Talk:Plimpton 322/Archive 2

This is an archive of past discussions about Plimpton 322. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

Archive 2

To any new editors interested in resurrecting the horse (Mansfield and Wildberger)

The editors who have taken the lead in blocking any mention of Mansfield and Wildberger's rational trig paper in a leading journal were editing this page before that article is published and I think it is reasonable to conclude that they don't agree with its findings. Some new editors have come along and expressed surprise that M and W don't get a mention but haven't got involved. I did get involved and opened a Request for Comment which was subsequently closed - I was one against several - and when another more experienced editor Prokaryotes tried to open another RFC this was denied. It's quite possible that other potential editors who have heard of M and W's thesis will come to the wiki article and be surprised they are not mentioned and turn to the talk page. If any should read this and think M and W should be mentioned please state so here and add this page to your watchlist, in future there may be a majority for mentioning them.Nine-and-fifty swans (talk) 16:34, 27 October 2017 (UTC)

I commented yesterday, but it's way up the page, so I'm restating it here. I would support adding the "exact trig table" interpretation to the page, since it is no less speculative than some other theories and has been around since at least the 1960's -- see the quote from de Solla Price included in my comment above. I do not see that it is necessary to use MW as the source here -- de Solla Price may be the first and has also been around long enough for sufficient vetting.Barryriedsmith (talk) 17:41, 27 October 2017 (UTC)

I have no particular objection to adding something about this sourced to de Solla Price. It's crediting old ideas to the new paper that I object to. —David Eppstein (talk) 18:06, 27 October 2017 (UTC)

[COI warning] I'm also happy with crediting de Solla Price. But I expect there will be disagreement on the phrasing. Friberg (1981) calls this statement on the intention of the tablet a conjecture, twice (p 287 and 288). Friberg goes on to interpret this conjecture without any reference to trigonometry. So while I'm happy that de Solla Price should be credited with the conjecture, historically the conjecture has not been interpreted in this way. Daniel.mansfield (talk) 23:29, 29 October 2017 (UTC)

Exactly. There are now twenty threads on this page, starting a mere six weeks ago, beating this dead horse. E Eng 18:15, 27 October 2017 (UTC)

I'd agree to Barryriedsmith's suggestion. If we can agree on an exact trig reference that is great progress, every type of approach should be traced back to its source, ie the person who originated it. Nine-and-fifty swans (talk) 19:33, 27 October 2017 (UTC) de Solla Price is mentioned several times in MW. If MW are to be mentioned they should be mentioned in context of the history of trignometrical interpretations. The article as it stands at present gives the impression that Robson's interpretation has displaced all others, which is not the case. Leaving aside the question of whether MW contribute anything new their paper is significant for showing the persistence of the trignometrical interpretation. BTW I haven't seen their press release, only the article, which as far as I read it didn't strike me as 'hype' apart perhaps from the very direct titleNine-and-fifty swans (talk) 20:49, 27 October 2017 (UTC)

@Nine-and-fifty swans: So go ahead! I'm one of those who object to the way some people try to protect articles from even mentioning certain interesting things, citing various Wikipedia policies. Eric Kvaalen (talk) 11:00, 15 February 2018 (UTC)

Eric, I am glad you agree with me, but ther e is no point in making changes, they would only be reverted. In fact quite a few people take the same few as us but nobody stays around to make a voting bloc. Still the good thing is that this page is a perfect example of one of the glaring defects of wikipedia and can be used to demonstrate this. Nine-and-fifty swans (talk) 15:58, 7 March 2018 (UTC)

The interpretation of Donald L. Voils (1975 et seq.)

The interpretation of Donald L. Voils (b. 1934) is worthy of note as being historically intermediate between that of Bruins (1949) and Robson (2001) (at current writing (September, 2010), the main article Plimpton 322 references, but otherwise does not mention, Bruins (1949), notwithstanding that Robson graciously acknowledges it as containing the thesis in Robson (2001) in a different guise). Robson expresses interest in, but ignorance of, Voil's interpretation based on passages in Buck (1980) and indeed one passage in particular resonates with the account of Robson's own thesis as described in the main article Plimpton 322.

Voils, then at the University of Wisconsin at Oshkosh, spoke at the April, 1975 meeting of the Wisconsin Section of the Mathematical Association of America (MAA) on the question Is the Plimpton 322 a Cuneiform tablet dealing with Pythagorean triples?, as reported in the issue of the American Mathematical Monthly for December that year (p. 1043). We may follow Robson (2001) in picking up the story from Buck (1980), p. 344, recalling that Buck was on the faculty at the University of Wisconsin at Madison:

Voils adds to this suggestion of Bruins the observation that the numbers A are exactly the results obtained at the end of the second step in the solution algorithm, (d/2)2, applied to an igi-igibi problem whose solution is x and xR. Furthermore, the numbers B and C can be used to produce other problems of the same type but having the same intermediate results in the solution algorithm. Thus Voils proposes that the Plimpton tablet has nothing to do with Pythagorean triplets or trigonometry but, instead, is a pedagogical tool intended to help a mathematics teacher of the period make up a large number of igi-igibi quadratic equation exercises having known solutions and intermediate solution steps that are easily checked [7].

Unfortunately, Buck's reference [7], apparently an item by Voils slated to appear in Historia Mathematica, was never published. Cooke (2005), pp. 163-164, in an extensive discussion of Plimpton 322, gives a sympathetic account of Voil's interpretation, but again based only on Buck (1980). Voils recalls the submission, written after taking a class in the history of Babylonian mathematics at the University of Wisconsin at Madison, was rejected on some technical ground and is now uncertain whether any copy survives, as he changed interests into computer science at about the same time. The class was taught by William D. Stahlman (1923-1975), who had taken his doctorate at Brown University under Otto Neugebauer.

This quotation from Buck (1980) also serves to remind us that, while this paper does discuss a trigonometic interpretation of Plimpton 322, as noted in the main article Plimpton 322, it was by no means confined to it, nor did it endorse it. Rather, in the light of Robson (2001), Buck's contribution seems to show an uncanny prescience of the limitations of the detective genre (Buck (1980), p. 345):

Unlike Doyle's stories, this has no final resolution. Any of these reconstructions, if correct, throws light upon the degree of sophistication of the Babylonian mathematician and breathes life into what was otherwise dull arithmetic. — Preceding unsigned comment added by 130.194.170.146 (talk) 00:14, 15 September 2010 (UTC)

Mathematical underpinnings and reconciliation of interpretations

It is a usual and customary part of the scholarly apparatus in the discussion of multiple interpretations to consider their underpinnings and possible reconciliation qua interpretations. Robson (2001) points the way and sets the standard in volunteering that the thesis being advanced already appeared in Bruins (1949) in a different guise. What is meant here by in a different guise, is that Robson recognises a broad ressemblence between the two interpretations (even if Bruins (1949) might not achieve the same elect state of perfection and grace accorded Robson (2001) in the main article Plimpton 322). But naturally there is no suggestion, nor should readers of Robson (2001) infer, that, say, Bruins thought the same way as Robson or would agree with this assessment. Examination of mathematical underpinnings and possible reconciliation tells us only how interpretations stand one to another, but is neutral on what is being interpreted.

Plimpton 322 has often been taken as the basis of claims that the Babylonians had some early acquaintance with a Pythagorean or diagonal rule, in keeping with the thesis ascribed to Neugebauer in the main article Plimpton 322. On the other hand, the thesis attributed there to Robson, but advanced previously in Bruins (1949) in a different guise, in recontextualizing Plimpton 322 within the corpus of Babylonian mathematics, removes it from this supporting role. It might be helpful to indicate (as the main article Plimpton 322 does not) that the claim to early acquaintance has a firmer, and certainly an independent, foundation in Db₂-146 = IM67118, a tablet from Eshnunna from about -1775, as discussed, for example, by Høyrup (2002). The tablet works a computation of the sides of a rectangle given its diagonal and area. The working prefigures a dissection of a square on the diagonal into a ring of four congruent right triangles surrounding a square of side the difference between the sides of the proposed rectangle. The general form of this dissection yields the Pythagorean rule on rearrangement of the pieces, although the working on the tablet skirts this observation (compare also the illustrated discussion in Friberg (2007), pp. 205-207). But, for good measure, the tablet also runs a check on the working by applying the Pythagorean rule to the sides to get back to the prescribed diagonal. (An updated listing of Babylonian appearances of the Pythagorean rule is given in Friberg (2007), pp. 449--451, building on an earlier listing by Peter Damerow as well as Høyrup (2002).)

The theses attributed to Neugebauer and to Robson are linked mathematically by two standard, age old tricks, taking the semi-sum (average) and semi-difference of two quantities coupled with difference of squares (notice that this internal link immediately gives a problem with Wikipedia's policy on references and sources, as this entry is currently flagged as open to challenge and removal). For, suppose that l, s and d stand in the Pythagorean relation l² + s² = d², so that l² = d² - s². Application of difference of squares then yields l² = (d + s)(d -s). Thus, taking x = (d + s)/l, we also have 1/x = (d - s)/l, and can then recover d and s from x and 1/x by the trick of taking the semi-sum and semi-difference: x + 1/x = 2d/l, x - 1/x = 2s/l. Consequently, we have solved the quadratic equation x - 1/x = c, where c = 2s/l, and for that matter also the quadratic equation x + 1/x = k, where k = 2d/l. The algebra is reversible, so starting from solutions to these quadratics, we can recover three quantities standing in the Pythagorean relation.

This mathematical exercise only tells us how the two theses are related (as the main article Plimpton 322 does not), not what skills the Babylonians possessed, still less what the purpose of the tablet might have been. However, as it happens, it is a commonplace of accounts of Babylonian mathematics that it exhibits a propensity to work with the semi-sum and semi-difference of a pair of quantities, as noted, for example, in Cooke (2005). But the mathematically careful account there misses the trick with the difference of squares, so fails to see that whenever solutions of certain quadratics are present so, too, are Pythagorean triads, and vice versa, although such fraility is itself a corrective in historical analysis. Nevertheless, turing back to Bruins (1958), we find the acknowledged progenitor of the thesis in Robson (2001) in a different guise reprising much of this mathematical exercise, with the claim that the approach was used by the Babylonians (Bruins (1958) is not cited in Robson (2001): it is a minor publication easily overlooked on account of its location; but that it appears in a popular journal makes it more accessible to a general reader in the tradition of Robson (2002)):

We begin by remarking that if we put one of the sides I of a right-angled triangle equal to unity, the Pythagorean relation between the remaining sides d and b is d² - b² = (d + b)(d - b) = 1. If therefore we set d + b = λ, then d - b = 1/λ. Now a reciprocal value can be calculated in Babylonian Mathematics only for numbers containing no prime factors other than 2, 3 and 5, i.e. for so-called regular numbers. Extensive tables of such reciprocals were calculated by the Babylonian mathematicians, and therefore by reference to such a table the numbers d = ½(λ+1/λ), b = ½(λ - 1/λ), l = 1, satisfying the Pythagorean relation, could be simply computed. Are there any indications that the Babylonians used this relation? Yes, there are. — Preceding unsigned comment added by 130.194.170.146 (talk) 06:59, 20 September 2010 (UTC)

References

Bruins, Evert M. (1949), "On Plimpton 322, Pythagorean numbers in Babylonian mathematics", Koninklijke Nederlandse Akademie van Wetenschappen Proceedings 52: 629–632 .
Bruins, Evert M. (1951), "Pythagorean triads in Babylonian mathematics: The errors on Plimpton 322", Sumer 11: 117–121 .
Bruins, Evert M. (1958), "Pythagoreans triads in Babylonian mathematics", Mathematical Gazette, 41: 25-28, http://www.jstor.org/stable/3611533
Buck, R. Creighton (1980), "Sherlock Holmes in Babylon", American Mathematical Monthly (Mathematical Association of America) 87 (5): 335–345, doi:10.2307/2321200, http://jstor.org/stable/2321200
Cooke, Roger L. (2005), The History of Mathematics: A Brief Course, 2nd ed., Wiley, pp. 159-164, ISBN 0-471-44459-6.
Conway, John H.; Guy, Richard K. (1996), The Book of Numbers, Copernicus, pp. 172–176, ISBN 0-387-97993-X
Friberg, Jöran (2007), A Remarkable Collection of Babylonian Mathematical Texts (Sources and studies in the history of mathematics and physical sciences; Volume 1 of Manuscripts in the Schøyen Collection: Cuneiform texts; Volume 1 of Manuscripts in the Schøyen Collection, Springer. ISBN 0-387-34543-4, ISBN 978-0-387-34543-7
Høyrup, Jens (2002), Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and its Kin (Sources and studies in the history of mathematics and physical sciences Sources and Studies in the History of Mathematics and The Graduate Texts in Mathematics), Springer, pp. 257-261, ISBN 0-387-95303-5, ISBN 978-0-387-95303-8
Knorr, Wilbur R. (1998), ""Rational Diameters" and the discovery of incommensurability", American Mathematical Monthly (Mathematical Association of America) 105 (5): 421-429, http://www.jstor.org/stable/3109803
Neugebauer, O.; Sachs, A. J. (1945), Mathematical Cuneiform Texts, American Oriental Series, 29, New Haven: American Oriental Society and the American Schools of Oriental Research .
Neugebauer, Otto (1969) [1957], The Exact Sciences in Antiquity (2 ed.), Dover Publications, ISBN 978-048622332-2
Robson, Eleanor (2001), "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math. 28 (3): 167–206, doi:10.1006/hmat.2001.2317, MR1849797 .
Robson, Eleanor (2002), "Words and pictures: new light on Plimpton 322", American Mathematical Monthly (Mathematical Association of America) 109 (2): 105–120, doi:10.2307/2695324, MR1903149, http://mathdl.maa.org/images/upload_library/22/Ford/Robson105-120.pdf .
Voils, Donald L. (1975), "Is the Plimpton 322 a Cuneiform tablet dealing with Pythagorean triples?". Talk listed by title only in Notice of April Meeting of the Wisconsin Section, American Mathematical Monthly (Mathematical Association of America) 82 (10): 1043, http://www.jstor.org/stable/2318291, allusion in Buck (1980). — Preceding unsigned comment added by 130.194.170.146 (talk) 07:09, 15 September 2010 (UTC)

Something wrong with this maths

The article says: If p and q are two coprime numbers, then $\scriptstyle (p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})$ form a Pythagorean triple, and all Pythagorean triples can be formed in this way.

Take the triangle with sides (12, 16, 20), which is both a Pythagorean Triple and is of the form $\scriptstyle (p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})$ where p = 4 and q = 2 which are not coprime. Therefore, this particular Pythagorean Triple cannot be formed with p, q both coprime and therefore the claim that all Pythagorean triples can be formed in this way is not true. Cottonshirt τ 11:30, 15 September 2012 (UTC)

The formula is valid for all primitive triples, ones that are not integer multiples of some smaller triple. Your example is a multiple of the (3,4,5) triple. I added a clarification. — Preceding unsigned comment added by David Eppstein (talk • contribs) 14:47, 15 September 2012 (UTC)

Superposition

Superposition exists if a positive integer right triangle, PIRT, is specified in an elementary fashion, (short side)^2+(long side)^2=(diagonal)^2. And use of this form ensures that the function (d/l)^2 will always lie between 1 and 2, the first two regular sexagesimal numbers.Thomas L. O"Donnell (talk) 02:17, 7 September 2018 (UTC)

Problems with the "problem list" theory

I've just read the 2002 paper by Eleanor Robson, after reading this Wikipedia article. There are a couple things that don't make sense to me. Supposedly this is a list of "teacher's notes" for solving $x-1/x=c$ , with the second and third columns being $(x-1/x)/2$ and $(x+1/x)/2$ respectively. But that's usually not the case. Take a look at the first row. We should have $(x-1/x)/2=119/120$ and $(x+1/x)/2=169/120,$ rather than just 119 and 169. Obviously if you subtract 119 from 169 you don't get the reciprocal of their sum. So how does this help the teacher in any way?

(Row 11 does work if you interpret 45 as 45/60 and 75 as 75/60. But that's the only row that can be made to work by "shifting the sexigesimal point".)

Secondly, if this is just a series of exercises, then why are the angles of the triangles spaced out by about 1°? There are many other examples that could have been used, based on similarly sized sexigesimal numbers, but they weren't used.

Note added later: I checked, and actually if you limit the range to the range of the tablet and use for X numerators and denominators that are regular numbers up to 125, then the Plimpton 322 tablet gives all of them except one, namely X=125/64. This should go between rows 11 and 12. I think it's pretty impressive. I used a spreadsheet, but the author didn't have Excel. Eric Kvaalen (talk) 21:13, 25 March 2018 (UTC)

And if it's a list of $x-1/x=c$ problems, why doesn't it give the answer (x)?

Have any of these questions been raised by the proponents of other theories, such as the trigonometry theory?

Eric Kvaalen (talk) 17:34, 25 March 2018 (UTC)

You are absolutely right that Columns 2 and 3 of the tablet do not contain

(x-1/x)/2

and

(x+1/x)/2

. A careful reading of Robson shows that she is not actually claiming that they do. According to the reciprocal-pair proposal there is an additional process that is carried out to produce the numbers in Columns 2 and 3 involving the successive removal of common regular factors. At the time you made your comment, the Wikipedia article did not mention this extra step, but it has now been added. The formulas the article gives for the numbers in Columns 2 and 3 now include an additional factor, a, that accounts for the effect of this additional process.

One other thing is that Robson is not, as often stated, claiming that the purpose of the tablet was to provide teacher's notes for solving x − 1/x = c type problems. Among her actual conclusions are (1) that the tablet most likely is some kind of teacher's notes, (2) that the numbers were produced by a process inspired by the x − 1/x = c problem, (3) that the problems being set by the tablet and the problems used to generate the tablet might not be the same, and (4) that the problems being set are uncertain but are likely "some sort of right-triangle problems". (See page 202 of "Neither Sherlock Holmes nor Babylon".) As an aside: from the description in Buck's article, it appears the Voils may actually have made the claim often attributed to Robson. Since Voils' paper was never published, this seems impossible to confirm now.

In answer to your question about why the tablet doesn't give the "answer" x, two comments: (1) if the problems set by the tablet aren't quadratic-equation problems, then x isn't actually the answer, and (2) Robson, in fact, does believe that the broken-off portion of the tablet did contain x and 1/x (not because they provide the answers but because of their role in generating the numbers on the tablet). Other scholars who subscribe to the reciprocal-pair theory, however, believe that it probably contained what our article calls v₁ and v₄ rather than x and 1/x.

Finally, observations similar to yours about restricting the numerators and denominators to regular numbers up to 125 have been made in quite a few published papers on the tablet, including those of de Solla Price and Friberg. A small section about this has now been added to the Wikipedia article.Will Orrick (talk) 15:48, 4 February 2019 (UTC)

For Row15 we get a sample of what the Teacher is teaching. p/q=7/2. And we need to realize that this p/q is in the complementary range, 45deg to 60deg, 2.4142<(p/q)<3.7321, and therefore we want the alternate set (which would be in the 45deg to 30deg range). Using p/q=7/2, our "reduced triple" alternate set becomes:

 a=1   b=((p/q)-(q/p))/2  c=((p/q)+(q/p))/2  with p/q=7/2
 a=1   b=((7/2)-(2/7))/2  c=((7/2)+(2/7))/2  and solving:
 a=1   b=45/28  c=53/28  and multiplying all sides of this "reduced triple" by 28:
 a=28  b=45  c=53, a primitive PIRT  Multiply by 2 and get:
 a=56  b=90 c=106  not a primitive

So this is the set to use when the PIRT is primitive and a is even, b is odd. And the Teacher would have pointed out that (b) should be a "regular" number, and it is.

Hope this helps. — Preceding unsigned comment added by 336sunny (talk • contribs) 17:56, 3 July 2018 (UTC)

The derivation for Row15 looks wrong because a p of (7) is not regular. But it's right because our test fundamentally is that "the long side must be regular". And the long side is now (p^2-q^2) which is (7^2-2^2)=(49-4)=45 which is indeed regular. Which brings up a very important point: The use of the short (s), and long (l) designations for the sides of the right triangle put it in superposition. Each one of two solutions for p&q pairs and for reduced triples gives exactly the same result. The orientation of the right triangle remains undefined until (a) and (b) labels are attached. Thomas L. O"Donnell (talk) 14:14, 9 August 2018 (UTC)

@Thomas: Well, actually that doesn't seem to address my questions. (I see from something you wrote below that PIRT means "positive integer right triangle". Does anyone call them that, rather than Pythagorean triangle?) Eric Kvaalen (talk) 22:38, 8 September 2018 (UTC)

1. And if it's a list of $x-1/x=c$ problems, why doesn't it give the answer (x)?

The answer is the beginning, x=p/q, the starting point. To work backwards, the complete reduced-triple equation must be used. It is at https://wiki.riteme.site/wiki/Number_theory#Dawn_of_arithmetic

The equation is (s/l)^2 + (l/l)^2 = (d/l)^2. (The left-hand column of P322). Dividing Row11 thru by 15, and dividing Row15 thru by 2 will get them back to primitive PIRTs. Substitute for (s/l) and solve for (x) in the primitive reduced-triple gets p/q for each of the 15 rows. (The resulting quadratic is solved by completing the square).

2. X numerators and denominators that are regular numbers up to 125.

Between 1 and 125 there are only 11 odd and 26 even regular numbers. How about that?

3. why are the angles of the triangles spaced out by about 1°?

Would you believe happenstance? With requirements of p and q beween 1 and 125 (nice regular numbers) and long sides a regular number and 15 PIRTS total, what fell out, put in (d/l)^2 decending order, is what we got. (With a few extras).

4. Who uses PIRT?

Everyone serious about Akkadian mathematics. Thomas L. O"Donnell (talk) 05:10, 11 September 2018 (UTC)

5.Plimpton 322 tablet gives all of them except one, namely X=125/64. This should go between rows 11 and 12.

That PIRT is s=11529, i=16000, d=19721.
Note 5 of the article says "See also Joyce, David E. (1995), Plimpton 322". Dr. Joyce in his paper asks the same question.
The answer will be posted on my (Talk) page.

Thomas L. O"Donnell (talk) 15:48, 24 September 2018 (UTC)

If you want to enter the debate about what this tablet could have been used for, you need to write up your ideas as an academic paper and get them published. Here, we can only report what the consensus of publications on this subject is, not our own speculations. —David Eppstein (talk) 05:32, 11 September 2018 (UTC)

That would seem very true for an edit of the article. TALK sounds like a discussion that might include scholarly speculation. The idea of superposition came from just such a discussion. David, aren't you being too restrictive?Thomas L. O"Donnell (talk) 11:02, 11 September 2018 (UTC)

To be honest, David Eppstein is being exactly the right amount of restrictive. Articles are written based on what is verifiable from reliable sources. Talk pages are used for discussing what should go in the article, not the subject of the article itself. While discussions about particular sources are fair game for talk pages, discussion about the subject itself isn't so much (unless it's about how to present some information). — Sasuke Sarutobi (push to talk) 12:59, 11 September 2018 (UTC)

@Will Orrick: Thank you for your response to me, and for the extensive work you have recently done on the article! Eric Kvaalen (talk) 08:27, 6 February 2019 (UTC)

Orphan sentence from edit war?

The article finishes its description of the content of P322 with the sentence

The sixty sexagesimal entries are exact, no truncations or rounding off

This is a peculiar description, since the numbers are rational. The use the of the words 'sexagesimal' and 'exact' is similar to the phrase 'exact sexagesimal trigonometry', from the title of Mansfield and Wildberger's controversial paper. 'Truncation' and 'rounding' are particular bugbears of Wildberger in his ideas on the non-existence of real numbers. I would suggest removal, not sure if anything is needed in its place; perhaps just a comment that the numbers are rational? — Preceding unsigned comment added by 129.127.37.121 (talk) 01:06, 16 July 2018 (UTC)

This statement was here in 2016, before Mansfield and Wildberger's controversial paper. This statement is accurate, and it's not trivial: even rational number could be rounded. Babylonians did rounded their numbers elsewhere (like in 1/59) but not in this table. There is no need to remove it. Alexei Kopylov (talk) 04:02, 16 July 2018 (UTC)

One could perhaps rewrite the sentence as

The entries should be regarded as exact values, that is rational numbers, rather than as truncated or rounded sexagesimal expansions approximating irrational real numbers.

129.127.37.121 (talk) 08:50, 23 July 2018 (UTC)

No, that would be incorrect. The distinction is not between rational and irrational, but between fractions with regular number denominators having terminating sexagesimal representations and the other rational numbers like 1/7 that don't. As far as I know the only ancient people to discover irrationality were the Greeks. We don't have any evidence that the Babylonians knew or cared whether numbers were irrational, but they definitely cared about regularity. —David Eppstein (talk) 15:58, 23 July 2018 (UTC)

Fair enough. Thanks for correcting my misunderstanding. 129.127.37.121 (talk) 03:56, 31 July 2018 (UTC)

Era style

"Either convention may be appropriate for use in Wikipedia articles depending on the article context." MOS:ERA The context isn't Christian. Rupert Loup (talk) 00:47, 7 November 2019 (UTC)

That makes sense, I suppose. But have some patience. Let's wait long enough for other editors here to weigh in, if they have opinions on such weighty matters as date formatting. —David Eppstein (talk) 07:40, 8 November 2019 (UTC)

I agree the context isn't Christian, nor in fact anything else that would augur for one or the other style; therefore we stick with the current style (see MOS:RETAIN). I'm sick of people wasting others' time and attention on era-style churning. EEng 07:48, 8 November 2019 (UTC)
I have to admit that the distinction between "before Christ" and "before the Christian era" makes little difference to me. You can say that one of those words is something else that starts with C, current or calendrical or something like that, but I don't think anyone really believes it. If they really wanted to avoid Christendom they would use AUC or SH or AM or one of those other calendars that wasn't numbered from the birth of Christ. In the meantime I prefer to use BC when I'm writing dates (and am not constrained by preexisting styles) because it's convenient, understood by everyone, has fewer characters, and because I don't care to be a snob about other people's religions. So I have a slight preference for retaining the existing style, but really it's more about avoiding pointless churn than about which is better. —David Eppstein (talk) 08:13, 8 November 2019 (UTC)
Exactly. (I actually think BCE stands for "Before Common Era" but even then it doesn't avoid the fact that this era is defined by ... you know.) E Eng 09:44, 8 November 2019 (UTC)
- This is a clear example of systemic bias in Wikipedia. Anno Domini is a Christian related term and Common Era is a culturally neutral term, pushing a Christian term in an unrelated article fails both WP:ERA and WP:NPOV. Rupert Loup (talk) 17:14, 8 November 2019 (UTC)

I agree with E Eng, this churn is a waste of time. The difference between BC and BCE is purely cosmetic since they both refer to the birth of Christ. If you want to avoid such a reference, use the Chinese calendar! Meanwhile, WP:RETAIN should be the policy that holds. --Bill Cherowitzo (talk) 18:57, 8 November 2019 (UTC)

First Wikipedia is not a democracy, consensus isn't meet by voting, second this is not a Chinese related article neither nor the Chinese calendar is mentioned in WP:MOS, so that is irrelevant. BCE doesn't refer to Christ, it doesn't stand as "before the Christian era". Wikipedia tries to achive neutrality in its articles, CE is the more neutral tone in relation with this content so stop POV pushing. Rupert Loup (talk) 19:52, 8 November 2019 (UTC)

Actually I think there is a significantly greater amount of pov-pushing in the notion that the "common era" began with Christ, as represented by the "BCE" expansion, than there is in the more factual and obvious-to-everyone notion that our calendar system is dated from the birth of Christ, true for both abbreviations but expressed more clearly in the "BC" expansion. —David Eppstein (talk) 21:53, 8 November 2019 (UTC)

Its article lead states that "In the later 20th century, the use of CE and BCE was popularized in academic and scientific publications as a culturally neutral term. It is also used by some authors and publishers who wish to emphasize sensitivity to non-Christians by not explicitly referencing Jesus as "Christ" and Dominus ("Lord") through use of the abbreviation[c] "AD"." It's more neutral than Anno Domini and as I said we should try to be the more neutral posible according with WP:NPOV. Since you, an Admin, keep trying to push your view I going to report this. Rupert Loup (talk) 23:40, 8 November 2019 (UTC)

I don't see the need for the POV tag on the article. One editor does not like the use of BC. That's why we have WP:RETAIN. This isn't a POV issue unless we're going to accept that any use of BC is a POV issue, which we have not done in the past. Meters (talk) 00:49, 9 November 2019 (UTC)

WP:ERA already specify when BC is justified as I already pointed out. The consensus on it is already there in that policy. Rupert Loup (talk) 04:46, 9 November 2019 (UTC)

(outdent) Agree with EEng; nothing compelling about the context merits overriding WP:RETAIN. OhNoitsJamie ^Talk 01:42, 9 November 2019 (UTC)

Of course you agree with me. It's in the rules. E Eng 02:46, 9 November 2019 (UTC)

My preference is for BCE. Yeah, yeah, there's RETAIN and ERA, but there's also IAR. "BC" is based on a value-laden title ("Christ") that should only be used in articles on/about Christianity—if it has to be used at all. "BCE", on the other hand, stands for "Before Common Era" and is neutral. Woodroar (talk) 02:51, 9 November 2019 (UTC)
In what sense is the implication that "common means Christian" in any way more neutral? —David Eppstein (talk) 05:24, 9 November 2019 (UTC)
Where is that stated? Who are you quoting? Rupert Loup (talk) 05:39, 9 November 2019 (UTC)
I'm not quoting anyone. The punctuation was just a way of setting a thought aside from its framing sentence. And that thought might not have been stated explicitly, but it's the obvious implication of a calendar name that continues to be dated from the birth of Christ but calls the time since that birth the common era. —David Eppstein (talk) 05:52, 9 November 2019 (UTC)
That is not stated in their article, is not their current usage according to the sourced content there. Right now CE is being used by several publications to maintain neutrality. So please explain how AD is related to this article. Give arguments, stop reverting my edits without explaining yourself and stop with the accusations of bad faith. Rupert Loup (talk) 06:40, 9 November 2019 (UTC)

Because "Common" is a different word than "Christ" and "The Lord"? We can all acknowledge that BCE/CE is a derivative of BC/AD, but it's a neutral derivative because it doesn't use non-neutral terms. Similarly, we can all acknowledge that Islamic honorifics exist without requiring that editors add "PBUH" after Muhammad. Woodroar (talk) 13:55, 9 November 2019 (UTC)
Using generic terms for non-generic topics is othering and is the opposite of neutral. It implies that anything else is special — in this case that Christianity is common and that everything else is uncommon. —David Eppstein (talk) 19:14, 9 November 2019 (UTC)
I don't follow. It's discrimination to prefer a neutral term over one that effectively says "I have faith in Jesus and the Christian god"? Is that what you're saying? Woodroar (talk) 19:39, 9 November 2019 (UTC)
What I am saying is that using the name "common era" to name the time period of Christianity (from Christ's birth until now) is an implicit endorsement of the view that Christianity is common and that everything else is uncommon, and is therefore unsuccessful as a way to make the naming scheme secular. As for the rest of your comment, I don't see how using the term "BC" denotes faith in anything, as it is merely saying "before this particular person existed". If you are referring to the technical issue of whether Jesus=Christ and saying that using the word Christ implies belief in that identity, then I suppose you might have some point, but I think that the distinction between Jesus and Christ is irrelevant to most people. For Christians they are two ways of referring to a single person / aspect of the trinity, and for everyone else they are two names for that person worshipped by the Christians. —David Eppstein (talk) 20:03, 9 November 2019 (UTC)
The "using the word Christ implies belief in that identity" issue is exactly the problem. He—or they, or "that person" as you said—is not the "Christ" or "Lord" to ~70% of the population on Earth. Yet if someone adds "BC" or "AD" to an article completely unrelated to Christianity, then that's what everyone has to use? Seems pretty ridiculous to me. Woodroar (talk) 20:31, 9 November 2019 (UTC)
Again, AD is off-topic here. And my impression is that to that 70% of the population, the word "Christ" is just a name for that guy, you know the one, not any kind of shibboleth of secret faith. —David Eppstein (talk) 22:03, 9 November 2019 (UTC)
That still doesn't explain how this article is related to AD/BC system and why we should use the Christian term instead of a neutral term. Rupert Loup (talk) 23:07, 9 November 2019 (UTC)
I support keeping BC/AD per WP:RETAIN. Paul August ☎ 04:01, 9 November 2019 (UTC)
This is not a democracy and WP:RETAIN is not a policy about the Era style, WP:ERAS is. Give a reason in how AD is related to this article please. Rupert Loup (talk) 05:39, 9 November 2019 (UTC)
Maybe you should actually read WP:ERAS yourself. Do not change the established era style in an article unless there are reasons specific to its content. Seek consensus on the talk page first, applying Wikipedia:Manual of Style § Retaining existing styles. Open the discussion under a subhead that uses the word "era". Briefly state why the style is inappropriate for the article in question. You edit warred the change, You opened the thread, There's no consensus to make the change. Meters (talk) 05:55, 9 November 2019 (UTC)
- I already give the specific reason to the change here, no conern was raised. The admin kept reverting my edits without engaign in the talk page, I waited for their comment an entire day. You still are not explaining how AD are related to this article. Rupert Loup (talk) 06:21, 9 November 2019 (UTC)
WP:ERA explicitly refers to WP:RETAIN. Twice, in fact. The project has a very low tolerance for this kind of time-wasting, because so very much time has been wasted on such things over the years, for no good reason. Unless there's some strong reason based on article subject matter to prefer one or the other era style, either is acceptable. Your personal hobbyhorse isn't a reason to change. E Eng 06:01, 9 November 2019 (UTC)
WP:ERA refers to MOS:VAR not WP:RETAIN, this is not about an English variety. I'm not who is pushing with any basis for a "personal hobbyhorse", you are who is stating opinion as facts contradicting what the AD and BC articles state. The context isn't Christian and there is no reason what so ever to maintain it. Explain how this AD is related to this article, please. Rupert Loup (talk) 06:32, 9 November 2019 (UTC)
Sorry, you're right: it's MOS:VAR aka MOS:STYLERET. I don't need to explain how AD is related to the article; you need to explain how CE is related. And as already explained, and now explained for the last time as I won't be responding further unless you say something new, unless there's some strong reason based on article subject matter to prefer one or the other era style, either is acceptable, and given that the article already exists we stick with whatever's there. Period. End of story. Give it up. E Eng 06:43, 9 November 2019 (UTC)

You have to, if you don't you are not presenting an argument, just voting. And Wikipedia is not a Democracy. The strong reason is that neutrality is one of the five pillars of Wikipedia, CE is currently used as a neutral term. AD is not. That what their articles state, this is not a Christian related article so we should strive for the more neutral tone on the issue. Rupert Loup (talk) 07:00, 9 November 2019 (UTC)

How did CE and AD come into this discussion? They are not used in the article, so whether being written in Latin is more neutral than being written in English (or whatever other similar argument one might want to make about them) is irrelevant. —David Eppstein (talk) 07:12, 9 November 2019 (UTC)

I'm refering to the article Anno Domini, the article that I presented before, you are not even reading what I'm writing. Rupert Loup (talk) 07:19, 9 November 2019 (UTC)

I'm aware of what AD stands for. But it's irrelevant to this discussion. Why are you suddenly bringing it up? —David Eppstein (talk) 07:29, 9 November 2019 (UTC)

Really, DE, stop responding to him. Get to bed early. E Eng 07:33, 9 November 2019 (UTC)

Oh, fine. It's not even that early any more. —David Eppstein (talk) 08:14, 9 November 2019 (UTC)

"The terms anno Domini[note 1][1][2] (AD) and before Christ[note 2][3][4][5] (BC) are used to label or number years in the Julian and Gregorian calendars. The term anno Domini is Medieval Latin and means "in the year of the Lord",[6] but is often presented using "our Lord" instead of "the Lord",[7][8] taken from the full original phrase "anno Domini nostri Jesu Christi", which translates to "in the year of our Lord Jesus Christ". " Rupert Loup (talk) 08:23, 9 November 2019 (UTC)

Article has many issues

The article is too technical, too lengthly-wordy-opinionated, relies too much on a handful of sources, and still omits roughly the past two decades of studies. For starters, a specific paper is continuously removed from mention, even though it is frequently discussed and won The Best Writing on Mathematics 2018 by the European Mathematical Society. Someone even moved the more recent talk page entries to the archive, why? prokaryotes (talk) 10:20, 22 April 2019 (UTC)

Hello @Prokaryotes:: I'm asking myself the same questions. And I do not find a valid answer... I do not understand why an article published in a first-class scientific journal on the subject can be hidden.--Ferran Mir (talk) 10:34, 22 April 2019 (UTC)

I'm responsible for many of the recent edits that increased the length and technical level of the article, and I'm aware that those are problems. Rather than give my own opinions about what should be done to address those issues, I'd be happy to hear what other people have to say first.

One thing I'd like to mention is that part of the reason for the added length was that I tried to include multiple points of view, so I'm especially curious about where you see the current article as too opinionated. The article now includes sources up to 2011, and makes extensive use of the post-2002 sources, but I'd be very interested to hear about other recent sources. The only two that I know about are Rudolf Hajossy's paper, "Plimpton 322: a universal cuneiform table for Old Babylonian mathematicians, builders, surveyors and teachers" and Mansfield and Wildberger's paper. There is also the preprint by Abdulaziz that the Wikipedia article cites but never actually uses. Apart from the rational trigonometry hypothesis in the Mansfield and Wildberger article, all three of these seem to make rather technical criticisms of previous work relating to interpretation of errors, selection criteria, and the feasibility of exact square-root calculations. I have to admit to not thoroughly understanding those arguments, but I'm not sure how much they change the big picture. If anyone has some insight, that would be very helpful. As for rational trigonometry, I was not involved in those discussions, and don't want to weigh in right now. I was not aware that the Mansfield and Wildberger paper had been frequently discussed, or that it had won an award. I will try to look for some of that discussion, but if you have references handy, that would also be helpful. Will Orrick (talk) 11:52, 22 April 2019 (UTC)

Hi Will, please take a look at Archive 1 then, though generally I don't take issue with length, as long it is properly referenced. Large parts without in-line references is the issue. Ofc there can be parts without reference too when explaining something, following a in-line reference. Instead of omitting every bit of mentioning the 2017 MW publication, we should add it as outlined here https://schaechter.asmblog.org/schaechter/2017/10/plimpton-v-muybridge.html I also suggest to align the tables to the right side, so that text can float along it, makes the visual appearing more compact. There could be a sub-section on interpretations. prokaryotes (talk) 16:36, 22 April 2019 (UTC)

Hi Prokaryotes. Now that you point it out, I see that the section "Content" contains no in-line references at all. I'll try to work on fixing that. Are there other sections that need more references? One thing that would slightly shorten the "Content" section would be if we could omit the controversy about the 1s in column 1. All recent papers I know of agree they were there, as is clear from the high-resolution CDLI image in our "External links" section. Britton, Proust, and Shnider say in their article that this should be a non-issue.

I also agree about having the text flow around the tables, and will see if I can figure out how to do that. The final section, "Purpose and authorship", covers interpretations, but hasn't been brought up to date in light of work after 2002. I've been intending to do that. Maybe it should be moved to a more prominent place in the article. I had actually looked through the talk page archives at one point, but I thought you meant that the MW article had been frequently discussed outside of Wikipedia. The most detailed response by an expert that I've found is "Le buzz de l’été autour de la tablette Plimpton 322" by Christine Proust (in French). Will Orrick (talk) 10:56, 23 April 2019 (UTC)

There are more recent items and on scholar including the mentioned entry in The Best Writing on Mathematics 2018. I have no opinion about the controversy you mentioned. prokaryotes (talk) 11:53, 23 April 2019 (UTC)

Just to be clear: the article reprinted in The Best Writing on Mathematics 2018 is not the Historia Mathematica paper, but the popular summary published in The Conversation. Will Orrick (talk) 16:16, 23 April 2019 (UTC)

prokaryotes please allow me to offer helpful advice. It's very easy to win an issue on Wikipedia if you point to a couple of good sources. However when there is a prior consensus, each attempt to pressure an issue while failing to present sources to adequately support your positions tends to have the effect of further solidifying opposition. If you have good sources, identifying them specifically is extremely effective. If you don't have the sources to point to, pressing the issue will only work against you.

I regret wasting my time digging through the Scholar link you provided. I didn't check all results there, but most turned out to be clearly frivolous search results. The only one of clear relevance was the reprint in Best Writing on Mathematics 2018. If anyone wants to reopen discussion on the Mansfield-Wildberger paper I suggest and request that you identify specific sources showing what impact or reception the concepts have had in the field. I would certainly support inclusion of any content that was demonstrably a "Hot topic" in the field. The Best Writing reprint is a bit interesting, but I don't think that's going to do it. Vague assertions won't work, and pointing to search results implying "there's gold in there somewhere" doesn't work very well a reasonable inspection turns up piles of slag. If I missed the gold, please identify it specifically. Alsee (talk) 21:24, 18 May 2019 (UTC)

For completeness, I've become aware of one more post-2011 source that I should have included on my list: Kazui Muroi's 2013 paper "Babylonian number theory and trigonometric functions: trigonometric table and Pythagorean triples in the mathematical tablet Plimpton 322" in Seki, Founder of Modern Mathematics in Japan: A Commemoration on His Tercentenary. I see that this paper was discussed at length on this Talk page (now in Archive 1). Also, I'm still interested in hearing other points of view about how our article can be improved. Will Orrick (talk) 11:56, 30 April 2019 (UTC)

In my view Plimpton 322 is a fascinating artifact, and I'm very much interested in making the Wikipedia article as good as it can be. I've been quite surprised at how the issue of whether to mention Mansfield and Wildberger's unverified and, to date, not-at-all-influential rational trigonometry hypothesis is, for many people, the most pressing concern facing the article when, in my opinion, there are far more consequential matters to discuss. There seem to be many knowledgeable people out there who care deeply about this topic. I'm hoping some of them might be willing to broaden the conversation. Will Orrick (talk) 15:08, 28 September 2019 (UTC)

It occupies those who consider that not including Mansfield and Wildberger's hypothesis is a violation of the second of the five Pillars of Wikipedia. >Wikipedia is written from a neutral point of view. We strive for articles in an impartial tone that document and explain major points of view, giving due weight with respect to their prominence. We avoid advocacy, and we characterize information and issues rather than debate them. In some areas there may be just one well-recognized point of view; in others, we describe multiple points of view, presenting each accurately and in context rather than as "the truth" or "the best view". All articles must strive for verifiable accuracy, citing reliable, authoritative sources, especially when the topic is controversial or is on living persons. Editors' personal experiences, interpretations, or opinions do not belong.< Multiple editors have commented on this and many readers who find no reference to M and W are likely to be puzzled. — Preceding unsigned comment added by 90.154.71.175 (talk) 18:34, 5 October 2019 (UTC)

No reader searching for information on M and W is going to be inconvenienced, by googling they will readily find information on this and Lamb's rebuttal. It would be better if they found this in the wikipedia article rather than by googling but if it encourages readers to think as to why certain things don't appear in wikipedia articles and to question how far wikipedia lives up to its aspiration to NPOV that is a good thing. — Preceding unsigned comment added by 90.154.71.175 (talk) 21:21, 5 October 2019 (UTC)

I'm not sure what the reaction of a reader seeking information on Mansfield and Wildberger is likely to be when they arrive at our article. I would guess that most such readers will come because they are intrigued by the idea that the Old Babylonians knew trigonometry over a thousand years earlier than the civilizations generally credited with its invention. But a second kind of reader may come because they have heard that the Old Babylonians knew a better, more accurate form of trigonometry than our modern one, a form that may soon revolutionize the practice and teaching of mathematics.

The first type of reader may leave satisfied, since they will read about the old observation that column 1 of the tablet contains the secant squared of a series of right triangles whose small angle decreases from about 45° to about 30° in roughly 1° increments. They will also learn that a leading expert in Mesopotamian mathematics regards this interpretation as "conceptually anachronistic", which I think is worth knowing. So they go away knowing that the trigonometric interpretation is a possibility that some researchers have proposed, but that, based on what we know of the Mesopotamian conceptual appartus, it is unlikely to be correct. Such a reader may be slightly puzzled if they remember that the trigonometric interpretation is supposed to represent a recent "breakthrough" in our understanding of Plimpton 322, whereas the sources cited in our article for the idea are rather old. That puzzlement may itself be a valuable lesson, as it illustrates something about the sensationalistic nature of university press releases and science journalism. More determined readers may be bothered enough by this to seek other sources of information. When they find that the M and W paper was indeed touted as a breakthrough in 2017, they probably are going to wonder what the breakthrough was, and what the difference is between the new hypothesis and the similar earlier ones. I, and I think many other editors of this page, share that puzzlement. I would ask those editors who feel it is important to mention M and W in Wikipedia to suggest how this should be explained to readers. I think, however, that it may be difficult to source such an explanation. When I first heard of M and W's paper, my first question was how their trigonometric hypothesis differed from the old trigonometric hypothesis, and I expected that the approach of most science journalists to the the story would be "discredited trigonometric hypothesis revived in light of new evidence". Not a single article I read actually took that tack; almost all presented the hypothesis as a brand new idea.

I really don't know what we can do for the second kind of reader. The notion that the hypothesized Old Babylonian trigonometry is more accurate than modern trigonometry is, to put it bluntly, promotional bullshit, not backed up by a single authoritative source. I don't think it's worth taking valuable article space to introduce and then shoot down such claims, especially since they are peripheral to an article about ancient Mesopotamian mathematics. Even more worthy of being ignored is the factually incorrect claim that only two fractions can be represented exactly in base 10, whereas many can in base 60.

The situation is truly a mess. I just performed a DuckDuckGo search on "Plimpton 322" and fully a third of the hits on the first couple of pages of search results were to stories about M and W's paper—far more than to any other single paper. Google gives similar results, but I am aware the Google tailors the results it shows to individual users. So by some criterion, the M and W hypotheses is the most widely publicized, the "latest and greatest". However, closer inspection reveals that not a single one of the articles in the search results is written by someone with expertise in ancient Mesopotamian mathematics, and none are suitable as sources for a Wikipedia article. By the way, if we ever do agree that M and W must be mentioned in Wikipedia, I would suggest using Christine Proust's "Le buzz de l’été autour de la tablette Plimpton 322" as the source, although the fact that it's in French might be seen as a problem. Proust makes an important point that I haven't seen made explicitly anywhere else, which is that the use of the tablet M and W propose is only "trigonometry" if one redefines that word, and that many readers get the wrong impression because they interpret "trigonometry" with its usual meaning, namely as a system relating length to angle, or equivalently, to arc length.

There actually is a really interesting story to be told about Plimpton 322 that has received almost zero publicity. (It is mentioned in Proust's article, however.) The reciprocal-pair explanation for how the numbers on the tablet were generated came into favor as a result of Robson's 2001 paper, although the idea appeared as far back as the original publication of Neugebauer and Sachs. The arguments of Robson and her predecessors were based on careful piecing together of many strands of indirect evidence. Then, in 2007, Friberg translated two tablets in which the precise kind of calculation proposed in this explanation is carried out in detail and its geometric purpose, namely to produce pairs corresponding to the short side and diagonal of a rectangle, is stated. It's rare enough in science that smoking-gun evidence for a theory shows up like this—much more so, I should think, in a field like history where one can't do controlled experiments and must take what evidence one finds. This truly dramatic discovery is, I think, one of the most exciting things to have happened in the field in a long time. I just wish that it had received even a tenth of the press coverage that M and W's paper got. I would not even have known about it were it not for the paper of Britton, Proust, and Shnider. Despite this confirmation of the reciprocal-pair idea, many mysteries remain, including how the sexagesimal fractions produced by the method get converted to the integers in columns 2 and 3, what the purpose of that was, and why such pairs were being tabulated in the first place. These developments, as of early this year, do appear in Wikipedia's article. Because I only gradually became aware of them as I was in the process of expanding the article, and because I was too timid to refocus the article entirely away from the reciprocal pair versus generating pair debate, the article is now a bit of a mishmash, I'm afraid. Will Orrick (talk) 17:59, 8 October 2019 (UTC)

>When they find that the M and W paper was indeed touted as a breakthrough in 2017, they probably are going to wonder what the breakthrough was, and what the difference is between the new hypothesis and the similar earlier ones. I, and I think many other editors of this page, share that puzzlement. I would ask those editors who feel it is important to mention M and W in Wikipedia to suggest how this should be explained to readers.< why explain? Better for readers to be puzzled by lack of explanation than lack of mention. For my own part I find M and W's exposition of the tablet so beautifully simple that whether it is historically accurate is neither here nor there. — Preceding unsigned comment added by 90.154.71.175 (talk) 05:59, 9 October 2019 (UTC)

I lost hope for wikipedia after M and W were excluded from this article. Never seen such a naked conspiracy to silence something or someone, but this sort of thing must happen in academia quietly all the time, so it's good it's out in the open here. I work round wikipedia biases and inform others as these. It's no more neutral than any newsmedia in any country.Nine-and-fifty swans (talk) 09:46, 20 August 2020 (UTC)