When I was somewhere between one and nine I brooded over the possibility of finding a new number, an integer between one and nine that had somehow been overlooked. Their names and shapes seemed so arbitrary, ten shapes out of a million trillion thousand hundred and eleventy possible arrays of lines and loops, so how on earth could the adult world, the world no longer in single digits, be so smugly sure it had got them all? This worry has not entirely gone. Like so many people who are good at sums, I turned out to have no aptitude for mathematics: the ciphers continue to haunt me, entering my dreams and my prayers and my obsessions.
Some of this anxiety (a glance at the appropriate textbooks suggests I am far from alone) is laid to rest by contemplating the way they order things elsewhere. Thomas Crump didn’t lead me to the lost digit, but my excessive attachment to numbers is not as hopeless as, for example, that of the Balinese, who operate ten separate simultaneous day-name cycles, a one-day cycle (in which presumably all days are Sundays), in addition to two, three, four, five, six, seven, eight, nine and ten-day cycles: you can see that that provides plenty of work for Bali’s twenty thousand temples, but it must he hard to know when the banks are open; nor yet as mad as the Japanese technique of Seimeigaku, ‘wholename science’, which involves endless takings away of the one you first thought of, to arrive at an auspicious name. Crump has soothed, stimulated, and occasionally transported me in the contemplation of numbers and mysteries. Frivolously perhaps, I would have preferred more wonders and fewer theories, more of a zoo and less of an anatomy museum.
The similarities in the way diverse peoples handle numbers is, according to taste, a matter for wonder, proof of divine order, evidence of a deep arithmetic to put beside deep grammar, or proof that similar needs elicit similar responses. It may again be a sampling artefact. Crump laments that descriptive linguists as well as ethnographers often skimp the number system. The reader notices that the same handful of names recur: the Kpelle of Liberia, the ’Are ’Are of the Solomon Islands, the Tikopia (who have managed to make darts complicated), the Balinese.
The disentangling, from the impenetrable mass of number itself, of written and spoken numerals, of cardinals and ordinals, of tallying, ordering and counting, is a matter for amazement. No people, no matter how impoverished their technical vocabulary, fail to observe some common factor between two sheep and two cows, or yet two dollars, though some can’t manage ‘two days’. But many are uneasy about it, as though it were irreverent to count gods as you count coconuts, and introduce a meditating group word: two head of cattle, three piece cooking-pots, four sharp-thing knives, five shiny-category gold rings. Some people do not count, though they can quantify small groups by the process called subitising, (the process we all use when we buy half a dozen apples and know that there are not five or seven); some cannot count but can only arrange things in order: conversely, the Ponam know only three places – first, middle, last; others, like the Weddah of Sri Lanka, do not count but only tally (the Mock Turtle’s arithmetic: one and one and one and one and one and one and ...), matching bulky or immobile things with handy and storable ones. The tallies may be sticks or stones or knots on a string, the Inca quipu or scratches on mesolithic bone – the oldest human inscriptions. From these primitive tally marks, or more likely from the representation of them on a hand or on an abacus, may have emerged a numeral system like the Roman: one one one one and the thumb laid obliquely thus: V; two thumbs crossed: X.
Quite impenetrably other is the process which leads the ancient tallyman to give voice as he tallies: no language actually uses repetition in its spoken numerals, though some Australian languages count ‘one, two, oneplustwo, twoplustwo, twoplustwoplusone’ until they reach the boundary of the knowable. Equally obscure is the process by which that sequence of words, ordinal at first, became the sequence of cardinal numbers. In the upper reaches of a counting system all kinds of exotic or elegant variation may occur: quatre-vingtdix-neuf, sylv og halv tres. (Mayans counted in blocks of twenty, kal, looking up, so the number after hunkal, ‘one score’, is not hunkalhun but huntukakal, ‘one towards two score’; and 381 is economically expressed as huntuhunbak ‘one towards bak’, a ‘score of scores’. The modern Mayans mix the decimal and the ancient systems so that bak may mean 400 or 1000, which would seem an intolerable confusion if we British didn’t tolerate two different billions). Roman numerals, a transcription of the abacus, are easy to manipulate; the Hebrew, Greek and Slavonic systems use the letters of the alphabet with numerical values, a quasi-decimal system in which the first ten letters represent one to ten, the next 20, 30, 40; in such a system 3 x 3 = 9 and 30 x 30 = 900 contain no common digits. Calculation is thus almost impossible, says Crump, suggesting that it set Greek mathematics back a thousand years. It lends itself instead to the magical or neurotic games of gematria, the basis of Kabbala, which concern themselves with the numerical value of words and the accidental meanings of numbers. (To avoid uttering the Holy Name, Hebrew replaces yod hai, 15, with tet var, 9 + 6.)
Most of Crump’s book, however, is concerned with applied number: in games, in song, in commerce, in architecture, in religion. If you haven’t built your society according to the formulae of civil engineers, then how much you use numbers is a matter of taste, or of climate: ‘the Balinese ... seem not to be able to do anything without numbers,’ while ‘the Bemba of Zambia would readily dispense with them altogether.’ The Balinese are a wet-rice growing people and, given fairly regular rains, their calendar of ritual ensures that the rice gets planted, irrigated and harvested at the right time, while the Bemba do not go in for bride-price, ritual obligations or usury: this is either the cause or the result of their scanty numeral system. Mayan and Hindu calendars count thousands, millions of days; at the other extreme, the Hopi allegedly do not count days at all (though they know when the next Snake Dance is due even if they won’t tell), and Australian aborigines indicate future engagements by mapping them on the palm of the hand, most portable of filofaxes.
Whatever numbers, whatever system of written and spoken numerals the adult ends with, the child’s method of acquisition appears to be the same: we all have the same cognitive paradigm and (by arguable implication) the same neurological method for number acquisition, a MAD to go alongside LAD. Crump and his sources are hard on Tsunoda, who argued in 1978 that special features of Japanese caused a unique reallocation of linguistic mechanisms between the two hemispheres (and explained the superiority of Japanese culture). Since Edelman’s Theory of Neuronal Group Selection, and the finding by Neville and Bellugi of a shift in cerebral localisation in profoundly deaf users of Sign, it no longer seems so improbable that the allocation of brain space might differ according to the chosen grammar. A machine to generate Chinese, which emerges in hard isolated rice-grains of meaning, might differ in its hardware from one dedicated to the production of Eskimo, which comes out as an endless noodle of utterance.
Even without a mathematics to grind lenses and measure the stars, numbers still shape a people’s universe. The family may he ranked, the tribe may be counted. Counting people ordinally may lead to ordering them about. To prevent this, the Chinese have an auxiliary cyclic counting system, kanshi, which avoids putting anybody first. The simplest case of a cyclical ordinal system is the Japanese (and now universal) game of janken: scissor cuts paper wraps stone blunts scissor, which yields winners, but no outright winner. The kanshi system is composed of a cycle of 12 ideograms combined with a cycle of five. But though there is no number one, each pseudo-number is laden with associations and secondary meanings, with good or ill luck. And this is the nature of number in pre-scientific societies: the modern fear of being just a number would make no sense to the Pythagorean or the Chinese. Balinese children are named by birth order, but the system is refreshingly un-hierarchical: the firstborn is called Wayan: after him come Njoman, Made and Ktut, but the fifth is called Wayan again, and so on. They have other names, less ambiguous: but the purpose of the names is not to identify, but to identify with – some natural fourfold recurrence.
You do not then need advanced mathematics to build your world on number. ‘If a distinction is to be made between traditional and modern cosmologies, the former tends to emphasise the applications, while the latter emphasises the mathematics. In a traditional cosmology the applications are sancrosanct, which is what the whole trial of Galileo was about.’ Arbitrary (and therefore holy) meanings given to numbers are hard to shift. I once tried explaining algebra to a child of ten. (I was six weeks older.) But I couldn’t get across the notion of x as an unknown, because he knew, stubborn little kabbalist, that x was 24. He would have burned me at the stake if he’d been allowed to play with matches.
In traditional cosmology, the math stays simple, but may be elaborate. Yin and yang are the archetypes of every bipolarity in the universe: but give rise to many other orders. If the to-ing and fro-ing of yin and yang be represented as a sine wave, then positive, negative and zero give a three-fold system; add the values half-way between zero and maximum and – behold! – the five elements, fire, water, earth, wood and metal; distinguish between half-way up and half-way down and you have the seven heavenly bodies. Multiples of three give the 12 zodiacal signs or the years of the Chinese cycle: multiply 12 by the five elements to get the 60 kanshi signs. Likewise from the simple duality of a line either whole or broken, come by permutation the eight trigrams and thence the 64 hexagrams of the I Ching.
The architecture of sacred buildings, West and East, betrays similar numerical harmonies (or obsessions). The divine dottiness of Sanjusangendo, with its 33 bays, sheltering 1001 Buddhas, each with 42 hands and 11 heads, is matched by the monastery of St Gall, whose design incorporates the sacred numbers three (of the Trinity), four (for the Gospel-makers), seven (for the days of creation), ten (for the Ten Commandments) and 12 (for the Twelve). Why 42 hands? ‘Each statue should in principle have 1000 hands, but in fact each hand is taken to represent 25,’ with a small deduction for the pair clasped in prayer and the pair holding a bowl. But when Crump compares 1001 with 10001, the binary representation of 33 (the number of bays you will recall), I fear he has become infected with the prevailing paranoia (which is what in mathematically sophisticated societies we call such fantasies of reference).
In traditional metrology, likewise, number often serves to maintain harmony rather than to solve problems. Each new Chinese Emperor literally set the tone for the whole of Chinese society by tuning to his liking the pitchpipe which tuned the yellow-bell from which all the instruments were tuned. From the length of the imperial pipe came measures, thoroughly decimal for millennia – the fen was a tenth of anything – of length, capacity and weight.
In other sections of this book, Crump deals with music (including the algorithms of change-ringing), sports (how to lay off your bets in a Balinese cockfight), government (Burmese state institutions based on the mathematical structure of the mandala), poetry and money (‘the peculiar significance of the state which the Tangu of New Guinea call mngwotngwotiki, a word which connotes a peculiar field of relations in which the individuals concerned arc temporarily unobliged to each other’ – roughly equivalent to that equally peculiar state called solvency.)
Contrary to the impression I have given, Crump’s purpose is not to provide a rumination rather than a thesis. ‘The question is what indeed is the numerical content of such artistic representations? What numbers can a tiled floor represent if its extent is indefinite. This is not the right question to ask.’ The reader may begin to feel harangued. Some of the statements are interesting but tough, as Huck Finn said. What does it mean to say that the Kpelle ‘do not think of length, weight or size as independent realities’? What is the next term in the series 1, 16, 24, 32, 1472, on which Borobudor is planned? (Is it a series?) And while I am bitching, in what sense is Stonehenge ‘non-viable’, why does he say that modular building implies rectangular units (did Buckminster Fuller live in vain?), the diagram of the dice-throws in the Amerindian game Patol seems to be incomplete, the Hindu aeon is a kalpa and it is twenty years too soon to resign oneself to the solecism ‘one pence’.
The general conclusion to this difficult but enormously engaging book (did you know that the Andamanese have a calendar of smells?) seems to be that everyone gets the mathematics she deserves. In the absence of higher maths comes an obsession with lower maths: magic squares, turning three times three. But since all races are equally capable of learning and using more complex mathematics, if they have a use for it and acquire the vocabulary, good number – which eventually means Arabic decimal zero-using place notation – drives out bad. A numerical institution of this kind ‘makes few demands of any local culture’.
Mathematics is the Trojan Horse of commercial civilisation: offering gifts of crop management and fertility or stock control, it brings a bellyful of cash-registers, junk bonds, census forms and schemes for proportional representation. Despite which Japan, age-old home of the pocket computer, prefers the translucency of the abacus. Champions can derive the cube root of 12-figure numbers in their heads, on an imaginary abacus.
And the obsessives? They dig their graves. The Aztec calendrical obsession ironically helped bring about their destruction. They anticipated the return of Quetzalcoatl on the day called Nine Wind in the year called One Reed. He would come from the east and wear black. 1519 was a One Reed year, and on the day Nine Wind, 22 April by his reckoning, Cortez consulted the calendar, observed that it was Good Friday, and piously donned black before setting out for the day’s genocide.