter space (such scrolls and eventually bound books, as we still use today). The […]

## Goedel’s Theorem for Dummies

When people refer to “Goedel’s Theorem” (singular, not plural), they mean the incompleteness theorem that he proved and published in 1931. Kurt Goedel, the Austrian mathematician, actually proved quite a few other theorems, including a completeness theorem for first-order logic. But the incompleteness theorem is the one for which he is most famous. To get […]

## The Mathematics of Hell

(Or Jonathan Edwards Meets Leonhard Euler) Visions of hell abound, not just in Christian literature but also in mythology. The Greeks, for instance, had Sisyphus, who was condemned to an eternity of futility, constantly rolling a rock up a hill, only to have it come tumbling back once he was almost at the crest. Or […]

## Potential Infinity vs. Actual Infinity

What is infinity and does it even exist? In our everyday experience, we find only finite things. A basket of eggs contains only a fixed number of eggs and no more. Our bodies are composed of particles (molecules, atoms, protons, quarks, etc.). But whatever particles describe our make up, we find only a finite number. […]

## Lesson of the Monty Hall Problem

On the television game show Let’s Make a Deal, Monty Hall, the show’s best known host, used to present contestants with the following situation: the contestant would be presented with three doors behind one of which was a big prize (say a brand new car). Behind the other two doors was a small prize (say […]

## The Bare Bones of Bayes’ Theorem

Thomas Bayes died over 200 years ago, but his legacy is still with us and provides some very useful insights into probability. What is his legacy? It is a probability formula that tells us how to update probabilities in light of new information. Suppose, for instance, you learn that Fred is a physical fitness fanatic […]

## The Most Interesting Number

Are all numbers interesting? Yes. And we can prove it using a “proof by contradiction.” Assume there are positive numbers that are not interesting. Therefore, there is a smallest number that is not interesting. Hey! That’s interesting! Therefore, the assumption in (2) is violated. We have a contradiction and have proved that all positive numbers […]

## Savvy Advice for Unsavvy Gamblers

Risk is an unavoidable feature of life, so in a sense all of us are all the time gambling. For instance, we are placing a bet when we invest money in a new business venture. Many business ventures don’t succeed. Why then do we do it? Most of the time, it’s because we think we’ve […]

## The Different Number Scales

When you encounter a number, what sort of numerical information is it giving you? Is the number functioning merely as a label? Or is it counting how many? Does it denote a magnitude or merely indicate an ordering relation? All such questions address the issue of scale. When you see a number, it belongs to […]

## The Bare Bones of Probability

How likely is it that an event will happen? Probabilities attempt to answer this question by assigning numbers to the likelihood of events. A probability is always a number between 0 and 1. The closer to 0 the probability, the less likely the event; the closer to 1, the more likely the event. An event […]