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

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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-

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

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:

[Description: Image of Figure 2]

In writing a number, these symbols could be repeated,

each up to nine times. An example is (Figure 3):

[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:

MMMDCCLXXXVIIII = 3789

(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 | 1 | 2 | … | 9 | |

ι | κ | λ | μ | ν | ξ | ο | π | ϙ | 10 | 20 | … | 90 | |

ρ | σ | τ | υ | φ | χ | ψ | ω | ϡ | 100 | 200 | … | 900 |

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

*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
system,* 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).

[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):

[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

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)

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

Dindje, a tribe of American Indians, have the following

meaning:

1 | the end is bent (little finger) |

2 | it is bent again (ring finger) |

3 | the center is bent (middle finger) |

4 | there is one left over |

5 | 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:

5 | whole hand; once my hand |

6 | 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 end. |

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:

1 | anuso | little finger (right) |

2 | doro | ring finger |

3 | doro | middle finger |

4 | doro | index finger |

5 | ubei | thumb |

6 | tama | wrist |

7 | unubo | elbow |

8 | visa | shoulder |

9 | 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 *a*0, *a*1,..., *an* are positive

or negative integers or zero. Let us see how further

types of numbers can be constructed by means of such

equations.

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:

*a*1*x* + *a*0 = 0. This equation may be understood as

the definition of the fraction -*a*0/*a*1 for if *x* = -*a*0/*a*1

it will satisfy the equation. Suppose, e.g., *a*0 = -1,

*a*1 = 2, then the equation would be 2*x* - 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

*a*2

*x*2 +

*a*1

*x*+

*a*0 = 0.

Supposing for simplicity that

*a*1 = 0, the basic type

of such a quadratic equation is

*a*2

*x*2 +

*a*0 = 0, whose

root we have to write in the form

*x*= √-

*a*0/

*a*2. If,

for instance,

*a*0 = 18,

*a*2 = -2 (-2

*x*2 + 18 = 0), the

numbers

*x*= 3 and

*x*= -3 will solve the equation,

and there is no further problem. If, however, we are

given

*x*2 - 2 = 0 (

*a*0 = -2,

*a*2 = 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*

numbersby the mathematicians. For any two unequal

numbers

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 *x*2 + 1 = 0, that is

*x* = √-1 which means *x*2 = -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 *x*4 - 6*x*2 + 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
numbers* (

*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-
finite;* 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

subset:

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

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-

zation.

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.

## *BIBLIOGRAPHY*

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.

Symbols: A Cultural History of Numbers

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).

Hindu-Arabic Numerals

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

*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-
ematics* (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).

CHRISTOPH J. SCRIBA

[See also Axiomatization; Infinity; Mathematical Rigor;Pythagorean....]

Dictionary of the History of Ideas | ||