Talk:Indian mathematics/Archive 3

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Archive 1 Archive 2 Archive 3

Request for comment:Indian Mathematics

Statement by Fowler&fowler

Indian mathematics is both poorly sourced and poorly written. Mostly, it is has long lists of tall claims. For the last two weeks I have tried to improve the article, by adding some real mathematics to the article as well as adding reliable sources (i.e. internationally recognized journals and books in mathematics or the history of mathematics). I have worked on three sections: Kerala Mathematics (only the introduction though), Charges of Eurocentrism, and Vedic Mathematics.

  • To give you some idea of changes I made, here is what the section on Kerala Mathematics (intro) looked like before I began to work on the article: Kerala Mathematics intro-Before and this is what it looked like after my edits: Kerala Mathematics intro-After. All the changes you see are mine, no one else has added to them.
  • Next, this is what the section Charges of Eurocentrism looked like before my edits: Charges of Eurocentrism-Before and this is the section after my edits: Charges of Eurocentrism-After.
  • I tried to be careful about adding only internationally recognized journals (like those published by the Mathematical Association of America) or well-known text books. Here are the Notes before my edits: Well, actually, there weren't any! and here are the notes after my edits: Notes-After All notes between 7 and 22 (except for the website 9) were added by me.
  • Finally, I've tried to edit the section on Vedic mathematics. I say tried, because the section was (and still is) so full of implausible claims that it reads more like a bad fantasy than a history of mathematics. At first I simply added a number of "citation needed" tags, but subsequently added a section which was first titled "Criticism of Vedic Mathematics," and was later changed to "Assessment of Mathematics of the Vedic Period." The idea of this section was to get independent assessment of these claims by experts in the fields. The last version of this section (that I edited) looked like this: Assessment of Mathematics of the Vedic Period. Finally I added some explanation (using Taylor's Theorem) for a formula in the Vedic Mathematics section here as well as to the Wikipedia page Sulba Sutras it was a summary of.
  • This is where user: Freedom skies comes in. I'm not sure what his goal or motivation is. Apparently he thinks I am anti-Indian or anti-Hindu. He began to revert my edits, accusing me—on the talk page—of distorting the views of the mathematicians I was quoting. In response, I began to post the exact quotes (in case you wondered why the "Assessment" section had so many quotes.) He then decided that the sources themselves were biased and began to remove the quotes one by one. See here and his next three edits. At that time, he left two paragraphs in (the section), but today, he removed the section altogether and, in addition, removed the "Charges of Eurocentrism" section (that someone else had created), and finally removed most of the "citations needed" tags in the Vedic Mathematics section and replaced them with references that not only inadequate, but, frankly, also are embarrassing. Here are his additions: Notes added by Freedom skies #3 through 16 and 19 through 22! Number 21 for example is a "homework help web site" and Numbers 11 and 20 are references to an undergraduate report by Ian G. Pearce, which I am sure is written by a very capable undergraduate, but nonetheless is not the best WP reliable source. (The only reason why Note 6 is a complete reference is that I completed it.) Most of his other references don't have a publisher (for example: "Know the Vedas By Raj Kumar (page 68)") ...

I am trying to be patient, but clearly this is not good for Wikipedia. I need some advice on how to deal with this. 05:26, 23 February 2007 (UTC)

PS. Since one of the comments below alluded to my inclusion of remarks by Amartya Sen, I thought I should add that the initial version of my section "Criticism of Vedic Mathematics" had "sociological" comments by Sen and others on the politics behind the resurgence of "Vedic mathematics." Here are the quotes from Sen and Staal, which I later removed and replaced with assessments of mathematics of the Vedic period. Fowler&fowler«Talk» 09:06, 24 February 2007 (UTC)
PPS The problem is not just the quality of the citations, but the claims in the first place. When the claim is not accurate, you will never find a reliable citation. To give an example, it is claimed in the section on Sulba Geometry (c. 800-200 BCE) that this period in India saw the first use of irrational numbers (i.e. anywhere in the world) and a citation is provided, See here. That citation is incomplete and the text doesn't say what exactly is claimed in the citation. One could look for better citations or more complete citations, but the problem is with the "fact" itself. For, I point out in the following subsection, that a Mesopotamian tablet from (and carbon dated for) the period 1900-1600 BCE, also had a formula for which is actually more accurate than the Vedic formula, although not as insightful, and so the "first use" claim would certainly belong to that more than Vedic Mathematics, since it is is dated a millennium earlier. See here. This in addition to the fact that the Babylonian tablet is hard copy as it were, whereas the Vedic citations are based on best estimates of the age of orally-transmitted texts that were finally only written down towards the end of the 1st millennium CE. Fowler&fowler«Talk» 20:34, 24 February 2007 (UTC)
PPPS. In light of last Wednesday's New York Times article, A History Department Bans Citing Wikipedia as a Research Source, we need to be ever more careful about the quality of the citations. Fowler&fowler«Talk» 21:13, 24 February 2007 (UTC)

Statement by user:Freedom skies

My statements will come shortly. I have been working through my examinations. I'll present my point of view in a few hours. It would be best if editors heard both sides of the story before resorting to judgement. Freedom skies| talk  16:36, 23 February 2007 (UTC)

I'll be as brief and concise about this as is possible:-

  • The Changes I've made
  1. I tried to replace the "citations needed" tag with actual citations. Compare my version to this one.
  2. I removed the "Charges of Eurocentrism" section; this is wikipedia not some soapbox. Indian laymen accuse everything to be Eurocentric but how that finds it's way into an encyclopedia is beyong my realm of understanding. If "EU as a potential superperpower" can be deleted then why shy away from flamebait sections such as these?
Instead of adding material to that section as most people seem to think of my edits I have been removing the section which violates the WP:ENC code of ethics.
  • My proficiency in the field
  1. Someone called my providing citations and removing undue sections as being a "drawback" of the "anyone can edit" policy. I'm a third year engineering student and I probably see more maths than most people; suddenly I'm summed up in the very ad hominem "zealous angry young man with an agenda."
  • My "agenda"
  1. I will continue to provide citations for every "citations needed" tag not only in this article but others as well. In my case I have provided the name of the book and the page number, which is a perfectly justifiable under Wikipedia rules.
  2. If anyone feels that some sources need correction then I will provide additional ones. I will not shy away from my duties as an editor for this encyclopedia. I am working through my examinations and I know that I'm going to be unduly harrasesed but then I'm doing this in any event.
  • My personal feelings about this
  1. Being a Hindu on Wikipedia is demoralizing at times. I asked for sources here and all I recieved were insults and people just kept reverting. I, of course operating alone, could not even as much as attach a tag and these people did not even care about providing explainations.
  1. Before my side of the story was heard, judements were passed on my being zealous and everything. I know most people probably find this amazing but I tried to talk to Fowler on this very page before editing. He, no doubt confident that he could recruit people to make me insignificant (notice the "why cannot they just leave good enough alone" response he gets), just left the discusion and initiated edit warring.
  1. I do see people disagreeing with part of my edits but indiscriminate reversion based on confidence in case of an edit war is not fair rationale. Reverting the entire, whole thing completely has been done in case of my edits, often cited as "rv vandalism". I did provide citations from the University of Michigan, British sources and the Stanford University; I'm sure mentioning them here must have slipped Fowler's mind.
  • Solution
  1. Point out the sources which violate the WP:RS and point out why do they violate WP:RS instead of reverting the whole thing indiscriminately. I'll bring additional sources. Being revert happy is not the solution. This solution should endure as I intend to stay longterm with this article and provide citations wherever needed.

Regards,

Freedom skies| talk  10:53, 24 February 2007 (UTC)

Comments by other editors

Comments by: Dbachmann (talk · contribs)

Fowler&fowler is a mature and academic editor proficient in the field, and Freedomskies is a zealous angry young man with an agenda. Just another instance of the well-known drawback of "anyone can edit" (vice-versa its many admitted boons). Enough said, really, the case couldn't be much more clear-cut. dab (𒁳) 09:47, 23 February 2007 (UTC)

Please refrain from making comments of a personal nature. =Nichalp «Talk»= 18:20, 23 February 2007 (UTC)
I agree with Nichalp. There is no doubt that Fowler & Fowler is mature/resposible.(I have also supported his viewpoints on many cases). It does not mean we have to support him on all cases blindly. There are many people who support him blindly on all issues without going through merit of actual incidents. We have to support issues based on contents and not based on the qualities of person who raised issues. I suggest Dab and DavidCbryant to address issue directly instead of eulogising F & F or withdraw their comments. (I am not commenting on this issue since I don't have knowledge on this subject.). If you want to appreciate contributions of individual then award Barnstar instead of supporting that person on all cases. --Indianstar 06:58, 24 February 2007 (UTC)
When the problem is user conduct, I will take the liberty to comment on user conduct. If this case isn't a no-brainer to any honest editor, Wikipedia has a problem. If there are genuine concerns with this article, let them be brought forward by someone who with a grasp of WP:ENC, responsibly and to the point. dab (𒁳) 21:49, 24 February 2007 (UTC)
looking at the article, there is so much childish nonsense hanging around that I wonder why we have to waste time with RfCs. Consult any encyclopedic treatment of the subject, and base the article on that. Editors trying to prevent this are in violation of WP:UNDUE at best (and simple trolls at worst), and should be penalized accordingly. dab (𒁳) 22:19, 24 February 2007 (UTC)
You do realize that I offered to remove every quote from the article and let the article be for Indian mathematics only, righ? How can the removal of the glowing Einstein and Lapalcae quotes from the intro result in "inflating India's role?" My notion of "encyclopedic treatement", which includes removal of all quotes of opinions and let the article be about Indian mathematics alone was rejected as well. Freedom skies| talk  01:23, 25 February 2007 (UTC)
Because outrageous, unattributed claims would have remained without qualification from the peer-reviewed, academic sources cited by Fowler&fowler.
That's how it would have resulted in "inflating India's role".
Playing dumb doesn't suit you, Freedom skies, and it's an insult to the rest of us.
CiteCop 02:14, 25 February 2007 (UTC)
Not quite, citecop. I removed quotes and provided citations. Provide diffs to where I inflated India's role in "Indian mathematics" and your ramblings may attain a semblence of truth. Freedom skies| talk  04:20, 25 February 2007 (UTC)
I have never seen a personal attack as blatant as this -- 121.246.11.93 20:48, 19 August 2007 (UTC)

Comments by: DavidCBryant (talk · contribs)

I have reviewed the edit history on this article, and I have looked at a sample of both Fowler&fowler's and Freedom_skies' contributions to Wikipedia. In my opinion, Fowler&fowler is a responsible, kind, decent, and hardworking contributor. Freedom_skies consistently pushes his own POV, does not respect guidelines, loses his temper frequently, and vandalizes Wikipedia with some regularity. DavidCBryant 14:12, 23 February 2007 (UTC)

Please refrain from making comments of a personal nature. =Nichalp «Talk»= 18:20, 23 February 2007 (UTC)
What was personal about it? I characterized FS' behavior based on the solid hour I spent reading his talk page, reviewing the many times he has recently been blocked from editing, and reading the incredible exchanges recorded on this talk page, above. In case you haven't noticed, FS placed a semi-protection template on Indian mathematics just 34 minutes after F&f opened the RfC. That was dishonest and unethical, in my opinion.
I also spent an hour reviewing F&f's talk page, saw that he was blocked once last summer, and read his responses to FS' wild accusations and incoherent babbling (above). I formed my opinion by reviewing the available evidence carefully. Everyone is fallible. I make mistakes. But I did quite a bit of research before I wrote my initial comments, and I still think they're fair. DavidCBryant 00:57, 25 February 2007 (UTC)

I suggest Dab and DavidCbryant to address issue directly instead of eulogising F & F or withdraw their comments. (Indianstar)

I did address the issue directly. I did not "eulogize" F&f – to the best of my knowledge, he's still breathing. I had never even heard of either of these gentlemen until this morning, about 14½ hours ago. I did my best to be fair, and I will not withdraw my comments. DavidCBryant 01:39, 25 February 2007 (UTC)

DavidCBryant, a user who obviously has little knowledge of the situation is quick to label freedom as an [sic] vandal (I was unaware adding references was vandalism, though on planet Glingxon I'm sure they think that way). (Bakasuprman)

It's true that I have little knowledge of this situation – I only spent two hours gathering facts before I reached a conclusion. How much time did you devote to it, Baka? FS removed entire sections of this article by blanking today. Removing well-sourced material is in fact vandalism. Doing it over and over again is an egregious violation of the norms of civil conduct. DavidCBryant 01:57, 25 February 2007 (UTC)
To Bakaman: Although I didn't know about David until earlier today, looking at his Wikipedia page, I can tell at a glance that he knows more than enough mathematics to judge the quality of the citations. See his contributions here. For example, quickly reading through one of them, I learned that quadratic equations can be numerically solved using continued fractions. I have of course known for a long time that continued fractions can be summed using quadratic equations, but didn't realize that there was a converse implication. Since the Vedic mathematics section claims that quadratic equations were solved in the Vedas, David even more than others, is superbly qualified to judge both the claims and the citations. Fowler&fowler«Talk» 03:49, 25 February 2007 (UTC)

Comments by: Grokmoo (talk · contribs)

I have noticed the very many incredible claims made here and on other pages with regards to Indian contributions to mathematics. I have also noticed in multiple cases edits where the author clearly does not even understand what it is he/she is actually claiming. Fowler&fowler is braver than I to try to make such sweeping changes to these articles, since even the smallest factual correction is often met with great resistance (in my personal experience). I have looked over Fowler&fowler's contributions, and the quality of his edits and the great number of reputable citations have greatly impressed me. I think his edits are a large step forward in the right direction of cleaning up this page. Grokmoo 21:50, 23 February 2007 (UTC)

Comment by: Sbandrews (talk · contribs)

Fowler&fowler, I was impressed by your section Assessment of Mathematics of the Vedic Period and feel that it deserves a place either in this article or perhaps the main article on vedic mathematics. However I also feel that in placing fact tags *all over* the vedic's section you have stirred up the current controversy more than was needful. If you place the new Assessment section in this article you are amply stating your position and can leave it up to the reader which side of the story to accept. Vedic mathematics may not be rigorous mathamatics - but it is mathematics due to its name - an uncomfortable truth perhaps. If the other party has exams I will comment more when he has put his case, Kind regards sbandrews 23:51, 23 February 2007 (UTC)

Freedom_skies, well, I had an open mind, and I am still inclined to keep one, but your last edit to this page, reverting all of Fowler&fowler's edits without discussion, counts against you IMO. Why do you feel it is innapropriate to edit now? Surely this is the *best* time to edit, while many impartial observers are here to help you work with Fowler&fowler to provide working consensus approach to this page. I see three options for you two -

  1. agree on areas of the article which are the sole preserve of the other - one take the vedic section, the other the critique of vedic - this has the advantage that the two of you need not communicate.
  2. start to communicate, stop deleting (Freedom_skies) and stop smothering with fact tags (Fowler&fowler) and perhaps, both of you, try to understand the others position.
  3. open warfare :), kind regards, sbandrews
I reverted to the version which had additional citations and was reverted in the first place. I realize that the consenseus here seems to be "better sources." I will provide them in a few hour's time and will try to have international sources and detailed information this time. I further would like to invite fowler to a workshop on this article, we can settle our disputes and come up with a source code there and then just copy/paste it. The whole process could be over in a few days if the parties involved cooperate with each other. I tried to hold a discussion with him in the first place, he just walked off and started edit warring. Freedom skies| talk  16:34, 24 February 2007 (UTC)
ok, however making reverts to a previous version like that in the middle of an edit war is seldom if ever a good policy - much better to go through by hand taking care not to undo the hard work of others. Moving back to a previous version is realy only good for undoing vandalism - not the case here, regards sbandrews 16:51, 24 February 2007 (UTC)
I have eliminated every single quote of either praise or critisism. The article had become a soapbox for citing people's opinions. Since this is an article about science I'll see that only the mentions of science and not personal opinions make it to the article. Having removed quoted opinion I'll sit down and work till verifiable WP:RS sanctioned sources are provided. Regards, and many thanks for keeping an open mind. I'll provide you with a finished version and links to both the earlier versions by me and Fowler for reasons of comparision and further recommendations. Might take me a few hours though (I'll try to keep it down to four or five), I have had to study a bit in real life as well. Freedom skies| talk  17:19, 24 February 2007 (UTC)
Sorry, I didn't see sbandrews comments when I reverted back to Freedom skies "default" version (in an effort to be conciliatory). But after reading the comments above, I agree that it is best to have everything out there, Freedom skies citations, my citations, and all the sections so that the editors commenting here can decide for themselves what is appropriate and what is not. So, I will revert to a version that is has all the additions and no deletions from the last few days. Fowler&fowler«Talk» 17:48, 24 February 2007 (UTC)

Comment by: Bakasuprman (talk · contribs)

I have noticed that Fowler quotes Amartya Sen as if Sen is some sort of Keith Devlin or Stephen Wolfram. Freedom skies seems to have produced refs which allude to and back up many of the assertions made on the page. The first two comments (by outsiders) only serve to prove that some users do show a rabid disdain for "Indian achievement" whether real or imagined. DavidCBryant, a user who obviously has little knowledge of the situation is quick to label freedom as an vandal (I was unaware adding references was vandalism, though on planet Glingxon I'm sure they think that way). Also Dbachmann's characterization of freedom skies as an "angry young man" indicates that he is stuck (or wholeheartedly believes) in a paternalistic mindset similar to White man's burden, where those providing a well-sourced antithesis to entrenched views in the Western World are branded with terms such as "communalist", "angry young man" and in India Hindutvavadi.Bakaman 01:32, 24 February 2007 (UTC)

Please refrain from making comments of a personal nature. CiteCop 02:41, 24 February 2007 (UTC)
as always when commenting on India related topics (he rarely comments on anything else), Bakaman is lost in wild-eyed paranoia, elevating his habitual playing of the "race-card" to some sort of fundamental editing principle. I have no idea why an editor with such an attitude is still with us at all. dab (𒁳) 22:34, 24 February 2007 (UTC)
Lost in wild-eyed paranoia? Its hilarious to see the great mischaracterizations dab creates. I guess when one is so busy patronizing trolls (BhaiSaab, TerryJ-Ho anyone?),and advocating for militant groups and ideologues, and hoarding sets of articles in a juvenile manner, one becomes quite adept at churlish behavior. I'm on wikipedia to add to the body of knowledge, a function I suggest dab make at least a miniscule effort to execute.Bakaman 23:27, 24 February 2007 (UTC)
Dear Bakaman, there's no need for inflammatory comments. Since you began the sniping in your comments above, you bear the primary responsibility. Regarding your comments about Amartya Sen, I have repeatedly told you that Sen was commenting on the politics and sociology of why Vedic mathematics was being promoted in India in the late 1990s, not discussing the specifics of the mathematics. Besides those comments are no longer a part of the page. As for you example of people, who you think are more authoritative, Keith Devlin and Stephen Wolfram, let me say that the former, who I enjoy listening to on the radio, is really now a popularizer of mathematics (and gave up his research career a while ago) and the latter was a mathematical physicist whose research career got side-tracked by Mathematica, and his recent efforts to revive it with A New Kind of Science have had mixed success. Neither is an authority on the History of Mathematics, like for example, B. L. van der Waerden or Dirk Struik, or a research mathematician like, for example, Pierre Deligne or Michael Atiyah. Fowler&fowler«Talk» 00:01, 25 February 2007 (UTC)
Dear Dab, Experienced editor and administrator like you could have avoided comments like "I have no idea why an editor with such an attitude is still with us at all.". Bakaman may differ from your views on some issues but he has definitely made many good contributions to India portal. --Indianstar 11:52, 26 February 2007 (UTC)

Comments by: CiteCop (talk · contribs)

In my experience, Freedom skies cites sources that do not meet WP:RS and, more egregiously, completely misrepresents them, citing them in support of statements they do not actually support. No matter what article he edits, Freedom skies' edits maximize the role of India whether or not reliable sources support his edits; I leave it to the reader to decide for himself whether this means Freedom skies has an agenda. Like Grokmoo, I have noticed the many incredible claims with regards to Indian contributions, not only in mathematics, but in a number of fields, as well as the great resistance that correcting these claims meets. I would also like to remind readers that implying that someone is a racist is no substitute for providing sources that meet the standards laid out in WP:RS. CiteCop 03:41, 24 February 2007 (UTC)

Freedom skies,
There is no "default" position. Editors are under no obligation to let statements lacking reliable sources remain while their contributors look for sources; quite the opposite, in fact: 'There seems to be a terrible bias among some editors that some sort of random speculative "I heard it somewhere" pseudo information is to be tagged with a "needs a cite" tag. Wrong. It should be removed, aggressively, unless it can be sourced.' Those are the words of Jimbo Wales, founder of Wikipedia; they can be found at WP:V.
Editors who apply the {{fact}} tag to your insufficiently sourced edits are actually doing you a favor, because they're not deleting your edits when they have every right to do so.
Fowler&fowler has cited sources that meet WP:RS in support of his material.
Now you must do the same, if you want your material to remain, whether in this Wikipedia article, or any other.
CiteCop 17:29, 24 February 2007 (UTC)
The last paragraph of this section is pure, unattributed speculation and ought to be deleted forthwith. CiteCop 21:37, 24 February 2007 (UTC)

Comments by: Jitse Niesen (talk · contribs)

Fowler&fowler says that Freedin skies added "references that not only inadequate, but, frankly, also are embarrassing". Well, I looked at the references added by Freedom skies in this edit, which seemed representative from a reading of the article history. The first references added are (quotes from Freedom skies' version, followed by my comments)

  • "The Modern Review edited by Ramananda Chatterjee. Original from the University of Michigan. Page 634" - This probably refers to the monthly magazine Modern Review, published in Calcutta. It doesn't seem to be peer-reviewed. It's unclear what Michigan has to do with it. The reference does not include a volume number, so it is impossible to find what is being referred to.
  • Replacing "Mathematical Expeditions: Chronicles by the Explorers by David Pengelley, Reinhard C. Laubenbacher" with "Toward a Global Science: Mining Civilizational Knowledge By Susantha Goonatilake (page 119)" - I don't really know which source is preferable, but I have my doubts about Goonatilake's book given that it seems to be written in order to argue that Indian's contributions to science are being ignored (anonymous (?) review) and that it got a rather negative review by Kavita Philip in Isis, Vol. 92, No. 1, pp. 247-248 (quote from the review: "The chapters on medicine and mathematics cannot therefore deal substantially with the provocative claims he [i.e., Goonatilake] puts forward.")
  • "Science in Ancient India By Narendra Kumar (page 9)" - I can find very little about this book, suggesting that it is not used often. It's unclear whether it can be considered reliable.
  • "Vedic Mathematics By Vasudeva Sharana Agrawala, Swami Bharati Krishna Tirtha" - don't know
  • "Vedic Mathematics for Schools Bk.1 By James Glover" - this is published by Motilal Banarsidass. It doesn't seem a scientific text, according from the description at Google Books

None of the references have full bibliographic information, as normal in references. Based on all this, it seems correct to consider the references inadequate.

Freedom skies says that he/she (sorry, I don't know) "removed the "Charges of Eurocentrism" section; this is wikipedia not some soapbox. Indian laymen accuse everything to be Eurocentric" However, the section had references to articles in journals as Historia Scientiarum (Japanese, peer-reviewed) and Mathematics Magazine (published by the Mathematics Association of America), so it's unclear what issues Freedom skies has and why the section is soapboxing. Freedom skies' solution ("Point out the sources which violate the WP:RS and point out why do they violate WP:RS instead of reverting the whole thing indiscriminately. I'll bring additional sources. ...") is a bit odd as it supposes that reliable sources exist, while that seems to be the whole problem.

On the wider issue, I agree with several people above that the Indian contributions in the history of mathematics are bigger on Wikipedia than in the other books I read. -- Jitse Niesen (talk) 14:30, 24 February 2007 (UTC)

J.T. Glover is a well-respected academic figure, fellow of the Institute of Mathematics and its Applications and head of the mathematics department at St. James Independent College. The Vedic Mathematics book he wrote received glowing reviews from other noted academic figure such a P.S. Charak (of GGM Science College in Jammu) and AK Vijayakumar.Bakaman 02:35, 25 February 2007 (UTC)
The back cover of that J.T. Glover book reads:
An elementary school textbook.
For eight- and nine-year-olds.
CiteCop 04:10, 25 February 2007 (UTC)
Do source which Glover book you are referring to, as I find no mention of any of the above in the one I have in front of me. You still have not refuted that he is a respected academic figure in Britain in the field of mathematics.Bakaman 17:47, 25 February 2007 (UTC)
Which Glover book? The only one cited in the article. CiteCop 17:53, 25 February 2007 (UTC)

An elementary school textbook. For eight- and nine-year-olds.

Which would make the information incorrect then? Freedom skies| talk  05:39, 25 February 2007 (UTC)

To me it seems unlikely that an elementary school textbook "reconstructed from ancient sources" is a reliable source of scholarly knowledge about the actual contents of those sources. One necessarily simplifies much in order to present things at an appropriate level for the readers of the book, and "reconstructed from" is not the same as a dispassionate presentation of what's actually in those sources. To put it bluntly: the purpose of citing sources is to convince your readers that you have thoroughly researched the subject and are fairly presenting it. Your insistance on using sources such as these instead convinces me that you are stretching, that solid sources are not available for what you want to claim and so you are citing flimsy ones instead. It makes me think there is a reason solid sources are unavailable. That is the opposite of what a source should be. —David Eppstein 05:45, 25 February 2007 (UTC)

Do take a look into this version Freedom skies| talk  05:48, 25 February 2007 (UTC)

Hi Freedom skies, providing ISBN numbers for references that previously didn't have them only fixes a minor part of the problem. The bigger problem that David alludes to above is that the claims themselves are often either false or unverifiable, and, therefore, the references cannot be reliable. Why do you think the authors you cite never have publications in internationally recognized journals or publications? Their publishers are all obscure Indian ones. How come the best known Indian publishing houses like Asia Publishing House, Oxford (India), Penguin (India), Harper-Collins (India), Roli Books, or Macmillan (India) never publish these authors? Internationally recognized journals like the Archives of the History of Exact Sciences or Journal of Indian Philosophy would love to publish a paper definitively showing that the Pythagoras theorem was discovered in India. (It would double their sales for the next two years!) Why do you think these authors have had no success there? I'm not trying to be confrontational, I'm just trying to point out that one has to look at the reliable sources first to decide what the claims should be, rather than treating the claims to be sacrosanct and then go about looking for sources in every nook and corner. I appreciate the effort you are putting in to find those sources, but, as you must have realized in this discussion, most commenting editors don't find your sources to be reliable. I think we can both collaborate on improving the article and will address it in a "proposal" subsection below. Thanks. Fowler&fowler«Talk» 15:57, 25 February 2007 (UTC)

Comments by: David Eppstein (talk · contribs)

This article is the center of a general pattern that I see in which the genuine accomplishments of ancient Indian mathematicians are artificially inflated so that they can be claimed as having precedence over similar ancient mathematics in Greece, Egypt, Babylon, and China. This pattern of inflation, paradoxically, makes India's contributions seem smaller than they were, because after seeing the exaggerated claims here it becomes difficult to believe any of the claims about Indian mathematics, even the ones that are legitimate.

As a case in point, the square root of 2. The Shulba Sutras give a rational approximation 1 + 1/3 + 1/(3*4) - 1/(3*4*34), which works out to 577/408, accurate to five digits of decimal accuracy; that much is the facts. However, the Shulba Sutra article and the article here go on to append a supposed explanation in terms of Taylor series; they don't outright say that these series were known to the ancients, but as far as I can see the only reason for stating this explanation is to hint at a greater ancient knowledge than can be supported by the facts (there's a much simpler explanation possible for the specific approximation given here, in terms of the side and diameter numbers known to the Pythagoreans contemporaneously to the Shulba Sutras). The claim in this article that the Shulba Sutras provided the "first use of irrational numbers", while it has a (not very verifiable) source listed, also seems difficult to swallow: for one thing, their claimed date range overlaps with the Pythagorean knowledge of irrationals; for a second, a much later (5th century AD) claim of Indian knowledge of irrationality is included in History of numerical approximations of π based only on a minor quirk of wording, while the Greeks gave explicit mathematical proofs of irrationality; for a third, "use of irrationals" regardless of mathematical proof that these numbers were irrational goes back much earlier to Babylonian uses of π.

In the dispute at hand, Fowler&fowler appear to be taking the side of restricting the article to what is known and documentable, providing solid accepted academic citations for all facts, and eliminating the speculation. Freedom skies, on the other hand, appears to be one of the principal perpetrators of the unencyclopedic exaggeration, adding speculative interpretations of what the ancients might have known, and badly sourcing things by leaving such claims undocumented, providing useless unverifiable documentation, or not taking care to distinguish sources that are accepted scholarly work from speculative popular-press writings. To the extent that this is true, I support Fowler&fowler.

Thanks, David, for you comments. Very well put. I have to confess that the Taylor series note was my handiwork! Alhough, I did say, "with hindsight," you are right that I (albeit unwittingly from my perspective) did leave the impression that there was more depth to the formula (because it fit a later (Taylor series) expansion). Actually, now that I think of it, I even implied it in my postscript to my statement above, where I implied that the Indian formula for was more "insightful" than the Babylonian formula, even as I was arguing for the Babylonian formula's older provenance! I didn't know about "side and diameter numbers" and will read that page. And I'll fix the Taylor series remark, and may take it out altogether once I have a better understanding of Pell number. Thanks again! Fowler&fowler«Talk» 22:04, 24 February 2007 (UTC)
Sadly, mathematics is far from the only topic on Wikipedia which suffers from this artificial inflation of India's role. CiteCop 23:46, 24 February 2007 (UTC)

Comments by:Blacksun (talk · contribs)

I have to support Fowler on this. It would be very nice to have a concise inventory of Indian contribution to mathematics without unverifiable claims and exaggerations. You can find a book to support almost anything nowadays. But it does not make it the truth. I am sure their are many things in the world of mathematics that are not really true in terms of who did what first and I personally never cared for such silly claims - but we have to stick with what is backed by the most rigorous methods of research. --Blacksun 22:41, 24 February 2007 (UTC)

Comments by: Wiki Raja (talk · contribs)

My take on this is that there should be proper sourcing and versions. As for entire reversions, I feel that information from both sides should be posted. I would not really recommend putting sources from homework sites. However, in regards to Freedom skies referenced sources from texts he puts the page number too which is really good, actually the best way for referencing your sources. Wiki Raja 23:19, 24 February 2007 (UTC)

Comments by: Abecedare (talk · contribs)

Freedom Skies, can you please clarify for me, which of the following references you still think should be used as reliable sources for the (respective) cited point in the article:

  1. The Modern Review edited by Ramananda Chatterjee. Original from the University of Michigan. Page 634
  2. Toward a Global Science: Mining Civilizational Knowledge By Susantha Goonatilake (page 119)
  3. Science in Ancient India By Narendra Kumar (page 9)
  4. Vedic Mathematics By Vasudeva Sharana Agrawala, Swami Bharati Krishna Tirtha (page 28)
  5. Vedic Mathematics By Vasudeva Sharana Agrawala, Swami Bharati Krishna Tirtha (page xiix)
  6. Vedic Mathematics for Schools Bk.1 By James Glover (page 1)
  7. Vedic Mathematics Teacher's Manual v. 3: Advanced Level By Kenneth R. Williams (page 125)
  8. Bulletin of the Calcutta Mathematical Society By Calcutta Mathematical Society. Original from Stanford University. (page 197)
  9. Hinduism, Its Contribution to Science and Civilisation By Prabhakar Balvant Machwe, Prabhākara Mācave
  10. Yajnavalkya Smriti: The Âchâra Adhyâya. - Page 358 by Yājñavalkya, Vijñāneśvara, Vaidyanātha Pāyagunde
  11. The Science of Empire: Scientific Knowledge, Civilization and Colonial Rule in India - Page 27
  12. http://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch4_2.html
  13. http://www.jiskha.com/mathematics/algebra/
  14. History of Mathematics By David Eugene Smith (page 288)

I have great appreciation for the effort you put in in locating these references, and assume that you added them to the article in good faith believing them to be relevant, reliable and adequate. I wonder, though, if your opinion about the adequacy of some of these references has changed after reading the comment of other wikipedia editors above. I think it would be very helpful if you could specify which references you now agree are not sufficiently reliable (as per WP:RS) for the purpose of this article, so that we can focus the discussion on remaining points of disagreement (if any!). In my opinion, it would be good to get beyond recriminations about past actions, comments and edits.
Finally, can you please add the relevant publisher, ISBN info (where missing) for the above references and clarify what is meant by "Original from the University of Michigan" and "Original from Stanford University", so that it is easier to verify the attributed statements (as per WP:V). Thanks in advance for your time and effort! Abecedare 04:52, 25 February 2007 (UTC)

Thanks for the kind tone and please do take a look into this version. I, for one, will continue to work in this article and will eagerly await suggestions from neutral, third parties. If you have some concerns about the newer version then kindly point them out. I'm not at my dilligent best right now but I'll find yet more additional sources in a few hours. Freedom skies| talk  05:46, 25 February 2007 (UTC)
Looking at the style of citations in the sources above, let me add to Abecedare's comments that the citations from the journals The Modern Review and Bulletin of the Calcutta Mathematical Society should also include the author and title of the article cited and the year, volume number, and pages of the article cited.
While we're looking for reliable sources, a quick bibliographic search turned up a number of articles on Indian mathematics in the Indian Journal of History of Science. The journal is published by the Indian National Science Academy, see [1]. --SteveMcCluskey 22:05, 25 February 2007 (UTC)

Comments by: Rayfield (talk · contribs)

I have not looked into detail in this article, but I have had experience with how User:Fowler&fowler deletes references in other articles. In the Indus Valley Civilization he deleted all references to the archaeologists B.B. Lal, Jane Mcintosh, V.N. Misra and S.P. Gupta. Some of these references were at least replaced with other references.after discussion on the talk page of Talk:Indus_Valley_Civilization.
And the reason for this censorship was apparently only because they were criticzed by some politicized and mostly non-archaeologist scholars [2] Every. scholar gets criticized, and if archaeolgists of the calibre of B.B. Lal, Jane Mcintosh, V.N. Misra and S.P. Gupta because of such political reasons, and this is called responsible and mature, I don't want to know what immature is.
Others above have said:"I agree with several people above that the Indian contributions in the history of mathematics are bigger on Wikipedia than in the other books I read." While the Indian contributions should not be maximized, they should equally also not be marginalized.--RF 02:31, 3 March 2007 (UTC)

Comments by: Skant (talk · contribs)

I am not sure what is happening. But here are a few free comments from another zealot that is "by me". Why the "irrational number" article doesn't mention that mesopotamia thing even now? As for citation, how do you decide what is good book, site or article or a person to refer to. I once wrote to Stanford that despite so many of western scholars having referred to Hindu philosophy, or Indian philosophy, having said so many flattering things about it, why you don't have anything on them at all on your pages. And the answer was excellent, they said they haven't found any author to do so!!! You have voltaire, schopenhauer, Mark Twain, Will Durant and so many German/American people who have been constantly writing about Hindu philosophy for more than a century, still you can't find good authors to write on Hindu philosophy. You have english people teaching Hinduism in Oxford, you have christian priests teaching Hinduism in america. What Amartya Sen says is right for him and not for Hindutva people, but that doesn't mean you can't decide antiquity of things, which is more important context. What is meant by "vedic" may differ for Hindutva people it can be something which was written or attributed to Vedas even in other later texts that are commentaries on Vedas. There is no truth without context, so you can change it to Indian or Hindu mathematics, instead of "Vedic" and add date related context to it.

I have been doing just one thing for quite sometime, adding "Hinduism" related sayings to people on wikipedia who said lot about Hinduism and the wikipedia doesn't even state once the word Hindu in those articles. What is the use of wikipedia? If wikipedia rules prevents article/idea to get close enough to facts (whatever they are), then you should change your rules. The articles on wikipedia have biases, that is true about any history as well, but soon the problem will become much worse, as wikipedia the whole citation-is-not-good thing is too subjective. By the way I don't want any one of you to tell me what this talk page is for, you can ban me if you don't feel the need to understand what i am saying, or if you find it too zealous.Skant 06:55, 5 April 2007 (UTC)skant

Proposals

Proposal by: Fowler&fowler (talk · contribs)

Thanks everyone for commenting and thanks, Freedom Skies, for participating in the RfC and providing your perspective. I think the RfC is definitely much better than the endless edit-warring of last week. To the extent that I contributed to the latter, I apologize. The Indian Mathematics page is clearly an important page; otherwise, so many people would not set aside their time to offer detailed comments. Reading through the comments, I feel a number of problems with the article are readily identifiable (and a majority of the editors seemed to agree on this).

  • The article has very little narrative. Mostly, it has long lists that discourage average readers to read on.
  • The claims are either exaggerated or unverifiable and consequently they cannot be reliably sourced. Average readers therefore do not infer credibility and, if they read on, are inclined to devalue all claims, even the ones that are true. This, as David Eppstein, observed above, has the opposite effect from what is intended when such claims are put in.

I feel there is a way out and in fact Freedom Skies, I, and others can work together on this. Here is my proposal (and some affirmations):

  1. Let us examine the claims together (say starting with the Vedic Period section) and look at reliable sources and see what they say. I will give you an example below. However, I cannot emphasize reliable enough. A primary-school text for mathematics cannot be a reliable source for an article on the history of mathematics. This also means that many sources (and sometimes the claims themselves) in the article will not pass muster and will need to be discarded. In particular, in the sources provided by Freedom Skies here, references, 2, 5, 8-11, 13-25, and 28 will all need to be replaced or the claims removed, since the sources are not reliable.
  2. Freedom Skies has to trust me that I am trying to improve the article and not undermining the achievements of Indian mathematics. Conversely, I will trust that he is making his effort in good faith.
  3. We now have a community of editors (many expert in mathematics or history of science) who have commented above, and who can be readily called upon to settle minor issues.
  4. I agree with Freedom Skies that in a well-written article the assessment and critique sections do not need to be separate sections, but rather should be blended into the text. However, I feel that they should be retained until such time as the rest of the article becomes well-written and well-sourced.
  5. For the Vedic Mathematics section, a parent page needs to be created. It turns out that there is a WP page Vedic mathematics, however, that page is about a book of the same title (describing a method of elementary computation—akin to the Trachtenberg system—that has been around since the mid-60s, but that got renewed publicity in the late 1990s. In any case, it would be more accurate to call the title of the parent page, "Mathematics of the Vedic period." However, there too, there is already a page with that name, i.e. Mathematics of the Vedic period, but, since 2005, it has redirected to the page Shulba Sutras. What about Shulba Sutras? Well, while it is true that most of the mathematics of the Vedic period is in the Shulba sutras, not all is. So the, MoVP page needs to be un-redirected. Once that is done and the page is available, more details could go into that page and only the summaries would need to be in the Indian mathematics page.

Example: Here is an example of how to create a reliably sourced statement. In the section on Vedic mathematics, it is claimed that the Vedic Period in India saw the first use of infinity (as a mathematical concept). Well, what are the facts here? The Isavasya Upanishad of the Yajur Veda begin with the following Sanskrit shloka:

This translates as:

In most critical interpretations, the lines are taken to describe the Universe, the Being, the Brahman, the Ultimate Reality, or the continuum. However, lately, some have interpreted these lines as evidence of knowledge of the mathematical in the Vedic period; in particular, the second stanza is taken to be a primary source for the mathematical statement: While there's always a chance that this was the case, we don't have any evidence of it. What options do we have for a WP article? We can either not make any statements about Vedic knowledge of or we can quote the stanza and the translation and make a statement, "In this period there was knowledge of the notion unboundedness or infinite measure or magnitude as evidenced in ..." provided we are not violating WP:NOR. But, we can't have any claims about first use of mathematical

As, I said, I am happy to work with Freedom Skies, provided we both restrict our efforts to reliable sources and provided we give each other the benefit of good faith. Fowler&fowler«Talk» 17:39, 25 February 2007 (UTC)

  1. ^ Swami Sivananda. 1998. The Principal Upanishads. Divine Life Society. 492 pages. ISBN 8170520010. page 4

Comments on the Proposals

Comments by: 192.8.198.65 (talk · contribs)

Basically the most important and readily accepted historical contribution by these guys is the invention of the decimal system so perhaps on should focus a lot of attention there.I mean the fact that everything can be represented in 9 or 10 symbols is brilliant and most of the article should focus on this(their most imp achievement) and the space allocated should be in the descending order of significance.Also a word limit on this indeed most articles of this time perhaps 2-3 pages as this is an encyclopedia not a phd thesis on Indian/greek/chinese etc etc maths. —The preceding unsigned comment was added by 192.8.198.65 (talk • contribs). at 09:50, 19 March 2007

Comments by:

Modern Indian mathematicians missing

Is this article intended to cover modern Indian mathematicians, such as Ramanujan? If not, it should be renamed to something like Medieval Indian mathematics. —The preceding unsigned comment was added by Pcu123456789 (talk • contribs) 06:14, 25 February 2007 (UTC).

I think there are two approaches:
  • Stop with the Kerala school (i.e. 16th century) and have no "modern period." The reasoning behind this is: (a) the other similar pages Greek mathematics, Chinese mathematics, etc. are really only about ancient mathematics, (b) the modern Indian mathematics page is more likely to be a list of mathematicians page, rather than a mathematics page, since there is no recognizably Indian school of mathematics or Indian area of mathematics in the modern era, and (c) what should be the cut-off date? No it can't be renamed "Medieval Indian mathematics" since "Indian mathematics" already includes both ancient and medieval (i.e. already includes more than medieval).
I'm not sure which approach I am more in favor of. Probably the first, but I'd like to hear from others. Fowler&fowler«Talk» 09:44, 25 February 2007 (UTC)
In fact much of this page's content should be at Hindu mathematics rather than Indian mathematics. I would agree that Tata and Indian Stat institute deserve a small mention here, but chandrasekhar didnt perform any math that was "Indian".Bakaman 16:11, 25 February 2007 (UTC)
Yes, Chandrasekhar, as I say above, didn't make any contributions to mathematics (and, in particular, as you say, to Indian mathematics). He was a theoretical astrophysicist. As for Hindu mathematics, I don't know what that means. It's not clear if there is anything Hindu about the mathematics, besides the page includes "Indus Valley mathematics," "Jaina mathematics," whose practitioners were not Hindus. The only reason why the appellation "Hindu mathematics" survives in the literature is that early Western writers in History of Mathematics used "Hindu" and "Islamic" to describe the mathematics of India and the Middle-East respectively. Fowler&fowler«Talk» 18:44, 25 February 2007 (UTC)
"should be in Hindu Mathematics rather than Indian Mathematics" is a funny comment. Assyrians changed "S" to "H" and called the people living in the Sindh valley as Hindus. It was anglicized to India....so whats the difference? are you saying that there are more "religious" connotations to this article and that it is not secular? Well, Hinduism is not a religion and it is a way of life of the people living in this subcontinent, and THAT, by definition makes every single contribution of theirs "Hindu". —Preceding unsigned comment added by 204.52.215.94 (talk) 22:08, 19 January 2008 (UTC)
I'm inclined toward the first possibility outlined above. Historically, the development of mathematics in India before say 1800 was largely distinct from developments in the western world. But as trade and communication increased, the boundary between "Indian" mathematics and "European" mathematics got fuzzier and fuzzier. AFAIK, there were no major mathematical developments in India during the 17th and 18th centuries, so cutting it off at the Kerala school makes a lot of sense. DavidCBryant 20:09, 25 February 2007 (UTC)
To me, first option makes lot more sense than second. --Blacksun 09:54, 26 February 2007 (UTC)

Comment by deeptrivia

(a) Why does this article read like such a brag list? (b) What is " could be expressed as " supposed to mean? Can't think of any interpretation of this which is a true statement! Most of this article needs a complete rewrite. deeptrivia (talk) 03:42, 28 February 2007 (UTC)

Well, there is one possible interpretation that would make the statement an approximate identity:
for .
However I agree with your larger point that the article right now is more of a list, rather than a well-written prose (+ math) article on Indian mathematics; and even if my above interpretation is true, the required assumptions should be clearly stated, and of course referenced. Abecedare 04:18, 28 February 2007 (UTC)
Alternately, if this was claimed in the Kerala school section, you could write:

for

Abecedare's solution is clearly quicker. Fowler&fowler«Talk» 14:03, 28 February 2007 (UTC)

PS. OK, I see, it was claimed in a section about 10th century CE mathematics. So, the reasoning would have been along the lines suggested by Abecedare. Fowler&fowler«Talk» 14:13, 28 February 2007 (UTC)
... or we could divide the two sides, take limit as w approaches w' (applying l'Hôpital's rule rule on the way) and show that it equals 1. So we have two alternatives:
(1) Starting from an incorrect mathematical expression; apply the necessary conditions to make it correct (), and then based on the above three derivations claim that Manjula (10th C) knew:
  • Trignometric identities (that Bhaskara II is also credited with 1-2 centuries later), or
  • Taylor series expansions (credited to Kerala school > 3 C later), or
  • Basic limit theorems and differential calculus
(2) Look at the reference from where this equation is sourced, and see if that has an explanation.
I have taken the second option, and added a "citation needed" tag (is there a better tag for our purpose?) to the claim. Abecedare 19:17, 28 February 2007 (UTC)
A geometric proof of Manjula's formula, where the "=" sign in the fourth line should be replaced by sign, since the two angles are only approximately equal.

Yes, the citation needed tag is what is needed. Of course this means that one has to look for a source! Meanwhile, here is another proof, probably the simplest, involving definitions. I'm attaching it in the accompanying image. Fowler&fowler«Talk» 22:52, 28 February 2007 (UTC)

PS I have ordered two books:
  • Thibaut, G. 1984. Mathematics in the Making in Ancient India: Reprints of "On the Sulva-sutras" and "Baudhyayana Sulva-sutras." (edited with an introduction by Debiprasad Chattopadhyaya). xxii + 134 pp., figs. Calcutta/Delhi: K. P. Bagchi.
  • Dutta, B. 1932. The Science of the Sulba: A Study in Early Hindu Geometry. (Readership Lectures for the year 1931), xvi+240p. University of Calcutta Press.

I should have the first book in a day or two, and since the book has the actual texts (the translations), we should be able to figure out which statement in the Vedic mathematics section is true and which not. Fowler&fowler«Talk» 22:56, 28 February 2007 (UTC)

Awesome! Thanks to all of you! We have a decent library here at Penn State. If you need me to look into any books for stuff, drop in a line. Regards, deeptrivia (talk) 23:39, 28 February 2007 (UTC)
OK, Good to know. Thanks for the offer! Fowler&fowler«Talk» 23:00, 1 March 2007 (UTC)
PS Abecedare, I know this is compulsive, but I can't resist myself. You don't really need L'Hospital, the equation:

is just another definition of the derivative. But, this means that the geometric proof on the right can be used to show that: The usual proof in text books involves a tedious geometric proof which first shows: I wonder if this is simpler? Fowler&fowler«Talk» 23:15, 1 March 2007 (UTC)

Good point ! By the way, I think your "geometric proof" is the most likely to have been discovered first. It would be good to get some citable "hard evidence" though. Let me know if I can be of help in looking up some specific reference. Abecedare 23:40, 1 March 2007 (UTC)
PS: Your geometric proof can be easily modified to show that , so I don't know whether this limit or is a more fundamental/easier-to-prove-result ...Abecedare 23:40, 1 March 2007 (UTC)
Wow, differentiating trig functions! Coincidentally, what I'm was just studying awhile ago in maths at school. GizzaChat © 09:07, 2 March 2007 (UTC)

Well, it turns out that Manjula's more common name (in the literature) is Munjala. He was a 10th century astronomer who also made contributions to mathematics. However, I haven't been able to turn much up yet on his mathematics. Let me know if you have better luck. Thanks. Fowler&fowler«Talk» 13:32, 3 March 2007 (UTC)

Merge proposal: Indian logic

It doesn't seem to be a good idea to merge Indian logic inside this article. It's not the same scope. Logic in Islamic philosophy is also separate. --RF 02:34, 3 March 2007 (UTC)

I do not support the merger proposal. Can someone who favors it please give us insight as why that would be a good idea? If there is overlap in some concepts, perhaps those can be ironed out with some steps short of a merger. Buddhipriya 02:50, 4 March 2007 (UTC)
Oppose merge - Bakaman 08:25, 4 March 2007 (UTC)

Lots of Problems

Hello, I've read over this article and many of the bullet points are simply false. For instance, it says under Brahmagupta that

  • "the first use of algebra to solve astronomical problems" which is plainly false, algebra had been used to solve astronomical problems for thousands of years. Just study ancient Mesopotamia and Egypt.

Many of the bullet points are also ambiguous, for instance again under Brahmagupta:

  • "Zero is clearly explained for the first time." Which is ambiguous since the concept of zero has a very long and complex history and zero was seen differently by different civilizations. Also the zero of Brahmagupta is different from the modern zero. Although Brahmagupta mentions properties of zero Brahmagupta wrote that 0 / 0 = 0, which is false.

and there are many more mistakes and ambiguities like this in this article. It is as if many of the people who have added to this article aren't even mathematicians.

This article also smells of Hinducentricism; it is as if someone has exaggerated every single little Hindu accomplishment. So I am placing the TotallyDisputed tag on the front page. Furthermore, there are many unreferenced claims and extraordinary claims require extraordinary evidence so I am placing the Unreferenced tag at the top of this article as well. selfworm - just downgraded to version 0.4B! 02:41, 4 March 2007 (UTC)

An RfC, just a few sections above, may be related to some of problems you are mentioning. Please see it. Thanks GizzaChat © 10:47, 4 March 2007 (UTC)

Quadratic Equations

I have removed the setence:

Since the source is actually one of the "Student projects on the history of mathematics" http://www-groups.dcs.st-and.ac.uk/~history/Projects/index.html .And a student is not a reputable historian of mathematics. Secondly this sentence is plainly false since the ancient Babylonians were solving quadratic equations over 4000 years ago (Boyer, A History of Mathematics Second Edition, p.32) _selfworm_ ( Give me a piece of your mind · Userboxes · Contribs )_ 21:01, 30 March 2007 (UTC)

I would say, for now, don't remove the sentence, just put a "citation needed" tag on it. The problem with "solving quadratic equations" statements is that they all refer to special cases, since neither the Babylonian, nor the Sulba Sutra Indians, had the notion of negative numbers firmed up. That notion was firmed up in India c. 5th century CE, but not at the time of the Sulba Sutras (6th century BCE). As I have mentioned below or above (can't remember now), I ordered two translations of the Sulba Sutras (they've now been sitting on my shelves for a few weeks) and will go through them (as soon as I can find the time) to figure out what exactly was done. Fowler&fowler«Talk» 21:22, 30 March 2007 (UTC)
I have placed the sentence back into the article, but I have replaced the source with a citation needed tag and I have removed the word "first". _selfworm_ ( Give me a piece of your mind · Userboxes · Contribs )_ 21:47, 30 March 2007 (UTC)

Indeterminate equations

I have removed the source from the sentence

and added a citation needed tag since the source does not mention anything of the sort. _selfworm_ ( Give me a piece of your mind · Userboxes · Contribs )_ 21:26, 30 March 2007 (UTC)

Comments: improving the article

I just came across this article and thought it could use some cleanup. After reading the discussion above I think I understand roughly how the article has been improving recently. If I may be so bold I'll offer some suggestions.

Major issues

  • I like the time period sections, but at the very least they could use a short prose introduction. Many people reading the article will not be familiar with the historical setting and could use an introduction.
  • Why are Jaina mathematicians apart from their contemporary counterparts? Unless there's a good reason I think these should be integrated.
  • I'll echo an earlier comment and suggest transitioning away from a listing by mathematician and to a more unified framework. The individual articles should handle the "by mathematician" part.
  • References are sparse in the latter two-thirds of the article. Along the same lines, any images of period mathematical texts (with pictures if at all possible) would be a real improvement to the article.
  • The list comprising "Fields of Indian mathematics" should be removed or replaced with a prose explanation, in my opinion. As it stands it's not very informative and invites POV arguments.

Minor issues

  • I disambiguated Apollonius to Apollonius of Perga; was that the right choice? I have no context to work from.
  • The Einstein quote seems somewhat out of place; it's actually not very flattering to Indian mathematicians, to whom we owe many early developments in math (as well as several major ones [not least of them AKS] after Einstein's death), to reduce their contribution to their numbering system.

Any comments or further suggestions are welcome.

CRGreathouse (t | c) 20:32, 20 March 2007 (UTC)

The "Samhita" section

it's out of place. There is really nothing in the Samhitas that would answer to a description "mathematics". The altar construction is described in the Brahmanas. The decimal numerals are just Indo-European terms for "ten", "twenty" etc.; The "purnam" thing is in the Isha Upanishad, which is a late addition to the Shukla Yajurveda Samhita. The only thing might be the "arithmetic sequences" of the Atharvaveda, which are unsourced. dab (𒁳) 10:04, 25 March 2007 (UTC)

it's not OUT of place .Pls see "Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita. Also it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times." Indian MathematicsBharatveer 10:42, 25 March 2007 (UTC)

Bharatveer (talk · contribs) took it upon himself to revert to the fantastic version including utterly unsourced claims like "Equations, such as 972x2 = 972 + m for example", apparently for no other reason than that he would like it to be true (factuality is for nerds and sissies?). This flies in the face of every WP policy and amounts to simple vandalism. The Shulbasutras are the Shulbasutras (discussed in their own section), and the Ishopanishad is the Ishopanishad, and both do not qualify as "Samhitas". And please let's spare ourselves the archaeoastronomical nonsense, that's simply insulting the reader's intelligence. The TS, however, is perfectly on-topic, and you are free to cite it directly: but, by the nature of the Black Yajurveda, unless the material is in the VSM, the material is from the Brahmana, and should be dated after 1000 BC. Only the mantra parts can be argued to reach into the 11th century. It is also pathetic to try and portray the Ishopanishad as "Yajurveda Samhita" -- the irony can only be appreciated if you are at all familiar with the topic of Vedic philology of course. dab (𒁳) 10:47, 25 March 2007 (UTC)
As usual , this user is trying to deliberately confuse the things here.Bharatveer 11:02, 25 March 2007 (UTC)
as usual, you are not even pretending to back up your version with sources. I will roll this back as vandalism until you condescend to present references.. Where in the Samhitas do we have " 972x2 = 972 + m"? Where do we have Pythagoras? Where do we have evidence for written numerals? dab (𒁳) 12:00, 25 March 2007 (UTC)

Bharatveer, We've just been through a long RfC, whose overwhelming consensus was that the article has fantastic claims, which make the reader skeptical even of the real mathematics that came out of India. We are in the process of cleaning up the article. I now have the actual translations of the Sulbasutra, and also the long 13 page article from Encyclopaedia Britannica on "South Asian mathematics." Please wait until the article has been cleaned up, then offer your objections, rather than simply reverting to Freedom Skies' version, when he himself has withdrawn from editing the article and put the matter in the hands of the RfC discussants. That would be the right thing to do, especially since you were nowhere to be seen during the RfC itself. Fowler&fowler«Talk» 15:24, 25 March 2007 (UTC)

Bharatveer, please refrain from adding unreferenced and/or nonsensical material. If you want to add specific things like the Taittiriya Samhita, do it, but don't do blanket reverts. deeptrivia (talk) 16:45, 25 March 2007 (UTC)

David W. Henderson reference

I noticed that Citecop recently reverted an edit by User:Bharatveer with the edit summary "unpublished papers from personal webpages are not adequate sources". I was, in fact, just about to revert the edit too, but for a different reason: the article Bharatveer added has been published, but it simply does not attest the claims that it was added as a citation for ! I will add the correctly formatted reference as a citation for facts that the publication does affirm. Regards. Abecedare 08:18, 27 March 2007 (UTC)

References

I love how somehow the Indian references don't seem to matter; the only good references being Western ones. That smacks of racism. (Jvalant 16:52, 16 April 2007 (UTC))

Which Indian references do you have in mind (that you think are reliable)? Fowler&fowler«Talk» 20:33, 16 April 2007 (UTC)

"The bigger problem that David alludes to above is that the claims themselves are often either false or unverifiable, and, therefore, the references cannot be reliable. Why do you think the authors you cite never have publications in internationally recognized journals or publications? Their publishers are all obscure Indian ones." This is the statement which I find racist. How does it matter who published it? —Preceding unsigned comment added by Jvalant (talk • contribs)

The reason I don't find the statement "racist" is because the emphasis is on "obscure", not Indian. Obscure sources of any nationality are to be deprecated, in this case they just happen to be Indian for understandable reasons.
And of course, the publisher of a work is highly relevant in judging a source's credibility, especially for non-experts such as wikipedia's editors and readers. See wikipedia guidelines on verifiabilty and reliable sources. And, yes, there are many reputable Indian publishers; for example besides the ones listed by F&F, one can also consider Orient Longman, Rupa and Co, Ananda etc. Not all these publish on history of mathematics, but if you would like to add "Indian" references to the article, I would highly recommend getting hold of this title published by Hindustan Books Agency.
  • "Contributions to the History of Indian Mathematics", Ed. G. G. Emch, R. Sridharan, M. D. Srinivas. Hind. Book. 2005. ISBN 81-85931-58-5
The aim should be to raise the standard of the article, not to lower the standard of sourcing. As has been pointed out before, a problem with adding poorly-sourced/unsourced dubious claims is that the whole article looses credibility and even genuine contributions of Indian mathematics (which are substantial) are overlooked. I look forward to your collaboration in improving the article. Abecedare 21:17, 18 April 2007 (UTC)

Priority claims spoil this article

The genuine case for Indian mathematics is definitely hurt by subtle priority claims inserted quietly into the article and its related articles. For example, the claim that Phythagoras learned his theorem from India has no value addition to the article other than make folks from India feel that they had some priority over Western mathematics. It would be better to state what is known about the Phythagoras theorem in Indian mathematics texts, and a possible dating of such a text, and leave it at that. Further, there should be more factual or substantive explanation of many statements. For example the note about the exact value of the time measure of an year by Aryabhatta seems incredible without more substantive explanation of how Indians measured time - how can the exact length of an year be stated in hours, minutes and seconds if the definition of hours/minutes/seconds itself might have come about much later? It would have been better if there was some substantiation of how Indians measured time in those days. I was left wondering for days how the measure of an year was calculated in modern time units before those units were invented in the first place..

The article on Indian Mathematics should not become a ground for making fantastic claims about Indian mathematics. This is an encyclopedia, not a cultural battleground. Thanks to those who are cleaning up the article for neutrality and accuracy.

Are there not enough folks from India who can scan the actual "sutras" into an image, translate these and let the readers interpret the significance for themselves? Not doing so will leave this article only as a series of claims. To cite an example to learn from, look at any of the physics articles, for example "Einstein's general theory of relativity" shows nice derivations of concepts and formulas involved - it does not just state the claims of General relativity. Is there no one from India who can create material based on genuine Indian mathematical texts, provide a translation so everyone can see the true worth of Indian math? If such cannot be done, then it leaves the reader only a list of claims to go through, which in itself is not worth any real knowledge..

135.245.8.33 18:35, 7 May 2007 (UTC)

—The preceding unsigned comment was added by We all humans (talk • contribs) 15:26, 7 May 2007 (UTC).

Revising the article

To all those who participated in the RfC above, sorry, I couldn't get around to working on the article on account of other involvements. I now have some time on my hands and also access to some literature (including two books on the Sulba Sutras that I had ordered at the time of the RfC). Please feel free to give feedback and participate in the revision. I am therefore removing the disputed and other tags (since we are all aware of them) and replacing them with the "under construction" tag. Thanks. Fowler&fowler«Talk» 00:56, 17 May 2007 (UTC)

Good luck in your renovation. I hope you convert those bullet points which dominate much of the article into well-written prose. GizzaChat © 10:26, 20 May 2007 (UTC)
You've set yourself a big task, but it's looking much improved so far. Thanks for all the effort and persistence! —David Eppstein 15:40, 20 May 2007 (UTC)
Thanks. Yes, it does seem daunting at times. Thanks (both) for your help. I'm creating a separate section below which has "roadmap" and "work completed" subsections. Hopefully, they will give readers an idea of the state of the revision. Fowler&fowler«Talk» 17:14, 20 May 2007 (UTC)

State of the revision

As I see it, the revision will proceed in three parts:

  1. Crude revision: Changing long lists to prose (in summary style), removing unsupported claims, and improving the citations.
  2. Mathematical revision: clarifying the mathematical points etc.
  3. Integration of math and prose; making prose more compelling.

Work Completed (up to end of section): Crude revision (Jaina mathematics), (Bakhshali Manuscript) Fowler&fowler«Talk» 21:38, 25 May 2007 (UTC)

Fowler&fowler«Talk» 17:14, 20 May 2007 (UTC)

Great job toll date, F&F! I'll try to join in the effort when you reach steps 2 and 3, since hopefully (parts) of that can be done without looking at the required references. If, in the meantime, you want to look up any specific book/article, let me know and I'll visit the library. Abecedare 05:20, 23 May 2007 (UTC)

Greek Ideas in Indian Trigonometry

Richard L. Thompson doesn't have any publications in peer-reviewed journals like Historia Mathematica, Archives of the History of Exact Sciences, etc. Getting a book published by any publishing house doesn't constitute recognition by the community. Why don't we simply stick to internationally-recognized peer-reviewed journals? Fowler&fowler«Talk» 19:35, 25 May 2007 (UTC)

Here are some of Thompson's notable publications Fowler&fowler«Talk» 19:50, 25 May 2007 (UTC)
For Greek ideas in Indian trigonometry please see:
  • Pingree, David. 2003. "The logic of non-Western science: mathematical discoveries in medieval India." Daedalus (American Academy of Arts and Sciences), 132(4):45-54. Quote: "In order to make my argument clearer, I will restrict my remarks to the first branch of Jyotihśātra—Ganita. Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century."
  • Hayashi, Takao. 2003. "Indian Mathematics" in Grattan-Guiness, Ivor (ed) Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. The Johns Hopkins University Press, 976 pages. ISBN 0801873967. "Moreover, no one can overlook Greek elements in Indian astronomy and horoscope astrology, such as epicycles, and a number of Sanskritizations of Greek astrological terms (Pingree 1978a, 1981). It might be natural, therefore, to hypothesize (with van der Waerden 1976) a Greek influence on the early stages of Indian mathematics proper, but we have yet to find satisfactory testimonies to that effect except for trigonometry, which was mostly confined (in the work of Indian mathematicans) to astronomical contexts."
  • From Encyclopaedia Britannica 2007 (article: mathematics, South Asian): "Greek mathematical models in astronomy and astrology appeared in India following the invasion of Alexander the Great. These models were integrated with existing Indian material to produce an extremely fruitful system of Sanskrit mathematical astronomy and astrology, known as jyotisa. The intellectual place of ganita, according to the canons of Sanskrit literature, was located within jyotisa, which in turn was identified as one of the six Vedangas (“limbs of the Veda”), whose purpose was to support the proper performance of Vedic rituals." (Please see the first citation above to see the connection with trigonometry.)

The point you have made regarding Thompson is not relevant to Wikipedia. Regardless of what other non-notable books he writes, the fact that he is a university researcher in the field of mathematics and his actual book being used here (Vedic Cosmography and Astronomy) was published by a leading publishing house (Motilal Banarsidass) is enough to merit inclusion on Wikipedia according to the Attribution policy:

"In general, the most reliable sources are books and journals published by universities; mainstream newspapers; and university level textbooks, magazines and journals that are published by known publishing houses."

There are also other scholars (such as G. Joseph and R. Billard) who also disagree with the possible Greek influences. As you may be aware from our previous dispute on Taxila, my position is to maintain a neutral point of view (i.e. presenting both sides of the argument), a policy you seem to be unwilling to follow. Jagged 85 12:15, 26 May 2007 (UTC)

Also keep in mind that I am more than willing to come to a comprimise with you, but only if you are actually willing to assume good faith and co-operate by building upon previous edits instead of just reverting everything you disagree with. Jagged 85 12:53, 26 May 2007 (UTC)

I see you've just reverted my edits without discussing it here first and even went as far as to accuse me of POV-pushing despite the fact that I was the one presenting both sides of the argument and you were the one choosing to present only your own opinion. What you just did right now is almost equivalent to vandalism. Jagged 85 13:00, 26 May 2007 (UTC)

"Mathematical Analysis" in Kerala

I have removed the text added by Jagged85 about the "development of mathematical analysis" by the Kerala School. Please see the article by Kim Plofker (expert on Kerala mathematics and author of the Britannica article on South Asian mathematics):

  • Plofker, Kim. 2001. "The 'Error' in the Indian 'Taylor Series Approximation' to the Sine". Historia Mathematica. 28(4):283-295. From the abstract: "It has been repeatedly noted, but not discussed in detail, that certain so-called “third-order Taylor series approximations” found in the school of the medieval Keralese mathematician Mādhava are inaccurate. That is, these formulas, unlike the other series expansions brilliantly developed by Mādhava and his followers, do not correspond exactly to the terms of the power series subsequently discovered in Europe, by whose name they are generally known.We discuss a Sanskrit commentary on these rules that suggests a possible derivation explaining this discrepancy, and in the process re-emphasize that the Keralese work on such series was rooted in geometric approximation rather than in analysis per se."

And, here some quotes from the conclusion of Plofker's paper:

"We have seen that the formulas generally and conveniently labeled “Taylor series approximations”

in the mathematics of the Kerala school appear to spring not from an investigation of calculus algorithms such as those studied by Taylor and Maclaurin, but rather from creative

manipulation of the geometry peculiar to sines and cosines."

"This reconstruction bears on the larger issue of what is sometimes called “Indian infinitesimal analysis.” It is not unusual to encounter in discussions of Indian mathematics

such assertions as that “the concept of differentiation was understood [in India] from the time of Manjula [or Mu˜nj¯ala, in the 10th century]” [Joseph 1991, 300], or that “we may consider Madhava to have been the founder of mathematical analysis” [Joseph 1991, 293], or that Bh¯askara II may claim to be “the precursor of Newton and Leibniz in the discovery of

the principle of the differential calculus” [Bag 1979, 294]... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in [Bag 1979, 285])... To speak of the Indian “discovery of the principle of the differential calculus” somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential “principle” was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here."

PS Here is a quote from Pingree, David. 1992. "Hellenophilia versus the History of Science." Isis. 83(4):554-563. "One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Madhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution."
A quote from Bressoud, David. 2002. "Was Calculus Invented in India?", The College Mathematics Journal (Math. Assoc. Amer.) 33(1):2-13. "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the 19th century. Gold and Pingree assert (4) that by the time these series were rediscovered in Europe, they had, for all practical purposes been lost to India. The expansions of the sine, cosine, and the arc tangent had been passed down through several centuries of disciples, but they remained sterile observations for which no one could find much use." Fowler&fowler«Talk» 21:50, 26 May 2007 (UTC)

Scholars such as G. G. Joseph, Rajgopal, J. O'Connor, and E. Robertson, regard Madhava to be the founder of mathematical analysis. If there are some scholars who disagree, then like I mentioned previously, both sides of the argument should be given to maintain a neutral point of view. However, your second source does not mention anything about analysis, so I don't see how it supports your view. Also, Katz never said anything about there being no "evidence of their results being transmitted outside Kerala". I am adding the actual quote from Katz and changing that part to a more neutral version. Jagged 85 12:53, 26 May 2007 (UTC)

What you are doing is POV-pushing. Pingree says that Madhava derived the series without the calculus. How, then, could he have invented Mathematical Analysis? Fowler&fowler«Talk» 13:03, 26 May 2007 (UTC)
  • John O'Connor and E. J. Robertson (See his papers) are professional mathematicians (with an obvious interest in the History of Mathematics) who, in their words, are the "maintainers of the MacTutor History of Mathematics Archive on the World Wide Web." They and their students have created a vast archive of the history of mathematics for undergraduate teaching that covers the gamut of mathematical history. "Indian mathematics: redressing the balance," for example, is an undergraduate student project. The archive is not the same thing as publications in peer-reviewed journals in the history of mathematics.Fowler&fowler«Talk» 14:07, 26 May 2007 (UTC)
  • G. G. Joseph is not a professional historian of mathematics. He is the author of a popular book "Crest of the Peacock," on the "non-European roots of western mathematics," but has no publications in the notable peer-reviewed journals in the history of mathematics. He has a number of (honorary) teaching appointments and has lately teamed up with D. Almeida (a lecturer in math education, with an interest in history of math) and written two speculative (and ideological) papers. I just finished reading one of them published in the journal Race and Class. I won't give my opinion of the paper, but here is a quote from it, "Given the existence of a corridor of communication between Kerala and Europe, especially from the sixteenth century onwards, and the crucial importance of calculus in the growth of modern mathematics, one would have expected that the possibility of the transmission of the Kerala mathematics westwards would be high on the agenda for historical investigation. That such an investigation has not yet been carried out may reflect, in our view, the strength and pervasive nature of the Eurocentrism in the history of science." This is hardly evidence for anything. Kim Plofker addresses these issues in the quotes above.
  • It is interesting too that in the paper above, they quote a paper of Rajagopal from 1944, when the knowledge about the Kerala school was still rudimentary. I just read the last paper of Rajagopal, publised in 1986 (a few years after his death). Rajagopal, C. T. and M. S. Rangachari. 1986. "On Medieval Kerala Mathematics. Arch. for History of Exact Sciences. 35(2):91-99. The paper makes no mention of calculus (only trigonometric series). Fowler&fowler«Talk» 22:31, 26 May 2007 (UTC)
I doubt you even have any idea what POV-pushing means. The fact that you are choosing to only present Plofker and Pingree's views and are ignoring the views of Joseph, Rajgopal, Robertson, O'Connor, etc. means you are the one that is POV-pushing. Now are you willing to co-operate or not? Or are you really persistent enough to continue this pointless edit war? Jagged 85 13:06, 26 May 2007 (UTC)
Also, weren't you the one who asked me to come here and contribute to the article? I thought you were doing a great job with the article, but some of your revertions are questionable. Like I said above, the article will make better progress if you are willing to assume good faith and actually co-operate with users you are in disagreement with. Jagged 85 13:14, 26 May 2007 (UTC)
Yes, I did ask you (as I did a dozen other people) and I value your and other people's contributions, but as I have mentioned above, I am still in the crude revision stage. Once that is done we can examine the literature. This is not the time for nitpicking about little details; however, since this is Wikipedia and free for any one to edit, I obviously can't legislate that ... Fowler&fowler«Talk» 22:31, 26 May 2007 (UTC)

Like I said above, I agree with most of the work you've done so far in the article, and it is certainly in a far better state right now than it was before your revision. My only contention is how the lead section completely agrees with the possibility of Greek influence in India but completely dismisses the possibility of Kerala influence in Europe, while suppressing the fact that there are other scholars out there who disagree with these views. Information suppression is simply not acceptable on Wikipedia.

As far as I am aware, there is no Wikipedia policy that says only peer-reviewed journals should be accepted. The fact that MacTutor has received an award from Britannica means that it is a reliable source. Joseph's book has been peer-reviewed by plenty of other scholars, which he states in a response to Pingree's review of his work:

George Gheverghese Joseph, David Pingree, S. I. Salem, F. Jamil Ragep (1994), "Letters to the Editor", Isis 85 (4), p. 668-670:

"Having had the book reviewed in over thirty publications of diverse interest, I am quite content receiving a few brickbats with a mountain of bouquets."

Are you implying that Rajgopal may have changed his views in his last work? It may be possible, but unless he explicitly stated that he believes there was no calculus or analysis involved, then it does not negate his earlier claims.

Jagged 85 00:10, 27 May 2007 (UTC)

I don't want to get into too much of an edit war, so I'll refrain from being the one to revert all your edits this time. But this article's subject is big enough, with enough political baggage, that you can find published sources for the craziest assertions. That is why it is especially important here to be strict about including only scholarly consensus, or, when consensus does not exist among scholars, to report on the controversy. By your claims here that web sites that have been given some awards, or popular press speculations, should receive equal status, you show yourself to me to be less interested in accurately reporting what scholars have determined about the past, and more interested in finding alternative sources who will tell you what you want to hear. That is not the way to build a good encyclopedia article. And moreover, giving way to speculation and questionable sources is a big part of the past problems with this article, as we went over in the recent RfC. It's not the only problem (the writing style full of lists instead of text is another) but I see your edits as unconstructive. —David Eppstein 00:49, 27 May 2007 (UTC)
As I've already stated in my previous posts, my main interests are to present a neutral point of view and counter any suppression of information, both of which are core Wikipedia policies (which both you and Fowler are not following). When did I ever say anything about giving MacTutor equal status to a peer-reviewed journal? I said it is a reliable source because of the fact that it was recognized as such by Encyclopedia Britannica. Even if you disagree with Britannica's view on MacTutor, I have already given other peer-reviewed publications also supporting O'Connor and Robertson's view on the possiblity of mathematical analysis. Also, if you believe there is some kind of consensus on the Kerala issues discussed here, then your claim of consensus will need to be sourced, and as far as I know, I have not seen any such scholarly consensus. You also forgot one more significant past problem with the article: a lack of neutrality (pro-Indian bias at the time). Jagged 85 01:41, 27 May 2007 (UTC)
Also, it seems to me that I am the only one here actually following Wikipedia policies. Both you and Fowler have only been giving your own personal opinions and throwing needless personal attacks against me, but neither of you have actually cited any Wikipedia policies to justify your POV-driven revertions. Jagged 85 01:46, 27 May 2007 (UTC)
I am not aware that Wikipedia says all sources need be treated on a equal footing. There is a difference between journals like Historia Mathematica, American Mathematical Monthly, Archives for the History of Exact Sciences and authors like B. L. van der Waerden and David Pingree, and an undergraduate student project from the MacArthur Tutor Archive. Fowler&fowler«Talk» 02:21, 27 May 2007 (UTC)

In case you missed it, I already said: "When did I ever say anything about giving MacTutor equal status to a peer-reviewed journal?" I was pointing out the simple fact that Britannica considers MacTutor to be reliable, therefore it is a reliable source. Ian Pearce's student project is not even used in the article, so it is irrelevant to the discussion. All of the other MacTutor articles are written by two mathematics professors: O'Connor and Robertson.

Anyway, I won't be un-reverting the article for now, mostly because I'll be busy for the next few days, and also because I wouldn't want this article to be locked because of the three-revert rule. I'll be back when I have more free time to counter the obvious systemic bias present here. For now, feel free to continue with your revision until then.

Jagged 85 11:25, 27 May 2007 (UTC)

Review of Joseph's Book

Here are some quotes from David Pingree's review of G. G. Joseph's book "Crest of the Peacock" . (The citation is: Pingree, David. 1993. "Reviewed Work(s): The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph". Isis (The History of Science Society), 84(3):548-549.) Since Jagged 85 has been citing and quoting from Joseph's book, not just on this page, but also on the Calculus, History of Calculus, and History of Mathematics pages, I feel editors should be aware of the quality of the source.

"Such scholars-and the general public-

should, however, be made aware of the fact that Joseph has entered into a field in which he has no particular expertise; the result is that, whenever he discovers a trustworthy secondary source (he uses no primary documentation, as he does not control the languages necessary to read them), his summaries are clear and cogent; but when, as all too often happens, he has come upon something less than trustworthy (e.g., the paper by ...), he can be very misleading. In my view, the misleading and just plain wrong statements in the book seriously affect the persuasiveness

of the author's arguments."

"This unfortunate tendency toward occasional

error is exacerbated by Joseph's biased stance against Eurocentrist bias. He is not a dispassionate scholar, weighing the historical evidence in an unemotional search for the "truth," but a polemicist fighting an imagined enemy, and all too willing to accept as true those parts of historical mythology, whether created in the past (e.g., the tales about the Presocratics on pp. 5-6 or about the Ethiopic origins of the Egyptians on p. 57) or in the present (e.g., ...), that coincide with his desire to aggrandize

the significance and acumen of non-Europeans."

One other way in which Joseph attempts to

arrive at the same result is by emphasizing the transmission of mathematical knowledge between cultures. Had he greater familiarity with the history of mathematical astronomy, where the pattern of transmission and transformation has long been established, he would have had an easier task. But his few remarks on mathematical astronomy demonstrate his lack of mastery in this field; and he resorts to rather extravagant speculations about the influence of Chinese and Indian mathematics on the West while attempting to minimize or deny the very real influence of Babylonian and Greek mathematical astronomy on India.

The weakest of the chapters are those about

the two non-European mathematical cultures that are best known, the Indian and the "Arab." Both chapters abound in inaccuracies of dating, of names, and of historical facts. It would be unproductive to speculate on the reasons for the discrepancy between Joseph's treatment of Mayan, Egyptian, Babylonian, and Chinese mathematics, on the one hand, and that which he accords to these two; I simply warn the prospective reader that they

should be approached with particular caution.

Fowler&fowler«Talk» 03:41, 27 May 2007 (UTC)

PS And here is a portion of Joseph's rambling reply (in the "Letter to the Editor" column) the following year.

"Let me state my position clearly. I am not

neutral when it comes to confronting Eurocentrism, which, in its extreme form, is nothing less than racism by omission. I am not in the business of writing for a small coterie or self-selected clique of academics who describe themselves variously as "historians of science, " "orientalists," "indologists, " etc. The audience that I seek is a broader public interested in the history of ideas, who wish to know the "truth" arrived at through serious scholarship, a global vision, and unbiased judgment, free of the imperialist assumptions of the past (and sometimes the present), when the contributions of the colonized peoples were ignored or devalued as part of the rationale

for subjugation and dominance."

Fowler&fowler«Talk» 04:11, 27 May 2007 (UTC)

I don't have much of substance to add here, but I think the quote who describe themselves variously as "historians of science, " "orientalists," "indologists, " etc. is amusing, though I'm not sure whether it tells more of Joseph's knowledge of academics or of changing fashions in academic nomenclature. "Orientalist" is very much not something the Asian studies scholars of my acquaintance would call themselves: it seems to be a dirty word among them, describing a Eurocentrist point of view that they see as a trap to be avoided. But I think they would view what Pingree accuses Joseph of, "desire to aggrandize the significance and acumen of non-Europeans", as part of the same trap. —David Eppstein 06:24, 27 May 2007 (UTC)
True. Well, Joseph's use of "orientalists" and "indologists" was already anachronistic at the time he used them in the letter. Edward Said's book, "Orientalism" (which got the ball rolling) had been published in the early 80s and most Asian studies departments (in the US at least) had changed their names from "Indian Studies" to ""South Asian Studies, or "Far Eastern Studies" to "East Asian Studies." By the 1990s, "Postcolonial" or "Subaltern" approaches were very much the vogue in these areas; however, their analyses (as I understand it) are more sophisticated than Joseph's—which is more like an old-fashioned and simplistic pseudo-Marxist rant—and their subject matter more recent (i.e. the 19th century). An example of a more sophisticated approach (and a paper I'm very much looking forward to reading) would be: Raina, Druv and S. Irfan Habib. 1990. "Ramchandra's Treatise through 'The Haze of the Golden Sunset': An Aborted Pedagogy" Social Studies of Science. 20(3):455-472. Here are the first few sentences of the abstract: "In 1850, Yesudas Ramchandra published a work entitled A Treatise on the Problems of Maxima and Minima. The inspiration behind the work lay in the nineteenth-century understanding that the Indian tradition of mathematics was essentially algebraic. As part of the task of 'revitalization' undertaken by the avant-garde of the Indian intelligentsia, Ramchandra sought to introduce the Indian people to the latest developments in calculus, in their 'native' idiom. In the paper, we discuss the conditions under which the work emerged, as well as the cultural grounding of this mathematical pedagogy." Fowler&fowler«Talk» 13:20, 27 May 2007 (UTC)

Re: Fowler&Fowler: You have only chosen to present this one negative review of Joseph and have clearly ignored all the other positive reviews on his book. I'll repeat what Joseph has stated regarding other reviews on his book:

George Gheverghese Joseph, David Pingree, S. I. Salem, F. Jamil Ragep (1994), "Letters to the Editor", Isis 85 (4), p. 668-670:

"Having had the book reviewed in over thirty publications of diverse interest, I am quite content receiving a few brickbats with a mountain of bouquets."

In other words, from over 30 reviews on his book, the vast majority of them were positive and only a small minority were negative. Pingree's review on his book was by far the most negative of them all, hence why Joseph felt the need to respond to him. It is manipulative to use only one negative review (from over 30 other reviews, which were mostly positive) to discredit Joseph's credibility. In fact, Joseph has also discredited Pingree in his above letter. One of these discreditations include:

"In fact, Pingree remains the Hellenocentrist par excellence".

The fact that Pingree has also been discredited by another scholar should mean that he is also an unreliable source according to your logic. If you're going to remove Joseph on the grounds that a minority have discredited him, then you might as well remove David Pingree on those same grounds. However, as you may have seen from the undue weight guideline Cronholm has given below, minority views should not be given the same weight as majority views. The fact that Joseph's book has been reviewed positively by the majority of peer-reviews means that he is a reliable source.

Jagged 85 11:25, 27 May 2007 (UTC)

You are taking a much too simplistic view on this issue. The problem with The Crest of the Peacock lies in the fact that it combines a tertiary source, summarizing the history of pre-European mathematics, with Joseph's original research. The three reviews published in academic journals (Pingree in Isis; Katz in CJM; Swetz in AMM) agree that the first part is good, although better and more specific (secondary) sources are available, but dismiss Joseph's conclusions because they lack evidence and note that they contradict accepted knowledge in many places. Therefore this work should only be used as a source only with great care: the sources he has drawn on for his overview are better cited directly, while his own research should probably not be cited at all. —Ruud 12:51, 27 May 2007 (UTC)
Well spoken. Fowler&fowler«Talk» 13:29, 27 May 2007 (UTC)
My simplistic view is based entirely on Wikipedia's policies, and not on any personal opinions. As Joseph has stated in his letter, a majority of reviews regarding his book were positive, hence the majority opinion should always be given more weight than the minority opinion according to the undue weight guideline. Jagged 85 13:32, 27 May 2007 (UTC)
No, determining if something is a reliable source is not as easy as determining if the "positive" reviews outnumber the "negative" ones. Not all reviews carry equal weight, nor does a positive review make the entire work reliable, or does a negative one make it useless. —Ruud 20:57, 27 May 2007 (UTC)
Let me add my view, having read Pingree's review of Joseph in Isis, Joseph's rebuttal, as well as Plofker's essay on "Taylor Series" in Historia Mathematica, which in its concluding passages, criticizes Joseph's presentation for its oversimplification of a complex historical reality:
It is not unusual to encounter in discussions of Indian mathematics such assertions as that “the concept of differentiation was understood [in India] from the time of Manjula [... in the 10th century]” [Joseph 1991, 300], or that “we may consider Madhava to have been the founder of mathematical analysis” [Joseph 1991, 293], or that Bhaskara II may claim to be “the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus” [Bag 1979, 294]. Such comparisons are an attempt to do justice to the breadth of the conceptual overlap between the Indian and European approaches to “calculating results produced by non-uniform continuous changes” [Bag 1979, 286], which in both traditions involved brilliant intuition and great acuity in approximation by means of small quantities. The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in [Bag 1979, 285]).
It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian “discovery of the principle of the differential calculus” somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context.
I think Pingree and Plofker have more than adequately undermined Joseph's well meaning, but false, presentation of non-Western mathematics. --SteveMcCluskey 20:32, 29 May 2007 (UTC)

NPOV clarification

Let me preface this with the fact that I would really rather not comment here, as I am no expert in this field, but I would like to point out a couple things about wiki policy.

Undue weight

Shortcuts

NPOV says that the article should fairly represent all significant viewpoints that have been published by a verifiable source, and should do so in proportion to the prominence of each. Now an important qualification: Articles that compare views should not give minority views as much or as detailed a description as more popular views, and may not include tiny-minority views at all. For example, the article on the Earth only very briefly refers to the Flat Earth theory, a view of a threedistinct minority.

Minority views can receive attention on pages specifically devoted to them — Wikipedia is not a paper encyclopedia. But on such pages, though a view may be spelled out in great detail, it must make appropriate reference to the majority viewpoint, and must not reflect an attempt to rewrite majority-view content strictly from the perspective of the minority view.

These two quotes come directly from the NPOV policy article here at WP, and I think that they speak directly to the issues being discussed here. Also, once these issues have been worked out, could someone here let me know so I can make the corresponding changes at the calculus articles. thanks so much--Cronholm144 05:23, 27 May 2007 (UTC)

The undue weight policy only applies if there is a consensus on an issue. As I have addressed above, such claims of consensus will need to be sourced. As there is no scholarly consensus on the Kerala issues discussed here (mathematical analysis and possible transmission), both sides of the arguments should be given equal weight. Jagged 85 11:25, 27 May 2007 (UTC)
This is quite simply not the case. If there are two disputing views on an issue, each with significant support, there is no consensus; nevertheless we should give each side due weight, proportionate to its support. Septentrionalis PMAnderson 18:50, 27 May 2007 (UTC)

Equal weight? The policy clearly says in proportion to the prominence of each source. Even if consensus has not been established, the point has been made that the prominence of peer reviewed articles definitely outweighs the two unreviewed sources referenced above. I would argue that multiple peer reviewed journal articles>>the other less rigorous sources quoted here. Therefore the article should similarly reflect that imbalance.--Cronholm144 11:48, 27 May 2007 (UTC)

In case you haven't been following the discussion, the published books and journal articles written by Joseph, Rajagopal, Rangachari, Almeida, Raju, Katz, etc. in support of Kerala analysis and/or transmission are all peer-reviewed, so your point is irrelevant. The fact remains that there is no majority scholarly opinion regarding the Kerala analysis/transmission issues, therefore both sides of the argument should be given equal weight. Jagged 85 12:27, 27 May 2007 (UTC)

I was referring to Joseph, who seemed to be the most recent point of contention. Sorry if I wasn't clear. Again I said I am no expert on this topic, I was trying to provide some neutral ground (WP policy) for dispute resolution. There is no need to be so dismissive. --Cronholm144 12:37, 27 May 2007 (UTC)

BTW Rajagopal has been addressed earlier as only mentioning trig functions not analysis in his most recent work. and Joseph has not appeared in a journal as far as I can tell. Rangachari, Almeida, and Raju seem to be new additions to the conversation, but again I am not qualified to argue with you.--Cronholm144 12:53, 27 May 2007 (UTC)

I was not dismissing all your points, but I apologize if it did sound like that. I do value your contributions to this discussion, but didn't agree with that one particular point you made.
As far as I am aware, if Rajgopal did not explicitly state that the Kerala school did not know analysis in his last work, then it does not negate what he stated in his earlier work. Joseph has appeared in several journals (I added one such journal article by Joseph and Almedia before the unwarranted revertions). His books have also been peer-reviewed (Crest of the Peacock has received mostly positive reviews from over 30 other publications) and have been published by reliable publishing houses such as Penguin Books and Oxford University Press. Crest of the Peacock certainly meets the criteria of a reliable source according to Wikipedia's policies. Jagged 85 13:18, 27 May 2007 (UTC)
In fact, you are probably the only user I've had a discussion with on this page who has actually cited a Wikipedia policy. Jagged 85 13:41, 27 May 2007 (UTC)
No, it doesn't meet the criteria. The book is not a scholarly work, but rather a popular book (now out of print) written by a newcomer to the field (who didn't have—nor has since accumulated—a track-record of scholarly work in the area). It is on the same level as "Nothing that is: a natural history of zero" by Robert and Ellen Kaplan and a host of other such popular books in mathematics. Penguin, BTW, usually doesn't publish scholarly work. Fowler&fowler«Talk» 13:47, 27 May 2007 (UTC)
Crest of the Peacock was also published by Princeton University Press, which only publishes scholarly work. The book is also not out of print where I live. Along with the fact that the majority opinion on the book is generally positive, it certainly meets Wikipedia's criteria of a reliable source. And this source is only one of many supporting the other point of view, therefore there is no justification for the suppression of such vital information on the possible Kerala analysis and/or transmission in the current lead section (no pun intended). Jagged 85 14:03, 27 May 2007 (UTC)
Princeton University Press also publishes, for example, the collected works of Carl Jung. Jung is eminently scholarly, in his field; but his positions should be described as his, either by "In Jungian psychology..." or "As Jung holds...". It seems quite likely that the same is true here. Septentrionalis PMAnderson 18:50, 27 May 2007 (UTC)
This discussion is quite time-consuming. Since I'll be busy over the next few days, I think I'll just give the discussion a rest for now, until I have more free time later on. Until then, feel free to have the last word for now. Jagged 85 14:52, 27 May 2007 (UTC)
Re: Pmanderson: I agree with your point. I was trying to present multiple points of view on certain issues in the article (which was eventually reverted due to systemic bias), but I prefer your suggestion of actually naming certain scholars when presenting their points of view in the article. Jagged 85 01:57, 1 June 2007 (UTC)

Bottom Line and Thanks

Dear Jagged 85, Thanks for your post above. I just looked at the extended history of this page and discovered that you single-handedly changed this page from a stub to full-length article. So, in a sense, we are all building on what you have done, which obviously must have taken an enormous effort. Sorry, I wasn't aware of your past input into the article (and the amazing extent of it). Rest assured that I am not deliberately trying to inject any kind of bias into the article. I will be mindful of your words above as I further revise the article. If you don't mind, I will ask you for clarification and feedback from time to time. Regards, Fowler&fowler«Talk» 16:05, 27 May 2007 (UTC)

Thanks. I really do regret neglecting the article for such a long time (I guess you could say I lost interest in Indian mathematics for quite some time), so I do appreciate your extensive efforts in improving the article. I'll look forward to see how your improved revision turns out (and then continue our discussion from where we left off after that). Take care. Jagged 85 22:24, 27 May 2007 (UTC)

State of the revision

Hi, The revision is taking a little longer because I came to realize that the ancient Indian style of presenting and transmitting mathematics is as important as the description of their results dressed in modern notation. This has resulted in two sections on oral and written transmission. The article will bulk up before it is pruned again, so please don't be too alarmed at the number of KBs. I have a little more integration to do before I attend properly to the classical mathematics section (which in many ways is easier since the math is more modern). Fowler&fowler«Talk» 14:58, 9 June 2007 (UTC)

Manjula

Hello, I decided to check on the accuracy of the citation for the line:

He understood the concept of differentiation after solving the differential equation that resulted from substituting this expression into Aryabhata's differential equation.

to find that the only thing written about Manjula at the indicated reference is the following:

Manjula (c. AD 930) was the first Indian astronomer to recognize that equation (9.8) could also be expressed as

u` - u = v' - v +- e(w'-w)cos w
since
(sin w' - sin w) = (w' - w)cos w
In modern notaiton, we could write equation (9.9) as
du = dv +- ecos w dw

Bhaskaracharya extended this result ...

It seems to me that the author does not say that Manjula solved differential equations or that he understood the concept of differentiation. Is it your opinion that the wiki-text correctly reflects the author's text? Thank you for your input. selfwormTalk) 03:29, 3 July 2007 (UTC)

Comments by:Vinay Jha (talk · contribs):

I read the whole argument between Fowler &c , and Freedom &c, and appreciated the proposal and the sober tone of Fowler. I think Wikipedians should strictly follow the principle that before changing the version of another editor (excepting those of explicit vandals), an editor should use talk page for puting forth his objections, and if the objections are not answered reasonably within a stipulated limit of time, then the editor can proceed and make whatever changes he deems fit. Otherwise, changing the version of another editor is an insult to that person which many Wikipedians should not resort to indiscriminately. Fowler is certainly a mature editor, but some other users have used objectionable epithets for others which should be avoided. In the present case, Freedom should have informed Fowler in talk pages about his objections and opinions, before changing Fowler's version. Or, alternatively, Freedom could have added his version in a separate para or separate section, without tampering with Fowler's version. If two versions differ, future discussions would automatically lead to some solution. Editors should respect each other. Reverting is an extreme step, which one should undertake under two situations only :(1)one's objections in talk page are not answered at all within ,say, three days; this should be the normal course,(2)if one feels that another editor's version is too bad or wrong for even a discussion on the talk page. I find that many editors are resorting to this second choice, and regard each other as vandals, which is not good for Wikipedia. This article addresses a problematic area and therefore should be more open to new suggestions, within the WIKI guidelines. [[User:Vinay Jha|VinJha ] 13:17, 18 July 2007 (UTC)

Interesting block log

See Of course, Aksi was merely doing his routine job. Dick's condescension looks obscene now. This log should suggest the value of common sense over rules. The moment one equates a scholarly editor with a revert warring troll, all the mass of policies is bunkum. IRA, now mostly a weapon in asinine admins' hands and going against the grain of common sense, should be made to take a U-turn.

Varna

To what varna did Indian Hindu mathematcians generelly belong? Kkrystian 16:18, 24 September 2007 (UTC)

Pāṇini and context-free grammars

I don't think that the grammar Pāṇini used in his Aṣṭādhyāyī is a context-free grammar, the grammar he used is a context-sensitive grammar. It seems a bit strange to claim that context-free grammar can be used for a definition of a natural language. Maybe it can be used for definition of certain parts, but not the complete structure of the language, which is what Pāṇini's work amounts to.

The BNF is just a notation in which one can write a definition of a certain formal language. Pāṇini's notation is equvivalent to the BNF (which I think should be emphasized in the article) and he used grammatical cases to express the position of the term in the substitution formula αAβ → αγβ: nominative expressed the 'γ', genitive the 'A', and ablative and locative the left and right context, 'α' and 'β'. Such substitution sūtra is usually read: 'In place of A (there) is (or comes) γ, if preceded by α and is followed by β' - the use of cases also metaphorically corresponds to the usual meaning they have in the grammar.

One simple example: Aṣṭādhyāyī VI.1.77:

iko yaṇaci or

ikaḥ, yaṇ, aci (without sandhi)

could be rewritten as "ik ac → yaṇ ac" in BNF. The sūtra describes one of the sandhis where the last vowel of the first word precedes another vowel of the subsequent word. 'aci' being locative of 'ac' is the right context, left context is not important in this case, ikaḥ is genitive of 'ik' and 'yaṇ' is the nominative. The technical terms 'ik', 'yaṇ' and 'ac' denote the sets of characters of the Sanskrit alphabet: ik = {i, u, ṛ, ḷ} (simple vowels, except a), yaṇ = {y, v, r, l} (semivowels) and ac = {a, i, u, ṛ, ḷ, e, o, ai, au} (any vowel). An example of this sandhi would be tri + ambaka → tryambaka (the three-eyed, an epithet of Śiva) - before an 'a' of 'ambaka', the final 'i' of 'tri' changes to 'y' (i + a → y a} NikNovi (talk) 02:44, 29 December 2007 (UTC)

Influences

On the Indian Mathematics category box at the bottom of the page, Islamic Mathematics is listed as an influence on Indian Mathematics... that's not true! Arab mathematicians were influenced by Indian mathematicians and there's more evidence of math flowing out of India from 400 - 1200 CE than into it... this needs to be changed or at least debated. --59.93.193.97 (talk) 09:57, 20 June 2008 (UTC)

They were an influence. Saying that doesn't mean that Indians didn't influence the Arabs; the influence went in both directions. Clearly, the decimal number system and some algebra went from India to the Middle East; however, for example, Ibn al-Haytham's formula for the sum of integral powers was used by Indian mathematicians. Fowler&fowler«Talk» 20:39, 3 September 2008 (UTC)

Bressoud's paper

This issue was discussed in Talk:Kerala school of astronomy and mathematics. User:Fowler , you know very well that those are Bressoud's personal opinions and not facts. He does not discuss this conclusion anywhere in his paper. Please revert your edits.-Bharatveer (talk) 05:02, 2 September 2008 (UTC)

Please see quotes from four scholars now footnoted in the statement. I have also corrected the text in the Kerala School page. Fowler&fowler«Talk» 20:31, 3 September 2008 (UTC)

Archived

The last discussion on this page was in September 2008 (apart from some reverted edits and Wildbot's templates). I've archived all those discussions into Talk:Indian mathematics/Archive 3. Shreevatsa (talk) 20:38, 25 June 2010 (UTC)

Archive 1 Archive 2 Archive 3
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