Monthly Archives: July 2016

Aryan Invasion Theory Violates Vedic Tradition

AITAVAIDIKA

For writers, editors, and scholars alike, it is critical to be honest not only about what the tradition says, but what they are competent to do. Per our tradition, there is a sharp division between Adhyatmika and Laukika. Those of us from Andhra know the division between Vaidiki and Niyogi for example. As such, it is imperative that those of us from the Laukika sphere, whatever may be our birth varna/jati, refrain from interfering (let alone corrupting) the Adhyatmika sphere. For far too long have self-appointed armchair acharyas attempted to play the role of Vedic seer, pushing as Shivoham has written, concocted model-based theories based purely on imagination.

The Vedas are apaurusheya, and thus, not the realm for “original thinking”, statistical and cliometric analysis, leave aside creative interpretation or reinterpretation. Only traditional Brahmanas who specifically lead the traditional way of life in agraharas and mathas have the authority to interpret what the Vedas actually say. Increasingly, Bharatiya Sanaatanis of all Varnas (all castes) have been granted Veda Adyapana and some have become Brahmanas not by birth but by Guna and Dharma, and this too is accepted in our era, provided they follow the traditional lifestyle and guidelines laid before them.

But foreign “indologists” and their native sepoys and their sycophants do not have such authority. Merely bearing a yajnopavita and performing rituals robotically does not make a Brahmana.Those are mere accoutrements.Preservation of the truth makes a Brahmana.

Brahmin or not, Initiate or not, Indian or not, those earning their living in foreign employ do not have authority to assert let alone pervert what is in the Vedas. Only degree-factory fools seek them out as some sort of “rishi”. Therefore, rather than presenting myself as some sort of authority, I too will follow this rule and simply report what an actual and public Adhyatmika Brahmana, Pandit Sri Kota Venkatachalam, has himself written.

PortraitChelam

Pandit Chelam was uniquely qualified, not only being born and brought up in an agrahara, but being competent in both traditional Vedic learning and Western history—particularly Indology. Pandit Chelam has categorically rejected & refuted the Aryan Invasion Theory (AIT) and diligently catalogued all the high crimes and misdemeanors of British Colonial Indologists and their sepoys—many of those comprador lineages exist among our ranks today.

Over the course of a lifetime until his retirement at age 72, this true Brahmana did his Dharma by preserving the truth when Bharat itself was prostrate and powerless under foreign colonial rule. The time has now come for his life’s work to be vindicated, and the Vedic Truth in all our hearts to be asserted, not through my scholarship, but his. My pranams to him. Here is what he has written [Emphasis and Proofing Ours]:

The following Excerpts are from various Books by Pandit Chelam


§

In the beginning, there was only one race, the Aaryan race. In the ancient times, when the Aaryans were spreading all over the continent of Bharat, the different regions and parts were named after the Kings that ruled over them. The people too were named by the names of these regions and came to be considered different races.

In those remote times in Eastern Bharat was known as ‘Praachyaka Desa’ and ruled by a king named Bali. After his death, several of his sons divided his kingdom, and each named his part after himself, one of them being Aandhra. The kingdom of prince Aandhra being known as Aandhra Desa and the Aaryans (of the four castes) inhabiting the region were called Aandhras.  Thus only one group or division of the Aaryans came to be known as Aandhras. The Aandhras were not a separate race from the Aaryans.

It is all one race known as Aaryans in the beginning, some of them later coming to be known as Aandhras from the name of the region inhabited by them. It is the same case with the Aaryans inhabiting the other different parts of Bharat, all of them of the same Aaryan stock but developing into various branches and coming to be considered different peoples and named after the different regions occupied by them.

Andhra-Kerala-Chola-Bharata

But all of the Aaryans of Bharat from the Himalayas to Cape Comorin [Kanyakumari] belong to the same racial (Aaryan) stock. This axiom should be kept steadily in mind in the study of the history of the Aandhras from the beginning of creation, attempted in this volume.

The Process of Creation

In the beginning the five elements evolved naturally [f]rom primordial nature or Prakriti, and from earth, of the five, living matter and living beings of all kinds. The first among the living creatures was Prajapathi. He is the first Aaryan. Rigveda 4 26 2-2, 2-11-18. He resolved on the creation of the human race and first created the ten Praja-pathis (the Devarishis). Then he himself residing in the region enclosed by the rivers Saraswati and Drishadvati, and cohabiting with his wife Sataruupa gave birth to two sons ‘sons Priyavrata and Utaana paada and three daughters Aakuuti, Devahuuti and Prasuuti. The region he first lived in came to be known as “Brahmavarta“.

The human race first appeared in Bharat only. To the west of the present Jamuna in North India there flowed in ancient times Sara-swati and to its west a tributary by name of Dru-shadvati. The region between these rivers Saraswati and Dri-shadvati was known as ‘Brahmavarta’ from time immmo-rial [immemorial]. The name indicates that the Swayambhuva Prajapati named Brahma resided there in gross physical form to cre-ate the human race on the earth.

At the beginning of every cycle of creation, this place where Swayambhuva Prajapati, the first man resides on the earth in his gross physical body, to create the human race is known as Brahmavartam’. In Rigveda-3-33-4 we hear ‘Yonim Deva Kritam’ and ‘Tam Deva Nirmitam Desam’ in Manu 2-17. This region is bound by the river Sara-swati on the east the junction of Sarasvati and Drushad-vati on the South, Drishadvati on the West and the Hima-layas on the North.

dakshinapatha

The First MigrationBrahmarshi Desa.

The Aaryans thus born in Brahmavarta left the place of their origin and inhabiting the region to the west of it gave it the name ‘Brahmarshi Desa’ (Manu 2-19). These migrations and colonisations were led by Brahmarshis of established spiritual eminence who settled down in the new regions with their disciples and hence it was called ‘Brahmarshi Desa.’In later times this region came to comprise the kingdoms of Kuru, Matsya, Panchala, Surasena & Uttara Madhura.

The Second MigrationMadhya Desa.

According to Manu, the region bounded by the Vindhyas in the South the Himalayas in the north, Allahabad [Prayag] in the east and the river Saraswathi in the West, was called Madhya Desa. (Manu 2-21). This was the region colonised by the second migration of Aaryans after the Brahmarishi Desa was fully occupied.

Aryavarta (The Third Migration)

Thereafter the Aryans, on the advice of the sages and under the leadership of the kings, started on the third migration and spread all over the plains between the Hima-layas and the Vindhyas and settled down in permanent homelands. At that time almost  all the surface of the earth was uninhabited and even in Bharat there were no people  other than the Aryans.

Fourth and Fifth Migrations.

Thereafter, a king by name of Videha Madhava, on the advice of his teacher Gautama Rahuguna, accompanied by the Aaryans who were rapidly increasing in numbers, orga-nised a great migration from the Brahmavarta and neigh-bouring regions and proceeded “to the east of Saraswati upto the river Ganges and established Aaryans settlements at several places. But confronted by the river Sadanira, the progress was halted and villages and towns were constructed all along the march up to the river Kubha or Kabul, and extended their settlements so far. These details of the migration are available  in the Satapatha Brahmana, the Rigveda and in the Manu Smriti

The land in which the Aaryans are born, grow and die  and are  born again is known as ‘Aaryavarta’. Thus it is clear the Aaryans were living in this region from the beginning of creation, according to the Manu Smriti.

The sixth migration “Dakshinapatha”

In those days this part of the country was uninhabited. After rendering habitable and fit for colonisation, the neighbourhood of the river Sadanira and proceeding through the regions to the east of it, Viz. Vanga, etc, they spread to the south along the coast. The south eastern coast lands of Bharat, which were thus occupied by the Aaryans gradually  down to modern Madras and below, were then known as ‘Prachyaka Desa’ and this region beyond further south to the sea ‘Dakshina Desa’ and the west coast and adjoining tracts ‘Paschima Desa’. Thus the Aaryans spread in course of time over the whole of the Southern peninsula and the Aryans who came  to occupy the whole of Bharat from the Himalayas in the north to the Indian ocean in the south were the followers of the Vedic culture and the social order of the fourfold division of society) which formed an integral part of it.

 

§

Many, may naturally, aver that while the Vedic Tradition asserts that humanity was born in the Brahmavarta (Sarasvati-Indus Valley) modern Science states that Africa is the origin. My response to that would be, that is fine. Let Science be Science and let Tradition be Tradition, rather than mix and mess up the two. The problem is when Science becomes Traditional Culture and Traditional Culture becomes Sciencethe result is Scientism.

Traditional culture provides values & historical memory that give guidance to a people. Science helps humanity understand the material world. Current Scientific Evidence does show the preponderant weight behind The Theory of Evolution, and the origin of humanity in Africa. But Science cannot dictate what our Tradition actually said.  There was only Aryan. Dravida was a subgroup of Arya.

Dravida

In fact, the Genetic Evidence for AIT is severely questioned, the Archaeological evidence for AIT non-existent, and the Vedic Tradition outright contradictory. Pandit Chelam’s own charges against the British for fabricating evidence to concoct the current Chronology (including outright destruction of evidence) only injure AIT even more. In fact, other than the compromised Academy, only a clique of casteists and their clueless sycophants (as well as a few naive, but well-meaning people) seek to preserve it. On what basis? Some blog ramblings? This article is not based on my work, but the scholarship of an actual and authentic Brahmin Pandit, Sri Kota Venkatachalam, who is also a western educated historian. His word on the Vedic Corpus of Texts, and the Puranas in particular, carries preponderantly more weight than social media personalities and cliques. Enough!

Aryan Invasion Theory is AVAIDIKA. The Aryan Invasion Theory violates Vedic Tradition. Those asking “what if”, the answer is “it’s not”. Those saying, “but I learned from such and such”, the answer is guru-moha is still moha. And those fools with a smattering of Vedic Sanskrit attempting to engage in “original thinking” putting forth noxious nonsense theories, and all and sundry, should be well-advised of the severe, multi-lifetime penalties of Smriti-vibhrama. Certainly, all true Brahmanas are apprised. No ritual will protect you from this paap.

That it has gained any credence at all is testament to the sad state of what passes for “intelligence” in ‘modern’ Hindus, who are spoiled Brats. As we’ve written before, the highest form of material intelligence is not some asinine, poodle pedantry or analysis-to-paralysis that offers no viable solutions. The highest form of material intelligence is strategic intelligence, because it understands how to efficiently and effectively deploy all other intelligences, even the over-emphasised quantitative intelligence. Quantitative intelligence and even good memory are important and have their applications, but memory tricks and math problems do not save civilizations, strategic intelligence and societal coherence does. The Vedic Tradition gave us such coherence through Dharma—do not violate it. All this is why time and again we have given example after example of why  Wisdom is more Important than mere Knowledge. But some are children in adult bodies, so they childishly and stubbornly refuse to get the message, because for all their book clippings, they are in fact like the very masses they condescend to, following only what is “popular”. Principle comes before Popularity.

Sanaatana Dharmikas have enough to deal with regarding foreign disinformation, misinformation, and inculturation. It is bad enough ridiculous things are said about Brahma and Sarasvati. Any real Hindu familiar with Ardha-Nareeshvara knows that that just as Shiva-Shakti are equal halves of Para-Brahman so too are Brahma and Sarasvati equal halves of the same soul. Thus Svaymbhuva Manu’s wife Satarupa is not his daughter but his other half, just as each human husband refers to his wife as his ‘other half’. This is not just an expression, but a statement of Dharmik philosophical reality. Monogamy is advocated for this precise reason.  But rather than making themselves useful, these fake acharyas and dushta brahmanas specialise in pedantry, which fools only rascals or rubes. The latter can be forgiven for foolishness, but the former must be punished. Murkha-panditas can be forgiven, dushta-brahmanas must be punished. If bahishkar (outcasting) existed, it is because of these dushta-brahmanas and their chamchas.

“Aryan Invasion Theory”, “Beef in Vedas”, “Dharmasastra is ok with Same-gender love”, “Rna determines Dharma” all these are the work of genuine casteists (courtesy their videshi paymasters to whom they are “rnis”), as these nincom poops are prepared to pay any price to fulfill their adharmic ambitions and assume videshi “rna”. True Brahmanas know that whatever their personal rivalries (inter-caste or intra-caste), it is mahapaapa to corrupt the Vedas, and therefore, they keep their Egos in check, in order to preserve the integrity of the inheritance bestowed upon them by their forefathers for intellectual guardianship.

Materialistic  fools and casteist frauds who defile sacred threads, do some ritual as show, and haughtily drop their gotras at the first opportunity, cannot hide behind their janeus when real adhyatmika Brahmin Pandits of authentic lineage have asserted what is actually in the Vedas.  Here is what one wrote about AIT, directly:

AITshahmat-hhi

Pandit Kota Venkatachalam, is an actual Acharya, and has spoken. Let there be no more confusion. AIT RIP…

Aryan Invasion Theory violates Vedic Tradition.

References:

  1. Kota, Venkatachalam Paakayaaji (Pandit). Chronology of Ancient Hindu History Part I. Vijayawada: AVG.  p.121-133

“An appeal to Young Indologists”

PKVCthePIIC

As we wrote previously, the Importance of History cannot be minimised in this era, let alone any other. A person, a people, a culture, a civilization, all derive their identity from history, sacred or otherwise. The critical lessons of history help politicians and military thinkers alike shape the course of their country’s destiny. But with a topic as powerful and as crucial as history, objectivity and dispassionate thinking are required. Scientific temper does not mean scientism. Ours is a spiritual civilization and our Vedas, a spiritual tradition. Therefore, before beginning to catalogue and disseminate True History, it is important to understand “True Indology”.

Instead today, mere regurgitations from social media and blog trivia are what pass for serious research and serious thinking. But serious people are driven by strategic thinking, not serial sycophancy and regurgitation of knowledge from self-appointed “acharyas”. They recognise that any nation that has been colonised must carefully review whether and how their society was tampered with. This is because…

What greater proof was there of this than British-colonised India?

Those wedded to scientism forget the true place of tradition, and how science exists to confirm tradition, rather than define or even pre-determine tradition. Fortunately, the modern and traditional are not always antipodal. There was one such true Pandit, indeed, a veritable “Bharata Charitra Bhaskara” who was learned not only in “western learning”, but our traditional Vedic and Pauranic learning as well. For those [b]raying for “true pandityam”, fine, let us then learn from a real Pandit, Sri Kota Venkatachalam.

Traditionally trained, but modern educated, he is the precise antidote to sage-imitating sepoys selling their knowledge to the highest bidder, while hiding behind sacred threads. Here is one actual Acharya of authentic lineage who actually deserved his yagnopavitham. And my pranams to him.

He wrote in the very era when Bharat’s history was being tampered with and painstakingly catalogued how our history was purposefully misrepresented, and archaeological evidenced destroyed. Here is what he had to say [emphasis ours]:

The following Post appeared on True Indian History on April 21, 2009


 

The history of India, particularly of the ancient period, as it is found in the Text Books of schools and colleges and in the writings of research scholars of Indology, requires thorough revision. European scholars, who attempted to construct our history, seriously erred in chronology.

  1. The false assumption that the Aryans came from outside India and the wrong identification of Chandra-Gupta-Maurya of 1534 B.C, with another Chandra-Gupta, the contemporary of Alexander(326 B.C.), led to several errors in chronology and other aspects of our history.
  2. The Puranas, which are a storehouse of historical information, were discredited as mere fiction. Several facts from the Puranas that do credit to our history and culture are entirely omitted in the historical writings of Europeans and their Indian followers.
  3. Some Indologists went to the length of interpolating in and otherwise tampering with the writings of ancient foreign visitors of India and with the Buddhist literature
  4. Many ancient inscriptions like the Kumbhalghar Inscription (V.S.1537) were destroyed.
  5. The genuine Inscription of Janamejaya ( Indian Antiquity pp333,334) dated Kali 89 or 3012 B.C. has been rejected as being spurious. Several other important ancient inscriptions between 4148 B>C. And 300 B.C., were destroyed.
  6. Some coins and inscriptions have been misread, mis-interpreted, misapplied and misrepresented and some are forged so as to be used for supporting the modern theories.
  7. The Aihole inscription and others that establish correctly the date of the Mahabharata War, 3138 B.C., have been neglected.
  8. Some important dates which are supposed to be the Anchor Sheets of Ancient Indian chronology have been arbitrarily determined, with no regard for or reference to ancient literature.

All this was to show that the historical literature of Bharat was unreliable as a document of history.

Although later researches by Indian Savants have brought to light several facts, the writings of these savants are not accepted by prominent Indologists for the simple reason that these writings do not fall in line with their modern theories. It is strange to expect that scholars that are bent upon showing the errors in the modern historians in the field should fall in line with the same writers. The interests of truth will heavily suffer if this attitude towards fresh research scholars of Indian history continues.

For about forty years I have been working in the field of historical research studying both Indigenous and modern histories and inscriptions etc., and during the last 9 years I have published genuine Historical facts in 24 books, some in Telugu and some in English running into 3000 pages. I have been sending my publications to research scholars and other prominent persons interested in the subject. Although the bulk of the scholars are too conservative even to examine my writings, some of them have accepted that my writings give a lead to the attempts for constructing a genuine history of Bharat. I am happy to note that there is a wide-spread desire in our country today, that our history should be rewritten so as to be nearer the truth.

I have done, through my writings, what I could towards the achievement of the legitimate wish of our people. I appeal to the younger generation to pursue the subject and do justice to the great culture and history of our country.

I have labored, long enough and am retiring in my 72nd year. I assure my young friends that as they proceed with the subject they will find in our ancient literature, inscriptions and coins, wonderful material that will enable them to construct history of our mother-land from 3138 B.C.. Beware of forged inscriptions etc.

This Ancient Hindu History consisting of two parts is the last of my works. In the first part of this book I have traced the dynasties of kings from 3138 B.C., the date of the Mahabharata War, to 1193 A.D., and I have also given historical accounts of these dynasties. This information is quite in accordance with the puranic accounts and genuine inscriptions. In this second part, I have proved that the genuine history of Bharat is to be found in the vast Sanscrit literature, that the so-called archaeological evidence cited by modern historians is full of misleadings, misrepresentations and misapplications and that this evidence besides being so very faulty has failed to help a correct reconstruction of ancient Hindu Chronology and has always tended to horribly curtail it.

My good wishes to all those interested in bringing the genuine history of our Bharat.

Kota Venkata Chelam

Author,

1-1-1957


Rajiv Malhotra has been shedding light on exactly how Western Indology is being used to Break India. Pandit Chelam showed precisely how history was and is still being used by Colonialists to confuse and disorient India. That is the danger of scientism–it fails to ask, cui bono?

bono

In the coming days and weeks, we will examine closely Pandit Chelam’s work. Many have heard of him, some are familiar with him, but it is time we study him. But study him we shall in his own example, and critically examine his statements to see exactly why the essential story, the core chronology, the true sheet anchor of history is in fact correct. Details here and there may be lost to time or uncertainty or require verification, but determining the correct chronology and place of origin properly defines the place of history and a people’s place in it.

Above all, someone of his calibre with knowledge of both realms clarifies precisely what our Vedic tradition actually says.

Emblem_of_India.svg


Our sincere thanks to G.D. Prasad garu, who is the grandson of Pandit Chelam for graciously granting permission to reprint this article, which reprints sections from Sri Venkatachalam’s work.

An Indic Perspective to Mathematics — 3

mahaviracharya

(This is the concluding part of the sequel to ‘Introduction to Ganita’)

Part 1 (Introduction) 

Part 2 (Ganita – Math Encounters)

Part 3 (below): Ganita prevailed over Math in their encounters, but what did it really win? While Ganita’s results were absorbed into Mathematics, the underlying pramana and upapattis were rejected. We explain why this happened, and its implications.

Digestion Of Ganita, the Needham Question, and the Road Ahead
Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less - Marie Curie.
The Digestion of Ganita

It appears the ancient Babylonians had something in common with the Indians: they were pattern-seekers. As far as trying to understand how the world around us works, Richard Feynman rejected (Greek) Mathematics in favor of what he recognized as a Babylonian method, as discussed in this lecture below. Despite this endorsement, it is the Greek approach that drives Mathematics today, while the Babylonian culture can be found only in famous museums today. Why?

It would not be a problem for any civilization to view and benefit from imported knowledge by employing a native lens, without denigrating and destroying the external source tradition, and based on mutual respect. However, when knowledge from another culture is deliberately cannibalized and appropriated as a predator, it is a serious problem. It turns into a process known as’digestion‘. We now describe how Ganita was digested into Mathematics after their encounters.

The Digestion of Ganita into Mathematics

This process of digestion has been laid out by Rajiv Malhotra in [4]. We apply this description step-by-step to see how Ganita was digested into Mathematics.

Step 1.The less powerful culture is assimilated into the dominant one in such a way that: the dominant civilization dismembers the weaker one into parts from which it picks and chooses which pieces it wants to appropriate“.  During their encounter, all the important results of Ganita, starting from the place value system with zero, to algebra, trigonometry, algorithms, combinatorics, … to calculus were accepted by Europe to obtain real-life benefits. However, the underlying epistemology and approach of Ganita that has worked so well for 2000+ years, and could be used to generate such astonishing results in the future were amputated from Ganita. Only the results were retained within Mathematics.

Step 2.appropriated elements get mapped onto the language and social structures of the dominant civilization’s own history and paradigms, leaving little if any trace of the links to the source tradition“. The formal Math rooted in the Greek tradition was enhanced and expanded so that the Ganita results could be systematically re-derived and reinterpreted in a compatible manner. Later, the beneficial features of the native Encuvati system of pedagogy was appropriated into the British teaching approach, and ‘undesirable’ features were deleted to ensure compatibility with ‘Christian values’ [15]. Once this process was complete, the source tradition of Ganita was expendable.

Step 3.the civilization that was thus mined gets depleted of its cultural and social capital because the appropriated elements are modified to fit the dominant civilization’s own history, and these elements are shown to be disconnected from, and even in conflict with, the source civilization“.

A. The credit for a re-engineered calculus was given to Newton/Leibniz and not Madhava and the Kerala School. We are taught the Pythagoras theorem and proof, not Baudhayana’s result and validation procedure. Fibonacci numbers, not Gopala-Hemachandra series. The IEEE journals recognize Arab numerals, not Hindu numbers, and so on. The list is long. In almost all these cases, the standard reason is that the Indians had not proved their results using the formal system devised by the west, even though each of these results were generated first by Ganita and also satisfactorily validated within the source tradition, often centuries earlier. The Ganita tradition was erased from the history of Mathematics.

B. On the other hand, the following types of claims are created:

  • Vedanga Jyotisha was full of astrology and religious mumbo-jumbo
  • Ganita was some kind of elite “Vedic Mathematics”
  • Hindu tradition was backward, caste-ridden, superstitious and incapable of producing such advanced scientific results.

Whereas, the exact opposite is true.

  • Vedanga Jyotisha is the science of time-keeping, and “the entire Jyotisa does not have a single sentence relating to astrology or prophecy” [1], whereas the main goal of European calendar reform was to advance the cause of organized religion [1]
  • Ganita was pragmatic and accessible to ordinary Indians including vegetable vendors who taught the greatest Arab scholar of their time [14], while today’s formal Mathematics is indeed the preserve of an elite few [1].
  • Hinduphobia is rampant in the Humanities departments of Western universities, which is subsequently exported to Indian universities, even as the digestion of Hindu science and technology results continues unabated [16].

Step 4. The final result is catastrophic for the source civilization: “the depleted civilization enters the proverbial museum as yet another dead culture, ceasing to pose a threat to the dominant one. After being digested, what is left of a civilization is waste material to be removed and destroyed.”  A mathematical monoculture was imposed on India during the colonial era after uprooting the ‘beautiful tree‘, India’s indigenous decentralized education system whose Ganita curriculum was sensitive to local requirements. Few students and teachers in Indian schools and universities today are aware of the source Ganita tradition. Among those who recognize the word,  few realize it is not an Indian neologism for Mathematics. Is this not an instance of cultural genocide?

How can we protect and revive the authentic and practical Ganita tradition that was the head of all the Indian sciences? To do this, we must identify the nature of the civilizational ‘Poison Pills’ within Ganita.

Civilizational Poison Pills

Rajiv Malhotra introduced the idea of civilizational poison pills from an Indian perspective in ‘Indra’s Net’. [13]. “Poison pills are those elements or tenets that cannot be digested into the DNA of a predator, because consuming them would lead to the destruction of the predator’s constitution. If a predator absorbs such an element, it will mutate so profoundly that it will lose its original identity and qualities.”  We now try to identify the poison pill in Ganita that needs to be preserved.

Ganita’s Poison Pill

The Indians achieved a smart reduction in uncertainty in calculations to a contextually admissible level, instead of beating themselves up trying to attain complete certainty. Ganita and Vedic thought recognizes that human understanding of the cosmos is never fully complete. In [4], the Indian and western mindset is compared thus: “Indians indeed find it natural to engage in non-linear thinking, juxtaposing opposites and tackling complexities that cannot be reduced to simple concepts or terms. They may be said even to thrive on ambiguity, doubt, uncertainty, multitasking, and in the absence of centralized authority and normative codes. Westerners, by contrast, tend by and large to be fearful of unpredictable or decentralized situations. They regard these situations as problems to be fixed. As we shall see, there is in fact some scholarly evidence that demonstrates this view of Western attitudes.” For a mindset that revels in perfection, this element of uncertainty that was acceptable within Ganita is a poison pill. This anxiety was evident in all stakeholders in Europe during the Ganita-Math encounters.

Western Fear of Uncertainty

Practically every Western point of view from the ultra-secular, to the religious during the Ganita-Math encounters was in conflict with Ganita’s poison pill:

  • In the abacus-algorismus battle, Ganita’s idea of ‘one manifesting as many’ in its place value system and the way it managed non-representability was suspect, given the scope for ‘chaos’ and ‘fraud’.
  • For a reasoning mind like Descartes, measuring the ratio of curved to straight lines involved an irreducible uncertainty, an understanding of which was beyond the human mind. This gave rise to the term ‘irrational numbers’ [1]. Not surprisingly, he rejected the idea of infinitesimals too.
  • Philosopher Thomas Hobbes was no friend of the Jesuits. But he too found the absolute, perfect order found in Euclidean geometry was its most appealing aspect and reflected his own perspective. As noted in [12] “in their deep structure, the Jesuit papal kingdom and the Hobbesian commonwealth are strikingly similar. Both are hierarchical, absolutist states where the will of the ruler, whether Pope or Leviathan, is the law.”
  • The Jesuits, Protestants, Eastern Orthodoxy, Anglicans, and a vast majority of Christian sects may have disagreed on some theological points, but all subscribed to the history-centric truth claims of the Nicene Creed [4]. At least three aspects of Mathematics would’ve appealed to them:
    • Calendar and time-keeping helped preserve history centric dogma and reestablish the importance of clergy.
    • The top-down, hierarchical perfect Eucliean order.
    • Proving theorems without need for empirical demonstration. History-centric Christianity treats the body as a vessel of original sin. Embodied knowing is problematic for this mindset.
  • Pioneer Jesuat monk Cavalieri underwent an inner struggle [12] after ingesting this poison pill, and all but disowned his Ganita-based idea of ‘indivisibles’.
  • Scientists who championed the cause of the infinitesimals, and their successors could never come to grips either. The Tagore-Einstein conversation is a good example. As mentioned in [4] “Not even Einstein was able to reconcile himself to the uncertainty inherent in quantum mechanics, prompting him to remark: ‘God does not play dice with the universe.’ But Shiva and Parvati, the Hindu cosmic couple, do happily play dice. Indian philosophy is receptive to the uncertainty theories of physics.

See Article 

However this poison pill does not negatively impact the Indian mindset. Why? Our Ganita Post discusses in detail, but we briefly summarize here for the sake of completion.

Ganita’s Comfort in Dealing with Uncertainty

The Indians were comfortable working with contextually accurate estimates for non-quantities like √2 and π, recognizing that the result could be improved upon.  Hindu society has no central authority that could ban innovation or the exploration of the realms of uncertainty. Its decentralized structure produced independent thinkers and innovators in every era. Dharma systems have built-in safeguards against Hobbesian/Church absolutism. As Rajiv Malhotra explains in [4] “Chaos is entrenched in the Vedas, the Puranas and Hinduism in general for a reason: its role is to counterbalance and dilute any absolutist tendencies as well as provide creative dynamism through ambiguity and uncertainty.” Ganita inherits all these features, and must retain all these properties for best results.

The inevitability of uncertainty was no cause for panic. It even opened up a degree of freedom for (dharmic, ethical) optimization using Yukti.  This comfort with uncertainty is visible right through Ganita’s storied history from Paanini‘s Ashtadhyayi before the common era, to the Aryabhatiya in the 5th century C.E, within the calculus results of Madhava in the 14th century, to Ramanujan in the 20th century. This perspective placed the Indian creation of all its algorithms, interpolations, calculus, etc. on solid epistemological ground. Let’s look at the Aryabhatiya, as an example.

Aryabhata‘s R-sine difference table shown below required an algorithmic package that managed uncertainty every step of the way in a transparent manner: one method for estimating square-roots, another for interpolation, and yet another non-mechanical exception step to generate an optimal final estimate for each value in the table. The Kerala Ganita experts extended such prior work to infinite series, including their own innovative exception terms [1].

Source: Indian lecture series on Mathematics [14]

Western mathematicians who reviewed Ramanujan’s notes found that he often used the terms “nearly” and “very nearly”[10]. Ramanujan came up with clever, non-mechanical approximations for specific quantities like π. Some of his approximations eventually lead to exact results. His exact infinite series for π triggered the most dramatic leap in accuracy since Madhava [14]. Some examples of his approximations are shown below [10].

ramanujan-1

ramanujan-1

The Indian approach seeks balance between chaos and order [4] and represents a dharmic optimization under uncertainty.

Eliminating uncertainty and deleting Yukti, Upapatti, and Pramana from Ganita to digest it, drains it of key features that make it a powerful and reliable approach for solving real-life problems. Furthermore, lack of Pramana can lead to pseudo-science and fraud, as we will see shortly. Preserving these features within an Indian approach to Mathematics has the twin benefits of recovering pragmatism and making the subject understandable and usable by everyone. It protects against further digestion and denigration of the source tradition.

Finally, How can Ganita preserve this poison pill while continuing to retain its open architecture [13] and confidently exchange knowledge with other cultures?

The need of the hour is a thorough and systematic purva paksha of Mathematics and Modern Science, employing an Indian lens.

We don’t have to be a Manjul Bhargava to experience some differences between Ganita and Math.  We can simply try out the basic instruments employed within each subject.

Indian Rope vs Euclidean Geometry Box

One of C.K. Raju’s most important contributions is his cogent argument for a fundamental change in the way math is taught in Indian schools and colleges.

Source: fastudent.com

The rope is a key entity in Ganita and the Darshanas. A fundamental feature of the rope is its flexibility, reflecting the idea of ‘one manifesting as many’. The night-time confusion between a rope and a snake is an example that has been used Dharmic seekers to communicate the deep ideas about the nature of ultimate reality.

Source: Library of Congress

The knotted rope is a critical component of the ancient Indian navigational instrument known as the rapalagai  or kamal [1]. The ‘Sulba’ in the Sulba Sutras means ‘cord/string/rope’, and the rope served as a measuring tool since ancient times. Consequently, as C.K. Raju notes, the circumference can be the independent quantity measured quite naturally using a rope, with the straight line radius derived from this. A mathematical mind measures the straight line (Euclidean distance) first. A geometry box consists of an assortment of rigid straight-edged tools, and each one is used for a specific operation.

source: Indian Mathematics Lecture Series [14]
A knotted string can measured curved lines. When it is stretched taut between pins, it becomes a straight line, and with one of the pins freed, it behaves like a compass. This strings-and-pins set can be used to construct squares, rectangles, circles, etc, i.e., its flexibility reproduces the functionality of an expensive geometry box at a fraction of the cost. It unlocks the creativity of Ganita and is available even to the poorest student.

Indian Nyaya versus Aristotelian Logic

From the Indian point of view, two-valued (Aristotelian) logic can play a supporting role (e.g. like tarka [22]) but does not enable a person to attain a higher level of consciousness [4]. Note that such reductive logic is different from the holistic logic of Nyaya, which accepts multiple pramanas. In fact, no major school of Indian thought directly mentions deductive logic as pramana [22]. On the other hand, all major Indian schools of thought accept pratyaksha pramana, which in rejected by Mathematics [1]. Misusing two-valued logic (that has no place for uncertainty) as pramana negates Ganita’s poison pill.

Mathematics in India Today

The  current approach to teaching mathematics in India appears to be a stressful  and boring mixture of bits-and-pieces of Ganita mashed up with partially understood formal Mathematics imported from the west. This digested teaching approach has been successful in confounding multiple generations of Indian students. The modern rote/mechanical mode is a distortion of the original approach of recollective memory, which was a distinct mode of learning that cultivated the amazing computational (Ganita) abilities of the Indians [15].

Repeat after me:

“An acre is the area of a rectangle

whose length is one furlong

and whose width is one chain” – Pink Floyd, The Wall.

The 2016 Hindi movie ‘Nil Battey Sannata’ (~ 0/0) dramatizes this state of confusion. The movie claims that Math is a natural enemy of girls (“Ladkiyon ko Maths se purani dushmani hain“). While this may or may not be true,  the daughters of Lilavati  should not experience any difficulty with Ganita. For the great Shakuntala Devi, Ganita was a bandhu, not an enemy. The sophisticated Ganita within Kolam designs attests to the embodied learning capability within women. Let us also not forget the women engineers of ISRO who mastered the Ganita of rockets and spacecraft (yes, Ganita’s calculus without limits can do this well [1]).

ISRO staff celebrating ‘Mangalyaan’ success. credit: www.aniruddhafriend-samirsinh.com

The intrepid mother in the movie tells her daughter that “maths yaad karne ki nahi, samajne ki cheez hai“, while the maths-savvy classmate advises: “ek baar maths se dosti karke dekho, usse majhedaar aur kuch nahi“. A key scene in the movie shows everyday, familiar objects from real life being used to convey this ‘samaj’ – clearly a Ganita rather than an Euclidean solution to an Indian problem [15].

In formal math, even something as simple as a point (Bindu) gets hairy. (Euclid: A point is that which has no part, then graduate to this).  A blind import of western approaches into the Indian classroom without subjecting it to a thorough purva paksha,  is a folly not just restricted to Ganita, but one that been repeated in different areas of study including social sciences, economics, religion, art, etc. The net result is years of misery for most Indian students followed by a trip to the west to get it straight from the horse’s mouth. S. Gurumurthy has repeatedly noted the negative impact and the poor track record of such a reductive mathematics in solving practical problems in the Indian economic context.  We close with a discussion on contemporary mathematics and the way forward.

The Needham Question
"With the appearance on the scene of intensive studies of mathematics, science,  technology and medicine in the great non-European civilisations, debate is likely to sharpen, for the failure of China and India to give rise to distinctively modern science while being ahead of Europe for fourteen previous centuries is going to take some explaining” - Joseph Needham.

Many Indian scholars have attempted to answer this complex question. However, virtually all of these responses that try to provide social/religious explanations offer little insight due to a shallow understanding of dharma and Ganita traditions, and the inability to do a systematic Purva Paksha of the western approach using an Indian lens. We quickly summarize three perspectives below noting that we are only scratching the surface here.

A. Several centuries of foreign occupation

This occupation of India ranked among the worst and longest-running genocides in history and was characterized by violence that specifically the Indian intellectuals. Such a strategy is likely to have taken a heavy toll on Indian R&D output and institutions. When there was a sustained break from this violence, e.g., the time period of  the Vijayanagara empire,  we observe that Ganita, Ayurveda, astronomy, and other sciences achieved significant progress.

B. Civilizational inertia: complacency or weariness?

The sharpest debates in India occurred internally, between the various darshanas, which may have shifted the focus away from the study of external cultures entering India. There appears to be no evidence of a thorough study of the axiomatic approach from a native perspective. The Indians may have identified the lack of integral unity in the western approach and rejected it without any further examination of possible useful features.  CK Raju notes in [1] that it was only in the 18th century that India got the Elements translated from Persian into Sanskrit (by Jai Singh). This lack of a systematic Purva Paksha is not limited to Ganita alone but is also seen in many other areas, as pointed out by Rajiv Malhotra [16], suggesting an overly inward focus, careless disunity against an external threat, and a lack of strategic thinking.

C. The Unreasonable Effectiveness of Mathematics

Roddam Narasimha’s analysis examines a question complementary to Needham’s: what are the reasons for a sudden European resurgence after 1400+ years of backwardness in science and technology? He cites a key reason for their resurgence in the 17th century: the mathematization of science. Galileo is his study of the motion of falling bodies, used the calculus (via Cavalieri) to came up with the ‘law of the parabolic fall’. This is considered the first ever quantitative representation of motion using mathematical equations [12].  Scientists thereafter began to develop effective quantitative models relating different physical quantities like velocity, momentum, etc. using abstract models and calculus.Newton titled his famous scientific work as ‘Principia Mathematica‘. These mathematical models, however ‘wrong’ they may be, helped in new discoveries.

Indian Ganita experts too may not have anticipated this unreasonable effectiveness of mathematics when they rejected it for centuries. Narasimha summarizes this in [17]:  “Modern science seems to have acquired, perhaps by fortunate accident, the property that the great Buddhist philosopher Nagarjuna called prapakatva: i.e., it delivers what it promises; it may not be the Truth, but it is honest“.

The Road Ahead

Ganita, in the more recent interactions with modern science and math has made positive contributions, e.g., Satyendranath Bose and Narendra Karmarkar. The Bose-Einstein statistics comes out of counting exercise and is a significant contribution to Quantum Mechanics[17]. Karmarkar is famous for inventing the first practically effective algorithm for solving linear programs that is also theoretically efficient. Karmarkar’s proof of convergence demonstrates Yukti in gradually reducing the level of uncertainty in the solution quality in way that is both practically viable, and theoretically rigorous (a teeny bit of uncertainty remains in the end but it can be safely ignored).  Clearly, interacting with and exchanging ideas with other cultures can be beneficial, provided it is done with eyes wide open.  Scientists and applied mathematicians today employ a variety of different methods, including deduction, induction, inference, etc., along with empirical validation, etc., to come up with new findings and inventions.

Per Roddam Narasimha, the Indians paid a price for rejecting the axiomatic approach, but their stance was vindicated later by the 20th century developments in Quantum/Classical Mechanics and Logic [17].  Furthermore, modern science is being increasingly plagued by a variety of harmful ‘viruses’ that would not affect a ‘Ganita OS’.

Unreasonable expectations from Mathematics

The mathematization of science has succeeded, but only when the order it brings is honestly balanced by the reality check of an unpredictable nature.  The unbalanced mathematization of economics has resulted in a series of spectacular failures when applied in real life. Indian thinkers like S. Gurumurthy have studied these economic models in depth, and opted for a balanced Ganita-like method, bringing in empirical validation and Yukti to determine practical solutions anchored in Indian reality. Western social science, which mimics the axiomatic approach is degenerating into a self-serving pseudo-science that offers little insight. A sizable proportion of results published in modern scientific journals are not reproducibleThis highly cited 2005 article discusses the implications.  And then there is the issue of fraud that is peculiar to the western modeling approach based on Aristotelian logic.

Falling for Supermodels
Without+photoshop+_305e904f954ae7c6b82bd7893278408d
Source: funnyjunk.com

Supermodels sell an advertising pitch, not reality. Yet the temptation of falling for the perfection of abstract math models and ignoring the uncertainty of the real world can be too strong. As [17] notes: “The history of Western science is shot through with the idea of theories and models and of fraud. Ptolemy himself has been accused of fraud; so in more recent times have Galileo, Newton, Mendel, Millikan and a great variety of other less well-known figures. I believe the reason for this can be traced to faith in two-valued logic.” All models approximate reality. When this gap gets too wide, it makes sense to reject that model. However, it is tempting to reject reality in favor of a pet model or preferred hypothesis by cherry-picking data, fudging results, or tweaking the model in ‘creative’ ways to ‘make’ it work (e.g. some ‘AIT’ models in the Indian context).

Ganita does not suffer from this issue. Why? As noted in [17] that when “observation is the starting point and one has no great faith in any particular physical model, which was the prevailing norm of Indian scientific thought, the question of fraud does not arise. Indian scientists, even classical ones, do not appear to have accused each other of fraud. This could not have been mere politeness, as they did make charges of ignorance or even stupidity against each other (as Brahmagupta did on Aryabhata, for example). We could say that fraud is the besetting sin of a model-making scientific culture“.

Synthetic unity has its advantages and has revolutionized modern science, but progress based on Integral unity is more sustainable.

Some western scientists and mathematicians may have sensed this lack of Pramana. Poincare explored the role of intuition and inference in his candid 1905 essay [18]. We even get a hint of integral unity here. Albert Einstein was aware of the limitations of Math when he noted “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” Contemporary mathematician Terrence Tao recognizes that there is more to mathematics than just rigor and proof [19]. Thus, we see a limited move by Math toward the Ganita position while remaining firmly grounded in its native western tradition. Ganita can reciprocate in mutual respect, anchored in its own epistemology. We conclude with an informal discussion on emerging technologies.

Digestion by Machine: Math versus Ganita

Ganita is well-suited for this era of decentralized internet, analytics, big data, and digital computing which is algorithm driven. The emerging world of Artificial Intelligence is also very interesting. We touched upon AI citing an important observation of Subhash Kak [20] in our post on Ganita. As AI becomes highly sophisticated, it will be able to automate many human capabilities. It may eventually master the axiomatic approach and digest the Euclidean mathematician.

On the other hand the Indian approach to knowledge is rooted in the correspondence principle of Bandhu. Potential fallibility is acknowledged. Machines cannot replicate embodied knowing since they lack Bandhus, and they will not have the ability to attain a higher state of consciousness. For example, machines cannot chant mantras. Next, this ‘Euclidean’ robot will be able to master scriptures, and emulate all text-prescribed functionality of a cleric. It can function as a virtual holy establishment by delivering impeccable discourses. It will become an expert of theology by encoding history-centric truth claims as axioms and applying two-valued logic. However, it cannot become a Yogi.  Learning Ganita and internalizing the Dharmic worldview offers job security in the world of robots!  India can lead the way forward by carefully reintegrating useful features of modern science and math into its Vedic framework [21].

References:
  1. Cultural foundations of mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE, C. K. Raju. Pearson Longman, 2007.
  2. Plato on Mathematics. MacTutor History of Mathematics archive. 2007.
  3. Plato’s Theory of Recollection. Uploaded by Lorenzo Colombani. Academia.edu. 2013.
  4. Being Different: An Indian Challenge to Western Universalism. Rajiv Malhotra. Harper Collins. 2011.
  5. Axiomatism and Computational Positivism: Two Mathematical Cultures in Pursuit of Exact Sciences. Roddam Narasimha. Reprinted from Economic and Political Weekly, 2003.
  6. Use and Misues of Logic. Donald Simanek. 1997.
  7. Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhasa. C. K. Raju. 2001.
  8.  American Veda: From Emerson and the Beatles to Yoga and Meditation How Indian Spirituality Changed the West. Phil Goldberg. Random House LLC. 2010.
  9. Logic in Indian Thought. Subhash Kak.
  10. Ramanujan’s Notebooks. Bruce Berndt. Mathematics Magazine (51). 1978.
  11. C. K. Raju. Teaching mathematics with a different philosophy. Part 2: Calculus without Limits. 2013.
  12. Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. Amir Alexander. Farrar, Straus and Giroux reprint / Scientific American. 2014.
  13. Indra’s Net: Defending Hinduism’s Philosophical Unity. Rajiv Malhotra. Harper Collins. 2011
  14. Mathematics in India – From Vedic Period to Modern Times: Video Lecture Series, by M. D. Srinivas. K. Ramasubramaniam, M. S. Sriram. 2013.
  15. Mathematics Education in India: Status and Outlook. Editors: R. Ramanujam, K. Subramaniam. Homi Bhabha Centre for Science Education, TIFR. 2012.
  16. The Battle For Sanskrit. Rajiv Malhotra. Harper Collins. 2016.
  17. Some thoughts on the Indian half of Needham question: Axioms, models and algorithms. Roddam Narasimha. Infinity Foundation. 2002.
  18. Intuition and Logic in Mathematics. English Translation of Essay by Henri Poincaré. 1905.
  19. The Pragnya Sutra: Aphorisms of Intuition. Subhash Kak. Baton Rouge, 2006.
  20. There’s more to mathematics than rigour and proofs. Terence Tao. 2009.
  21. Vedic Framework And Modern Science. Rajiv Malhotra. Swarajya Magazine. 2015.
  22. Epistemology and Language in Indian Astronomy and Mathematics. Roddam Narasimha. Journal of Indian Philosophy, 2007.
  23. The Math Page. Plane Geometry: An Adventure in Language and Logic based on
    Euclid’s Elements. Lawrence Spector, 2016.
  24. Continuity and Infinitesimals. Stanford Encyclopedia of Philosophy. 2005, substantive revision 2013.
  25. The Indian Origins of the Calculus and its Transmission to Europe Prior to Newton and Leibniz. Part II: Lessons for Mathematics Education. C. K. Raju, 2005.
  26. Why Write: Legos, Power, and Control.  F. D. Poston. Johns Hopkins School of Education.
  27. Indo-Portuguese Encounters: Journeys in Science, Technology, and Culture. Edited by Lokita Varadarajan. Indian National Science Academy. 2006.
  28. The Kerala School, European Mathematics and Navigation. D. P. Agarwal. Infinity Foundation Mandala website.
Acknowledgments: I'm deeply grateful to the ICP blogger and editor for their constructive comments, review, and feedback.

An Indic Perspective to Mathematics — 2

This is the Second in a Set of Posts as a follow up to our ‘Introduction to Ganita’.


AlgorithmGanita
Source: IIT Lecture Series on Indian Mathematics [14]
This Set of Posts on the Indic Perspective to Mathematics is the third installment of our continuing Series on Ganita.  Our first article in the Series celebrated Srinivasa Ramanujan. The Second provided an Introduction to Ganita. Emphases within quotes are ours.

Topic Outline

Part 1: (Introduction) 

In Part 1 of this Set of Posts on the Indic Perspective to Mathematics, we provided a background on the historical paradigms that drive the engines of Ganita and Western Mathematics respectively.

Part 2: (below) Ganita-Math Encounters. Ganita and Math came face-to-face when Indian Algorithms and Calculus traveled to Europe to help solve two critical problems: calculating with big numbers and managing the infinitely small. In a tense battle, Ganita’s balance of order and chaos prevails over the top-down Euclidean order backed by the church. We become aware of the massive contribution of the Vijayanagara empire to modern science.

Part 3: We adopt an Indic civilizational perspective of the Math-Ganita encounters. This gives rise to  interesting questions like ‘What was lost when Mathematics digested Ganita?’. We also look ahead, exploring the importance of Ganita and its Indian approach in a futuristic world.

Ganita-Mathematics Encounters
Experts have their expert fun
ex cathedra 
telling one 
just how nothing can be done. - Piet Hein.

In the Introduction to this Set of Posts, we studied the Greek origins of ‘Mathematics’. The abstract nature of Mathematics resulted in a drastically reduced practical output and Europe plunged into a 1000+ year dark era. During this period, Ganita contributions from Dharma thought systems helped keep math practically relevant in other parts of the world, right up to the 17-18th century CE. In particular, this injection of Ganita helped resolve two Math crises in Europe [1]. For the purpose of this post, we oversimplify and classify these problems as the ‘big’, and ‘small’ number’ crises. By helping resolve these crises, Ganita played a leading role in the birth and progress of modern science.

Big Number Crisis (Abacus vs Algorismus)

Here is example of a 10-digit Hindu number and its Roman numeral representation.

large numbers
credit: http://forbrains.co.uk/free_online_tools/convert_to_roman_numerals

There are several such websites that allow us to perform this conversion and three aspects stand out. First is the reference to ‘Arab numbers‘ in many sites. Second, is a maximum limit on the input. Third, ‘0’ or negative numbers are not valid input. The idea of ‘Arab numbers’ is of course, deep-rooted in the western STEM community to this day (IEEE journal publication guidelines still refer erroneously to ‘Arabic numerals’) since a large body of Ganita knowledge made it to the west via Arab translations of Sanskrit works. As can be gauged from the conversion tool, the Roman system is cumbersome for doing actual calculations. Its representation is additive in nature and there is no place value for zero, and the idea that placing a ‘0’ after a number would increase its value was befuddling. The west relied on the abacus / counting board, which was adequate for simple arithmetic calculations (the Indians did most of their routine arithmetic mentally). The introduction of ‘algorismus’ from India via Arab sources  around the 11th-12th century CE provided the merchants of Florence with an incredibly advanced way of quickly and accurately performing all kinds of numerical calculations [1].

Although traders found it to be practically useful, resistance to the alien method was stiff and it was several centuries (16th century) before the Hindu system gained unanimous acceptance. Well, almost. The British treasury preferred to place their money in the ‘secure’ hands of the abacus and held out until the 17th century [1].  By that time, the second math crisis in Europe was well underway.

Source: wikimedia.org.

Smiling Boetius‘ works with Hindu numerals to prevail over his opponent, Pythagoras, who is sadly stuck with a counting board abacus. This depiction of the victory of ‘algorismus’ is on the cover of Gregor Reisch’s Margarita Philosophica (1508) [1].

Aside from the suspicion of an Arab source in a crusading world, a technical reason for the distrust appears to be Ganita’s approximation techniques combined with the fear of zeroes being added to make sums bigger. To a mind accustomed to the perfection of Euclidean math, not even the tiniest quantity could be discarded. Such unacceptable imperfections could open the door to fraud and chaos [1]. The Indian approach, since the Sulba Sutras, recognized the non-representability of certain quantities (e.g. √2) and employed pragmatic and epistemologically secure approximation methods without anxiety, in order to reduce uncertainty (round-off error) to within an acceptable level [1]. ‘Algorismus’ was absorbed into European practice in order to resolve real-life calculations, but not the underlying pramana and empirical rationale (e.g. upapatti).  Why?

Small Number crisis (Infinitesimals and the Indian Origin of Calculus)

The Indian Background Story

Source: HaindavaKeralam| Zenith of Vijayanagara Empire
Brothers Harihara and Bukka, with the blessings of Rishi Vidyaranya, laid the foundation for one of the most important empires in Indian and world history in 1336 CE. In particular, the global scientific community owes the Vijayanagara empire a debt of gratitude.

While most regions of 14th century India reeled from the attack of fundamentalist invaders who had already destroyed India’s top universities and institutions, the Vijayanagara Empire became an oasis that protected and nurtured the Dharma. In particular, a school of Ganita was etablished in Kerala thanks to the prosperity and security enjoyed by the region during the Vijayanagara period, between the 14th and 16th century CE. An important member of this Ganita tradition was Madhava of Sangamagrama (~1350-1425 CE). This school produced a illustrious line of scholars who were the genuine adhyatmic and intellectual successors of Aryabhata, Bhaskara, and other great seekers. A major part of the foundation for modern science was laid by the Kerala school and the Ganita tradition they carried forward.

Recall that Aryabhata had already come up with finite difference equations for interpolation by 499 CE to generate fine-grained sine values. His practical approach essentially translates into Euler’s  18th century method for solving ordinary differential equations (ODEs). These results were subsequently improved upon by Brahmagupta (his second order interpolation result is known as ‘Stirling’s Interpolation Formula‘ today),  Bhaskara-2, and others [1]. Today, Indians are familiar with the phrase ‘Tatkal booking’ of train tickets. The ancient Ganita experts had developed algorithms  to calculate the Tatkalika gati of planets, their instantaneous velocity (an important quantity in Newtonian physics), as shown below.

Source: Lecture Series on Indian Mathematics [14]

We can observe a continual progress in India toward calculus, right from Aryabhata [1]. For all practical purposes, the Ganita school in Kerala during the Vijayanagara period can rightfully claim to be the developers of Calculus (from a formal mathematics perspective, western historians credit them for ‘pre calculus’). C.K. Raju has demonstrated the all-around practical viability of this epistemologically secure calculus without the use of ‘limits’ [11].

Madhava gave the world some beautiful and important results in infinite series by 1375 CE, centuries before Newton/Leibniz/Gregory/Taylor/McLaurin & Co.

madhava_collage
Source: Indian Mathematics, An Overview (https://youtu.be/p2WankcGP3Q)

In the derivation of these calculus results we can observe a smart management of the non-representability of infinitesimals based on order counting, along with a judiciously chosen exceptional / end-correction term (right side of the picture above). This is a really cool and important innovation that serves twin purposes, as explained by C. K. Raju below [1].

correction_term

There are many other novel ideas and instances of such Yukti within the Indian approach.  The interested reader can refer to [1] for a detailed description of the techniques employed.

It is worth comparing the meaningful Sanskrit non-translatable abhiyukti (expressing, or translating one’s Yukti in action) to its nearest English counterpart ‘algorithm’. The latter from the Latin ‘algorismus’, which in turn came from Al-khwarizmi who had translated Sanskrit texts of Ganita (see the picture at the top of this post). Jyesthadeva published the Ganita Yuktibhasa around 1530 CE in Malayalam, which provides the detailed mathematical rationale validating the Calculus results[1].

Why was Calculus Important to India?

Madhava’s infinite series with end-correction terms, allowed him to quickly calculate estimates for trigonometric values and π (pi) to very high levels of accuracy. For example, Madhava was able to calculate π to 11 decimal places, which represents both a quantitative, and methodological leap over prior brute-force type approaches (the next such dramatic leap was also due to Ganita, via Ramanujan) [14].  A natural follow-up question is: why were precise trigonometric values useful? Isn’t calculating π to many decimal places purely an academic exercise?  We summarize the reasons below, referring the interested reader to [1] for a detailed description.

Agriculture and Trade were key contributors to an Indian economy that played a dominant role on the world stage from 0 CE (and earlier) through 1750 CE.

Agriculture
  1. Krishi was and is a dominant component of the Indian economy. It was (and still is) dependent on a successful rainy season, which means that accurately calculating the arrival time of monsoons is important. A couple of weeks ago, the Indian government announced a $60M supercomputer project to better predict monsoons.
  2. Vedanga Jyotisha is primarily a science of time keeping that has numerous applications and has been recognized by researchers as a key source of knowledge in the ancient world [1]. It enabled the Indians to maintain an accurate calendar. Thus, from a Krishi perspective, the Ganita of Jyotisha acted as a decision support system for planning and scheduling key agricultural activities.
  3. The Indian calendar date and time was calculated with respect to the prime meridian at Ujjain (long before Greenwich), which was then re-calibrated to obtain local times at locations all over Bharatvarsha that covered a vast area (ancient India was united by time too!). This local re-calibration:
    • ⇒ required the calculation of the local latitude and longitude (lat-long)
    • ⇒ which (in the Indian approach) used the size of Earth as input
    • ⇒ this required a value for π
    • ⇒ trigonometric values were also needed for lat-long calculations
    • Precise numerical values were required since tiny errors get magnified after multiplication by big numbers (in the order of the Earth’s radius). Thanks to the Ganita tradition, the Indians had access to good estimates that were continuously improved upon.
ujjainmeridian
Source: builtheritageconservation| The Ujjain Meridian
Overseas Trade

India has a culture of calculation and embodied knowing that goes back thousands of years. Many ordinary Indians even today take pride in their ability to think and calculate on their feet, or pull off some Jugaad without the aid of electronic devices. The pattern-seeking Indian nature is visible in their traditional approach to navigation, reflecting an ability to discover sufficient order even within an ocean of chaos. The metaphor of the Samudra Manthana truly comes alive here.

  1. India, thanks to its manufacturing and technological prowess, had established lucrative trading relationships as a net exporter with several countries, from ancient Rome to the far east. Much was this was done through open sea routes, and not just sailing close to the coast [1].
  2. Prior to the 11th century CE, accumulated navigational knowledge included seasonal wind patterns (‘wind lore’), nature of ocean currents (‘current lore’), etc., and the empirical wisdom of sea-craft. The ancient Tamizh seafarers made use of the Saptharishi mandalam (Ursa Majorin the southern hemisphere. This database of seafaring wisdom and best practices were preserved, improved upon, and transmitted from generation to generation via the oral traditions of the seafaring Jatis [27].
  3. Thus, the Indian sailors had already established a tradition of navigation and deep sea voyage without written charts (they rejected the method of dead reckoning‘ in order to stay alive). Their approach included an empirical understanding of ocean patterns, Ganita, and instrumentation like the rapalagai (kamal) for celestial observations. Tamizh navigators deciphered currents using a simple device known as mitappu palagai [27].
  4. Such historical data further debunks the theory that oral traditions were ‘pre-rational’ and the sole preserve of Vedic scholars. Hinduphobic Indologists like Sheldon Pollock are dismissive of such priceless oral traditions [16]. The western universal idea of history begins with written text and it is tough for this mindset to imagine open-sea navigation without written charts.
  5. Accurately determining the local lat-long using celestial observations (solar altitude at noon, pole star at night, etc.) was part of this approach.
  6. More reliable navigation in the open seas is possible if the 3L: latitude, longitude, and loxodrome can be accurately obtained for any given location. These were indeed calculated in multiple ways by the Indians using trigonometric values [1].
chola sea route pic
Source: Indo-Portuguese Encounters [27] | Chola Sea Route
Continual Progress in Calculating Accurate Trigonometric Values
  1. Aryabhata’s astounding publication of his R-sine difference table along with an interpolation method stepped away from the geometrical approach that was employed until then [1]. The Aryabhatiya was a prized intellectual property of its time. It significantly improved the accuracy of trigonometric values (given the sine value of an angle, one can use elementary identities to calculate all other trig values).
  2. Aryabhata’s work paved the way for Calculus. Over the next 1000 years, the Indians steadily improved upon prior estimates.
  3. Calculus was a natural outcome of this process of deriving ever more accurate trigonometric values. The Kerala school’s calculus extended the finite series based trigonometric results to a highly accurate infinite series based approach.

We refer the reader to this essay [28] by D.P. Agarwal for his summary of the Kerala School, European Mathematics, and Navigation. It is highly likely that this Ganita knowledge traveled to Europe via European missionaries in Kerala and played a key part in revolutionizing physics and mechanics via Newton’s Principia Mathematica and other works.  This story serves as background for the question: why did the idea of ‘infinitesimals’ which was a non-issue in the Ganita world, spark a crisis in Europe?

The European Background Story

Ancient Greek math hit a roadblock after encountering paradoxes tied to infinitely small quantities. Mathematics could not deal with the irritating uncertainty around infinitesimals and the problem of non-representability: For example, an infinite number of threads of minuscule but nonzero length, joined end-to-end should yield an infinitely long thread. On the other hand, combining even an infinite number of threads of ‘zero’ length would only yield zero. Aristotle believed that continuum could be divided endlessly and could not be made up of ‘indivisibles’.

A famous paradox (which used to be popular among those preparing for engineering school entrance exams in India) is that of Achilles and the Tortoise. Around 500 BCE,  Zeno of Elea came up with several such paradoxes that exposed the gaps in a seemingly perfect mathematics and two-valued logic. Unable to satisfactorily resolve such contradictions and deal with non-representability of certain quantities (a fundamental requirement for numerical calculations), Greek progress halted. The dark ages robbed the west of native expertise and appears to have hurt them in key areas including, but not limited to [1, 27]:

  • Astronomy, Navigation, Instrumentation
  • Calendrical Systems, Ship Building
  • Medicine and Botany

After more than a thousand years, between the 12th-16th century, we can observe the emergence of a new kind of Mathematics in Europe, which was fundamentally different in its epistemology from the Euclidean approach. This knowledge first arrived via Arab/Persian translations of Ganita works in Sanskrit, and later through Missionaries who had direct access to Ganita’s latest results in Sanskrit and local Indian languages. We kick off this discussion using the European calendar as a case study.

Trick question: What came after Thursday, October 4th, 1582 in Europe?

The answer is Friday, October 15th. The European (Julian) calendar was slow by about 11  minutes per year for about 1200 years across their dark age. Church and Biblical dogma reigned supreme from the time the Nicene creed was formalized in 325 CE. This dogma can be best understood as an instance of history-centrism [4], and a key to preserving the credibility of this ‘history’ of unique divine intervention is proper time keeping and dating of these events. This was a key motivation behind the European quest for a better calendar.

The Indians had maintained accurate calendars since ancient times thanks to Vedanga Jyotisha for use within multiple applications, and Buddhists even helped with calendars in China [1] (helping the Chinese is an old Indian habit…). The Roman Church realized in 1582 that their calendar was trailing the correct date by 11 whole days. This key project of calendar reform was taken up by Christopher Clavius (1538-1612 CE), a Jesuit priest. Thanks to his painstaking work, Pope Gregory was able to press the fast-forward button on the calendar (thereafter named after him), recommend a leap year correction, and the rest is history.

Milanese artist Camillo Rusconi’s sclupture, 18th century. Pope Gregory is on top of an urn depicting the 1582 promulgation of the Gregorian calendar. Source: http://vminko.org/ under GNU Free Documentation License 1.3.

C.K. Raju has uncovered the Indian source of this calendar bug fix [24, 1]: “Jesuits, like Matteo Ricci, who trained in mathematics and astronomy, under Clavius’ new syllabus [Ricci also visited Coimbra and learnt navigation], were sent to India. In a 1581 letter, Ricci explicitly acknowledged that he was trying to understand local methods of timekeeping from “an intelligent Brahmin or an honest Moor”, in the vicinity of Cochin, which was, then, the key centre for mathematics and astronomy, since the Vijaynagar empire had sheltered it from the continuous onslaughts of raiders from the north. Language was hardly a problem, for the Jesuits had established a substantial presence in India, had a college in Cochin, and had even started  printing presses in local languages, like Malayalam and Tamil by the 1570’s.“. The Jesuits have continued to exercise their influence on the Indian education system to this day. They also played a key role in the second Math-Ganita tussle.

Jesuit (Euclidean) Order versus Indian (Ganita) Chaos

The Jesuits are members of the Society of Jesus, an organization founded by St. Ignatius of Loyola (1491-1556) and rooted in Roman Catholicism. Per [12] “In the broadest sense, imposing order on chaos was the Society’s core mission, both in its internal arrangements and in its engagement with the world.

Sir Peter Paul’s ‘The Miracles of Saint Ignatius of Loyola’ (Source: wikimedia.org)

This painting of Ignatius of Loyola is richly symbolic. It depicts the victory of a perfect top-down hierarchical order over chaos. Loyola and his back-robed Jesuits are in the middle, watched over by angels at the top. Loyola is calmly performing an exorcism, expelling the chaotic evil spirits possessing the bodies of terrified people at the bottom of the picture. [12] provides an insightful description of this picture, noting the role of the black robed Jesuits of Loyola’s Society of Jesus: “They are Ignatius’s army, there to learn from their master, follow his directions, and ultimately take over his mission of turning chaos into order and bringing peace to the afflicted. For that was indeed the “miracle” of St. Ignatius and his followers. Like no one else, they managed to restore peace and order in a land torn apart by the challenge of the Reformation.“.

The Church gained immensely via this decisive mathematical triumph of calendar reform, and Clavius who played an instrumental role, realized the benefits obtainable by investing in mathematics. This was a time period characterized by fissures and dissent in Christianity, with several alternatives and reformations (e.g., led by Calvin) cropping up that challenged the exclusive authority of the catholic church. In this climate, Clavius felt that the top-down hierarchical perfection within Euclidean geometry would be a great fit for the Jesuit curriculum, and in sync with the primary goal of their founder St. Ignatius of Loyola.

As mentioned in [12] “It was clear to Clavius that Euclid’s method had succeeded in doing precisely what the Jesuits were struggling so hard to accomplish: imposing a true, eternal, and unchallengeable order upon a seemingly chaotic reality. Just as Ganita was recognized as the foremost of the sciences in India since ancient times, Euclidean Mathematics became a most important subject in Europe after the calendar reform. The Society of Jesus embraced Math and all was well for a while. The focus had shifted to other pressing topics. For example, navigational challenges had to be overcome in order to ‘discover‘ reliable sea routes to new lands.

The Indivisibles

Calculus created a rather sudden splash into Europe within 50 years of the calendar reform [1]. By that time, the calculus, which was rooted in Indian epistemology had already been developed and studied for two centuries.  Bonaventura Cavilieri (1598-1647), a Milanese Jesuat monk and a student of Galileo was an early adopter. While the Jesuits were more like a MNC, the Jesuats were a local group of Italian monks lower in the pecking order. However, Galileo’s endorsement boosted Cavalieri’s profile significantly. Cavalieri introduced the ‘method of indivisibles’, in which “planes and solids had an indeterminate number of indivisibles” and authored the book Geometria indivisibilibus (Geometry by Way of Indivisibles) in 1635 [12].

While the idea of indivisibles was embraced by the Galileans, the Jesuits were not as welcoming. Those who worked with infinitesimal quantities did so for its practical value in generating realistic new results and could not really establish any logical consistency needed to prove infallible theorems. Unlike Euclid’s Elements which used top-down deductive logic to prove specific theorems from axioms, the use of infinitesimals required the ground-up Ganita approach: to start from physical reality and work toward generalized results, which could lead to innovation and potentially unpredictable discoveries. Clearly, Yukti was not welcomed by the church whereas Galileo’s methods were more compatible with Ganita.

Galileo Galilei (1564 – 1642 CE)

Galileo had become a formidable opponent by that time. He had earlier discovered the moons around Jupiter, and as a prashasthi [16] to a rich grand duke who ruled Florence, named the moons after him and his family. In return, he was rewarded with benefits that included the post of ‘Chief Mathematician’ to the Duke in 1611, which also freed him up to pursue his work as an independent researcher. As [12] notes, “The Galileans also sought truth, but their approach was the reverse of that of the Jesuits: instead of imposing a unified order upon the world, they attempted to study the world as given, and to find the order within.” This started a conflict between the Galileans and the Jesuits.

For the church, the idea that matter could be broken down into infinitely small indivisible atoms was unacceptable. The archives of the Society of Jesus in Rome records for posterity the ruling of their leaders in 1632 on infinitesimals [12]:” Judgment on the Composition of the Continuum by Indivisibles”. …The permanent continuum can be constituted of only physical indivisibles or atomic corpuscles having mathematical parts identified with them. Therefore the said corpuscles can be actually distinguished from each other.” The church basically ducked the question of non-representability and banned the idea and the mathematical study of ‘indivisibles’.

Among the critics of these indivisibles was Thomas Hobbes, the philosopher author of the Leviathan, who deeply influenced Western thought. Hobbes was also an excellent mathematician and a devotee of the Euclidean approach. He was bitterly opposed in this battle of the infinitesimals by John Wallis of England, one of the founders of the Royal Society, the new science academy [12]. Wallis had little time for eternal proofs, and was firmly rooted in what we can unmistakably recognize as the pragmatic Ganita approach for solving real-life problems. Hobbes had tried in vain for several years to prove that he could ‘square the circle‘, and each attempt in this futile exercise was eagerly demolished by Wallis and exploited to the hilt in their public feud [12]. Eventually, Wallis’ team ‘won’ the contest (possibly in terms of cultural and scientific acceptance) and Newton came up with his famous work Principia Mathematica that relies heavily on calculus. Interested readers can refer to [12, 1] for a detailed discussion.

The Ghosts of Departed Quantities

It is worth noting some logical inconsistencies in the positions of both sides in this battle. The church was fighting to save their dogmatic belief in an infallible and orderly Euclidean math against a group injecting a practically useful but poorly-understood imported concept into this math. Every researcher seemed to have his own pet model showing how the math of the infinitely small worked.  In an important and devastating piece of satirical writing, Anglican church bishop Berkeley ridiculed the questionable fluxions of Newton, and Leibniz’s ‘infinitesimal change’ as “the ghosts of departed quantities”. CK Raju concludes (as do others) that this calculus was not on firm epistemological ground.

The European approach appeared to be mechanical and did not, for example, employ the end-correction terms that had helped keep Indian derivation transparent and anchored in a valid pramana [1]. Mathematicians could not accept, understand, or were unaware of the Ganita rationale behind the amazing calculus results derived by the Kerala School. For example, it is known that “Newton later became discontented with the undeniable presence of infinitesimals in his calculus, and dissatisfied with the dubious procedure of “neglecting” them” [24].  Mathematics was enhanced so that calculus was eventually placed on a firm formal foundation in the 20th century [1].

Transmission of Calculus from India to Europe

The etymology of ‘calculus‘ (17th century CE, Latin) relates to ‘reckoning’ and ‘accounting’. This focus is entirely empirical and on calculation, far away from the Euclidean world of theorems and proofs. On the other hand, it is directly corresponds in meaning, intent, and usage to Ganita. So far, research has uncovered three kinds of evidence linking Indian Calculus transmission to Europe: documentary, circumstantial, and epistemological. The interested reader is referred to [24, 1] for details. A primary, initial motivation for appropriating Ganita’s calculus results appears to be the practical problem of navigation: to obtain accurate trigonometric values required to calculate the 3L mentioned earlier [1].

A note in [24] on the circumstantial evidence is worth stating: “Unlike India, where the series expansions developed over a thousand-year period 499-1501 CE, they appear suddenly in fully developed form in a Europe still adjusting to grasp arithmetic and decimal fractions“. The 1400+ year discontinuity in the study of infinitesimals  in Europe was followed by a sudden upsurge in results in the 16th-17th century [12], right after Ganita’s documented achievements in Kerala and the establishment of European missions along the west coast. In fact, this was also a period when results from Ayurveda and Siddha began traveling to Europe giving birth to modern Botany, and similarly revolutionizing western medicine, health-care, and sanitation.

Epistemological Evidence

The epistemological evidence is fascinating to read [1]. A barrier in the western mindset as far as dealing with uncertainty manifests itself clearly in both the first and second math crises. As noted in [24]: “The European difficulty with zero did not concern merely the numeral zero, but related also to the process of discarding or zeroing a “non-representable” during the course of a calculation—similar to the process of rounding. Though the Indian method of summing the infinite series constituted valid pramana, it was not understood in Europe; the earlier difficulty with non-representables zeroed during a calculation reappeared in a new form. This was now seen as a new difficulty—the problem of discarding infinitesimals… In both cases of algorismus and calculus, Europeans were unable to reject the new mathematical techniques because of the tremendous practical value for calculations (required for commerce, navigation etc.), and unable also to accept them because they did not fit in the metaphysical frame of what Europeans then regarded as valid“.

Another instructive story (see page 3 of this essay), highlighting the outcome and unintentional humor caused by a borrow-copy-paste of Ganita without fully understanding its epistemology, is about how ‘sine’ and ‘cosine’ entered Europe. These mistranslated terms destroy the insight behind the original Sanskrit terms jya and kojya [1], baffling generations of Indian students studying Trigonometry.

To this day, neither organized religion and its theology, nor secular mathematicians, have been able to fully embrace the epistemology and validation procedure of Ganita. Why is this? And examining this question from the other direction, why did the Indians not take Euclidean math seriously for two thousand years? What is the future of Ganita? We study these civilizational perspectives in the third and concluding Post of this Set.

Selected References
  1. Cultural foundations of mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE, C. K. Raju. Pearson Longman, 2007.
  2. Plato on Mathematics. MacTutor History of Mathematics archive. 2007.
  3. Plato’s Theory of Recollection. Uploaded by Lorenzo Colombani. Academia.edu. 2013.
  4. Being Different: An Indian Challenge to Western Universalism. Rajiv Malhotra. Harper Collins. 2011.
  5. Axiomatism and Computational Positivism: Two Mathematical Cultures in Pursuit of Exact Sciences. Roddam Narasimha. Reprinted from Economic and Political Weekly, 2003.
  6. Use and Misues of Logic. Donald Simanek. 1997.
  7. Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhasa. C. K. Raju. 2001.
  8.  American Veda: From Emerson and the Beatles to Yoga and Meditation How Indian Spirituality Changed the West. Phil Goldberg. Random House LLC. 2010.
  9. Logic in Indian Thought. Subhash Kak.
  10. Ramanujan’s Notebooks. Bruce Berndt. Mathematics Magazine (51). 1978.
  11. C. K. Raju. Teaching mathematics with a different philosophy. Part 2: Calculus without Limits. 2013.
  12. Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. Amir Alexander. Farrar, Straus and Giroux reprint / Scientific American. 2014.
  13. Indra’s Net: Defending Hinduism’s Philosophical Unity. Rajiv Malhotra. Harper Collins. 2011
  14. Mathematics in India – From Vedic Period to Modern Times: Video Lecture Series, by M. D. Srinivas. K. Ramasubramaniam, M. S. Sriram. 2013.
  15. Mathematics Education in India: Status and Outlook. Editors: R. Ramanujam, K. Subramaniam. Homi Bhabha Centre for Science Education, TIFR. 2012.
  16. The Battle For Sanskrit. Rajiv Malhotra. Harper Collins. 2016.

(A full list of references will be published along with Part-3).

Acknowledgment: Big thanks to the ICP blogger and the editor for their constructive feedback, patience, and comments that helped shape and improve this post.

An Indic Perspective to Mathematics — 1

This is the first of a 3-part set of Posts that follows our ‘Introduction to Ganita’


baudhayanatheorem
Pythagorean or Baudhayana Theorem? (from Bhaskara’s Lilavati)
Topic Outline

This Post studies from an Indic perspective, the path taken by Mathematics from ancient Greece to reach its present form. We compare and contrast Math with Ganita (introduced in our previous post) and in this process, also gain a better appreciation for Ganita. In some places, oversimplifications are employed for ease of understanding, and to bring into focus certain latent aspects of the discourse. All emphases within quotes are ours.

For convenience, this Post has been divided into a set of three, to be published consecutively. The first part is presented today, but the entire set is previewed below:

Part 1: We study the origins and motivations of Math and the pivotal roles of Plato, Aristotle, and Euclid (via Elements) in shaping the initial course of Mathematics. We compare the Indian and Greek logic, noting the non-universality of logic. To each civilization and culture, their own: Pramana versus Proof. A fundamentally different understanding of the nature of ultimate reality guides the Math and Ganita approaches: The integral unity underlying Ganita versus a synthetic unity in which Math lives as a separately independent component.

Part 2: We observe and learn what happens when Ganita encountered Math. Sparks fly in a tussle between order and chaos when two sharply different approaches clash.

Part 3: We adopt an Indic civilizational perspective of the Math-Ganita encounters. This gives rise to  interesting questions like ‘What was lost when Mathematics digested Ganita?’. We also look ahead, exploring the importance of Ganita and its Indian approach in a futuristic world.

Part 1:Introduction
Dolores Umbridge: It is the view of the Ministry that a theoretical knowledge will be sufficient to get you through your examinations, which after all, is what school is all about.

Harry Potter: And how is theory supposed to prepare us for what's out there?

(Harry Potter and the Order of the Phoenix, by J. K. Rowling).

Mathematics is the ‘science of learning’ that originated in ancient Greece, and comes from the Greek root mathesiz, or learning [1]. Plato’s Republic (~375 BCE) mentions the five specific disciplines of mathematics as: Arithmetic, Astronomy, Plane and Solid Geometry, and Harmonics [2]. Plato founded the Academy in Athens and gave Western (Greek) philosophy to the world.  ‘Learning’ had a specific meaning in this philosophy. His ‘theory of recollection’ indicates that ‘mathesiz’ is all about a soul recollecting the knowledge it has forgotten. We cannot learn anything new, and only recall what we forgot [3]. His teacher was Socrates, and Aristotle was his famous pupil.  Plato took as ideal that which was perfect, unchanging, abstract, even spiritual, and regarded the phenomenal world riddled with uncertainty as inferior. He favored the rational over the empirical, and the goal of uplifting the soul as superior to the task of performing mundane calculations. For example, when it came to arithmetic, his views as the narrator in the Republic were pretty clear [2]:

I must add how charming the science of arithmetic is! and in how many ways it is a subtle and useful tool to achieve our purposes, if pursued in the spirit of a philosopher, and not of a shopkeeper!’

‘How do you mean?’, he asked.

‘I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the mind to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument.”

Several elegant results came out this Greek approach which can be broadly viewed as a sequence of axiom/model followed by the use of deductive logic to prove an infallible theorem [5]. The exemplar for this approach is Elements, the treatise on geometry attributed to Euclid (~300 BCE), and this ancient work played a very powerful role in shaping the course of Mathematics. The impact of Euclidean geometry is visible to this day. However, progress in the realm of practical application and calculation was curtailed by the downgrading or even the elimination of the empirical.  While logic and deductive reasoning are indispensable in detecting inconsistencies in arguments and help in viewing existing ideas more clearly, scholars have recognized the limitations of logic when it comes to understanding the nature of ultimate reality:

  1. Logic can be misused when it is employed to find Truth. About Aristotle [6]: “it was, for him, a tool for finding truth, but it didn’t keep him from making the most profound errors of thought. Nearly every argument and conclusion he made about physical science was wrong and misguided. Any tool can be misused, and in these pre-scientific days logic was misused repeatedly“.
  2. Deductive reasoning can help us analyze existing ideas better and lead us to a different way of tackling a problem, but in itself cannot lead us to new knowledge.  “deduced conclusions are just restatements and repackaging of the content contained in the premises. The conclusions may look new to us, because we hadn’t thought through the logic, but they contain no more than the information contained in the premises. They are just cast in new form, a form that may seem to give us new insight and suggest new applications, but in fact no new information or truths are generated. This is especially noticeable in mathematics…“[6].

This Mathematics lived in an abstract infallible world divorced from reality.  One cannot also overemphasize the impact of Aristotle’s ‘law of the excluded middle’ on western thought – a law that leaves no room for uncertainty. The intellectual ideas of Greece were eventually digested [4] into Christianity via the so-called ‘Hellenic-Hebraic’ synthesis. This should come as no surprise given the motivation for the studying mathematics included ideas of absolute perfection and ‘uplifting of the soul’. Mathematics thus became intertwined with the theology of an organized religion. A comparative study of the Indian and the Greek approach bring out the sharp differences between the Ganita and the Mathesiz approaches. Ganita, the integral science of computing, is not the same as mathematics. Unlike the five categories of Mathematics laid out by Plato, Ganita is all pervasive.

via @Calvinn_Hobbes

In [4], Rajiv Malhotra comments on the influence of Aristotle on western thought: “The Law of the Excluded Middle dictates that the principle ‘P or not-P’ separates one thing from another in an absolute sense. All physical and logical entities are invariant units, mutually exclusive of each other. This is not just a pragmatic criterion for distinguishing one thing from another; it is the very nature of reality in both concrete and abstract realms. The law eliminates the possibility of things being mutually dependent, interrelated and interpenetrated. It is diametrically opposed to the intertwined and fluid relationships characteristic of integral unity…”.

There appears to have existed a state of tension between the fallible-and-real and the infallible-and-perfect domain in the western thought since the time of Plato, which manifests itself today as the anxiety-filled binary of ‘religion versus science’. Since this gap was never breached, only a synthetic unity was ever possible [4], and the resultant western approach is reductionist. The independent parts have to be subsequently synthesized to achieve unity. For example, we read in  [25] that “much of Western civilization is based on separating the parts. One date is separate from another, history separate from math which is separate from biology. It’s a world view we inherited from Newton and Descartes, so useful in many ways and disastrous in others. However, there has always been an alternative view of the universe as a single, totally interconnected system. You’ll find that in Eastern traditions.“. To this day, Mathematics and Science are treated and taught as two different school subjects. A key tussle here is between the ‘lower’ empirical world we can experience, and the ‘higher’ abstract-theoretical domain, with the latter being considered superior. This western view is even being taken as the universal approach to knowledge.

Western Universalism

Today, we can observe the promotion of the notion of a western universalism that traces its origin to the intellectual tradition of ancient Europe. For example, the choice of the logo for UNESCO, a world body, reflects a desire to preserve the memory of Parthenon in ancient Greece, which was damaged in wars eons ago. Key buildings in several prominent universities in the United States are designed to remind viewers of the glory of ancient Rome and Greece.

The UNESCO logo (Credit: wikimedia.org)

The belief in the dominance of Euclidean Mathematics is reflected in the argument between the ancient Greeks and Epicureans.

The Epicurean Ass

The Epicureans opposed the followers of Euclid who, from their perspective, appeared to be proving obvious results. For example, consider the following proposition in Elements as discussed in [23]:

Any two sides of a triangle are together greater than the remaining side.

In other words, a straight line is the shortest distance between two points!

If anyone wanted to ridicule mathematics for its insistence on the axiomatic method of orderly proof, this theorem offers a wide target. In fact, the Epicureans (those Athenian free-thinkers, who defined philosophy as the art of making life happy) did exactly that. They said that this theorem required no proof, and was known even to an ass. For if hay were placed at one vertex, they argued, and an ass at another, the poor dumb animal would not travel two sides of the triangle to get his food, but only the one side which separated them.”

C. K. Raju explains both sides of the argument [7]: “Proclus replied that the ass only knew that the theorem was true, he did not know why it was true. The Epicurean response to Proclus has, unfortunately, not been well documented. The Epicureans presumably objected that mathematics could not hope to explain why the theorem was true, since mathematics was ignorant of its own principles..” In the end, the Greek response cites the authority of Plato that mathematics “takes its principles from the highest sciences and, holding them without demonstration, demonstrates their consequences. [7].

Let us now introduce an Indic perspective.

In contrast with this Greek view, all Indian schools of thought accept empirical means of verification (e.g., pratyaksha pramana [1, 22]) while acknowledging the potential fallibility. All darshanas would reject any axiomatic approach that lacked valid pramana. The use of empirical rationale has existed in India since ancient times, including the Sulba Sutras (800 BCE or earlier) and is different from the axiom-theorem approach. C. K. Raju puts this in perspective: “Because no proof was stated it does not, of course, follow that the authors of the sulba sutras did not know why the result was true. But the method of proof that convinced them may well have  differed from the current definition of proof. Thus, it is incorrect to assert that the constructional methods used in the sulba-sutras implicitly lead to a proof in a formalistic sense. It is incorrect because the rationale for the formula for a right-angled triangle, from the constructional methods of the sulba-sutras right down to the 16th century Yuktibhasa, explicitly appeals to the empirical“. [7]

The Epicurean Ass argument has been kept alive in some form or the other to this day in a western worldview. From an Indian point of view, a Ganita expert like Srinivasa Ramanujan too was deemed a ‘wizard’ [14, Lecture 1] who did not know why his results were true, despite his point that he employed his own valid method, which produced so many astounding new and true results. He had to move from Kumbakonam to work in the U.K. to prove his results to the satisfaction of the formal math community in order to gain acceptance.

Indian Gurus, Yogis, Siddhas, and Tantriks who, through years of practice and sadhana, demonstrated amazing results in transcendental meditation, mind sciences, and medical sciences are sometimes labeled pre-rational Indian ‘mystics’ [4] as opposed to western ‘scientists’ who came up with sophisticated instrumentation that subsequently confirmed these results. Universities like Harvard periodically comes out with a research report ‘proving‘ prior findings in Yoga and Ayurveda from the Dharma traditions, which have been practically employed for centuries.

Public intellectuals like Rajiv Malhotra also ask: How often are these Hindu and Buddhist monks, who are the primary producers of this knowledge, credited as co-authors in the journal papers? This bias is propagated subtly by western scholars who study Hinduism. For example, Phil Goldberg who teaches at Loyola Marmount University, an institution rooted in the Jesuit Catholic tradition, compares ‘Indian philosophy and Western science’ in [8]. He also endorses the rejection of the ‘orange’ [saffron] robe of Dharma in favor of the authoritative western scientific garb of a ‘white lab coat’ in order to increase the credibility of Yoga and meditation techniques in the minds of westerners. Note the approach is one of extracting the benefits, and then rejecting/denigrating the Dharma source. Such biased attitudes have also helped feed an increasing Hinduphobia within western academia.

Two-valued logic is not universal. India had not one but several different schools of thought that also studied logic [22], including Nyaya and Navya Nyaya, as well the Buddhist Catuskoti, and Jaina Syadavada. In fact, the Buddhist understanding of integral unity as encapsulated in Nagarjuna’s brilliant arguments has been recognized as nothing short of a “death-blow to all synthetic unities that start with different essences and then look for unity” [4].

Indian Logic vs Greek Logic

There are several papers available that discuss the Indian approach to logic. For example see this work of Subhash Kak [9] and this discussion of Indian and Greek logic. In the popular textbook example for Indian syllogism versus that of Aristotelian logic, the first thing we notice are the ‘five steps’ in the Indian approach versus three in the Greek template [22]. The steps in the Indian rules of inference are not redundant and serve as a reality-check based on the correspondence principle of Bandhu [9], whereas the Greek argument is restricted to the infallible abstract domain. As Roddam Narasimha notes in [5] where he compares Greek Axiomatism and Indian Computational Positivism, the Indian distrust of deduction-based logic “appears to have been based on the conviction that the process of finding good axioms was a dubious enterprise. Note that logic in itself was not something that was shunned in India; without going into a detailed discussion of Indian systems of logic, it is enough to note here that time and again Indians use deductive logic to demonstrate inconsistencies or to refute the positions of an adversary in debate, rather than to derive what western cultures have long sought through that method – namely, certain truth.“.

The intellectual prowess of the ‘deductive logician’ has been promoted in popular western culture. For example, Sherlock Holmes is recognized foremost for his superb deductive reasoning, and is considered the most portrayed literary human character in history. However, an analysis of his stories show that Holmes relied a lot on anumana (inference) including the so-called abductive and inductive methods, and Conan Doyle did consider Holmes’ methods to be fallible, which resembles a Ganita approach to sleuthing!

Sherlock Holmes Portrait Paget.jpg
‘Sherlock Holmes’ By Sidney Paget (1860-1908) , Public Domain. Credit: Wiki Commons

CK Raju [1] calls out some flaws in the claim to universality of two-valued logic. First, the Hindu darshanas, Buddhist Catuskoti, and Jaina Syadavada offer solid alternatives from a different culture. These alternatives have always been compatible with the latest developments in science at every point in time, including Quantum Mechanics. We do not find any serious ‘religion vs science’ problem in India [4]. Even the materialist Charvaka school would reject this reductive logic for not accepting a Pratyaksha Pramana [1, 22]. Finally, it is tough to justify two-valued logic citing empirical evidence if its claim to dominance lies in its empiricism-free perfection [1].

A remaining argument in favor of a universality of two-valued logic and axiomatism is the endorsement by ‘higher authority’, representing a distorted version of Sabda pramana [22]. Indeed some proofs published in journals today are so abstract and technical that they can only be decoded by top formal mathematicians. The remainder of the global math community take it as truth based on the verbal authority of an elite few.

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true - Bertrand Russell.
Vignette: Demotion of a Theorem

In middle school geometry, we learn about the congruence of triangles and come across the side-angle-side (SAS) postulate [23]:

“The fundamental condition for congruence is that two sides and the included angle of one triangle be equal to two sides and the included angle of the other.”

This result can be easily verified using empirical rationale (proof-by-superposition, as Euclid himself did), and would be perfectly acceptable in Ganita, but not in mathematics. This is because superposition involves moving one triangle and placing it on top of the other, which is considered a ‘fallible’ process. The SAS result is difficult to prove using logic alone and thus the SAS theorem was demoted to the status of an unproven postulate.

We conclude Part 1 by delineating a key, irreconcilable difference between Ganita and Mathematics. This difference also manifests in virtually every other field of study.

Summary: Fundamental Difference between Ganita & Mathematics

The ancient Indians recognized Nyaya (logic) and employed Tarka (reasoning) and even mastered it, but did not put it on a pedestal because of certain limitations. Results in Ganita, like all other Indian disciplines, are tied to a valid Pramana and rooted in reality, rather than an axiom-based proof operating in a separate abstract domain. The empirical approach can elevate the practitioner to a higher state of consciousness (The Bhagavad Gita recognizes it as a valid way to transcendental knowledge [4]).

Subhash Kak summarizes the Indian approach to acquiring knowledge based on bandhus [9]: “The universe is viewed as three regions of earth, space, and sky which in the human being are mirrored in the physical body, the breath, and mind. The processes in the sky, on earth, and within the mind are taken to be connected. The  universe is mirrored in the cognitive system, leading to the idea that introspection can yield knowledge“.  It is worth repeating what has been said before: In nature, the western civilization is intellectual, the Chinese civilization is philosophical, and the Indian civilization is spiritual (adhyatmic).

Ganita is rooted in an integral unity whereas Mathematics exists as a separately independent part of a synthetic unity.

This integral approach produced some of the most important contributions, from Hindu numerals, place value system with zero, to symbolic language for managing equations [5]  and calculus. On the other hand, the abstract nature of Mathematics resulted in a drastically reduced practical output while Europe drifted into a 1000+ year Dark Age. During this entire period, Ganita contributions from all Dharma thought systems proved to be crucial in keeping mathematics practically relevant in other parts of the world, up to the 17-18th century CE. We discuss these Ganita-Math encounters in the upcoming second part of this set of Posts.

Selected References
  1. Cultural foundations of mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE, C. K. Raju. Pearson Longman, 2007.
  2. Plato on Mathematics. MacTutor History of Mathematics archive. 2007.
  3. Plato’s Theory of Recollection. Uploaded by Lorenzo Colombani. Academia.edu. 2013.
  4. Being Different: An Indian Challenge to Western Universalism. Rajiv Malhotra. Harper Collins. 2011.
  5. Axiomatism and Computational Positivism: Two Mathematical Cultures in Pursuit of Exact Sciences. Roddam Narasimha. Reprinted from Economic and Political Weekly, 2003.
  6. Use and Misues of Logic. Donald Simanek. 1997.
  7. Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhasa. C. K. Raju. 2001.
  8.  American Veda: From Emerson and the Beatles to Yoga and Meditation How Indian Spirituality Changed the West. Phil Goldberg. Random House LLC. 2010.
  9. Logic in Indian Thought. Subhash Kak.

(The complete list of references will be published along with part 3).

Acknowledgments: I would like to thank the ICP bloggers for their constructive feedback and the editor for his incisive comments and ideas.