Mathematical proofs can be established by various means, including induction, contradiction, construction, and exhaustion. My favourite is proof by intimidation.
If you’re a gambling man, you better believe in God. So suggested Pascal’s wager, one of the first applications of decision theory to philosophy.
What is the smallest boring number? There’s no such thing, because the title of smallest boring number automatically makes that number interesting.
Statistics are tricky. Consider this: of two treatments for kidney stones, Treatment A is better on average for large stones and small stones. But consider all stones together and Treatment B is better. This is Simpson’s paradox.
The gambler’s fallacy is the belief that random independent events “even out” over time. In Monaco in August 1913, this belief cost casino gamblers millions because of an extraordinary streak at a roulette table.
What are the practical limits of computation? Hans-Joachim Bremermann theorized a computer the size of the world, as old as the world. It could process 10^93 bits.
Can mathematics be beautiful? Mathematicians often describe proofs in aesthetic terms – they are elegant, sublime, ineffable; in a word, they are beautiful.
The Aboriginal languages of southeast Australia have an ingenious counting system – there’s a physical mnemonic built directly into the language.
A magic square is a grid of numbers in which any row, column, or diagonal adds up to the same total. They look complex, but it’s actually easy to design your own using the Siamese method.
0.999… is the same number as 1. It’s counterintuitive, but mathematics says that it’s true. Whether you believe it or not is another matter.