Journey into Maths Country

Frank Benford observed that the number one seems to pop up a lot in both in the supermarket and on tax bills. In understanding this law, it helps to simply look at the world from a different perspective.

Speed is such a common term that it's easy to forget how much of a role maths plays in understanding it. Until three or four centuries ago, there was no speed at all. It was only since the Renaissance that the concept of movement crept into the world of mathematics, thanks to calculus and Isaac Newton.

A circle is also a triangle and a triangle is a square. Sound impossible? Not in the realm of topology, which even applies to 3-dimensional spaces and anything from donuts, UFOs and potatoes. The discipline's inventor, Henri Poincaré, created one of the most difficult problems in mathematics.

In this episode of our travels in the land of maths, we are heading towards Infinity. And even beyond, because infinity comes in many sizes. That may seem strange, but Georg Kantor and his set theory will help us come to terms with this maths concept of dizzying scale.

In this episode, we look at the relationship between maths and truth. Maths is meant to be certain, either right or wrong. Turns out, it’s not that simple. For Gödel's theorem has proved that there are “undecidable” theories, which one can neither prove nor disprove.

Two prisoners must choose between cooperation and betrayal without consulting each other. This famous prisoner's dilemma that will take us to the heart of game theory. We think mathematically about a very philosophical question: is it in our interest to collaborate with others?

The Monty Hall paradox, named after a game show from the 60s, concerns the way in which information acquired during the course of a game modifies (or not) the winning statistics. Theoretically solved, the question is so disturbing to our worldview that it continues to be the subject of passionate debate to this day.

Statistics seem, almost by their very nature, to convey a positivist message. They are, in fact, a formidable tool in the attempt to master the complexity of the real world... But numerous "biases" threaten any discourse that refers to them without care: an over-simplistic reading of the figures can lead us - for example - to confuse correlation with causation... And more complex phenomena (notably Simpson's paradox) can distort conclusions that seem objective.

For centuries, geometry was based on Euclid's postulates, which seemed eternal and irrevocable. However, one of the postulates (the fifth) has always seemed "a little less natural" than the others, and hundreds of mathematicians have tried in vain to do without it by deducing it from the other postulates. In the mid-19th century, Bernhard Riemann came up with a novel idea: let's imagine it's false! This was the birth of "non-Euclidean geometries", which would later have major applications in physics.

A tessellation is a way of covering a plane with a repeating pattern... Basically, it's like creating wallpaper. In 1975, Marjorie Rice (1923- 2017), a mother and amateur mathematician, read an article by Martin Gardner in Scientific American that listed ALL possible "pentagonal tessellations" in the plane. A mathematician had just proved that the list was complete. Except that Marjorie, working alone at home, found 4 new ones... The theorem was wrong!

The question is how to make a network that is both "economical" and "robust" without taking up too much space. This is a theoretical question worked on by the great Russian mathematician Andrey Kolmogorov (1903-1987). But this theoretical question also conditions the way in which we can build a computer network or... a human brain: to be intelligent without having a big head, you need a neural network that is efficient BUT ALSO compact! Mention Szemeredi lemma?

To begin with, there are the five "Platonic solids" beloved of geometers: the cube, the tetrahedron, the octahedron, the dodecahedron and the icosahedron. But why stop at the 3 dimensions of ordinary space? Alicia Boole Stott has devoted her life to finding regular solids in dimension 4... and she's found them! A journey into unsuspected mathematical regions.