The Story Of Our Numbers

The history of mathematics is a vast topic which can never be studied perfectly since much of the work of ancient times remains undiscovered or has been lost through time. However, there is also much that is known and many important discoveries have been made over the past 150 years that debunk the theory that mathematics is a European “invention”. The truth is being restored in academic circles about the historical development of mathematics and India’s contribution to it.

Once the dots are connected, it is not difficult to imagine the continuation of the same pursuit of knowledge and excellence of solutions by Indus, Vedic and post-Vedic Indian minds. When seen in this thread, it seems all but natural to recognise the advancements of the Indus Valley civilization (standardized weights in binary sequence, the world’s first measuring ruler, proto-dentistry, advanced metallurgy) to those of Baudhayana in 800 BCE (the so-called Pythagoras theorem), Pingala in 300 BCE (combinatorics, the mathematics of finite or countable dicrete structures), Aryabhatta in 499 CE (trigonometry) and Madhava in 1400 CE (calculus) with many in between.

While some will always turn a blind eye to facts staring in the face, there are many who would like to know the reality but can’t access or comprehend the greatness of the work. I am not adding any original research of mine to the history of math, and have drawn a lot from the wonderful and pivotal book, The History of Ancient Indian Mathematics by C.N. Srinivas Iyengar. My humble contribution to this discussion is merely to present the relevant facts in a simple way and connect the dots.

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Labelling Science

The labels used in the study of history of science or math are Egyptian, Greek, Mesopotamian, Indian, European, Hindu, Islamic or Christian. I think it is unfair to use any religion’s name with scientific development. None of the book-based religions fostered or encouraged science; in fact, they hindered and quashed science as is evident from the history of Europe.

While it is okay to refer to Greek or European math and Arabic or Indian math, it is not right to say Islamic or Christian math. Only in India do you find that the pursuit of faith and the pursuit of fact went side by side amicably.

Giving Credit Where It Is Due

One might also wonder if it makes any difference to world hunger and global warming if this record is set straight. While academic researchers know the true history, does it matter if our school and college students know it as well? After all, these are trivial things like basic trigonometry or Pythagoras theorem or zero. Math has come such a long way ahead, who cares if it was Pythagoras or Baudhayana?

But, if it really doesn’t matter, then why do we even call it Pythagoras theorem, Euclid’s geometry or Newton’s power series for sine? We should simply call them by their function, like the Hypotenuse theorem, geometry, or Power Series for sine.

Why should we take the trouble to remember the people who discover or do something for the first time, in science or in life?  Even in divine matters, we keep comparing who said what and when, and who said it first! Maybe we do so to give credit where it is due? And human achievements inspire us?

Why should India not have its inspiration? And have its credit where it is due? Everyone with a basic education in sciences should know about the great Indian minds. Why should some Indian names not become household names, at least in India, through the education system? The first step towards that is to understand, realize and appreciate how our ancestors pursued knowledge.

Indian scholars made vast contributions to mathematical astronomy and thus contributed mightily to the development of arithmetic, algebra, trigonometry and secondarily geometry (although this topic was well developed by the Greeks) and combinatorics.

Perhaps most remarkable were the developments in the fields of infinite series expansions of trigonometric expressions and differential calculus. Surpassing all these achievements however was the development of decimal numeration and the place value system, which, without doubt, stand together as the most remarkable developments in the history of mathematics. The decimal place value system allowed the subject of mathematics to be developed in ways that, to put it simply, would not have been possible otherwise. We would still be a ‘humanind’ as depicted (so frightfully entertainingly) in Asterix comics.

The Eurocentric Worldview

The standard of evidence required to claim transmission of knowledge from East to West is greater than the standard of evidence required for knowledge traveling from West to East. In the Eurocentric view, the ancient Greeks are the epitome of logic, scientific temperament, and mathematical achievement, whereas Indians “indulge in flights of fancy with airplanes in Vedic times or rishis travelling in space”. One of the most talked about recent examples in the media is the Pythagoras theorem and India’s claim to it. This is also the simplest one to understand.

Pythagoras

Most of the information about Pythagoras (570-495 BCE) was written down centuries after he passed away, so very little reliable information is known about him. No texts by Pythagoras are known to have survived.

Diogenes Laërtius reported that Pythagoras had undertaken extensive travels, having not only visited Egypt but also “journeyed among the Chaldaeans and Magi”, for the purpose of collecting all available knowledge and especially looking for information concerning the secret or mystic cults of the gods. Around 530 BCE, he moved to Croton, in Magna Graecia, and set up a religious sect. His followers pursued the rites and practices developed by Pythagoras and studied his philosophical theories. He was more known for his philosophy than his mathematics in his own times.

In time, Pythagoras became the subject of elaborate legends. Aristotle described Pythagoras as a wonder-worker and somewhat of a supernatural figure, attributing to him such aspects as a golden thigh, which was a sign of divinity. Some ancients believed that he had the ability to travel through space and time and to communicate with animals and plants. Another legend claims that Pythagoras asserted he could write on the moon.

Euclid

The long-lasting nature of The Elements must make Euclid (4th and 3rd century BCE) the leading mathematics teacher of all time. However, little is known of Euclid’s life except that he taught at Alexandria in Egypt. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward. The few historical references to Euclid were written centuries after he lived, by Proclus, 450 CE and Pappus of Alexandria, 320 CE.

A detailed biography of Euclid is by Arabian authors, mentioning, for example, a birth town of Tyre. But this story of Euclid’s life is generally believed to be completely fictitious. Because the lack of biographical information is unusual for the period, some researchers have proposed that Euclid was not, in fact, a historical character and that his works were written by a team of mathematicians.

Proclus, the last major Greek philosopher, who lived around 450 CE wrote: “Euclid, who put together The Elements, arranging in order many of Eudoxus’s theorems, perfecting many of Theaetetus’, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man lived in the time of the first Ptolemy (of Egypt).”

Probably no results in The Elements were first proved by Euclid but the organisation of the material and its exposition are certainly to his credit.

Connecting The Dots

How can we say that Indian mathematicians actually had an impact, that too a transforming impact, on the world stage? Let us simply look at some verifiable facts from history (turn the page and look at at the chart Mathematical Milestones), and you can connect the dots. Most of this information is also easily available on the net as well.

But there is an important question that needs to be asked.

Why is it that there was no advance in math in Europe from 230 BCE to 1572 CE, a full 18 centuries? And why is it that all of a sudden from 17th century onwards, we see hundreds of new mathematicians in Europe?

Here is what was happening in another part of the world—India and Arabia—during this period between Eratosthenes and Bombelli. Only major achievements are mentioned, a lot more happened between the lines.

1st century BCE (some date it to 1st century CE), Lalita Vistara mentions the Buddha enumerating to a mathematician Arjuna the multiples of 100 starting from 107 to 1,053.

Taking this as a first level, he then carries on and gets eventually to 10,421 (the Ramayana, a much earlier work, has names for number as high as 1,055  in Yuddha Kanda, sarga 28, where Ravana’s spy reports about the strength of Rama’s army)

 499 CE Aryabhatta of Kusumpura (modern Patna) composes Aryabhatya, gives best value of pi to 5 decimal places, trigonometric tables accurate to 0.03 per cent, gives formulae for calculating sine for intermediate angles, develops word numerals, kuttaka method of solving indeterminate equations; solves cube roots.

 628 CE Brahmagupta composes BrahmasphutaSiddhanta. Uses zero, negative numbers, solves so-called Pell’s (1668) equation Nx2+1=y2, geometric progressions, formula for finding “Pythagorean” triplets and methods for calculating sine of intermediate angles from the sine tables.

762 CE Baghdad city founded by Abu Jafar al-Mansur with its famed “House of Wisdom” centre of learning.

770 CE Kanka, a scholar from Ujjain, invited to Baghdad to explain Hindu arithmetic and astronomy. Brahmasphuta Siddhanta translated by Al Fazari into Arabic and named Sind Hind (for siddhanta). Another translation of Indian astronomy, Alarkand, may have been that of Surya Siddhanta.

820 CE al-Khwarizmi writes The Book of Addition and Subtraction According to the Hindu Calculation (Kitab al-Jam wa-l-tafriq bi-hisab al-Hind). The original Arabic does not survive.

About 830 al-Khwarizmi writes The Compendious Book on Calculation by Completion and Balancing (al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala). From the word al-jabr, we get the word algebra.

850 CE Mahavira writes Ganita Sara Sangraha, works on unit fractions, combinatorics and fractions.

900 CE Sridhara is first mathematician to give a rule to solve a quadratic equation.

About 1120 CE Adelard of Bath translates Sridhara’s book in Latin, which survived.

1150 CE  Bhaskara writes Siddhanta Shiromani with works on arithmetic, algebra, spheres and astronomy; advances operations on zero and concept of infinity; advances knowledge on permutations and combinations; first sure signs of differential calculus and Rolle’s theorem (1691) when finding instantaneous speed of a planet; leaps in trigonometric formulae; refines kuttaka method; solves so-called Pell’s equation using his chakravala method;

12th century European scholars travel to Spain and Sicily seeking scientific Arabic texts, including al-Khwarizmi’s The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester.

1202 CE Fibonacci publishes Liber Abaci and introduces the Indian number system to Europe. He finds no takers.

About 1400 Madhava of the Kerala school of mathematicians gives what later will be called Gregory series (1667), Newton power series (1665), Maclaurin series (1740), Leibniz power series for pi (1673), Euler series (1727), Taylor series (1715), all important results of calculus and series.

The first person in modern times to realize that the mathematicians of Kerala had anticipated some of the results of the Europeans on the calculus by nearly 300 years was Charles Whish in 1835. But Whish’s paper, published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland, went unnoticed by historians of mathematics. Only 100 years later, in the 1940s, did the world look in detail at the works of Kerala’s mathematicians and find that the remarkable claims made by Whish were essentially true.

As Dr George Gheverghese Joseph of the University of Manchester puts it:

“Europe had a 500-year-old tradition of importing knowledge and books from India and the Arab world. There was plenty of opportunity to collect the information as European Jesuits were present in Kerala at that time. They were learned with a strong background in math and were well versed in the local languages. And there was strong motivation: Pope Gregory XIII set up a committee to look into modernizing the Julian calendar.

“On the committee was the German Jesuit astronomer/mathematician Clavius who repeatedly requested information on how people constructed calendars in other parts of the world. The Kerala School was undoubtedly a leading light in this area. Large prizes were offered to mathematicians who specialized in astronomy. Again, there were many such requests for information across the world from leading Jesuit researchers in Europe. Kerala mathematicians were hugely skilled in this area.”

Ian Pearce of St Andrews University, Scotland, explains (emphasis mine):

“The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. The Greek mathematical achievements were based on geometry. Although Euclid’s Elements contains a book on number theory, it is based on geometry. In other words, Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines.

“The [Indian] system also spread to Spain in the 12th century. It took much longer for the system to be accepted in mainland Europe, but eventually, by the 16th century, it was widely used. That said, both prejudice and suspicion continued to be widespread, while orthodoxy also played its part in the continued use of Roman numerals.

“Cardan solved cubic and quartic equations without using zero. He would have found his work in the 1500s so much easier if he had had a zero but it was not part of his mathematics.”

What should one do after knowing all this? This is already part of school textbooks, and elementary for today’s math.

The best thing modern India can do with this knowledge of its history is perhaps this. Understand the original manuscripts with the help of math scholars and Sanskrit pundits to see if there are any insights to new ways of looking at things available.

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Get a sense that the Indian thought system and society not only produced great minds but also accepted the pursuit of truth without any conflict with or oppression by “religious” thinkers. Take inspiration from the glorious pursuits of our forefathers, just like you would from Newton, Gauss, Feynman or Einstein, but much closer to home. And students can also take some hope and pride to pursue pure sciences as well, rather than just go to IT in droves. Awaken. Arise. Advance.

You can do it. They already did.