No one shall be able to drive us from the paradise that Cantor has created for us. (David Hilbert)

I would say, `I wouldn't dream of trying to drive anyone from this paradise.' I would do something quite different: I would try to show you that it is not a paradise — so that you'll leave of your own accord. I would say, `You're welcome to this; just look about you.' . . .

(For if one person can see it as a paradise . . ., why should not another see it as a joke?) (Ludwig Wittgenstein)

At one level, nobody could fail to be impressed by Cantor's work. It showed mathematical craftsmanship of the very highest calibre. But there was great room for debate about its significance. The infinite seemed at last to have been subjected to precise mathematical scrutiny. But perhaps Cantor had been indulging in technical flights of fancy. Perhaps, as I suggested in the last chapter, he had actually confirmed some of the most deeply entrenched prejudices about the infinite with his talk of inconsistent totalities.

Certainly it was too much to expect that the infinite would now uncritically be taken on board as an object of mathematical enquiry in just the way that Cantor had represented it. There was some immediate opposition and hostility to his work, as we saw in the last chapter. Later thinkers reacted against it at a somewhat deeper level, recognizing the importance and merit of what they were reacting against but using this to think through some of the most fundamental questions about the nature of the infinite (and of mathematics). It is to their work, and related work, that we turn in this chapter.

1 Intuitionism¹

The Dutch mathematician L.E.J. Brouwer (1881-1966) founded what was to become one of the most influential schools in the philosophy of mathematics, intuitionism. One of his main contentions was that mathematical statements had content: they were about something. Mathematics was not just a game in which meaningless symbols were manipulated in various ways. It had to confront mathematical experience, and answer to it. Picking up on an idea that had been present in both Aristotle and Kant, Brouwer argued that the experience to which all of mathematics answered (including geometry) was the experience which we each enjoyed of the pure structure of time. Starting with this, each of us was able to `construct' the subject matter of mathematics. How so?

Time was a continuous (seamless) whole, but we could separate it into parts — past and future, say — in our thought. By doing this we arrived at what Brouwer called `the basal intuition of mathematics', namely `the intuition of the bare two-oneness'. (He also referred to this as `the falling apart of moments of life into qualititively different parts, to be reunited only while remaining separated by time.'²) We could then indefinitely repeat this process, over time. We thus arrived at the fundamental idea of a progression. This in turn gave rise to two of the structural concepts integral to mathematics, infinitude by addition and infinitude by division.

This led Brouwer to re-embrace an Aristotelian conception of the infinite. The infinite was something that had to be given, or (better) constructed, over time. Its existence was potential, never actual.

Cantor's work had run directly counter to this. It had therefore, in Brouwer's view, lost touch with mathematical experience — the experience of the structure of time. Any interest it had was of a purely formal kind. It showed some of the technical tricks that could be played with (finite) mathematical symbols, but it did not truly relate to the infinite. For example, Cantor's principal defence for treating as a completed whole was that it was consistent to do so. But for Brouwer, mere consistency did not make pronouncements true, or even meaningful. To have meaning, they needed to answer to mathematical experience. Brouwer could not accept that there was any such thing (completed whole) as ; no such totality could be constructed from mathematical experience. What could be constructed were individual real numbers. This was done through the specification of laws determining their decimal expansions (say). But there was no way of constructing the real numbers in toto. (With it was different. There was a single principle whereby successive natural numbers were constructed, that of adding 1. Talk of as a whole could be legitimated by our grasp of this principle.) Brouwer's attitude to the diagonal argument was somewhat like Poincaré's (see above, Chapter 8, § 3). It did not show that one thing was bigger than another. What it showed was that, given any progression of real numbers, there was a general recipe available for constructing a real number not in the progression.

If this critique is right, as many intuitionists still believe, then much of Cantor's edifice crumbles. But so too, significantly, does much of so-called classical mathematics, mathematics of the kind that has been accepted ever since antiquity. Why is this?

Consider the natural numbers. It is impossible for them all to be constructed within time. But we are inescapably immersed in time; and, for intuitionists, mathematical statements have to derive their meaning from what we, so immersed, are capable of constructing. How then is arithmetic possible? There is no problem with statements about particular natural numbers, for example that 7 + 5 = 12. We can always construct a large enough finite selection of natural numbers to effect the relevant calculations. But there is a problem with (true) generalizations about the natural numbers, for example that each one is the sum of four squares (in the way that, say, 21 = 16 + 4 + 1 + 0). The intuitionist solution to this problem is as follows. They argue that, in the case of a generalization of this kind, what we can construct is a proof, which trades on the principle whereby successive natural numbers are constructed (the principle of adding 1), and which enables us to see, in advance, why every natural number must have the property in question. For example, we might be able to see that 0 has it; and we might be able to see that, if some arbitrary natural number n has it, then n + 1 must have it too; we can then conclude, by the principle known as mathematical induction, that every natural number must have it. What is unintelligible on this view is the idea of an `infinite co-incidence', whereby it so happens that every natural number has some property but there is no proof of this kind to show why. For this to be intelligible it would have to be possible to construct all of the natural numbers, in a finite time, and then to determine, by brute force, that they were alike in the relevant respect. But this is precisely what is not possible.

But why does this cast doubt on classical mathematics? This is best answered in terms of an example. Consider Goldbach's conjecture, that every even number greater than 2 is the sum of two primes (as, say, 8 = 5 + 3). At the time of my writing this, no counterexample to the conjecture has been discovered: but neither has the conjecture been proved. Still, it is part and parcel of classical mathematics to assume that either there is a counterexample or the conjecture is true. In recoiling from the possibility of an infinite co-incidence, intuitionists cannot share this assumption. It is not that they have in mind a third alternative. Their position needs to be subtler than that. It is rather that they are not prepared to invoke the assumption, or others like it, in the course of their mathematical reasoning. Their Aristotelian conception of the infinite thus poses a challenge to some of the most elemental presuppositions of classical mathematics.³

2 Finitism⁴

If Brouwer had thrown down the gauntlet, then the great German mathematician David Hilbert (1862-1943) was prepared to pick it up. He famously described Cantor's creation as a `paradise'.⁵ Though well aware that it was not without its problems, he felt that it was worth every effort to defend, along with the framework of classical mathematics within which it had been presented. Given all its critics, and given also the newly discovered paradoxes, this was a daunting challenge. What was needed, Hilbert thought, was a careful critique of the infinite, something that would serve to explain and to justify just what Cantor had been up to when he represented the infinite as an object of mathematical study. A similar critique of infinitesimals had already shown how talk of the infinite was sometimes just a harmless façon de parler, a fact that had been recognized before by Leibniz and indeed by the medievals. Perhaps the same might prove true with respect to Cantor's work.

The first thing to note, Hilbert said, was that, given recent scientific developments, there was less and less empirical backing for the concept of infinity. The empirical evidence suggested — and still does — that the physical world was only finitely big and only finitely divisible. (Kant had been wrong to suppose that no empirical evidence could ever bear on this question (see above, Chapter 6, § 3).) Likewise, perhaps, with space and time themselves.⁶ For example, the possibility that space was finitely big had been mooted by the German physicist Albert Einstein (1879-1955) in the context of his celebrated theory of relativity. The idea was that space might be finite but unbounded, in something like the way in which the space of two-dimensional beings on the surface of a sphere would be finite but unbounded; finite, because any sufficiently long journey `in a straight line' away from a given point would bring them back to it.⁷ (The possibility of finite but unbounded space undercuts Archytas' inference from the universe's having no edge to its being spatially infinite (see above, Chapter 1, § 5).)

It seemed, then, that any defence of the infinite, as construed by Cantor, needed to be internal to mathematics. The way to proceed, Hilbert argued, was to start with those finitary parts of mathematics that were incontrovertible; then, renouncing all intuitions concerning the infinite (for by now, these were all inevitably suspect or question-begging), to see how talk about the infinite might nevertheless fit in. What were incontrovertible were simple numerical equations such as `7 + 5 = 12' and the like. Unlike Brouwer, Hilbert did not think that the meaning of such equations was to be accounted for in terms of our experience of time. Rather, such equations reported the results of combining and manipulating sequences of perceptible signs in various ways. To explain this, he envisaged a crude system of numerals whereby each positive natural number was represented by the same number of strokes. When such equations were spelt out in this symbolism, they very nearly were their own meaning. For example, instead of `7 + 5 = 12', we could write

               1111111 + 11111 = 111111111111,

and then see by inspection that it was true. Equations of this kind Hilbert referred to as finitary propositions. But there were also various generalizations concerning the natural numbers, references to as an infinite whole, and suchlike. These he referred to as ideal propositions. It was clear, in Hilbert's mind, that ideal propositions played a very important role in mathematics. His task now was to account for this.

We have already seen the account that Brouwer gave. But Hilbert wanted something that would do less violence to classical mathematics and indeed to Cantor's work. He knew he could not proceed by explaining the meaning of ideal propositions in terms of how they hooked up with some (infinite) feature of mathematical reality. To do that would have been to disregard the very problems that had prompted his enquiry in the first place. Instead he argued that ideal propositions had no meaning. They did not correspond to anything in mathematical reality in the way that finitary propositions did. Rather they were supplementary devices of a purely formal kind that were brought in to facilitate proofs and to make for greater elegance and perspicuity. Thus instead of proving in laborious detail that every natural number less than a million was the sum of four squares, we could take a short-cut via the ideal proposition, `Every natural number is the sum of four squares.' The concept of the infinite was therefore something like a Kantian Idea, an a priori concept applied beyond the realm of experience, to guide and regulate mathematical practice. There was no such thing as the infinite, which is why Hilbert's position is often called finitism, but we could proceed as if it existed. Hilbert himself put it as follows:

Nowhere is the infinite realized; it is neither present in nature nor admissible as a foundation in our rational thinking. . . . The role that remains to the infinite is, rather, merely that of an Idea — if, in accordance with Kant's words, we understand by an Idea a concept of reason that transcends all experience and through which the concrete is completed so as to form a totality.⁸

What Hilbert needed to do now was to show that this vindicated Cantor's transfinite mathematics, with all its classical presuppositions. Of course, since there was no question of the ideal mathematics having to be true, it could be rigged to precisely this end. But there needed to be some guarantee that we could not then use it to prove things that we did not want to prove (say, that 7 + 5 = 13); otherwise it would have lost its rationale. It did not have to be true. But what issued from it did. However, Hilbert was confident that suitable ratification could be provided. He argued that this would involve carrying out a two-part programme, which subsequently became known as Hilbert's programme.

Hilbert's programme:

(i) The new mathematics, including all its ideal elements, would have to be formalized, by being cast axiomatically.

(ii) Its consistency (with finitary mathematics) would have to be established, without — question-beggingly — presupposing any of its ideal methods of proof.

(ii) would have to involve investigating the purely formal properties of these ideal methods in a way that would be tantamount to doing straightforward finitary mathematics. Completing the programme would have the added bonus of putting paid to any worries we might still have about the possibility of further paradoxes lurking, of the kind that had already come to light.

Hilbert presented all of these ideas in a paper in which he also sketched how the programme might be completed.⁹ Towards the end of the paper he `played a last trump'. He gave an outline of how the theory he had presented, also in outline, could be used to settle Cantor's unanswered question about the size of (see above, Chapter 8, § 3). He attempted to show that was the next infinite size up from .

However, he used assumptions that very few later mathematicians were prepared to accept. Not that that was by any means the greatest setback to his efforts. Much more serious was Gödel's work, mentioned at the end of the last chapter. This appeared only six years later, and it seemed to show conclusively that in fact neither part of his programme could be satisfactorily carried out. I shall discuss this in further detail in Part Two (see below, Chapter 12, § 3).

Prior to Gödel's discoveries, however, there were many who embraced Hilbert's finitism. The German philosopher Hermann Weyl (1885-1955), having at one stage sided with Brouwer, later came round to the view that we did have access to infinite wholes — in the sense licensed by Hilbert, namely that we could use symbols as if they stood for them, in ideal propositions.¹⁰

The Austrian philosopher Felix Kaufmann (1895-1949) likewise accepted much of Hilbert's critique. He denied that there were strictly any infinite sets, but insisted that we could talk about as a convenient façon de parler, for this made it easier to discuss finite sets. (It was like talking about the average parent: there was no such person but it made it easier to discuss real parents.) However, unlike Hilbert, Kaufmann did not feel that the whole of Cantor's transfinite mathematics could be vindicated in this way. For it was too far removed from reality, that is the reality of the natural numbers as described in simple equations such as `7 + 5 = 12'. (This was reminiscent of Kronecker (see above, Chapter 8, § 3).) For example, like Brouwer, Kaufmann denied the existence of and took the diagonal argument to show merely how, given a recipe for filling in an `infinite square' of digits, we could construct a real number whose decimal expansion was not on any row of the `square'.¹¹

Although these reactions pre-dated Gödel's work, Hilbert's ideas have continued to be of great interest, and to exert considerable influence in the philosophy of mathematics. There are those who think many of them are still viable. They were also an important spur to Wittgenstein.

3 Wittgenstein

Ludwig Wittgenstein (1889-1951) was an Austrian who spent much of his life in England. For me he stands with Heidegger as one of the two giants of twentieth-century philosophy. It is customary to divide his work into two phases, and, despite profound continuities between the earlier phase and the later phase, this is entirely apt. (It is said that one of the things that initiated the later phase, by reawakening his interest in philosophy after a long period of philosophical inactivity, was attending a lecture by Brouwer on the foundations of mathematics.) I shall be making special use of his earlier work in Part Two (see below, Chapter 13). But the main focus of this section is his later work, which included much on the topic of infinity.

One of the ideas that dominated his later work was that the meaning of a word was a matter of how it was used. Words were like tools. To grasp a concept was to be clear about the use of the words and phrases governing it. As soon as words were wrenched from their proper use and mishandled, (needless) philosophical perplexity arose. For it became possible to frame all sorts of pseudo-questions which posed as philosophical problems but which, in the nature of the case, we did not have the wherewithal to answer. For example, we could imagine mishandling a phrase like `the average parent'. While it makes good sense to say, `The average parent has 2.4 children,' it does not make sense to say, `The average parent is expecting another child.' A philosophical `problem' might arise about what would happen to all ordinary parents if the average parent found herself (himself? itself? theirselves?) expecting another child. Once we returned to the proper use of the phrase, and thus to a correct grasp of the concept, this `problem' would be dissolved. Wittgenstein saw something similar in traditional discussion of the infinite.

He believed that the correct use of terms such as `infinity' was to characterize the form of finite things and, relatedly, to generalize about the endless possibilities that finite things afford. (We shall see more clearly in a little while what this amounts to.) It was incorrect to apply such terms directly to what we encounter in experience. And it was incorrect to use them to describe anything as being actually infinite. So, for example, we could say that there were infinitely many numbers. But this must mean that however many numbers we had counted we could always count more (and not, so to speak, because there was no last number, but because the phrase `last number' made no sense.¹²) Again, we could say that space and time were infinite. But this must mean that it was part of the form of a spatiotemporal object to have various unlimited possibilities of movement: however far such an object had travelled, there would be space and time enough for it to travel still further. There was no question here of an `infinite reality'.¹³ So too we could say that space was infinitely divisible. But this must mean:

Space isn't made up of individual things (parts) . . . [It] gives to reality an infinite opportunity for division.¹⁴

(Of course, there were empirical issues here that were not to be prejudged. But Wittgenstein was only talking about what made sense.)

To describe something as actually infinite, then, was not just a mistake. It was a mishandling of the language. It was like saying, `The average parent is two months pregnant,' or, `It is 5 o'clock on the sun.'¹⁵ And it was Wittgenstein's belief that once this fact was properly cognized, then philosophical perplexity about the infinite would at last be dissipated. A good example was the perplexity surrounding the paradox of the divided stick. For Wittgenstein, there was an incoherence in the very setting up of this paradox; no genuine situation had been described. It made sense to say, `This stick is infinitely divisible.' It did not make sense to say, `This stick is (has been) infinitely divided.'¹⁶ Moreover, if we did say, `This stick is infinitely divisible,' it was important that we should be clear as to precisely what sense it made. As we saw in our discussion of Aristotle, such sentences are ambiguous. To construe it in the wrong way (as meaning that a situation could be brought about in which this stick was divided into infinitely many pieces) was to start mishandling the language again, reopening the possibility of ill-begotten philosophical conundrums. What it meant was that however much the stick had been divided it could always be divided more.

Relatedly, we had to resist the idea that infinity was something like a natural number, only much bigger (so that, for example, three was somehow closer to infinity than two). For proper uses of `infinity' were very different from proper uses of `two' or `three'. Even if it made sense to say that a path was infinitely long, it made a very different kind of sense from saying that the path was three miles long. An infinitely long path was, as Aristotle would have said, a path that could never be traversed — that is, a path with no end, not a path with an end infinitely far away. Again, an infinite set was a completely different kind of thing from a set with three members. `Set' was hardly even univocal in the two cases.¹⁷

Many of Wittgenstein's conclusions were reminiscent of those of his great predecessors. Aristotle too would have found the wrong construal of `This stick is infinitely divisible' unintelligible. Kant would have licensed various claims about the inifinite whole, but then insisted that we be clear as to what sense they made: they were to be understood as injunctions (involving a regulative use of our Idea of the infinite whole) and not as ordinary assertions. But the way Wittgenstein arrived at his conclusions, that is via careful scrutiny of the use of words, was in fact as reminiscent of the medievals as of anyone. It called to mind the categorematic/syncategorematic distinction, an item of essentially grammatical categorization. Wittgenstein himself was for ever talking about the `grammar' of words, when appealing to their proper use. It was almost as if he was saying, or was committed to saying, that the `grammar' of `infinity' meant that it could be used only syncategorematically.

But what about its use in mathematical contexts? What about `Cantor's paradise'?

Wittgenstein felt very strongly that it was not his business, as a philosopher, to interfere with mathematical practice.¹⁸ His task was carefully to observe mathematical practice, to gain a clear view of how mathematical expressions were used, and then, where appropriate, to try to combat their misuse.

Did he then have to regard Cantor's work as sacrosanct, or at least as immune to philosophical criticism?

Not exactly. For one thing, mathematics was not a completely isolated discipline. Mathematical treatment of the infinite was not, and could not be, independent of how it was treated in other contexts. In any case, as we saw when looking at the early history of the calculus, it is not impossible for mathematicians themselves to mishandle their own apparatus and to import conceptual confusion into their own discipline. This raised a problem of circularity, though. How was Wittgenstein to know what to observe in the first place? Being able to distinguish between cases of legitimate mathematical practice and cases of mathematicians themselves going astray seemed to require the very discernment that was supposed to be acquired by observation of legitimate mathematical practice. If this circularity was not to be vicious, Wittgenstein did after all need to approach mathematics with a degree of humility. This, I think, helps to explain an ambivalence in his attitude to Cantor's work. On the one hand he felt pressure not to challenge it in any way. That, for better or worse, was what mathematics was now like. On the other hand his own views concerning the infinite meant that he did not like the tenor of the work at all. Still, there was a perfectly reasonable way out for him. What he did was to let the work stand but to remonstrate strongly against certain attitudes towards it. He was struck by Hilbert's view, shared by many, that Cantor had created a mathematical paradise. For Wittgenstein it could just as well be seen in a quite different light. `Imagine set theory's having been created by a satirist,' he wrote, `as a kind of parody on mathematics.'¹⁹

But it was not principally at Hilbert's attitude that Wittgenstein took umbrage. It was at an attitude quite foreign to Hilbert, though prevalent elsewhere, namely that transfinite mathematics served to describe a kind of super-physical landscape with all its bumps and nooks and crannies, populated by objects of various different sizes. This attitude was very much Gödel's for example. Gödel did not express it exactly like that. But he did have a robust sense of mathematical reality. In a discussion of Cantor's unanswered question about the size of , the question that Hilbert took himself to have settled but only by a kind of fiat, Gödel said that we perceived mathematical objects in something like the way in which we perceived physical objects; certainly they were just as real, and Cantor's question was a genuine question about something quite independent of us, not a question to be settled by fiat.²⁰ This whole view of mathematics was an anathema to Wittgenstein.

Wittgenstein believed that when we scrutinized transfinite mathematics, what we saw was a variety of formal techniques, proof-procedures, and the like, but that, in a sense, was all there was to it. There was no `landscape' being described. There were no `objects' being perceived. And certain ways of couching the results were to be deplored insofar as they encouraged the idea that there were. `The dangerous, deceptive thing about the idea: [` is bigger than '] ...,' he wrote, `is that it makes the determination of a concept . . . look like a fact of nature.'<sup>21</sup> Similarly, he was highly suspicious of the kind of account of the relationship between real numbers and rational numbers that I gave above in Chapter 4, § 2, whereby real numbers were seen as filling the `gaps' between rationals.²² By the same token, there was, for Wittgenstein, nothing mysterious or transcendent about transfinite mathematics, any more than there was about chess, or noughts and crosses. There was a kind of heady pleasure that we got from discovering that some infinite sets were bigger than others, like the pleasure of discovering that space was curved — what Wittgenstein would have called a `schoolboy' pleasure.²³ This had to be resisted. There was nothing more to the diagonal proof than the technique actually set down on the page (that is, the technique for specifying a sequence of digits different from all those listed); and there was nothing more to the result than to the proof.

Uses of `infinity' and related terms were quite straightforward. We had to learn to take them at face value. For instance, the three dots in

were not an abbreviation for something too long to write down. They were themselves part of the mathematical symbolism with a perfectly precise, specifiable, unmysterious use.²⁴ The symbolism seemed puzzling and enigmatic only when we tried to look beyond it to what it was pointing to. It was not pointing to anything. It was the mathematical reality.

One very important consequence of these views was that Wittgenstein, like Brouwer, wanted to challenge the idea of an infinite co-incidence. Attention to the `grammar' of generalizations about infinite totalities, based on close inspection of the relevant mathematical techniques and proof-procedures, revealed that the idea of an infinite co-incidence was unintelligible. What gave a true generalization about (say) its meaning was that we could recognize a proof of it. (We were seduced into thinking otherwise by the false picture of a determinate mathematical landscape out there, independent of what we could or could not prove about it.)²⁵

It was here that Wittgenstein felt prepared to question standard mathematical practice — to see mathematicians as mishandling their own apparatus. For, like Brouwer, he believed that they made assumptions that they were not entitled to make once the idea of an infinite co-incidence had been rejected. They assumed, for example, that unless every natural number had a given mathematical property, there must be a counterexample.

As we can see, Wittgenstein's route to these intuitionist conclusions was quite different from Brouwer's. Something that helps to reinforce this point is the work of the English philosopher Michael Dummett (born 1925). Dummett has done as much as anybody to show how broadly Wittgensteinian considerations about language and meaning can sustain conclusions very like those arrived at by Brouwer, though in a way that runs directly counter to much of what Brouwer himself believed. This is an apt point at which to begin our discussion of current thought about the infinite.²⁶

4 Current thought

The mathematical experience on which Brouwer laid so much emphasis was essentially private and incommunicable. Dummett, by contrast, has taken as his starting point the essential publicity and communicability of mathematical ideas, and has tied this in with the Wittgensteinian tenet that the meaning of a mathematical expression must ultimately be a matter of how it is used in mathematics, just as the power of a chess piece is a matter of how it is used in chess. Were the meaning not something open to public view in this way (for example, were it some range of occult images or indeed were it the kind of thing Brouwer took it to be), it would be impossible for anybody ever to have learned it or to demonstrate that they had done so; and this in turn would make communication impossible. (Brouwer thought that communication was impossible, in any ideal form.) It follows, in Dummett's view, that no mathematical sentence could be true in a way that essentially outstripped any capacity we had to recognize it or prove it to be such, for its meaning would not then be relevantly dependent on the kind of use to which we put it, or indeed could put it. So, as we saw with Wittgenstein, the idea of an infinite co-incidence comes under threat again. Although Dummett's argument is radically opposed to Brouwer's, he is taking something for granted that marks a vital point of contact between them: the fact that we are inescapably immersed in time. If we could somehow escape from time, and thereby survey a totality like in its entirety, the idea of an infinite co-incidence might not be so suspect. But we cannot. The upshot is that Dummett, like Brouwer, and like Wittgenstein, has urged us to take a much more critical attitude to some of the most elemental presuppositions of classical mathematics.²⁷

All three have placed emphasis on our capacities — what we are capable of constructing, what we are capable of recognizing as a proof, and the like. But consider: if our immersion in time is taken to place severe constraints on these, then why not similarly our physical limitations? After all, there is a perfectly good sense of `could' in which none of us could construct, or survey, a finite segment of that was so big that it included more members than the number of atoms in the known universe, or more members than the number of milli-seconds that will have elapsed by the time the earth has been swallowed up by the sun. So cannot these arguments casting doubt on the idea of an infinite co-incidence be extended to cast analogous doubt on the idea of a truth concerning some sufficiently large natural number?

Wittgenstein may have thought that they could. He sometimes verged on a correspondingly extreme position.²⁸ Others have recently explored the position more or less sympathetically.²⁹ But it finds no place in either Brouwer or Dummett. Indeed Dummett has argued that it is incoherent.³⁰ The problem that Brouwer and Dummett both thereby face (of steering a middle course) has thus become a focus of much debate. Brouwer could insist that what matter are our capacities insofar as they relate to the pure structure of time, because it is that which ultimately furnishes mathematics with its content. Dummett, for his part, has only ever wanted to maintain an open-minded scepticism about classical mathematics. He may be at liberty to continue to do so. A different, but related, problem is this: if physical limitations are not relevant here, then can we not, after all, construct, or survey, the whole of in a finite time, by starting with 0, then dealing with 1 twice as quickly, then dealing with 2 twice as quickly as that, and so on ad infinitum? This question was raised by Russell, who famously declared that the impossibility of performing infinitely many tasks in a finite time was merely `medical'.³¹ I shall return to this issue in Part Two (see below, Chapter 14, § 6).

One thing that it helps to show is that the same old puzzles and preoccupations are as relevant as they ever were to discussion of the infinite. A survey of the current literature reveals a continuing concern with all the perennials: the distinction between the actual infinite and the potential infinite; the relationship between the infinite and time; Zeno's paradoxes; the paradoxes of thought about the infinite; and so forth. But current debates have the advantage of being informed by recent empirical and mathematical discoveries — if it can be called an advantage: these discoveries have in many cases served only to exacerbate and to set in sharper relief some of the old problems, something that the new paradoxes of the one and the many illustrate only too well. (This is a point that I shall try to develop in Part Two.) There is perhaps also more emphasis now, in the wake of Wittgenstein, than there used to be on questions of meaning and linguistic understanding. But still the main concern is how to understand our own finitude and our relation to the infinite.

Two American writers in particular deserve mention for having produced fascinating work on the infinite that helps to bear this out: José Benardete (born 1928) and Rudy Rucker (born 1946). Both have offered enlightening surveys of the current state of the art.³² Benardete, whose lively sense of the paradoxical nature of the infinite has graced our drama more than once — the paradox of the gods, for example, is due to him (see above, Introduction, Section 1) — has argued, as against Wittgenstein, that saying that there are infinitely many stars is much the same sort of thing as saying that there are a trillion. Rucker, whose discussion of the paradoxes of the one and the many is especially gripping, has followed a route from them, and from the paradoxes of thought about the infinite, to a kind of mysticism. Some of his conclusions are very close to those that I shall try to defend in Part Two.

A final twist comes in the work of another American, the philosopher and logician W.V. Quine (born 1908). He has argued that it is only because there are infinitely many things that we need to operate with the fundamental notion of a thing at all. For this notion is used principally in making generalizations, for example when we say that everything is thus and so. But if there were only finitely many things, we could make such generalizations by spelling out, one by one, what each was like.³³ Of course, the `could' here is interesting. In what sense could we do this? We are reminded of the way in which Brouwer and Dummett were prepared to prescind from our physical limitations where they were not prepared to prescind from our temporality. Likewise, it seems, Quine.

Our historical drama has now finished. Not one of its protagonists was prepared to accept the infinite unconditionally. There was always a caveat against some conceptual aberration with which it might be associated, conflated, or confused, whether this was the actual infinite (Aristotle), or the unconditioned physical whole (Kant), or the mathematical infinite (Hegel), or inconsistent totalities (Cantor), or the idea of a super-physical mathematical landscape (Wittgenstein). Each of them might simply have rejected the idea of the infinite itself as a conceptual aberration. But none of them did. There was a momentary insurrection among the British empiricists, but otherwise this never looked like a serious option. I submit that it still does not. How then are we to view the infinite?