A Presentation on Hindu Contribution to Maths & Science – Archna Sahni
Dear All: Below, a set of slides painstakingly created by Archna Sahni on the Hindu contribution to Science and Mathematics..Please do share widely – especially with the younger ones in your family…
I need volunteers to further develop these slides (and create similar ones on Hindu Contribution to the Arts, Archeology etc).
Please let me know if you have the time and the interest.
Pl also read: The Hindu Contribution to Mathematics Part 1 and Part 2
Indian Contribution to Technology
and these category of posts on Science and Technology in Ancient India. Happy Reading (and sharing!). Jai Hind, Jai Bharat!
Hindu Contribution to Mathematics & Science – Archna Sahni
View more PowerPoints from B Shantanu
Dear Shantanu, While I appreciate her painstaking efforts and good intention, it is somewhat naive, and rife with mistakes. Here is a simple :
1. Scholarly evidence does not suggest that science came to Europe late – no one can refute that Euclid’s elements, Archimides’ work etc. (because their works have survived to this date we know) contain a lot of science and math a lot of which are not found in any Indian text.
2. This statement of some Raju about “During the crusades Greek origin of science was fabricated” is wrong again. We have Islamic sources themselves that quoted Greek works (and they also quoted Indian works). Note that it is from old Greek works again that we know a lot about ancient Indian history – so at least those works could not have been fabricated.
3. Slide 4 : There is a lot of science that doesn’t require large numbers, e.g., plane geometry etc. Even calculation-wise Eratosthenes’ work on the circumference of the earth can be done with numbers which were well expressible during Greek times.
Slides 5 and 6 are fine.
4. India was not the first to discover 0. Mesopotamians discovered it before us (record of 3rd century BC), though our invention was most likely independent of that.
5. Decimal system is not a “contribution”. There is nothing too natural or convenient about the base being 10. It is the place value notation,a nd not decimal system per se, that is good.
6. Brahmagupta : Newton’s discovery of gravity is not the statement that “earth attracts”; rather the statement that the differently defined physical quantities known as weight and mass are related by a simple equation. So Brahmagupta doesn’t get much credit for that notion.
7. There is some possibility that Mesopotamians discovered Pythagoras theorem before us.
8. It is a bit misleading to relate Lilavati’s results to calculus. Calculus was the insight that the inverse process of differentiation gave areas of regions (a priori differentiation and computation of area are two different things, the relation between them was nontrivial). Indian texts did do both of these processes without knowing they were related, but then so did many other cultures.
9. Saying that everything is built of small particles is not the same as atomic theory. And saying that there are 5 elements (in the physical as opposed to spiritual world) etc. is nonsense.
10. The approach of Yoga is systematic, but not scientific. Mind is too subtle a thing to perform controlled lab experiments with, and nothing is quantifiable – in the modern times partial work in this direction has been done though. Yoga can be made scientific, but the traditional yoga texts themselves did not follow any scientific method, due to the sheer complexity of such an endeavor (they were busy with their own spiritual development).
11. Finally, some of the most landmark achievements in Indian mathematics, such as the first ever infinite series for pi, and infinite series expansion for some trigonometric functions, came from the Kerala school. Why is that being ignored in these slides?
I agree with Froginthewell’s comments. Particularly the contribution of the Kerala School of Astronomy and Math. As Elst says, it is this kind of shoddy research that reinforces the charge of pamphleteering and intellectual bankruptcy against the Hindu viewpoints.
J
Dear Froginthewell: Thank you for your comments…Agree with almost all except perhaps the chronology re. invention of “zero” i.e. whether Mesopotamia predates us or not (but this is tied to the larger question around the chronology of Indian History).
The Kerala school is a significant omission..(which I have written about before – here and here) – that needs to be fixed – no questions about that; Also I noticed some references may need to be reinforced – or changed – ideally by someone who knows/ understands Mathematics..
So my question is: do you have time to refine/revise and review these slides and produce a revised version? I am certain Archna would not mind and we shall all be grateful..Pl let me know – either by leavinga comment here – or dropping me an email at JaiDharma AT gmail.com
**
@Jay: I think it is a bit harsh to dismiss this as shoddy research…There may be (are some) inaccuracies but I think the effort ought to be appreciated..particularly when there is so little available in the public domain – written in simple words – and easily accessible – on this topic.
The point about Kerala school – and others highlighted by froginthewell are well taken, though.
Pl await an updated version.
Thanks
Hi Shantanu, Thanks for suggesting about revision, but I do happen to be a bit busy for making with slides; it is far easier to sit in front of a computer and pass comments on someone else’s original work, which unfortunately is all I can do, being busy.
For now, let me just make a few suggestions. First, some references :
(i) http://www-history.mcs.st-and.ac.uk/Indexes/Indians.html
(ii) http://www.youtube.com/watch?v=KZgvi0bcMVY and
http://www.youtube.com/watch?v=Xw2Kmc9Re84
There is also a book by Kim Plofker that I haven’t read, but I read a review by a great mathematician (actually a fields medalist) praising it.
I would also suggest to be careful with works like this Mr. Raju’s that she has quoted. For all I know, Raju may be committed and honest/sincere, but when one quotes opinions like Greek science being fabricated one has to be really careful with fact-checking.
Then there is this subtle point one has to try to be careful about; it is about the scientific spirit of writing. e.g., consider the one about gravity. Archana ji cannot be faulted for what she wrote, because most people are told in school that discovery of gravity just referred to earth attracting objects. Galileo-Newton-types’ work is much subtler than that; it implies for instance that the acceleration due to gravity is constant during the fall – see here for how difficult an experiment Galileo had to do to figure that out. It implies that a feather and stone would fall down with similar velocity, if you account for air resistance (which is not obvious at all). Saying one entity attracts another is rather vague; it is the quantification (GMm)/r^2 that gives meaning to it; bringing in the fundamental notion called mass or “quantity of matter” (a priori it is not clear it has anything to do with attraction, it is a non-trivial fact that it has). Today we take all these notions for granted (aside : for this reason, I don’t think school education almost anywhere in the world is as scientific as it should be, the so called atheists are only making a religion out of science), but it took very painstaking efforts, “cutting edge research” of those days, to figure that out.
All this said, the ancestors still need to get some credit – for instance there is this quote (I haven’t verified it is authentic, but for now let us trust it) in these slides that celestial bodies cannot fall since they attract each other. This already reflects a non-trivial understanding (being less obvious than objects seeming to be attracted to the earth). A good treatment should bring out the correct difference between the various notions, giving our pUrvAcAryas their deserved share of credit, yet not conflating with western scientists in a way that will make the writing less credible in the eyes of scientists.
And yes – your point about Mesopotamia. Yes, I agree, chronology in Indian history needs to be all examined more carefully; I was being sloppy in taking the dates given by our “white masters” for granted. Also, the link about Mesopotamia that I gave you – I got very irritated by its claim that zero made its way from Mesopotamia to India, ignoring even the possibility that Indians invented zero independently. Unfortunately, many of these westerners make such unsubstantiated claims and deny India credit, and yet if we question these assumptions they won’t even consider our questions because they will take it for granted that we are asking the question because of being nationalist.
@frog
“There is also a book by Kim Plofker that I haven’t read, but I read a review by a great mathematician (actually a fields medalist) praising it.
I would also suggest to be careful with works like this Mr. Raju’s that she has quoted. For all I know, Raju may be committed and honest/sincere, but when one quotes opinions like Greek science being fabricated one has to be really careful with fact-checking.”
Your comment really shows who you are!
Not just a frog in the well ! A well grounded EUROCENTRIC frog! Keep it up. (Oe suggestion Trying reading CKR’s books instead of reviews!
“1. Scholarly evidence does not suggest that science came to Europe late – no one can refute that Euclid’s elements, Archimides’ work etc. (because their works have survived to this date we know) contain a lot of science and math a lot of which are not found in any Indian text.”
As a starter, Please try to list out ‘primary sources’ used by the western scholars for the creation of ‘myth’of EUCLID.
S – first, I don’t mind discussing with an uncouth attacker like you, but I suspect Shantanu would like higher standards in the comment thread, without personal attacks. In fact, people like you are as anti-Hindu was Wendy Doniger and co, because by means of such uncouth attacks [like your calling me a “well grounded EUROCENTRIC frog”], you dent the credibility of the pro-Hindu side.
To answer your questions :
1. I did not rely on just a review of CKR’s book – I relied a quote from it. If I see someone quoting from a book somewhere that 1 + 1 = 3, I would suggest caution with that book (now this is an analogy, don’t take the wrong side of it).
2. Greek works are well documented in the *arabic* literature already, which date back to before the Arabic contact with Europeans. I have also read Al-Biruni quoting Varahamihira as praising Greeks for their scientific knowledge.
@manduka,
“Greek works are well documented in the *arabic* literature already” – This argument will not work, Show us the ‘primary sources by the so called western scholars to create the myth of Euclid, instead of resorting to ‘appeal to emotion’ techniques.
manduka bhai,
“….no one can refute that Euclid’s elements, Archimides’ work etc.”
Read CKR first, you might get cured of your ‘eurocentric’ bias.
Read CKR –
“Euclid is one of the pillars of the story that mathematics originated in Greece. But Euclid is a fake. Ask yourself: do you really know of any evidence of Euclid?
A few years ago, the late David Fowler, an authority on Greek mathematics, candidly and publicly admitted in response to my post, on the Historia Matematica discussion list, that “nothing†is known about Euclid.
The racist element of the Euclid story is brought out by Indian school texts which display images of Euclid (and various other “Greek†mathematicians) as Caucasian stereotypes. Note that even these sources now admit that Euclid was from Alexandria in Africa. So how did they determine the color of Euclid’s skin? Such racist images from “reputed†Internet history sites like McAndrews lend a concrete reality to “Euclid†in the minds of young children. This conditions them lifelong to look askance at anyone who questions the existence of Euclid. This is the crux of propaganda–to make a large number of people believe something without evidence, and to make them believe that anything to the contrary is absurd.” (http://ckraju.net/blog/?p=16)
Read CKR’s another article on the same topic – http://ckraju.net/papers/MathEducation1Euclid.pdf
Zero has come from the great debate of Shunyavad and Mayavad.
Shunyavad means merging into Nirvan (void) (Brahman) and the cessation
of consciousness. thus losing all identity of EVERYTHING and existing
like a drop of water merges in the ocean and looses it’s identity and
becomes nir-guna.
Zero has come in India through the famous philosophical line “Nothing ever happened”. Indian philosophical schools have been intense in the debate on zero from time immemorial.
Now what is Mayavad. This is where the advaita of snake and rope comes in which has been acknowledged by none other than the father of Quantum Mechanics. Erwin Schrodinger (ES)
“From the early great Upanishads the recognition ATHMAN = BRAHMAN upheld in (the personal self equals the omnipresent, ll-comprehending eternal self) was in Indian thought considered, far from being blasphemous, to represent the quintessence of deepest insight
into the happenings of the world. The striving of all the scholars of Vedanta was, after having learnt to pronounce with their lips, really to assimilate in their minds this grandest of all thoughts”.
and then
“Even in the that what seems to be a plurality is merely a series of different personality aspects of this one thing, produced
by a deception (the Indian MAJA); the same illusion is produced in a gallery of mirrors, and in the same way Gaurisankar and Mt Everest
turned out to be the same peak seen from different valleys”.
MAJA is Maya here. Germans pronounce it that way which is MAJA.
http://whatislife.stanford.edu/LoCo_files/What-is-Life.pdf
Please keep this pdf file as a personal copy . It is must for an inquring mind. The above quote two quotes in inversted commas is from
the cited pdf booklet of ES.
Now again , one may say that Adi Sankara was from Kerala who reignited this advaitic ( Maya ) philosophy. Yes , we all agree. But Adi Sankara in His wonderful Dakhshinamurthi verses acknowledges all Rishis for these great thoughts( Sarva Muni Jananam… ) which goes to show His humility. If one goes back further Maya was propounded first by Sage Vashista to Rama , in the masterpiece Yoga Vashista when Rama is perpelexed, when told to go to forest for 14 yrs.
In essence
Advaitavad is ALL NIRAKAR CHIT – it is ” white ” shunya
Shunyavad is ALL NIRAKAR ACHIT – it is ” black ” shunya
some even give this above interpretation.
Zero is defintely an Indian discovery through intense metaphysical debates over centuries thru different schools.
In the pdf booklet, the quotes are from the epilogue section of ES’s great book cited. Please read it patiently.
Thanks for the link. I am planning write a blog on a similar subject. Hindu/Indian original contribution ranges in the following topics.
1. Number system
2. Pythagoras Theorem
3. Surds
4. Value of Pi
5. Plane geometry
6. Solid Geometry
7. Astronomy
8. Evaluation of Series
9. Continuous fractions
10. Trigonometry
I suggest the following references
1. A history of Ancient Indian Mathematics – C N Srinivasaiengar, The World Press Private Ltd. Calcutta. (1967)
2. http://www-history.mcs.st-andrews.ac.uk/Indexes/Indians.html
3. Journey Through Genius – William Dunham, Penguin Books 1990
4. Tantra Sangraha of Nilakantha Somayaji – K.Ramasubramanian ans M S Sriram, Hindustan Book Agency – 2011.
You may also see my blog
http://lvnaga.wordpress.com/2012/04/12/tamilindian-solar-calendar/
L V Nagarajan
@S : If you can discuss without talking about “curing me” of eurocentrism etc., we can continue. Otherwise I am not interested. You are of course free to think whatever you want of me.
@manduka,
“Galileo-Newton-types’ work is much subtler than that;”
Regarding newton, read this – http://www.indianscience.org/essays/t_es_agraw_kerala.shtml
“‘There is, however, one issue, which remains puzzling, even from a purely Eurocentric perspective. In what sense did Newton invent the calculus? Clearly, the calculus as a method of calculation preceded Newton, even in Europe. Clearly, also, the calculus/analysis as something epistemologically secure, within the formalist frame of _mathematics as proof_, postdates Dedekind and the formalist approach to real numbers. While Newton did apply the calculus to physics, that would no more make him the inventor of the calculus than the application of the computer to a difficult problem of genetics, and possible adaptations to its design, would today make someone the inventor of the computer. Doubtless Newton’s authority conferred a certain social respectability on the calculus. The credit that Newton gets for the calculus depends also upon his quarrel with Leibniz, and the rather dubious methods of “debate” he used in the process. But none of this convincingly establishes the credit for calculus given to Newton, even within the Eurocentric (as distinct from Anglocentric) frame. So what basis is there to give credit to Newton for originating the calculus, while denying it, for example, to Cavalieri, Fermat, Pascal, and Leibniz?”
”
In his recent talk (2000) Raju emphasised that the calculus has played a key role in the development of the sciences, starting from the “Newtonian Revolution”. According to the “standard” story, the calculus was invented independently by Leibniz and Newton. This story of indigenous development, ab initio, is now beginning to totter, like the story of the “Copernican Revolution”. The English- speaking world has known for over one and a half centuries that “Taylor” series expansions for sine, cosine and arctangent functions were found in Indian mathematics/astronomy/timekeeping (jyotisa) texts, and specifically the works of Madhava, Neelkantha, Jyeshtadeva etc. No one else, however, has so far studied the connection of these Indian developments to European mathematics.”
“Europeans encountered difficulties in using these precise sine value for determining longitude, as in Indo-Arabic navigational techniques or in the Laghu Bhaskariya, because this technique of longitude determination also required an accurate estimate of the size of the earth, and Columbus had underestimated the size of the earth to facilitate funding for his project of sailing West. Columbus’ incorrect estimate was corrected, in Europe, only towards the end of the 17th c. CE”
For my Europhile friends-
Read CKR’s article – http://drckraju.blogspot.com/2009/03/yellow-learned-journalism.html
“My book Cultural Foundations of Mathematics constructs a new philosophy of mathematics. Construction is always accompanied by destruction, and the book has destroyed not only the Western philosophy of mathematics, but incidentally also undermined most Western philosophy since the 12th c. Crusades.
Here is one very simple and very robust argument: Western mathematics and philosophy is founded on the belief that deductive inference is certain and universal. However, deductive inference varies with the logic used, and 2-valued logic is neither (a) culturally universal, nor (b) empirically certain”
Manduka bhai, please don’t forget to read this:
“Here, the reviewer takes up only one point: namely that Euclid did not exist. Note that this is a historical issue not a philosophical one, but the reviewer engages with it anyway. He seemingly concedes the point, only to quibble that some other persons or group of persons could have systematised the Elements before the Christian era! That is, if Euclid did not write the book, Euclid1 could have written it! (And if Euclid1 did not exist, then Euclid2 could have written it, or maybe it was Euclid1+Euclid2 and so on!) Presumably, as a formal logician the reviewer is immensely satisfied with such postulates and quibbles, and regards them as a way to work around my argument without addressing any evidence. This is doubtless a genuine failure to comprehend a very simple argument. The reason I regard it as a genuine failure is that another “professor” on the philomathes list had displayed the same level of aql, and put forward a similar simplistic counter-argument, which distinctly crosses the borderline of the facetious: the accumulation of hypotheses (a stock theological tactic) is bad enough, but here even the accumulated hypotheses do not even remotely meet my argument!
So, let me explain things at the kindergarten level. My first claim is that Euclid was concocted during the Crusades in order to concoct a theologically correct early-Greek origin for the Elements, obtained from Arabs (since neither Arabs nor late Greeks such as Porphyry, Theon, and Proclus were theologically correct). In addition to a theologically-correct origin, the fear of heresy during the Inquisition made it mandatory to provide a theologically correct interpretation of any book for it to be acceptable. So the second claim is that the Elements was reinterpreted (exactly like the Bible) to suit post-Crusade Christian theology. The attribution of the text to a Euclid about whom nothing is known facilitated this reinterpretation (which could be passed off as the philosophy of the unknown Euclid). Such concoctions are a common trick used by priests to lend authority to their theological interpretations. The question is why should anyone believe an iota of this history put forward by religious fanatics during the Crusades and the Inquisition, and developed by later-day colonial and racist historians?
As emphasized in my book (p. 25) we can can believe neither in Euclid NOR in his purported philosophy. The purported evidence for both claims is the same single forged remark from a late text:
“In particular, irrespective of whether Euclid was real or invented, the Monacensis remark about his alleged philosophy of “irrefragable demonstration†is obviously a later-day interpolation.”
The circumstances of this forgery are very suspicious.
There is no other evidence that the philosophy of mathematics (attributed to Euclid) existed prior to Proclus. From Plato to Proclus there prevailed an understanding of geometry that was explicitly linked to their religious beliefs about the soul, virtue, and so on.”
Try to read completely. you would know more about worthies like Newton, Russell, Hawking etc.
The response in 10 is what I agree with. I have been a Ramanarishi devotee for many yrs . I come from Canada to visit once in two yrs the abode of Ramanarishi at T’vannamalai Hills in Tamil Nadu.
Zero and the number system is from India. Zero is due to philosophical debates. I agree with this premise of 10. I gather that in India , there was a oral tradition of passing over knowledge to future generations , which is very evident , the way even a shopkeeper does his grocery calculations (mental) in villages in India. I have visited Iraq ( Messopotamia) when peaceful , I did not find this speed of doing mental arithmetic calculations in Iraq as done by a shopkeepers in India. You may find this argument stretched, but nowhere I have found this oral way of remembering and calculating numbers in their native language as done in India by the common folk.
I feel that due to the philosophical atitude and the conquering of ego aspect, the philosophers just do the work and not claim credit. The oral tradition and keeping their records in palm leaves, is surely not full proof record keeping. If available still ( there must be), these must be digitised.
I would like to end here saying that modern neurobiologists have come to the conclusions of the Self as well as ” Who Am I?” as ancient Indian seers thought and reemphasized by Ramanarishi in 20th century.
Discovery of calculus doesn’t refer to the discovery of differentiation or integration. Rather it refers to the discovery, with proof, that these were inverse processes. In our schools the integral is taught as anti-derivative. Historically that was not the case. Integral was thought of as signifying area under a graph, it was not at all clear that the area can be calculated using anti-derivative. The discovery that you could do so created a revolution in math because now you could integrate so many functions just by finding another function whose derivative is the given function.
Example : finding the integral of x^n dx from 0 to 1 was a tricky affair – because you actually needed to subdivide the interval [0, 1] into small pieces, get an expression for the sum of areas of several rectangles and take the limit as n goes to infinity. If you try to work this out, you will get an expression involving 1^n + 2^n + … + m^n, and there is no simple formula for it. You might have worked out the cases n = 1 or 2 or 3 or may be even 4 in school. But people came up with clever tricks to get around this issue and compute the integral of x^n dx from 0 to 1. After “Newton’s discovery of calculus”, anyone could compute this without such clever tricks, just by finding an anti-derivative, namely x^{n + 1}/(n + 1), and evaluating the difference between the values of this function at 0 and 1.
As far as the previous issue is concerned, my point is about Greek math and science existing before Arabic communication with Europe. I don’t care if “Elements” was written by some one called Euclid or not (I agree that the name of the author is disputable), but it is a product of Greece, and we have a manuscript dating back to 888 BC which you can access here). There is a fragment here, from 100 BC. The oldest available manuscript for the present day version of Mahabharata, in contrast, is from 1200s.
BTW you are now alleging that I am a europhile, which is better than eurocentric (which is why I responded to you), but I will tell you I am not that either. You can refer to one of my comments above to see my opposition to mainstream western historians’ positions. You can look at my blog and decide if it looks like the blog of a europhile. If you want to have sensible discussion with others, stop calling others names.
@froginthewell
“I don’t care if “Elements†was written by some one called Euclid or not (I agree that the name of the author is disputable), but it is a product of Greece”
I think you should pay attention to myth of Euclid, because that will give insight to these goras psyche and also see similar process going on with respect to Yoga. Even if these documents are Greek, one should consider the fact that Greece had civilizational contact with Persian and Indic civilizations. So there is high probability of exchange of ideas too not just goods.
Newton is a historical figure, however he was not working in vacuum.
Western historians deliberately fabricate the chronology of Indian history, texts for nefarious reasons. They do so just to ensure that the goras came up with this ‘scientific discovery’ even though there is much evidence that Jesuits were translating and transmitting many texts to Europe. Now there is a concerted effort to change the accepted birth/death date of Buddha.
Here is an excerpt from an article by D.P Aggarwal regarding exchange of mathematical ideas from India(Kerala school of Mathematics) to Europe.
“Raju (personal communication) explains that Whiteside, while conceding Madhava’s priority for the development of infinite series, distorts the dates of both Madhava and the Yuktibhasa, by about a century in each case. (Madhava was 14th-15th c. CE,not 13th, while the Tantrasangraha [1501 CE] and Yuktibhasa [ca. 1530 CE] are both 16th c. CE texts, not 17th.) In fact, in the 16th c. CE Jesuits were busy translating and transmitting very many Indian texts to Europe; during the 16th c. CE, their activities were especially concentrated in the vicinity of their Cochin College, where they were teaching Malayalam to the local children (especially Syrian Christians) whose mother tongue it was, and where copies of the Yuktibhasa and several other related texts were and still are in common use, for calendar-making for example.
After the trigonometric values in the 16th and early 17th c. CE, exactly the infinite series in these Indian texts started appearing in the works, from 1630 onwards, of Cavalieri, Fermat, Pascal, Gregory etc. who had access in various ways to the Jesuit archives at the Collegio Romano. Since Whiteside has a copy of the printed commentary on the Yuktibhasa, he could hardly have failed to notice this similarity with the European works with which he seeks to make the Yuktibhasa contemporaneous!
Raju has no doubt that in the course of “the fabrication of ancient Greece” (in Martin Bernal’s words), some Western historians acquired ample familiarity with this technique of juggling the dates of key texts. Having anticipated this, the evidence for the transmission of the calculus from India to Europe is far more robust than the sort of evidence on which “Greek” history is built – it cannot be upset by quibbling about the exact date of a single well-known manuscript like the Yuktibhasa.”
http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_kerala.htm
“BTW you are now alleging that I am a europhile, which is better than eurocentric (which is why I responded to you)”
You will respond even if I call you Urophile.
Btw, I like your change of opinion from
“Scholarly evidence does not suggest that science came to Europe late – no one can refute that Euclid’s elements, Archimides’ work etc. (because their works have survived to this date we know) contain a lot of science and math a lot of which are not found in any Indian text.â€
to
“I don’t care if “Elements†was written by some one called Euclid or not (I agree that the name of the author is disputable), but it is a product of Greece”.
But it still requires more correction .
Malavika : I am aware of claims that some math traveled from Kerala to Europe. I haven’t examined those arguments carefully, hence do not have an opinion on that one way or another. Just similarity in the formulas derived is not enough to convince me because all too often people do discover things independently. I can’t take Raju on his word because as I have noted above Raju doesn’t even know what “discovery of calculus” means.
@S : By talking of “Urophile” you have resorted to degrading language again, so I stop. First learn to be a human being, everything else comes after that.
BTW Malavika : Even if these documents are Greek, one should consider the fact that Greece had civilizational contact with Persian and Indic civilizations. So there is high probability of exchange of ideas too not just goods.
Sure. I am not arguing against the possibility that Greeks borrowed from Indians in mathematics too, I am only saying science did not reach Europe late. And here is a verse of our great Varahamihira from the Brihatsamhita : मà¥à¤²à¥‡à¤šà¥à¤›à¤¾à¤ƒ हि यवनाः तेषॠसमà¥à¤¯à¤• शासà¥à¤¤à¥à¤°à¤‚ इदं सà¥à¤¥à¤¿à¤¥à¤‚ |
ऋषिवतॠतेऽपि पूजà¥à¤¯à¤¨à¥à¤¤à¥‡ किं पà¥à¤¨à¤ƒ दैवविदॠदà¥à¤µà¤¿à¤œà¤ƒ ||
Just to convince you it is a genuine quote, it is in (numbered) verse 14, chapter 2 from this online source
It says “This shaastra being well established among the Greeks, though they are mlecchas; even they are worshipped as Rishis, wouldn’t it be even more so about a braahmaNa who is “daiva-vid”?”
Of course I trust you will agree that a Rishi like Varahamihira was not being eurocentric.
@froginthewell – the quotes from Varahamihira can be challenged on various grounds. Firstly the reference itself seems to be a backhanded complement to the subject which is apparently referred to as ‘mlecchas’. Secondly this does not provide any information that Varahamihira is referring to their scientific or mathematical prowess in comparison to his own. But finally and most damningly the hasty equivalence of ‘yavanas’ with greeks is something that derives from colonial indology. Many people active in the effort to decolonize India’s history have questioned these claims first made by indologists. They point out that yavanas were not greeks and every reference to yavanas does not automatically point to greeks. Indeed if assumptions of colonial indology were bypassed – yavanas are shown to be renegade kshatriyas – tribes expelled in the unrecorded from India proper – references to whom keep appearing sporadically in indian texts. So in conclusion it is certainly not Varahamihira who is eurocentric rather it is the eurocentrics who have interpreted him as such to us.
froginthewell said:
“Sure. I am not arguing against the possibility that Greeks borrowed from Indians in mathematics too, I am only saying science did not reach Europe late.”
Ok, there is some agreement.
The bone of contention is that Indian mathematics does not get the credit it deserves. Also, European mathematics was stuck because of its Roman numerical s. Try doing decimals with Roman numbers! And Indian calender was far more accurate than the Gregorian calender.
It was the genius of Brahmagupta who came up with rules for computations for zero. And was first to conceptualize negative numbers. And he was several hundred years(nearly a millinium) ahead of Fermat.
“Furthermore, he pointed out, quadratic equations (of the type x2 + 2 = 11, for example) could in theory have two possible solutions, one of which could be negative, because 32 = 9 and -32 = 9. In addition to his work on solutions to general linear equations and quadratic equations, Brahmagupta went yet further by considering systems of simultaneous equations (set of equations containing multiple variables), and solving quadratic equations with two unknowns, something which was not even considered in the West until a thousand
years later, when Fermat was considering similar problems in 1657.”
http://www.storyofmathematics.com/indian_brahmagupta.html
“I am only saying science did not reach Europe late”
This is a very vague statement Indian science was far superior to ‘Western’ science in the medeival period. However in the 18, 19, 20 centuries West was far ahead of Asia.
froginthewell said:
“Just similarity in the formulas derived is not enough to convince me because all too often people do discover things independently. I can’t take Raju on his word because as I have noted above Raju doesn’t
even know what “discovery of calculus†means.”
You can believe whatever you want. But, unless there is evidence to support that either(i) Jesuits did not translate Indian mathematical texts or (ii) that Cavalieri, Fermat, Pascal, Gregory did not have access to these texts at Jesuit archives at the Collegio Romano there is no evidence to claim independent discovery. That is just a belief.
I think there is some epistemological quibbling about origin of Calculus. What Indians were doing can’t possible be calculus!
@Moonclan : Well you are right that I haven’t fact-checked that Varahamihira’s usage of the term Yavana refers to Greeks. However, I should tell you that I first learnt the quote from reading Al-Biruni who interprets it as Greeks. So I didn’t get it from any colonialists.
@Malavika : Gregory did not have access to these texts at Jesuit archives at the Collegio Romano there is no evidence to claim independent discovery. That is just a belief.
Wait, did you read my comment properly. I said I just don’t have an opinion on the issue. I didn’t claim westerners made any independent discovery. You are the believer here, if there is any; I am an agnostic.
The bone of contention is that Indian mathematics does not get the credit it deserves.
That is not a bone of contention at all. I too believe that Indian mathematics does not get the credit it deserves. Comment 5 above refers to an attitude of westerners towards Indian math that I was criticizing.
I think there is some epistemological quibbling about origin of Calculus. What Indians were doing can’t possible be calculus!
There is no epistemological quibbling; to mathematicians, discovery of calculus always meant the *fundamental theorems of calculus* – the first and the second. This Cavalieri that you mentioned – he preceded Newton, and gave an expression for the integral of x^n dx on a closed interval. Yet, even Westerners don’t credit him with inventing calculus. Derivatives of trigonometric functions were well known in the west before Newton, and that is not considered discovery of calculus even by westerners. Why? Because the discovery of calculus refers to not integration or differentiation but to the first and second fundamental theorems.
As per ‘Frog’ –
“And here is a verse of our great Varahamihira from the Brihatsamhita : मà¥à¤²à¥‡à¤šà¥à¤›à¤¾à¤ƒ हि यवनाः तेषॠसमà¥à¤¯à¤• शासà¥à¤¤à¥à¤°à¤‚ इदं सà¥à¤¥à¤¿à¤¥à¤‚ |
ऋषिवतॠतेऽपि पूजà¥à¤¯à¤¨à¥à¤¤à¥‡ किं पà¥à¤¨à¤ƒ दैवविदॠदà¥à¤µà¤¿à¤œà¤ƒ ||
It says “This shaastra being well established among the Greeks, though they are mlecchas; even they are worshipped as Rishis, wouldn’t it be even more so about a braahmaNa who is “daiva-vidâ€?â€
Now Swami Vivekananda says _
”
On a single Sanskrit Shloka —
मà¥à¤²à¥‡à¤šà¥à¤›à¤¾ वै यवनाः तेषॠà¤à¤·à¤¾ विदà¥à¤¯à¤¾ पà¥à¤°à¤¤à¤¿à¤·à¥à¤ िता। ऋषिवतॠतेऽपि पूजà¥à¤¯à¤¨à¥à¤¤à¥‡ . . .
— “The Yavanas are Mlechchhas, in them this science is established, (therefore) even they deserve worship like Rishis, . . .†— how much the Westerners have indulged their unrestrained imagination! But it remains to be shown how the above Shloka goes to prove that the Aryas were taught by the Mlechchhas. The meaning may be that the learning of the Mlechchha disciples of the Aryan teachers is praised here, only to encourage the Mlechchhas in their pursuit of the Aryan science.”
& ‘Frog’ says ‘it’ is not ‘Eurocentric’!!!!
Again I did not see the mention of Anantha / infinity as an indian concept.Strangely I have had a discussion with a boy in Iran where I was talking about the cyclical yugas repeating themselves. The boy a MUSLIM opened ny eyes by saying eternity and infinity can exist only in circle and not in any other form. Call it naive but I was impressed. Anantha existed long before any other civilisation.
Similarly I went to London to study a new course in toxicology there principles were so familiar .Amithey Amrutham visham and the corollary that if you dilute a poison you have the property of a drug coming out.I think it is an Indian contribution from ancient times ie the prerecorded times. Maybe the Frog in the well can enlighten me
@froginthewell
“to mathematicians, discovery of calculus always meant the *fundamental theorems of calculus* – the first and the second. ”
The formal study of calculus combined Cavalieri’s infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term.[8] The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1670.
These ideas were systematized into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton.[9] He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for manipulating infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts.
Leibniz and Newton are usually both credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today
http://en.wikipedia.org/wiki/Calculus
Newtons gravitational theory is different(has mistakes) compared to the current gravitational theory. Should this be a reason to discount Newtons contributions?
Also, I think Yuktibhasha and Kerala school of mathematics were pioneers of calculus. And I am not alone in having such an opinion.
“n the 14th Century Indian mathematician Madhava of Sangamagrama and the Kerala school of astronomy and mathematics stated many components of calculus such as the Taylor series, infinite series approximations, an integral test for convergence, early forms of differentiation, term by term integration, iterative methods for solutions of non-linear equations, and the theory that the area under a curve is its integral. Some consider the YuktibhÄá¹£Ä to be the first text on calculus”
from wiki
I’m impressed with the ongoing debate/discussions, and hope to not disturb it, yet add my 2 cents.
I only want to point out that the decimal system (“base 10” system) is natural. It is so because we are humans having ten fingers. The number ten becomes important because of that fact alone.
That is not to be confused with the invention of the place-value system. The representation of ten as “10” (a one in a separate place and nothing or zero in the units place) is really the core of the place-value system under a decimal base. Easy arithmetic operations follow from that.
The place value system is useful even in non-decimal arithmetic (eg. binary arithmetic where the only numbers are 1 and 0; this being used inside computer ALU’s. Here, two is 10, three is 11, and four is 100 etc and 11+1=100 generating a carry into the 4th’s place, for instance.)
I surmise that the inventors of a symbol for zero would probably have invented the place-value system around the same time. The need for a zero is acutely felt if you are going to have place-value notation.
@Malavika : First : WHEN DID I SAY MADHAVA’S CONTRIBUTIONS SHOULD BE DISMISSED? I just meant what he did is different from what is referred to as “discovery of calculus”. I know the source of your confusion, so let me try to explain the difference again for you.
1. When people talk of doing calculus as in differentiation/integration, this is a concept that dates back at least to what is known as Archimedes’ integration (regardless of whether Archimedes was historical etc. – I don’t care for those issues). May be even before that, and may be the first place it was done was India. I don’t know.
2. When people talk of the “discovery of calculus”, they don’t refer to the sense of the word calculus in point 1, they refer to what is known as the “fundamental theorems of calculus”.
There is some unfortunate conflict of terminology here, and you are (unwittingly) exploiting that. This is how people evolved their language, it is not perfect, yet we have to live with it. Let us not spend much time this.
One can certainly call Madhava a pioneer of calculus as you say. I am only saying that his work is not “discovery of calculus” in the commonly understood sense of that, what shall I say, idiomatic expression?
Dear All: Thank you very much for a truly educative, enlightening and sober discussion on the various aspects..I am struggling under work-load at present but have downloaded all the comments and hope to read them up on an upcoming flight…
Please keep the discussion going…
Thanks again..
I would like to suggest Michel Danino’s excellent book, “Indian Culture and India’s Future”, and my review of the book – http://blog.abhinavagarwal.net/2012/01/review-of-indias-culture-and-indias.html
@froginthewell – Al Biruni was removed from Varahamihira by 400+ years (by his own reckoning) and by about 1000 years (per puranic chronology). Accuracy of his alleged references to ‘Greeks’ must be understood in this context. Also, presumably your reference to Al-Beruni’s work is not of ‘Tarikh Al-Hind’ in the original but the far more accessible and widely available ‘Al Biruni’s India’ by Eduard Sachau the noted German Orientalist. Here translated references to ‘Greeks’ are inferential and heavily depend on prevalent scholarly consensus of his age, which happened to be heavily eurocentric. While stating this no value judgement is intended, rather the intent is only introduce a note of caution into any discourse positing proof of ‘Greek Priority’ (which itself is a product of the last few hundred years of european historiography).
@froginthewell – A little nitpicking here ! The date for Bodleian Euclid is 888 AD (not BC) – a mistake ( no doubt inadvertent ) but something which greatly inflates the antiquity of the manuscript. This manuscript is self referentially 1100 years after the purported date of the quasi-historical Euclid (300 BC) and is derivative in character and not an original. Furthermore there is no surviving original text of the Elements. Only latter day ostensibly translated manuscripts exist. Transmitted manuscripts of earlier ages are invariably accretive and were updated time and again. There are inherent difficulties in attributing such works to definite quasi-historical figures (like Euclid) and cultures (like Greeks), or assigning definitive dates to the contents.
Also, the manuscript in question miraculously appeared in Bodleian Library in the early 19th century (a time by which the myth of euclid was well established) and Europeans were already weaving tales attributing to the Greeks priority in all areas of knowledge.
http://www.rarebookroom.org/pdfDescriptions/eucmsd.pdf
Now this is how the not-much-loved wikipedia describes references to Euclid from within the elements itself
“There is no mention of Euclid in the earliest remaining copies of the Elements, and most of the copies say they are “from the edition of Theon” or the “lectures of Theon”, while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.”
In line with my previous comment on exercising caution regarding ‘Greek achievements’ one must also mention Martin Bernal’s ‘Black Athena’ set of works which arguably establishes synthetic nature of the construction of the ‘Greeks’. He demonstrates how a unified Greek identity entirely divorced from its geographical surroundings (eastern-mediterranean and middle east) was constructed during the last two centuries. Thus was born the Hellenism that has been the one of the important cornerstones of western identity for the past two hundred years at least. He goes on to show how the purported Greek miracle was in reality majority contributed by non-greeks (most notably egyptians). It is therefore not inconcievable that late accretive manuscripts
have been lately and wrongly attributed to Greeks when they were contributions of various people and cultures in the first place. Perhaps the only thing conclusively Greek about the manuscript is that it is written in Byzantine Greek (note not ancient greek).
Given this background it is very important (from the perspective of balanced historiography) that we view with skepticism European narratives about the Greeks. This will serve us well in understanding our own history as well.
“In line with my previous comment on exercising caution regarding ‘Greek achievements’ one must also mention Martin Bernal’s ‘Black Athena’ set of works which arguably establishes synthetic nature of the construction of the ‘Greeks’. He demonstrates how a unified Greek identity entirely divorced from its geographical surroundings (eastern-mediterranean and middle east) was constructed during the last two centuries. Thus was born the Hellenism that has been the one of the important cornerstones of western identity for the past two hundred years at least.”
Read (http://2ndlook.wordpress.com/2011/11/07/manufacturing-history-euro-style/)
The “Big Powers†of the time — Britain, France and Russia — appointed a Bavarian prince as Greece’s first king – Otto. He arrived in his new kingdom with an entourage of German architects, engineers, doctors and soldiers — and set out to reconfigure the country to the romantic ideal of the times.
The 19th century had seen a resurgence of Europeans’ interest in ancient Greece. Big names such as Goethe, Shelley, Byron, Delacroix and many other artists, poets and musicians sought inspiration in classical beauty. They marveled at the white marble and solemn temples of Hellas, and longed for a lost purity in thought, aesthetics and warm-blooded passion. Revisiting the sensual Greece of Orpheus and Sappho was ballast to the detached coolness of science or the dehumanizing onslaught of the Industrial Revolution.
Otto saw to it that modern Greece lived up to that romantic image. Athens, at that time a small hamlet of a few goatherds, was inaugurated as the new national capital.
===
From delanceyplace.com’s review of the book Title: 1616: The World in Motion by Thomas Christensen (2012)-
we now think of Galileo’s proof that the sun was the center of our solar system as the dawn of the scientific age, and he is often called “The Father of Modern Science.” However, all scientific discovery is a continuum-the result of the work of many people. Though it is commonly known Galileo relied on Poland’s Nicolaus Copernicus, we now know that Copernicus’s work owed a debt to the Muslim astronomers Ibn al-Shatr and Nasir al-Din al-Tusi, whose works were published by Cardinal Ferdinand de Medici. Galileo owed an even greater debt, one he never acknowledged, to the meticulous work of the German Johann Kepler and his mentor, the Dane Tycho Brahe:
“Despite his personal eccentricities Tycho Brahe (1546-1601) had compiled deÂcades of data from his celestial observations that were far more meticulous and precise than anything previously available. His elevation of observaÂtion over speculation was one of the key developments leading to the modÂern concept of inductive scientific investigation. Kepler (1571-1630), though appreciaÂtive of the benefit of accurate data, was less capable of obtaining it directly. A bout of smallpox in childhood had left him frail and sickly, with a severe visual handicap: he was short-sighted and had double vision in one eye. Nor did he have the means to construct a large observatory like Tycho’s Uraniborg. So he depended on Tycho for the data he needed to elaborate his theories of celestial harmony. …
“When the published version of Kepler’s The Cosmographic Mystery [the first published defense of the Copernican system] appeared, Kepler was still convinced that his connection of the planetary orbits to the five regular solids was a fundamental breakthrough toward discovering God’s plan for the universe. With great enthusiasm he mailed copies to all of the influential people he could think of who had an interest in astronomical topics or might assist him in his career, but the results of these mailings would be disappoint- ing. Still, among the recipients was a thirty-three-year-old professor of mathematics at the University of Padua [Galileo Galilei], who wrote back to confess that he too was a Copernican, subscribing to the radical notion that the earth orbited around the sun. But he was afraid, he said, to state that belief publicly. Kepler responded by urging him to speak out, but Galileo did not acknowledge this second letter- in fact, he would not be in touch with Kepler again for thirteen years. The reason, according to Albert Einstein, was vanity, which he considered a failing of many great scientists. ‘It has always hurt me to think,’ he wrote in a letter to a friend, ‘that Galileo did not acknowledge the work of Kepler.’ …
“Kepler’s working calculations on the orbit of Mars take up nine hundred pages in a minuscule hand. How many more pages would he have needed if he had had to make his calculations using Roman instead of Arabic numerals? It is no lonÂger possible to see the Scientific Revolution as a self-contained European phenomenon; exchange of ideas between Islamic West Asia and Christian Europe was a lively and vital component of the new scientific discoveries.
“Kepler had been an enthusiastic Copernican since his student days, when his embrace of heliocentrism was probably more intuitive than rational. GalÂileo was a more reluctant Copernican, who tried to avoid addressing the isÂsue until led to confront it through his astronomical observations and other research. … Copernicus (1473-1543), who had pub- lished his groundbreaking book On the Revolutions of the Celestial Spheres just before his death in 1543, had been significantly influenced by Islamic astronomical research made centuries before his lifetime.
“In 1957 Otto Neugebauer, a scholar researchÂing Copernicus, happened on some diagrams by the fourteenth-century astronomer Ibn al-Shatr, and he recognized that they were identiÂcal to some in Copernicus’ work. Later he found that Copernicus had also relied on the work of Nasir al-Din al-Tusi, an even earlier astronomer, who had tried to revise traditional Ptolemaic astronomical theory to make it better conform to actual observation. … It was subsequently discovered that Copernicus had even used the same letters as al-Tusi to designate the points in a key diaÂgram, removing any lingering doubt that Copernicus had access to the work of Muslim astronomers. (Evidence for Copernicus’s reliance on the work of early Islamic astronomers is summarized by George Saliba in his Islamic Science and the Making of the European Renaissance). …
“A remarkable Arabic publishing operation was funded by Cardinal FerÂdinand de Medici, Duke of Tuscany, in Italy in the late sixteenth century. The Medici Oriental Press-relying on the library of a Turkish scholar who had fled a dispute in his homeland, arrived at Venice around 1577, and conÂverted to Christianity-published a number of Arabic-language books. Among those publications was one based on the work of Nasir al-Din al-Tusi, one of the astronomers whose work Copernicus drew upon.”
Note the bolded parts. the author of the book use the same argument of C.K.R regarding ‘usefulness’ of greek numerals.Also note connection of Muslim mathematicians & Medici to India.
@froginthewell
Scientists from IIT who examined the Meherauli iron pillar of Delhi believed to have been built in the 5 th Century CE have opined that the technology used in construction of the pillar was not known to modern world till the 1930’s. What is your view about that?
All,
Please read The Origin of Mathematics by Dr.V.Lakshmikantham and S.Leela. Dr.Lakshmikantham is a Emeritus Mathematics professor in Florida Institute of Technology.
In this book, the author presents with proofs for various mathemtical ideas that originated from India Pythagaros theorem and the idea of Sine of an angle, long before the Greek dates. Proofs are well presented and there are numerous references too.
Sumitra
Placing these here for the record, a 57-min video on “Ancient India’s Engineering Skills”
From Vedic Mathematics and Boeing 787 dreamliner by umeshdas, August 9, 2014:
…..
When we say talk about Indian contribution to Mathematics, It somehow seems to be known that we invented 0. That is the first and the last word. Most people dont know, that we calculated pi.
Also, nobody knows that we had calculated infinity. I guess our humanity background students must be confused by now. How does one calculate infinity. The answer is, Mathematics can do a lot. Vedic Mathematics allows us to have the ability to conceptualize ,postulate and prove values for infinity.
Here is a video which explains How infinity was calculated.
https://www.youtube.com/watch?v=pElvQdcaGXE
This was done by a 12th century Mathematician known as Bhaskara 2 . Fairly great Mathematician.
Then in 14th century we had a another Mathematician,who was known as Madhava. He was a pioneer in the field of differential Mathematics and series expansions. There is comprehensive proof that, his work was taken to Europe via a port called Muziris.
Considering the progression.
we have Vedic Mathematics -> postulation of infinity -> Creation of series expansions -> differential calculus -> Integral calculus.
At this point , it is commonly believed that series expansions were a western creation. Indian students are not familiar with the logic behind creation of infinity. differential and integral calculus are associated withe Leibnitz and Newton.
They might be right about Integral calculus.