‘It’s nothing special’, I said. ‘It’s the ability to distinguish one, two, and three – in short, I’m talking about number and counting. I mean, isn’t it the case that every branch of expertise and knowledge is bound to have some involvement with numbers and with counting?’ (VII, 522c)
The third interpretation of the Myth of the Cave takes seriously Plato’s talk of order and proportion, as well as the timelessness of the reality with which the elite is acquainted. The suggestion is that the transcendental world is fundamentally that of mathematics, or at most the world of mathematical objects suitably extended, to include, inevitably, the good and the beautiful.
It is almost impossible to over-emphasize the place that mathematics, particularly the theory of harmony and the theory of geometry, occupied in Plato’s conception of wisdom, or insight into the scheme of the cosmos. Plato is supposed to have visited southern Italy, and learned what he could from the Pythagoreans installed at their base there in Metapontum, on the Gulf of Taranto. To these the whole universe ran according to a mathematical scheme, just as the sequence of musical notes obeyed the mathematical laws of harmony. Plato’s description of the education of the elite flows from mathematics (VII, 525a–531d).
The mathematical model makes a good deal of sense. The subject-matter of arithmetic and geometry is timeless. Furthermore, the understanding we attain by using them is often timeless as well: the laws governing musical harmony, for instance. Better than that, we find here for the first time a kind of defence of the view which I pilloried in chapter 9, that the domains of knowledge and belief are different, and that the first concerns what is permanent and unvarying while the second does not. Suppose we think not in terms of knowledge but of understanding. Then there is something right about the claim that however often we are faced with change, we only come to understand it by bringing to bear something unchanging, a constant that can be seen throughout the process, or that can be brought up alongside it, like a ruler, in order to make it intelligible. And perhaps this is the vital nerve of Plato’s system.
Thus faced with a changing system, science itself requires the application of laws, themselves unchanging, governing the changes of the magnitudes defining the system, and its state at one time and another. These laws, and an initial set of magnitudes, govern the state of the system at a later time. What is fixed may be mass, or temperature, or energy, or combinations of these or complex functions of them, or even more ethereal quantities such as probabilities. But something there must be, to feed the equations that describe the evolution of state – the nature of changes in systems, whether they are mechanical, chemical, gravitational, thermodynamic, or quantum-mechanical – and thence make scientific understanding of change possible. Where one constant after another flickers out, for example as we consider times nearer and nearer to the ‘singularity’ which we dub the big bang, we get closer and closer to the point at which our understanding gives out as well.
On this account, it is the scientist and mathematician who ‘sees eternity in a grain of sand’ – or in the starry night sky, or the intricacies of the atom – by finding the changeless laws or forms of growth underlying temporal becoming. It would of course be entirely unhistorical to credit the Pythagoreans, or Plato himself, with an understanding of the shape that scientific explanation would eventually take. Even late into the seventeenth century, when scientists such as Newton were finally coming up with equations governing simple mechanical and gravitational interactions, the prevailing view was that this was all very well, but it did not really amount to ‘science’, the holy grail of rational, mathematical insight into not just the way things are, but the way they have to be. It was only in the eighteenth century that philosophers reconciled themselves to the empiricist dispensation, that the kind of thing Newton did was all we were ever going to get. It would be absurd to see Plato as anticipating those centuries. But all that we really require for this kind of interpretation of the enlightenment of the philosopher is that Plato and Pythagoras knew they were ‘on to something’, so that the triumph represented by a mathematical modelling of musical harmony was a paradigm to follow. Hence the massive concentration upon mathematics in the education of the elite, in the ideal republic.
The mathematical/scientific model has another advantage. Modern epistemologists fall over themselves denouncing the ‘spectator theory of knowledge’, or the view that they find in Plato which thinks of moments of especially intense vision (‘illuminations’) as the best exemplars of knowledge. But in Plato (and still more obviously in Aristotle) the relevant ‘spectator’ is not what Plato would have called a mere sightseer, but rather what we would call the theorist: the person who can deploy an understanding of what he sees and thereby literally sees something different from the untutored eye, however intense the latter may be. In fact, the Greek word theoria originally meant a spectator or spectacle in the rather refined context of the expert or official observer of a sacred event: someone who understands what is going on (compare the expert spectator of a cricket match, or the expert listener to a jazz quartet, beside the casual sightseer or listener).1
It may well be that Plato himself conceived the application of mathematics to nature in a more exotic way than any gained by measuring magnitude of different properties of systems, and establishing by observation and conjecture, trial and error, how they change through time. He probably had something much more occult and ‘mystical’ in mind, like persons fascinated by numerology, cryptic understandings, golden sections, Fibonacci series, and the mathematics of prime numbers. He can hardly have had our own post-Kantian understanding of the distinction between applied and pure mathematics, or in other words between the brute empirical fact and the purely a priori abstract structures of numbers and sets. Francis Bacon, quoted in the Introduction, clearly saw it that way, regarding Plato as having ‘contaminated and corrupted’ any chance of Greek natural science by an admixture of speculation and theology, whereas only the revolution he was himself advocating would find the right mix:
Those who have treated of the sciences have been either empiricists or dogmatists. Empiricists, like ants, simply accumulate and use; Rationalists, like spiders, spin webs from themselves; the way of the bee is in between: it takes material from the flowers of the garden and the field; but it has the ability to convert and digest them.2
However, a more positive spin can be put on Republic than Bacon suggests. It clearly envisages the use of mathematics for military matters, but also for the theory of music and of astronomy (VII, 530b), and these subjects might conform to ‘the way of the bee’. Apart from that it offers two central examples of mathematics in action in Plato’s mind, and unfortunately they are less reassuring. At VIII, 546, he is explaining how even a well-ordered community ruled by philosopher-kings could degenerate. This could happen because one of the rulers’ prime responsibilities is to time the conception and births of children. An elaborate foray into the mathematics of cubes, diagonals, harmonies, and so forth produces the so-called ‘nuptial number’ or 12, 960,000, although what that has to do with the timing of births remains entirely obscure. However, if the guardians are unaware of this number and its application, they ‘pair men and women sexually on the wrong occasions’ and the culture spirals downwards. Plato’s reasoning was proverbially incomprehensible, even in antiquity, and moderns have to confess pretty much total bafflement.3 The impression, certainly, is of a Pythagorean confidence that the laws relating to numbers also govern the universe, and then that sufficient knowledge of those laws, and perhaps skill in applying them, will enable the wise to order things. In this case what they are ordering is the eugenic policy of the society. But it is unintelligent for us to let our sense of revulsion at this destroy our admiration for the fact that Pythagoreans and Plato were indeed on to something. They were on to the mathematicization of nature, the discovery of properties and relations that can be ordered by magnitude, generating data that, once fed into the right (eternal) equations, determine how things fall out. Plato’s eugenic application may be mysterious, and fanciful, and perhaps partakes more of astrology than astronomy. But the ambition may have been wholly laudable.
The other exercise in mathematics, in a lighter vein, occurs at IX, 587e. Here Plato is calculating how much happier a philosopher-king must be than a dictator. By assuming three kinds of ruler, and three kinds of pleasure, with the philosopher at the top and the dictator and his pleasures at the bottom, Plato comes up with the figure of 729 (93 or 36) as the ratio of a philosopher’s happiness to that of a dictator. Again, there seems to be a misdirected enthusiasm for the spurious appearance of accuracy and ‘science’ that numbers provide in contexts such as this. But Plato was neither the first nor the last to fall into this particular trap. The philosopher Jeremy Bentham’s notorious ‘felicific calculus’ was just a generalization of the idea, and arguably substantial parts of the discipline of welfare economics is a temple to it. In fact, it remains a difficult logical exercise to tease out exactly the properties which enable us to provide a scale, either ordering magnitudes or doing the more exacting job of associating cardinal numbers with them. Pleasure is probably more like ‘hardness’, which can be ordered (on the Moh scale of minerals, hardness is indexed from 1 to 10, according to whether the substance in questions scratches or is scratched by comparison substances) but not assigned magnitudes among which arithmetical ratios make any sense. It makes no sense to say that topaz is twice as hard as quartz for example. In Republic the mathematical ratio of the philosopher’s pleasure to the dictator’s may be offered somewhat tongue-in-cheek, while the first, eugenic exercise, seems to have been entirely serious.
The obvious advantage of the mathematical/scientific interpretation of the ascent from the Cave is that it accords with the pre-eminent place of mathematics in the education of the philosopher-king. The ruler is not given a spiritual training in religious mysticism. Nor is he given a Romantic development saturated in poetry and imagination – on the contrary, these are to be banished from the well-ordered state and hence by analogy, from the well-ordered mind. Instead, he or she, having been proved exceptional in the years of primary education and military training, up to around age twenty, is then offered ten years of mathematics, followed by five years of dialectic or logic: the kind of immersion in argument that the liberal Socrates of the earlier dialogues represents. Mathematics is near the copestone of the Platonic arch. Naturally, philosophy has that honour. But mathematics is certainly the supporting intellectual buttress.
There are other pleasant features of this interpretation. One is that Plato often associates the fact that the world is changing with the fact that people come to different opinions about it, as two very similar obstacles to understanding (e.g. V, 479a–d). At first sight this is puzzling. Change is one thing, but the different perspectives different witnesses may take on what things are like, whether they are big or small, sweet or sour, and so on, is quite another thing. Again, however, science gives us a useful entrée into this aspect of Plato’s thought. It transcends relativity, not by going off to discuss something else entirely, but by finding what is invariant behind the different judgements. The perspective according to which the finger is bigger than the moon, and that according to which the moon is bigger than a finger, are alike intelligible in terms of different points of view on the one invariant set of spatial relations.4 We come to understand different perspectives, when we do, not by succumbing to the relativistic idea of utterly incommensurable verdicts, made by people living in separate worlds, but by finding what is invariant, and therefore explains the emergence of difference itself.
Relativism in judgement can be comprehended by distinguishing the invariant way in which the one reality acts upon observers who themselves occupy different places or times or who have different senses or different historical and cultural experiences. The theory of relativity itself substitutes one invariance (distance in space-time) for two magnitudes hitherto thought invariant (distance in space, distance in time) but which prove to vary with the velocity of the observer. That is progress, and in broad outline it follows what we said about the scientific understanding of change. We comprehend variety in judgement, just as we comprehend change, by finding solid rock under the shifting sands.
The other, even more important feature is that this interpretation solves the problem of relevance. As we have seen in the previous two chapters, on more transcendental, otherworldly, interpretations of what is required, there is the severe problem of getting back to earth. How should knowledge of another world qualify one to behave well, govern well, or indeed have anything at all to do with this one? It is fine to see eternity in a grain of sand, but what if that amounts to seeing eternity instead of the grain of sand? That is not much use to the builder or geologist. On the scientific model, the answer is immediate. The scientist alone understands the unchanging within the changing. He alone is not a mere ‘sightseer’ or theatre-goer. But he does not see the unchanging instead of the changing. He sees the changes and evolutions of this grain, here, now, in front of him, but in the light of known constancies. For this reason he can predict and explain what happens to this very grain. His knowledge is not otherworldly, but this-worldly – yet it goes beyond that of the mere sightseer.
Where the mathematical-scientific model breaks down is, of course, with the final five years of dialectic, equipping the philosopher with the kind of understanding of men and events that entitle him or her to govern. As VII, 519a, reminds us, most knowledge may be used for good or ill, and that is certainly true of scientific knowledge. The final illumination somehow transcends that limitation. It is easy to feel that it has to be something beatific: a vision in as much as, like sight, it generates understanding and ‘oneness’ with the order of nature, and beatific in that once achieved it guarantees the goodness of the subject who has gained it. But we are not told how. If Hume is right, we can never be told how, since the idea of a state of mind which counts both as some kind of understanding of things, and also and at the same time one which of necessity points its subject towards the good and away from the bad, is a delusion. Noble perhaps, but a delusion none the less. Also, unfortunately, a profoundly dangerous delusion, substituting as it does an illumination open only to the elite, and incommunicable to the vulgar, for anything more ordinary. In this it belittles the human, democratic, shared processes of communicating, decreasing fear and mistrust, increasing humanity, and enlarging sympathy – the actual routes through which moral improvement emerges.
Do we want to settle for just one of the religious, the poetical, and the mathematical-scientific interpretations of the Myth of the Cave? I do not think so. Like other myths it weaves its own spell, and is sufficiently capacious to include almost any journey of increased understanding, whatever local form it may take. This is, of course, part of its power, but also part of what may make it dangerous.