Globalization and A Mathematical Journey

Eiko Tyler
Chaminade University, Hawaii, USA

In this paper I will explain how Islamic mathematicians solved cubic equations by using geometry. Similar achievements and other examples of brilliant Islamic mathematical and scientific works were later introduced to Europe. During the Dark Ages in Europe, a substantial amount of the brilliant work of the Greek and other European mathematicians and scientists were lost. However, with the introduction to Europe of the work of the Islamic mathematicians and scientists, ideas that were developed by peoples who embrace different religions, were thus merged. The mathematics and sciences of different civilizations are, in fact, dependent of each other. Ideas developed in one part of the world, inevitably influence scientific thinking in other parts of the world.

This paper intends to show that the concept of zero connects Buddhism, Islam and Christianity. The early globalization that occurred in the 13^th century did not last long. In the 21^st century, we have to learn from the past and realize that we are all interconnected and that we must maintain an open interfaith dialogue between and among different religions to cultivate a global culture whose aim should be the prevention of violence and advocacy of world peace.

By showing the interrelationship between various cultures and religions through mathematics, this paper will attempt to promote the cause of peace by promoting mathematics education along with hope, love, compassion, and social justice for all the people that share the planet.

Through an explanation of the concept of "sunyata", I will attempt

to demonstrate how the concept of zero and other aspects of Indian mathematics influenced Islamic mathematicians who, in turn, influenced European mathematics. I will show this inter-relationship in the context of the history of Silk Road.

The Silk Road generally refers to a historical East-West trade route that connected China, Western Asia, and the Adriatic Sea region. The eastern end of the Silk Road was established in the 2^nd century BC during The Han Dynasty (206BC-AD 220). The Silk Road established not only economic relations between Eastern and Western civilizations, but it also acted as a vehicle for cross-cultural exchanges between these regions. [SJS] The Silk Road consists of three routes; the Steppe route in the north (at about 50 degrees north), the Sea route in the south, and the Oasis route in central Asia, connecting scattering oasis’s skirting the Taklamakan Desert (at around 40 degrees north). The north route crossed Kyrgyzstan and Uzbekistan between the cities of Tashkent and Samarqand.

The name Silk Road came from the fact that the silk produced in China was transported to India Western Asia, and even to Rome. A German geographer, Ferdinand von Richtofen named the route "Seidenstrasse" in 1870. Later, Swedish archeologist Sven Hedin also used the term "Silk Road". In addition to silk, other goods such as spices, metals, and fur were transported along this route.

The western end of the trade route started to develop after the Iranian empire of Persia was conquered and colonized by Alexander the Great in about 330 BC. Thus the Greek language, mythology, and arts were brought into the south of the Hindu Kush and Karakorum ranges. This region became a crossroad of Asia where the cultures of Persian, Indian and Greek merged.

In the 2^nd century BC, Emperor Wudi of the Han Dynasty sent the general Zhang Quian to recruit the Yuezhi people from the Northern borders of the Taklamakan Desert. The Yuezhi settled in Northern India after the Xiongnu tribe took their traditional homeland in the 2^nd century BC. In the 1^st century AD the descendents of the Yuezhi people, who were Buddhists and called Kushan, moved into this crossroads area. The Kushan people generally adapted to the Greek culture that existed in the area and produced the interesting sub-culture called Gandhara. The Gandhara culture fused Greek and Buddhist art into a unique style. Before the Kushan people no one had expressed the Buddha in human form. The sculptures of Buddha, Tathagata (a person who has attained Buddhahood) and Bodhisattva (a Buddhist saint) have strong remembrances to the Greek mythological figures. By the 1^st century BC, the Ghandhara civilization was well linked to the trading centers of Khotan where economic, cultural, and religious exchange between east and west took place. One of the ancient provinces of Khotan is called Turkistan, which was situated along the east-west passage connecting China with Kushan and Gandharai in Afghanistan and Pakistan. The Greco-Buddhist Gandharan culture was later propagated to Japan through Turkistan and Khotan. [ATH]

In 125 BC, Zhang Quian returned to China with information about unknown states to the west, and exciting news of seeing larger horses, which could be trained for use by the Han cavalry. In addition to many objects that Zhang Quian and his man brought back from the crossroad regions, they also brought a faith known as Buddhism. The Han people were especially appreciative of the religious artwork from Gandhara. An expedition under the leadership of Zhang Quian, opened the route to the west. For this reason, Zhang Quian is often referred to as the "father of Silk Road". [ATH]

During the Tang Dynasty (AD618-907) the network of routes that made up the Silk Road thrived. In the 10^th century, the success and influence of Qur'an began to decline and the once flourishing Buddhist temples were replaced by Islamic places of worship. The success of the land route known as the Silk Road began to decline as well.

From the 11^th century to the 12^th century, various trade goods as well as aspects of culture and religion were exchanged between the East and West along the four routes, which connected the Chin dynasty (AD1115-1234) in China, the Karakitai Dynasty in Western Liao, Khorezm kingdom in the central Asia and Rome. A monetary system using silver and paper currency was established. The prosperous commerce that developed along the northern route of the Silk Road encourages the establishment of the South Sea route that also became prosperous at about this same time.

Established about 1270, the ocean route of the Silk Road became an increasingly important trading route. From the East, trading ships departed the port of Quanzhou, China sailing to India via the Strait of Malacca. From the West trading ships traveled from Basra and Hormuz to India. Within twenty years, the ocean trafficking had developed into one of worldwide scale and heavy cargo such as china was transported in mass quantity. Marco Polo visited Khanbalik (Beijing) to attend the court of Kubili Khan. He traveled along the land route of the Silk Road, passing through Turkistan in 1271, during the Yuan Dynasty. In 1292, Marco Polo returned by the ocean route to Venice from Quanzhou, China.[ATH]

From the latter half of the 13^th century to the 14^th century, the Eurasian continent, from the Adriatic Sea in the west to the Japan Sea in the east, was recognized as one worldwide region and an exchange of cultures, and institutions occurred, which is unprecedented in history. Thus world trading began to occur in the Eurasian world. Through the trades routes of the Silk Road, the Islamic religion was spread not only along northern Silk Road from Central Asia to the Western China, but also along the southern sea route to various island nations in Asia. Through this early example of globalization, Islam had contact with Christianity in the west, Hinduism in the south and Buddhism and Confucianism in the east. As a result of the early form of globalization, the development of Islamic civilization surpassed any other civilization in the world. [SAK]

The Silk Road witnessed the rise and fall of nations and the progress of civilization. When the Ming Dynasty (AD 1368-1644) closed China to outsiders, it effectively ended the centuries-old Silk Road connection, which had connected Imperial China with Imperial Rome.

The definition of a natural number is given from Peano's Axiom (1858-1932), which contains five propositions as following, [YOS]

Let N be a set with the following properties:

           III If (N) does not contain 1.

           V If f is a subset N`of N has the following properties (i) and (ii),

In the 5^th century BC, Panini (who is called the father of computing machine) formulated the Sanskrit grammar rules and wrote the Astadhyayi. In his book metarules, transformations and recursions are discussed. The original purpose of this book was to systematize the grammar of Sanskrit. In 300BC, The Sanskrit word ‘sunya" was used for referring the concept of ‘void".

In 300BC, the Brahmi numerals came into being in India. These Brahmi numerals were symbols for the numbers from 1 to 9. However, the Brahmi numeral did not include the concept of the place-value of numbers system Various symbols originated from the Brahmi numerals started to develop and in the 11th century, Islamic mathematician, al-Birumi who wrote 27 works rgarding India wrote: [SAH]

The Brahmi one, two, three and four are written as follows:

About 150 BC, the Jaina mathematicians in India wrote the "Sthananga Sutra", which contains theory of numbers, arithmetic operations, geometry, quadratic equations, and cubic equations, and permutations and combinations.

During the 3^rd to 1^st century BC, Pingala wrote the treatise of a prosody called "Chhandah-shasfra". It contains the first use of zero as a digit, which was indicated by a dot, the Fibonacci numbers (called maatraameru), and Pascal’s triangle. The notation by dots for zero was used as follows: [CAJ]

Sanskrit numerals were also created around this time. Between the 4^th century AD to the 6^th century AD, the Gupta numerals were developed from Brahmi numerals. In 628 AD, an Indian mathematician, Brahmagupta, wrote the Brahma-Sphuta-Siddhanta wherein the concept of zero is explained. Zero was added as a tenth positional digit in his work. [UTK]

Around the 7^th century, Gupta numerals changed into Nagari numerals, which contained zero as a number. The Nagari numeral is sometimes called the Devangari numerals, which means the "writing of the gods". The Nagari numerals were then introduced to the Islamic people. The words for the Devangari numerals in Sanskrit are given as follows: [MAN]

1. sunya
2. 2eka
3. dva
4. tri
5. catur
6. pancan
7. sas
8. saptan
9. astan
10. navan

The positional number system we use today was developed in India. Some may wonder why this number system was not developed in the West. There are theories to explain why the establishment of a positional number system occurred in India. Among them, three of the noteworthy conjectures are as follows:[SAH]

Names of large numbers are found in the Buddhism sutra called Buddha-avatamsaka-nama-maha-vaipulya-sutra. In this voluminous sutra, which forces the practices of a bodhisatova, names of large numbers are mentioned in section 30 of Asamkhya (innumerable) of volume 45. [ASO]

大方廣佛華嚴經卷第四十五

　　阿僧祇品第三十

The Hindu Arabic numerals for the first 19 units and the last 7units in section 30 of Asamkhya (innumerable) of volume 45 are listed below: [AIR]

The concept of zero may have been facilitated by the Buddhism concept of Sunyata. Sunyata is often translated as emptiness, void, or nothingness. However the Buddhist concept of sunyata is relativity. Sunyata does not deny the concept of existence as such, but holds that all existence and the constituent elements, which make up existence, are dependent upon causation. In Hinayama, the concept of sunyata primarily indicates the impossibility of having an independent atman (self) but Mahayana goes one step further, in denying the possibility of a self-existing nature within the dharma, which make up the material world. All things (dharmas) are sunyata, i.e., relative, and hence dependent. The teaching of Buddha explains sunyata as following: [TEA]

The early contributions of Islamic mathematicians affected all branches of western mathematics. The word ‘algebra" first used in Arabia in the 9^th century AD. Around the middle of this century, Baghdad became the center of world culture and learning under a Moslem caliphate.

Although the concept of zero was developed in India, the name "zero" derives from the Arabic word "sifr". Zero was discovered in India in the 7^th century and it was used as both a placeholder and as a number between positive numbers and negative numbers. With this event, the modern place-value numeral system was fully developed. This development was recorded in the Brahma-sphuta-siddanta.

Meanwhile in the early 7^th century, the beginning of Islamic Empire was about to occur. In the Arabian Desert, Muhammad established the religion of Islam. He escaped from his hometown of Mecca to Media in 622 AD. Later, that date became the first date of the Muslim calendar year. After he returned to Mecca after 8 years of exile, the religion of Islam had begun to spread rapidly to the north of the Arabian peninsula to areas such as Syria, Iraq, Egypt, to all of north Africa, to Persia and even to the borders of India. Eventually, the religion of Islam spread to the borders of China in the east and to Spain in the west. In twenty years, Muhammad successfully achieved the conversion all of Arabia to the ideas embraced in the Qur’an. Within a hundred years, the religion of Islam spread to the borders of India, to North Africa and even to Spain. After the death of Muhammad, the Umayyad family ruled the empire. The empire from the period of Muhammad to that of the Umayyad family is called the Arabian Empire. The empire that lasted from the time of Muhammad to the end of the Abassid rule is sometimes called Islamic Empire. Combined, the Arabian Empire and the Islamic Empire is sometimes referred to as the Sasanian empire.

In 751 AD, the military of the Tang dynasty of China and that of the Islamic empire had a battle along the River Tales. The Islamic armies achieved a victory and took many Chinese soldiers as prisoners of war. Among them were papermaking experts, who later showed their captors how to make paper. With the knowledge gained from these Chinese papermakers, papermaking was established in the Islamic empire. The first paper mill was built in Samarqand, and the name "Samarqand Paper " spread and it was known even by the people of Persia and as away far as Spain. Samarqand paper was made by using a sweet bay or mulberry. By 900 AD, papermaking technology had spread to Egypt, the birthplace of papyrus scrolls. By 1040 AD, it spread to Africa and in 1100 AD papermaking technology reached Morocco. From there, papermaking technology was propagated to France in 1189.

Before the Islamic empire did not know the method of papermaking, sheepskin was used as written medium.

From the 7^th century, books from India along with mathematics books in Greek were translated into Arabic. In 773 AD, about twenty years after the introduction of papermaking to the Islamic people, the Brahma-sphuta-siddanta, written by Brahmagupta was brought to Baghdad. The Indian numeral system along with the table of sine was contained in the Brahma-sphuta-siddanta, which was translated into Arabic and those new concepts were introduced to the people of Islam. By this time, the Abassid family had already replaced the Umayyad family, and the religion of Islam had spread from Spain to the borders of China. The empire ruled by the Abassid family is called the Islamic Empire. The Abassid ruled from 750 AD to 1258 AD. This period was the golden age of Islamic Sciences and mathematics. [MAA] During the 9^th century, Baghdad became the world center of learning. The culture of Islam expressed a great degree of tolerance [NAG] toward the people who lived in the empire. Rulers promoted learning and scholarship eagerly. Noteworthy is the effort of the Caliph Al Ma’mun who began a state project of translating Greek, Syrian, Pahlavi, and Sanskrit texts into Arabic. Mohammed ibn Zakariya al-Razi (865-925) wrote: [MAA]

Even though arguments occurred against such heresies, Mohammed ibn Zakaria al-Razi lived a long life.

From 750 AD to 1450 AD, Islamic civilization produced a series of brilliant mathematicians, among whom, Al-Khwarizmi, Al-Biruni, Umar al-Khayyami, and Al-Kashi are the most well known.

Al-Kwarizmi (780-850) was the greatest Muslim mathematician in medieval Islam. His works range from arithmetic, algebra and trigonometry to astronomy. al-Kwarizmi served Caliph al-Mamun in the House of Wisdom. He introduced the Hindu methods of arithmetic to the Islamic world through his arithmetic work "The Book of Addition and Subtraction According to the Hindu Calculation" His other arithmetic book, "al-jabr wal-muqabala (The book of restoring and balancing) contains solutions for quadratic equations, geometric proofs and trigonometry. The first Arabic arithmetic book translated into Latin is the one written by al-Kwarizmi and that was "al-jabr wal-muqabala". From "al-jabr", the west came to use the ward "algebra". Al-Biruni (973-1048) was a central Asian scholar. He wrote a number of books, most noteworthy among them are books called "the determination of the Coordinates of the Location" and "India". In "India", the comparisons of Islam with Hinduism are discussed. He translated Islamic mathematics into Sanskrit. Umar al-Khayyami a Persian poet and mathematician who found a solution of a cubic equation by using a pair of intersecting conic sections. This was truly the golden age of Islamic mathematics. Ghiyath al-Din Jamshid Al-Kashi was born in Kashan. He wrote the "Aryabhatiya Bhasya", which contains works on infinite series expressions and spherical geometry.

Change began to occur in Islamic culture in the 12^th and 13^th century AD. The Mongols started to gain power and threatened the Islamic Empire. The Muslim Seljuk Empire, which captured Constantinople in 1453, defeated the Christian Crusaders.

The first European who introduced the Hindu numeral system in Latin was Gerbert d’Aurillac (940-1003). In 1202, Leonardo Fibonacci introduced Hindu-Arabic numerals to Europeans in his book "Abacus". Through this book many people in the west learned about Islamic mathematics for the first time. These ideas contributed to the development of mathematical studies in Europe. He also introduced Fibonacci numbers:

In the thirteenth century, there were many great mathematicians who were are also theologians, such as Albertus Magnus, Robert Grosseteste, Thomas Aquinas (1226-1274), and Roger Bacon (1214-1294). In the 1470s, Luca Pacioli became a Franciscan friar and he was a mathematics tutor to Leonardo Da Vinci. Pacioli wrote several books on mathematics. Most noteworthy among them are the "Summa de arithmetica, geometrica, proportion et proportionalita" (written in Venice in 1494), "De viribus quantitatis (On the Poers of Numbers), "Geometry", and "De divina proportione, which contained the subject of the golden ratio and its applications.

The mathematics of medieval Islam was propagated to Europe by the commercial route, which connected Constantinople and Vienna via Balkan nations. [BER]

I believe that there is a strong possibility that the concept of zero was developed out of the Buddhism concept of Sunyata. Though Sunyata is often translated as emptiness, void, or nothingness, zero was understood as a number between positive numbers and negative numbers. That is the Buddhist concept of sunyata, which is relativity. When the Indian numerical system was developed after the introduction of zero, it was quickly propagated to various civilizations. The swift propagation was possible, because papermaking was introduced to the Muslim world by Chinese civilization. Muslims embraced the Indian numerical system, which has zero or sunya in Sanskrit, and in turn Christians accepted it. Muslims translated various texts from the Greeks, Indians, Persians and other civilizations. In doing so, they preserved the important works of the past. They also advanced the works of the Indians and Greeks and helped modern mathematics to develop. Clearly, there exists an interrelationship between the mathematics of India, Islam and Europe and those cultures and religions. As a result of the early form of globalization brought about by contacts from the Silk Road, Islamic culture prospered and the works of the scientists and mathematicians of medieval Islam influenced the cultures of the Eurasian continent. At its height, the Islamic empire was one of the most brilliant civilizations in history. It truly pursued intellectual Excellencies.

All the world religions share basic moral and spiritual values. They all affirm that all of life is sacred. By recognizing the moral and spiritual values of each religious tradition being bridges of peace and justice, and affirms the sacredness of life, we can work together to promote a non-violent path to conflict resolution and peace building.

I am grateful to Brother Jim Faccett of the Chaminade University Marianist Community of Honolulu, Hawaii and Archbishop Ryoukan Ara of Tendai school of Buddhism in Honolulu, Hawaii for their in the completion of this paper. I thank Bro. Jim for his encouragement, help with getting books for research by giving me rides to the library, discussing many topics and giving me insight on the mission of the Marianist Community, commenting on the manuscript and buying me tortillas. I thank Ara sensei for his guidance in the understanding of the concept of "sunya" and ‘ichinenn sanzen" of Mahayana Buddhism and for loaning me numerous books on Buddhism.

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[KLE] H. A. Klein, The World of Measurements, Simon &Schuster, 1974

[CAJ] Florian Cajori, A History of Mathematical Notations: Two Volumes Bound As One, Dover Publications, Inc., New York, 1928-1929

[SAK] Taichi Sakaiya: Prophetic dream of Taichi Sakaiya- Not a crush but a merging of civilizations (in Japanese), Weekly Asahi April 11, 2003

[SMI] David Eugene Smith, A Source Book in Mathematics: 125 Selections From the Classic Writings of Pascal, Leibniz, Euler, Fermat, Gauss, Descartes, Newton, Riemann, and many others, Dover Publications, Inc., New York, 1959

[HOF] Paul Hoffman, The Man Who Loved Only Numbers, Hyperion Books, 1998

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[ASO] http://www.buddhist-canon.com/SUTRA/DCX/EHuaYan/T100237b.htm

[SJS] Kathleen Cohen, A World of Art and Culture,

[ATH] http://www.athenapub.com/9khotan1.htm

[YOK]

[WNP] http://wordnet.princeton.edu/perl/webwn  

Eiko Tyler holds a Ph.D. in Mathematics.

Department of Mathematics

Division of Natural Sciences and Mathematics

Chaminade University

3140 Waialae Ave.

Honolulu, HI 96816