Talk:Binomial theorem

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Examples

In the examples section there are some with fractional or negative exponent, but the generalized binomial theorem that deals with those cases is only mentioned later. Up to this point only non-negative integer values of $n$ have been considered and the summation is always from 0 to $n$. — Preceding unsigned comment added by Pacosantosleal (talk • contribs) 12:24, 25 November 2015 (UTC)

Meaning of n choose k

$Does{n \choose k}$ mean n_k ? — Preceding unsigned comment added by Skk146 (talk • contribs) 15:49, 27 November 2016 (UTC)

History

Joel B. Lewis I think it's better to cite Rashed here, because the source used is St Andrews, but while O'Connor and Robertson quote Rashed's statement, and apparently agree with it, the statement itself is Rashed's alone, and would be best cited directly to the source where he makes it. Not doing so when the source is readily available online is very poor scholarly practice. Best regards.---Wikaviani (talk) 00:37, 5 May 2018 (UTC)

recent edits

To help settle this recent conflict, I'm starting this talk page section. To the IP who keeps edit warring their preferred version in, there are a lot of inappropriate spacing changes you're making, as well as things like changing the capitalization of piped links (the link part, not the visible part); this should not be done. There are also a lot of invisible wikicode changes you're making. These are fine in and of themselves, but edits should not be made just to do that. See WP:COSMETICBOT for further info about these. WP:BRD is a pretty good idea. If someone reverts your changes from the status quo, it's really up to you to try to justify the changes, not make snarky retorts that don't address the disagreement. So again, please discuss here rather than trying to re-add. –Deacon Vorbis (carbon • videos) 21:05, 28 April 2019 (UTC)

The only reply to your only point, which is to assert "inappropriate", is to assert "appropriate". I am not going to further engage in pointless rhetoric with people want to claim that some standard should be met, never stating that standard, and moving it when it pleases them.80.65.247.112 (talk) 22:57, 28 April 2019 (UTC)

If you're unwilling to discuss, then you have no basis to continue to insist on your changes. If there's objection to those changes, you need to pause and resolve that objection rather than just plowing ahead. I've made some specific points here which should be addressed. Since you're unhappy with my use of "inappropriate", I'll elaborate on the spacing. You're using little-used manual spacing templates that aren't well known, aren't needed, produce inconsistent results, run contrary to MOS:MATH, and thus make maintenance more difficult. You've also added delimiter sizing commands that aren't needed – like $(x^{n})$ vs. $\left(x^{n}\right).$ This produces no difference in rendered output. A small handful of the changes you made were reasonable, mainly display mode vs. text mode for binomial coefficients. I reinstated those (which you took back out in your rush to revert me), but the vast majority were either unnecessary or actively harmful. I don't know why you feel such a need to see these edits stand, but there are good reasons why they shouldn't. –Deacon Vorbis (carbon • videos) 23:18, 28 April 2019 (UTC)

I notice that 80.65.247.112 doesn't want to say much on this Talk page, and doesn't leave edit summaries to give some hint as to his motivation. Neither of these habits is compatible with the way we do things on Wikipedia. Conversely, Deacon Vorbis not only shows a willingness to use the Talk page constructively, he shows an enthusiasm for doing so. Therefore I endorse the standard advocated by Deacon Vorbis. The editor at 80.65.247.112 should either adopt the Wikipedia way of doing things, or get used to the idea that all his hard work will one day amount to nothing. It should be an easy decision. Dolphin (t) 13:16, 29 April 2019 (UTC)

Multiple issues

^{NOTE: The version of the page being discussed}

Currently only a history is provided with enough citations and the rest is poorly sourced. Would be nice of someone adds a couple of links out there. DAVRONOVA.A. ✉ ⚑ 08:27, 17 June 2019 (UTC)

That's only one issue :p. Abstractly you are certainly right; concretely, are there any particular places you think are suspect (either, might be wrong, or with inappropriate weight, or original research)? Because "please make the article better" alone is not very constructive. --JBL (talk) 14:28, 17 June 2019 (UTC)

Original Research in substitution of of e^ax and e^bx in general Leibniz formula section of generalizations

Hi! The substitution of e^ax and e^bx into the general Leibniz formula seems to have no sources and thus seems to be original research. Although I understand that this substitution is correct mathematically, but there are no published sources having the same idea. Please take a moment to review my edit and verify my changes. The two changes I made are:

• Added [original research?] tag to the e^ax, e^bx substitution claim
• Added [verification needed] to the source "Calculus in One and Several Variables" by Robert Seeley since the book does not contain the Leibniz formula at all. The "Calculus for One Variable" can be found here, available for digital borrowing. This is single-variable calculus part of the combined book. Therefore, the Leibniz formula is expected to be in here. However, after extensive searching, no record of the Leibniz formula was found here.

I understand that a similar issue was addressed under the "Multiple issues" section, but I wanted to keep a concrete discussion just for this change. If a citation is found, please feel free to enter a citation. Otherwise I will be forced to remove the content according to the Wikipedia: Original research policy. Thank you. --Dh*Phoenix (talk) 17:40, 20 June 2020 (UTC)

This isn't OR; I'm having trouble finding a good ref for it, but a quick google search finds a lot of message-board type expositions of this fact. While a source would be ideal, the ease with which one finds non-RS discussion + WP:CALC probably lets this one stand. As for the Leibniz rule, there's another source at the main article that you could probably use if you think it's better. You could always just cite Abramowitz & Stegun too, or probably hundreds of other Calculus textbooks. –Deacon Vorbis (carbon • videos) 18:54, 20 June 2020 (UTC)

Ref added for the binomial theorem as a corollary of the Leibniz rule. –Deacon Vorbis (carbon • videos) 19:38, 20 June 2020 (UTC)

Yes, I'm sorry: this is not original research. Thank you for the source! And yes, I find the main article Leibniz rule citation better: I'll correct that.--Dh*Phoenix (talk) 05:47, 21 June 2020 (UTC)

Applications

The section on the infinite series for e is alright up until "This indicates that e can be written as a series:", at which point rigor and accuracy is lost. See Rudin, Priniciples of Mathematical Analysis, Section 3.3.1, for a proof. The mistake in the page as it exists is that it is not sufficient to say that the in the limit kth term is 1/k! implies that the entire series of 1/k! terms is equal to the original series. — Preceding unsigned comment added by 192.80.55.86 (talk) 18:20, 29 November 2021 (UTC)

History section

@Jacobolus: Hi, I just came to see your recent edits. In my humble opinion, sources like Amulya Kumar Bag, Biggs ... cannot challenge the views of prominent experts of the history of mathematics like Roshdi Rashed, Robertson, O'Connor as per WP:WEIGHT. According to these sources, Al-Karaji discovered the binomial theorem. please let me know if you think I'm missing something. Best.---Wikaviani ^{(talk) (contribs)} 21:18, 2 December 2024 (UTC)

You can read the historical Indian sources or their translations directly for yourself. It is not at all controversial to claim that what is described is binomial coefficients / "Pascal's triangle". I haven't looked at Rashed's book (paper?) but this sounds like a miscommunication.

It is entirely plausible though that these topics were developed independently in India, the Islamic world, and China. Who counts as "discoverer" or "inventor" seems like a pointless argument; we should just try to describe precisely what various historical scholars did, and let readers draw their own conclusions. –jacobolus (t) 21:46, 2 December 2024 (UTC)

I have checked your sources, honestly, nothing very impressive, some obscure mathematicians of Indian origin, mainly. Also, the historical section should give a fair representation of the history of this theorem, the discovery of it might be pointless for you, but not for our readers. The page you're looking for is 63 in Rashed's work. Here, another source repeating what Rashed says :"THE BINOMIAL THEOREM : A WIDESPREAD CONCEPT IN MEDIEVAL ISLAMIC MATHEMATICS" (PDF). core.ac.uk. p. 401. Retrieved 2019-01-08..---Wikaviani ^{(talk) (contribs)} 22:00, 2 December 2024 (UTC)

The authors involved here are world class experts in the history of Indian mathematics. I find your dismissive tone pretty insulting to be honest, and elevating one or another religious or ethnic group for point-scoring purposes by erasing the contributions of other groups is one the things I find personally most unpleasant and unfortunate about discussions of math history both off and especially on Wikipedia. Mathematics is a great human achievement which should be celebrated rather than fought over. Rashed does good work, and is well worth citing as an authority on developments among medieval Islamic mathematicians, but it's also not like he's omniscient. –jacobolus (t) 23:44, 2 December 2024 (UTC)

It's about discussing the quality of the sources, not about making it personal, and that's a legit concern of mines. So according to you, "Amulya Kumar Bag" is a world class expert ? Wow, so how many influencial books has this guy published ? with what publisher ? how many awards has he ? how about his academic career ? The source number 3 that you use repeatedly in the section about Indian maths is a 60 years old paper with barely any information about its publisher.

"Mathematics is a great human achievement" : agreed, so what ?---Wikaviani ^{(talk) (contribs)} 10:27, 3 December 2024 (UTC)

I'm going to stop engaging with this now, because I find your sarcasm very unpleasant. I recommend you adopt a more agreeable attitude before engaging in Wikipedia talk page discussion. Don't delete paragraphs of material from this page without consensus. –jacobolus (t) 16:23, 3 December 2024 (UTC)

So convenient, you refuse to achieve a consensus here, ok then, I'll revert back to the status quo version. This is not "sarcasm" but legit concerns about the quality of the sources you cite as I told you already above. Also, even Amyula kumar Bag says that some sources say that the so-called Pascal triangle found by Indian mathematicians has nothing to do with it. Best.---Wikaviani ^{(talk) (contribs)} 18:26, 3 December 2024 (UTC)

@JayBeeEll Do you want to take this one on? I'm not in the mood to deal with persistent rudeness today. @Wikaviani, please read Wikipedia:Civility, Wikipedia:What Wikipedia is not § Wikipedia is not a battleground and Wikipedia:Here to build an encyclopedia before making further edits or comments. –jacobolus (t) 18:38, 3 December 2024 (UTC)

Well, I'm quite baffled to see an editor who has been editing here for 20 years blatantly throwing baseless accusations of rudeness while faced with an editor who tries to discuss the quality of the sources ... How about complying with WP:CONSENSUS ? WP:UNDUE ? WP:BURDEN ? WP:RS ? WP:ASPERSIONS ? and so on ? You have made many edits to the article without any consensus and while some of them are cosmetic, others aren't. @JayBeeEll: I would appreciate your input about the reliability of Amyula kumar Bag for this topic. Thanks. Best.---Wikaviani ^{(talk) (contribs)} 22:05, 3 December 2024 (UTC)

"Tries to discuss" would be a more convincing summary if you had skipped the unjustified content removal, edit warring, repeated insults, sarcasm, air quotes scare quotes, and unexplained purity tests about which historians are "true" enough.

From a content perspective, please just be straight forward about your goals. My speculation is that you don't think the history of combinations (n choose k) and Pascal's triangle should be discussed in a page about the binomial theorem. You can try to build consensus for this position but you should explain yourself clearly and give some clear rationale for your position, instead of just blanking content and revert warring about it.

The sources are clearly fine: Bag's paper in the Indian Journal of History of Science was written by a professional historian in a respectable peer reviewed journal, has been widely cited since, and unquestionably meets WP:RS. Bibhutibhushan Datta, Radha Charan Gupta, Kripa Shankar Shukla were all celebrated and decorated mathematical historians. I'm sure if you put in some effort could find yet further secondary and tertiary sources discussing this topic, including by European/American authors if you don't like Indian historians. –jacobolus (t) 23:50, 3 December 2024 (UTC)

• "would be a more convincing summary if you had skipped the unjustified content removal, edit warring, repeated insults, sarcasm, air quotes, and unexplained purity tests about which historians are "true" enough." : Uh ? are you aware of WP:PA ? you must know that baseless accusations qualify as personal attacks ? where are my so-called insults ? please clarify or this will end up at ANI.
• "The sources are clearly fine: Bag's paper in the Indian Journal of History of Science was written by a professional historian in a respectable peer reviewed journal, has been widely cited since, and unquestionably meets WP:RS. Bibhutibhushan Datta, Radha Charan Gupta, Kripa Shankar Shukla were all celebrated and decorated mathematical historians. I'm sure if you put in some effort could find yet further secondary and tertiary sources discussing this topic, including by European/American authors if you don't like Indian historians" : I never said I don't like Indian historians, I just said that Bag has not the expertise to challenge prominent historians of maths like Robertson, Rashed etc baseless accusations of yours, again ... Interestingly, you have not included Bag in your blue links lists above by the way ... Also, please read WP:ONUS, I don't need to achieve consensus since you are the one who make repeated changes to the article, not me, thus you need to achieve consensus, not me. I will step out this article for now as you are not in the mood to have a constructive discussion. I will wait for more input from other editors.

---Wikaviani ^{(talk) (contribs)} 08:55, 4 December 2024 (UTC)

We got into this because you blanked a whole relevant and RS-sourced paragraph, based on the complaint that "none of the cited authors is a prominent expert of this topic", because Jean-Claude Martzloff is a mathematical historian primarily focused on China rather than India and Norman L. Biggs is a professional mathematician (though in this case writing a peer-reviewed paper in a top mathematical history journal).

That blanking was inappropriate, so was quickly reverted by @JayBeeEll, but you edit-warred to re-blank the paragraph, this time saying "sorry, but I insist" because you don't consider Martzloff or Biggs to be a "true historian of maths". In my opinion these sarcastic comments about both Martzloff and Biggs were insulting and out of line.

But whatever, fair enough. Since you didn't like these quasi-tertiary sources, I expanded the section adding closer secondary sources by subject experts. I don't know much about Bag – he is a professional mathematical historian specializing on the history of Indian mathematics who wrote a good number of widely cited papers in the 60s–70s which are still being cited today, and remains active. The other cited authors included Bibhutibhushan Datta and Radha Charan Gupta. In an edit summary you sarcastically and insultingly called Bag's paper a "'source'" with gratuitous air quotes scare quotes, and then in your comment here you insultingly summarized all of these sources as "nothing very impressive, some obscure mathematicians of Indian origin". Neither Datta nor Gupta is obscure: these are two of the most famous, celebrated, and prolific historians of Indian mathematics.

I was at that point quite unhappy with your repeated rude comments, but what put you over the line was "So according to you, 'Amulya Kumar Bag' is a world class expert? Wow, so how many influencial books has this guy published? with what publisher? how many awards has he? how about his academic career? The source number 3 that you use repeatedly in the section about Indian maths is a 60 years old paper with barely any information about its publisher." – Here you put more sarcastic air scare quotes, plus this time a string of sarcastic and insulting rhetorical questions. (To answer them though: Yes Amulya Kumar Bag is a world-class expert on the history of Indian mathematics and science. He doesn't have a Wikipedia article about him yet (feel free to write one), but here's a Google Scholar page. According to a quick web search he's a fellow of the Indian Academy of Sciences and was the editor of the Indian Journal of History of Science from 2002–2018. In particular his 1979 book Mathematics in ancient and medieval India has been very widely cited. I'm not sure which awards he has won, I'll leave you to research that. Both Datta and Gupta won multiple awards for their mathematical history work (e.g. Gupta won the Kenneth O. May Prize in 2009; Rashed won the same prize in 2017), and Datta's book History of Hindu Mathematics (with Singh) is one of the most influential ever written about the subject. As for that "its publisher", we are talking about the Indian Journal of History of Science published by the Indian National Science Academy.) I found your tone here to be completely unacceptable, insulting both to me personally and to these professional scholars. Since you are now making threats, I demand that you retract your insults.

After that, we have more sarcastic and exaggerated language from you which I found to be insulting: "So convenient," "baffled [...] blatantly", "Uh?" " Interestingly,". Please cut that out now.

––jacobolus (t) 17:38, 4 December 2024 (UTC)

I guess you and me don't have the same definition of the word insult. Let me be clear about that, if my comments were insults, go ahead and report me, otherwise, stop casting aspersion by labelling as "insults" legit concerns of another editor about that 60 years old source. Also, for your information, a mathematician is not a historian of maths, Biggs and Bag may be respectable mathematicians, but they are not expert in the field of history of maths. You seem to consider that since a publisher is reliable, that's enough to make the source reliable, this is wrong and you probably know that. Let me remind you what is a reliable source here, for Wikipedia (WP:SOURCEDEF) :

"When editors talk about sources that are being cited on Wikipedia, they might be referring to any one of these three concepts:

• The piece of work itself (the article, book)
• The creator of the work (the writer, journalist)
• The publisher of the work (for example, Random House or Cambridge University Press)

Any of the three can affect reliability. Reliable sources may be published materials with a reliable publication process, authors who are regarded as authoritative in relation to the subject, or both. These qualifications should be demonstrable to other people."

This is cristal clear, being published by a good publisher is not enough to make a source reliable, thus, while the Indian journal of history of science might be a great publisher, the author (Bag) is all but a world class expert and quite outdated since the sources that come after him, like Rashed or Robertson don't share his views (WP:AGEMATTERS). Besides, even if Bag was a world class expert like you claim, his views should not be given such an undue weight since they are not supported by the mainstream of reliable sources that are cited just after, especially when that source makes such an extraordinary claim about about the discovery of something equivalent to the triangle of Pascal in the 10th century. By the way, you keep editing the article while there is an ongoing dispute here, which is all but a constructive behaviour.---Wikaviani ^{(talk) (contribs)} 08:59, 5 December 2024 (UTC)

I don't intend to "report" you, which seems like a waste of time and energy. Let's assume (per WP:AGF) that your repeated inaccurate, sarcastic, condescending (and in my opinion quite insulting both to me personally and to the scholars cited here) comments were an honest mistake. I am merely asking you to please stop now. Specifically: I expect no more air scare quotes; no more sarcasm or condescension (along the lines of "so convenient", "interestingly", or most recently "for your information"); no more comments like "nothing very impressive"; no more sarcastic rhetorical questions. If you can cut the attitude and engage respectfully, we have no problem.

Amulya Bag is a professional historian of mathematics and science who was for an extended time the editor of a respected journal of mathematical history.

"extraordinary claim" Well let's just look at the translated 10th century text directly, shall we:

"After drawing a square on the top, two squares are drawn below (side by side) so that half of each is extended on either side. Below it three squares, below it (again) four squares are drawn and the process is repeated till the desired pyramid is attained. In the (topmost) first square the symbol for one is to be marked.. Then in each of the two squares of the second line figure one is to be placed. Then in the third line figure one is to be placed on each of the two extreme squares. In the middle square (of the third line) the sum of the figures in the two squares immediately above is to be placed; this is the meaning of the term pūrṇa. In the fourth line one is to be placed in each of the two extreme squares. In each of the two middle squares, the sum of the figures in the two squares immediately above, that is, three, is placed. Subsequent squares are filled in this way.

What is described here is precisely Pascal's triangle – indeed, even in a more "modern" form than Pascal himself used.

"You seem to consider that ..." – You are putting words in my mouth, but you are also setting up a standard that has nothing to do with WP:RS. Scholarly papers about mathematical history written in reputable mathematical history journals clearly meet the wikipedia reliable source standard.

I think you are somewhat mixing up what these various sources are claiming (and in particular the claims we are repeating here); to be specific, I think you are interpreting claims made about the history of combinations (which are numerically the same as binomial coefficients) as claims about the binomial theorem per se. Binomial coefficients, as numbers, occur in multiple contexts in mathematics. One of the context where they appear is in combinatorics, in counting the number of subsets of size ⁠$k$⁠ of a set of size ⁠$n$⁠. The paragraph about Indian examples is mainly about this combinatorial occurrence of these numbers. A distinct context is the algebra of polynomials, where these numbers appear as the coefficients of a binomial multiplied by itself some number of times.

There's really no question that there are multiple Indian sources ranging over a wide time period discussing binomial coefficients as numbers, especially in a combinatorial context. The Bhagavati Sutra describes combinations up through ⁠$n=4$⁠. (Here's O'Connor & Robertson on this since you consider them an acceptable source: "Permutations and combinations are used in the Sthananga Sutra. In the Bhagabati Sutra rules are given ... in the cases where n = 2, 3, and 4. The author then says that one can compute the numbers in the same way for larger n."). Pingala's description of making verses with various metres is in my understanding somewhat cryptic, but the elaboration given by his 10th century commentator describes exactly Pascal's triangle with the standard method of generation. (O'Connor and Robertson's summary: "In a commentary on this third century work in the tenth century, Pascal's triangle appears in order to give the coefficients of the binomial expansion." [O'Connor and Robertson make a mistake here: the commentary in question is of Pingala, not of the Bhagabati Sutra – I think they probably accidentally mixed up nearby sections of whatever secondary source they were working from]) The fraction-multiplication rule for finding n choose k is very clear and explicit in multiple sources from the 8th–9th century (here's a page by Ian G Pearce on O'Connor and Robertson's website discussing Mahavira's version).

Where there's some amount of ambiguity is the question of what counts as the "binomial theorem" per se. Amulya Bag makes the claim that Pingala's rules for verses should be considered as binomial expansion, because they involve forming metres as arrangements of two types of syllables, so for instance with length three we get all of the syllable patterns (aaa, baa, aba, aab, bba, bab, bba, bbb), which can be grouped by how many a's and b's they have and then counted, yielding the counts (1, 3, 3, 1), the 3rd row of Pascal's triangle. For Bag, this is a form of binomial expansion. However, there's an argument that could be made that these aren't binomials in the sense of sums of variable numerical quantities ⁠$x+y$⁠ to be expontiated like $(x+y)^{3}={}\!\!$$xxx+yxx+xyx+xxy+{}\!\!\!$$yyx+yxy+xyy+yyy={}\!\!$$x^{3}+3x^{2}y+3xy^{2}+y^{3}$. (Partly for this reason, I didn't repeat Bag's more opinionated claim in the article, but only his clearly factual claims. I think discussion in detail of this point is too in the weeds, and should be relegated to a more detailed history of binomial coefficients article if it is to be discussed at all.)

Per Rashed, the oldest known description of the the binomial theorem per se, i.e. taking powers of a binomial sum of numbers, at least for exponents greater than 3, can be found in al-Samawʾal's book from the 12th century, credited by al-Samawʾal to al-Karajī.

These different claims are not contradictory, and we don't need to reject the claim that Indians were working on combinatorics or expressed Pascal's triangle in order to accept Rashed's claim that the earliest version of something pretty close to the binomial theorem as we think of it today is al-Samawʾal/al-Karajī. Frankly we don't even need to reject Bag's claim that Pingala's verses are a kind of binomial expansion to also accept Rashed's claim. These claims are just not in any kind of conflict.

As for your concern about sources though: These sources are clearly "reliable" by Wikipedia standards because they are written by reputable mathematical historians in reputable peer-reviewed history journals or published in scholarly books, have been widely cited by other scholars, and are based directly on primary historical sources whose translations we can easily read directly and understand. We don't need to rely on anyone's subjective interpretation: the texts are right there in black and white. –jacobolus (t) 19:10, 5 December 2024 (UTC)

This wall of text is mainly your own interpretation of this topic, what I see on my end, is an editor who is not capable to provide serious sources for the claim "The Chandaḥśāstra by the Indian lyricist Piṅgala (3rd or 2nd century BC) somewhat crypically describes a method of arranging two types of syllables to form metres of various lengths and counting them; as interpreted and elaborated by Piṅgala's 10th-century commentator Halāyudha his "method of pyramidal expansion" (meru-prastāra) for counting metres is Pascal's triangle.".

Bag is not an expert historian of maths and is a bit outdated while Jayant shah's field of expertise is "computer vision" (Jayant shah is source number 12). I will remove this sentence but leave in the rest of your work since it is quite well-sourced. Best.---Wikaviani ^{(talk) (contribs)} 08:27, 9 December 2024 (UTC)

Bag is not an expert historian of maths – This is a falsehood which you now know to be false because we have been over this several times. Bag is a professional historian of mathematics who spent his career in the field and was the editor of a respected history of mathematics journal.

editor who is not capable to provide serious sources – I demand that you retract your insults and desist from further personal attacks against me or other people.

If you remove this perfectly fine sentence you will be reverted. Your personal understanding of sourcing policy does not accord with WP:RS and your behavior and comments here continue to well outside Wikipedia policy and norms. –jacobolus (t) 15:54, 9 December 2024 (UTC)

"If you remove this perfectly fine sentence you will be reverted. Your personal understanding of sourcing policy does not accord with WP:RS and your behavior and comments here continue to well outside Wikipedia policy and norms." Said the guy who already refused to comply with WP:RS, WP:CON, WP:ASPERSIONS, WP:NPA, WP:WEIGHT and so on ...

What I see is a list of sources (mainly with no precise page numbers) who are either unreliable or outdated.

• Alsdorf Ludwig 90 years old source, outdated
• Bag : outdated and unreliable
• Shah Jayant : unreliable (field : computer vision, not historian of maths)
• Bose : unreliable (he was a physicist)
• Edwards A. W. F. unreliable (British statistician and geneticists)
• Fowler : historian of Greek maths
• Ranjan Roy : mathematician, not historian
• Divakaran P. P. : Mathematician

Not a single historian of maths specialiced in the history of maths in Asia and no precise page numbers to allow me to verify what the sources really say. Please stop edit warring and self revert until a consensus is found here.---Wikaviani ^{(talk) (contribs)} 11:21, 10 December 2024 (UTC)

Not a single historian of maths specialiced in the history of maths in Asia – this is outright false. Amulya Kumar Bag is and Samarendra Nath Sen was a career expert on the history of South Asian mathematics and science. Sen's obituary was titled "The Doyen of Research in the History of Science in India".

You've replaced the ordinary WP:RS standard with a new "Wikaviani must personally vouch for every source" standard. You were upset that sources were too old, so I added sources from a wide range of dates, but you're not happy with newer sources either.

Your pick-and-choose criterion lets you reject any and all sources you want: this one is out because it is by a mathematician, the next one is out because it is by a historian but their specialty is the wrong country, the third one is out because it is by a historian specializing in the right topic but it was published in a book edited by a physicist, etc.

And now you made a formal complaint at Wikipedia:Administrators' noticeboard/Incidents § Jacobolus and WP:ASPERSION about my asking you repeatedly to stop using sarcastic insulting language.

My interpretation is that you pre-determined what outcome you want, and now you're going to make up whatever excuse you need or (mis)use whatever tool you can to obtain that outcome, irrespective of truth or community norms. –jacobolus (t) 15:58, 10 December 2024 (UTC)

My goal is to find a compromise, looking at your last comment, you sound like you own this article.---Wikaviani ^{(talk) (contribs)} 16:52, 10 December 2024 (UTC)

I threw more sources in. We can keep piling more sources on, but I really don't see the point. –jacobolus (t) 18:11, 9 December 2024 (UTC)

I added that the indian method was equivalent to Pascal's triangle. This is in accordance with the sources cited at Pascal's triangle, namely :

• Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (Helaine Selin) : "Other, lost works of al-Karaji's are known to have dealt with inderterminate algebra, arithmetic, inheritance algebra, and the construction of buildings. Another contained the first known explanation of the arithmetical (Pascal's) triangle; the passage in question survived through al-Sama'wal's Bahir (twelfth century) which heavily drew from the Badi."
• The Development of Arabic Mathematics Between Arithmetic and Algebra (Roshdi Rashed) : "The first formulation of the binomial and the table of binomial coefficients, to our knowledge, is to be found in a text by al-Karaji, cited by al-Sama'wal in al-Bahir."
• From Alexandria, Through Baghdad (Nathan Sidoli and Glen Van Brummelen) : "However, the use of binomial coefficients by Islamic mathematicians of the eleventh century, in a context which had deep roots in Islamic mathematics, suggests strongly the table was a local discovery - most probably of al-Karaji."

Hope this compromise is ok for you. Best.---Wikaviani ^{(talk) (contribs)} 22:10, 10 December 2024 (UTC)

I don't really see the difference, but sure.

Rashed is making a point about a "formulation of the binomial" which doesn't say anything one way or the other about earlier examples of Pascal's triangle interpreted in other fashions.

Len Berggren (1985), the paper you are quoting, which was reprinted in the book edited by Sidoli & Van Brummelen, says: "some had suggested that the table was a Chinese import. However, the use of binomial coefficients by Islamic mathematicians of the eleventh century, in a context which had deep roots in Islamic mathematics, suggests strongly the table was a local discovery—most probably of al-Karajī." This is only making the point that binomial coefficients as they appear in Islamic work were an independent discovery, not based on earlier work elsewhere, and the evidence for this is that the context related to topics/methods which had "deep roots" in other algebra work from the Islamic world – this is not the same as a claim that nobody else ever explored the same numbers in any other context. It's clear that the same table of numbers has been independently invented and reinvented repeatedly in a wide variety of cultures by a wide range of authors (in India, China, Persia, Italy, etc.).

Jacques Sesiano's encyclopedia entry about al-Karajī (in the encylopedia edited by Selin) says that the "first known explanation" of the arithmetical triangle is from a lost work by al-Karajī. I don't know what Sesiano means by an explanation.

(Aside: note Berggren, while being a fine mathematical historian, was professionally employed by a math department and as far as I can tell primarily taught pure math courses. All of your criticisms above about mathematicians not being historians would apply equally here (which is to say, are just as inapplicable in either case). Selin, the editor of the encyclopedia you mentioned, was a librarian. Many people have a career split between fields and manage to do good professional work in an area outside their job title.)

None of Sesiano, Berggren, or Rashed is an expert in ancient Indian mathematics, so it's not surprising that they wouldn't go out of their way to discuss it in survey sources focusing on Islamic contributions. –jacobolus (t) 02:53, 11 December 2024 (UTC)

Brummelen, Rashed, Berggren and Selin are all historians of science and reliable for this topic. Your above quote of Rashed is incomplete, he says "The first formulation of the binomial and the table of binomial coefficients, to our knowledge, is to be found in a text by al-Karaji, cited by al-Sama'wal in al-Bahir." meaning that the binomial coefficients and Pascal's triangle were first found in a work by al-Karaji. I'm going to remove Pascal's triangle from this article and move it to the relevant article. All views should be fairly represented with sentences like some sources claim that Pascal's triangle first appeared in indian works (cite sources) while others claim that it first appeared in al-Karaji's work (cite sources). Since I am not conviced by the reliability of many of the sources you mentioned, I will exclude those (like Shah jayant, Edwards etc) who, in my opinion, obviously have no expertise for this specific topic, feel free to take them at WP:RSN in order to include them. However, keep in mind WP:CITEOVERKILL since many sources are already in the article. Best.---Wikaviani ^{(talk) (contribs)} 08:03, 12 December 2024 (UTC)

I'm going to remove Pascal's triangl I don't support this at all. The history of these topics is closely intertwined. What we could do however is summarize the whole history section in a more compressed way, and elaborate on it at a dedicated article called something like History of binomial coefficients, as I suggested below.

My take is that you are backing away from your supposed "compromise" toward your previous maximalist (and in my opinion inappropriate and unjustified) content removal of before.

obviously have no expertise – this is, once again, nonsensical and extremely insulting. Please stop. –jacobolus (t) 13:02, 12 December 2024 (UTC)

I kept the sentence about Pascal's triangle (I already added it to the article about the triangle) but reworded it in order to represent the different views of the sources about this claim, I strongly advise you to desist from reverting again with no legit reason. Also, you should stop from making baseless comments about me insulting the sources while I only say that they are not expert for this specific topic (i.e. the history of sciences), as it is starting to become disruptive. your mass revert of all my work is all but constructive, you could just add back the content about Pascal's triangle without undoing all my work.---Wikaviani ^{(talk) (contribs)} 14:17, 12 December 2024 (UTC)

In my opinion it is awkward, not according with expected encyclopedic tone/style, and unhelpful to throw a bunch of "this is contradicted" caveats to the end of every sentence of the paragraph about combinatorics in India which mention the topic of al-Samawʾal, since the latter topic is described clearly in the immediately following paragraph, which the reader is expected to immediately encounter. The resulting prose reads like a confused author with multiple personalities is having an argument with himself and the reader is being involuntarily dragged along as a spectator. The source provided (Rashed's book) does not address combinatorics in India at all, and I think you are unjustifiably putting words in Rashed's mouth by implying his work says something which it very clearly does not. If I were Rashed I would be unhappy to have my work mischaracterized in this way. –jacobolus (t) 20:29, 12 December 2024 (UTC)

Aside: I started a discussion at WikiProject Mathematics asking for some help resolving this dispute from uninvolved editors. –jacobolus (t) 20:32, 12 December 2024 (UTC)

Ping also @Slawekb, who somewhat expanded this section in July 2015 and may be interested to weigh in / may have other recommended sources. –jacobolus (t) 01:23, 4 December 2024 (UTC)

Apologies @Jacobolus and Wikaviani: I am aware of this discussion but I am travelling and so I can't log in and don't have the time to digest and engage properly. --JBL writing from 158.144.178.11 (talk) 15:06, 12 December 2024 (UTC)

No need for apologies as far as I'm concerned, enjoy your trip JBL!---Wikaviani ^{(talk) (contribs)} 15:12, 12 December 2024 (UTC)

No worries, there's really no rush here. –jacobolus (t) 17:43, 12 December 2024 (UTC)

Wikaviani's edits such as this one, use sources claiming al-Karaji to have produced the first copy of Pascal's triangle as the basis for adding text to the article that casts doubt on unrelated claims of the appearance of binomial coefficients in earlier Indian mathematics. My feeling is that this is a serious violation of both WP:SYN and MOS:DOUBT. Those sources cannot be used to back up statements that Indian mathematics did not do these things, because they are not about Indian mathematics at all. They cannot be used to back up statements that the Indian accomplishments contradict scholarly consensus, because they do not make that comparison. To back up such statements we need sources explicitly comparing the Indian and Persian contributions, and we do not have such sources. We cannot point to the difference between what the sources say and say ourselves that it is a contradiction; that is WP:SYN. To be blunt, Wikaviani: please stop making these badly-sourced claims of contradiction unless you can back it up with reliable sources that make the same claims.

Taking a step back, part of the issue may be that the binomial theorem, binomial coefficients, multiplicative method of computing binomial coefficients, additive method of computing binomial coefficients, Pascal's triangle, and combinations from sets, are six different but closely related things. Even when we have a statement saying that the first appearance of Pascal's triangle was in one place, and even if we take that statement at face value as accurate, it implies nothing about the first appearance of any of the other aspects of the subject. —David Eppstein (talk) 03:22, 13 December 2024 (UTC)

Well, don't get me wrong, I have much respect for you as an admin here, but with all due respect, I'm quite speechless when i read you. First, that 200 years old source you posted at Wiki project maths and now your above post, saying that so many prominent historians of sciences are wrong when they deal with the history of maths while we, editors, know better than them. I hope I'm missing something but I don't know what exactly. Best.---Wikaviani ^{(talk) (contribs)} 15:34, 13 December 2024 (UTC)

Ok, so you are digging in. But you are wrong when you use a source about the Persians and the binomial theorem to make inferences about the Indians and binomial coefficients. Inferences are what our sources do. You should not be doing them here. Stop. —David Eppstein (talk) 17:41, 13 December 2024 (UTC)

Ok, let's say that the sources I provided are not reliable (even if I'm not convinced by that) but how about the sources like Jayant Shah, Bose, Edwards etc ? They seem to be good in their fields (i.e. maths, physics etc), but they are not really historians and they are about Indian maths, not Islamic maths.---Wikaviani ^{(talk) (contribs)} 08:07, 16 December 2024 (UTC)

Roshdi Rashed's 1972 paper is an excellent source about the content and relevance of the works of al-Samawʾal and al-Karajī, which is why it has been widely cited and its content has been incorporated into later surveys of Islamic mathematics and the history of algebra and combinatorics. But it, and sources drawing on it, aren't relevant about topics they never discussed.

Likewise, papers by Bag and Shah are excellent sources about the works of Piṅgala and later Indian combinatorialists, but would not be relevant to cite about topics they did not mention. Aside: Bose was the chief editor of a book (not an author), and S. N. Sen, the author of that broad survey (also another editor for the whole book), was a celebrated historian of Indian mathematics. Edwards's book has been cited hundreds of times (including by historians) because it is one of the best high level surveys about Pascal's triangle. –jacobolus (t) 12:53, 16 December 2024 (UTC)

If all involved editors agree with that, then it's all good for me. Howrver, while Rashed's 1972 work does not discuss Indian maths, sources like Bag, Sen or Edwards don't discuss about Islamic sciences either. Best.---Wikaviani ^{(talk) (contribs)} 19:41, 16 December 2024 (UTC)

Do you have a point to this, or is this just culture warring? Your comment comes across as a total non-sequitur, because we are not using any of those people in the sources to the paragraph about Islamic mathematics. —David Eppstein (talk) 19:51, 16 December 2024 (UTC)

No, David, this is not "culture warring", I'm just talking about what you said above :"Those sources cannot be used to back up statements that Indian mathematics did not do these things, because they are not about Indian mathematics at all. They cannot be used to back up statements that the Indian accomplishments contradict scholarly consensus, because they do not make that comparison.". I just say that if the Islamic sources don't make a comparative study with India, the Indian ones don't do a comparative study with Islamic sciences either.---Wikaviani ^{(talk) (contribs)} 20:41, 16 December 2024 (UTC)

That's precisely the point. So the Wikipedia article shouldn't speculate about relaions between the two or set up a direct comparison. The standard thing Wikipedia articles do in this kind of situation is to mention both topics, separately and independently. For Wikipedia to misattribute Wiki editors' speculation to scholars is a serious problem, and especially so when the speculation is an (implicit or explicit) criticism of sources that the cited source never mentioned. –jacobolus (t) 21:16, 16 December 2024 (UTC)

Leaving aside what your goal is, you keep saying things that are wrong. Edwards's book discusses both of these topics. The part discussing developments in India is pages 27–33, while the part mentioning al-Karajī is on page 52:

"[...] In India, Brahmegupta (A.D. 628) gave ⁠$(a+b)^{3}$⁠ in his Arithmetic [6]. ¶ It is to Persia, however, that the European thread can be traced back. The Al-bahir of Al-Samawal (died about 1180) is reported [7] as containing a calculation of the coefficients, resulting in the Binomial Triangle, which had been discovered by Al-Karaji some time soon after 1007. It is possible that Al-Karaji was inspired to make his discovery by hearing of Brahmegupta's result for the cube of a binomial, for it is believed that Brahmegupta's work had been brought to Baghdad in the eighth century, and Al-Karaji, who worked in Baghdad, drew much else from Hindu sources [8]. ¶ Al-Kashi [9], who died in Samarkand [...]"

I don't have any idea whether or not Al-Karaji was inspired by Brahmagupta, and I'm not sure such speculation is worth mentioning in this article, whose history section should remain relatively concise in my opinion. We probably should mention Jamshīd al-Kāshī and Naṣīr al-Dīn al-Ṭūsī, however. –jacobolus (t) 20:21, 16 December 2024 (UTC)

Perhaps,but if we are talking about A. W. F. Edwards, I see "Anthony William Fairbank Edwards, FRS (born 1935) is a British statistician, geneticist and evolutionary biologist. Edwards is regarded as one of Britain's most distinguished geneticists, and as one of the most influential mathematical geneticists in the history." Again, this scholar has no expertise in the field of history of sciences, thus, I'm not convinced by what he can say in this field.---Wikaviani ^{(talk) (contribs)} 20:53, 16 December 2024 (UTC)

You keep saying "X, therefore not Y", first in your "Persia, therefore not India" edits and now in your "mathematician, therefore not historian of mathematics". It is fallacious, it is ad hominem, and it is unconvincing. Focus on the work, not on its author. Its reviews in e.g. MR930876, Zbl 0641.01004, and Zbl 1032.01013 treat it seriously and positively; there is no reason to consider it as unreliable. And if you read our article on Edwards more carefully than just skimming its lead and looking for gotchas, you would see that "He has also written extensively on the history of genetics and statistics" (which apparently he considers binomial coefficients to fall under, since he also published about their history earlier than his book in International Statistical Review, focusing on the significance of Pascal's work on his triangle in the later history of probability and statistics). —David Eppstein (talk) 21:07, 16 December 2024 (UTC)

You are putting words in my mouth David, I never said such things like "Persia therefore not India" or "mathematician therefore not historian of mathematics", rather the opposite. Again, if all involved editors agree with the reliability of the sources proposed by Jacobolus, then it's all good for me.---Wikaviani ^{(talk) (contribs)} 21:58, 16 December 2024 (UTC)

@David Eppstein: There are 8 sources in number 11, this is not WP:OVERKILL while the 4 sources are ? If the formatting was your concerns, you should have formatted it in a way that is ok for you. Anyway, I think we're done here.---Wikaviani ^{(talk) (contribs)} 22:32, 17 December 2024 (UTC)

Let me be more clear since you apparently failed to understand my edit summary. There were three reasons why your edit adding a reference was bad.

1. It duplicated a reference that was already in the article. If the footnote was kept, it should have re-used the existing footnote using named references, instead of making another footnote with another copy of the reference.
2. It was very badly formatted. But it should not have been reformatted because of the duplication issue.
3. You added it as a footnote to a sentence that already had three footnotes, each reliable and with a detailed supporting quote. We do not need a fourth footnote on that one sentence. Putting another footnote there does not improve the article in any way.

Because of the third problem, I undid your addition rather than reformatting your new footnote or reusing the old one. —David Eppstein (talk) 22:53, 17 December 2024 (UTC)

I think the issue is the number of consecutive footnotes rather than the number of sources, but the previous claim really also does not need so many sources. The reason I kept adding more was because you kept objecting to sources one after another (for what I found to be inappropriate and mysterious reasons), but if we're all okay with the paragraph about Indian combinatorics we can thin out some of those. In case we want them later, I'm taking out:

• Alsdorf, Ludwig (1991) [1933]. "The Pratyayas: Indian Contribution to Combinatorics" (PDF). Indian Journal of History of Science. 26 (1): 17–61. Translated by S. R. Sarma from "π Die Pratyayas. Ein Beitrag zur indischen Mathematik". Zeitschrift fur Indologie und Iranistik. 9: 97–157. 1933.
• Sen, Samarendra Nath (1971). "Mathematics". In Bose, D. M. (ed.). A Concise History Of Science In India. Indian National Science Academy. Ch. 3, pp. 136–212, esp. "Permutations, Combinations and Pascal Triangle", pp. 156–157.
• Fowler, David H. (1996). "The Binomial Coefficient Function". The American Mathematical Monthly. 103 (1): 1–17, esp. §4 "A Historical Note", pp. 10–17. doi:10.2307/2975209. JSTOR 2975209.

Alsdorf was one of the earliest sources about this topic. Fowler's paper might be worth citing elsewhere in this article, either for the historical note included or for his more mathematical content. –jacobolus (t) 00:18, 18 December 2024 (UTC)

Re "one of the earliest sources about this topic": I mentioned this also on WT:WPM, but see Burrow, Reuben (1790), "A Proof that the Hindoos had the Binomial Theorem", Asiatick Researches: 487–497

I don't think we should cite it here but it may be worth keeping if we ever split out a history of binomial coefficients article. —David Eppstein (talk) 00:27, 18 December 2024 (UTC)

Yeah, this one is focused on an (anonymous?) historical source with similar content to that found in Śrīdhara or Mahāvīra which we currently mention here, notably the formula ⁠${\tbinom {n}{k}}={\tfrac {n}{1}}\times {\tfrac {n-1}{2}}\times \cdots \times {\tfrac {n-k+1}{n-k}}$⁠. Though that historical source does also mention ⁠$\textstyle \sum _{k=1}^{n}{\tbinom {n}{k}}=2^{n}-1$⁠, which could also be worth mentioning in a dedicated history article. By "one of the earliest sources" I meant specifically about the counting of poetic metres. But you are right that Burrow's paper is quite a bit older than Alsdorf. –jacobolus (t) 00:53, 18 December 2024 (UTC)

Perhaps a dedicated history article would be helpful

It might be useful to have an article History of binomial coefficients to consolidate Binomial coefficient § History and notation, Pascal's triangle § History, and Binomial theorem § History. That would leave more room to describe specific historical versions and their context in more detail and list more historical examples without unduly burdening these articles which otherwise are in a hurry to get to the mathematical content. This article could then make do with a more concise summary of 2–3 paragraphs. I don't have the motivation to write such a thing in the near future, but if someone feels inspired I'd be supportive of that effort. –jacobolus (t) 17:46, 3 December 2024 (UTC)