Early Number Systems and Symbols
To think the thinkable—that is the mathematician’s aim.
C. J. K E Y S E R
1.1 Primitive Counting
A Sense of Number
The root of the term mathematics is in the Greek word math-
emata, which was used quite generally in early writings to
indicate any subject of instruction or study. As learning ad-
vanced, it was found convenient to restrict the scope of this
term to particular elds of knowledge. The Pythagoreans
are said to have used it to describe arithmetic and geometry; previously, each of these
subjects had been called by its separate name, with no designation common to both. The
Pythagoreans’ use of the name would perhaps be a basis for the notion that mathematics
began in Classical Greece during the years from 600 to 300 B.C. But its history can be
followed much further back. Three or four thousand years ago, in ancient Egypt and
Babylonia, there already existed a signi cant body of knowledge that we should describe
as mathematics. If we take the broad view that mathematics involves the study of issues
of a quantitative or spatial nature—number, size, order, and form—it is an activity that
has been present from the earliest days of human experience. In every time and culture,
there have been people with a compelling desire to comprehend and master the form of
the natural world around them. To use Alexander Pope’s words, “This mighty maze is
not without a plan.”
It is commonly accepted that mathematics originated with the practical problems of
counting and recording numbers. The birth of the idea of number is so hidden behind
the veil of countless ages that it is tantalizing to speculate on the remaining evidences of
early humans’ sense of number. Our remote ancestors of some 20,000 years ago—who
were quite as clever as we are—must have felt the need to enumerate their livestock,
tally objects for barter, or mark the passage of days. But the evolution of counting, with
its spoken number words and written number symbols, was gradual and does not allow
any determination of precise dates for its stages.
Anthropologists tell us that there has hardly been a culture, however primitive, that
has not had some awareness of number, though it might have been as rudimentary as
the distinction between one and two. Certain Australian aboriginal tribes, for instance,
counted to two only, with any number larger than two called simply “much” or “many.”
South American Indians along the tributaries of the Amazon were equally destitute of
number words. Although they ventured further than the aborigines in being able to count
to six, they had no independent number names for groups of three, four, ve, or six. In their counting vocabulary, three was called “two-one,” four was “two-two,” and so on.
A similar system has been reported for the Bushmen of South Africa, who counted to
ten (10 D 2 C 2 C 2 C 2 C 2) with just two words; beyond ten, the descriptive phrases
became too long. It is notable that such tribal groups would not willingly trade, say,
two cows for four pigs, yet had no hesitation in exchanging one cow for two pigs and a
second cow for another two pigs.
The earliest and most immediate technique for visibly expressing the idea of number
is tallying. The idea in tallying is to match the collection to be counted with some easily
employed set of objects—in the case of our early forebears, these were ngers, shells,
or stones. Sheep, for instance, could be counted by driving them one by one through
a narrow passage while dropping a pebble for each. As the ock was gathered in for
the night, the pebbles were moved from one pile to another until all the sheep had
been accounted for. On the occasion of a victory, a treaty, or the founding of a village,
frequently a cairn, or pillar of stones, was erected with one stone for each person present.
The term tally comes from the French verb tailler, “to cut,” like the English word
tailor; the root is seen in the Latin taliare, meaning “to cut.” It is also interesting to note
that the English word write can be traced to the Anglo-Saxon writan, “to scratch,” or
“to notch.”
Neither the spoken numbers nor nger tallying have any permanence, although nger
counting shares the visual quality of written numerals. To preserve the record of any
count, it was necessary to have other representations. We should recognize as human
intellectual progress the idea of making a correspondence between the events or objects
recorded and a series of marks on some suitably permanent material, with one mark
representing each individual item. The change from counting by assembling collections
of physical objects to counting by making collections of marks on one object is a long
step, not only toward abstract number concept, but also toward written communication.
Counts were maintained by making scratches on stones, by cutting notches in wooden
sticks or pieces of bone, or by tying knots in strings of different colors or lengths. When
the numbers of tally marks became too unwieldy to visualize, primitive people arranged
them in easily recognizable groups such as groups of 5, for the ngers of a hand. It
is likely that grouping by pairs came rst, soon abandoned in favor of groups of 5,
10, or 20. The organization of counting by groups was a noteworthy improvement on
counting by ones. The practice of counting by ves, say, shows a tentative sort of
progress toward reaching an abstract concept of “ ve” as contrasted with the descriptive
ideas “ ve ngers” or “ ve days.” To be sure, it was a timid step in the long journey
toward detaching the number sequence from the objects being counted.
Notches as Tally Marks
Bone artifacts bearing incised markings seem to indicate that the people of the Old
Stone Age had devised a system of tallying by groups as early as 30,000 B.C. The most
impressive example is a shinbone from a young wolf, found in Czechoslovakia in 1937;
about 7 inches long, the bone is engraved with 55 deeply cut notches, more or less equal
in length, arranged in groups of ve. (Similar recording notations are still used, with
the strokes bundled in ves, like . Voting results in small towns are still counted in
the manner devised by our remote ancestors.) For many years such notched bones were interpreted as hunting tallies and the incisions were thought to represent kills. A more
recent theory, however, is that the rst recordings of ancient people were concerned with
reckoning time. The markings on bones discovered in French cave sites in the late 1880s
are grouped in sequences of recurring numbers that agree with the numbers of days
included in successive phases of the moon. One might argue that these incised bones
represent lunar calendars.
Another arresting example of an incised bone was unearthed at Ishango along the
shores of Lake Edward, one of the headwater sources of the Nile. The best archeological
and geological evidence dates the site to 17,500 B.C., or some 12,000 years before the
rst settled agrarian communities appeared in the Nile valley. This fossil fragment was
probably the handle of a tool used for engraving, or tattooing, or even writing in some
way. It contains groups of notches arranged in three de nite columns; the odd, unbalanced
composition does not seem to be decorative. In one of the columns, the groups are
composed of 11, 21, 19, and 9 notches. The underlying pattern may be 10 C 1, 20 C
1, 20 -
1, and 10 -
1. The notches in another column occur in eight groups, in the
following order: 3, 6, 4, 8, 10, 5, 5, 7. This arrangement seems to suggest an appreciation
of the concept of duplication, or multiplying by 2. The last column has four groups
consisting of 11, 13, 17, and 19 individual notches. The pattern here may be fortuitous
and does not necessarily indicate—as some authorities are wont to infer—a familiarity
with prime numbers. Because 11 C 13 C 17 C 19 D 60 and 11 C 21 C 19 C 9 D 60, it
might be argued that markings on the prehistoric Ishango bone are related to a lunar
count, with the rst and third columns indicating two lunar months.
The use of tally marks to record counts was prominent among the prehistoric peoples
of the Near East. Archaeological excavations have unearthed a large number of small
clay objects that had been hardened by re to make them more durable. These handmade
artifacts occur in a variety of geometric shapes, the most common being circular disks,
triangles, and cones. The oldest, dating to about 8000 b.c., are incised with sets of parallel
lines on a plain surface; occasionally, there will be a cluster of circular impressions as if
punched into the clay by the blunt end of a bone or stylus. Because they go back to the
time when people rst adopted a settled agricultural life, it is believed that the objects are
primitive reckoning devices; hence, they have become known as “counters” or “tokens.”
It is quite likely also that the shapes represent different commodities. For instance, a
token of a particular type might be used to indicate the number of animals in a herd,
while one of another kind could count measures of grain. Over several millennia, tokens
became increasingly complex, with diverse markings and new shapes. Eventually, there
came to be 16 main forms of tokens. Many were perforated with small holes, allowing
them to be strung together for safekeeping. The token system of recording information
went out of favor around 3000 b.c., with the rapid adoption of writing on clay tablets.
A method of tallying that has been used in many different times and places involves
the notched stick. Although this device provided one of the earliest forms of keeping
records, its use was by no means limited to “primitive peoples,” or for that matter, to
the remote past. The acceptance of tally sticks as promissory notes or bills of exchange
reached its highest level of development in the British Exchequer tallies, which formed an
essential part of the government records from the twelfth century onward. In this instance,
the tallies were at pieces of hazelwood about 6–9 inches long and up to an inch thick.
Notches of varying sizes and types were cut in the tallies, each notch representing a xed
amount of money. The width of the cut decided its value. For example, the notch of £1000
was as large as the width of a hand; for £100, as large as the thickness of a thumb; and
for £20, the width of the little nger. When a loan was made the appropriate notches were
cut and the stick split into two pieces so that the notches appeared in each section. The
debtor kept one piece and the Exchequer kept the other, so the transaction could easily
be veri ed by tting the two halves together and noticing whether the notches coincided
(whence the expression “our accounts tallied”). Presumably, when the two halves had
been matched, the Exchequer destroyed its section—either by burning it or by making
it smooth again by cutting off the notches—but retained the debtor’s section for future
record. Obstinate adherence to custom kept this wooden accounting system in of cial use
long after the rise of banking institutions and modern numeration had made its practice
quaintly obsolete. It took an act of Parliament, which went into effect in 1826, to abolish
the practice. In 1834, when the long-accumulated tallies were burned in the furnaces that
heated the House of Lords, the re got out of hand, starting a more general con agration
that destroyed the old Houses of Parliament.
The English language has taken note of the peculiar quality of the double tally stick.
Formerly, if someone lent money to the Bank of England, the amount was cut on a
tally stick, which was then split. The piece retained by the bank was known as the foil,
whereas the other half, known as the stock, was given the lender as a receipt for the sum
of money paid in. Thus, he became a “stockholder” and owned “bank stock” having the
same worth as paper money issued by the government. When the holder would return,
the stock was carefully checked and compared against the foil in the bank’s possession;
if they agreed, the owner’s piece would be redeemed in currency. Hence, a written
certi cate that was presented for remittance and checked against its security later came
to be called a “check.”
Using wooden tallies for records of obligations was common in most European
countries and continued there until fairly recently. Early in this century, for instance,
in some remote valleys of Switzerland, “milk sticks” provided evidence of transactions
among farmers who owned cows in a common herd. Each day the chief herdsman would
carve a six- or seven-sided rod of ashwood, coloring it with red chalk so that incised
lines would stand out vividly. Below the personal symbol of each farmer, the herdsman
marked off the amounts of milk, butter, and cheese yielded by a farmer’s cows. Every
Sunday after church, all parties would meet and settle the accounts. Tally sticks—in
particular, double tallies—were recognized as legally valid documents until well into the
1800s. France’s rst modern code of law, the Code Civil, promulgated by Napoleon in
1804, contained the provision:
The tally sticks which match their stocks have the force of contracts between persons who
are accustomed to declare in this manner the deliveries they have made or received.
The variety in practical methods of tallying is so great that giving any detailed
account would be impossible here. But the procedure of counting both days and objects
by means of knots tied in cords has such a long tradition that it is worth mentioning. The
device was frequently used in ancient Greece, and we nd reference to it in the work of
Herodotus ( fth century B.C.). Commenting in his History, he informs us that the Persian
king Darius handed the Ionians a knotted cord to serve as a calendar:
The King took a leather thong and tying sixty knots in it called together the Ionian tyrants
and spoke thus to them: “Untie every day one of the knots; if I do not return before the
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