Infinities come in different sizes Raymond Depardon/Magnum Photos
With infinity, we made a monster. Our minds demand that it should exist – only to rapidly melt at the consequences of a concept that is, by definition, too big for our brains.
The pleasure and the pain start when we write out the whole numbers: 1,2,3,4… There is no obvious end point to this sequence – so we call it infinite. But now write out the squares of those numbers: 1,4,9,16… This sequence gets bigger a lot faster, so it must reach infinity faster, right? Not so. Every whole number has a square, so there are as many square numbers as whole numbers – infinitely many.
So infinity is infinity is infinity – except it isn’t. Take the real numbers: the whole numbers plus all the rational and irrational numbers in between (1.5, π, the square root of 2 and so on). There are also infinitely many of these – except you can show that this infinity is a bigger number. “In fact there is an infinite set of infinities and however far you go you can always get to a bigger one,” says mathematician of the University of Warwick, UK.
What makes this infinitely troubling is that the whole logic of arithmetic rests on the existence of these logic-defying infinities. In fact, there’s very little in mathematics that works smoothly without manipulating the infinite and its obverse, the infinitesimal. Defining a perfect circle requires the infinite digits of π; calculating smooth motions requires chopping time into infinitesimally small chunks (click on the diagram to…
