Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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VII. | NUMBER |

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Dictionary of the History of Ideas | ||

#### NUMBER

For modern man it is impossible to conceive of a world

without numbers. If we were unable to distinguish

between 1 and 2, between 10 and 12, between one

thousand and one million, our whole culture and civili-

zation would collapse. No policeman could stop us for

passing the speed limit, for this limit must be fixed in

terms of numbers, provided of course that it would

be possible to build automobiles without being able

to count the number of wheels or doors to be built

into them. Whatever we think about in our daily life

and surroundings is in one way or another dependent

on our ability to count. In this sense, if in no other,

certainly the old Pythagorean saying is true: “All is

number.”

Considering for a moment the number system in

common use today, probably the most remarkable fact

about it is that the whole of civilized mankind, with

very few exceptions, is using the same kind of system

and symbols. Though we speak many languages and

write in different scripts, the number of different num-

ber systems still in use today all over our planet is far

more limited. And for all scientific work there is in

fact only one system—the one Westerners have all

known since their childhood. Consisting of ten symbols

1, 2, 3, 4, 5, 6, 7, 8, 9 and 0, it is so highly developed

that all other numbers are expressible by means of these

two handfuls of signs. A remarkable achievement, if

one stops to think about it for a moment.

The story of our numbering system has two aspects.

It is the story of the names given to numbers, and it

is the story of the symbols representing numbers. Both

have, in various degrees, contributed to the concept

of number itself and the systematic structure of our

present number system.

Besides the spoken number sequence, the number

words, and the written number sequence, the number

symbols, there once existed a third way of communi-

cating the meaning of a number from person to person:

the use of gestures. By different positions of the ten

fingers one may convey various numbers. Methodically

developed, this can be extended to rather large num-

bers. Thus, medieval manuscripts and early printed

books contain pictures indicating how by different

positions of the ten fingers it is possible to represent

any number up to 9999.

In the eighth century the Venerable Bede, an English

monk of the order of Saint Benedict, for the first time

in history recorded the gestures for numbers in his work

on the ecclesiastical calendar. While Bede described

the method in detail, let it be sufficient here to say

that the three outer fingers of the left hand had to

represent the units from 1 to 9, the index finger and

thumb of the same hand the tens from 10 to 90, the

same two fingers of the right hand the hundreds, and

the outer three fingers of the right hand the thousands.

Thus, for the person facing a man who signalized a

number this way, the four digits would appear in

increasing order from right to left. In fact the meaning

of “digit” here is derived from the Latin word for

finger: *digitus* (Figure 1).

While it is impossible to say definitely where and

when these “finger numbers” were invented for the

first time, it seems very likely that they arose from

the needs of commerce; they are a language of trades-

men. A similar way of representing numbers by means

of fingers can still be observed in certain Arabic and

East African marketplaces. Seller and buyer will

touch and rub each other's hands under a cloth so that

onlookers are unable to find out for what price the

bargain is completed. This method works even when

the traders do not speak a common language—they

selves. Even in our modern industrialized world there

still exists a place where finger gestures are used to

transmit numbers: at the stock exchange. The system,

however, is adapted to the special need of the brokers.

In general, finger numbers are no longer a common

medium for the conveyance of numbers.

Finger gestures are a mode of silent communication

about numbers. They are by nature short-lived and

transitory, not suitable for keeping a permanent record.

The same holds for the spoken number word, unless

it is remembered and thus kept alive in a human mind.

For a permanent record, numbers must be written

down or stored in some other convenient way. Modern

computers, for instance, may store numbers on a mag-

netic tape which can be “read” again by the computer

though not directly by the human eye. Primitive men,

too, invented procedures of storing numbers. Some of

these do work on a very elementary, and yet, as we

shall see, very basic, principle, not needing any signs

or script.

The Wedda on the island of Ceylon, when counting

coconuts, used to take a bundle of sticks and assigned

one stick to each coconut, always saying “this is one.”

In this way they obtained just as many sticks as there

were coconuts; nevertheless they had no number

words. But they were able to keep a record: if a coco-

nut was stolen, one stick was left over when the assign-

ment of sticks to coconuts was repeated.

Mathematically speaking, what the Wedda do is

establish a *one-to-one correspondence* between the ob-

jects to be counted and an auxiliary set of objects. This

is the most basic principle of counting of all, here

applied in its most elementary way. One coconut—one

stick, another coconut—another stick, still another

coconut—still another stick; one stick for each coconut,

but never more; hence also: one coconut for each stick,

and not one less. It may come as a surprise to some

that it is possible to count without having numbers,

yet, as we just saw, it can be done. It is inconvenient,

of course, since the sticks have to be carried and kept,

and the process of counting is slow. To inform a fellow

about a number, one has to show a set of auxiliary

objects of the same number of items.

Awkward as it may seem we do sometimes employ

the same elementary process. Think of a teacher who

is sent with his pupils into another classroom. If he

wants to know whether there is a sufficient number

of chairs for his students he need not first count the

students, then the chairs; he will just ask the class to

sit down and observe if somebody will be left without

a chair. The one-to-one correspondence will solve his

problem, not a single number word or number symbol

being required.

Number systems are nothing else but such *auxiliary
sets* of a very special kind. First of all, these sets do

not consist of hard objects. The real objects are re-