Infinity and the Unlimited

Infinity and the Unlimited

By Richard Baron, Independent Philosopher, United Kingdom

We all have a general idea of the infinite: something bigger than any specific thing. But if that is how we think of the infinite, we have a problem. If no specific thing is big enough to be infinite, whatever has infinite size cannot be a specific thing. Yet it is only specific things that have sizes. So how big is infinity?

We need to shift our focus. Infinity cannot be a particular size. If it were, there would be something bigger than it. We would just add a bit to whatever we already had. Instead we should think of a particular feature of things, and imagine that this feature is not there. The feature is having a limit. A planet, or a galaxy, could be fitted into a box. Even such enormous things have their limits. And we must imagine that there is no such limit. The same goes for enormous numbers, such as the number of possible games of chess. This has been estimated as something like 1 with 120 zeros after it. For any enormous number, there will be a bigger one. The bigger number is like a box: the enormous number we started with fits inside it. We have to imagine that there is no bigger number.

We can see this idea of no limit in action by looking at the usual numbers 0, 1, 2, 3 and so on. We know how to continue the sequence as far as we like. We might carelessly say that it goes on to infinity, but actually it will never get there. What we can say is that there is no limit, no stopping point. Indeed, the formal rules that mathematicians lay down to define these numbers include the rule that every number has a successor. 1 is the successor of 0, 2 of 1, and so on. So wherever we have got to, we can always go one more step. This makes the idea of no limit precise.

We can also use the idea of no limit when we think about smaller and smaller moves which never take us very far. Take the sequence of fractions 1/2, 3/4, 7/8, 15/16, and so on. This sequence will never take us above 1. We will never even reach 1, and we will never take all the steps in the sequence we could take. But the sequence can go on without limit, and the distance from 1 can be made as small as we like although it will never reach zero. The notion of no limit on the number of steps we take replaces the notion of an infinite number of steps. The notion of no limit on smallness replaces the notion of infinitely small. And we need that replacement notion because for any particular tiny distance from 1, there would always be a smaller one.

That is mathematics, but what about life? Nobody's life is infinite. And while we can see ourselves as taking an unlimited number of smaller and smaller steps towards the end, that does not make the duration of a life unlimited.

Fortunately, there is a more positive application to life. What we do influences what happens in the future. We must start with our choices, and then bring in the idea of the unlimited.

Different people make different choices. Suppose that the sight of a rainbow inspires one person to write a poem, and another to do some gardening. The different choices would reflect both their different personalities and their different situations. The choices will not have been random, but sensible and explicable. And they will have been the choices of the individuals, even if everything they did was ultimately determined by factors outside themselves. Perhaps the personalities and the situations of our two individuals were the products of events in the world that led to their parents meeting, their being brought up in particular countries, economic forces which determined which careers were available to them, and so on. It would still have been their personalities which led to their responses to the rainbow.

To see the personalities of individuals as leading to their choices, so it is really they who leave their marks on the world, is important. What someone does leaves a mark on the world that is distinctively their own. It is not a trace left by nobody in particular. Other people in the same situation would have left different marks. But this does not in itself overcome the limit to anyone's life.

The next step is to follow the mark that someone leaves forward in time. It will change the situation for someone else. Let us go back to the poet. Someone who reads their poem will be prompted to have their own thoughts and take actions they would otherwise not have taken. If the poet had decided to go for a walk instead of writing, the second person would not have had the thoughts that the poem in fact inspired. So their next actions would have been different. And so on. Some physicists talk about the universe being split and following two different future paths every time a choice is presented. We do not need to go that far to see a choice made today as leading down a road which opens up more choices, and different future choices depending on which choice is made today. The paths through the range of possible lives and worlds fork rapidly.

This rapid forking means that a choice today, even a trivial one, will have some consequences or other an unlimited distance in the future. People may not know that the consequences are attributable to the choice. And indeed the consequences cannot be attributed to that choice alone. Other choices will have mattered too. We cannot even say that it was only the original choice that opened up the possibility of some consequence or other. There might have been other possible routes to the consequence, even in the short term. If our poem had not been written, the reader of it might have been inspired to the same thoughts by some other prompt. But what we can say is that a choice someone actually makes will have its impact on the future. It will not in itself determine what happens next, but it will play a part in setting the range of options at the next point of choice. And whatever is chosen then will do the same again. And so on. Thus even a small choice may leave its mark into the indefinite future. And this is so even if the determinists are right and we could not in fact make any choices other than the ones we in fact make.

Here we have the unlimited. We do not live for ever. And even our most significant actions cannot on their own be seen as having significant consequences very far into the future. But paths through the range of possible lives and worlds fork so rapidly that we cannot draw any limit to the span of time over which our choices will have impacts, or to the range of those impacts. The impacts may be small, and after a few decades impossible to trace, but they will be many, and they will far outlive us.

Can art mirror life in this respect, as it can in many other ways? Yes, it can. We should start by going back to mathematics. There are curves called fractals, defined mathematically and also to be found on posters and place-mats. A famous example is the boundary of the Mandelbrot set. These curves have unlimited fine detail. As you zoom in, looking more closely at particular stretches of the curve, more and more detail is revealed. Most remarkably, patterns you see before zooming in, such as the fern leaves and seahorse tails in the boundary of the Mandelbrot set, re-appear at smaller and smaller scales. There is no limit to the fineness of detail we can see. And however closely we look, we never reach all of the detail that lies in wait for us.

The physical works of art we see can only approximate this idea. Eventually the detail runs out, when we hit the level of one pixel or one tiny dab of ink. But works of art can still indicate the idea.

Take for example this work by Dr Gindi’s In Reverie. There are large indentations, and smaller ones within them, creating an intricate pattern of ridges. The artist could have gone on, right up to the limit of the metal. And we can imagine her going further and further, without the physical limit of atoms.

Finally, there is more to explore. We started by listening to mathematicians and thinking about the absence of limits, rather than about something bigger than anything else. That saved us from trying to assign any particular size to infinity. But mathematicians actually work with several different infinities, and discover that some are bigger than others. It looks as though they are assigning sizes to infinities, when we started by saying that this would be a mistake. In fact they change our notion of size to make it work among infinities.

If you have a heap of knives and a heap of forks, you can establish that they are the same size by pairing them off. If there is one knife for each fork, and one fork for each knife, the heaps are the same size, and you can say that even if you have not counted the knives or the forks. Likewise you can take the infinite sets of numbers 1, 2, 3, 4, ... and 2, 4, 6, 8, ... , and pair off the numbers: 1 with 2, 2 with 4, 3 with 6, 4 with 8, and so on. So these two infinities are the same size in our new sense of size. But we still have no size in our everyday sense for either set of numbers. So we cannot complain that the mathematicians must be wrong because the set 2, 4, 6, 8, ... really ought to be half the size of the set 1, 2, 3, 4, ... . Size in the everyday sense is still not there, and we should not use that everyday notion. Sometimes we understand the infinite better by not trying to grasp it in everyday terms.

The Portuguese writer Fernando Pessoa had this to say: To reach the infinite – and I believe it can be reached – we need to have a sure port, just one, from which to set out for the indefinite. (The Book of Disquiet, section 251)
Fortunately, Pessoa was a man who rejoiced in paradox. He would not have objected to our saying that a willingness to let go of everyday terms, to step forward without the usual solid ground under our feet, was the key to understanding the infinite. Our essential starting point, our sure port, is a willingness to manage without a sure port.

Richard Baron is an independent philosopher who lives in Cambridge. He was educated at the University of Cambridge and has worked in adult education. He works mainly on the theory of knowledge, particularly knowledge in different academic disciplines from physics to history, and has also written on ethics.

November 27, 2023