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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas

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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

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
of a very special kind. First of all, these sets do
not consist of hard objects. The real objects are re-
placed by symbols written on paper or made visible
in some other way. Secondly, the objects or elements
of the auxiliary set are not all alike. Both these facts
are real advances over the primitive method applied
by the Wedda. Both are related to the invention of
the art of writing, although the second distinction is
not limited to written symbols.

Consider an ancient way of counting soldiers. Passing
through a gate in single file, a pebble was dropped
into a box as each soldier passed. When ten soldiers
had passed, the ten pebbles were taken out of the box
and one pebble was put into a second box instead. For
each of the following soldiers one pebble was placed
in the first box until again ten men had passed. Then
the ten pebbles were taken out of this box and another
pebble was placed into the second instead. When the
second box received its tenth pebble, these ten were
interchanged for one pebble in the third box, and so


on. After all soldiers had passed, their total number
could be determined almost instantly.

This story exemplifies another principle in counting:
the introduction of a collective unit. One pebble in the
second box represents ten pebbles in the first, one
pebble in the third box is valued as much as ten pebbles
in the second, etc. Although there is, in this example,
only one kind of pebbles, the value assigned to each
depends on its position, on its being placed in a certain
box. Another way of introducing collective units would
have been possible. Using, for instance, small pebbles
to count the individual soldiers, medium-sized pebbles
to represent ten small ones, and large pebbles to repre-
sent ten medium-sized ones, only one box would have
been necessary. We see: when collective units are
introduced, this can be done in two ways. If there is
only one type of objects (or symbols) at hand, the
distinction must be made by help of the position; if
on the other hand different objects (or symbols) are
available for the various collective units, position does
not matter. As we continue our study of number sys-
tems, this will lead to important consequences.

An example of a number system in which collective
units are used in regular fashion is the Egyptian hiero-
glyphic one, dating from about 3000 B.C. Except for
the symbol for one, a simple stroke, there are no other
symbols but six collective units, for ten and its powers:
illustration [Description: Image of Figure 2]
In writing a number, these symbols could be repeated,
each up to nine times. An example is (Figure 3):
illustration [Description: Image of Figure 3]
The order of the symbols does not matter, they could
be arranged in horizontal as well as in vertical direc-
tions. That is, position is irrelevant; each sign carries
its meaning in a unique way. We call such a system
a tallying system, since the individual number symbols
are marked or tallied as often as required.

The Roman number system essentially is a tallying
system, too. It is distinguished from the old Egyptian
one in that it employs collective units not only for the
powers of ten (I, X, C, M = 1, 10, 100, 1000 respec-
tively), but also for the quintuples of these
(V, L, D = 5, 50, 500). There is no essential difference;
the addition of the latter symbols only makes the num-
bers more readable since at most four symbols of one
kind are necessary, against nine in the Egyptian mode
of writing numbers:
(The use of IX for nine, instead VIII, or XC for
LXXXX, etc., is a later development.)

Both the Egyptian and the Roman system are con-
structed by rule, in that all powers of ten (up to a
certain limit) are assigned new symbols as collective
units. The number ten therefore is called the base of
the system. Not all number systems have base ten; in
fact, not all have a base at all. In our present time
measurement, for example, 60 seconds are equivalent
to one minute, 60 minutes to one hour, but 24 hours
to one day, 30 days to one month, and 12 months to
one year. This system has no base, therefore.

The systems discussed so far operate with relatively
few signs which, if required, must be repeated several
times. Mankind also invented systems that in principle
do not demand any repetition of symbols. Such for
instance is the Greek method of taking the letters of
the alphabet as number signs (Figure 4):

α β γ δ ε ς ζ η ϑ … 
ι κ λ μ ν ξ ο π ϙ 10 20 … 90 
ρ σ τ υ φ χ ψ ω ϡ 100 200 … 900 
While repetition of symbols is eliminated in this
Alexandrian system, and hence numbers become much
shorter and more easily readable, the disadvantage
clearly lies in the fact that a very large number of signs
is necessary. Even the Greeks had to add a few Semitic
signs to their alphabet in order to have at least 27
symbols (9 for the units, 9 for the tens, and 9 for the
hundreds). For thousands, they repeated the signs for
the units, distinguishing them by a little stroke. It
would have been more consistent to use entirely differ-
ent characters. There is no tallying in the Alexandrian
number system since each number has its own code


symbol; we therefore call it a code system. Again, order
or position of the symbols within a number does not
really matter as each sign carries only one value. The
handicap lies in the quantity of symbols necessary to
extend the system far enough.

Let us summarize our observations. Counting, we
saw, is based on the principle of one-to-one corre-
spondence between the objects to be counted and the
elements of an auxiliary set. In the simplest case these
elements are indistinguishable sticks or strokes. In a
more advanced case there are some different kinds of
elements in the auxiliary set, e.g., those representing
the first powers of ten, or other collective units. In
the extreme case each element of the auxiliary set is
different from all others; a long alphabet would be an
example, in which no two letters were alike. Which
of the three cases could serve as an ideal number
system? The first has the advantage of providing an
infinitely large auxiliary set (stroke after stroke without
end), but the elements are not distinguishable, and
reading of large numbers becomes cumbersome. The
last allows for easy reading as each number has its own
character, but the sequence cannot be extended to
cover “all” numbers since nobody can remember in-
finitely many different signs. What is needed for an
ideal number system obviously is an arrangement of
some, but not too many code symbols with repetition
after a given pattern. How this pattern can be formed
was suggested in the story about counting soldiers: it
is the position of symbols that must be used in addition
to their immediate meaning.

Such a system we have in our Hindu-Arabic number
as it is usually called. It combines the advan-
tages of the various systems that have been discussed
in this article: there is no tallying since code symbols
for each of the numbers from one to nine are provided.
Beginning with ten these code symbols are employed
again with a new meaning indicated by the position
in which they are standing. This repeated employment
is taking place completely regularly: the system has
a base ten. Thus, the symbol 3 may stand for three,
but also for thirty, three hundred, three thousand, etc.;
only its position within the number fixes its value
exactly. The base being ten, all collective units are
multiples of ten. It is therefore possible to extend the
system as far as necessary; even if one should run out
of number words, the number symbol for any number
however so large can immediately be written down.

Such a positional system contains one logical diffi-
culty which does not occur in a tallying system with
collective units. Consider the number three hundred
and six in Roman numerals: CCCVI. There are no tens
in this number, hence the symbol X does not appear.
It is very simple. Not so in a positional system. We
cannot simply write 36 but need only indicate that
the place for the tens is empty, i.e., we need a place-
holder as it is sometimes called. This is of course the
symbol 0 for zero. Logically this presents an immense
difficulty: one writes something to indicate that there
is nothing. That must have sounded queer to many an
early student of our positional number system! It was
one of the really great steps in the historical develop-
ment of number systems that such a sign was intro-
duced. Without it, our mode of numeration would be
far less perfect.

There does indeed exist a way to evade, so to speak,
the invention of zero. As example let us consider the
basic numbers of the Chinese; they have several num-
ber scripts. A decimal system with base ten, it consists
of a mixture of code symbols for the units from 1 to
9 and collective units for the powers of ten (Figure 5).
illustration [Description: Image of Chinese Numbers]
We may call the collective units “labels” for they serve
to label the positions within a number. In other words:
the code symbols, taken by themselves, carry the values
from one to nine, but when they are combined with
a label they multiply the latter's value by their own.
No symbol for zero is required. If, e.g., there are no
hundreds in a number, the label for hundred is omitted,
as in Figure 6 (to be read from top to bottom):

illustration [Description: Image of Chinese Numbers]

This Chinese system therefore is a positional system
with labels.
Its distinction from a positional system
without labels—such as ours—is to be found in two
points: (1) the suppression of the labels, the meaning
of each position being understood as self-evident ac-
cording to its natural sequence; (2) the introduction


of a place holder, i.e., a symbol for zero which then
becomes necessary. With these two steps we can con-
struct the singularly efficient positional system without
labels. Logically, there only remains the choice of the
most convenient base for it. Historically this base came
to be the number ten, but from time to time arguments
have been aired in favor of twelve which would allow
for easier divisibility of numbers into the most common
small fractions.

We need not tell the history of our Hindu-Arabic
number system in detail, but we may take a glance
at the highly essential concept of zero. It was not
until the seventeenth century that zero was accepted
as a “real” number. In the second century B.C. a little
circle appeared in Greek astronomical texts as a place
holder, most probably an abbreviation of the Greek
word oudén (“not one,” “nothing”). It may well be the
same little circle that we meet again in a Hindu in-
scription (ninth century A.D.) where the number 270
is represented in this form: 70. The inscription
is written in Brahmi script, the very number script
which, with some variations, was taken over by the
Arabs and by them transmitted into Europe, and named
the Hindu-Arabic number system. As for the spoken
word: the original Indian term for the little circle as
place holder was sunya (“empty”). It was translated
by the Arabs as aṣ-ifr (“emptiness”), which word was
taken over into Latin as cifra or zefirum. Our cipher
is derived from the former word; our zero from the
latter. It is strange to observe that cipher, once the
name for zero only, became the term for all “ciphers,”
i.e., for the figures from 1 to 9, too. This is a witness
to the great difficulties that were encountered when
the strange characters of the present number system
were first introduced into Europe. Another Latin name
for zero was nulla figura (“no figure”), the origin for
the German Null and identical in meaning with the
English nought = nothing. In a formal way pre-
scriptions for the handling of zero had been given in
late antiquity, but it was not before the sixteenth and
seventeenth centuries respectively, that zero was ad-
mitted as coefficient or as root of an algebraic equation.
Only with A. Girard and R. Descartes did the symbol
0 gain full equality of rights as a number.

Apart from zero unity too was for centuries not
considered a number. The ancient Pythagoreans (fifth
century B.C.) were the first to philosophize about the
nature of number. Their statement, “All is number”
expresses their belief that numbers are the essence of
all existing things. Hence to understand a thing one
had to know its number. As Philolaus remarked: “All
things which can be known have number; for it is
impossible for a thing to be conceived or known with-
out number” (Diels [1934], 47 B 1). Unity itself, how
ever, was not a number but the principle from which
all (further) numbers were generated. This view per-
sisted beyond medieval scholasticism, but again, and
no later than Descartes, all distinctions between 1 and
the rest of the integers had completely vanished. At
the same time, and not least by the influence of
Descartes, the modern mathematical point of view
towards numbers was gaining ground. Here numbers
are nothing but abstract entities that can be produced
according to certain rules and that serve to describe
order and quantity. This is the number concept of the
mathematician who does not know the difficulties
mankind had to overcome before this abstract idea
could be formed, before it could be molded into a
rigorous logical framework.

The historical process, in large parts not recon-
structible and hence for ever open to speculation,
nevertheless has left some marks. Use of the names for
0 illustrates an important step in the construction of
a written number system. The vast store of names for
other numbers, in particular for the first positive inte-
gers, in many living and dead languages, makes it
possible to draw further conclusions about the history
of the idea of number. One of the main insights one
gains from a comparative study of number words in
various languages, particularly those of primitive peo-
ple, is the fact that in the early stages of counting
numbers have much in common with adjectives. That
is to say, numbers are seen in very close relation with
the objects they count.

In some cases, the number concept may even merge
with the noun to make a special grammatical form,
as in Greek, where besides the singular and the plural
there exists a special dual:

ho philos the friend to philo the two friends
hoi philoi the friends (more than two)
In other cases, a number word has several forms ac-
cording to the gender of the noun to which it belongs.
Thus, in Latin “one” has three forms, “two” and
“three” have two forms, and only from “four” onwards
the number words are indeclinable. In still other cases,
number words may not be used with any arbitrary
object but only in connection with items of a special
kind or class. A tribe of American Indians had special
number words for living objects, for round objects, for
long objects, and for days. Even the English language
today contains several expressions for the number 2
which can only be applied with respect to certain
situations: a yoke, a pair, a couple, a duet, twins.

While all these examples show the close relationship
between early numbers and the things they count,
other number names reveal that in many parts of the
earth counting began with the help of fingers, and


sometimes toes, too. The number words of the Déné-
Dindje, a tribe of American Indians, have the following

the end is bent (little finger) 
it is bent again (ring finger) 
the center is bent (middle finger) 
there is one left over 
my hand is finished 

Not often is the relation between the names for the
first numbers and the finger gestures so clear as in this
example. The following number words are collected
from various cultures:

whole hand; once my hand 
one on the other hand; other one 
10  both hands; both sides; two hands die (i.e., all ten
fingers are bent) 
11  one on the foot 
16  one on the other foot 
20  my hands, my feet; a man; man brought to an

Where fingers (and toes) formed the first auxiliary
set for counting and provided a ready source for the
first number names, a decimal (or vigesimal) system
was the natural outcome if the system was later ex-
tended in regular pattern. It is therefore not surprising
that decimal systems are widespread among the spoken
number sequences, or that they are mixed with vigesi-
mal elements, as we see in French:

10  dix 
20  vingt 
30  trente 
40  quarante 
50  cinquante 
60  soixante 
70  soixante-dix 
80  quatre-vingts (four times twenty) 
90  quatre-vingt-dix 
100  cent 

Reference to the human body might go beyond the
use of fingers and toes as the number sequence of a
Papuan tribe demonstrates:

anuso  little finger (right) 
doro  ring finger 
doro  middle finger 
doro  index finger 
ubei  thumb 
tama  wrist 
unubo  elbow 
visa  shoulder 
denoro  ear 

10  diti  eye 
11  diti  eye (left) 
12  medo  nose 
13  bee  mouth 
14  denoro  ear (left) 
15  visa  shoulder 
16  unubo  elbow 
17  tama  wrist 
18  ubei  thumb 
19  doro  index finger 
20  doro  middle finger 
21  doro  ring finger 
22  anuso  little finger 

Unfortunately there cannot be given such a simple
and instructive explanation for our own number words.
They are modifications of the Anglo-Saxon ones, which
in turn are of old Germanic origin. All Germanic lan-
guages show similarities in their spoken number se-
quences but the original meaning is not clear. Here
we leave historical considerations and turn to the mod-
ern mathematical viewpoint.

Formally, if he considers zero and the positive inte-
gers 1, 2, 3,... to be given, the mathematician may
construct further numbers as roots of equations, whose
coefficients are taken from these integers. The equation
x + 1 = 0 for instance will produce the “root” or
solution x = -1, since -1 + 1 = 0. Similarly, all
other negative integers may be produced. We may
hence assume the general form of an algebraic equation
to be
anxn + an-1xn- + an-2xn-2 + ... + a1x + a0 = 0
where all coefficients a0, a1,..., an are positive
or negative integers or zero. Let us see how further
types of numbers can be constructed by means of such

The simplest type that is contained in the general
form above is the so-called linear equation in which
the unknown x appears only in the first degree:
a1x + a0 = 0. This equation may be understood as
the definition of the fraction -a0/a1 for if x = -a0/a1
it will satisfy the equation. Suppose, e.g., a0 = -1,
a1 = 2, then the equation would be 2x - 1 = 0 with
the root x = 1/2. Hence we have “generated” the
fraction 1/2. In an obvious way all other fractions,
positive or negative, may be thus constructed. It is the
task of the mathematician to show that these fractions
do obey the common laws to which all numbers have
to be subjected, and that in particular the elementary
operations (addition, subtraction, multiplication, and
division) can be carried out in a meaningful and non-
contradictory way.

New types of numbers may occur when equations


of the second degree are studied: a2x2 + a1x + a0 = 0.
Supposing for simplicity that a1 = 0, the basic type
of such a quadratic equation is a2x2 + a0 = 0, whose
root we have to write in the form x = √-a0/a2. If,
for instance, a0 = 18, a2 = -2 (-2x2 + 18 = 0), the
numbers x = 3 and x = -3 will solve the equation,
and there is no further problem. If, however, we are
given x2 - 2 = 0 (a0 = -2, a2 = 1), the root can be
written in the form x = √2, but it cannot be expressed
as an integer or a fraction. Such a number is called
irrational since it does not form a ratio or fraction (in
the sense of 1/2 = 1:2 or 5 = 5:1). It was a funda-
mental discovery of far-reaching consequences, made
first by the Pythagoreans, that not all numbers are
rational numbers, that is, fractions or integers. As a
consequence they had to reconstruct a great deal of
their mathematics. It is possible to show that these
irrational numbers can be incorporated into the num-
ber system without difficulty. In fact, without irrational
numbers the number system would be incomplete. By
the way, more complicated roots such as ∛7 or
⁵√7 + ∛2 are not of an essentially different type.
Rational and irrational numbers together are called real
by the mathematicians. For any two unequal
real numbers it is possible to decide which one is
greater than the other. In consequence, all real num-
bers may be ordered according to magnitude and
represented on the real number line (Figure 7):

There remains one case to be considered: the square
root of a negative number. The most simple case would
be the solution of the equation x2 + 1 = 0, that is
x = √-1 which means x2 = -1. Now there is no
real number—whether integer, fraction or irra-
tional—whose square is -1. This fact has baffled
mathematicians since the sixteenth century; before that
time they would simply say that this equation has no
root at all. Slowly they learned to accept the new,
“imaginary” type of number for the reason that it was
possible to operate with it in the usual way. Writing
i = √-1 for brevity's sake it became clear that all
numbers which may arise as roots of algebraic equa-
tions are either real, or imaginary, of the complex form
a + bi, where a and b are real numbers and i = √-1.
For instance, the equation x4 - 6x2 + 25 = 0 has a
root x = 2 + i (a = 2, b = 1).

A different problem was how to realize or represent
this new type of number which no longer fitted into
the linear arrangement on the line of real numbers.
It was only about 1800 that independently of one
another C. Wessel, C. F. Gauss, and J. R. Argand
conceived the possibility of representing these complex
(a + bi) in a plane, the complex number plane.
Later in the century W. R. Hamilton developed a more
abstract introduction of complex numbers as pairs
(a, b) of real numbers. He also showed that no
further extension of the number system is possible if
all the usual laws of the four elementary operations
+, -, ×, ÷ are to remain valid.

An extension of the number system of quite another
kind was given by G. Cantor in 1874. Galileo had
discussed the question whether the number of squares
(1, 4, 9, 16,...) is to be reckoned the same as the num-
ber of positive integers (1, 2, 3, 4,...). The problem
required a quantitative treatment of the actually in-
neither Galileo nor other mathematicians be-
lieved such treatment to be possible. Cantor, much
against his will, was forced to the following conclu-
sions. If a one-to-one correspondence is taken as the
essential principle of counting, the infinite set (or col-
lection) of all positive integers has exactly as many
elements as the infinite set of all square numbers. The
obvious correspondence is a one-to-one matching of
each element of the whole set, to each element of the

Introducing the concept of “power,” mathematicians
say with Cantor that two sets have the same power,
if they can be matched element for element. For infi-
nite sets there is an immediate consequence: it is no
longer true that a subcollection or part is less than the


whole set, and the whole may have the same power,
as in the example just given.

All infinite sets that can be arranged in one-to-one
correspondence with the set of positive integers are
said to be denumerable. For instance, the set of all
positive rational numbers (i.e., fractions) is seen to be
denumerable by the following arrangement:

Not all infinite sets, however, are denumerable. If
they were so, the concept of power would be useless.
Cantor was able to show that the set of all positive
“real” numbers—which include irrational numbers,
like √2—is not denumerable, i.e., it is impossible to
match all such reals with the positive integers: there
are just “too many” of the former. Hence the set of
real numbers has a power greater than that of the
integers. It is called the power of the continuum. One
may construct infinite sets whose power is greater than
that of the continuum. Indeed, the sequence of these
powers or transfinite numbers, which so to speak count
the orders of infinity, is itself infinite. This is truly
beyond the powers of imagination of any human being;
it can only be established by strict logical reasoning.

When one surveys the whole development of the
idea of number from its earliest cultural origins to the
abstract modern concepts, one becomes aware of the
close relations and mutual interdependence between
the course of this development and the growth of
science and technology that has taken place since the
Renaissance. In those days the study of nature turned
away from the Aristotelian world view with its em-
phasis on qualitative change and teleological reasoning.
The basic question of the philosopher of old: “Why
does this happen?” was replaced by the more restricted
question of the modern scientist: “How does this hap-
pen?” Galileo recognized that the answer to this last
question could only be expressed in the language of
quantity, that is, in mathematical form. Geometry and
algebra in the time of Galileo and Descartes offered
the patterns according to which the new science of
mechanics could be modeled. As mechanics grew, new
branches of mathematics began to blossom: differential
and integral calculus, probability theory and statistics,
differential geometry, the theory of differential equa-
tions and the calculus of variations, and a host of other
mathematical disciplines. The great success of analyti-
cal mechanics and its applications during the seven-
teenth and eighteenth centuries inaugurated a mathe-
matization of more and more physical, natural, and
social sciences during the past two centuries, which
seems to be still far from its peak. Thus number in
one way or another has conquered our whole culture.
As concept, it is everywhere present, materialized in
thousands or millions of computers (which begin to
become the secret rulers of all our life), it has opened
the door to a new scene of our technological civili-

It is a generally observable fact in the history of
human ideas, particularly of ideas capable of develop-
ment to a high degree of abstraction, that progress
towards logical clarification and abstract formulation
has to be paid for by loss of the close connection with
the original cultural descent of these ideas. While with-
out an abstract and rigorous building-up of the number
concept modern science and technology which is based
on mathematical theories would be impossible, without
the first intuitive steps in numeration made by primi-
tive man in prehistoric time no number system could
have been developed. Those finger gestures, spoken
number words, and written number symbols of ages
long gone by mark the beginning of a development
which resulted in the present-day highly sophisticated
mathematical number concept. A few of the aspects
of the early beginning and of later improvements have
been touched upon in the present article, showing how
our number system and number concept are rooted
in the general cultural soil which nourished the his-
torical growth and unfolding of all human ideas.


The most complete treatment of the historical develop-
ment of number systems and elementary arithmetic pub-
lished in recent years is: K. Menninger, Zahlwort und Ziffer.
Eine Kulturgeschichte der Zahl,
2 vols., 2nd ed. (Göttingen,
1957-58); trans. P. Broneer as Number Words and Number
Symbols: A Cultural History of Numbers
(Cambridge, Mass.
and London, 1969). This outstanding work contains an
extensive bibliography of primary literature. The same
subject is dealt with on a much more restricted scale in
the little book by D. Smeltzer, Man and Number (London,
1958). Good introductions are also the following: D. E.
Smith, Number Stories of Long Ago (Washington, 1919; repr.
1951); D. E. Smith and J. Ginsburg, Numbers and Numerals
(Washington, 1937); D. E. Smith and L. C. Karpinski, The
Hindu-Arabic Numerals
(Boston, 1911).

T. Dantzig, Number, the Language of Science, 4th ed.
(New York and London, 1954) emphasizes the mathematical
development up to and including the Cantorian transfinite


numbers. C. J. Scriba, The Concept of Number (Mannheim
and Zurich, 1968) was written as a text for a graduate course
offered at the Ontario College of Education; it deals with
the origins of number systems, the development of elemen-
tary arithmetic, algebra, and number theory, and includes
nineteenth-century contributions to the number concept.

Also recommended are Carl B. Boyer, A History of Math-
(New York, 1968), and P. E. B. Jourdain, trans. and
ed., Contributions to the Theory of Transfinite Numbers
(Chicago and London, 1915; also reprint). The Diels refer-
ence is to H. Diels, Die Fragmente der Vorsokratiker...,
5th ed. (Berlin, 1934).


[See also Axiomatization; Infinity; Mathematical Rigor;