Hilbert, a young German mathematician, boldly set out what he believed are the twenty-three most important problems for mathematicians to crack. Professor Marcus du Sautoy, The Story of Maths IV: To Infinity & Beyond
Before Cantor no-one had really understood infinity ... There are different infinities, some bigger than others. ibid.
[Kurt] Godel lived in Vienna in the 1920s and 1930s ... The Incompleteness Theorum. Godel proved that within any logical system for mathematics there will be statements about numbers which are true but which you cannot prove. ibid.
With the help of colleagues, Julia [Robinson] developed what became known as the Robinson hypothesis. ibid.
The Americans were right: Nicolas Bourbaki does not exist at all and never has; Boubaki is in fact the nom de plume for a group of French mathematicians led by Andre Weil. ibid.
Mathematics now pervades every aspect of our lives. ibid.
Until we can prove it [Reimann Hypothesis] there will still be doubt. ibid.
Hidden in this cathedral are clues to a mystery. Something that could help answer one of humanity’s most enduring questions. Why is the world the way it is? Marcus du Sautoy, The Code I: Numbers BBC 2011
The precise proportions of this magnificent cathedral: to the medieval clergy these divine numbers were created by God. But to me they are evidence of something else: a hidden code that underpins the world around us, a code that has the power to unlock the doors that govern the universe. ibid.
The code: an abstract world of numbers that has given us the most detailed description of our world we’ve ever had. For centuries, people had seen significant numbers everywhere. An obsession that has left its mark on the stones of this medieval cathedral. ibid.
For Saint Augustine, numbers had to come from God. ibid.
I don’t share their religious beliefs; I can’t help feeling something in common with the people who built this place. I share their awe and wonder at the beauty of numbers. For them those numbers brought them closer to God. But I think they’re important for another reason: because I believe they are the key to making sense of our world. ibid.
Numbers have given us an unparalleled ability to understand our universe. And in places this code emerges from the ground. ibid.
Unlike other numbers primes can only be divided by themselves and one. ibid.
Pi is the essence of circle-ness distilled into a language, a code. ibid.
Just as the circle appears everywhere in nature, so Pi crops up again and again in the mathematical world. ibid.
Some numbers are so strange they don’t seem to make sense as numbers. ibid.
We’ve given this number a name and it’s called i – part of a whole class of new numbers called imaginary numbers. ibid.
1.08 ... The Nautilus is growing at a constant rate. So every time the Nautilus builds a new room the dimensions of that room are 1.08 times the dimension of the previous one. ibid.
Behind the world we inhabit there’s a strange and wonderful mathematical realm. ibid.
It’s only through the code that we can understand the laws that govern the universe. ibid.
Gravity ... It will accelerate to land at a constant rate of 0.98 meters per second per second. This is a fundamental law on our planet. ibid.
This is the Giant’s Causeway at the northern tip of northern Ireland. And it’s famed for these strange angular rocks. There are forty thousand of them ... These stones tell of a hidden geometric force that underpins and pervades all Nature. And if we can uncover that force it will tell us the shape of everything. Professor Marcus du Sautoy, The Code II: Shapes
The bees’ honeycomb is a marvel of natural engineering. The bees have made an identical pattern to the columns on the Giant’s Causeway. Each cell is exactly like the others ... It’s as if the hexagon is built into the bees’ DNA. ibid.
It’s the hexagons that use the least amount of wax. ibid.
This is Nature’s code at work and the bees are in tune with it. ibid.
The soap bubble reveals something fundamental about Nature – it’s lazy; it tries to find the most efficient shape, the one using the least energy, the least amount of space. ibid.
Frei Otto started something of a revolution in architecture. The sweeping curves of the Olympic Stadium are echoed in countless modern structures. ibid.
The mainstay of Greek geometry was the discovery of five perfect shapes now called the Platonic solids ... The tetrahedron with its four faces, the cube with its six faces, the octahedron with its eight faces, the dodecahedron twelve faces, and the most complicated shape of all the icosahedron with its twenty faces. Today these are more commonly known as dice ... These are the only five shapes like this that can possibly exist; they are the only perfectly symmetrical solids. ibid.
The world clearly isn’t just built from simple geometric shapes. ibid.
We now know that the movement of the planets is incredibly predictable. By understanding the code we can model their orbits far back into the past and thousands of years into the future. Marcus du Sautoy, The Code III: Prediction
We are so addicted to patterns that we let them seep into almost everything we do. ibid.
Very few of us are aware of the patterns we leave behind. ibid.
The future rarely turns out exactly as we planned. ibid.
Our lives are controlled by the strangest code of all – the code of chaos. ibid.
The average guess to the nearest bean is 4,515 [4,510] ... Collectively we get something which is just 0.1% away from the real number of beans in there. So as individuals the guesses are just that – guesses, but when you take them collectively they become something else entirely. ibid.
The accuracy of the group is far greater than the individual – we call this the wisdom of the crowd. ibid.
Everything has mathematics at its heart. When everything is stripped away all that remains is the code. ibid.
They’re ruling our lives: algorithms. Algorithms are everywhere. These bite-sized chunks of maths have become essential to our daily lives. Marcus du Sautoy, The Secret Rules of Modern Living: Algorithms, BBC 2015
They are strangely beautiful, tapping into the mathematical order that underpins how the universe works. ibid.
A series of step by step instructions. ibid.
Algorithms are extremely old. ibid.
Numbers are the fabric of mathematics. But not all numbers are the same. There are numbers and there are prime numbers … They crop up at random getting larger as the numbers get rarer. Horizon: A Mathematical Problem, BBC 1984
It doesn’t matter how many primes you multiply together, by adding one you’ll either discover a new Prime number or a number that’s divisible by a Prime that’s not on your list. Horizon: Alan and Marcus Go Forth and Multiply, Marcus du Sautoy, BBC 009
Is the universe infinite? Might every event repeat again and again and again ... Is the Earth just one of uncountable copies tumbling through an unending void? Horizon: To Infinity & Beyond, BBC 2010
A google-plex has so many zero that there is not enough space in the observable universe just to write the number down. ibid.
Graham’s number is so big it even made it into the Guinness Book of Records. ibid.
In an infinite universe anything that is possible has to happen. ibid.
While no-one likes the idea of space coming to an end, the consequences of an infinite universe are even more bewildering. ibid.
The numbers didn’t lie: we did. Penn & Teller, Bullshit s4e9: Numbers, Showtime 2006
Numbers by themselves aren’t vicious; just the jerks who manipulate them. ibid.
We assume that somebody has counted something. That it’s really really true. Professor Joel Best, author Damned Lies & Statistics
When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be. Lord Kelvin, 1824-1907