Science and Marxism

What is infinity?

What does Woods mean by infinity? In the section, Does the Infinite Exist? Woods suggests that:

What does Woods mean by infinity? In the section, Does the Infinite Exist? Woods suggests that:

The idea of the infinite seems difficult to grasp, because, at first sight, it is beyond all human experience. (p353)

An infinite universe would indeed be “beyond all human experience”. As the physicist Brian Greene says: “Experimenters never measure an infinite amount of anything. Dials never spin round to infinity. Meters never reach infinity. Calculators never register infinity.” (The Fabric of the Cosmos,p335)

But this raises precisely the question we will address in this survey: how did the universe come to be reckoned to be infinite?

After all, science, which is instinctively materialist, bases itself on human experience (including, of course, through the use of scientific instruments of all kinds), not on what is “beyond all human experience”. This raises a second question: how can Woods’ claim that the universe is infinite be a materialist claim?

We can all envisage an unending series of numbers, a series of numbers that continually grows greater in an infinite repetition of some additional amount. No matter how large the number gets, we can always add one more. A simple repetitive task, we imagine, at least in principle, can always be repeated in an infinite process that need never stop. In this sense, we cannot agree with Woods’ claim that the infinite seems “difficult to grasp”.

But it is important to realise that this infinite process of addition or repetition will never reach actual infinity. The number of repetitions, however large, will be a finite number, and will remain finite.

This infinite process is not the kind of infinity that Woods is talking about. “Infinity, by its very nature cannot be counted or measured.” (p353) “The idea of infinity cannot begin with one, or any other number. Infinity is not a mathematical concept.” (p218)

Potential and actual infinity

So we begin to see that, contrary to Woods’ assumption, there are two contrasting concepts of infinity which may concern us here. (We will glance at George Cantor’s contribution to this subject later.) The first is the familiar one, which can, in fact, begin with one, or any other number, where we can always imagine adding one more in an infinite process.

Woods does not accept this, but this is what Aristotle, the Greek philosopher of the fourth century BCE, termed “potential” infinity. It is a process that never leaves the finite – you never reach infinity – yet, at any particular stage under consideration, it is an infinite process. It is the only type of infinity which science recognises (in the real world, as opposed to mathematical methods such as calculus). The dial never reaches infinity even if the process appears to be infinite.

In this way, as we shall see later, Engels at one point envisaged a universe rolling out indefinitely in time and space, in an infinite process comprised only of finites, and never becoming infinite (Anti-Dühring, Part V, p67). Elsewhere, Engels envisaged the death of the universe, pointing out that at a certain point all the stars must exhaust their fuels.

The second concept of infinity – the only one that Woods recognises – is “beyond all human experience”. Aristotle calls this second type of infinity “actual infinity”. Woods claims that the infinite is quite distinct from the finite. Yet Aristotle and many philosophers and scientists, through the ages to the present day, have explained that the ‘actual infinite’, the infinite that is “beyond all human experience” according to Woods, is an ideal with ‘potential’ but no ‘actual’ reality.

Engels’ views

Before we start our historical survey let us address directly the question: is the universe – or for that matter, the ‘multiverse’ – infinite?

Friedrich Engels (1820-1895)

We answer this question in the course of this discussion, but to jump ahead, it may be useful to take a glimpse at Engels’ remarkable insight on this question. We will argue that among these insights, guided by dialectical considerations, are some that approximate to the position of modern science today.

As we will show, Engels discusses the coming into being of our universe and says that there must have been a cause to this event even if, at present, we have no idea what it is. In today’s terminology, science assumes that there must be a cause to the Big Bang and is searching for it.

But if one was to ask whether there must be an infinity of previous causes to the cause of the Big Bang, at another point Engels replies: infinity is a contradiction, and is full of contradictions (Anti-Dühring, PartV, p66). Engels was well aware that Aristotle had shown that the actual infinite does not exist. It was common knowledge. It was also common knowledge that Aristotle discovered contradictions in the concept of actual infinity, and others, beginning with Galileo, have discovered many more. One such contradiction is called the infinite replication paradox, and simply follows from the fact that infinity can contain within it an infinite number of infinities.

Consider an infinity of people. With more than six billion people in the world, there are bound to be people who look very much like you. Famous people sometimes employ look-alikes to pretend to be them. Since antiquity it was understood that if the universe was infinite – a Newtonian universe or a multiverse of modern conception – there must be an infinite number of worlds (or universes) of every possible type, since even the most improbable worlds occur infinitely given an eternity of time and an infinity of space. Among them there will be an infinite number of worlds like ours, and even an infinite number of people like us living on these worlds – in fact, there must be an infinite number of people exactly like us, on these infinite worlds, doing exactly what you and I are doing right now.

As materialists, we must leave the actual infinite for what it is, a contradiction. As scientists emphasise, we have no material evidence for an infinite universe – just a sense that there must always be an endless chain of causes. Let us now place our discussion in its proper historical context.

Next: The dialectic of ‘becoming’ in ancient Greece