Infinities come in different sizes Raymond Depardon/Magnum Photos
With infinity, we made a monster. Our minds demand that it should exist 鈥 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鈥 There is no obvious end point to this sequence 鈥 so we call it infinite. But now write out the squares of those numbers: 1,4,9,16鈥 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 鈥 infinitely many.
So infinity is infinity is infinity 鈥 except it isn鈥檛. Take the real numbers: the whole numbers plus all the rational and irrational numbers in between (1.5, 蟺, the square root of 2 and so on). There are also infinitely many of these 鈥 except you can show that this infinity is a bigger number. 鈥淚n fact there is an infinite set of infinities and however far you go you can always get to a bigger one,鈥 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鈥檚 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 蟺; calculating smooth motions requires chopping time into infinitesimally small chunks (click on the diagram to…
