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
do not need a language as the gestures speak for them-
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-
