Number Words and Number Symbols

A Cultural History of Numbers

Karl Menninger

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486319773
• Topic: Mathematics
• Subtopic: History & Philosophy of Mathematics
• Index: Mathematics

''The historian of mathematics will find much to interest him here . . . while the casual reader is likely to be intrigued by the author's superior narrative ability.` — Library Journal
This book is not only a fascinating introduction to the concept of number and to numbers themselves, hut a multifaceted linguistic and historical analysis of how numbers have developed and evolved in many different cultures. Drawing on evidence from history, literature, philosophy and ethnology, noted German scholar Karl Menninger. recounts the development of numbers both as they are spoken (and written as words) and as symbolic abstract numerals that can he readily manipulated and combined.
Despite the immense erudition the author brings to the topic, he maintains a light tone throughout, presenting much of the information in anecdotal form. Moreover, almost 300 illustrations (photographs and drawings) and many comparative language tables serve to enhance the text. The author begins with a lucid treatment of number sequence and number language, including the formation of number words in both Indo-European and non-IndoEuropean languages, hidden number words and the evolution of the number sequence. He then turns to written numerals and computations: finger counting, folk symbols for numbers, alphabetical numerals, the `German` Roman numerals, the abacus and more. The final section concerns the development of our modem decimal system, with its place notation and zero, based on the Indian number system, and its introduction to the West through the work of the Italian mathematician Fibonacci. The author concludes with a review of spoken numbers and number symbols in China and Japan.
`The book is especially good on early counting and calculating devices: primitive tally sticks, the knotted cords of ancient Peru, the elaborate finger symbols once used for numbers, counting boards with movable counters, and of course the abacus.` — Martin Gardner, Book World

WRITTEN NUMERALS AND COMPUTATIONS

Preface

The next section of our cultural history of numbers deals with those written numerals and computations that have been used by ordinary people in simple domestic calculations. The main subject of this section is the development of the numerals and computations used in Western Europe. From time to time, however, some light may be shed on this development by occasional glimpses of the numerals and computational operations of other peoples, whose cultures may be either closely or distantly related to our Western civilization.

The Indian numerals which we use today in writing and computation will be discussed before we take up the Roman numerals and counting boards. The history and influence on written computations of the now forgotten counting board or abacus will be the subject of a thorough treatment, more exhaustive than any other published thus far, to my knowledge. It is not only an absorbing segment of the history of numbers and mathematics, but also and above all it forms the background for the fascinating events which accompanied the introduction of Indian place-value notation into the Western world. We come upon this seldom-trod path in the midst of the intellectual revolution from which Western man emerged to “take the world into his hands.” Numbers have been one of the most important means by which he learned to master his environment.

The previous section dealt with number words — the number sequence and spoken numbers; this section concerns itself with numerals. Each section, representing spoken numbers or written numerals, can stand alone, although many themes are common to both and many developments and events will not become fully clear until light has been cast on them from both directions.

Introduction

When we speak of Roman or Egyptian or even our own numerals, we almost always mean the “official” numerals in which a state’s records are kept and which all individuals are thus obliged to know. Most people are completely unaware of any other kinds of written numbers.

Yet historically every “official” system of numerals is preceded by some set of primitive number symbols, signs which individuals or associations or villages have used to “write” down numbers, in the form of notches carved in a piece of wood or of knots tied into a string cord, or in some other popular manner. A study of these pre-“official” recorded numbers will acquaint us with a multitude of now generally forgotten things that are still alive in language and custom, and above all will provide valuable insights into the “early” concepts which underlie or antedate written representations of numbers. These insights are important, moreover, because even the formal numerals frequently embody early forms which can be understood only with a knowledge of various folkways. A surprising amount of light is thrown on Roman numerals, for example, by notched tally sticks.

To “write” a number is to make it visible and record it permanently, in contrast to speaking it. Thus finger counting, the representation of numbers by various signs made with the hands and fingers, makes numbers visible but nevertheless evanescent, in a sort of intermediate stage between spoken and written numbers. These “finger numbers” are not merely child’s play: they had great importance and significance in earlier times, as they still have today in many parts of the world.

These two small bits of information about finger counting and primitive number symbols already suggest that a people’s formal written numerals do not weigh as heavily in the scales of culture as spoken numbers do. The written numerals are usually foreign imports; only seldom, as in the case of Egyptian and Chinese, do they grow out of the same native soil as the spoken numbers.

We should suspect from this fact that letters have also been used as number symbols. The Goths wrote alphabetical numerals in imitation of the Greeks, and so did virtually all peoples within the sphere of influence of the (Semitic) alphabetical form of writing. For many centuries they were the numerals used by Greek mathematicians.

These letter numerals and the so-called “early” numerals, whose systems were based only on the laws of ordering and grouping, were used merely to record numbers; they were not well suited to computations. For their numerical calculations people used the counting board or abacus, whose now forgotten history we shall also look into. We shall see its use in the world of the ancient Greeks and Romans, we shall find special forms of the abacus still in use in China, Japan, Russia, and even in our own culture, and then we shall turn our attention to the medieval counting boards used in Western Europe: to their different forms, to their remarkable ancillaries, the calculi, apices, and other counters, and finally to the main procedures used in making computations on the abacus.

Thus we shall have gained some insight into the way people during the Middle Ages were working with numbers at about the time the “new” Indian numerals began to penetrate into Europe by way of Italy, around 1500.

Where did the Indian numerals come from and how did they develop? What was new about them? By what route were they brought into the lands of the western Mediterranean and from there to northern Europe? The answers to these questions will unfold a vast drama of civilization extending from India through the Arab world into the West, from the quiet monastic cells of the early Middle Ages to the progressive offices of the Italian and German merchants and beyond, up to the threshold of modern times. We shall take part in this fascinating pageant of intellectual history, and thus understand how and why the Indian numerals came to be the only written numbers used by all the major nations of the world. These, and only these, enabled man to extend his capacity for calculations to undreamed-of heights. For better or for worse, they have subjected the world to the tyranny of numbers.

After this broad view we shall finally focus our attention on our narrower subject, spoken numbers and written numerals, this time in the Far East, in China and Japan. The culture of the Chinese and the Japanese is so unique and so isolated from the rest of the world that it is useful in two ways: it serves both as a comprehensive review of the main threads by which we guided ourselves through the complicated cultural history of numbers, and also as a counterpart, an analogue to our own world of numbers, from which we can learn what is extraneous or peculiar to our own numbers and what is unique and intrinsic to numbers and numerals themselves.

The common threads which we have followed through different times and civilizations will thus remind us that the common spirit of mankind first once began to evolve a single set of numbers, and then struck out along different paths represented by various separate cultures, until it finally arrived at the most highly perfected system, that of the Indian numerals. As these spread out over the whole inhabited world and came to be universally used, pushing aside all other numerals, they have become a symbol of the essential unity of mankind, which is manifested not at the outset, but only at the end of a long course of evolution.

Finger Counting

Finger Counting

Nunc mihi iam credas fieri quod posse negatur:

octo tenes manibus, si me monstrante magistro

sublatis septem reliqui tibi sex remanebunt.

“Now you shall believe what you would deny could be done:

In your hands you hold eight, as my teacher once taught;

Take away seven, and six still remain.”

A Roman riddle, which was never solved during the Middle Ages (see p. 204)

Language uses words to capture numbers. But words are ephemeral. From time immemorial, mankind has tried to find some way of making words and numbers permanent. The answer to this problem was writing.

But to fix spoken words by means of a pictorial or phonetic form of writing is a very difficult and laborious task, which only a few peoples in the world have succeeded in accomplishing. Even we, for example, though we speak a language of our own, write it with symbols that we did not invent. For “our” letters are actually Roman, and the development of alphabetical writing goes back further than the Romans, the Greeks, and the Phoenicians, all the way back to the ancient Egyptians.

With numerals, however, the story is a little different. A truly native system of writing numbers which all members of the culture had to learn and use, and which was thus an “official” set of numerals, was also developed by only a small number of peoples; the numerals used by the vast majority of cultures have been foreign imports. The digits which we commonly use today originated in India; before these, we wrote numbers in the Roman fashion. But before and quite often along with such “official” numerals, people from the earliest times have always...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Preface
• List of Phonetic Symbols
• Contents
• NUMBER SEQUENCE AND NUMBER LANGUAGE
• WRITTEN NUMERALS AND COMPUTATIONS
• Index
• Back Cover