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| ISBN-13: | 9780486319773 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/10/2013 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 512 |
| File size: | 17 MB |
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Number Words and Number Symbols
A Cultural History of Numbers
By Karl Menninger, Paul Broneer
Dover Publications, Inc.
Copyright © 1992 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-31977-3
CHAPTER 1
The Number Sequence
The Abstract Number Sequence
"... decem ... Hic numerus magno tunc in honore fuit. Seu quia tot digiti, per quos numerare solemus."
"... ten ... This number was of old held high in honor, for such is the number of fingers by which we count." Ovid, Fasti III
How do we count today?
Before we go into the historical development of our words for numbers, let us first determine how and what we actually count and what "counting" really is.
Before us lies a heap of peas, which we wish to count. How do we go about this? We arrange the peas in a row, physically or mentally, touch the first one and say "one," then touch the second and say "two," touch the next and say "three," ... touch the last and say "twenty-two"; there are 22 peas in all. What have we actually done? We have assigned a word to each individual pea. Counting thus constitutes assigning words to things.
To what are these words assigned? To the things we are counting — in this case to peas. At other times we may be counting houses, trees, people, or fingers. Can we also count things of different natures: a pen, a desk, and a cat, for example? Yes, these are three "objects." Can intangible things be counted as well, such as the conclusions of a proof, or the thoughts embodied in a thesis? Yes. Even a person's traits: intelligent, slender, lively, generous, and so on, can be enumerated. In short, any distinguishable entity, tangible or intangible, identical or different, can be counted. These distinguishable things, considered together, constitute a set and are themselves the elements of this set.
Thus, we now say: A set can always be counted by assigning number words to its elements.
But the number words themselves form a set, the elements of which are the words "one," "two," "three," and so on. In the process of counting, the elements of the set of number words or of the number sequence, whichever we prefer to call it, are assigned uniquely to the elements of the set of things to be counted: uniquely, because only a single number word is attached to each pea.
If we think of the separate terms of the number sequence as small boxes, labeled 1, 2, 3, and so on, we can conceive of the counting process as follows: In each box, starting with the first, we place a single pea, the first in box 1 and the last in box 22. Then 22 boxes of our number sequence are full, while all the remaining boxes from 23 on remain empty.
Now we can explain the heading, "The Abstract (Empty) Number Sequence." So long as there is no counting, it is merely there, detached from all concrete objects, unused but ready. But as soon as we count, then according to our first image the number words become assigned to the objects, and according to the second one the objects are placed in the empty boxes of the number sequence. The last number word (or the last box) indicates the cardinality, or number, of the set.
This insight is as important as it is simple. We shall see, in fact, that the "abstraction" of the number sequence from the things counted created great difficulties for the human mind. We need only ask ourselves: How would we count if we did not possess this sequence of remarkable words, "one," "two," "three," and so on? Yet there was a time when it did not exist!
Thus one achievement of our number sequence is its independence of the things themselves. It can be used to count anything.
But can it also be used to count arbitrarily large sets, even the sands of the sea? Yes, even these sets "without number" can be counted by using our number sequence — this is its other achievement. To each successive grain of sand it assigns a number word, tirelessly and inexhaustibly. And when the last particle of sand has been counted, it still has "infinitely many" number words with which to go on counting.
The number sequence could go on counting, even though we cannot. Yet we know for certain that it would do so correctly and in the proper sequence. We hear of three million inhabitants of a city: does anyone count them one after another, 1, 2, 3? Still we are quite sure that if this were to be done, we would eventually come to inhabitant No. 2,999,974, No. 2,999,975, No. 2,999,976 ..., and finally to No. 2,999,999 and No. 3,000,000.
Whence this certainty, which we never gained from experience? We know that our number sequence embodies the law of infinite progression; we know that every number has a successor; and we also know how that successor is formed from its predecessor.
Hence, our number sequence is not a motley collection of words arbitrarily gathered together, but an ordered creation of the mind. It incorporates the law of infinite progression, by force of which we acknowledge the countability of sets even though we ourselves cannot actually do the counting.
A finite and amazingly small quantity of number words is enough for this purpose, for the number sequence uses these words over and over again, in their proper order and context. And it is completely independent of the objects it counts: it is abstract. Therefore it can count anything.
This is our modern number sequence, the number sequence in its highest state of development. And now that we are familiar with it, the question becomes especially absorbing: was it not always so?
The Number Sequence Used Concretely
Haven't people always counted as we do today?
We shall find the answer to this question if we descend the ladder of culture down to the very lowest steps, scarcely above the level where mind could not rise above its environment. Early man counted, too, whenever he merely gathered fruits or hunted, whether he grew his own food by more or less primitive methods of cultivation or drove his herds from pasture to pasture or whether, like many tribes living near the coast, he sought to earn his living by trade. His way of life taught him to count, the nature of his economy determining the extent of his number sequence. Why should a pygmy people, living in isolation in the primeval forest, need to count beyond 2? Anything over that is considered "many." But the cattlebreeder must count his herd, head by head, up to 100 or even more. For him "many" is something far greater, something that no longer has economic meaning to him. Thus early man's environment determined his thinking and actions, and also his counting.
So that we can understand his number sequences, which we shall now consider, let us dwell for a moment on the way primitive man perceives the world around him. It still impinges on him directly, in all its myriads of colors and forms. Things have not yet been "cooled off" for him by his intellect, which sifts them and orders them and separates them, filing their elements away in the gray, colorless pigeon-holes of concepts. On the contrary, in their immediate, hot- blooded, many-colored uniqueness they touch his innermost heart. Thus they are not objects to him, things which are alien to him and stand "outside" himself — here am I and there is the world — rather they are completely absorbed in his own life. He is a part of them, as they are of him. He is woven into the very fabric of the universe by powerful strands of religion; he does not, like "modern" man, like ourselves, stand before it in wonderment, in calculation, or indifference.
Yet some remaining fragments of that early perception of the world still loom up in our own. Many a superstition, many an oddity, survives unrecognized in the midst of the intense consciousness of our own culture. Who today knows or cares much about the number 7? Though for us it has lost its supernatural content, though it offers not the slightest advantage in measuring and reckoning time, the seven-day week still governs our whole external life. From that early interpenetration of man with the world arises the infusion of objects and numbers with mystic significance, and hence the "holiness" of the numbers 3 and 7 and the auspiciousness or bad luck of the number 13. It is the task of mythology to uncover the concepts that led early man to impart supernatural significance to certain numbers.
Our own purpose, however, is to understand how early man gave expression to things and events, with all their profusion of kaleidoscopic detail, in his own primitive language. "A man has killed a rabbit" — an American Indian would never say this in such a colorless way. His statement, broken down into its verbal component, says: "The man, he, one, animate, standing, has purposely killed by shooting an arrow at the rabbit, animate, him, one, sitting." The Indian does not put it thus because he wants to express the event in a specially picturesque manner, as our highly developed speech could also do by adding words and phrases; he cannot say it any other way, because that is how he has experienced the event, and he cannot free himself from its uniqueness. Generalization into pale concepts is completely foreign to him. His language proves this, since it achieves its colorful expression not as does ours, by using auxiliary words and phrases, but rather through the inflection of its words and through particles prefixed, suffixed, or incorporated in them. Just as we (in German) can indicate only the tense and the mood of a verb (gibt, gab, gäbe — "gives," "gave," "would give") by inflection and phonetic shift (i—t, ä), so can the Indian express the gender, number, intention, and detailed manner of the act of killing in one single, inflected word. Whereas we, especially in scientific language, stress only the essential aspects of an observation, shedding all incidentals and compressing the main point into a general concept, primitive man puts as many of the details observed by him as he can into his speech. We would have to reassemble a large number of general concepts in order to express the "death of the rabbit" as the Indian does. The abundant inflectional potentialities of early language and its completely different vocabulary testify to early man's keener observation and to his more intimate involvement with the world. The Lapplander has twenty different words for "ice" and twice as many again for "snow"; he can also describe thawing and freezing by a single word in almost as many various ways. How dull, by contrast, are the word forms of English, for example! The four grammatical cases Mann, Mannes, Manne, Mann which we express in German by inflecting the word, are all simply "man" in English and must be specified by the auxiliary prepositions "of" or "to," as in Chinese. Chinese has no inflections at all and therefore expresses the relationships among words almost exclusively by their position in the sentence. But Chinese is by its very nature an uninflected language, whereas English over the course of time abandoned its inflections, just as inflected languages generally lose their inclination and eventually even their ability to inflect.
It is also worth noting, however, that while Lithuanian, for instance, has different words for the "gray" of geese and of horses, or wool, of human hair, and so forth, it has no separate word for the generic concept of "gray," which is abstract, or "empty," and must be embodied or "filled" by actual, concrete objects. So powerfully does the idea of the unique, the real, thing persist in the mind of early men.
After this brief linguistic discussion, let us now consider the early number sequences.
NUMBERS WITHOUT WORDS
It was related by a missionary to the Abipones, a tribe of South American Indians compelled by a shortage of food to migrate (in the 18th century): "The long train of mounted women was surrounded in front, in the rear, and on both sides by countless numbers of dogs.From their saddles the Indians would look around and inspect them. If so much as a single dog was missing from the huge pack, they would keep calling until all were collected together again. I have often since wondered how they, without knowing how to count, would tell at once, in spite of the confused throng, that one dog was missing." Yet they had only three number words and showed the strongest resistance to learning the number sequence from white men. They would indicate the size of a herd of horses by stating how much space the horses occupied when standing next to each other.
We can understand both these phenomena if we remember the far closer relationship of these people with the world around them: the keen observation that unhesitatingly notes the absence of a single animal and can say which one is missing, and the translation of a number that cannot be visualized into a clearly perceived spatial form.
The term number sense may be applied to the first of these manifestations. Animals show it when they immediately detect the absence of one of their young. Men also have this latent sense and can develop it. Many a teacher distinctly senses the absence of a pupil when he faces a class doing calisthenics.
NUMBERS AS ATTRIBUTES
Here I must ask the reader to focus his attention on a more subtle distinction expressed in language and offering a wealth of insight into the origin of number words.
Numbers as attributes — is a number not, indeed, an attribute, a trait? "Two cows" — "two" precedes the word "cows" just like, for example, the adjective "beautiful." But we must not let ourselves be deceived by this. "Two" is not a characteristic of the cows themselves, for one cow cannot be two; the Two could be at best an attribute of the entity "two-cows." If, however, we regard "two-cows" as a single unit, we of course no longer need to feel the "two-ness" as a particular attribute, since it is part of the essence of the concept "two-cows." Thus we see that Two is not an attribute in the same sense as "beautiful." Hence, Two is not an adjective. What is it, then? It is a special kind of word — a number word.
Nevertheless, primitive man at first always felt the number to be an adjective. We shall now demonstrate that this is so.
The Number Word in the Noun
Some primitive peoples have completely fused the number and the object into a single entity. The Fiji Islanders, for example, call 10 boats bola, 10 coconuts koro and 1000 coconuts saloro. Naturally this does not hold for any arbitrary number (such as 5 nuts or 23 nuts); yet in contrast to other authorities I see in these words a designation of quantity, although admittedly, tied to the object counted. In German it would be like saying ein Malter, a "bushel," or eine Mandel (15), thereby implying potatoes in the first instance and eggs in the second. We have a parallel example in the German word Faden, "thread." One would think that this was something in the nature of a filament, but it is not: It is a measure — "full fathoms five ..." — applied to yarn. It is as much as can be encompassed by a man's outstretched arms. Today a Faden has come to mean a yarn of about this length, and hence is a measure coupled to a specific object.
The examples given show that the primitive people of the Fiji Islands have no number sequence, at least not an extensive one, that has been consciously and clearly detached from objects and thus become abstract.
The Grammatical Double Number (the Dual)
The absorption of the number into the object led to remarkable word forms for the double (the dual), the triple (the trial) and, in some languages of the South Pacific islands, even the quadruple (the quaternal) number. Besides the singular, the German language has only the indefinite plural (the multiple number): der Mann — die Männer. If, to make up a hypothetical example, Manna (in German) were to mean "two men," then this would be a grammatical dual, an inflected word form indicating the specific number two. This embodiment of the number word into the noun itself is reminiscent of the primitive incorporation of all the details of the "rabbit death" (see p. 10) into a single word. Specific grammatical number words, such as "dual," thus belong to an early stage of civilization.
(Continues...)
Excerpted from Number Words and Number Symbols by Karl Menninger, Paul Broneer. Copyright © 1992 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
PrefaceList of Phonetic Symbols
NUMBER SEQUENCE AND NUMBER LANGUAGE
Introduction
The Number Sequence
The Abstract Number Sequence
The Number Sequence Use Concretely
NUMBERS WITHOUT WORDS
NUMBERS AS ATTRIBUTES
NUMBERS AS ADJECTIVES
Expansion of the Number Sequence by Means of Supplementary Quantities
NUMBERING BY FINGERS AND TOES
Principles of the Number Sequence
SUCCESSION AND GROUP
GROUPINGS OF HIGHER RANK
GRADATION BY STEPS
GROUPINGS IN THE GRADUATED NUMBER SEQUENCE
"20-GROUP "MAN"
NUMBER LANGUAGE AND NUMBER SYMBOLS
Historical Gradations
REPRESENTING THE GRADATION
NUMBER SYSTEMS BASED ON GRADATIONS OF 20
VIGESIMAL GRADATIONS IN EUROPEAN NUMBER SEQUENCES
THE AINU NUMBER SEQUENCE
Laws Governing the Formation of Number Words
SUCCESSION OF MAGNITUDES
BACK-COUNTING
OVERCOUNTING
ADDITIONAL WAYS OF FORMING NUMBER WORDS
THE FIRST STEPS BEYOND TEN
Our Number Words
Indo-European Family Languages
SURVEY OF LANGUAGES AND NUMBER WORDS
THE ORIGINAL INDO-EUROPEAN LANGUAGE
THE INDIVIDUAL INDO-EUROPEAN LANGUAGES
The Non-Indo-European Languages
The Meaning of Our Number Words
IMAGES BEHIND THE CONCEPTS OF SIZE AND NUMBER
SEMANTIC FADING
OUR INDO-EUROPEAN NUMBER WORDS
Was There a Babylonian Influence on Our Number Sequence?
THE BREAK AFTER SIXTY
THE ROLE PLAYED BY THE NUMBER SIXTY IN CULTURAL HISTORY
GREAT HUNDRED
THE NUMBER TWELVE AS THE BASIC UNIT OF THE GREAT HUNDRED
THE ROMAN DUODECIMAL FRACTIONS
THE BABYLONIAN SEXAGESIMAL SYSTEM
THE BREAK AFTER SIXTY IN THE GERMANIC NUMBER SEQUENCE
Hidden Number Words
ONE
TWO
THREE
FOUR
FIVE
SIX
SEVEN
NINE
TEN
TWENTY
FORTY
FIFTY
SEVENTY
HUNDRED
THOUSAND
NUMBER
The Evolution of the Number Sequence
FOREIGN LOAN WORDS
WRITTEN NUMERALS AND COMPUTATIONS
Preface
Introduction
Finger Counting
Finger Counting
THE VENERABLE BEDE AND HIS FINGER COUNTING
FINGER COUNTING IN ANTIQUITY
FINGER COUNTING IN ARABIC AND EAST AFRICAN COMMERCE TRADE
ROMAN FINGER COUNTING IN THE WEST
COMPUTATIONS ON THE FINGERS
SOME FORMS OF FINGER COUNTING USE BY PEOPLES OF OTHER CULTURES
Folk Symbols for Numbers
Tally Sticks
EARLY WRITING AND READING
TALLY STICKS WERE UNIVERSAL
VARIETIES OF TALLY STICKS
THE BRITISH EXCHEQUER TALLIES
THE NUMBERS ON THE TALLY STICKS
ROMAN NUMERALS
CHINESE HAN STICKS
Peasant Numerals
Knots Used as Numerals
Alphabetical Numerals
The Gothic Numerals
Letters and Numbers
HISTORY OF THE ALPHABET
The Two Greek Sets of Numerals
Some Other Connections Between Letters and Numbers
"The "German" Roman Numerals"
Roman Numerals in Cursive Form
MEDIEVAL NUMBERS
THE GRADUAL PENETRATION OF THE NEW INDIAN NUMERALS
THE ACCOUNT BOOKS OF THE IMPERIAL FREE CITY OF AUGSBURG
WRITTEN NUMERALS AND COMPUTATIONS
The Abacus
The Nature of the Counting Board
The Counting Board in Ancient Civilizations
THE SALAMIS TABLET
DARIUS VASE
THE ETRUSCAN CAMEO
THE ROMAN HAND ABACUS
THE HAND ABACUS IN ASIA
THE ROMAN COUNTING BOARD WITH LOOSE COUNTERS
The Counting Board in the Early Middle Ages
THE WEST
THE MONASTIC ABACUS
The Counting Board in the Later Middle Ages
THE EVIDENCE FOR ITS USE
THE NEW RECKONING BOARD
THE NAMES OF THE COUNTING BOARDS
"RECKONING "ON THE LINES"
THE COUNTING BOARD IN EVERYDAY LIFE
THE COUNTING BOARD AND WRITTEN NUMERALS
The Counters
Our Own Numerals
Place-Value Notation
The Antecedents of Our Numerals
THE KHAROSTHI NUMERALS
THE BRAHMI NUMERALS
PLACE-VALUE NOTATION
The Westward Migration of the Indian Numerals
THE ZERO
ALEXANDRIA
THE INDIAN NUMERALS IN ARAB HANDS
THE FAMILY TREE OF INDIAN NUMERALS
The Indian Numerals in Western Europe
ITALY
LEONARDO OF PISA
THE GERMAN ARITHMETICIANS
"THE "NEW" NUMERALS"
RETROSPECT
Spoken Numbers and Number Symbols in China and Japan
Far Eastern Number Systems
Spoken Numbers
THE CHINESE NUMBER WORDS
JAPANESE NUMBER WORDS
KOREAN NUMBER WORDS
SUMMARY
WRITTEN NUMERALS
RETROSPECT: SPOKEN LANGUAGE?WRITING?WRITTEN NUMERALS
Index