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Beyond number - the role of infinity in understanding the universe
Aristotle conceived it, Galileo ran away from it. Professor John Barrow considers the role of infinity in our understanding of the universe.
Does infinity exist? Is it really possible for something to go on and on without end? It is an ancient question that has huge implications for maths, physics and cosmology. And yet it is also a question that has reality in everyday life – a question that even young children grapple with as they learn to count: 1, 2, 3 ... 100 ... 1000 ... 1 million ... the biggest number you can think of? One gazillion?
Aristotle, the first person recorded to consider the issue of infinity, distinguishes between two varieties of infinity: potential and actual infinity. Potential infinity characterises an unending universe or an unending list – for example the natural numbers 1, 2, 3, 4, 5 and so on. These lists or expanses have no end or boundary: you can never reach the end of all numbers by listing them, or the end of an unending universe by travelling in a spaceship.
Aristotle was relatively happy with potential infinities, content that they didn’t create any conflict with his understanding of the universe. In contrast, Aristotle banned actual infinities. Defined as a measurable, local item – such as the density of a solid, the brightness of a light, or the temperature of an object – that becomes infinite at a particular place or time. Encountering this infinity locally in the universe was, as far as Aristotle was concerned, an impossibility. His only permissible actual infinity was the divine – and this philosophy underpinned Western and Christian thought for several thousand years.
But towards the end of the 19th century, mathematician Georg Cantor developed a more subtle way of defining mathematical infinities. Cantor recognised that there were ‘smaller’ and ‘larger’ types of infinity, and he declared countable infinity a minor infinity. Countable infinity can literally be counted – put in one-to-one correspondence with the natural numbers. This takes in the unending list of natural numbers (1, 2, 3, 4, 5 and so on), but also any other series that can be counted without limit.
This idea had some funny consequences. For example, if you make a list of all the even numbers, you have a countable infinity. Intuitively you might think there are only half as many even numbers (2, 4, 6, 8…) as natural numbers (1, 2, 3, 4, 5, 6, 7, 8…) because that would be true for a finite list. But when the list becomes unending that is no longer true.
This fact was first noticed by Galileo (although he was counting the squares 1, 4, 9, 16 and so on, rather than even numbers), who thought it was so strange that it put him off thinking about infinite collections of things any further. He thought there was just something dangerously paradoxical about them. For Cantor, though, this feature of being able to create a one-to-one correspondence between a set of numbers and a subset of them was the defining characteristic of an infinite set. Cantor then went on to show that there are also other types of infinity that are in some sense infinitely ‘larger’ because they cannot be counted in this way. One such infinity is characterised by the list of all real numbers. These cannot be counted; there is no recipe for listing them systematically. This uncountable infinity is also called the continuum.
But finding this infinitely bigger set – the real numbers – wasn’t the end of the story. Cantor showed that you could find infinitely bigger sets still, all the way upwards forever: there was no biggest possible infinite collection of things. If someone presented you with an infinite set A, you could create a bigger one that wasn’t in one-to-one correspondence with A just by finding the collection of all the possible subsets of A. This neverending tower of infinities pointed towards something called absolute infinity — an unreachable summit of the tower of infinities.
Another type of infinity arises in gravitation theory and cosmology. Einstein’s theory of general relativity suggests that an expanding universe (as we observe ours to be) started at a time in the finite past when its density was infinite – the big bang. Einstein’s theory also predicts that if you fall into a black hole you will encounter an infinite density at the centre. These infinities, if they do exist, would be actual infinities. People’s attitudes to these infinities differ.
A big crunch and black holes
Cosmologists who come from particle physics and are interested in what string theory has to say about the beginning of the universe would tend to the view that these infinities are not real, but are rather just an artefact of the unfinished character of the theory. There are others who think that the initial infinity at the beginning of the universe plays a very important role in the structure of physics. But even if these infinities are an artefact, the density of those false infinities is stupefyingly high: 1096 times bigger than that of water*. For all practical purposes that’s so high that we’d need a description of the effects of quantum theory on the character of space, time and gravity to understand what was going on.
Something very odd happens if we assume that the universe will eventually stop expanding and contract back to another infinity, a big crunch. That big crunch could be non-simultaneous because some parts of the universe, where there are galaxies and so on, are denser than others. The places that are denser will run into their future infinities before the low-density regions. If we were in a bit of the universe that had a greatly delayed future infinity, or even none at all, then we could look back and see the end of the universe happening in other places – we would see something infinite. You might see evidence of space and time coming to an end elsewhere.
It is hard to predict exactly what you will see if an actual infinity arises somewhere. The way our universe is understood at the moment implies a curious defence mechanism. A simple interpretation of things suggests that there is an infinite density occurring at the centre of every black hole, which is just like the infinity at the end of the universe.
But a black hole creates a horizon around this phenomenon: not even light can escape from its vicinity. So we are insulated, we cannot see what goes on at those places where the density looks as though it’s going to be infinite. And neither can the infinity influence us. These horizons protect us from the consequences of places where the density might be infinite and they stop us seeing what goes on there, unless of course we are inside a black hole.
There is no beyond
Another question is whether our universe is spatially finite or infinite. I think we can never know. It could be finite but of a size that is arbitrarily large. But to many people the idea of a finite universe immediately raises the question of what is beyond. There is no beyond – the universe is everything there is.
To understand this, let’s think of two-dimensional universes because they are easier to envisage. If we pick up a sheet of A4 paper we see that it has an edge, so how could it be that a finite universe doesn’t have an edge? But the point is that the piece of paper is flat. If we think of a closed 2D surface that’s curved, like the surface of a sphere, then the area of the sphere is finite: you only need a finite amount of paint to paint it. But if you walk around on it, unlike with the flat piece of paper, you never encounter an edge. So curved spaces can be finite but have no boundary or edge.
To understand an expanding two-dimensional universe, let’s first think of the infinite case in which the universe looks the same on average wherever you go. Then wherever you stand and look around you, it looks as though the universe is expanding away from you at the centre because every place is like the centre. For a finite spherical universe, imagine the sphere as the balloon with the galaxies marked on the surface. When you start to inflate it the galaxies start to recede from one another. Wherever you stand on the surface of the balloon you would see all those other galaxies expanding away from you as the rubber expands. The centre of the expansion is not on the surface, it is in another dimension, in this case the third dimension. So our three-dimensional universe, if it is finite and positively curved, behaves as though it is the three dimensional surface of an imaginary four-dimensional ball.
Einstein told us that the geometry of space is determined by the density of material in it. Rather like a rubber trampoline – if you put material on the trampoline it deforms the curvature. If there is a lot of material in the space, it causes a huge depression and the space closes up. So a high-density universe requires a spherical geometry and it will have a finite volume. But if you have relatively little material present to deform space, you get a negatively curved space, shaped like a saddle or a potato crisp. Such a negatively curved space can continue to be stretched and expand forever. A low-density universe, if it has a simple geometry, will have an infinite size and volume. But if it has a more exotic topology, like a torus, it could also have a finite volume.
One of the mysteries about Einstein’s equations is that they tell you how you can work out the geometry from the distribution of matter, but his equations have nothing to say about the topology of the universe. Maybe a deeper theory of quantum gravity will have something to say about that.
A version of this article first appeared in Plus Magazine, a free online magazine about maths, published by the University and aimed at a general audience. This version first appeared in CAM - the Cambridge Alumni Magazine, edition 74. Find out how to receive CAM.
About the author
John D Barrow FRS is Professor of Mathematical Sciences and a Fellow of Clare Hall. He is the director of the Millennium Mathematics Project, of which Plus is a part. His latest book is 100 Essential Things You Didn’t Know About Maths and the Arts (Bodley Head).
* In transferring the text of this article from the print edition of CAM to the web, we inadvertently changed 1096 to 1096. Thank you to the eagle-eyed readers who picked up on this and alerted us to the magnitude of the error.