The Universal History of Numbers: From Prehistory to the Invention of the Computer

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"Ifrah's book amazes and fascinates by the scope of its scholarship. It is nothing less than the history of the human race told through figures."--International Herald Tribune

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Overview

"Monumental [and dazzling. A wonderful gift."--Kirkus Reviews

"Georges Ifrah is the man, and this book, quite simply, rules. . . . It is outstanding . . . a mind-boggling and enriching experience."--Guardian (London)

"Monumental . . . a fascinating journey taking us through many different cultures."--The Times (London)

"Ifrah's book amazes and fascinates by the scope of its scholarship. It is nothing less than the history of the human race told through figures."--International Herald Tribune

A riveting history of numbers from the time of the cave dwellers to the twentieth century, this landmark international bestseller is the first complete, universal study of the invention and evolution of numbers the world over. Georges Ifrah brings numbers to thrilling life, explaining their development in human terms, the interesting situations that made them necessary, and the brilliant achievements in human thought that they made possible. The reader is taken through the numbers story from Europe to China, via ancient Greece and Rome, Mesopotamia, Latin America, India, and the Arabic countries. Exploring the many ways civilizations developed and changed their mathematical systems, Ifrah imparts a unique insight into the nature of human thought--and into how our understanding of numbers and the ways they shape our lives has changed over thousands of years. The engaging text is illustrated with over 150 figures.

George Ifrah (France) is an independent scholar and former math teacher. He has been called the "Indiana Jones" of numbers.

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Editorial Reviews

Booknews
Ifrah is a former math teacher who put his book together as a result of the prodding simplicity of student questions like "Where do numbers come from?" and "Where does zero come from?" Translated from the French edition (1994), this 27-chaptered volume examines topics like the ethnological and psychological sources of numbers, how Cro-Magnon man counted, how the Sumerians did their sums, Greek and Roman numerals, alphabetic numbers, magical numbers, and generally, numbers throughout the ancient and modern worlds. Ifrah's research has led him to assert that "all societies learned to number their own bodies and to count on their fingers; and the use of pebbles, shells and stick is absolutely universal." Ifrah should have just said "We are all one." Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9781860463242
  • Publisher: Random House Adult Trade Publishing Group
  • Publication date: 1/1/1998
  • Language: French
  • Pages: 633

Meet the Author

Georges Ifrah is an independent scholar and former math teacher.

David Bellos, the primary translator, is Professor of French at Princeton University.

Sophie Wood, cotranslator, is a specialist in technical translation from French. Ian Monk, cotranslator, has translated the works of Georges Perec and Daniel Pennac.

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Read an Excerpt




Chapter One


EXPLAINING THE ORIGINS


Ethnological and Psychological Approaches
to the Sources of Numbers


WHEN THE SLATE WAS CLEAN


There must have been a time when nobody knew how to count. All we can surmise is that the concept of number must then have been indissociable from actual objects — nothing very much more than a direct apperception of the plurality of things. In this picture of early humanity, no one would have been able to conceive of a number as such, that is to say as an abstraction, nor to grasp the fact that sets such as "day-and-night", a brace of hares, the wings of a bird, or the eyes, ears, arms and legs of a human being had a common property, that of "being two".

    Mathematics has made such rapid and spectacular progress in what are still relatively recent periods that we may find it hard to credit the existence of a time without number. However, research into behaviour in early infancy and ethnographic studies of contemporary so-called primitive populations support such a hypothesis.


CAN ANIMALS COUNT?


Some animal species possess some kind of notion of number. At a rudimentary level, they can distinguish concrete quantities (an ability that must be differentiated from the ability to count numbers in abstract). For want of a better term we will call animals' basic number-recognition the sense of number. It is a sense which human infants do not possess at birth.

    Humans do not constitute the onlyspecies endowed with intelligence: the higher animals also have considerable problem-solving abilities. For example, hungry foxes have been seen to "play dead" so as to attract the crows they intend to eat. In Kenya, lions that previously hunted alone learned to hunt in a pack so as to chase prey towards a prepared ambush. Monkeys and other primates, of course, are not only able to make tools but also to learn how to manipulate non-verbal symbols. A much-quoted example of the first ability is that of the monkey who constructed a long bamboo tube so as to pick bananas that were out of reach. Chimpanzees have been taught to use tokens of different shapes to obtain bananas, grapes, water, and so on, and some even ended up hoarding the tokens against future needs. However, we must be careful not to be taken in by the kind of "animal intelligence" that you can see at the circus and the fairground. Dogs that can "count" are examples of effective training or (more likely) of clever trickery, not of the intellectual properties of canine minds. However, there are some very interesting cases of number-sense in the animal world.

    Domesticated animals (for instance, dogs, cats, monkeys, elephants) notice straight away if one item is missing from a small set of familiar objects. In some species, mothers show by their behaviour that they know if they are missing one or more than one of their litter. A sense of number is marginally present in such reactions. The animal possesses a natural disposition to recognise that a small set seen for a second time has undergone a numerical change.

    Some birds have shown that they can be trained to recognise more precise quantities. Goldfinches, when trained to choose between two different piles of seed, usually manage to distinguish successfully between three and one, three and two, four and two, four and three, and six and three.

    Even more striking is the untutored ability of nightingales, magpies and crows to distinguish between concrete sets ranging from one to three or four. The story goes that a squire wanted to destroy a crow that had made its nest in his castle's watchtower. Each time he got near the nest, the crow flew off and waited on a nearby branch for the squire to give up and go down. One day the squire thought of a trick. He got two of his men to go into the tower. After a few minutes, one went down, but the other stayed behind. But the crow wasn't fooled, and waited for the second man to go down too before coming back to his nest. Then they tried the trick with three men in the tower, two of them going down: but the third man could wait as long as he liked, the crow knew that he was there. The ploy only worked when five or six men went up, showing that the crow could not discriminate between numbers greater than three or four.

    These instances show that some animals have a potential which is more fully developed in humans. What we see in domesticated animals is a rudimentary perception of equivalence and non-equivalence between sets, but only in respect of numerically small sets. In goldfinches, there is something more than just a perception of equivalence — there seems to be a sense of "more than" and "less than". Once trained, these birds seem to have a perception of intensity, halfway between a perception of quantity (which requires an ability to numerate beyond a certain point) and a perception of quality. However, it only works for goldfinches when the "moreness" or "lessness" is quite large; the bird will almost always confuse five and four, seven and five, eight and six, ten and six. In other words, goldfinches can recognise differences of intensity if they are large enough, but not otherwise.

    Crows have rather greater abilities: they can recognise equivalence and non-equivalence, they have considerable powers of memory, and they can perceive the relative magnitudes of two sets of the same kind separated in time and space. Obviously, crows do not count in the sense that we do, since in the absence of any generalising or abstracting capacity they cannot conceive of any "absolute quantity". But they do manage to distinguish concrete quantities. They do therefore seem to have a basic number-sense.


NUMBERS AND SMALL CHILDREN


Human infants have few innate abilities, but they do possess something that animals never have: a potential to assimilate and to recreate stage by stage the conquests of civilisation. This inherited potential is only brought out by the training and education that the child receives from the adults and other children in his or her environment. In the absence of permanent contact with a social milieu, this human potential remains undeveloped — as is shown by the numerous cases of enfants sauvages. (These are children brought up by or with animals in the wild, as in François Truffaut's film, The Wild Child. Of those recaptured, none ever learned to speak and most died in adolescence.)

    We should not imagine a child as a miniature adult, lacking only judgement and knowledge. On the contrary, as child psychology has shown, children live in their own worlds, with distinct mentalities obeying their own specific laws. Adults cannot actually enter this world, cannot go back to their own beginnings. Our own childhood memories are illusions, reconstructions of the past based on adult ways of thinking.

    But infancy is nonetheless the necessary prerequisite for the eventual transformation of the child into an adult. It is a long-drawn-out phase of preparation, in which the various stages in the development of human intelligence are re-enacted and reconstitute the successive steps through which our ancestors must have gone since the dawn of time.

    According to N. Sillamy (1967), three main periods are distinguished: infancy (up to three years of age), middle childhood (from three to six or seven); and late childhood, which ends at puberty. However, a child's intellectual and emotional growth does not follow a steady and linear pattern. Piaget (1936) distinguishes five well-defined phases:

1. a sensory-motor period (up to two years of age) during which the child forms concepts of "object" out of fragmentary perceptions and the concept of "self" as distinct from others;
2. a pre-operative stage (from two to four years of age), characterised by egocentric and anthropomorphic ways of thinking ("look, mummy, the moon is following me!");
3. an intuitive period (from four to six), characterised by intellectual perceptions unaccompanied by reasoning; the child performs acts which he or she would be incapable of deducing, for example, pouring a liquid from one container into another of a different shape, whilst believing that the volume also changes;
4. a stage of concrete operations (from eight to twelve) in which, despite acquiring some operational concepts (such as class, series, number, causality), the child's thought-processes remain firmly bound to the concrete;
5. a period (around puberty) characterised by the emergence of formal operations, when the child becomes able to make hypotheses and test them, and to operate with abstract concepts.

    Even more precisely: the new-born infant in the cradle perceives the world solely as variations of light and sound. Senses of touch, hearing and sight slowly grow more acute. From six to twelve months, the infant acquires some overall grasp of the space occupied by the things and people in its immediate environment. Little by little the child begins to make associations and to perceive differences and similarities. In this way the child forms representations of relatively simple groupings of beings and objects which are familiar both by nature and in number. At this age, therefore, the child is able to reassemble into one group a set of objects which have previously been moved apart. If one thing is missing from a familiar set of objects, the child immediately notices. But the abstraction of number — which the child simply feels, as if it were a feature of the objects themselves — is beyond the child's grasp. At this age babies do not use their fingers to indicate a number.

    Between twelve and eighteen months, the infant progressively learns to distinguish between one, two and several objects, and to tell at a glance the relative sizes of two small collections of things. However, the infant's numerical capabilities still remain limited, to the extent that no clear distinction is made between the numbers and the collections that they represent. In other words, until the child has grasped the generic principle of the natural numbers (2 = 1 + 1; 3 = 2 + 1; 4 = 3 + 1, etc.), numbers remain nothing more than "number-groupings", not separable from the concrete nature of the items present, and they can only be recognised by the principle of pairing (for instance, on seeing two sets of objects lined up next to each other).

    Oddly enough, when a child has acquired the use of speech and learned to name the first few numbers, he or she often has great difficulty in symbolising the number three. Children often count from one to two and then miss three, jumping straight to four. Although the child can recognise, visually and intuitively, the concrete quantities from one to four, at this stage of development he or she is still at the very doorstep of abstract numbering, which corresponds to one, two, many.

    However, once this stage is passed (at between three and four years of age, according to Piaget), the child quickly becomes able to count properly. From then on, progress is made by virtue of the fact that the abstract concept of number progressively takes over from the purely perceptual aspect of a collection of objects. The road lies open which leads on to the acquisition of a true grasp of abstract calculation. For this reason, teachers call this phase the "pre-arithmetical stage" of intellectual development. The child will first learn to count up to ten, relying heavily on the use of fingers; then the number series is progressively extended as the capacity for abstraction increases.


ARITHMETIC AND THE BODY


The importance of the hand, and more generally of the body in children's acquisition of arithmetic can hardly be exaggerated. Inadequate access to or use of this "counting instrument" can cause serious learning difficulties.

In earliest infancy, the child plays with his or her fingers. It constitutes the first notion of the child's own body. Then the child touches everything in order to make acquaintance with the world, and this also is done primarily with the hands. One day, a well-intentioned teacher who wanted arithmetic to be "mental", forbade finger-counting in his class. Without realising it, the teacher had denied the children the use of their bodies, and forbidden the association of mathematics with their bodies. I've seen many children profoundly relieved to be able to use their hands again: their bodies were at last accepted [...] Spatiotemporal disabilities can likewise make learning mathematics very difficult. Inadequate grasp of the notions of "higher than" and "lower than" affect the concepts of number, and all operations and relations between them. The unit digits are written to the right, and the hundred digits are written to the left, so a child who cannot tell left from right cannot write numbers properly or begin an operation at all easily. Number skills and the whole set of logical operations of arithmetic can thus be seriously undermined by failure to accept the body. [L. Weyl-Kailey (1985)]


NUMBERS AND THE PRIMITIVE MIND


A good number of so-called primitive people in the world today seem similarly unable to grasp number as an abstract concept. Amongst these populations, number is "felt" and "registered", but it is perceived as a quality, rather as we perceive smell, colour, noise, or the presence of a person or thing outside of ourselves. In other words, "primitive" peoples are affected only by changes in their visual field, in a direct subject-object relationship. Their grasp of number is thus limited to what their predispositions allow them to see in a single visual glance.

    However, that does not mean that they have no perception of quantity. It is just that the plurality of beings and things is measured by them not in a quantitative but in a qualitative way, without differentiating individual items. Cardinal reckoning of this sort is never fixed in the abstract, but always related to concrete sets, varying naturally according to the type of set considered.

A well-defined and appropriately limited set of things or beings, provided it is of interest to the primitive observer, will be memorised with all its characteristics. In the primitive's mental representation of it, the exact number of the things or beings involved is implicit: it resembles a quality by which this set is different from another group consisting of one or several more or fewer members. Consequently, when he sets eyes on the set for a second time, the primitive knows if it is complete or if it is larger or smaller than it was previously. [L. Lévy-Bruhl (1928)]


ONE, TWO ... MANY


In the first years of the twentieth century, there were several "primitive" peoples still at this basic stage of numbering: Bushmen (South Africa), Zulus (South and Central Africa), Pygmies (Central Africa), Botocudos (Brazil), Fuegians (South America), the Kamilarai and Aranda peoples in Australia, the natives of the Murray Islands, off Cape York (Australia), the Vedda (Sri Lanka), and many other "traditional" communities.

    According to E. B. Tylor (1871), the Botocudos had only two real terms for numbers: one for "one", and the other for "a pair". With these lexical items they could manage to express three and four by saying something like "one and two" and "two and two". But these people had as much difficulty conceptualising a number above four as it is for us to imagine quantities of a trillion billions. For larger numbers, some of the Botocudos just pointed to their hair, as if to say "there are as many as there are hairs on my head".

    A. Sommerfelt (1938) similarly reports that the Aranda had only two number-terms, ninta (one), and tara (two). Three and four were expressed as tara-mi-ninta (one and two) and tara-ma-tara ("two and two"), and the number series of the Aranda stopped there. For larger quantities, imprecise terms resembling "a lot", "several" and so on were used.

    Likewise G. Hunt (1899) records the Murray islanders' use of the terms netat and neis for "one" and "two", and the expressions neis-netat (two + one) for "three", and neis-neis (two + two) for "four". Higher numbers were expressed by words like "a crowd of ..."

    Our final example is that of the Torres Straits islanders for whom urapun meant "one", okosa "two", okosa-urapun (two-one) "three", and okosa-okosa (two-two) "four". According to A. C. Haddon (1890) these were the only terms used for absolute quantities; other numbers were expressed by the word ras, meaning "a lot".

    Attempts to teach such communities to count and to do arithmetic in the Western manner have frequently failed. There are numerous accounts of natives' lack of memory, concentration and seriousness when confronted with numbers and sums [see, for example, M. Dobrizhoffer (1902)]. It generally turned out much easier to teach primitive peoples the arts of music, painting, and sculpture than to get them to accept the interest and importance of arithmetic. This was perhaps not just because primitive peoples felt no need of counting, but also because numbers are amongst the most abstract concepts that humanity has yet devised. Children take longer to learn to do sums than to speak or to write. In the history of humanity, too, numbers have proved to be the hardest of these three skills.


PARITY BEFORE NUMBER


These primitive peoples nonetheless possessed a fundamental arithmetical rule which if systematically applied would have allowed them to manipulate numbers far in excess of four. The rule is what we call the principle of base 2 (or binary principle). In this kind of numbering, five is "two-two-one", six is "two-two-two", seven is "two-two-two-one", and so on. But primitive societies did not develop binary numbering because, as L. Gerschel (1960) reminds us, they possessed only the most basic degree of numeracy, that which distinguishes between the singular and the dual.

    A. C. Haddon (1890), observing the western Torres Straits islanders, noted that they had a pronounced tendency to count things in groups of two or in couples. M. Codrington, in Melanesian Languages, noticed the same thing in many Oceanic populations: "The natives of Duke of York's Island count in couples, and give the pairings different names depending how many of them there are; whereas in Polynesia, numbers are used although it is understood that they refer to so many pairs of things, not to so many things." Curr, as quoted by T. Dantzig (1930), confirms that Australian aborigines also counted in this way, to the extent that "if two pins are removed from a set of seven the aborigines rarely notice it, but they see straight away if only one is removed".

    These primitive peoples obviously had a stronger sense of parity than of number. To express the numbers three and four, numbers they did not grasp as abstracts but which common sense allowed them to see in a single glance, they had recourse only to concepts of one and pair. And so for them groups like "two-one" or "two-two" were themselves pairs, not (as for us) the abstract integers (or "whole numbers") "three" and "four". So it is easy to see why they never developed the binary system to get as far as five and six, since these would have required three digits, one more than the pair which was their concept of the highest abstract number.


THE LIMITS OF PERCEPTION


The limited arithmetic of "primitive" societies does not mean that their members were unintelligent, nor that their innate abilities were or are lesser than ours. It would be a grave error to think that we could do better than a Torres Straits islander at recognising number if all we had to use were our natural faculties of perception.

    In practice, when we want to distinguish a quantity we have recourse to our memories and/or to acquired techniques such as comparison, splitting, mental grouping, or, best of all, actual counting. For that reason it is rather difficult to get to our natural sense of number. There is an exercise that we can try, all the same. Looking at Fig. 1.1, which contains sets of objects in line, try to estimate the quantity of each set of objects in a single visual glance (that is to say, without counting). What is the best that we can do?

    Everyone can see the sets of one, of two, and of three objects in the figure, and most people can see the set of four. But that's about the limit of our natural ability to numerate. Beyond four, quantities are vague, and our eyes alone cannot tell us how many things there are. Are there fifteen or twenty plates in that pile? Thirteen or fourteen cars parked along the street? Eleven or twelve bushes in that garden, ten or fifteen steps on this staircase, nine, eight or six windows in the façade of that house? The correct answers cannot be just seen. We have to count to find out!

    The eye is simply not a sufficiently precise measuring tool: its natural number-ability virtually never exceeds four.

    There are many traces of the "limit of four" in different languages and cultures. There are several Oceanic languages, for example, which distinguish between nouns in the singular, the dual, the triple, the quadruple, and the plural (as if in English we were to say one bird, two birdo, three birdi, four birdu, many birds).

    In Latin, the names of the first four numbers (unus, duos, tres, quatuor) decline at least in part like other nouns and adjectives, but from five (quinque), Latin numerical terms are invariable. Similarly, Romans gave "ordinary" names to the first four of their sons (names like Marcus, Servius, Appius, etc.), but the fifth and subsequent sons were named only by a numeral: Quintus (the fifth), Sixtus (the sixth), Septimus (the seventh), and so on. In the original Roman calendar (the so-called "calendar of Romulus"), only the first four months had names (Martius, Aprilis, Maius, Junius), the fifth to tenth being referred to by their order-number: Quintilis, Sextilis, September, October, November, December.

    Perhaps the most obvious confirmation of the basic psychological rule of the "limit of four" can be found in the almost universal counting-device called (in England) the "five-barred gate". It is used by innkeepers keeping a tally or "slate" of drinks ordered, by card-players totting up scores, by prisoners keeping count of their days in jail, even by examiners working out the mark-distribution of a cohort of students:

    Most human societies the world has known have used this kind of number-notation at some stage in their development and all have tried to find ways of coping with the unavoidable fact that beyond four (IIII) nobody can "read" intuitively a sequence of five strokes (IIIII) or more.

    And yet others, like the Greeks, the Manaeans and Sabaeans, the Lycians, Mayans, Etruscans and Romans, came up with an idea (probably based on finger-counting) for a special sign for the number five, proceeding thereafter on a rule of five or quinary system (6 = 5 + 1, 7 = 5 + 2, and so on).

    There really can be no debate about it now: natural human ability to perceive number does not exceed four!

    So the basic root of arithmetic as we know it today is a very rudimentary numerical capacity indeed, a capacity barely greater than that of some animals. There's no doubt that the human mind could no more accede by innate aptitude alone to the abstraction of counting than could crows or goldfinches. But human societies have enlarged the potential of these very limited abilities by inventing a number of mental procedures of enormous fertility, procedures which opened up a pathway into the universe of numbers and mathematics ...


DEAD RECKONING


Since we can discriminate unreflectingly between concrete quantities only up to four, we cannot have recourse only to our natural sense of number to get to any quantity greater than four. We must perforce bring into play the device of abstract counting, the characteristic quality of "civilised" humanity.

    But is it therefore the case that, in the absence of this mental device for counting (in the way we now understand the term), the human mind is so enfeebled that it cannot engage in any kind of numeration at all?

    It is certainly true that without the abstractions that we call "one", "two", "three", and so on it is not easy to carry out mental operations. But it does not follow at all that a mind without numbers of our kind is incapable of devising specific tools for manipulating quantities in concrete sets. There are very good reasons for thinking that for many centuries people were able to reach several numbers without possessing anything like number-concepts.

    There are many ethnographic records and reports from various parts of Africa, Oceania and the Americas showing that numerous contemporary "primitive" populations have numerical techniques that allow them to carry out some "operations", at least to some extent.

    These techniques, which, in comparison to our own, could be called "concrete", enable such peoples to reach the same results as we would, by using mediating objects or model collections of many different kinds (pebbles, shells, bones, hard fruit, dried animal dung, sticks, the use of notched bones or sticks, etc.). The techniques are much less powerful and often more complicated than our own, but they are perfectly serviceable for establishing (for example) whether as many head of cattle have returned from grazing as went out of the cowshed. You do not need to be able to count by numbers to get the right answer for problems of that kind.


ELEMENTARY ARITHMETIC


It all started with the device known as "one-for-one correspondence". This allows even the simplest of minds to compare two collections of beings or things, of the same kind or not, without calling on an ability to count in numbers. It is a device which is both the prehistory of arithmetic, and the dominant mode of operation in all contemporary "hard" sciences.

    Here is how it works: You get on a bus and you have before you (apart from the driver, who is in a privileged position) two sets: a set of seats and a set of passengers. In one glance you can tell whether the two sets have "the same number" of elements; and, if the two sets are not equal, you can tell just as quickly which is the larger of the two. This ready-reckoning of number without recourse to numeration is more easily explained by the device of one-for-one correspondence.

    If there was no one standing in the bus and there were some empty seats, you would know that each passenger has a seat, but that each seat does not necessarily have a passenger: therefore, there are fewer passengers than seats. In the contrary case — if there are people standing and all the seats are taken — you know that there are more passengers than seats. The third possibility is that there is no one standing and all seats are taken: as each seat corresponds to one passenger, there are as many passengers as seats. The last situation can be described by saying that there is a mapping (or a biunivocal correspondence, or, in terms of modern mathematics, a bijection) between the number of seats and the number of passengers in the bus.

    At about fifteen or sixteen months, infants go beyond the stage of simple observation of their environment and become capable of grasping the principle of one-for-one correspondence, and in particular the property of mapping. If we give a baby of this age equal numbers of dolls and little chairs, the infant will probably try to fit one doll on each seat. This kind of play is nothing other than mapping the elements of one set (dolls) onto the elements of a second set (chairs). But if we set out more dolls than chairs (or more chairs than dolls), after a time the baby will begin to fret: it will have realised that the mapping isn't working.

    This mental device does not only provide a means for comparing two groups, but it also allows its user to manipulate several numbers without knowing how to count or even to name the quantities involved.

    If you work at a cinema box-office you usually have a seating plan of the auditorium in front of you. There is one "box" on the plan for each seat in the auditorium, and, each time you sell a ticket, you cross out one of the boxes on the plan. What you are doing is: mapping the seats in the cinema onto the boxes on the seating plan, then mapping the boxes on the plan onto the tickets sold, and finally, mapping the tickets sold onto the number of people allowed into the auditorium. So even if you are too lazy to add up the number of tickets you've sold, you'll not be in any doubt about knowing when the show has sold out.

    To recite the attributes of Allah or the obligatory laudations after prayers, Muslims habitually use a string of prayer-beads, each bead corresponding to one divine attribute or to one laudation. The faithful "tell their beads" by slipping a bead at a time through their fingers as they proceed through the recitation of eulogies or of the attributes of Allah.

    Buddhists have also used prayer-beads for a very long time, as have Catholics, for reciting Pater noster, Ave Maria, Gloria Patri, etc. As these litanies must be recited several times in a quite precise order and number, Christian rosaries usually consist of a necklace threaded with five times ten small beads, each group separated by a slightly larger bead, together with a chain bearing one large then three small beads, then one large bead and a cross. That is how the litanies can be recited without counting but without omission — each small bead on the ring corresponds to one Ave Maria, with a Gloria Patri added on the last bead of each set of ten, and a Pater noster is said for each large bead, and so on.

    The device of one-for-one correspondence has thus allowed these religions to devise a system which ensures that the faithful do not lose count of their litanies despite the considerable amount of repetition required. The device can thus be of use to the most "civilised" of societies; and for the completely "uncivilised" it is even more valuable.

    Let us take someone with no arithmetical knowledge at all and send him to the grocery store to get ten loaves of bread, five bottles of cooking oil, and four bags of potatoes. With no ability to count, how could this person be trusted to bring back the correct amount of change? But in fact such a person is perfectly capable of carrying out the errand provided the proper equipment is available. The appropriate kit is necessarily based on the principle of one-for-one correspondence. We could make ten purses out of white cloth, corresponding to the ten loaves, five yellow purses for the bottles of cooking oil, and four brown purses, for the bags of potatoes. In each purse we could put the exact price of the corresponding item of purchase, and all the uneducated shopper needs to know is that a white purse can be exchanged for a loaf, a yellow one for a bottle of oil and a brown one for a bag of potatoes.

    This is probably how prehistoric humanity did arithmetic for many millennia, before the first glimmer of arithmetic or of number-concepts arose.

    Imagine a shepherd in charge of a flock of sheep which is brought back to shelter every night in a cave. There are fifty-five sheep in this flock. But the shepherd doesn't know that he has fifty-five of them since he does not know the number "55": all he knows is that he has "many sheep". Even so, he wants to be sure that all his sheep are back in the cave each night. So he has an idea — the idea of a concrete device which prehistoric humanity used for many millennia. He sits at the mouth of his cave and lets the animals in one by one. He takes a flint and an old bone, and cuts a notch in the bone for every sheep that goes in. So, without realising the mathematical meaning of it, he has made exactly fifty-five incisions on the bone by the time the last animal is inside the cave. Henceforth the shepherd can check whether any sheep in his flock are missing. Every time he comes back from grazing, he lets the sheep into the cave one by one, and moves his finger over one indentation in the tally stick for each one. If there are any marks left on the bone after the last sheep is in the cave, that means he has lost some sheep. If not, all is in order. And if meanwhile a new lamb comes along, all he has to do is to make another notch in the tally bone.

    So thanks to the principle of one-for-one correspondence it is possible to manage to count even in the absence of adequate words, memory or abstraction.

    One-for-one mapping of the elements of one set onto the elements of a second set creates an abstract idea, entirely independent of the type or nature of the things or beings in the one or other set, which expresses a property common to the two sets. In other words, mapping abolishes the distinction that holds between two sets by virtue of the type or nature of the elements that constitute them. This abstract property is precisely why one-for-one mapping is a significant tool for tasks involving enumeration; but in practice, the methods that can be based on it are only suitable for relatively small sets.

    This is why model collections can be very useful in this domain. Tally sticks with different numbers of marks on them constitute so to speak a range of ready-made mappings which can be referred to independently of the type or nature of the elements that they originally referred to. A stick of ivory or wood with twenty notches on it can be used to enumerate twenty men, twenty sheep or twenty goats just as easily as it can be used for twenty bison, twenty horses, twenty days, twenty pelts, twenty kayaks, or twenty measures of grain. The only number technique that can be built on this consists of choosing the most appropriate tally stick from the ready-mades so as to obtain a one-to-one mapping on the set that you next want to count.

    However, notched sticks are not the only concrete model collections available for this kind of matching-and-counting. The shepherd of our example could also have used pebbles for checking that the same number of sheep come into the cave every evening as went out each morning. All he needs to do to use this device would be to associate one pebble with each head of sheep, to put the resulting pile of pebbles in a safe place, and then to count them out in a reverse procedure on returning from the pasture. If the last animal in matches the last pebble in the pile, then the shepherd knows for sure that none of his flock has been lost, and if a lamb has been born meanwhile, all he needs to do is to add a pebble to the pile.

    All over the globe people have used a variety of objects for this purpose: shells, pearls, hard fruit, knucklebones, sticks, elephant teeth, coconuts, clay pellets, cocoa beans, even dried dung, organised into heaps or lines corresponding in number to the tally of the things needing to be checked. Marks made in sand, and beads and shells, strung on necklaces or made into rosaries, have also been used for keeping tallies.

    Even today, several "primitive" communities use parts of the body for this purpose. Fingers, toes, the articulations of the arms and legs (elbow, wrist, knee, ankle ...), eyes, nose, mouth, ears, breasts, chest, sternum, hips and so on are used as the reference elements of one-for-one counting systems. Much of the evidence comes from the Cambridge Anthropological Expedition to Oceania at the end of the last century. According to Wyatt Gill, some Torres Straits islanders "counted visually" (see Fig. 1.30):

They touch first the fingers of their right hand, one by one, then the right wrist, elbow and shoulder, go on to the sternum, then the left-side articulations, not forgetting the fingers. This brings them to the number seventeen. If the total needed is higher, they add the toes, ankle, knee and hip of the left then the right hand side. That gives 16 more, making 33 in all. For even higher numbers, the islanders have recourse to a bundle of small sticks. [As quoted in A. C. Haddon (1890)]

    Murray islanders also used parts of the body in a conventional order, and were able to reach 29 in this manner. Other Torres Straits islanders used similar procedures which enabled them to "count visually" up to 19; the same customs are found amongst the Papuans and Elema of New Guinea.

(Continues...)

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Table of Contents

Foreword
List of Abbreviations
Introduction: Where "Numbers" Come From
Ch. 1 Explaining the Origins: Ethnological and Psychological Approaches to the Sources of Numbers 3
Ch. 2 Base Numbers and the Birth of Number-systems 23
Ch. 3 The Earliest Calculating Machine - The Hand 47
Ch. 4 How Cro-Magnon Man Counted 62
Ch. 5 Tally Sticks: Accounting for Beginners 64
Ch. 6 Numbers on Strings 68
Ch. 7 Number, Value and Money 72
Ch. 8 Numbers of Sumer 77
Ch. 9 The Enigma of the Sexagesimal Base 91
Ch. 10 The Development of Written Numerals in Elam and Mesopotamia 96
Ch. 11 The Decipherment of a Five-thousand-year-old System 109
Ch. 12 How the Sumerians Did Their Sums 121
Ch. 13 Mesopotamian Numbering after the Eclipse of Sumer 134
Ch. 14 The Numbers of Ancient Egypt 162
Ch. 15 Counting in the Times of the Cretan and Hittite Kings 178
Ch. 16 Greek and Roman Numerals 182
Ch. 17 Letters and Numbers 212
Ch. 18 The Invention of Alphabetic Numerals 227
Ch. 19 Other Alphabetic Number-systems 240
Ch. 20 Magic, Mysticism, Divination, and Other Secrets 248
Ch. 21 The Numbers of Chinese Civilisation 263
Ch. 22 The Amazing Achievements of the Maya 297
Ch. 23 The Final Stage of Numerical Notation 323
Ch. 24 Part I: Indian Civilisation: the Cradle of Modern Numerals 356
Ch. 24 Part II: Dictionary of the Numeral Symbols of Indian Civilisation 440
Ch. 25 Indian Numerals and Calculation in the Islamic World 511
Ch. 26 The Slow Progress of Indo-Arabic Numerals in Western Europe 577
Ch. 27 Beyond Perfection 592
Bibliography 601
Index of Names and Subjects 616
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