There are entries relevant to this chapter in the blog for these lecture notes
In our last lecture we studied the development of place-value number systems and how the concept of zero was essential to make them functional. We saw that fully developed place-value number systems existed in China since about 200 BC and in Central America since about 400 AD. The Chinese number system is still in use today; the Mayan number system was wiped out during the Spanish conquest.
This lecture addresses the question: When was a place-value number system introduced in Europe?
We already know that Babylonian scientists had developed a place-value number system around 1800 BC and had added the "space" symbol for the zero in about 400 BC. We also noted that this effort to save the first place-value number system did not overcome its other problems and that the rise of Alexandria spelled the end of the Babylonian number system and its cuneiform numbers.
The important role of Alexandria in the development of science will be the topic of Lecture 10. It is remarkable that the rise of a civilization as advanced as Alexandria also meant the end of a place-value number system in Europe for nearly 2000 years. Neither Egypt nor Greece or Rome had a place-value number system, and throughout medieval times Europe used the absolute value number system of Rome. This held the development of mathematics in Europe back and meant that before the period of Enlightenment of the 17th century the great mathematical discoveries in the discipline of arithmetic were made in East Asia and in Central America.
When we talk about the numerals of today's decimal number system we usually refer to them as "Arabian numbers." Their origin, however, is in India, where they were first published in the Lokavibhaga on the 28th of August 458 AD. Many changes had occurred in India since the rise and decline of the Indus Civilization. For several hundred years life had returned to small villages, but a second period of urbanization had developed at about 1500 BC. Shortly before that time the Aryans, a nomadic people, had entered India from the Iranian region. They introduced cattle breeding into the fertile river valleys and established a new civilization in the Ganges River valley.
The arrival of the Aryans coincided with the introduction of Vedic, an early form of Sanskrit and the first Indian script. The earliest examples of Indian literature, the Vedas, originate from this time, and the Ganges civilization is therefore often called the Vedic period. It lasted until about 500 BC.
The first empire that comprised nearly all of the Indian subcontinent and unified both the Indus and Ganges River valleys was established by the emperor Candra Gupta between 321 and 297 BC. Another 600 years later the so-called Gupta Period (c. 320-540) established the characteristics of the modern Indian civilization in the areas of literature, art, architecture, and philosophy.
The Indian numerals are elements of Sanskrit and existed in several variants well before their formal publication during the late Gupta Period. In contrast to all earlier number systems the Indian numerals did not relate to fingers, pebbles, sticks or other physical objects. Originally the numerals 1, 2 and 3 could still be recognised as sticks. They first appeared in 300 BC. 4 and 5 existed in two alternative forms. By 100 AD the numerals 1 to 9 had evolved to
The close relationship with our own numerals is obvious, particularly for the numerals 6, 7, 8 and 9. A dot or small circle for zero was added at a later stage, certainly before about 700 AD, as can be seen in the work of the Indian buddhist scholar Gautama Siddharta, who lived in China and worked at the astronomical college in the capital of the T'ang dynasty. In his collection of astronomical and astrological texts Khai-yuan Chan Ching, which he published under his Chinese name Chhütan Hsi-ta between 718 and 729 AD, he wrote: "When one of the nine numerals reaches ten it is placed in a field in front of the other numerals, and a dot is inserted every time an empty field occurs in the row, to indicate it symbolically." (Needham, 1959)
The great intellectual achievement of the Indian number system can be appreciated when it is recognized what it means to abandon the representation of numbers through physical objects. It indicates that Indian priest-scientists thought of numbers as an intellectual concept, something abstract rather than concrete. This is a prerequisite for progress in mathematics and science in general, because the introduction of irrational numbers such as π ("pi", the number needed to calculate the area inside a circle) or the use of imaginary numbers is impossible unless the link between numbers and physical objects is broken.
Because every Indian numeral is represented by a single symbol there is no need for an auxiliary base. But the need arises what to call them. The way Indian numerals were pronounced adds to the evidence that the Indian mathematicians did not associate numbers with pebbles or sticks. Each numeral was pronounced through association with words that did not relate to quantities and were written totally differently. The number 3710, for example, would be "spelled" or pronounced "fire-mountains-moon-sky".
It is likely that the association of 3 with fire, 7 with mountains, 1 with moon and 0 with sky had religious roots, but it does not have to be taken literally in the sense that Indians would think of mountains when they see the numeral 7. In the English language the word "Saturday" may trigger thoughts of "weekend", but the association with Saturn as a Roman god or a planet does not usually enter our consciousness. In the same way an Indian mathematician did not consciously think "mountains" when he said "fire-mountains-moon-sky" to indicate the number 3710.
Indian mathematicians used their revolutionary number system to advance human knowledge at great speed. The Sthananga Sutra, a religious work from the second century AD, contains detailed operations that involve logarithms to the base 2. Modern texts credit the discovery of logarithms to the Scottish mathematician John Napier, who published his discovery in 1614. Indian knowledge of logarithms thus precedes Napier's discovery by more than 1000 years.
Harmonic functions (sine and cosine) were known in India before the 7th century AD. The Brahmasphutasiddhanta ("The Opening of the Universe"), a work of the priest-mathematician Brahmagupta written in 628, contains the interpolation formula to compute values of sines.
With the expansion of the Muslim empires during the 7th and 8th centuries news of the ingenious Indian number system came to the Middle East. Muslim scholars used it to advance the classical Greek mathematical knowledge in several famous texts. Their contribution to the development of modern mathematics lives on in the words algorithm and algebra:
Christian monks heard about the new number system and adopted it from its use in Arabic words. Because at that time Latin was the official written language in Europe, the first occurrence of "Arabian numbers" is in Latin manuscripts:
We recognize again the numerals 6, 7, 8 and 9 and also some of the other numerals, which appear upside down. In other manuscripts some numerals are on their sides - unlike Arabian manuscripts, which always showed the numerals correctly, medieval Latin manuscripts could not agree on their correct orientation.
Arabian scholars were always prepared to give Indian scientists credit for their number system. An early Arabian work states that "we also inherited a treatise on calculation with numbers from the sciences of India, which Abu Djafar Mohammed Ibn Musa al-Charismi developed further. It is the most comprehensive, most practical, most comprehensible method and requires the least effort to learn; it testifies for the thorough intellect of the Indians, their creative talent, their superior ability to discriminate and their inventiveness." (Woepcke, 1863).
The same attitude was not shared by some Greek scholars, who regarded Greek science as the only source of wisdom. Their attitude caused bishop Severus Sebokt of Syria in 662 to claim the superiority of Syrian science. But he, too, did not negate the achievements of India: "I shall not speak here of the science of the Hindu, who are not even Syrians, and not of their subtle discoveries in astronomy that are more inventive than those of the Greeks and of the Babylonians; not of their eloquent ways of counting nor of their art of calculation, which cannot be described in words - I only want to mention those calculations that are done with nine numerals." (Nau, 1910) It is indeed remarkable that the Indian number system was known in Syria in the 7th century already but did not come into use in Europe until centuries later.
There is plenty of evidence that the Arabian mathematicians used the system inherited from India to make significant progress in mathematics. They derived the binomial coefficients nearly 100 years before China's mathematicians published them in 1303. The introduction to al-Khwarizmi's master piece al-jabr Wa'l mukabala proovides another example how science develops in resp[onse to society's needs:
"The imam and emir of the believers, al-Ma'mun, encouraged me to write a concise work on the calculations al-jabr and al-muqabala, confined to a pleasant and interesting art of calculation, which people constantly have need of for their inheritances, their wills, their judgements and their transactions, and in all the things they have to do together, notably, the measurement of land, the digging of canals, geometry and other things of that kind." (Benoît and Micheau, 1995)
The fact that the European monks depicted Indian numerals in a variety of orientations is clear evidence that they did not understand the usefulness of place-value number systems. Calculations in Europe were made on calculation boards. Among the first uses of the Indian system in Europe was the introduction of Indian numerals for checker board calculations by Gerbert of Aurillac, who became pope Sylvester II in 999. When he encountered Indian numerals in Arabic manuscripts held in a Spanish monastery he introduced round tokens with Indian numerals to his calculation board.
Using numbered tokens defeats the purpose of the board: Instead of shifting marbles around very quickly, tokens now have to be removed and replaced during calculations, which slows down the operation. Because the tokens can be placed in any orientation, the correct orientation of the Indian numerals was soon forgotten, and in written manuscripts they could appear in any orientation.
Use of the calculation board and of the abacus coexisted with the Indian number system for centuries. Because most people in medieval Europe were illiterate and the Indian calculation method requires the writing down of numbers, the abacus remained the preferred tool in commerce and administration. Science, on the other hand, adopted the Indian place-value number system early.
As already remarked in the previous lecture, the parallel use of competing systems for calculation and measurement is not an unusual occurrence. The use of the Fahrenheit temperature scale by the public of the USA and the Celsius temperature scale by the scientists of the USA is another current example. Scientists like Copernicus, Brahe and Kepler would have understood the superiority of the Indian number system over the Roman numbers and used it for their detailed observations and calculations. Medieval publications demonstrate the use of the Indian method parallel to the use of the abacus and calculation boards during their time.
The language of medieval European science was Latin. As Latin was no longer spoken by the general public it underwent no further development during the centuries. In contrast, Arabic science was written in a living language, and the writing of its numbers evolved with the Arabic alphabet. This may explain why the "Arabian numbers" used in the European civilization of today show much more similarity to their Sanskrit original than the same numbers in modern Arabic:
When James Cook in 1776 planned the voyage that brought him to Australia, the financial commitment was comparable to the commitment made by the USA and the USSA to get a man to the moon. Yet the Colonial Office prepared his budget with tokens on a checker board. A reminder of the practice is retained in the title of the British minister of finance as the "Chancellor of the Exchequer". The use of the abacus or calculation board for administrative purposes continued in Europe until 1791, when the French National Assembly, which was set up through the French Revolution two years earlier, adopted the Indian calculation method for France and banned the use of the abacus from schools and government offices. Government offices in England continued to calculate taxes on calculation boards for another decade.
A study of the history of numbers is an exploration of human ingenuity. Humans recognize order in the natural world. Numbers give structure to that order. Several civilizations developed number systems in different continents. Some discovered the place-value number system; the Indians perfected it.
What does the history of numbers tell us about the human mind? Great inventions have often been used to value one civilization more than others. It is then only a small step from ranking different civilizations to ranking groups and individuals and claiming that some races are superior to others.
The history of numbers shows us that racism has no basis in fact and is the attitude of misguided minds. It is doubtful, and indeed most unlikely with the lowest of probabilities, that any of us would be capable of inventing a place-value number system or of developing the concept of zero. We all start out in life counting on our fingers; but once we are told how to use a place-value number system we have no problem using it from an early age.
Take the example of the Papua New Guinea highlands, where an old society embraces the modern world. Until about 1930 the people of the highlands had been living in the late stone age without contact to any civilization. They cultivated their gardens with stone tools, used bows and arrows for warfare. In the 1930s, when the first white explorers appeared, a typical man from the highlands spoke at least three languages (the language of his village and the languages of two neighbouring tribes) and counted on his fingers. Today his children speak five languages: the language of their village, the languages of two neighbouring tribes, Pidgin (the lingua franca of modern Papua New Guinea) and English; his daughter uses the Indian number system to run a coffee plantation, his son runs a garage and uses the metric system to look after his clients' cars.
Indigenous Americans provide another example of an ancient society taking up ideas within one generation. Before the arrival of the Europeans, American Indians had no need for a script; all their instruction was given and preserved through practical demonstration and oral tradition. When the need arose to deal with the invaders, the usefulness of a script became obvious as the settlers could exchange letters, demand signatures under treaties and produce bills for payment. Sequoia, a Cherokee, developed a script for the Cherokee language that was so easy to comprehend that within a decade most Cherokees could read and write. In 1828 the Cherokee nation established its own printing press and newspaper, the Cherokee Phoenix and Indian Advocate, that developed into an organ of all native Americans. (Ballantine, 1993) The Cherokee established their own nation, adopted a constitution and established a republic in Oklahoma. So successful were they in taking up the ideas of freedom and independence, ideas the invaders themselves had promulgated during the War of Independence of 1775 - 1783, that the settlers burnt the Cherokee printing house to the gound and forced the closure of the Cherokee Phoenix. (As a true phoenix it rose from the ashes a century later and continues to be published today.)
These examples show that ideas, inventions and discoveries made millennia ago can be taken up in the time span of one generation, as soon as the information about them becomes available. The capacity of the human brain to make use of the ideas and inventions of others is indeed unlimited, but the development of peoples' capabilities depends on access to other peoples' ideas. It follows that racism is nothing but prejudice; its ideas are not supported by facts.
The history of numbers provides support for our thesis that science develops only in response to the needs of society: When people had no need to count, numbers did not exist. The need for numbers arose with the development of private property.
This completes our survey of the development of the modern number system. Was it sensible to spend four lectures on something so elementary? It is difficult to compare later achievements of science with the development of our number system, but the amount of ingenuity that went into it was certainly not less than the amount of ingenuity that went into the development of nuclear physics, relativity or the mapping of the genome. The question remains: Why did people invent number systems? Even the richest king would not have been able to amass a herd of cattle of such size. The answer becomes obvious when the next lecture turns to another intriguing science problem, the calendar.
Ballantine, B. and I., eds. (1993) The Native Americans. Turner Publishing, Atlanta.
Benoît, P. and F. Micheau (1995) The Arab Intermediary. In: M. Serres (editor): A History of Scientific Thought, Elements of a History of Science. Blackwell, Oxford, 191 - 221. (Translation of Éléments d'Histoire des Sciences, Bordas, Paris, 1989)
Nau, F. (1910) Notes d'astronomie indienne. Journal Asiatique 10 Ser. 16, 209 - 228.
Needham, J. (1959) Science and civilization in China vol. 3: Mathematics and the Sciences of the Heavens and the Earth. Cambridge. (quoted after Ifrah, G. (2000) Universal History of Numbers John Wiley & Sons, New York.)
Woepcke, F. (1863) Mémoires sur la propagation des chiffres indiens. Journal Asiatique 6 Ser. 1, 234 - 290 and 442 - 529.