Enlightening Symbols
A Short History of Mathematical Notation and Its Hidden Powers
By Joseph MazurPRINCETON UNIVERSITY PRESS
Copyright © 2014 Joseph Mazur
All rights reserved.
ISBN: 978-0-691-15463-3
Contents
Introduction, ix,
Definitions, xxi,
Note on the Illustrations, xxiii,
Part 1 Numerals, 1,
1. Curious Beginnings, 3,
2. Certain Ancient Number Systems, 10,
3. Silk and Royal Roads, 26,
4. The Indian Gi, 35,
5. Arrival in Europe, 51,
6. The Arab Gi, 60,
7. Liber Abbaci, 64,
8. Refuting Origins, 73,
Part 2 Algebra, 81,
9. Sans Symbols, 85,
10. Diophantus's Arithmetica, 93,
11. The Great Art, 109,
12. Symbol Infancy, 116,
13. The Timid Symbol, 127,
14. Hierarchies of Dignity, 133,
15. Vowels and Consonants, 141,
16. The Explosion, 150,
17. A Catalogue of Symbols, 160,
18. The Symbol Master, 165,
19. The Last of the Magicians, 169,
Part 3 The Power of Symbols, 177,
20. Rendezvous in the Mind, 179,
21. The Good Symbol, 189,
22. Invisible Gorillas, 192,
23. Mental Pictures, 210,
24. Conclusion, 216,
Appendix A Leibniz's Notation, 221,
Appendix B Newton's Fluxion of xn, 223,
Appendix C Experiment, 224,
Appendix D Visualizing Complex Numbers, 228,
Appendix E Quaternions, 230,
Acknowledgments, 233,
Notes, 235,
Index, 269,
CHAPTER 1Curious Beginnings
No one knows precisely when humans first began to deliberately leave marks for communication with others. Surely it was in that misty period of time, when herds of woolly mammoth freely wandered Europe, and all sorts of living creatures were following the northward spread of food and vegetation from the plains of Africa. The ice of Europe had been receding for centuries in the slow ending of one of the great climate changes of all time. Most of the human population was still in southern Asia.
That was between fifty and thirty thousand years ago, when humans had to think about day-to-day necessities for survival. Deep ontological thoughts—such as where did I come from? and why do I exist at all?—that could be shaped only by strength of language and metaphor potential were not likely. Even without a richly developed language, they must have had our natural urge to tell stories, the impulse to relay to others those pictures in the mind. These may have been fancies about thunderstorms, darkness, beasts, or even the puzzlement of dreams, but such is the nourishment needed to push language further.
As language developed, so did contemplations of the experience of being alive. The twentieth-century preeminent folklorist Joseph Campbell told us that humans have always been "seeking an experience of being alive so that the life experiences that we have on the purely physical plane will have resonances within, that are those of our own innermost being and reality, so that we actually feel the rapture of being alive."
Humans could have survived in their severe, menacing environments by some combination of instinct and intelligence, just as most mammals did and still do, without the spark exploding natural human language. They could have survived the freezing winters and scorching summers in an oral world without a written record, in a world without marks, signs, symbols, or paintings. Monkeys did; so did caribou.
What possessed those Neolithic cave painters to ignore the dangers of daily life while they sat, etched, scribbled, or painted? More than 40,000 years ago, the dwellers near the Cave of El Castillo in Spain bothered to stencil their hands against cave walls by blowing pigment. For tens of thousands of years, humans had been leaving signification marks in their surroundings, gouges on trees, footprints in hard mud, scratches in skin, and even pigments on rocks.
A simple mark can represent a thought, indicate a plan, or record a historical event. Yet the most significant thing about human language and writing is that speakers and writers can produce a virtually infinite set of sounds, declarations, notions, and ideas from a finite set of marks and characters. Animals may have their languages, but they cannot produce an infinite number of communication signifiers from a finite number of sounds and gestures.
From pigment-sketched mammoths on rocks to alphabets, writing developed through transitional stages. Pictures were clues to picture writings, which in turn were clues to ideograms, and so on through modifications, all the way to early metaphorical poetry and modern writing. A "pictogram" is a picture that resembles what it intends to mean. In Asia, such writing became the foundation for modern Chinese hanzi and Japanese kanji. In today's world, a picture of a knife and fork might represent a restaurant. A slash across the knife and fork would be an "ideogram": it might signify no eating allowed. Whereas a pictogram depicts objects, an ideogram expresses meaning through similarity or analogy. To signify the word "home" by ideogram, for example, in early domestic China, you would combine the pictogram for "roof" with the pictogram for "pig" to make the word "home." For at least thirty thousand years, stories have been told through pictures, and as the years progressed, the stories became more elaborate.
Some years ago, a friend returned home from Thailand and gave me a gift of a Hmong embroidered "story cloth" that was bought from a weaver living in a refugee camp. It depicted the story of daily life during the Vietnam War. From it one could "read" the cycle of life. There is the birth of a child, work in the fields, falling in love, a wedding, and a new birth—a whole story without a written word.
Pictograms are easy to understand in a simple world of simple stories. The problem comes when the storytelling is more complex. Imagine the Odyssey "written" in pictogram characters. Who would fully understand it? It would be too elaborate and too laborious to absorb, and quite possibly too inflexible for the metaphoric complexities of serious poetry. Far better is to have the characters represent the phonemes of speech, so one utterance is distinguished from another—a for "ah," b for "be," and so on.
It's one thing to have words and quite another to think about the words themselves. Writing sentences is altogether different from talking; it must have come after a great deal of social growth, after the first civilizations, after kings and emperors, and long after adventurous tribesmen started wandering beyond the familiar for adventure and trade.
If you ask a person in the street what he or she thinks is the most important invention in the history of civilization, you are likely to be given the proverbial answer: the wheel. Surprisingly, the wheel didn't come into existence before the late Neolithic Age and possibly as late as the early Bronze Age. That would put it somewhere between 6,000 and 3,500 BC. The earliest depiction of a wagon with wheels can be seen on a ceramic pot that was excavated in Bronocice, Poland, in 1976. The Bronocice pot dates back to ca. 3500–3350 BC. But with the new agriculture of that period, the wheel should have been an obvious invention—after all there was circular pottery, and slices of tree trunk must have given good clues to the enormous utility advantages of a rolling disk. Surely rolling logs were used before the true simplicity of the wheel. But the wheel is not just a rolling disk. It involves the relatively complex concept of wheel and axle, combined.
What about the alphabet? It is surely a contender. I would argue that most of the other significant inventions that have made our lives possible could not effectively exist without the alphabet, or at least some other clever way of writing the words we speak. True, that person on the street might argue that the great pyramids of Egypt could not have been built without the wheel in the form of rolling logs to help the slaves, and that the tall stone buildings of the world could not have been built without the wheel and axle. The wheel would have come to the world sooner or later, but some form of writing the sounds we make trumps all.
Modern alphabetic writing is a rough mimic of spoken language. Before any evidence of an alphabet, there was Sumerian picture writing, where each syllable of the Sumerian language was a distinct picture impressed by a wedge stylus in clay. Originally, the impressions were meant to be pictures of objects with the same syllabic sound of the word that was to be conveyed. This was a different sort of writing than that of mere pictographs. Fortunately, spoken Sumerian was a language of words made from many syllables, and o en the syllables themselves were the names of concrete objects. The writing consisted of marks, each denoting a syllable. For example, a picture of a house being held in a hand might signify "household."
Hieroglyphic picture writing was used in the Mediterranean area around Egypt at roughly the same time as Sumerian picture writing, which went through several transitional stages before slipping away from its pictorial character to evolve into a pure sound-sign system, and eventually something alphabetic. By the time the Phoenician alphabet was introduced, sometime before the first millennium BC, numerous cultures in almost every part of the world had developed some form of representational writing using pictorial symbols. This gave those cultures the means of immediate communication and a means of leaving a record of knowledge for future generations as well.
Unlike the phonetic writing that we have today, in which the symbols of each word represents the sounds of the spoken words, pictorial writing was an indicator of the meaning of the spoken word, not the sound. By the middle of the first millennium BC, however, pictorial writing was replaced by phonetics.
Pictures can be used to represent words through their sounds. In English, for example, you could write "I believe" by juxtaposing the pictures of an eye, a bee, and a leaf. Meaning in hieroglyphics was represented through context, just as it is in phonetic writing. Phonetic writing, however, has at least one important advantage over pictorial: it can express far more combinations of thoughts and ideas. One might also argue that writers can work in a much freer playing field to invent richer metaphors.
It is not surprising that the need to write came from the need to record memories, not stories. The earliest documents are of accounts, names, recipes, and itineraries. As the skill of writing spread, so did the reasons. One can imagine graffiti on public buildings, secret notes and magic formulas passed to other people, writings to help one's memory, or epitaphs for one's tombstone. Such memories and epitaphs "call men and women to a deeper awareness of the very act of living itself, and they guide us through trials and traumas from birth to death."
At first, writing was limited to the initiated, mostly the priestly sects or special classes who were trained; once it settled to some standards, however, its power had profound effects on spoken language. Educated peoples from distant lands and roughly similar languages were soon able to share a common written language, thus fixing the verbal traditions and creating a common bond of experiences between foreign lands and separated times.
The beginnings of civilizations and cities coincide strikingly with the construction of temples and the rise of priestly classes, which attracted bright recruits from the common populace. Primitive agrarian life slowly included a temple life with priest kings who built their empires. This may have been a result of the growth of agrarian cultures, which depended on calendars that were understood by the priests and held by the temples for seasonal rituals. Thus priests, human representatives to the gods, governed the earliest civilizations. Their temples were observatories, libraries, clinics, museums, and treasure houses. Though the Babylonians had relatively extensive star catalogues by 1200 BC, it was the Egyptian priests who—believing the sky divine—mapped out the stars and constellations as early as 3000 BC. The complexities of star map calculations, along with land surveys and taxes, required writing numbers beyond the simple low numbers that were useful in accounting for sheep in the fields.
Primitive humans had simple needs. At first, counting was limited to very low numbers. The shepherd could know that a sheep was missing from the flock without needing to count. Any ape could do that—that is, know that a member of the family is missing. To know that something is missing is a qualitative, rather than quantitative, notion of sets. Facets of primitive life didn't require any real sense of number. No one needed to know what number is.
Yet still, for some wonderful reason that seems almost inexplicable, humans—even primitive humans—have always had an uncanny ability to recognize numbers beyond the values for which they had words. Children today are taught to recite numbers in preschool to get a sense of the words associated with quantity. They can easily recite the numbers from 1 to 10. Reciting numbers, however, is not the same as understanding what those numbers actually mean. A three-year-old may be able to count to 5 without understanding the one-to-one correspondence between the words "one," "two," "three," "four," "five," and the five fingers on one hand. That correspondence, whenever it occurs in child or human development, is a gargantuan leap of cerebral maturity. We don't notice the moment of that leap. There doesn't seem to be any "aha!" experience at that moment. Having five fingers on each hand does not seem to naturally suggest a one-to-one correspondence with the first ten numbers. Until the middle of the last century, several aboriginal tribes in Australia had no words for numbers, but could count by making marks in the sand. Curiously, there were—at least before the last century—several indigenous tribes of Australia, the Pacific islands, and the Americas that had no words for numbers beyond four, suggesting that the modern concept of numbers as one-to-one counters had not yet matured.
In both the East and the West, mathematical writing predates literature by more than a thousand years. It even predates the oldest surviving written story, The Epic of Gilgamesh, a Sumerian poem that was written more than a thousand years before the Iliad. We have no direct evidence as to where or when numerical writing first occurred, just as we have no direct evidence as to where or when writing first began to develop. Some would attribute the earliest concepts of numerical writing to the Chinese, as far back as the Early Stone Age. That seems doubtful. But it appears reasonable that it coincides more or less with cuneiform Sumerian number writing dating back to 3400 BC.
Like the art found in the caves of southern France and northwestern Spain, number writing came about through the human endeavor to record. One of the world's oldest extant written records (German Archeological Institute Museum number W 19408,76+) seems to be an exercise in calculating the areas of two fields, written sometime in the late fourth millennium BC. It is a collection of fragmented clay tablets found among the reused building rubble of the city of Uruk. Its carbon date (ca. 3350–3200 BC) predates any known evidence of writing, at least of writing that we agree is phonetically associated with a spoken language.
Traces of Sumerian number writing on clay tablets with numbers as large as 10,000 have been found in caves from Europe to Asia. Egyptian hieroglyphics had a distinct symbol for the number 10,000. By 1600 BC, the algebra problems in the famous Rhind (or Ahmes) papyrus presented simple equations without any symbols other than those used to indicate numbers.
CHAPTER 2Certain Ancient Number Systems
Call them what you wish—Babylonians, Sumerians, or Akkadians. We have heard their stories before. Almost every history of early Western mathematics begins with the Babylonian conception of number, a so-called sexagesimal (base 60) system for writing large numbers, formulations of multiplication tables, and ideas for astronomy. But who were those Babylonians, and why were they the ones to first come up with human civilization, culture, art, and science?
To answer, examine the geographical region of the Fertile Crescent, that crescent-shaped region between the Eastern Mediterranean and the Persian Gulf, and running through southeastern Turkey to Upper Egypt. It happens to be a unique area responsible for the spread of wild emmer wheat, wild einkorn, and wild barley, and therefore an exceedingly favorable area for the birth of local agriculture. Within the Fertile Crescent lies an area near the Tigris-Euphrates valley. Generally, the term "Babylonian" refers to things related to far more than just the city of Babylon, and essentially to a wide geographic area that today includes southern Iraq, Kuwait, and parts of western Iran. It is an area near and between two great rivers that converge close to modern Baghdad, then diverge and zigzag until they meet at Al Basrah in southern Iraq, just north of Kuwait, before pouring into the Persian Gulf. If you look at a map of these great rivers, you cannot avoid being impressed by their meanderings. The Tigris wanders south of Baghdad as if it were a water snake that cannot make up its mind whether to go southwest or northeast. In some places—near Suwayrah, for instance (figure 2.1)—it can take two hours to navigate the Tigris by boat only to find that a ten-minute walk over land will bring you to the same spot. In other places, a half-hour walk will bring you to the same spot it would take a boat six hours to reach. This means that the land between two relatively long lengths of the river may be easily irrigated. Even today much of the banks of the Tigris is undeveloped farmland. There are few long sharply twisting rivers in the western world. Rivers generally go places from high elevation to lower. There are rivers in northern Europe that have sharp meanderings—for instance, the thousand-kilometer Elbe—but northern climates were not terribly welcoming to winter crops. Though the Tigris-Euphrates valley terrain was not ideal for farming, the great rivers, with their many tributaries and canals running slowly through, were outstanding for irrigation. Small villages grew along the rivers that cut through the moderately flat countryside south of Baghdad to collectively become the first urban centers in the West. Many of the ancient tributaries and canals along the alluvial plain south of Baghdad that were around when the first settlements carpeted the region are now gone, dry.
(Continues...)Excerpted from Enlightening Symbols by Joseph Mazur. Copyright © 2014 Joseph Mazur. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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. 
