The Infinite

A.W. Moore

• Publisher: Routledge
• Year: 2018
• Print ISBN: 9781138504257
• eBook ISBN: 9781351381253
• Edition: 3
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

We are all captivated and puzzled by the infinite, in its many varied guises; by the endlessness of space and time; by the thought that between any two points in space, however close, there is always another; by the fact that numbers go on forever; and by the idea of an all-knowing, all-powerful God.

In this acclaimed introduction to the infinite, A. W. Moore takes us on a journey back to early Greek thought about the infinite, from its inception to Aristotle. He then examines medieval and early modern conceptions of the infinite, including a brief history of the calculus, before turning to Kant and post-Kantian ideas. He also gives an account of Cantor's remarkable discovery that some infinities are bigger than others. In the second part of the book, Moore develops his own views, drawing on technical advances in the mathematics of the infinite, including the celebrated theorems of Skolem and Gödel, and deriving inspiration from Wittgenstein. He concludes this part with a discussion of death and human finitude.

For this third edition Moore has added a new part, 'Infinity superseded', which contains two new chapters refining his own ideas through a re-examination of the ideas of Spinoza, Hegel, and Nietzsche. This new part is heavily influenced by the work of Deleuze. Also new for the third edition are: a technical appendix on still unresolved questions about different infinite sizes; an expanded glossary; and updated references and further reading.

The Infinite, Third Edition is ideal reading for anyone interested in an engaging and historically informed account of this fascinating topic, whether from a philosophical point of view, a mathematical point of view, or a religious point of view.

Part I

The history

Philosophers have traditionally concerned themselves with two quite disparate tasks: they have, on the one hand, tried to give an account of the origin and structure of the world and, on the other hand, they have tried to provide a critique of thought. With the concept of the infinite, both tasks are united. Since the time of Anaximander to apeiron has been invoked as a basic cosmological principle. And the conceptual change that occurs as to apeiron of the Presocratics is refined and criticized by Plato and Aristotle, to the development of Cantor’s theory of the transfinite and its critique by Brouwer, is one of the great histories of a critique of reason.

(Jonathan Lear)

1 Early Greek thought

It is incumbent upon the person who treats of nature to discuss the infinite and to enquire whether there is such a thing or not, and, if there is, what it is… [And] all who have touched on this kind of science in a way worth considering have formulated views about the infinite.

(Aristotle)

1.1 Anaximander and to apeiron¹

The Greek word ‘peras’ is usually translated as ‘limit’ or ‘bound’. ‘To apeiron’ denotes that which has no peras, the unlimited or unbounded: the infinite.

To apeiron made its first significant appearance in early Greek thought with Anaximander of Miletus (c.610 BC to shortly after 546 BC). Its role was very different from that which it tends to play in modern thought. It was introduced in response to what was then (and has remained) a basic intellectual challenge: to identify the stuff of which all things are made. What, as the Greeks would have put it, is the ‘principle’ of all things? Thales had earlier proposed that it is water. Perhaps he had been impressed by the natural processes whereby the sea evaporates under the influence of the sun, then forms clouds, dissolves in the form of rain, and soaks into the earth, moistening the food by which living things are nourished. Still, why single out water in this way as anything more than just one of the many forms that basic stuff could take? There was something arbitrary about this. So Anaximander’s proposal was that the primal substance of which all things are made is to apeiron. This he conceived as something neutral, the boundless, imperishable, ultimate source of all that is.

But it was not just that. It was also something divine, something with a deeper significance. Given the processes whereby substances change into one another, the losses and compensating gains, it made good metaphysical sense to suppose that there was an underlying changeless substratum. But for Anaximander it made as much ethical sense. He saw, in the multifarious activity that surrounds us, disharmony and imbalance. He held that opposites were in continual strife with one another (hot with cold, dry with wet, light with dark,…); they were continually encroaching on one another (day giving way to night, night giving way to day,…) and continually committing injustice against one another. He believed that they must, in time, return to to apeiron in order to atone. There they would lose their identity, for where there is no peras there are no opposites; and all strife would be overcome.

Anaximander’s concerns were at once scientific, philosophical, and ethical. He would not have recognized the modern distinction between empirical hypotheses about the physical nature of the world and a priori reflections on how things must or ought to be. We shall see later Greek thinkers to some extent disentangling these strands. Aristotle, in particular, put an empirical gloss on many of these ideas, though he continued to recognize their theological connotations. But Anaximander was simply interested in knowing what the world was like, in the most general sense.

One consequence of this is that it is hard for us to know how seriously to take the materiality of to apeiron. To apeiron could not be identified with water, or gold, or anything else of such a specific kind: these were at most limited and determinate aspects of it. But could it even be identified with matter? Or was being material and occupying space already a way of being limited and determinate and having a peras? If to apeiron was not material, but something utterly transcendent, then it was signalling what was to become a pervasive and characteristic feature not only of Greek thought but of much subsequent philosophy: the idea of a radical distinction between appearance and reality, where the former includes all that we ever actually encounter and the latter is what underlies and makes sense of it. But it is not clear whether such a radical metaphysics was Anaximander’s. He talked of to apeiron as ‘surrounding’ us. He may have meant this quite literally.

Given this uncertainty, we find that, when we return to the distinction drawn in the introduction between the metaphysical and the mathematical, we cannot say definitely that Anaximander was working in either territory. For example, given that we cannot even be sure that to apeiron was spatial, we certainly cannot be sure that it was mathematically infinite. Later on in this chapter we shall see clear early signs of the polarization between the mathematical and the metaphysical, with concepts of both kinds beginning to filter through into Greek consciousness. But at this early stage in the story we do best not to press the categories. After all, to apeiron was radically indeterminate: it was supposed to resist any easy classification.

What does emerge from Anaximander’s thinking is a keen awareness of our own finitude and of the finitude of the ephemera around us, characterized by their generation and decay. I suggested in the introduction how such an awareness might at the same time be an awareness of the infinite. Anaximander, certainly, could make no sense of such finitude except in terms of that which is unlimited and unconditioned; that which suffers neither generation nor decay, so ensuring that the patterns of change that we observe never give out; that into which the ephemeral is destined, ultimately, to be cast back. As he himself put it, in what is the oldest surviving fragment of western philosophy:

The principle and origin of existing things is to apeiron. And into that from which existing things come to be they also pass away according to necessity; for they suffer punishment and make amends to one another for their injustice, in accordance with the ordinance of time.²

1.2 The Pythagoreans

By the time of the Pythagoreans there had been a remarkable turn-about. Pythagoras (born c.570 BC) was an Ionian. He founded a religious society in Crotona in southern Italy. Central to the outlook of its members was a passionate belief in the essential goodness of what had seemed to Anaximander essentially bad. Where he saw disharmony, imbalance, and strife, they saw harmony, order, and beauty. The regular cycles of the planets, the recurring patterns in nature, the finely proportioned structures in the physical world – these all betokened, for the Pythagoreans, rhyme and reason; that which is comprehensible and good; that which has a peras. To apeiron, by contrast, was something abhorrent.

It was now unquestionably being conceived also as something spatial. More specifically it was a dark, boundless void beyond the visible heavens. They believed that because it had no end in the sense of limit (peras), it equally had no end in the sense of purpose or destiny (telos). It was senseless, chaotic, indeterminate, and without structure, simply waiting to have a peras imposed upon it. For they quite generally assimilated what has a peras to what is good (and to what is one, or odd, or straight, or male, among other things); and they correspondingly assimilated to apeiron to what is bad (and to what is many, or even, or curved, or female, among other things). These assimilations were part of a table of opposites that they recognized, whose two fundamental principles, or heads, were Peras and Apeiron. They believed that the world was the result of an imposition of the former on the latter, a planting of the seed of Peras into the void of Apeiron. What issued was a beautifully structured, harmonious whole whose parts were held together in unity precisely because of their limitedness and finitude. And the world continued to ‘breathe in’, and at the same time to subjugate, the surrounding apeiron: by doing so, it structured it, and ordered it, and gave it definite shape.³

Integral to this picture were the natural numbers (see above, §0.2). These for the Pythagoreans were the key to everything. For it was in their terms, most characteristically in terms of finite numerical ratios, that the imposition of Peras on Apeiron was to be understood. The Pythagoreans were particularly impressed, for example, by Pythagoras’ own (alleged) discovery that musical harmony can be understood in such terms. If the ratio of the lengths of two tuned strings is 2:1, then the shorter sounds an octave higher than the longer; if 3:2, then a fifth; and if 4:3, a fourth. In radical contrast to Anaximander, the Pythagoreans believed that everything that ultimately made sense made such sense as this. (There is a connection here with the very fact that we use the word ‘ratio’ as we do, and talk of ‘rational’ numbers (see above, §0.2): ‘ratio’ in Latin means reason.⁴)

Natural numbers took on a mystical significance for the Pythagoreans. For example, the first four, those involved in the musical intervals – they discounted 0 – add up to 10, which they held to be a perfect number. This sum is illustrated in the symbol known as the tetractys (see Figure 1.1), a symbol which they believed to be sacred and by which they swore. Indeed not only were natural numbers the key to everything in this scheme, ultimately they were everything. They were everything because in the last analysis there was nothing else to which intelligible reference could be made. When Peras was imposed on Apeiron, numbers were what resulted. The world was a system of structures built within a void, each definable in numerical terms and together constituting a glorious musico-mathematical whole. (This idea, in a suitably modern guise, still has adherents. Mathematics plays a crucial role in the most fundamental scientific theories; and it is still possible to cherish the hope of being able to account for all physical phenomena by appeal to their formal or structural properties, in essentially mathematical terms.)

Figure 1.1

But the Pythagoreans’ veneration of the natural numbers, and their abhorrence of to apeiron, were to receive a rude shock.

Pythagoras himself, as is well known, is said to have discovered that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. This means, f...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Preface to the second edition
• Preface to the third edition
• Introduction Paradoxes of the infinite
• Part I: The history
• Part II: Infinity assessed
• Part III. Infinity superseded
• Glossary
• Bibliography
• Index