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I do not believe that the cited sources support the assertion that:
References
Here is what these cited sources say:
None of these sources are asserting—as a fact—that the Shang dynasty numerals were the origin of the Hindu-Arabic numeral system. The most we can say, based on these sources, is that they might be.
Unless better sources are provided I'm going to delete this sentence. Paul August ☎ 17:26, 17 October 2023 (UTC)
Lam Lay-Yong (1986, 1987, 1988) hypothesizes that the rod-numerals were ancestral to the Hindu positional numerals. Her evidence for this hypothesis is that the rod-numerals are positional and decimal, and there was considerable cultural contact between China and India in the 6th century AD, around the time when positionality developed in India. Because the rod-numerals were used in computation and commerce, she asserts that it is inconceivable that the Indians would not have learned of this system from the Chinese, and, since it is so practical, they obviously would have borrowed it (Lam 1988:104). From this, she asserts that the rod-numerals are the ultimate ancestor of the Western numerals. While Lam's hypothesis is plausible, I am deeply skeptical of its validity. Two immediate objections are that the Indian positional numeral-signs are those of the earlier Brahmi numerals, not of the rod-numerals, and that the rod-numerals have no zero-sign (whereas the Indian system does). To the first objection, Lam responds that "since six of the nine digits in rod numeral notation were strange to them, they would naturally have preferred their own numerals" (Lam 1986: 193). The notion that the rod-numerals were so foreign to the Indian mind as to require the total abandonment of its signs is unacceptable; who cannot comprehend the use of vertical and horizontal strokes? To the question of the zero, Lam replies that the abandonment of the alternating zong and heng positions required that the Indians develop a sign to fill the blank space (Lam 1986:194). I do not think this follows; a blank space would have served just as well as a zero-sign in either system, and if the abandonment of the alternating positions created such difficulty, why would the Indian mathematicians have done it? Even more damaging to Lam's argument are two structural differences between the rod-numerals and the Indian numerals that she ignores entirely: the rod-numerals have a quinary sub-base that the Indian numerals lack, and the rod-numerals are intraexponentially cumulative whereas the Indian positional numerals are ciphered. Moreover, no Indian texts of the period mention rod-numerals or any other Chinese numeration. Indeed, as I will discuss below, the Indian positional numerals were seen as remarkable in China in the early 8th century AD, suggesting that the Chinese traders who hypothetically transmitted the rod-numerals to India were entirely unaware of the result of their transmission. Lam's theory is so weak that it is equally plausible that the Greco-Roman counting board, which was also quinary-decimal, cumulative-positional, and used in the Middle East, was an ancestor of the Indian numerals - that is, it is not very plausible at all.
... In most recent times, this claim has been made most succinctly in the works of Wang Ling and Joseph Needham. In a 1958 paper delivered in Adelaide, Wang presented a detailed case for a Sino origin of the "Hindu-Arabic" numerals and pointed to the strong possibility of westward transmission to India. Wang's theory was further amplified in his collaborative work with Joseph Needham. Science and Civilization in China, vol. 3, devotes several pages (pp. 146-150) to this very issue and the phenomena of "stimulus diffusion". Needham's work clearly indicates the need for further research clarification as to the status of early Hindu mathematics and the possibility of cultural transmissions. It is exactly this research that must be undertaken to strengthen the claim for a Chinese genesis of our numeral system and, unfortunately, it is exactly this research that is lacking in Fleeting Footsteps. What was the status of ancient Indian mathematics during the Warring States period of (Chinese history? How were the numerals used in ancient India? Could the Chinese have obtained their mathematical knowledge from India? after all, Buddhism was an intellectual import from China's western neighbor. These are some of the issues and questions that must be addressed in positing a claim of a Chinese origin for the "Hindu-Arabic" numeral system and they remain missing footsteps in the path this book has taken. Despite the inability to develop and strengthen its major premise, Fleeting Footsteps is a valuable resource for understanding early Chinese mathematics. ...
References
Further, and perhaps more interesting, is the conjecture by historians of mathematics such as Wang Ling, Joseph Needham, and Lam Lay Yong and Ang Tian Se that our contemporary numeral system is derived from rod placements [Xu 2005]. They suggest that, as rod numerals were recorded and copied over centuries, scribes became complacent and hastened their writing process, gradually slipping into more cursive forms as illustrated below. What do you think? Perhaps our numeral system could more correctly be designated as the ‘Sino–Hindu–Arabic’ numeral system. Such a title might be more encompassing and historically revealing.M.Bitton ( talk) 10:44, 19 October 2023 (UTC)
Some historians of mathematics have conjectured that ...should do. M.Bitton ( talk) 17:55, 19 October 2023 (UTC)
I bring to your notice that I insist on adding an image of the Hindu Numeral Image that has reference to the evolution from the Brahmi Script to Gwalior to Devanagari Script. I also insist on adding the names of the books of Al-Khwarizmi and Al-Kindi in their native language. I also insist on bringing the Evolution of the Number System Image to the top section. GurkhanofAsia ( talk) 20:23, 7 January 2024 (UTC)
The recent edits by Jacoblus are factually incorrect and problematic
1) He claims that positional decimal system was invented in India around 1st Century AD, but the world's earliest positional decimal system was used by the Chinese in rod calculus. This system is older than the Indian positional decimal system.
2) He classified all those mathematicians, including Helaine Selin, who discussed the Chinese origin of number system as "scinologists"
3) He claims that the research of these "scinologists" are disputed, and are based on speculations, but he hasn't provided any reliable source for this. WP:OR
4) He has removed other sources like Campbell Douglas and J.C. Huggins without any explanation
Overall, he has edited this article in a way so as to give the reader a "purely Indian origin" while completely neglecting others as "scinologists" whose research, he described, is based on "speculations." Hu741f4 ( talk) 12:34, 23 February 2024 (UTC)
this is apparently a copyright violationthis what you claimed after removing it. M.Bitton ( talk) 16:22, 23 February 2024 (UTC)
Are you saying that I removed it under false pretenses?you reasserted the same accusation:
"this is apparently a copyright violation" this what you claimed after removing it. If the copyright issue could be resolved then I would be willing to consider adding the image back. But that would still leave the issue I've raised twice above, but you've yet to address, which is that that image
seems to assert as a fact the descent from Shang numerals. Paul August ☎ 17:02, 23 February 2024 (UTC)
"If one views a popular schematic of the evolution of our modern system of numeration and places the Chinese system in the appropriate chronological position, an interesting hypothesis arises, namely that the numeration system commonly used in the modern world had its origins 34 centuries ago in Shang China."
@ Paul August, let met try to give a summary for you. The original speculation comes from Joseph Needham (Science and Civilisation in China, vol. 3), and is pretty thin. Needham valuably piles up a lot of material related to China (it would be great if Wikipedia could cover some of this in greater detail), but in his eagerness to credit China with "more advanced and scientific" mathematics than other contemporary civilizations throughout history, unfortunately mischaracterizes the numerical systems and mathematical accomplishments of ancient Mesopotamia and the Mediterranean (and perhaps Egypt), and of medieval India and the Arab world, even as they were known 60 years ago; in the past half-century these have all been studied further, and his presentation is now quite dated. Needham is also pretty free mixing speculation with fact, which sometimes makes it hard to figure out precisely what he is claiming is known vs. conjectured. It makes a potential good starting point, but should be read with some healthy skepticism.
Here's most of the specifically relevant bit:
We are free to consider the possibility (or even probability) that the written zero symbol, and the more reliable calculations which it permitted, really originated in the eastern zone of Hindu culture where it met the southern zone of the culture of the Chinese. What ideographic stimulus could it have received at that interface? Could it have adopted an encircled vacancy from the empty blanks left for zeros on the Chinese counting-boards? The essential point is that the Chinese had possessed, long before the time of the Sun Tzu Suan Ching (late +3rd century) a fundamentally decimal place-value system. It may be, then, that the 'emptyness' of Taoist mysticism, no less than the 'void' of Indian philosophy, contributed to the invention of a symbol for sunya, i.e. the zero. It would seem, indeed, that the finding of the first appearance of the zero in dated inscriptions on the borderline of the Indian and Chinese culture-areas can hardly be a coincidence.
You can see that this doesn't present any particular evidence, and is explicitly speculative.
Lam Lay Yong, I believe at some point a student of Needham's, took the idea and elaborated it into a paper (or a few?) and a book Fleeting Footsteps with coauthor Ang Tian Se, which includes an English translation of the Sunzi Suanjing (a Chinese book of unknown original date and unknown original author, the main part of which probably dates from the 5th century or before and which was eventually included among the canon of "ten mathematical manuals" during the Tang dynasty in the 7th century). Lam analyses the counting-rod-based arithmetical procedures described in the available copy of the Sunzi Suanjing, and finds that a variety of arithmetical algorithms are similar to those found in medieval arithmetic sources written in Arabic. Noting that there was cultural contact between India and China at the time, e.g. the transmission of Buddhism from India to China, Lam therefore claims that Indian arithmetic must have come from China.
The problem with the book is that (a) Lam does not have any direct evidence whatsoever for her main theory, and (b) Lam does not address the several most obvious criticisms of her theory:
First, Chinese counting rods are in many ways similar to counting boards used in Egypt, Greece, Rome, and Mesopotamia, which were even older than Chinese counting rods, and used across a wide geographical area over long periods of time by civilizations with even more cultural contact with India. Unfortunately our knowledge of the precise methods used with these is somewhat scanty because we don't have any contemporary written manuals. I'm not enough of an expert to immediatly do it myself but I'd love to see improvement of our articles abacus and counting board, among others. Lam does not consider that Indian and/or Chinese arithmetic may have been influenced by these other tools/systems, a speculative conjecture which also has no direct evidence but is just as plausible as Lam's own. Positional numbering is also similar in many ways to the Mesopotamian positional (base sixty) system, to the Greek numeral system, etc.; the Chinese counting rod system is not particularly more similar to Indian arithmetic than these other previous systems, and there's nothing obvious about the relationship between Indian and Chinese systems.
Second, Lam doesn't consider, as Martzloff points out, that the copy of the Sunzi Suanjing she translated might have been modified or extended by copyists later than its original date. Indeed there is explicit historical documentation of such modification in the 7th century, not discussed by Lam's book ("the biography of Li Chunfeng (602-670) [...] states that Li Chunfeng and others reedited the SZSJ because 'the text was very erroneous (or contradictory) from the point of view of the principles'"). It is hard to firmly date the various pieces of the Sunzi Suanjing or figure out precisely when the methods described developed, but it is entirely plausible that methods described by Lam were even brought to China from outside. (It is also entirely plausible that they developed independently.) We have explicit evidence of the use of Indian numerals in China as early as the 8th century, and considering the cultural contact Lam stresses, it is entirely plausible that their influence was felt centuries earlier than that. But again this is all pure speculation, and any comments about it in Wikipedia should be described as such.
Third, Lam does not address the clear structural differences between Indian and counting-rod number systems. The counting-rod system uses a 5 × 2 structure also found in the Roman counting board but not found at all in the Indian arithmetic system. Lam puts great emphasis on the use of a blank space for zero in computing with counting rods, but this is a prominent feature of every kind of counting board. Lam hangs much of her theory on the claim that counting rods were used in a strictly positional way, unlike Greek, Mesopotamian, Egyptian, or other < 0 BCE written numeral systems, which involved various irregularities. She assumes that this purported feature couldn't have developed independently. But as Marzloff points out, this is not even an accurate summary of the actual use of counting rods, noting that for example blank spaces for "zeros" were not uniformly preserved, and that "irregular forms [...] were rather common in [Chinese] mathematical texts. [...] in practice, the counting-rod system was not as perfectly decimal and positional as the descriptions in Fleeting Footsteps would imply." The counting rod system is very concrete, almost identical in basic format to the Mayan (base twenty) or Sumerian (base sixty) systems, whereas Indian arithmetic uses symbols which are substantially abstract, more comparable to Greek alphabetic numerals. Neither counting rods nor Chinese written numbers of the time were discernably similar to the symbols adopted in India, which likely developed indigenously from earlier Indian symbols; nobody has even speculated that the symbols themselves originated in any Chinese system. Martzloff's summary is that "examples of Chinese written numerals from sources anterior to the tenth century do exist, and the structure of these militates against the idea of the Chinese origin of the Hindu-Arabic numerals."
Finally, Lam does not sufficiently address the lack of any direct or textual evidence, and does not surface any with her book. No numerically related words were adopted from Chinese into Indian languages (however, several were transmitted in the other direction). No extant Indian sources mention Chinese number systems. There are no examples of counting rods found in India. Etc. Etc.
Note again that Marzloff is one of the preeminent historians of Chinese mathematics; this is not some kind of politically motivated position. Indeed, I imagine the journal editors chose to have Martzloff write a review specifically to forestall any accusation of political motivation, as might arise if a scholar focusing on e.g. Greece, Persia, or Southeast Asia wrote a review. – jacobolus (t) 19:25, 23 February 2024 (UTC)
I intend to restore certain edits as I see fit and I would like to discuss each such restored edit here in each in its own section. Paul August ☎
Note: I tried to move the discussion of this edit (see above) here but I was reverted by M.Bitton. Paul August ☎ 17:44, 23 February 2024 (UTC)]
the issue with the image for me is not just copyrightin that case, I will wait until this is sorted before making a derivative. M.Bitton ( talk) 19:28, 23 February 2024 (UTC)
I've restored a Jacobolus's rewrite to the first three sentences of the "Origins" section, which adds content, with what seem to me to be better sources, and which also does not change the current text concerning the Shang numerals. Does anyone have any issues with these edits? Paul August ☎ 18:08, 23 February 2024 (UTC)
Please cite your sources that say these were "positional"
On the other hand the Sanskrit grammatic system of Pāṇini (c. 500 BC) has been claimed recently to contribute to the concept of zero in mathematical sense (i.e. involving positional analysis, operation of subtraction, process for going from maximum to minimum). It is even said that "he was the first man to use mathematical concept of 'zero' before mathematicians accepted it". His conception is presented in three forms, namely the linguistic zero, the it zero, and the anuvṛtti-zero, but his idea of 'absence' (lopa, etc.) cannot be truly compared with a zero in a place-value system.
According to Sadguru-śisya, the prosodist Piṅgala was a younger brother of Pāṇini, but usually Piṅgala is taken to flourish about 200 BC. For computing 2n, he gave a set of four sūtras one of which reads (VIII, 29 in his Chandah-śāstra), rūpe śūnyam, or "(Place) a zero (śūnya) when unity is subtracted (from index or power)". ¶ So it is believed that India possessed a zero symbol at that time (but śūnya may mean blank space.).
The word 'thibuga' used by Bhadrabāhu (c. 300 BC) has been found in a quoted gāthā and interpreted by Hemacandra to mean bindu. Some scholars try to see 'zero' of place-value notation in this. The Jaina canonical work Anuyogadvāra-sūtra (c. 100 BC) is said to provide the "earliest literary evidence" of the use of the word of place-value notation in this (see sūtra 142). Now credit for inventing the place-value system (with zero) is also being given of Kundakunda (between 100 BC and 100 AD) who may be the possible author of relevant works (Parikarma and Saṃta-kamma-paṃjiya) which are relevant.
That the decimal place-value system was in use then in India is clear from reference to it by Vasumitra (first century AD) to illustrate that ‘things are spoken in accordance with their states’. He says “When the clay counting-piece is placed in the place of Units, it is denominated 'one', when placed in the place of Hundreds, it is denominated 'hundred', and in place of Thousands it is denominated a 'thousand'. Vasumitra was a Buddhist. Similar counting process is mentioned in ancient Jaina works. In such positional process, the circular symbol (representing empty pit) would automatically denote zero. The use of zero symbol to fill the blank space66 is also found in Mahābandha (c. AD 100).
Wikipedia isn't a place for pursuing original research and giving judgements.
"Victor J Katz, Carl Boyer, Lam lay young, Joseph Needham, Douglas A Campbell, Helaine Selin, Frank Swetz"
Does anyone have recommended sources to examine while trying to draw a replacement for scans of Menninger's glyph evolution image? I mentioned Chrisomalis (2010) Numerical Notation: A Comparative History as one fairly comprehensive scholarly survey by a career expert. Are there other relatively recent surveys with decent coverage of this topic? – jacobolus (t) 02:10, 25 February 2024 (UTC)
@ Paul August has reverted my edit https://en.wikipedia.org/?title=Hindu–Arabic_numeral_system&diff=prev&oldid=1209835548 because he has some issues with that edit. Please discuss your issues here, because I don't see any issue. All the contents were supported by multiple reliable sources. Hu741f4 ( talk) 19:59, 23 February 2024 (UTC)
"The world's earliest positional decimal system was the Chinese rod calculus."– This is not an accurate statement. Decimal counting boards were used in Ancient Egypt many centuries before that. For example Herodotus in the 5th century BCE explicitly described how the (positional, decimal) counting boards used in Egypt and Greece were oriented in opposite directions; by that time this was an old technology firmly embedded in culture and language. – jacobolus (t) 20:09, 23 February 2024 (UTC)
The only Greek scholar to have researched the subject is Lang, in a series of publications of fundamental value, published in the journal Hesperia between 1957 and 1968, to which I shall refer below. T. L. Heath, A history of Greek mathematicsP is still of some value. At any rate, while many questions on this issue are still open, there is no doubt that, in the ancient Mediter- ranean, calculations were frequently made by moving counters on a surface known as the 'abacus'. We therefore need to look at the ancient, or western abacus. In M. L. Lang's original publication in the field, "Herodotus and the abacus", 14 abaci were listed from the classical Aegean world. In a later publication, "Abaci from the Athenian Agora", the same author added two more from the Athenian Agora. A. Scharlig extends the list to 30 objects, with largely the same pattern of distribution: nearly all from the Aegean world, most from Attica. (The furthest afield seems to be SEG XXIII 620, a third-century B.C. abacus found in Cyprus.)
In her original publication from 1957, as well as two later articles, Lang went on to argue that some arithmetic features in calculations preserved in the literary tradi- tion of classical texts may be accounted for by assuming operations on the abacus. Finally, while no ancient source discusses the abacus as such, there are many passing references that take it for granted. Based on this archeological and literary evidence, a coherent picture of the physical shape of the ancient, western abacus and its usage may be suggested. [...]
Like Arabic numerals (and their Babylonian antecedents) the abacus is essentially positional: hence follows a certain abstraction. Just as it makes no difference, for pen-and-pencil operations, which absolute value the positions have (to add 1.345 and 1.678 is the same as to add 1345 and 1678), so it makes no difference, for the abacus, whether we move from 'fives' to 'tens' or from 'fifties' to 'hundreds'. If only for this reason, it makes clear sense to avoid marking the lines. It is true that the abacus is not as totally homogenous as are the positions of Arabic numerals: one must distinguish odd, 10n, from even, 5 × 10n positions. But such an alternate marking may easily be inserted on an ad hoc basis. We thus find that the western abacus has very little substance: really, no more than a row of scratches. The abaci listed by Lang were identified because, if not on the lines themselves, they had numbers marked at some other position of the abacus (perhaps to keep records during the operation). In the Greek world (unlike the Roman case) no counters were ever identified as "abacus counters", and there is no reason to suppose any existed. Ordinary pebbles would do and, as we shall note below, the Greek world had a profusion of other counters of all kinds, all useable on the abacus. Further, while the extant abaci (with a few exceptions, e.g. two abaci scratched hastily on roof-tiles) tend to be made with marble, in ordinary circumstances a mobile board would have been more useful. Most probably, abaci were mostly made with wood, but this is pure guesswork, as naturally none survives. Ultimately, indeed, the very notion of the abacus as a clearly defined artifact is misleading. While scratches are useful, the lines can very well be imag- ined, perhaps referring to whatever irregularity the surface at hand may have. Thus any surface will do. The abacus is not an artifact: it is a state of mind. The western abacus was wherever there were sufficiently flat surfaces - as well as sufficiently many objects that the thumb and fingers could grasp. Probably more designated abaci can be found if we look for them with more attention. But perhaps designated abaci are less important than the skills that make them so easy to construct and use on an ad hoc basis.
@ Jacobolus: as I explained in the edit summary, the non-positional Brahmi numerals have their own article (that's what wikilinks are for), the origin of the ciphered-positional system is what matters here. What part of that do you disagree with? M.Bitton ( talk) 16:24, 25 February 2024 (UTC)
transmission of ciphered-positional numerals (again, this what the subject is about) is obviously relevant– the section is explicitly titled "origin". Material about the later evolution and transmission of the system does not "obviously" belong there; indeed it is directly out of the scope implied by the title. – jacobolus (t) 18:19, 25 February 2024 (UTC)
Refocus on content disagreement: M.Bitton, can you please explain, clearly and in some detail, what you think the appropriate scope is for an "origin" section, and how you feel it fits with the rest of the article? My personal opinion is that any section under this title should clearly and somewhat completely discuss the origin of the Hindu–Arabic numeral system, as implied by the title. You clearly disagree, as evidenced by your removal of content directly addressing that question, and have by your editing behavior, edit summaries, and comments implied a scope which in my opinion does not match the section heading "Origin". I don't really understand what you think the scope should be. Under the scope you have set out as I best understand it currently, the section seems largely redundant and unnecessary, and I would also be happy to just merge it into the existing "history" section. – jacobolus (t) 19:09, 25 February 2024 (UTC)
"BRD is not a justification for imposing one's own view or for tendentious editing. ¶ BRD is not a valid excuse for reverting good-faith efforts to improve a page simply because you don't like the changes. ¶ BRD is never a reason for reverting. Unless the reversion is supported by policies, guidelines or common sense, the reversion is not part of BRD cycle". I'd really rather not derail this to further meta-discussion here though. Can you try to focus on the content? – jacobolus (t) 19:49, 25 February 2024 (UTC)
your patterns of editing behaviorNot only have you repeatedly attacked me personally for no reason other than to provoke me, but you're still doing it and doubling down on it. I will just ignore you until the admins that I pinged weigh in. M.Bitton ( talk) 19:58, 25 February 2024 (UTC)
none of my comments are intended as a personal attackis that some kind of joke? I asked you repeatedly to refrain from casting aspersions and personally attacking me, but to no avail. In fact, you doubled down on them. In any case, I will await the admins' input (because this has been going on for months and it needs to stop, regardless of the intentions behind it). M.Bitton ( talk) 20:24, 25 February 2024 (UTC)
References
have ever been questioned by other reputed scholars– this statement is outright false. I recently linked to 2 very strong criticisms of Lam Lay Yong and Ang Tian Se's book by some of the worlds' foremost experts on this topic, and can find at least another 1–2 if you like. Campbell and Douglas are two separate people who have not weighed in on this topic; Frank Swetz's paper is a game-of-telephone repetition of Lam's claims which in my opinion does not accurately convey them and is thus not a great source. If you want to litigate either the counting rods -> positional Indian numbers or the Shang numerals -> Brahmi numerals theories (which are entirely distinct and should not be conflated), can we make a new discussion topic about it though? It's getting a bit cramped in this one. – jacobolus (t) 20:53, 25 February 2024 (UTC)
The Brāhmī numerals, a ciphered additive decimal numeral system, developed in the Indian subcontinent probably sometime around the 3rd century BCE, spreading through the Maurya Empire. The ultimate origin of the Brāhmī numerals is unresolved, but they have been theorized to have evolved from Greek, Chinese, or most plausibly Egyptian numerals, or to have developed indigenously.
"Stephen Chrisomalis isn't a reputed authority in history of Mathematics"– He wrote his PhD and later this book, published by Cambridge University Press, about the comparative history of numeral systems. His book has hundreds of citations in the scholarly literature. Here are some reviews:
@ Jacobolus: Can you please explain why you chose to not use the short form cites I introduced into the "Origins" section? Paul August ☎ 15:22, 26 February 2024 (UTC)
I added a "dubious" flag to the sentence about "Shang numerals". This sentence is currently inaccurate and also not reflective of the linked sources. The referenced ~1400 BCE numerals found on oracle bones are an interesting topic in their own right and it would be great if Wikipedia had an article about them, but they are not "positional" in the sense meant by this article. To the best of my understanding, Lam's claim in her papers and book is that the feature from Chinese arithmetic adopted in India was the positional idea (from counting rods), not anything about written numerals from more than a millennium year earlier. I'll try to re-read Lam's papers; it's been a few months since I looked at them. – jacobolus (t) 15:49, 26 February 2024 (UTC)
"Can we please not make any substantive changes to this section which are undiscussed on the talk page."– jacobolus (t) 15:57, 26 February 2024 (UTC)
There are 2 different subjects which may be worth addressing in this article, but if either or both is to be addressed, it should be accurate, moderately complete, and well sourced.
We should be clear to separate these two topics, since they are mostly unrelated.
– jacobolus (t) 15:57, 26 February 2024 (UTC)
The Brāhmī script came to prominence in the mid third century BC [...]. Brāhmī script was probably derived from a Semitic prototype (Aramaic, South Semitic, or Phoenician), although many South Asian scholars still support the theory that the script was indigenously developed (Salomon 1996: 378–379). [...]
The question of the ultimate origin of the Brāhmī numerals – specifically, whether or not they constitute a case of independent invention, and if not, on which ancestor(s) they were modeled – is unresolved, and is made more complex by the politicization of the matter. [...]
One set of theories regarding the origin of the Brāhmī numerals derives them from existing representational systems used in South Asia. Borrowing from the letters of the Brāhmī script to create an alphabetic numeral-system, while once a popular theory, is not really sustainable (Prinsep 1838, Woepcke 1863, Indraji 1876, Datta and Singh 1962 [1935], Gokhale 1966, Verma 1971). [...] The derivation of the Brāhmī numerals from the Kharoṣṭhī letters is even more improbable [...]. Finally, a more recent set of theories derives the Brāhmī numerals from those of the Indus Valley civilization (Sen 1971, Kak 1994), but there are no examples of any writing from India between the latest Harappan inscriptions (around 1700 bc) and the first Brāhmī inscriptions [...].
If not derived from any South Asian system, the Brāhmī numerals could have developed independently. Woodruff (1994 [1909]: 53–60) speculated that both the Chinese and Brāhmī numerals derived from a hypothetical ancient set of cumulative tally signs for 1 through 9, which would then have spread to both China and India. [...]
Finally, a number of theories argue for a foreign origin of the Brāhmī numerals. Falk (1993: 175–176), noting structural and paleographic resemblances between the Brāhmī and the earliest Chinese (Chapter 8) numerical notations, argues for a Chinese origin. However, there is little evidence of contact between the two regions at this period, and the only paleographic similarity between the systems is the common use of horizontal strokes for 1, 2, and 3. It has occasionally been proposed that the Greek alphabetic numerals inspired the Brāhmī numerals, given their appearance following the Alexandrine period, the strong trade ties with the Greco-Iranian kingdoms of Parthia and Bactria, and the structural similarities between the two systems. However, the evidence for the “alphabeticity” of the Brāhmī numerals is weak at best (see the previous discussion), and there is no paleographic correspondence between the Greek and Brāhmī numerals.
It is most plausible that the Brāhmī numerals are derived from the Egyptian hieratic or demotic numerals. Burnell (1968 [1874]) argued for a demotic origin, while Bühler’s (1963 [1895]) much more prominent analysis argued for a hieratic origin. The three systems are structurally similar: they are all decimal, hybrid ciphered- additive/multiplicative-additive systems, and represent 200, 300, 2000, and 3000 by adding quasi-multiplicative strokes to the signs for 100 or 1000. There are resemblances in around one-third of the sign-forms, and very close resemblances for a few, such as 9 (Bühler 1963 [1895]: 115–119; Salomon 1995, 1998). While there was not tremendous Egypto-Indic cultural contact, Ptolemaic traders reached as far as the city of Muziris (modern Cranganore) on the Malabar Coast, and Aśoka is known to have sent Buddhist missionaries to Alexandria (Basham 1980: 187). Of the two Egyptian systems, I believe the demotic to be a more likely ancestor, because in the Ptolemaic period the use of hieratic numerals was very limited. Thus, although the demotic and Brāhmī systems differ in both the power at which multiplication is used and the direction of writing, I believe that a demotic origin should be adopted as a working hypothesis.
To help facilitate discussion, here is the current text under discussion:
According to some sources, this number system may have originated in Chinese Shang numerals (1200 BC), which was also a decimal positional numeral system.
Here is some new text proposed
here by
jacobolus:
According to some scholars, the Hindu–Arabic number system originated in Chinese counting rods, also a positional decimal number representation.
Paul August
☎ 16:00, 26 February 2024 (UTC)
Oops sorry, the proposed changes by jacobolus here were:
Some scholars have theorized that the positional concept in the Hindu–Arabic number system may have originated in Chinese counting rods, also a positional decimal number representation.
Paul August ☎ 16:11, 26 February 2024 (UTC)
According to some scholars, the Hindu–Arabic number system originated in Chinese rod numeral system, also a positional decimal number representation.
According to some sources, this number system may have originated in Chinese Shang numerals (1200 BC), which was also a decimal positional numeral system.
Some scholars have theorized that the positional concept in the Hindu–Arabic number system may have originated in Chinese counting rods, also a positional decimal number representation.
According to some scholars, the Hindu–Arabic number system originated in Chinese rod numeral system, also a positional decimal number representation.
Paul August ☎ 16:37, 26 February 2024 (UTC)
Here is what the sources currently cited in our article say:
Paul August ☎ 16:56, 26 February 2024 (UTC)
Lam Lay-Yong (1986, 1987, 1988) hypothesizes that the rod-numerals were ancestral to the Hindu positional numerals, because the rod-numerals are positional and decimal, and because there was considerable cultural contact between China and India in the sixth century AD, when positionality developed in India. Because the rod-numerals were used in computation and commerce, she asserts that it is inconceivable that the Indians would not have learned of this system from the Chinese, and, since it is so practical, they obviously would have borrowed it (Lam 1988: 104). Yet the Indian positional numeral-signs are those of the earlier Brāhmī numerals, not of the rod-numerals, and the rod-numerals have no zero- sign (whereas the Indian system does). Moreover, the rod-numerals have a quinary sub-base that the Indian numerals lack, and the rod-numerals are intraexponentially cumulative, whereas the Indian positional numerals are ciphered. No Indian texts of the period mention rod-numerals or any other Chinese numeration.
Adding two more quotes:
#Helaine Selin, 2008,
p. 198: "this fact, together with other evidence supports the thesis that the Hindu-Arabic system has its origin in the Chinese rod numeral system." this tertiary source is already mentioned above.
M.Bitton ( talk) 17:16, 26 February 2024 (UTC)
"Some scholars have theorized that the positional concept in the Hindu–Arabic number system may have originated in Chinese counting rods, also a positional decimal number representation."is nothing more than a temporary stopgap, and I do not consider it an acceptable medium-term solution to the problems with the current article. I would like to arrive at a mutually acceptable solution which we can build on without constant edit warring.
From what I can tell,
M.Bitton and I can agree that the
§ Origins section in its current place at the very top of this article, isn't really enough space to flesh out possible theories about the origin/early evolution of the Hindu–Arabic number system. To quote
M.Bitton: "I have no problem with mentioning everyone's view in details, but it has to be everyone (quotes and all) with no cherry picking. That comes at a price: we'd have to create a whole section about the Chinese origin."
Paul August, does moving/merging the 'origins' section seem okay with you also?
If so, I'd like to propose, as a way forward, that I will try to write more text in the next few days about both Lam's counting rod theory and also other competing theories, in a draft page somewhere (e.g. in user space), and then will merge it in as a subsection of the § History section after discussion here. M.Bitton: if you would like to write your own competing draft for that section, that would be fine, then we can bring them together and compare. As a separate subsection, I'll try to flesh out the section § Predecessors, possibly retitled to "Brāhmī numerals", and include a few sentences inline about theorized origins for the Brāhmī numerals there, including a theorized Chinese origin. – jacobolus (t) 18:13, 26 February 2024 (UTC)
"make the least change necessary to make the sentence about Chinese counting rods minimally accurate"– this is a temporary stop-gap, not a "proposal". But I would appreciate it if you would address my proposal in this section and answer the several direct questions I have asked you, instead of derailing yet another conversation by aggressive off-topic rhetorical flourishes. – jacobolus (t) 18:32, 26 February 2024 (UTC)
To be clear, my "proposed change" is what I consider the minimal change necessary to make this statement not outright false / grossly misleadingis also what you wrote after the bold edits. M.Bitton ( talk) 18:46, 26 February 2024 (UTC)
So you think that you've been perfectly polite here, and haven't being at all rude?That was a sincere question on my part. I think we could all be more polite. I was trying to see if you also thought you could have been more polite. Did you think my question was itself rude? I'm sorry if you did, that was certainly not my intension. But whatever, I will discuss all of this at length anywhere else but not here.
The earliest extant physical examples of decimal place value numerals are found in inscriptions from around the middle of the first millennium CE, written in scripts derived from Brāhmī. At present, the first such inscription known in an Indian source may be the one on a certain copper plate from Gujarat [... from] about 595 CE.7 Decimal place value numbers are also found in some inscriptions from Indianized cultures in Southeast Asia around the same time. [...] We can certainly infer that if the decimal place value system had been incorporated into epigraphic styles over much of South and Southeast Asia by this time, it must have originated quite a bit earlier.
footnote 7: However, it has been persuasively argued that this particular record is spurious and was actually inscribed at a later date; [...]
But we do not need to rely only on such inferences to push back the date of origin of decimal place value beyond the time of its earliest known inscriptional records. The content of some older textual sources includes hints about the writing of numbers that suggest a place value system, although of course the texts themselves are physically recorded only in much later manuscript copies. For example, a commentary from probably the fifth century CE on an ancient philosophical text, the famous Yoga-sūtra of Patañjali, employs the following simile about the superficial “changes of inherent characteristics” (Yoga-sūtra 3.13):
Just as a line in the hundreds place [means] a hundred, in the tens place ten, and one in the ones place, so one and the same woman is called mother, daughter, and sister [by different people].
Even earlier, the Buddhist philosopher Vasumitra in perhaps the first century CE used a similar analogy involving merchants’ counting pits, where clay markers were used to keep track of quantities in transactions. He says, “When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred.” Such statements clearly expect the audience to be familiar with the concept of numerical symbols representing different powers of ten according to their relative positions. Due to the brevity of their allusions and the ambiguity of their dates, however, they do not solidly establish the chronology of the development of this concept.
A different representation of decimal place value is revealed by a verbal notation called by medieval authors bhūta-saṅkhyā or “object-numbers,” here designated the “concrete number system.” Its function is to provide synonyms for ordinary number words such as “three” or “twelve.” Recall that in Sanskrit, at least after the Vedic period, even technical treatises were most often composed in verse. [...]
The concrete number system, to judge from all its extant examples, has apparently always been a place value system, representing large numbers with strings of words that stand for its individual digits or groups of digits, in order from the least significant to the most significant. Thus, if we encounter, say, the verbally expressed concrete number “Veda/tooth/moon,” we translate it as “four [for the four Vedic collections]/thirty-two/one,” and write it as 1324. These concrete numbers are not combined with number words signifying powers or multiples of ten, so their only unambiguous interpretation is as pure decimal place value. Hence the idea of a positional system for numerals must have been commonplace by the time the concrete number system was invented.
A firm upper bound for the date of this invention is attested by a Sanskrit text of the mid-third century CE, the Yavana-jātaka or “Greek horoscopy” of one Sphujidhvaja, which is a versified form of a translated Greek work on astrology. Some numbers in this text appear in concrete number format, as in its final verse [...] So it corresponds to [269/270 CE]. Evidently, then, positional decimal numerals were a familiar concept at least by the middle of the third century, at least to the audience for astronomical and astrological texts.
Exactly how and when the Indian decimal place value system first developed, and how and when a zero symbol was incorporated into it, remain mysterious. One plausible hypothesis about its origin links it to the symbols used on Chinese counting boards as early as the mid-first millennium BCE. These counting boards, like the Indian counting pits mentioned above, had a decimal place value structure: they were divided into columns representing successive powers of ten, with units on the right. Small rods were arranged in regular patterns in the columns of the board to designate numbers from 1 to 9, and a column left blank signified a zero. Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the same concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion.
We will see in section 3.3 that there are textual indications of a written symbol for zero in India even before the start of the Common Era, but it is not clear whether the symbol was part of place value notation at that time. The use of zero in decimal numerals and its characteristic round shape may have been reinforced by the round zero markers in sexagesimal place value numerals introduced to India in Greek astronomical and astrological texts.
Exactly how and when the Indian decimal place value system first developed, and how and when a zero symbol was incorporated into it, remain mysterious. One plausible hypothesis about its origin links it to the symbols used on Chinese counting boards as early as the mid-first millennium BCE. These counting boards, like the Indian counting pits mentioned above, had a decimal place value structure: they were divided into columns representing successive powers of ten, with units on the right. Small rods were arranged in regular patterns in the columns of the board to designate numbers from 1 to 9, and a column left blank signified a zero. Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travellers, or they may have developed the same concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion.
Not seeing any specific reasons claimed why this proposal wouldn't be an effective way of resolving this dispute, I am going to work on making those draft changes under the assumption that other editors will broadly support the reorganization once it's ready. If anyone would still like to see a different kind of organization, please make clear and concrete proposals now and we can discuss them: I'm not aiming for a fait accompli here. – jacobolus (t) 19:16, 26 February 2024 (UTC)
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I do not believe that the cited sources support the assertion that:
References
Here is what these cited sources say:
None of these sources are asserting—as a fact—that the Shang dynasty numerals were the origin of the Hindu-Arabic numeral system. The most we can say, based on these sources, is that they might be.
Unless better sources are provided I'm going to delete this sentence. Paul August ☎ 17:26, 17 October 2023 (UTC)
Lam Lay-Yong (1986, 1987, 1988) hypothesizes that the rod-numerals were ancestral to the Hindu positional numerals. Her evidence for this hypothesis is that the rod-numerals are positional and decimal, and there was considerable cultural contact between China and India in the 6th century AD, around the time when positionality developed in India. Because the rod-numerals were used in computation and commerce, she asserts that it is inconceivable that the Indians would not have learned of this system from the Chinese, and, since it is so practical, they obviously would have borrowed it (Lam 1988:104). From this, she asserts that the rod-numerals are the ultimate ancestor of the Western numerals. While Lam's hypothesis is plausible, I am deeply skeptical of its validity. Two immediate objections are that the Indian positional numeral-signs are those of the earlier Brahmi numerals, not of the rod-numerals, and that the rod-numerals have no zero-sign (whereas the Indian system does). To the first objection, Lam responds that "since six of the nine digits in rod numeral notation were strange to them, they would naturally have preferred their own numerals" (Lam 1986: 193). The notion that the rod-numerals were so foreign to the Indian mind as to require the total abandonment of its signs is unacceptable; who cannot comprehend the use of vertical and horizontal strokes? To the question of the zero, Lam replies that the abandonment of the alternating zong and heng positions required that the Indians develop a sign to fill the blank space (Lam 1986:194). I do not think this follows; a blank space would have served just as well as a zero-sign in either system, and if the abandonment of the alternating positions created such difficulty, why would the Indian mathematicians have done it? Even more damaging to Lam's argument are two structural differences between the rod-numerals and the Indian numerals that she ignores entirely: the rod-numerals have a quinary sub-base that the Indian numerals lack, and the rod-numerals are intraexponentially cumulative whereas the Indian positional numerals are ciphered. Moreover, no Indian texts of the period mention rod-numerals or any other Chinese numeration. Indeed, as I will discuss below, the Indian positional numerals were seen as remarkable in China in the early 8th century AD, suggesting that the Chinese traders who hypothetically transmitted the rod-numerals to India were entirely unaware of the result of their transmission. Lam's theory is so weak that it is equally plausible that the Greco-Roman counting board, which was also quinary-decimal, cumulative-positional, and used in the Middle East, was an ancestor of the Indian numerals - that is, it is not very plausible at all.
... In most recent times, this claim has been made most succinctly in the works of Wang Ling and Joseph Needham. In a 1958 paper delivered in Adelaide, Wang presented a detailed case for a Sino origin of the "Hindu-Arabic" numerals and pointed to the strong possibility of westward transmission to India. Wang's theory was further amplified in his collaborative work with Joseph Needham. Science and Civilization in China, vol. 3, devotes several pages (pp. 146-150) to this very issue and the phenomena of "stimulus diffusion". Needham's work clearly indicates the need for further research clarification as to the status of early Hindu mathematics and the possibility of cultural transmissions. It is exactly this research that must be undertaken to strengthen the claim for a Chinese genesis of our numeral system and, unfortunately, it is exactly this research that is lacking in Fleeting Footsteps. What was the status of ancient Indian mathematics during the Warring States period of (Chinese history? How were the numerals used in ancient India? Could the Chinese have obtained their mathematical knowledge from India? after all, Buddhism was an intellectual import from China's western neighbor. These are some of the issues and questions that must be addressed in positing a claim of a Chinese origin for the "Hindu-Arabic" numeral system and they remain missing footsteps in the path this book has taken. Despite the inability to develop and strengthen its major premise, Fleeting Footsteps is a valuable resource for understanding early Chinese mathematics. ...
References
Further, and perhaps more interesting, is the conjecture by historians of mathematics such as Wang Ling, Joseph Needham, and Lam Lay Yong and Ang Tian Se that our contemporary numeral system is derived from rod placements [Xu 2005]. They suggest that, as rod numerals were recorded and copied over centuries, scribes became complacent and hastened their writing process, gradually slipping into more cursive forms as illustrated below. What do you think? Perhaps our numeral system could more correctly be designated as the ‘Sino–Hindu–Arabic’ numeral system. Such a title might be more encompassing and historically revealing.M.Bitton ( talk) 10:44, 19 October 2023 (UTC)
Some historians of mathematics have conjectured that ...should do. M.Bitton ( talk) 17:55, 19 October 2023 (UTC)
I bring to your notice that I insist on adding an image of the Hindu Numeral Image that has reference to the evolution from the Brahmi Script to Gwalior to Devanagari Script. I also insist on adding the names of the books of Al-Khwarizmi and Al-Kindi in their native language. I also insist on bringing the Evolution of the Number System Image to the top section. GurkhanofAsia ( talk) 20:23, 7 January 2024 (UTC)
The recent edits by Jacoblus are factually incorrect and problematic
1) He claims that positional decimal system was invented in India around 1st Century AD, but the world's earliest positional decimal system was used by the Chinese in rod calculus. This system is older than the Indian positional decimal system.
2) He classified all those mathematicians, including Helaine Selin, who discussed the Chinese origin of number system as "scinologists"
3) He claims that the research of these "scinologists" are disputed, and are based on speculations, but he hasn't provided any reliable source for this. WP:OR
4) He has removed other sources like Campbell Douglas and J.C. Huggins without any explanation
Overall, he has edited this article in a way so as to give the reader a "purely Indian origin" while completely neglecting others as "scinologists" whose research, he described, is based on "speculations." Hu741f4 ( talk) 12:34, 23 February 2024 (UTC)
this is apparently a copyright violationthis what you claimed after removing it. M.Bitton ( talk) 16:22, 23 February 2024 (UTC)
Are you saying that I removed it under false pretenses?you reasserted the same accusation:
"this is apparently a copyright violation" this what you claimed after removing it. If the copyright issue could be resolved then I would be willing to consider adding the image back. But that would still leave the issue I've raised twice above, but you've yet to address, which is that that image
seems to assert as a fact the descent from Shang numerals. Paul August ☎ 17:02, 23 February 2024 (UTC)
"If one views a popular schematic of the evolution of our modern system of numeration and places the Chinese system in the appropriate chronological position, an interesting hypothesis arises, namely that the numeration system commonly used in the modern world had its origins 34 centuries ago in Shang China."
@ Paul August, let met try to give a summary for you. The original speculation comes from Joseph Needham (Science and Civilisation in China, vol. 3), and is pretty thin. Needham valuably piles up a lot of material related to China (it would be great if Wikipedia could cover some of this in greater detail), but in his eagerness to credit China with "more advanced and scientific" mathematics than other contemporary civilizations throughout history, unfortunately mischaracterizes the numerical systems and mathematical accomplishments of ancient Mesopotamia and the Mediterranean (and perhaps Egypt), and of medieval India and the Arab world, even as they were known 60 years ago; in the past half-century these have all been studied further, and his presentation is now quite dated. Needham is also pretty free mixing speculation with fact, which sometimes makes it hard to figure out precisely what he is claiming is known vs. conjectured. It makes a potential good starting point, but should be read with some healthy skepticism.
Here's most of the specifically relevant bit:
We are free to consider the possibility (or even probability) that the written zero symbol, and the more reliable calculations which it permitted, really originated in the eastern zone of Hindu culture where it met the southern zone of the culture of the Chinese. What ideographic stimulus could it have received at that interface? Could it have adopted an encircled vacancy from the empty blanks left for zeros on the Chinese counting-boards? The essential point is that the Chinese had possessed, long before the time of the Sun Tzu Suan Ching (late +3rd century) a fundamentally decimal place-value system. It may be, then, that the 'emptyness' of Taoist mysticism, no less than the 'void' of Indian philosophy, contributed to the invention of a symbol for sunya, i.e. the zero. It would seem, indeed, that the finding of the first appearance of the zero in dated inscriptions on the borderline of the Indian and Chinese culture-areas can hardly be a coincidence.
You can see that this doesn't present any particular evidence, and is explicitly speculative.
Lam Lay Yong, I believe at some point a student of Needham's, took the idea and elaborated it into a paper (or a few?) and a book Fleeting Footsteps with coauthor Ang Tian Se, which includes an English translation of the Sunzi Suanjing (a Chinese book of unknown original date and unknown original author, the main part of which probably dates from the 5th century or before and which was eventually included among the canon of "ten mathematical manuals" during the Tang dynasty in the 7th century). Lam analyses the counting-rod-based arithmetical procedures described in the available copy of the Sunzi Suanjing, and finds that a variety of arithmetical algorithms are similar to those found in medieval arithmetic sources written in Arabic. Noting that there was cultural contact between India and China at the time, e.g. the transmission of Buddhism from India to China, Lam therefore claims that Indian arithmetic must have come from China.
The problem with the book is that (a) Lam does not have any direct evidence whatsoever for her main theory, and (b) Lam does not address the several most obvious criticisms of her theory:
First, Chinese counting rods are in many ways similar to counting boards used in Egypt, Greece, Rome, and Mesopotamia, which were even older than Chinese counting rods, and used across a wide geographical area over long periods of time by civilizations with even more cultural contact with India. Unfortunately our knowledge of the precise methods used with these is somewhat scanty because we don't have any contemporary written manuals. I'm not enough of an expert to immediatly do it myself but I'd love to see improvement of our articles abacus and counting board, among others. Lam does not consider that Indian and/or Chinese arithmetic may have been influenced by these other tools/systems, a speculative conjecture which also has no direct evidence but is just as plausible as Lam's own. Positional numbering is also similar in many ways to the Mesopotamian positional (base sixty) system, to the Greek numeral system, etc.; the Chinese counting rod system is not particularly more similar to Indian arithmetic than these other previous systems, and there's nothing obvious about the relationship between Indian and Chinese systems.
Second, Lam doesn't consider, as Martzloff points out, that the copy of the Sunzi Suanjing she translated might have been modified or extended by copyists later than its original date. Indeed there is explicit historical documentation of such modification in the 7th century, not discussed by Lam's book ("the biography of Li Chunfeng (602-670) [...] states that Li Chunfeng and others reedited the SZSJ because 'the text was very erroneous (or contradictory) from the point of view of the principles'"). It is hard to firmly date the various pieces of the Sunzi Suanjing or figure out precisely when the methods described developed, but it is entirely plausible that methods described by Lam were even brought to China from outside. (It is also entirely plausible that they developed independently.) We have explicit evidence of the use of Indian numerals in China as early as the 8th century, and considering the cultural contact Lam stresses, it is entirely plausible that their influence was felt centuries earlier than that. But again this is all pure speculation, and any comments about it in Wikipedia should be described as such.
Third, Lam does not address the clear structural differences between Indian and counting-rod number systems. The counting-rod system uses a 5 × 2 structure also found in the Roman counting board but not found at all in the Indian arithmetic system. Lam puts great emphasis on the use of a blank space for zero in computing with counting rods, but this is a prominent feature of every kind of counting board. Lam hangs much of her theory on the claim that counting rods were used in a strictly positional way, unlike Greek, Mesopotamian, Egyptian, or other < 0 BCE written numeral systems, which involved various irregularities. She assumes that this purported feature couldn't have developed independently. But as Marzloff points out, this is not even an accurate summary of the actual use of counting rods, noting that for example blank spaces for "zeros" were not uniformly preserved, and that "irregular forms [...] were rather common in [Chinese] mathematical texts. [...] in practice, the counting-rod system was not as perfectly decimal and positional as the descriptions in Fleeting Footsteps would imply." The counting rod system is very concrete, almost identical in basic format to the Mayan (base twenty) or Sumerian (base sixty) systems, whereas Indian arithmetic uses symbols which are substantially abstract, more comparable to Greek alphabetic numerals. Neither counting rods nor Chinese written numbers of the time were discernably similar to the symbols adopted in India, which likely developed indigenously from earlier Indian symbols; nobody has even speculated that the symbols themselves originated in any Chinese system. Martzloff's summary is that "examples of Chinese written numerals from sources anterior to the tenth century do exist, and the structure of these militates against the idea of the Chinese origin of the Hindu-Arabic numerals."
Finally, Lam does not sufficiently address the lack of any direct or textual evidence, and does not surface any with her book. No numerically related words were adopted from Chinese into Indian languages (however, several were transmitted in the other direction). No extant Indian sources mention Chinese number systems. There are no examples of counting rods found in India. Etc. Etc.
Note again that Marzloff is one of the preeminent historians of Chinese mathematics; this is not some kind of politically motivated position. Indeed, I imagine the journal editors chose to have Martzloff write a review specifically to forestall any accusation of political motivation, as might arise if a scholar focusing on e.g. Greece, Persia, or Southeast Asia wrote a review. – jacobolus (t) 19:25, 23 February 2024 (UTC)
I intend to restore certain edits as I see fit and I would like to discuss each such restored edit here in each in its own section. Paul August ☎
Note: I tried to move the discussion of this edit (see above) here but I was reverted by M.Bitton. Paul August ☎ 17:44, 23 February 2024 (UTC)]
the issue with the image for me is not just copyrightin that case, I will wait until this is sorted before making a derivative. M.Bitton ( talk) 19:28, 23 February 2024 (UTC)
I've restored a Jacobolus's rewrite to the first three sentences of the "Origins" section, which adds content, with what seem to me to be better sources, and which also does not change the current text concerning the Shang numerals. Does anyone have any issues with these edits? Paul August ☎ 18:08, 23 February 2024 (UTC)
Please cite your sources that say these were "positional"
On the other hand the Sanskrit grammatic system of Pāṇini (c. 500 BC) has been claimed recently to contribute to the concept of zero in mathematical sense (i.e. involving positional analysis, operation of subtraction, process for going from maximum to minimum). It is even said that "he was the first man to use mathematical concept of 'zero' before mathematicians accepted it". His conception is presented in three forms, namely the linguistic zero, the it zero, and the anuvṛtti-zero, but his idea of 'absence' (lopa, etc.) cannot be truly compared with a zero in a place-value system.
According to Sadguru-śisya, the prosodist Piṅgala was a younger brother of Pāṇini, but usually Piṅgala is taken to flourish about 200 BC. For computing 2n, he gave a set of four sūtras one of which reads (VIII, 29 in his Chandah-śāstra), rūpe śūnyam, or "(Place) a zero (śūnya) when unity is subtracted (from index or power)". ¶ So it is believed that India possessed a zero symbol at that time (but śūnya may mean blank space.).
The word 'thibuga' used by Bhadrabāhu (c. 300 BC) has been found in a quoted gāthā and interpreted by Hemacandra to mean bindu. Some scholars try to see 'zero' of place-value notation in this. The Jaina canonical work Anuyogadvāra-sūtra (c. 100 BC) is said to provide the "earliest literary evidence" of the use of the word of place-value notation in this (see sūtra 142). Now credit for inventing the place-value system (with zero) is also being given of Kundakunda (between 100 BC and 100 AD) who may be the possible author of relevant works (Parikarma and Saṃta-kamma-paṃjiya) which are relevant.
That the decimal place-value system was in use then in India is clear from reference to it by Vasumitra (first century AD) to illustrate that ‘things are spoken in accordance with their states’. He says “When the clay counting-piece is placed in the place of Units, it is denominated 'one', when placed in the place of Hundreds, it is denominated 'hundred', and in place of Thousands it is denominated a 'thousand'. Vasumitra was a Buddhist. Similar counting process is mentioned in ancient Jaina works. In such positional process, the circular symbol (representing empty pit) would automatically denote zero. The use of zero symbol to fill the blank space66 is also found in Mahābandha (c. AD 100).
Wikipedia isn't a place for pursuing original research and giving judgements.
"Victor J Katz, Carl Boyer, Lam lay young, Joseph Needham, Douglas A Campbell, Helaine Selin, Frank Swetz"
Does anyone have recommended sources to examine while trying to draw a replacement for scans of Menninger's glyph evolution image? I mentioned Chrisomalis (2010) Numerical Notation: A Comparative History as one fairly comprehensive scholarly survey by a career expert. Are there other relatively recent surveys with decent coverage of this topic? – jacobolus (t) 02:10, 25 February 2024 (UTC)
@ Paul August has reverted my edit https://en.wikipedia.org/?title=Hindu–Arabic_numeral_system&diff=prev&oldid=1209835548 because he has some issues with that edit. Please discuss your issues here, because I don't see any issue. All the contents were supported by multiple reliable sources. Hu741f4 ( talk) 19:59, 23 February 2024 (UTC)
"The world's earliest positional decimal system was the Chinese rod calculus."– This is not an accurate statement. Decimal counting boards were used in Ancient Egypt many centuries before that. For example Herodotus in the 5th century BCE explicitly described how the (positional, decimal) counting boards used in Egypt and Greece were oriented in opposite directions; by that time this was an old technology firmly embedded in culture and language. – jacobolus (t) 20:09, 23 February 2024 (UTC)
The only Greek scholar to have researched the subject is Lang, in a series of publications of fundamental value, published in the journal Hesperia between 1957 and 1968, to which I shall refer below. T. L. Heath, A history of Greek mathematicsP is still of some value. At any rate, while many questions on this issue are still open, there is no doubt that, in the ancient Mediter- ranean, calculations were frequently made by moving counters on a surface known as the 'abacus'. We therefore need to look at the ancient, or western abacus. In M. L. Lang's original publication in the field, "Herodotus and the abacus", 14 abaci were listed from the classical Aegean world. In a later publication, "Abaci from the Athenian Agora", the same author added two more from the Athenian Agora. A. Scharlig extends the list to 30 objects, with largely the same pattern of distribution: nearly all from the Aegean world, most from Attica. (The furthest afield seems to be SEG XXIII 620, a third-century B.C. abacus found in Cyprus.)
In her original publication from 1957, as well as two later articles, Lang went on to argue that some arithmetic features in calculations preserved in the literary tradi- tion of classical texts may be accounted for by assuming operations on the abacus. Finally, while no ancient source discusses the abacus as such, there are many passing references that take it for granted. Based on this archeological and literary evidence, a coherent picture of the physical shape of the ancient, western abacus and its usage may be suggested. [...]
Like Arabic numerals (and their Babylonian antecedents) the abacus is essentially positional: hence follows a certain abstraction. Just as it makes no difference, for pen-and-pencil operations, which absolute value the positions have (to add 1.345 and 1.678 is the same as to add 1345 and 1678), so it makes no difference, for the abacus, whether we move from 'fives' to 'tens' or from 'fifties' to 'hundreds'. If only for this reason, it makes clear sense to avoid marking the lines. It is true that the abacus is not as totally homogenous as are the positions of Arabic numerals: one must distinguish odd, 10n, from even, 5 × 10n positions. But such an alternate marking may easily be inserted on an ad hoc basis. We thus find that the western abacus has very little substance: really, no more than a row of scratches. The abaci listed by Lang were identified because, if not on the lines themselves, they had numbers marked at some other position of the abacus (perhaps to keep records during the operation). In the Greek world (unlike the Roman case) no counters were ever identified as "abacus counters", and there is no reason to suppose any existed. Ordinary pebbles would do and, as we shall note below, the Greek world had a profusion of other counters of all kinds, all useable on the abacus. Further, while the extant abaci (with a few exceptions, e.g. two abaci scratched hastily on roof-tiles) tend to be made with marble, in ordinary circumstances a mobile board would have been more useful. Most probably, abaci were mostly made with wood, but this is pure guesswork, as naturally none survives. Ultimately, indeed, the very notion of the abacus as a clearly defined artifact is misleading. While scratches are useful, the lines can very well be imag- ined, perhaps referring to whatever irregularity the surface at hand may have. Thus any surface will do. The abacus is not an artifact: it is a state of mind. The western abacus was wherever there were sufficiently flat surfaces - as well as sufficiently many objects that the thumb and fingers could grasp. Probably more designated abaci can be found if we look for them with more attention. But perhaps designated abaci are less important than the skills that make them so easy to construct and use on an ad hoc basis.
@ Jacobolus: as I explained in the edit summary, the non-positional Brahmi numerals have their own article (that's what wikilinks are for), the origin of the ciphered-positional system is what matters here. What part of that do you disagree with? M.Bitton ( talk) 16:24, 25 February 2024 (UTC)
transmission of ciphered-positional numerals (again, this what the subject is about) is obviously relevant– the section is explicitly titled "origin". Material about the later evolution and transmission of the system does not "obviously" belong there; indeed it is directly out of the scope implied by the title. – jacobolus (t) 18:19, 25 February 2024 (UTC)
Refocus on content disagreement: M.Bitton, can you please explain, clearly and in some detail, what you think the appropriate scope is for an "origin" section, and how you feel it fits with the rest of the article? My personal opinion is that any section under this title should clearly and somewhat completely discuss the origin of the Hindu–Arabic numeral system, as implied by the title. You clearly disagree, as evidenced by your removal of content directly addressing that question, and have by your editing behavior, edit summaries, and comments implied a scope which in my opinion does not match the section heading "Origin". I don't really understand what you think the scope should be. Under the scope you have set out as I best understand it currently, the section seems largely redundant and unnecessary, and I would also be happy to just merge it into the existing "history" section. – jacobolus (t) 19:09, 25 February 2024 (UTC)
"BRD is not a justification for imposing one's own view or for tendentious editing. ¶ BRD is not a valid excuse for reverting good-faith efforts to improve a page simply because you don't like the changes. ¶ BRD is never a reason for reverting. Unless the reversion is supported by policies, guidelines or common sense, the reversion is not part of BRD cycle". I'd really rather not derail this to further meta-discussion here though. Can you try to focus on the content? – jacobolus (t) 19:49, 25 February 2024 (UTC)
your patterns of editing behaviorNot only have you repeatedly attacked me personally for no reason other than to provoke me, but you're still doing it and doubling down on it. I will just ignore you until the admins that I pinged weigh in. M.Bitton ( talk) 19:58, 25 February 2024 (UTC)
none of my comments are intended as a personal attackis that some kind of joke? I asked you repeatedly to refrain from casting aspersions and personally attacking me, but to no avail. In fact, you doubled down on them. In any case, I will await the admins' input (because this has been going on for months and it needs to stop, regardless of the intentions behind it). M.Bitton ( talk) 20:24, 25 February 2024 (UTC)
References
have ever been questioned by other reputed scholars– this statement is outright false. I recently linked to 2 very strong criticisms of Lam Lay Yong and Ang Tian Se's book by some of the worlds' foremost experts on this topic, and can find at least another 1–2 if you like. Campbell and Douglas are two separate people who have not weighed in on this topic; Frank Swetz's paper is a game-of-telephone repetition of Lam's claims which in my opinion does not accurately convey them and is thus not a great source. If you want to litigate either the counting rods -> positional Indian numbers or the Shang numerals -> Brahmi numerals theories (which are entirely distinct and should not be conflated), can we make a new discussion topic about it though? It's getting a bit cramped in this one. – jacobolus (t) 20:53, 25 February 2024 (UTC)
The Brāhmī numerals, a ciphered additive decimal numeral system, developed in the Indian subcontinent probably sometime around the 3rd century BCE, spreading through the Maurya Empire. The ultimate origin of the Brāhmī numerals is unresolved, but they have been theorized to have evolved from Greek, Chinese, or most plausibly Egyptian numerals, or to have developed indigenously.
"Stephen Chrisomalis isn't a reputed authority in history of Mathematics"– He wrote his PhD and later this book, published by Cambridge University Press, about the comparative history of numeral systems. His book has hundreds of citations in the scholarly literature. Here are some reviews:
@ Jacobolus: Can you please explain why you chose to not use the short form cites I introduced into the "Origins" section? Paul August ☎ 15:22, 26 February 2024 (UTC)
I added a "dubious" flag to the sentence about "Shang numerals". This sentence is currently inaccurate and also not reflective of the linked sources. The referenced ~1400 BCE numerals found on oracle bones are an interesting topic in their own right and it would be great if Wikipedia had an article about them, but they are not "positional" in the sense meant by this article. To the best of my understanding, Lam's claim in her papers and book is that the feature from Chinese arithmetic adopted in India was the positional idea (from counting rods), not anything about written numerals from more than a millennium year earlier. I'll try to re-read Lam's papers; it's been a few months since I looked at them. – jacobolus (t) 15:49, 26 February 2024 (UTC)
"Can we please not make any substantive changes to this section which are undiscussed on the talk page."– jacobolus (t) 15:57, 26 February 2024 (UTC)
There are 2 different subjects which may be worth addressing in this article, but if either or both is to be addressed, it should be accurate, moderately complete, and well sourced.
We should be clear to separate these two topics, since they are mostly unrelated.
– jacobolus (t) 15:57, 26 February 2024 (UTC)
The Brāhmī script came to prominence in the mid third century BC [...]. Brāhmī script was probably derived from a Semitic prototype (Aramaic, South Semitic, or Phoenician), although many South Asian scholars still support the theory that the script was indigenously developed (Salomon 1996: 378–379). [...]
The question of the ultimate origin of the Brāhmī numerals – specifically, whether or not they constitute a case of independent invention, and if not, on which ancestor(s) they were modeled – is unresolved, and is made more complex by the politicization of the matter. [...]
One set of theories regarding the origin of the Brāhmī numerals derives them from existing representational systems used in South Asia. Borrowing from the letters of the Brāhmī script to create an alphabetic numeral-system, while once a popular theory, is not really sustainable (Prinsep 1838, Woepcke 1863, Indraji 1876, Datta and Singh 1962 [1935], Gokhale 1966, Verma 1971). [...] The derivation of the Brāhmī numerals from the Kharoṣṭhī letters is even more improbable [...]. Finally, a more recent set of theories derives the Brāhmī numerals from those of the Indus Valley civilization (Sen 1971, Kak 1994), but there are no examples of any writing from India between the latest Harappan inscriptions (around 1700 bc) and the first Brāhmī inscriptions [...].
If not derived from any South Asian system, the Brāhmī numerals could have developed independently. Woodruff (1994 [1909]: 53–60) speculated that both the Chinese and Brāhmī numerals derived from a hypothetical ancient set of cumulative tally signs for 1 through 9, which would then have spread to both China and India. [...]
Finally, a number of theories argue for a foreign origin of the Brāhmī numerals. Falk (1993: 175–176), noting structural and paleographic resemblances between the Brāhmī and the earliest Chinese (Chapter 8) numerical notations, argues for a Chinese origin. However, there is little evidence of contact between the two regions at this period, and the only paleographic similarity between the systems is the common use of horizontal strokes for 1, 2, and 3. It has occasionally been proposed that the Greek alphabetic numerals inspired the Brāhmī numerals, given their appearance following the Alexandrine period, the strong trade ties with the Greco-Iranian kingdoms of Parthia and Bactria, and the structural similarities between the two systems. However, the evidence for the “alphabeticity” of the Brāhmī numerals is weak at best (see the previous discussion), and there is no paleographic correspondence between the Greek and Brāhmī numerals.
It is most plausible that the Brāhmī numerals are derived from the Egyptian hieratic or demotic numerals. Burnell (1968 [1874]) argued for a demotic origin, while Bühler’s (1963 [1895]) much more prominent analysis argued for a hieratic origin. The three systems are structurally similar: they are all decimal, hybrid ciphered- additive/multiplicative-additive systems, and represent 200, 300, 2000, and 3000 by adding quasi-multiplicative strokes to the signs for 100 or 1000. There are resemblances in around one-third of the sign-forms, and very close resemblances for a few, such as 9 (Bühler 1963 [1895]: 115–119; Salomon 1995, 1998). While there was not tremendous Egypto-Indic cultural contact, Ptolemaic traders reached as far as the city of Muziris (modern Cranganore) on the Malabar Coast, and Aśoka is known to have sent Buddhist missionaries to Alexandria (Basham 1980: 187). Of the two Egyptian systems, I believe the demotic to be a more likely ancestor, because in the Ptolemaic period the use of hieratic numerals was very limited. Thus, although the demotic and Brāhmī systems differ in both the power at which multiplication is used and the direction of writing, I believe that a demotic origin should be adopted as a working hypothesis.
To help facilitate discussion, here is the current text under discussion:
According to some sources, this number system may have originated in Chinese Shang numerals (1200 BC), which was also a decimal positional numeral system.
Here is some new text proposed
here by
jacobolus:
According to some scholars, the Hindu–Arabic number system originated in Chinese counting rods, also a positional decimal number representation.
Paul August
☎ 16:00, 26 February 2024 (UTC)
Oops sorry, the proposed changes by jacobolus here were:
Some scholars have theorized that the positional concept in the Hindu–Arabic number system may have originated in Chinese counting rods, also a positional decimal number representation.
Paul August ☎ 16:11, 26 February 2024 (UTC)
According to some scholars, the Hindu–Arabic number system originated in Chinese rod numeral system, also a positional decimal number representation.
According to some sources, this number system may have originated in Chinese Shang numerals (1200 BC), which was also a decimal positional numeral system.
Some scholars have theorized that the positional concept in the Hindu–Arabic number system may have originated in Chinese counting rods, also a positional decimal number representation.
According to some scholars, the Hindu–Arabic number system originated in Chinese rod numeral system, also a positional decimal number representation.
Paul August ☎ 16:37, 26 February 2024 (UTC)
Here is what the sources currently cited in our article say:
Paul August ☎ 16:56, 26 February 2024 (UTC)
Lam Lay-Yong (1986, 1987, 1988) hypothesizes that the rod-numerals were ancestral to the Hindu positional numerals, because the rod-numerals are positional and decimal, and because there was considerable cultural contact between China and India in the sixth century AD, when positionality developed in India. Because the rod-numerals were used in computation and commerce, she asserts that it is inconceivable that the Indians would not have learned of this system from the Chinese, and, since it is so practical, they obviously would have borrowed it (Lam 1988: 104). Yet the Indian positional numeral-signs are those of the earlier Brāhmī numerals, not of the rod-numerals, and the rod-numerals have no zero- sign (whereas the Indian system does). Moreover, the rod-numerals have a quinary sub-base that the Indian numerals lack, and the rod-numerals are intraexponentially cumulative, whereas the Indian positional numerals are ciphered. No Indian texts of the period mention rod-numerals or any other Chinese numeration.
Adding two more quotes:
#Helaine Selin, 2008,
p. 198: "this fact, together with other evidence supports the thesis that the Hindu-Arabic system has its origin in the Chinese rod numeral system." this tertiary source is already mentioned above.
M.Bitton ( talk) 17:16, 26 February 2024 (UTC)
"Some scholars have theorized that the positional concept in the Hindu–Arabic number system may have originated in Chinese counting rods, also a positional decimal number representation."is nothing more than a temporary stopgap, and I do not consider it an acceptable medium-term solution to the problems with the current article. I would like to arrive at a mutually acceptable solution which we can build on without constant edit warring.
From what I can tell,
M.Bitton and I can agree that the
§ Origins section in its current place at the very top of this article, isn't really enough space to flesh out possible theories about the origin/early evolution of the Hindu–Arabic number system. To quote
M.Bitton: "I have no problem with mentioning everyone's view in details, but it has to be everyone (quotes and all) with no cherry picking. That comes at a price: we'd have to create a whole section about the Chinese origin."
Paul August, does moving/merging the 'origins' section seem okay with you also?
If so, I'd like to propose, as a way forward, that I will try to write more text in the next few days about both Lam's counting rod theory and also other competing theories, in a draft page somewhere (e.g. in user space), and then will merge it in as a subsection of the § History section after discussion here. M.Bitton: if you would like to write your own competing draft for that section, that would be fine, then we can bring them together and compare. As a separate subsection, I'll try to flesh out the section § Predecessors, possibly retitled to "Brāhmī numerals", and include a few sentences inline about theorized origins for the Brāhmī numerals there, including a theorized Chinese origin. – jacobolus (t) 18:13, 26 February 2024 (UTC)
"make the least change necessary to make the sentence about Chinese counting rods minimally accurate"– this is a temporary stop-gap, not a "proposal". But I would appreciate it if you would address my proposal in this section and answer the several direct questions I have asked you, instead of derailing yet another conversation by aggressive off-topic rhetorical flourishes. – jacobolus (t) 18:32, 26 February 2024 (UTC)
To be clear, my "proposed change" is what I consider the minimal change necessary to make this statement not outright false / grossly misleadingis also what you wrote after the bold edits. M.Bitton ( talk) 18:46, 26 February 2024 (UTC)
So you think that you've been perfectly polite here, and haven't being at all rude?That was a sincere question on my part. I think we could all be more polite. I was trying to see if you also thought you could have been more polite. Did you think my question was itself rude? I'm sorry if you did, that was certainly not my intension. But whatever, I will discuss all of this at length anywhere else but not here.
The earliest extant physical examples of decimal place value numerals are found in inscriptions from around the middle of the first millennium CE, written in scripts derived from Brāhmī. At present, the first such inscription known in an Indian source may be the one on a certain copper plate from Gujarat [... from] about 595 CE.7 Decimal place value numbers are also found in some inscriptions from Indianized cultures in Southeast Asia around the same time. [...] We can certainly infer that if the decimal place value system had been incorporated into epigraphic styles over much of South and Southeast Asia by this time, it must have originated quite a bit earlier.
footnote 7: However, it has been persuasively argued that this particular record is spurious and was actually inscribed at a later date; [...]
But we do not need to rely only on such inferences to push back the date of origin of decimal place value beyond the time of its earliest known inscriptional records. The content of some older textual sources includes hints about the writing of numbers that suggest a place value system, although of course the texts themselves are physically recorded only in much later manuscript copies. For example, a commentary from probably the fifth century CE on an ancient philosophical text, the famous Yoga-sūtra of Patañjali, employs the following simile about the superficial “changes of inherent characteristics” (Yoga-sūtra 3.13):
Just as a line in the hundreds place [means] a hundred, in the tens place ten, and one in the ones place, so one and the same woman is called mother, daughter, and sister [by different people].
Even earlier, the Buddhist philosopher Vasumitra in perhaps the first century CE used a similar analogy involving merchants’ counting pits, where clay markers were used to keep track of quantities in transactions. He says, “When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred.” Such statements clearly expect the audience to be familiar with the concept of numerical symbols representing different powers of ten according to their relative positions. Due to the brevity of their allusions and the ambiguity of their dates, however, they do not solidly establish the chronology of the development of this concept.
A different representation of decimal place value is revealed by a verbal notation called by medieval authors bhūta-saṅkhyā or “object-numbers,” here designated the “concrete number system.” Its function is to provide synonyms for ordinary number words such as “three” or “twelve.” Recall that in Sanskrit, at least after the Vedic period, even technical treatises were most often composed in verse. [...]
The concrete number system, to judge from all its extant examples, has apparently always been a place value system, representing large numbers with strings of words that stand for its individual digits or groups of digits, in order from the least significant to the most significant. Thus, if we encounter, say, the verbally expressed concrete number “Veda/tooth/moon,” we translate it as “four [for the four Vedic collections]/thirty-two/one,” and write it as 1324. These concrete numbers are not combined with number words signifying powers or multiples of ten, so their only unambiguous interpretation is as pure decimal place value. Hence the idea of a positional system for numerals must have been commonplace by the time the concrete number system was invented.
A firm upper bound for the date of this invention is attested by a Sanskrit text of the mid-third century CE, the Yavana-jātaka or “Greek horoscopy” of one Sphujidhvaja, which is a versified form of a translated Greek work on astrology. Some numbers in this text appear in concrete number format, as in its final verse [...] So it corresponds to [269/270 CE]. Evidently, then, positional decimal numerals were a familiar concept at least by the middle of the third century, at least to the audience for astronomical and astrological texts.
Exactly how and when the Indian decimal place value system first developed, and how and when a zero symbol was incorporated into it, remain mysterious. One plausible hypothesis about its origin links it to the symbols used on Chinese counting boards as early as the mid-first millennium BCE. These counting boards, like the Indian counting pits mentioned above, had a decimal place value structure: they were divided into columns representing successive powers of ten, with units on the right. Small rods were arranged in regular patterns in the columns of the board to designate numbers from 1 to 9, and a column left blank signified a zero. Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the same concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion.
We will see in section 3.3 that there are textual indications of a written symbol for zero in India even before the start of the Common Era, but it is not clear whether the symbol was part of place value notation at that time. The use of zero in decimal numerals and its characteristic round shape may have been reinforced by the round zero markers in sexagesimal place value numerals introduced to India in Greek astronomical and astrological texts.
Exactly how and when the Indian decimal place value system first developed, and how and when a zero symbol was incorporated into it, remain mysterious. One plausible hypothesis about its origin links it to the symbols used on Chinese counting boards as early as the mid-first millennium BCE. These counting boards, like the Indian counting pits mentioned above, had a decimal place value structure: they were divided into columns representing successive powers of ten, with units on the right. Small rods were arranged in regular patterns in the columns of the board to designate numbers from 1 to 9, and a column left blank signified a zero. Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travellers, or they may have developed the same concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion.
Not seeing any specific reasons claimed why this proposal wouldn't be an effective way of resolving this dispute, I am going to work on making those draft changes under the assumption that other editors will broadly support the reorganization once it's ready. If anyone would still like to see a different kind of organization, please make clear and concrete proposals now and we can discuss them: I'm not aiming for a fait accompli here. – jacobolus (t) 19:16, 26 February 2024 (UTC)