Mathematics

Claims and Evidence: A Joke

The other day a friend posted the following old joke about the level of rigor that mathematicians usually require. (Disclaimer: if you take the joke as a serious claim about the standards of quality in the other fields referenced in the joke, it is an obviously unfair characterization of both astronomy and physics.) A Mathematician, […]

Turning Distances into Distributions

Deriving Distributions from Distances Several of the continuous univariate distributions that frequently come up in statistical theory can be derived by transforming distances into probabilities. Essentially, these distributions only differ in terms of how frequently values are drawn that lie at a distance \(d\) from the mode. To see how these transformations work (and unify […]

Understanding the Pseudo-Truth as an Optimal Approximation

Understanding the Pseudo-Truth as an Optimal Approximation

Introduction One of the things that set statistics apart from the rest of applied mathematics is an interest in the problems introduced by sampling: how can we learn about a model if we’re given only a finite and potentially noisy sample of data? Although frequently important, the issues introduced by sampling can be a distraction […]

Twitter Math Puzzle and Solution

Yesterday I posted a very simple math puzzle to Twitter that I found in Jonathan Baron’s book, Thinking and Deciding. The puzzle is the following: Show that every number of the form ABC,ABC is divisible by 13. The puzzle comes up in Baron’s book as an example of an “insight problem” in which one goes […]

The Price of Calculation

In a world in which the price of calculation continues to decrease rapidly, but the price of theorem proving continues to hold steady or increase, elementary economics indicates that we ought to spend a larger and larger fraction of our time on calculation.1 Over the next ten years, I hope that more and more mathematically […]

Algebra and Education

Algebra and Education

Recently a friend asked me to help her learn enough math to take the GRE’s. My response was to give her the first problem that I thought she should be able to solve before we discussed anything else. It was a very simple problem from the perspective of a mathematician, but one that is not […]