Introduction

A 150-year colonial assessment of a five-thousand-year-old Indian civilization as steeped in ‘superstition’ and ‘primitive’ in science and technology before they came became a benchmark for post-independent India. The political leaders and the scholarship that gained ground in influential institutes, instead of countering the view, persisted with the colonial narrative. The decolonizing project never took off as we continued with the typical Eurocentric claims of a primitive Indian past in need of progress to reach the standards set by modern and progressive Europe.

The linear progression of history from a primitive past to an advanced future, deeply entrenched in western philosophy, embeds itself in Indians even today as a classic case of ‘colonial consciousness’. Dr. S. N. Balagangadhara deeply explains this phenomenon in his writings. There have been great attempts, like the writings of Dharampal, to deconstruct these narratives of an absence of science and technology in pre-colonial India. However, it has not permeated into the general awareness of the country, especially for those who are exclusive products of an English education.

A most important book in recent times is the book under review by Bhaskar Kamble, a theoretical physicist from IIT Kanpur and presently a data scientist in Germany. He has done a brilliant job of describing the history of mathematics, a much ignored subject in our educational systems,  and tracing the roots of many subjects to the genius mathematicians of ancient and mediaeval India. There is an inherent resistance amongst not only Indians but across the world to the idea that pre-colonial India had any contributions towards advancements in science. Hence, any author, especially if he happens to be an Indian, needs to take extreme care to convince others of the truth of this thesis. The author does so admirably.

The words Hindu and Indian are used interchangeably by the author. Again, the connotation of the word ‘Hindu’ is much broader, more in line with geographical and cultural contexts, which includes mathematicians in the Vedic, Buddhist, and Jain traditions.

The Roots and the Philosophical Underpinnings

The author shows how most of the mathematics we take for granted has its strongest roots in India. Of course, we all know vaguely that India discovered zero, but there is a lot more to mathematics than the concept of zero. Aryabhata, Brahmagupta, Bhaskara 1, Bhaskara 2, Mahavira, and the Kerala mathematicians laid the foundation, some ideas continuing uninterrupted till date, in the fields of arithmetic, algebra, geometry, trigonometry, combinatorics, linguistics, and calculus starting from the 5^th century of the Common Era. Their past roots go further back to Vedic times. Many of the ‘inventions’ and ‘discoveries’ of European mathematics finally, as the author shows, amount to who first and best plagiarised Indian knowledge systems. The Indian mathematical developments had many applications in the construction of buildings and temples, doing astronomy with great precision, and conducting navigation to distant lands by sea route.

He begins the book by discussing the basic metaphysical underpinnings of Hindu mathematics. Indian philosophy talks of para vidya and apara vidya, the knowledge of the higher Self and the knowledge of the external material world, respectively. It is one of the fundamental tenets of Indian knowledge systems that these two are not antagonistic to each other and are manifestations of a single unity in the form of Brahman. In such a stance, there is an intense spiritualization of every single aspect of apara vidya (all the ‘secular activities’) dealing with the material world. The antagonism between the ‘word of God’ and the ‘word of science’, present in the Abrahamic traditions, never existed in Indian knowledge systems.

As Indian intellectuals like Sri Aurobindo or Ananda Coomaraswamy say, the separation of the ‘sacred’ and the ‘profane’ profoundly fails to make sense in our culture. Mathematics can also be a route to the divine. Most of the secular activities are finally in search of the unity that binds the para and the apara. In contrast, science, as popularised in western culture, seeks unity only in the apara, or material realm.

Arithmetic

Hindu texts starting in the 5^th and 6^th centuries established many concepts of mathematics. Nearly all of the teaching today in primary schools across the world originates from Hindu mathematics, says the author. In their texts, they described zero, fractions, negative numbers, decimal systems, squares and cubes of numbers, square roots and cube roots, series, the value of pi, rules for carrying out arithmetic operations, and so on. The Roman numeral system was very complex and incapable of doing any sort of mathematical operation.

As Michel Danino points out in his articles, zero might have been known by the Chinese or even the Mesopotamians (as some sceptics try to downplay Indian contributions), but it was only the Indians who developed the decimal notations and gave a place value to zero. Infinities and large numbers were practically unknown in other parts of the world. The transfer of knowledge occurred through Arabic translations of Sanskrit texts. In the 12th–13th centuries, the Hindu number systems made their appearance in Europe through further translations of Arabic works into Latin. Europe, until that time, had only a positive number system that required an abacus for its calculations. Khwarizmi and European mathematicians like Fibonacci acknowledged the Hindu number systems, though the deep prejudice of the Church towards anything pagan led to some state decrees against the use of Hindu numerals and even termed them ‘infidel’ numbers.

Algebra

Algebra, the generalisation of arithmetic using symbols, attained high achievement in the works of Aryabhata (500 CE) and Brahmagupta (598–668 CE). By their time, algebra had become a ‘central pillar’ of Hindu mathematics. Symbols, linear equations, quadratic equations, integral solutions, progressions, series, and so on found ample expression in Hindu texts centuries before they found their way into Europe. For example, Jayadeva (10^th century CE) and Bhaskara had worked out Pell’s equation (17^th century CE) many centuries earlier in India.

Al-Khwarizmi, an important Arab mathematician, wrote Hisab al-jabr wal-muqabla. Algebra derives from the Arabic word al-jabr, and thus began the great story of the Arab origin of algebra. It was an integral part of Hindu mathematics long before Khwarizmi, who does not cite his sources. However, the influence of Hindu mathematics is obvious since Hindu mathematicians had developed many of the ideas discussed in his book, and translations of Sanskrit mathematical texts by Brahmagupta were readily available in the Arab world.

Thus, though there is a grudging acknowledgement of Hindus for the decimal system (even as Khwarizmi himself calls it Hindu numbers), algebra becomes an Arabic contribution in a classic case of mistaken attribution. Peculiarly, the algebra of Khwarizmi was rhetorical and devoid of all symbolism. Similarly, there is no reason to believe Diophantus, a Greek, was the originator of algebra. This follows the tendency of historians of Europe to attribute the origins of most of its scientific and philosophical achievements to the Greeks rather than the East.

Geometry, Trigonometry, Astronomy, and Calendars

The author describes how Indian roots in the fields of geometry, trigonometry, and astronomy remain astounding. The Shulba Sutras (1700 BCE) laid down deep geometric principles in designing the Vedic fire-altars of unique shapes. The Baudhayana Shulba sutra describes the Pythagorean theorem a few centuries before Pythagoras, and there is a small movement to rename it the Baudhayana-Pythagoras theorem. The fantastic and exquisite architecture of temples, forts, and monuments in ancient and mediaeval India is based on the Shulba Sutra principles.

Aryabhata, Bhaskara, Varahamira, and Brahmagupta were all great mathematicians and astronomers. For example, Aryabhata gave the sidereal period of the Earth as 23 hours, 56 minutes, and 4 seconds. His cosmology described Mahayugas of 4,320,000 years with four equal parts of 1,080,000 years. Trigonometry, of a highly evolved type, had a central place in astronomy. In the 5th and 6^th centuries, mathematicians created sine tables up to the 5^th and 6^th decimal places.

Calendar making (Panchanga) and eclipse predictions were an evolved science in ancient and mediaeval India. The best of mathematical and astronomical skills come to use here. Astronomical tables instead of trigonometric calculations simplified the Panchanga calculations. Brahmagupta spoke of iterative processes (successive approximation) until the solutions are convergent, persisting even today, to calculate the solar eclipses. The Islamic invasions forced the intellectual output to move south. The most famous was the Kerala School of Mathematics and Astronomy, which started in the 14^th century and was vigorous until as late as the 17^th century.

Combinatorics, Linguistics, and Computer Science – the Role of Sanskrit

Combinatorics has a key role in computer science, cybersecurity, cryptography, statistics, and probability theory. In ancient India, combinatorics had importance in music, dance, linguistics, prosody, philosophy, and medicine. This is in close association with the spirit of computer science and the development of algorithms, says the author.

Sanskrit is the most evolved language, with almost perfect grammar rules. Panini’s Ashtadhyaya (500 BCE) is a most significant work on Sanskrit grammar that assumes significance in Natural Language Processing (NLP). Scholars believe that the history of linguistics begins not with Plato or Aristotle but with Panini’s grammar treatise, the Ashtadhyaya. This rule-based grammar of Sanskrit stood unrivalled until the modern period in its ability to describe a natural language by rules alone.

European scholars discovered Panini in the 19th century, and the latter’s work inspired a generation of modern linguists like Bopp, Saussure, and Frits Staal. The works of Panini also presage semiotics—the study of signs and symbols for communication. It also represents the world’s first formal universal grammatical and computing system, much before the 19th-century development of mathematical logic.

Panini’s ‘auxiliary symbols’ technique to mark syntactic categories and control grammatical derivations, rediscovered by the logician Emil Post, became a foundational method in the design of computer programming languages. Mary Boole writes that the Indian logic system was central to the development of machine theory, vector analysis, and modern mathematics. Yet, malicious narratives by the likes of Sheldon Pollock make Sanskrit a dead and exploitative language!

Calculus – Mistaken Attributions

Calculus is an extremely important branch of mathematics, supposedly ‘invented’ by Newton and Leibniz almost simultaneously in the early 17^th century. The author shows substantial evidence of the development of calculus a few centuries earlier by Hindus. Aryabhata created sine tables, and Bhaskara had calculated volumes and areas of different geometrical shapes.

Madhava (1340–1425) initiated the Kerala school of mathematics, which had an unbroken lineage up to the 17^th century. These included Parameshvara, Brahmagupta, Bhaskara, Nilakantha, and Jyeshthadeva. Their contributions included handling infinities and infinitesimals; notions of differentiation and integration; series expansions; infinite summation; the creation of trigonometric tables accurate to one part in a million; spherical trigonometry; calculating the pi value to seventeen decimals; and so on. These had deep connections to astronomical tables and calendar-making.

This most sophisticated mathematics, a work of many successive generations in ancient and mediaeval India, emerged in Europe ‘suddenly’ in the works of Newton and Leibniz three centuries later. The author shows that a huge transfer of knowledge happened through the Jesuit priests, trained in mathematics, who were in Kerala at the time. Many derivations, such as Kepler’s calculation of spherical areas, Rolle’s theorem, Taylor’s series expansions, and the Gregory-Leibniz series for pi, were in the texts of Hindu mathematicians a few centuries before. European historians credit it as a ‘coincidence’, but the weight of evidence shows a direct transfer of knowledge, as the author elegantly demonstrates in his book.

The Myth of the Greek Miracle

Greece takes pride of place for Europe, which takes credit as the ‘ancestor’ of its knowledge. However, Greece had an intimate connection with India in ancient times. There were many common points in the areas of philosophy and mathematics, especially. Europe (especially the Renaissance) appropriated the Greeks as their intellectual ancestors by taking what suited their own ideas. The Greek philosophy had more connection to Indian philosophy than modern European philosophy, which still struggles with its epistemology, ontology, and explanation of consciousness. The Greeks were deficient in arithmetic and algebra, but geometry showed a similarity. However, the evidence is clearly in favour of an Indian origin and a Greek destination. However, scholars attempt to attribute an earlier common origin to both Greek and Indian mathematics, invoking the convenient but unproven Aryan theory.

Thomas Aquinas (12^th century CE) started the appropriation of Greek philosophy and moulded it to fit Christian theology. The Greeks Diophantus gets the credit for being the ‘father of modern algebra’ and Viete for ‘introducing symbolic algebra’, but this was just denial on the part of the old Europe, which continues to this day to deny the contributions of ‘pagan’ societies to the development of knowledge. However, the author shows that the near-universal consensus in western academia on the Greek origins of mathematics has, in fact, little to show by way of evidence.

Most claims remain tenuous, even as the existence of Euclid himself is in doubt. Many scholars have commented on the elusive nature of Euclid. The Greeks, having no access to a convenient number system, could not handle complicated arithmetical operations. It is also puzzling to scholars that Greek knowledge of algebra was rudimentary despite their advanced skills in geometry. It appears that most of the Greek mathematics had Arabic and Indian sources or came from later Greek mathematicians like Theon and Proclus (4th–5th centuries). The sweeping claims for Greek origins have their basis in a single manuscript and not from multiple cross-verified sources, as would be the normal process in ascertaining primary authorship, says the author. Church-led propaganda that could not accept pagan knowledge was primarily responsible for creating these Greek myths.

The Criticism of Improper Methodology

There is a large body of sceptics, both western and Indian, who claim that Indians did not have a proper methodology while doing mathematics. This is an oft-repeated claim to show the superiority of western mathematics over Indian. The author demonstrates that such claims are either based on ignorance, prejudice, or both. The Indian metaphysics has deep principles of the acquisition of knowledge (epistemology), which western philosophy is still to acquire or formulate apart from its definition of knowledge as ‘justified true belief.’

Yet, these claims are ironical since it was only after contact with Hindu mathematics that Western mathematicians progressed further and started dealing with irrational numbers, negative numbers, zero, infinities, and many other concepts. The Roman numbers hardly had the capacity to carry numerical operations, and for a long time, Europe was using the abacus to count, and that too only for positive integers. Though the formulas and original equations were terse and short, there was a tradition of detailed commentaries that explained the equations and formulas. These proofs in commentaries were an important component of Indian mathematics, but the critics conveniently ignored them.

The Diffusion Of Indian Knowledge Systems Into Europe

How did Hindu mathematics reach the European world? There were two routes. The first route involved, through the 8^th to 13^th centuries, the transfer of Indian knowledge systems (numbers, algebra, trigonometry, and astronomy) through Arabic translations to Baghdad, the centre of the Arab intellectual world. Later, there were further Latin translations of the Arabic translations, mainly in Spain, which had a significant Arab presence. This was during the 11^th to the 16^th centuries of the common era. In this whole journey, the Arabs had the numerals and algebra attributed to them instead of the Hindus in India.

Though Europe tries to appropriate the Greeks as its intellectual ancestors, it was severely critical of Greek ‘paganism’ during the ancient and mediaeval periods. The Church persecuted the Neoplatonist school of Alexandria, which had many similarities to Indian philosophy (like the ideas of ‘spirit’ and ‘reincarnation’), and fanatic mobs lynched Hypatia, a prominent lady intellectual. However, the ‘pagan’ Hindu knowledge that appeared to be useful in calendars and navigational systems led to the sending of many Jesuit missionaries trained in mathematics to Kerala. This was the second route.

From the 16th to 18th centuries, there was a remarkable splurge of inventions and discoveries in Europe in a series of ‘coincidences’ a few years after these missionaries returned from India: reform of the Gregorian calendar (1582); Tycho Brahe’s planetary model (Nilakantha’s planetary models predated this by a century); sine tables of Christoph Clavius (1607), which were remarkably similar to those of Nilakantha in the 14^th century; and the single-handed inventions of calculus by the superhuman Newton (1642–1726) and Leibniz (1646–1716).

The legend of Newton or Leibniz inventing calculus, however, deeply entrenches itself in our education system. Both fought a bitter battle to claim the honour, but for the Kerala mathematicians, the question might have been who was the best plagiarizer. As a side note, Newton was an extremely mean person on a personal level. The arrival of Jesuits in Kerala also coincided with a surge of interest in infinities, infinitesimals, and infinite series. Even Galileo (1564–1642) borrowed directly from Jesuit sources in the Collegio Romano. The instances are too numerous (the author lists them in the latter part of the book), which clinches the point that European mathematics was anything but original.

Why Did The Intellectual Output Stop And Why Is Our Knowledge Of History So Deficient?

In the final chapters, the author describes the various reasons why intellectual output stopped and why ignorance about a glorious Indian past is still widely prevalent among its citizens. Indian culture had, in the past, a rich education system consisting of the gurukula system, deeply embedded in the ashrama dharma of its people. The gurukula, where the student stayed mostly in the ashram itself, was a place to generate and propagate both secular (laukika or apara vidya) and spiritual (alaukika or para vidya) knowledge.

A rich intellectual tradition started moving south in the 11^th and 12^th centuries with the start of Islamic invasions. From the generation of knowledge, the efforts of the civilization started towards protecting the knowledge systems from annihilation. Following this, continuing till date, is a long list of factors that saw to it that India could not regain its rightful place in the field of knowledge.

In the pre-independence period, colonialism by different European powers (Portuguese, English, French, and Dutch), missionary propaganda, and Macaulay’s educational system first destroyed the education system and then created narratives with far-reaching consequences. The colonial-missionary stories like ‘Brahmanical denial of knowledge’ to other groups were false, as Dharmpal shows elegantly in his writings. Non-Brahmins had equal access to primary education and higher education, as the British records themselves show.

The initial sense of awe and wonder that the English colonials had for Indian knowledge systems underwent a radical change in the need to show a ‘primitive’ culture in need of progress. Access to the Vedas ‘restricted’ to a certain group was more in the nature of concepts like swadharma, guna, and adhikara, but it became a narrative of rights, exploitation, and denial to other groups. The apara vidya related to the material realm was open to all varna categories. Aryabhata was a non-Brahmin.

Post-independent India did not change much in its perceptions of an ancient and mediaeval India. A dominant Marxist scholarship that set our academic narratives, a deeply prevalent colonial consciousness, and English education continued the same disbelief in the richness of the Indian past. That a self-sufficient country stood strong for such a long time without any scientific or technological achievements is a firm idea to most Indians. Everything we had was either fake or borrowed from the Greeks, Chinese, Babylonians, Mesopotamians, or Arabs. Our achievements in science and technology remained footnotes in our textbooks. In a few crucial generations, our education system could deracinate most of the Indians successfully.

Concluding Thoughts

India has a great intellectual heritage, and a huge corpus of texts (the broad five groups of Vedas, Upavedas, Vedangas, Puranas, and Darshanas) covering all fields of human activity is a testimony to the capacity of Indians to create knowledge by reflecting on their own experiences without any foreign influences. Michel Danino quotes David Pingree as saying that India has at least 30 million surviving ancient manuscripts in Indian libraries, repositories, and private collections dealing with the widest variety of topics. Learning the Vedas is a very small aspect. The linear historical narrative of the past, or primitive, equaling India, and the future, or golden, equaling the West, does not seem to budge from our educated minds.

Secularism, a solution applicable to the Christendom of Europe at a specific time in its history, was the biggest violence against Indian culture and prevented the education system from regaining our rightful place on the world stage. In a culture where every secular activity was in intimate connection with the spiritual, where every secular activity (like arts, music, literature, sciences, or politics) was a way to the spiritual and thus intensely ‘spiritualized,’ where ‘spiritual’ meant to gain the highest ideal of moksha, and where spiritual teaching did not mean ‘othering’ or ‘hating’ another, the idea of separation of the ‘sacred’ from the ‘profane’ failed to make sense in a most profound manner. The conversion of the most wonderful philosophies and metaphysics available in our Vedas, Upanishads, Darshanas, Mahabharata, Ramayana, Bhagvad Gita, Puranas, and countless other texts as ‘religion’ and then excluding them from study at the school level for the sake of secularism has been the single most important cause of the lack of pride in our culture.

SN Balagangadhara writes: Colonialism is a supreme educational project because the belief of civilizational superiority contains the messages of the ignorance and the immorality of the colonized. … Colonization was not merely a process of occupying lands and extracting revenues. It was not a question of us aping Western people and trying to be like them. It was not even about colonizing the imagination of a people by making them ‘dream’ that they, too, would become ‘modern’, developed, and sophisticated. It goes deeper than any of these. It is about denying peoples and cultures their own experiences; of rendering them aliens to themselves; of actively preventing any description of their own experiences except in terms defined by the colonizers. Colonialism generates a set of attitudes and beliefs- a feeling of shame about their own culture; the conviction of backwardness; and the desire to learn from the colonizer.

Colonial consciousness imbibes a framework of civilizational superiority, which the colonised articulate. Thus, the most immoral consequence of colonialism is the colonial consciousness generated in the colonised long after the colonisers have left. This consciousness precisely prevents most educated Indians from admitting or even looking at the possibility that India has had great achievements in the past.

English education is a major factor in generating and perpetuating this colonial consciousness. More than a century ago, Ananda Coomaraswamy wrote:  A single generation of English education suffices to break the threads of tradition and to create a nondescript and superficial being deprived of all roots—a sort of intellectual pariah who does not belong to the East or the West, the past or the future. The greatest danger for India is the loss of her spiritual integrity. Of all Indian problems, the educational is the most difficult and most tragic. The statement makes sense even today.

Balagangadhara rues that when we want to decolonize, there should be our own narrative to replace the colonial story. Otherwise, decolonization does not make sense. This book, ideally a compulsory reading at high school level, is an important narrative from our own side that replaces the colonial story of a western or Arabic origin of mathematics and, at the same time, reclaims our place of pride in the field of mathematics. Compliments to the author and the editor (Sankrant Sanu) for putting across many ideas in an extremely easy language while keeping the detailed explanations in the appendix for the more interested or the more sceptical.