Infinity

1. Early Greek thought
2. Aristotle
3. The rationalists and the empiricists
4. Kant
5. Post-Kantian metaphysics of the infinite
6. Modern mathematics of the infinite
7. Human finitude

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 in the sixth century bc (see Anaximander §2). He thought of it as the boundless, imperishable, ultimate source of everything that is. He also thought of it as that to which all things must eventually return in order to atone for the injustices and disharmony which result from their transitory existence.

Anaximander was something of an exception, however. On the whole, the Greeks abhorred the infinite (as the old adage has it). More typical of that era were the Pythagoreans, a religious society founded by Pythagoras (see Pythagoreanism §2). They believed in two basic cosmological principles, Peras and Apeiron, the former subsuming all that was good, the latter all that was bad. They held further that the whole of creation was to be understood in terms of, and indeed was ultimately constituted by, the positive integers 1, 2, 3,...; and that this was made possible by the fact that Peras was continuously subjugating Apeiron (the integers themselves, of course, are each finite). The Pythagoreans were followed to some extent in these beliefs by Plato, who also held that it was the imposition of limits on the unlimited that accounted for all the numerically definable phenomena that surround us.

However, the Pythagoreans soon learned to their dismay that they could not simply relegate the infinite to the role of cosmic villain. This was because of Pythagoras' own discovery that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. Given this theorem, the ratio of a square's diagonal to each side is . There are some good approximations to this ratio: it is a little more than , for example, and a little less than . Indeed there are approximations of any desired degree of accuracy. Nevertheless, given the basic tenets of Pythagoreanism, it ought to be exactly , for some pair of positive integers p and q. The problem was that they discovered a proof that it is not, which they regarded as nothing short of catastrophic. According to legend, one of them was shipwrecked at sea for revealing the discovery to their enemies. The Pythagoreans had stumbled across the 'irrational' within mathematics. They had seen the limitations of the positive integers, and had thereby been forced to acknowledge the infinite in their very midst.

At around the same time, Zeno of Elea was formulating various celebrated paradoxes connected with the infinite (see Zeno of Elea §6). Best known of these is the paradox of Achilles and the tortoise: Achilles, who runs much faster than the tortoise, cannot overtake it in a race if he lets it start a certain distance ahead of him. For in order to do so he must first reach the point from which the tortoise started, by which time the tortoise will have advanced a fraction of the distance initially separating them; he must then make up this new distance, by which time the tortoise will have advanced again; and so on ad infinitum. Such paradoxes, as well as having a profound impact on the history of thought about infinity, did much to reinforce early Greek hostility to the concept.

Aristotle's understanding of the infinite was an essentially modern one in so far as he defined it as the untraversable or never-ending. But he perceived a basic dilemma. On the one hand Zeno's paradoxes, along with a host of other considerations, show that the concept of the infinite really does resist a certain kind of application to reality. On the other hand there seems to be no prospect of doing without the concept, as the Pythagoreans had effectively realized. As well as , time seems to be infinite, numbers seem to go on ad infinitum, and space, time and matter all seem to be infinitely divisible.

Aristotle's solution to this dilemma was masterly. It has dominated all subsequent thought on the infinite, and until very recently was adopted by almost everyone who considered the topic. Aristotle distinguished between the 'actual infinite' and the 'potential infinite'. The actual infinite is that whose infinitude exists, or is given, at some point in time. The potential infinite is that whose infinitude exists, or is given, over time. All objections to the infinite, Aristotle insisted, are objections to the actual infinite. The potential infinite is a fundamental feature of reality. It is there to be acknowledged in any process which can never end: in the process of counting, for example, in various processes of division, or in the passage of time itself. The reason why paradoxes such as Zeno's arise is that we pay insufficient heed to this distinction. Having seen, for example, that there can be no end to the process of dividing a given racecourse, we somehow imagine that all those possible future divisions are already in effect there. We come to view the racecourse as already divided into infinitely many parts, and it is easy then for the paradoxes to take hold.

Even those later thinkers who did not share Aristotle's animosity towards the actual infinite tended to recognize the importance of his distinction. Often, though, Aristotle's reference to time was taken as a metaphor for something deeper and more abstract. This in turn usually proved to be something grammatical. Thus certain medieval thinkers distinguished between categorematic and syncategorematic uses of the word 'infinite'. Putting it very roughly, to use the word categorematically is to say that there is something with a property that surpasses any finite measure; to use the word syncategorematically is to say that, given any finite measure, there is something with a property that surpasses it. In the former case the infinite has to be instantiated 'all at once'. In the latter case it does not.

The categorematic-syncategorematic distinction heralds another distinction, whose importance to the infinite is hard to exaggerate. This is the distinction between saying that there is something of kind X to which each thing of kind Y stands in relation R, and saying that each thing of kind Y stands in relation R to something of kind X (not necessarily the same thing each time). This is referred to below as the 'Scope Distinction' (see Scope).

But Aristotle himself was not thinking in these very abstract terms. He took the references to time in his own account of the actual-potential distinction quite literally, and this gave rise to his most serious difficulty. He held that time (unlike space) is infinite. He also held that time involves constant activity, as exemplified in the revolution of the heavens. When our attention is focused on the future, there is no obvious problem with this. Past revolutions, however, because they are past, seem to have an infinitude which is by now completely given to us, and hence which is actual. This difficulty, in various different guises, has been a continual aggravation for philosophers who have wanted to see the infinite in broadly Aristotelian terms.

For over two thousand years Aristotle's conception of the infinite was regarded as orthodoxy. Often this conception was motivated by a kind of empiricism: the actual infinite was spurned on the grounds that we can never encounter it in experience. But does the potential infinite fare any better in this respect? Is experience of an infinitude that is given over time any less problematic than experience of an infinitude that is given all at once? The more extreme of the British empiricists were hostile to the infinite in all its guises. Where Aristotle had felt able to accept that space and time were infinitely divisible, Berkeley and Hume denied even that. They thought that the concept of the infinite was one that we could, and should, do without (see Empiricism).

This was partly a backlash against their rationalist predecessors. The rationalists had argued that we could form an idea of the infinite, even though we could neither experience it nor imagine it. They thought that this idea was an innate one, and that it constituted, or helped to constitute, a vital insight into reality. They did not see any difficulty in this view. As Descartes put it, the fact that we cannot grasp the infinite does not preclude our touching it with our thoughts, any more than the fact that we cannot grasp a mountain precludes our touching it (see Rationalism §2).

Descartes believed that our idea of the infinite had been implanted in our minds by God (see Descartes, R. §6). Indeed this was the basis of one of his proofs of God's existence. Only a truly infinite being, Descartes argued, could have implanted such an idea in our minds. Note here the assimilation of the infinite to the divine: this was a legacy of medieval thought which is nowadays quite commonplace. But when the assimilation was first made, at the end of antiquity - most famously, by the Neo-Platonist Plotinus-it marked something of a turning point in the history of thought about the infinite (see Plotinus §§3, 4). Until then there had been a tendency to hear 'infinite' as a derogatory term. Henceforth, it was quite the opposite.

The empiricists, meanwhile, needed to defend their rejection of the infinite against the charge that it invalidated contemporary mathematics. They had more or less sophisticated ways of doing this, though in the case of geometry, where the problem was at its most acute, Hume took the rather cavalier step of simply denying certain crucial principles which mathematicians took for granted. (Berkeley's chief concern was with the use of infinitesimals in the recently invented calculus. In fact, his reservations were perfectly justified: it was a century before they were properly addressed.)

Kant played his characteristic role of conciliator in the debate on the infinite (see Kant §§2, 5, 8). He had an empiricist scepticism about the infinite, based on the fact that we cannot directly experience it. Nevertheless, he sided with the rationalists by insisting that there are certain formal or structural features of what we experience, which are accessible a priori and which do involve the infinite. Thus he thought that space and time were infinite (both in the sense of being infinitely extended and in the sense of being infinitely divisible): it is written into the form of whatever we experience that there can also be experience of how things are further out, further in, earlier or later. These, on Kant's view, were mathematical truths, a priori and unassailable.

But there is a question about how the topology of space and time can be a priori. Kant's celebrated reply was that space and time are not features of 'things in themselves'; they are part of an a priori framework which we contribute to our experience of things. What then of the contents of space and time, the physical universe as a whole? This was different. Kant did not think that what was physical was constructed a priori. Nor, on the other hand, did he think that it was ultimately real, that is to say real in a way that transcends any possible access we have to it. It had no features, on Kant's view, that exceed what we are capable of grasping through experience. So here the concept of the infinite did resist application. It still had what Kant regarded as a legitimate regulative use. That is, we could proceed as if the physical universe as a whole were infinite, thereby encouraging ourselves never to give up in our explorations. But we ultimately had no way of making sense of such infinitude. Kant was forced to take an extreme empiricist line by denying that the physical universe as a whole is infinitely big, that it has infinitely many parts and, going this time beyond Aristotle (thus bypassing the difficulty that had beset Aristotle himself), that it is infinitely old.

However, there was a dilemma. Kant was also forced to deny that the physical universe is finite in each of these three respects. Apart from anything else, to postulate infinite, empty space or time beyond the confines of the physical universe is itself to postulate that which exceeds what we are capable of grasping through experience.

This dilemma looks acute. Kant himself presented it in the form of a pair of 'antinomies'. These antinomies consisted of the principal arguments against the physical universe's being infinite in each of the specified respects, and the principal arguments against its being finite. But he believed that the dilemma contained the seeds of its own solution. If what is physical is not ultimately real - if there is no more to it than what we are capable of experiencing - then we are at liberty to deny that there is any such thing as the physical universe as a whole. There are only the finite physical things that are accessible to us through experience. The physical universe as a whole is neither infinite nor finite. It does not exist.

Kant's solution involved him in a direct application of the Scope Distinction (see §2 above). On the one hand he affirmed that any finite physical thing is contained within something physical. On the other hand he denied that there is something physical within which any finite physical thing is contained. Both of these, the affirmation and the denial, were grounded in the fact that there is nothing we can identify in space and time such that we cannot identify more. This is fundamentally a fact about us: the fact that we are finite. Our identifications are always incomplete. What Kant added, in an idealist vein, was that what we cannot identify does not exist. Here, as in so many other places, we see how deeply involved with human finitude Kant's philosophy was, and how seriously he took it.

Metaphysical thought about the infinite since Kant has continued to be just as deeply involved with human finitude. Existentialists in particular have been greatly exercised by it, especially in its guise of mortality. But they have also for the most part recognized an element of the infinite within us. This too is Kantian. Kant believed that we are free rational agents, and that when our agency is properly exercised, it has an unconditioned autonomy that bears all the hallmarks of the truly infinite. For Kant, this was something which exalts us. But for many of the existentialists, still preoccupied with the fundamental fact of human finitude, it is something which is responsible for the deepest tensions within us, and thus for the absurdity of human existence (see Existentialism §2).

Hegel agreed with Kant that the truly infinite is to be found in the free exercise of reason (see Hegel, G.W.F. §8). But he took this further than Kant. He argued that reason is the infinite ground of everything. Everything that happens, on Hegel's view, can be understood as the activity of a kind of world-spirit, and this spirit is reason.

This led Hegel to a very non-Aristotelian conception of the infinite. For Hegel, the infinite was the complete, the whole, the unified. Aristotle's conception of the infinite as the never-ending was in Hegel's view quite wrong. He explained this conception as arising from our finite attempts to assimilate the truly infinite. And he described Aristotelian infinity as a 'spurious', or 'bad', infinity - a mere succession of finite elements, each bounded by the next, but never complete and never properly held together in unity. Such 'infinitude' seemed to Hegel at turns nightmarish, then bizarre, then simply tedious, but always a pale, inadequate reflection of the truly infinite.

Despite Kant's influence on Hegel, and despite his own commitment to infinite reason (as well as to infinite space), Kant certainly helped to propagate the Aristotelian tradition of treating the actual infinite with hostility and suspicion. As this tradition prevailed, the actual infinite came increasingly to be understood in the more general, non-temporal sense indicated in §2 above. Eventually, exception was being taken to any categorematic use of the word 'infinite'. The most serious challenge to this tradition, at least in a mathematical context, was not mounted until the nineteenth century, by Cantor, whose mathematical contribution to this topic is unsurpassed (see Cantor, G.).

Objections to the actual infinite had tended to be of two kinds. The first kind we have already seen: objections based on the fact that we can never encounter the actual infinite in experience. Objections of the second kind were based on the paradoxes to which the actual infinite gives rise. These paradoxes fall into two groups. The first group consists of Zeno's paradoxes and their variants. By the time Cantor was writing, however, the calculus (which had then reached full maturity) had done a great deal to mitigate these. Of more concern by then were the paradoxes in the second group, which had been known since medieval times. These were paradoxes of equinumerosity. They derive from the following principle: if (and only if) it is possible to pair off all the members of one set with all those of another, then the two sets must have just as many members as each other. For example, in a non-polygamous society, there must be just as many husbands as wives. This principle looks incontestable. However, if it is applied to infinite sets, it seems to flout Euclid's notion that the whole is greater than the part. For instance, it is possible to pair off all the positive integers with those that are even: 1 with 2, 2 with 4, 3 with 6 and so on.

Cantor accepted this principle. And, consistently with that, he accepted that there are just as many even positive integers as there are positive integers altogether. Far from being worried by this, he defined precisely what is going on in such cases, and then incorporated his definitions into a coherent, systematic and rigorous theory of the actual infinite, ready to be laid before any sceptical gaze.

It might be expected that, on this understanding, all infinite sets are the same size. (If they are, that is not unduly paradoxical.) But much of the revolutionary impact of Cantor's work came in his demonstration that they are not. There are different infinite sizes. This is a consequence of what is known as Cantor's theorem: no set, and in particular no infinite set, has as many members as it has subsets. In other words, no set is as big as the set of its subsets. If it were, then it would be possible to pair off all its members with all its subsets. But this is not possible. Suppose there were such a pairing and consider the set of members paired off with subsets not containing them. Whichever member was paired off with this subset would belong to it if and only if it did not belong to it (see Cantor's theorem).

In the course of developing these ideas, Cantor laid down some of the basic principles of the set theory which underlay them; he devised precise methods for measuring how big infinite sets are; and he formulated ways of calculating with these measures. In short, he established transfinite arithmetic.

Even so, there are many who remain suspicious of his work and who continue to think of the infinite in broadly Aristotelian terms. Cantor himself was forced to admit that there are some collections, including the collection of all things, which are so big that they cannot be assigned any determinate magnitude (their members cannot be given 'all at once'). Concerning such collections, he even sometimes said that they are 'truly' infinite. There is in fact a real irony here: Cantor's work can in many ways be regarded as corroborating Aristotelian orthodoxy.

Brouwer believed that Cantor had gone wrong in not showing sufficient respect for the first kind of objection to the actual infinite: that we cannot encounter it in experience. All Cantor had done, in Brouwer's view, was to demonstrate certain tricks that can be played with (finite) symbols, without addressing the question of how these tricks answer to experience. The relevant experience here - the experience to which any meaningful mathematical statement must answer, according to Brouwer and other members of his intuitionistic school - is our experience of time. It is by recognizing the possibility of separating time into parts, and then indefinitely repeating that operation over time, that we arrive at our idea of the infinite. And such infinitude is potential, not actual - in the most literal sense (see Intuitionism §1).

There was a very different critique of Cantor's ideas in the work of Wittgenstein, though it led to similar results (see Wittgenstein §14). Wittgenstein believed that insufficient attention had been paid (at least by those interpreting Cantor's work, if not by Cantor himself) to what he called the 'grammar' of the infinite, that is to certain fundamental constraints on what could count as a meaningful use of the vocabulary associated with infinity. In effect, Wittgenstein argued that the word 'infinite' could not be used categorematically.

Problems about the infinite, we have seen, are grounded in our own finitude. On the one hand our finitude prevents us from being able to think of anything, including the whole of reality, as truly infinite. On the other hand it also prevents us from being able to think of anything finite - anything to that extent within our grasp - as the whole of reality. One way to reconcile these would be to deny that there is any such thing as the whole of reality and to argue that there are only bits of reality, each a part of some other. Here once again we see application of the Scope Distinction: every bit of reality is a part of something, but there is nothing of which every bit of reality is a part. Aristotle, Kant and even to an extent Cantor played out variations on this theme.

But one of the most pressing questions of philosophy still remains: in what exactly does our finitude consist? Some of the most striking features of that finitude are conditioned by our temporality. In particular, of course, there is the fact of our death. How are we to view death? Among the many subsidiary questions that this raises, there are two in particular which are superficially equivalent but between which it is important to distinguish. Putting them in the crudest possible terms (their refinement would be a large part of addressing them): (1) Is death a 'bad thing'? (2) Would immortality be preferable to mortality?

It can easily look as if these questions must receive the same answer. True, no sooner does one begin refining them than one sees all sorts of ways in which a full, qualified response to one can differ from a full, qualified response to the other. But it is in any case important to see how, even at this crude level, there is scope for answering 'yes' and 'no' respectively. Putting it very roughly, death is a bad thing because it closes off possibilities, but immortality would not be preferable to mortality because mortality is what gives life its most basic structure and, therewith, the possibility of meaning.

To answer 'yes' to (1) and 'no' to (2) in this way is once again to invoke the Scope Distinction. It is to affirm that at each time there is reason to carry on living for longer, while denying that there is reason to carry on living forever. Meaning, for self-conscious beings such as us, can extend further than any given limits. But it cannot extend further than them all.

If it is true that, in some sense, at some level and with all the myriad qualifications that are called for, the answer to (1) is 'yes' and the answer to (2) is 'no', then, coherent though that is, it points to a basic conflict in us: while it would not be good never to die, it is nevertheless never good to die. That conflict is one of the tragedies of human existence. It is also a version of the original conflict which underlies all our attempts to come to terms with the infinite. In thinking about the infinite, we are thinking, at a very deep level, about ourselves.

See also: Continuum hypothesis, the §1; Death