Read Book Online : http://www.archive.org/stream/algebrawitharith00brahuoft#page/n5/mode/2up

Download pdf Book : http://www.archive.org/download/algebrawitharith00brahuoft/algebrawitharith00brahuoft.pdf

The principal Indian writers on algebra and arithmetic generally are Aryabhatta in the fifth century of our era, Brahmagupta who wrote about AD 628 and Bhaskara in the twelfth century. More at : http://www.brahmagupta.net/Early_arithmetic_and_astronomy.html

Hi Amita, Hi Amita - with all due apologies, but isn't 4000 years a bit of a stretch? The Rig Veda is the oldest surviving 'document' (for want of a better word) that we have, and while some of its hymns may be 4000 years old, the maths in it is fairly rudimentary, at least as far as I have been able to tell. Of course, I could be wrong - and would therefore appreciate any further information on this. Thanks.

RBSI - thanks for this!

Thanks for the book, gives an opportunity to compare with Al Beruni's writings :-) Since the Rig Veda is dated as compiled by abt 1000 BC, and if we accept the writings of Al Beruni about Hindu mathematics c 1030 AD, the cause of the degradation of so-called Vedic Mathematics to the mumbo jumbo of the medieval era must be a fascinating story in itself :-).

Amit - yes the story of the development of Indian mathematics IS fascinating. There is surpisingly little 'mumbo jumbo', and a steady growth of mathematical thought - Vedic, late Vedic, Jain and Buddhist. And then came August 25, 458 AD - and the first RECORDED use of zero. :) That changed the history of mankind. By the time Aryabhat, Varahamihira, Brahmagupta and then Bhaskara II did their work, mathematics had become distinct from religion. From the time of Brahmagupta, there was Arab interaction - which added a whole new dimension to mathematical thinking. I could go on and on...but I think I should stop right here!!:)

Interesting information Rohini....Thanks ! Wish you could go on...

Rohini ...just wondering ,,where does Calculus, Trig,, Differential and Difference Equations, Lagrangians , Fourier and Laplace Transforms . Matriices ..and indeed the whole range of Manipulative and Basic mathematics ..fit into all this...Or does it ..either substantially or peripherally ..?

@ Rohini: Do go on...what led to Vedic Mathematics with landmark achievements like the system of numericals (now called Arabic system), zero, planetary observations and calculations decay to the point of being called " consisting of particles of sand with some glittering crystals" by Al Beruni? The Arabic translations and findings (from India and early Europe- esp Greece and Rome)led to the European Renaissance. By the time of Newton and Leibnitz "Vedic Mathematics" was a subject for only Indologists. Except for the celebrated Ramanujam, "a pearl beyond any price"Indian Mathematics languished in the darkness of obscurity. Why?

And pls...my question is to improve my understanding and not to provoke :-)

Amit - yours is an extremely valid question, and one which I am trying to explore. So far, from what I have been able to understand, mathematical development in most of India came to a halt after BhaskaraII, as did most other scientific development. Scholars attribute this to invasion, conquest and resulting political instability. Maths then retreated to Kerala, and the Kerala school of Mathematics flourished, with mathematicians of the stature of Madhava and Nilkantha. In 1835, after Macaulay's infamous Minute on Education and the full fledged introduction of western education, indigenous thinking more or less died. That really was the end. After that, whatever mathematical advances took place did so in the western tradition. In the course of my research, I have noticed one more thing - Western scholars accepted the contribution of the Arabs, but not that of India at all. All ancient advances in mathematics were attributed to the Greeks, and British indologists like Kaye have written reams and reams showing how Indian thought was either non-existent or borrowed from the Greeks. Now why this was so - there are any number of theories. I won't go into those here. That, so far, has been my understanding. If anyone - Amit, Amita, Jaacob, Deepak - has a different perspective, I would love to hear it. What I know is limited, and I am still very much in the process of research and understanding.

Rohini - I was particularly fascinated by the fact that...."Maths then retreated to Kerala, and the Kerala school of Mathematics flourished, with mathematicians of the stature of Madhava and Nilkantha"....Since the south was fairly insulated from oppressive invasions, I would presume that the scientific learning would have continued until 1835. Do you also imply that Tanjore, Andhra and other southern dominions follow the Kerala School ?

Jaacob - at this point, I can't answer your question with specifics. From what I know Calculus is generally attributed to the Greeks. I suspect that some of what you mention will definitely fit in peripherally, but I cannot say so with any degree of certainty yet. Maybe in a year's time I will have more knowledge. :) I am plodding through old texts, medieval commentaries, modern interpretations in an attempt to understand this - and believe me, it is a slow process. The trouble with Indian mathematics has been a) the texts that have come down to us are in many cases only partially preserved, b) the method of discussion employed in those texts is often obscure, so that only the initiates would understand; the texts have therefore needed a lot of study and interpretation in modern times, and their often ambiguous nature has led to controversies. It is almost impossible to understand an original piece of work without the help of commentaries.

@ Rohini and Amita: Probably the manuscripts on leaves and barks did not survive or were destroyed. But fact remains that Indian Mathematics which allowed intricte structures to be built got side stepped into astrology and astronomy. Was this the effect of invasions and drying up of royal patronage? Or was this ossification into the wasteland of soothsaying and divination? Hope you can find some answers, I suspect there are major truths and discoveries lying in wait :-)

Amit - I don't think maths got 'side-stepped' into astronomy. Maths developed initially out of a practical need - that of constructing fire altars for sacrifice and performing other rituals accurately. Maths was not a discipline in its own right at all, not for centuries.

RBSI - I don't know enough about Tanjore and Andhra to make any comment on this. I will have to find out.

Amita - thanks for the details. One of my problem with ancient Indian maths, and with much of ancient Indian history really, is the difficulty of establishing a precise chronology. Another issue with Indian maths is the interpretation, as I have mentioned before somewhere. Very little is stated directly, and that which is is stated obscurely. Also, the sutra form of writing makes interpretation difficult - not just of maths, but other 'sciences' as well. Take Panini for instance - he is almost impossible to understand without his commentators. So I am wary of making definitive statements either about dates, or about the precise extent of mathematical knowledge.

Thank you Amita Roy....most interesting information and references !!

I think we have the basis for a most interesting story :-)

Rohini... May there is enough stuff for an interesting book? written in a style and fluidity that even i can understand?

Rohini...it's remiss of me no to have chanced upon your reply until now. Thank you very much. I'd certainly love to know how Ancient Indian Mathematics ..led in or indeed paved the way for the two principal streams i.e. " Basic Mathematics ' and 'Manipulative Mathematics'. Sadly , although I did Physics past the Post Grad . level , that was over 40 years ago ..and anyway never really had any kind of penchant for it . Just a dilettante's fancy , truth be told.. This stands in stark contrast to two of my friends from yesteryear viz. Sheelkant Sharma ( awarded the Distinguished Alumnus Gold Medal from I.I.T Bombay circa 2006 ) and Christopher Rego ( ex.Maths Fellow at T.I.F.R - who was sadly forced to abandon his research and take over the family tile business in Mangalore ) .

In this conext, we must not forget the Sri Chakra (also called Sri Yantra), a complex drawing of geometry with bindu in the centre and ine interlocked triangles with an Avarana (outer enclosure). It is the oldest sacred drawing. Its origin is unknown but it is being worshipped since thousands of years. If Indian Rishis knew how to draw such a complex diagram, they must have mastered the entire branch of mathematics.

@Rohini : Thank you for invitaion to respond. Elsewhere I recently commented that developments in technology and science may follow different paths. Here also (post Bhaskara), techology continued to advance (irrigation, roads, architecture, metallurgy, etc.) but not so with science (maths, medicine) which "retreated" to the South. Perhaps, relative tranquility in the South could have accounted for this but I am not very sure. Have a look - http://www.ambedkar.org/jamanadas/WhyScience.htm This may not be exactly a scholarly treatise (please ignore some of the incoherent strands) but it does present a viewpoint that intuitively appears to have some force - that the decline of Science in India coinsided with the decline of Buddhism and rise of brahminical Hindu orthodoxy. Debiprasad Chattopadhyaya in Science & Society in Ancient India argues, albeit in the context of ancient Indian medicine, that divisions in society, especially the caste system formed obstructions in the way of scientific development in India. I think it might be worthwhile to explore these arguments.

Deepak, thank you. Yes, I would agree that the caste system was obstructive to scientific development. Knowledge was the preserve of the Brahmins, and they worked hard to keep it so. Which is one reason that the texts were composed in a manner so obscure that the uninitiated couldn't understand. Jaacob, thank you for your reply. If I learn anything of interest, I will definitely pass it on.

There is the risk of making virtue out of the mystic. Care should be taken that we do not read more than what is indicated. For example, names for numbers such as dwadash or treeshat should not be confused with the placeholder value of zero or something else for logarithmic tables. Ability to construct geometrical shapes or yantras do not make anyone great mathematicians. Mathematics like any other science must have developed from generation to generation. We cannot attribute knowledge to this oor that one individual even if he is a Isaac Newton. As Rohini Chowdhuri suggests we should be sure of what we are saying and then we cannt find great mathematics in the past. All we can do is assert that we were with the times or even ahead of the times; not through tall claims but by accepting shortcomings as lack of rigour, lack of symbolic language so perfected by mathematics of today, obscurantism of our predecessors and above all, the break in tradition which may have happened because of the colonial disruption in our history or our own social and philosophical faults as Deepak has suggested above.

The discussion as I see above is very interesting. I am really happy that Rohini is researching Math development in India. Though one must read what the earlier Indologists of Western origin have written, I feel we must do what Rohini is doing. Go to the original sources and try to understand them. Well that is not an easy task. Whatever knowledge was being preserved was done in the form of sUtras.This mode of preserving had become the standard form of encoding scientific knowledge in India in the post-Vedic age. It is tru that the sUtras are so cryptic that many of them are not intelligible without a commentary. Rohini had referred to Panini's AShTAdhyaayi, but I must add here that the aShTAdhyAyI is by far the clearest f the sUtras to interpret. Just a vR^itti is sufficient to understand the main import of the sUtras. If at all it is the Vedanta Sutras that are ambiguous, but the reason therfor is not in the sutras themselves but in the indeterminacy that is essential part of knowledge that is not sensory at all! In a science like Math as in Shulba sUtras etc., there is always an external rel world reference to see and interpret the real meaning of the sutra so is not that difficult.Once a person learns one or two subjects through the sutra literature it becomes easyto understand others. For instance learning to interpret the Vyakarana sutras with the help of commenataries I have found understanding yogasutras easier. The real problem is the sutras also assume quite a bit of interdisciplinary knowledge. That is why it is said a deep knowledge in one of the Vedangas and some superficial or elementary knowledge of others is necessary and sufficient. Coming to Nilakantha and other Kerala school Mathematicians who have dealt with Differential Calculus and some methods of Numericl Integration I am sure they are not in sutra form but may be as poems. In fact Bhartruharis VakyapadIyaM is not in sUtra form at all. It is only in the vedic and post vedic age the sUtra forms were popular.

The slow pace of progress of sciences in India, is due in part to external causes like external aggression, social instability and to continual wars; many here would fault the caste system and the restriction of knowledge to Brahmins. Yes, the restriction of knowledge to one community must have shrunk the talent pool, so to say. But one other reason for not developing rigor in sciences like Math is the Indians' penchant for knowledge that can be applied, to some real world requirement, only meriting the consideration of the Mathematicians. Except in a vague field like Vedanta wherein right or wrong can never be decided with any certainty, we have never exercised the need for "hair splitting". Look at the well developed Science of Grammar - there has been a lot of hair splitting. Whereas the current stance would be to leave language at some point! The same amnount of effort in Math or other Real World Sciences would have given our sciences the much needed rigor.

The Caste system might have been oppressive and detrimental to the society in the last few centuries. But I have actually felt that it was one of the most ingenious methods to preserve, refine and improvise technology. Almost all the castes are vocation based...and from that perspective caste system has existed in every culture...all over the world. A blacksmith's son would continue (not mandatorily) as a blacksmith. Same for a carpenter, mason, priest etc. It ensured a continuity and improvisation of a skill which was basically learnt on the job. It ensured job security, it ensured distribution of skill-sets neeeded for a society...Basically everyone knew what to do and everyone seeked everybodys help....ensuring a kind of harmony within the society.

RBSI - yes, the caste system preserved. but it did not allow for innovation. inquiry, curiosity, growth - all were stifled, or became the preserve of the few. so beyond a point, whatever there was began to get lost. for any branch of knowledge to flourish and retain its vigour, it must be freely available to all. it must be open to inquiry, investigation, scepticism, research. Not be the preserve of the privileged few. We don't have to look far - see how the internet has made open source a reality. A concept such as open source was impossible in ancient India - it could not be. Even in medieval Europe, as long as learning remained confined to the cloisters, it didn't go anywhere. Learning as we all know, is that strange treasure - the more we share it, the more we have of it, the more we hoard it, the less it becomes.

The roti-beti taboos are fast vanishing at the same time one is seeing an upsurge in communal (religious, linguistic and ethnic) intolerance aided by some small, well-to-do, but shrill and vocal sections of our population.

I think the success of a society depends on the size of its collective brain. The more people you leave out, the weaker you become. Historians and fundamentalists often fail to see the importance of the collective brain. The common mistake we all make is to believe that a sutra is the work of one individual or an invention is the feat of only a brilliant mind.

Caste system did not represent just division of labour. Besides being inequitable in terms of distribution of rewards, it also entailed a duly sanctified ban on lateral movements and hierarchical rigidities. Please also see my comment on RBSI photo 245. Development of Technology and development of Science are distinct processes. Technology may produce immediate results and bring prosperity whereas development of Science has long term implications in terms of building healthy society. Perhaps, Shekhar's observations above point at a disconnect in this regard at present.

Thank Shekhar for another succinct observation (... work of one individual ...). An analogy from recent times that readily comes to my mind - "Nelson's Textbook of Pediatrics", a bible for Pediatricians, was originally written in 1933 or thereabouts but even after 20 editions gets published under the same name although scores of pediatricians have contributed to its updations and revisions from time to time. The works of ancient learning (maths, medicine...) attributed to various authors probably are of similar nature - contibutions of several men (and women) over a period of time. Considering the (solwer) pace of developments in those times, the period would be spread over several centuries (as compared to 70-80 years in case of Nelson). They represent original schools of thought rather than individuals.

True Deepak. This is the tradition in India too. Or I would say more so in the ancient Indian Guru parampara. The students do append to the knowledge but it issues forth in the name of the Guru. Take Shankaracharya for instance. Many works are attributed to him. But later day researchers only are able to concentrate on and find subtle clues as to why a particular work could not have been attributed to Shankara. Likewise the other authors. Further, education never became the "state subject " that it is today hence even the monarchs who ruled did not bother to make it "compulsory" for all. This perhaps was the most short-sighted act of all. Further in th Indian tradition if a student finds out by research a fact that contradicts the teacher's view point, stating it openly would mean next to blasphemy. The teacher would condemn the student on reasons other than logic to put him down. Example: Ramanujacharya and Yadava Prakasha. It was only in time that Ramanujacharya could establish himself. Contrast this with the student-teacher relationship between Tyco Brahe and Johannes Kepler! This is one of the other reasons for slow advance of Science. What India can feel happy about is Religion or Clergy did not interact with "heresy" the Galileo was treated by the Church.

Remember Pythagoras and what was done to the student who discovered the irrational number sq rt of 2 !

Indeed! Technology now holds the possibilities of greater collaborative effort and large scale participation in knowledge building cutting across all man made boundaries. Wikipedia is an example of such collaboration, its weaknesses notwithstanding. Incidentally, "Network Dynamics of Knowledge Creation in the Wikisphere" is the topic of Myshkin's dissertation at IIM Kolkata FPM program.

@Shekhar : Our posts crossed. My "Indeed...." was in response to MSS post.