** **

*Li* — what is *li *? The basic sense seems to be *“the pre-established harmony and unity of the universe” *(Chu Hsi).

There are three ideas here. Firstly, Chu Hsi states that this ‘order’ is *pre-established*, it is not something that we would like to come about (such as the Millennium) nor something that was once but is now out of reach (like Eden) , it is simply there and always will be. Secondly, there is the idea of *harmony*, which implies variety and movement — one could hardly speak of the ‘harmony’ of Nirvana or the Platonic solids. Finally, there is mention of *unity *, which implies, amongst other things, that there is no fundamental difference between man and the natural world. *Li *also has the more mundane sense of ‘*rites’*, *‘ritual action’* and again ‘*propriety*’, ‘*ceremonial behaviour’*. Confucian thought traditionally divides up the cosmos into the triad **Earth, Man, Heaven**. The latter, *Ch’ien*, **Heaven**, is above all ‘order’ : it has permanence and stability and most traditional Chinese thinkers seem to assume, without question, that this underlying order is ‘good’ — ‘good’ in the sense of ‘desirable’ or ‘as it should be’ rather than in the sense of ‘benevolent’. **Man** has limited free will : he can align himself or not according to the ‘rule of Heaven’ though ultimately he/she will be brought back within the larger scheme even if he rebels against it. If his actions in life, his own *li*, mirror or embody the overriding heavenly *li*, he will attain contentment — “the good life consists in attunement to *li*” (Chu Hsi). Within the human sphere *li* is thus behaviour which is aligned to the ‘rule of Heaven’, the harmonizing of the here-and-now and the eternal (which is the essential aim of ritual). Courtly ceremonies, politeness, suitable dress and music (to which Confucius gave much importance) are all means to the same end. The following passage from a philosopher of the Chinese classical era shows the connection between *li* as ‘rational behaviour’ and ‘celestial harmony’ — the spirit is very close to that of the Enlightenment French philosophers who, of course, greatly admired Chinese civilization. “In all matters relating to the functioning of the body and the mind, if they are in keeping with *li* , there will be a far-reaching self-control; if they are not, there will be a disordering of the rhythm of living. Thus in eating and drinking, in clothing and housing, in [the alternation of] energetic action and stillness, if these matters are in keeping with *li* , then there is the harmony of moderation : if they are not, then there is physical collapse and disease. In matters of outward appearance and bearing, in meeting and parting with people, in one’s style of walking, if these are in keeping with *li* , then there is the beauty of refinement about them : if they are not, then they show arrogance, surliness, vulgarity and a barbarous spirit. Thus it is that without *li* man cannot live, nor his business in life succeed, nor his states and families abide.” (Hsun Ch’ing)

All this, however, is a strictly Confucian approach, stressing as it does *deliberate* behaviour. The Taoist approach is quite the opposite though the underlying aim is identical : to get human behaviour in tune with the rhythms of the universe. For the Taoist, and even more so for the Zen Buddhist, only a spontaneous response to a situation can have *li *(or ‘be *li*’?). Hence the development of ‘grass-style’ calligraphy (*‘ts’ao-shu-fa’*) which supposedly imitates the movement of grass bending in the wind, or *‘hsieh-i’ *brush painting which is done in a flash ideally without the brush even leaving the paper — the great modern exponent of this style is Ch’i Pai-shih. It should be noted that the aim is not self-expression and spontaneity for its own sake, though there were undoubtedly painters, particularly in the latter Sung era, whose work does sink to this level, much as our own ‘modern’ art has done. The underlying purpose is to give expression to latent energies and event-patterns within the individual and within nature :

“In his paintings Ch’i attempts to express the hidden order of things, to depict their substance as a general symbol and to catch the rhythm of life in nature. (…) Ch’i’s art is a synthesis of the concrete and the spiritual; it is an expression of balance between the objectivity of the world and the subjectivity of the creator.”

Josef Hejzlar, *Chinese Watercolours*

**Appropriateness **In a number of contexts *‘li´ *simply means ‘behaviour appropriate to the circumstances’ which sounds at first sight rather commonplace but is actually a very far-reaching conception. The notion is almost completely absent from the Western cultural tradition which, mainly because of Plato and his influence on Christian theology via Saint Augustine, focuses on absolutes. From the Chinese point of view, the cosmos is in a perpetual state of flux — or, more correctly perhaps, *is *perpetual flux — although there are certain recognizable repeating event-patterns ; in consequence behaviour which is a proper response to the situation at one moment may well be quite misguided a moment later. Hence the value of the *Y Ching *(‘The Book of Changes’)* *which was originally regarded as a *technical* work, on a par with a treatise on hydraulics — this is why the first Ch’in Emperor, when he ordered the burning of all unnecessary books, spared the *Y Ching* considering it to be, not a philosophical work, but a work of practical utility. The *Y Ching* purports to tell the enquirer what the ‘world-situation’ is at that precise moment, and what it is most likely to evolve into. Thus, by responding appropriately to the situation the individual can “be harnessed by, and harness for himself the changing state of nature” (“The Fortune Teller’s *Y* *Ching*”).

If we take this idea of *li* as appropriateness and apply it across the board, we end up with some interesting conclusions. Moral virtues are, then, not absolutes in the sense of being praiseworthy whatever the circumstances. Modesty is undoubtedly a virtue but when inappropriate is indistinguishable from cowardice. This approach does not necessarily lead to moral nihilism or relativism since we can still hold to certain *general* principles while being at all times ready to adapt them to the passing moment. Indeed, fitting actions to the circumstances is itself a general principle, while opportunism is a debased form of *li *since the notion of a higher order and an objective standard of ‘rightness’ is completely lacking. Photographs of buildings can be beautiful in themselves, but to be successful as buildings, churches or houses must harmonize with their surroundings. A baroque cathedral requires a baroque city. The eighteenth-century architects who found it necessary to make vast changes in the landscape when they designed and built a country house were, from the point of view of *li*, completely right : the building needed an appropriate setting. Of course, one could equally well take the Romantic view that the natural setting should dictate the style of architecture. To be a ‘world-historical figure’, it is not enough to have exceptional abilities : one must be in tune with the underlying (but not necessarily the apparent) *Zeitgeist*. A genius in the wrong place and time will achieve nothing : if he had not lived during the ferment of mid-seventeenth century England, Cromwell, who had no military training whatsoever and no military interests even, would have remained an obscure country squire. Einstein’s genius fitted his time (1905-1920) — but only just. Twenty years earlier his ideas would have been too novel (thus not *li*), while twenty years later Einstein found himself fighting a desperate rearguard action against Quantum Mechanics in the name of classical physics as he conceived it.

What of pure mathematics? Without a doubt contemporary pure mathematics is not *li*. The idea of producing a proof so long-winded that it requires a computer to print it out, let alone check it (Four Colour Theorem) is just plain ludicrous — and what is ludicrous is by definition not *li*, is ‘anti-*li*’. Judged in terms of appropriateness, Fermat’s Last Theorem has not yet been proved and quite possibly never will be since a result in Elementary Number Theory should, to be *li*, only use the methods of Elementary Number Theory. On the other hand, Wiles’s approach is perfectly acceptable as a means of establishing the Shimura-Tanayama Conjecture since the latter concerns modular forms, a very modern branch of mathematics.

*“Here is a branch that is short, and here is a branch that is long” *(Ts’ui-wei) I was relieved when I first read that it had been proved that no formula will ever give the complete distribution of the primes : this is how it should be. If he were alive in our scientific and mathematical era, Ts’ui-wei might well have written, *“Here is a number that is prime and here is a number that is composite”. *Several people have remarked on the agreeable combination of apparent deep structure and randomness that the distribution of the primes exhibits. But this is exactly what nature when left to itself exhibits almost everywhere! To be sure, we do not expect this combination within number theory and its presence at the very heart of the natural number system is highly significant : it suggests that the distribution of the primes is ‘natural’ in a way that man-made distribution functions are not. Although I believe there must be some physical/mathematical constraints for there to be a universe at all, *living *Nature does not seem to adhere to them with much consistency. I am not referring to the unpredictable element introduced into evolution by chance mutation — though certainly this is an extremely important fact of life. On a more mundane level, just look around you at the extraordinarily diverse and convoluted forms of plants, trees and grasses. One might have thought that the ‘laws of physics’ combined with the ceaseless ‘struggle for existence’ would have left in place only a very few mathematically correct shapes which maximized certain parameters. If we assume an upright stem (or trunk), and the periodic production of leaves and branches around this axis according to a *single* interval fixed in advance, it can be shown that a distribution based on the angle *360° /Φ** ^{2}* (roughly

*137.5°)*is the most advantageous since it keeps successive branches well spaced out while allowing them all to receive the light of the sun. Having worked this out in the study, I went out armed with callipers and protractor to see how many plants and trees actually employ this angle (sometimes known as the Golden Angle). The answer was none at all as far as I could make out.

^{1}In reality shrubs and trees don’t need to bother about all this since their branches, being flexible, can easily curve round to avoid each other. One even comes across plants making the elementary mistake of using an angle of 180° : clearly they have not yet heard of irrational numbers. The moral is that although certain features are fixed in advance, in the genes, plenty of other features are deliberately left unspecified with the result that the plant can adapt to varying environments and improvise its responses (which a man-made mechanical device is incapable of doing). The planned features give the feeling of underlying order, the unplanned the sense of randomness : what you

*never*get in the animal kingdom is shapes taken from a textbook of Euclidian geometry. If you want to find

*li*in this sense of ‘order + randomness’ your best bet is to go somewhere untouched by man, a deserted beach, a wilderness. In practice few trees and shrubs have a single upright trunk anyway and the arrangement of leaves and branches is pretty haphazard — a complete mess mathematically speaking. If you don’t believe me, take a walk in the park.

^{2}

In the fascinating section on

*li*on his website, Dr Watkins and the authors he quotes emphasize the mobile, ‘flowing’ aspect of

*li*. Dr Watkins himself defines

*li*as “the order of flow, the wonderful dancing pattern of liquid” while Alan Watts refers to

*li*as ‘a watercourse’ and David Wade says that

*li*“are essentially dynamic formations”. Now, if the patterns to be found in the ripples of sand-dunes, in the cell-structure of a nettle-stalk, in the protuberances on the bark of trees and so forth, are in some sense ‘residues’ or ‘relics’ of a deeper level of reality which is the Chinese view, it follows that this ‘ground-swell’ of existence, the ‘order’ which is of Heaven rather than of Earth, is

*in motion*. David Wade speaks of the observed patterns as “frozen moments” and Dr Watkins relates that he was at one time haunted by the idea of “the prime numbers as moving particles…eventually coming to rest when they achieved dynamic equilibrium” (

*Prime Numbers, the Zeta function and Li*).

*I emphasize this because it runs completely counter to the entire Western philosophic and mathematical tradition which has always viewed the Absolute as essentially motionless. Plato’s Ideas are static and were intended to be : by Plato’s time the Athenians had had enough of change since the disastrous Peloponnesian war with Sparta and subsequent political upheavals were in everyone’s memories. These beautiful eternal Forms that man could approach only by way of geometry were utterly removed from the conditions of earthly existence*

“War, death, disease could not affect them and their truth

Did not depend on trial or experiment,

Each step self-evident, demonstrable and sure”.

Sebastian Hayes, *The Initiates *

We fare no better if we jump nearly two thousand years to Descartes’ Co-ordinate Geometry. The algebraic formula of a curve *y = f(x) *includes all the points along it and it is ‘our fault’ if you like that we have to laboriously work out particular features — to the eye of the omniscient mathematician, God, all these features and doubtless many more not apparent to us are immediately present. The sixteenth and seventeenth centuries saw the birth of dynamics but in reality motion is always presented as a succession of stills — how could it be otherwise since *“notre intelligence ne se représente clairement que l’immobilité”*? As Bergson pointed out, the trajectory of the moving particle is a set of points : the moving arrow is never in motion. Newton, perhaps following some sort of a mystic intuition comparable to his intuition of the universality of Attraction, groped towards a true mathematics of motion in his theory of fluxions ^{3} but even he was unable to make it into a coherent doctrine and he found to his annoyance, his version of the Calculus short-changed by that of Leibnitz which dealt in final ratios between infinitesimal quantities. We may, in fact, ask whether mathematics is, or can be, *li* in the sense that it reflects and embodies in its operations and formulae features of a transcendental ‘order’ (that of Heaven, *Ch’ien*). The Platonic answer is, “Yes”, and almost all pure mathematicians in all eras are either open or covert Platonists. The vision of Kepler and Newton was of a Creator God who decided once and for all what the rules governing the universe were to be; moreover, these rules were mathematical in nature and only mathematicians could hope to detect and decipher them. Even today, although most mathematicians have long since dispensed with a Creator God, they hold firm to a strong belief in mathematics, not as an aid to industry and science, but as the nearest one can get to certainty in this world. “If there is another world, then it must be mathematical” is the unspoken (and occasionally outspoken) assumption. However, mathematics deals in truths which are essentially unchanging and has the greatest difficulty representing movement — much greater difficulty than music or even painting — so this means that the transcendental realm, if it exists at all, must be static. My personal feeling is that there is indeed a higher level of reality but that it is *not *static, and so for this and other reasons is *un mathematical. *The Eastern traditions, inasmuch as one can generalize, tend to view the ‘beyond’ as being ‘in motion’, as, for example, the

*dharma*of Hinayana Buddhism, a flux of evanescent point-instants, or the ceaselessly changing but endlessly recurring event-currents studied by the

*Y Ching*. Mathematics aims at finality and by and large achieves it — which is why it is so impressive. Generally speaking, art does not. A painting, however, well-executed and inspired is hardly going to stop someone else trying his or her hand at the same theme and each generation finds itself obliged to produce its own love songs, funeral dirges and tales of adventure. But once someone has stated that if

*p*is prime,

*a*, that is it. The theorem can be generalized and proved in different ways but, for all that, it stands there as unchanged and unchangeable as a rock. Mathematics tends to advance by accretion, by building on what has been already established, and for this and other reasons appears to be timeless. Successive generations of mathematicians simply uncover different portions of a gigantic sphinx buried in the sand.

^{p-1}= 1 (mod p) There is, however, a serious limitation to this approach : precisely because mathematics aims at finality and logical consistency it cannot tolerate anything the slightest bit random or subjective. Thus it cannot be *li *in the ‘Order + randomness’ sense : it is ‘Order + Order + Order + ….’. The beauty of Euclidian constructions or modern algebraic formulae is a completely different beauty from that of ripples on sand dunes : it is an *unnatural *beauty. It is not necessarily the worse for that but one cannot have everything and there is something deeply offensive in statements such as Bertrand Russell’s :

“Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world must conform” .

Is this true? I say it is not — at any rate not without very serious qualifications. Go out into Nature and you receive an impression that is far, far closer to the Taoist vision of the casual combination of the haphazard and the constrained than it is to the strait-jacket of modern or ancient mathematics. The ‘region of absolute necessity’ of which Russell speaks is essentially a figment of his imagination, a projection, since it neither corresponds to the reality ‘down here’ nor ‘up there’. Unpredictability has made an astonishing come-back into science during the latter twentieth century though rather few people have learned useful lessons from this. I have sometimes wondered whether it would be possible to introduce randomness into a mathematical system without wrecking it completely — as far as I know noone has tried. It may be that only a very different type of mathematics, one that precisely does allow a certain degree of randomness and subjectivism, will be able to cope adequately with the shifting realities of the quantum domain which lies below the sharply defined particle-like level of reality we are more familiar with.

*Sebastian Hayes *

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**Notes **

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^{1} I have since then found some lilies and hollyhocks that use something approaching the Golden Angle.

^{2 }Phyllotaxis in plants does exist but it seems to have more to do with ‘close packing’ at the tip of the growing plant than with optimizing air and sunlight.

^{3} “I consider mathematical Quantities in this Place not as consisting of very small Parts; but as described by a continued Motion. Lines are described, and thereby generated not by the apposition of Parts, but by the continued Motion of Points; … Portions of Time by a continual Flux : and so in other Quantities. These Geneses really take place in the Nature of things….” Newton *De Quadratura*

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