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.
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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 see how this happens).
But is any of this truly real? Take those whole numbers: you could never actually write all of them out. “You’d snuff it before you did,” says Stewart. Even if someone else took over, a finite universe would eventually run out of paper to write them down on – and information to encode them with.
Such practical considerations make many physicists wary of infinity as a really existing thing – and developing new and better theories is often a case of finding ways of eliminating it. “An infinity generally means there is something wrong with your theory,” says Stewart.
Often that has been a game of whack-a-mole – lose one infinity, and another one pops up in its place. Take our attempts to find a theory of quantum gravity, which would wind us back closer to our universe’s origin. The big bang theory portrays this beginning as a “singularity” of infinite density and temperature, but the mathematical description breaks down when you pack too much gravity in too small a space. But replacements such as string theory often conjure up an infinitude of universes, making it impossible to work out why ours looks as it does.
A small minority of physicists and mathematicians eschews infinity completely, claiming it has no place in a finite universe. Infinity’s endless uses make that a step too far for most people, says Stewart, but it’s probably a concept we don’t want to stare at too hard. “Infinity is incredibly useful, but only if you can make sure it doesn’t blow up in your face,” he says.
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