In her book Beyond Infinity, Dr Eugenia Cheng explores the inner workings of infinity, a mysterious concept that we all know about from childhood, but never truly claim to understand. There are infinite facts in the book (well, quite a lot at any rate), but here’s just a fraction for now.
Infinity plus one
Imagine an infinite hotel with rooms numbered 1, 2, 3, 4 and so on forever. Even if it was full you could always fit in another guest – just ask all the old guests to move up one room, leaving room 1 free for the new guest. However, this does cause an infinite hassle, as all the guests have to move rooms. It would be no hassle if the extra guest arrived first. This shows that one plus infinity is not the same as infinity plus one. Shakespeare knew this when he said “Forever and a day”. He must have known that this is longer than merely “forever”.
Bigger than infinity?
Some infinities are bigger than others. The smallest infinity is how many whole numbers there are: 1, 2, 3, 4 and so on forever. If we include fractions there are infinitely many more numbers. In fact there are infinitely many fractions in between each whole number. But overall there aren’t more numbers unless we include the irrational numbers, the “decimals that go on forever”. There are two to the power of infinity of those, that is, 2 x 2 x 2… multiplied infinitely many times. The mathematician Georg Cantor proved that this is bigger than infinity, whatever kind of infinity you start with.
A day has only a finite number of hours and a finite number of minutes, but you do infinitely many things every day. Even just to walk over to the fridge you cover an infinite number of distances: first you have to cover half the distance, then half the remaining distance, and half the remaining distance, and so on forever. Fortunately, you can cover those infinitely many distances in finite time, otherwise you’d be infinitely hungry. This is Zeno’s paradox, and wasn’t really resolved until the invention of calculus a couple of thousand years after Zeno died.
You might think 1/0 is infinity. But it isn’t. But also it is. How can those both be true? It depends what mathematical world you’re in. In the world of ordinary numbers, dividing by zero can’t be defined. If 1/0 had an answer, then everything would collapse to zero. But there is a mathematical world called the extended complex numbers in which we can define 1/0 to be infinity without everything collapsing. This shows us that maths isn’t all about right and wrong, but about investigating different possible worlds in which different things can be true.
If we were immortal we could procrastinate forever. There is actually a mathematical version of this, which is a theorem I have proved. My research is in higher-dimensional category theory, which involves studying relationships between things, relationships between relationships, relationships between relationships between relationships, and so on. In finite dimensions you have to stop at some point and decide which relationships count as equivalent. Those decisions are mathematically difficult. Whereas in infinite dimensions you can put off the decision forever. This means that the infinite dimensional category is easier to work with than the finite one. It is satisfyingly weird.