During his Hamilton lecture in Dublin recently, Fields medalist Martin Hairer made a passing mention of the “Two Envelopes Paradox”. This is a well-known problem in probability theory that has led to much misunderstanding. It was originally developed in 1912 by the leading German number theorist Edmund Landau (see Gorroochurn, 2012). It is frequently discussed on the web, with much misunderstanding and confusion. I will try to avoid adding to that.

## Archive for November, 2018

### Gravitational Waves & Ringing Teacups

Published November 22, 2018 Occasional ClosedTags: Astronomy, Relativity, Wave Motion

Newton’s law of gravitation describes how two celestial bodies orbit one another, each tracing out an elliptical path. But this is imprecise: the theory of general relativity shows that two such bodies radiate energy away in the form of * gravitational waves* (GWs), and spiral inwards until they eventually collide.

### The “Napoleon of Crime” and The Laws of Thought

Published November 15, 2018 Irish Times ClosedTags: History, Logic

A fascinating parallel between a brilliant mathematician and an arch-villain of crime fiction is drawn in a forthcoming book – *New Light on George Boole –* by Des MacHale and Yvonne Cohen. Professor James Moriarty, master criminal and nemesis of Sherlock Holmes, was described by the detective as “the Napoleon of crime”. The book presents convincing evidence that Moriarty was inspired by Professor George Boole [TM151, or search for “thatsmaths” at irishtimes.com].

Continue reading ‘The “Napoleon of Crime” and The Laws of Thought’

### Listing the Rational Numbers III: The Calkin-Wilf Tree

Published November 8, 2018 Occasional ClosedTags: Arithmetic, Number Theory

The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. In previous articles we showed how the rationals can be presented as a list that includes each rational precisely once. One approach leads to the Farey Sequences. A second, related, approach gives us the Stern-Brocot Tree. Here, we introduce another tree structure, The Calkin-Wilf Tree.

Continue reading ‘Listing the Rational Numbers III: The Calkin-Wilf Tree’

### Johannes Kepler and the Song of the Earth

Published November 1, 2018 Irish Times ClosedTags: Astronomy, Music

Johannes Kepler, German mathematician and astronomer, sought to explain the solar system in terms of divine harmony. His goal was to find a system of the world that was mathematically correct and harmonically pleasing. His methodology was scientific in that his hypotheses were inspired by and confirmed by observations. However, his theological training and astrological interests influenced his thinking [TM150, or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Johannes Kepler and the Song of the Earth’