Talk:History of mathematics/Archive 1

Archive 1

Notation

I think a section on the history and etymology of mathematics notation would be interesting, and appropriate. I didn't find an article on this topic, and I would be willing to put one together if the idea is supported. Thoughts? --Monguin61 02:52, 10 December 2005 (UTC)

Extensions of complex numbers

Despite what the passage on complex numbers says, surreal numbers and quaternions are not extensions of complex numbers; at least, they are not _field_ extensions, and that is pretty important. IIRC, it is, in fact, mathematically provable that the complex numbers has *no* superfield; assuming one exists leads to a contradiction. While quaternions and surreals may be spiffy in their own way, they do not supercede complex numbers.— Preceding unsigned comment added by Saalweachter (talk • contribs) 01:06, 17 December 2005 (UTC)

Jitse Niesen suggested I visit this page.

Since the introductory paragraph doesn't even mention the history of mathematics, yes, I think some work needs to be done. Rick Norwood 17:01, 21 December 2005 (UTC)

It seems clear to me that none of the current section heads are appropriate. I would like to suggest a structure for the article:

Introduction; prehistoric mathematics (megalithic structures); early mathematics (Mesopotamia, Egypt, China, India); Greek mathematics; Arab mathematics; Renaissance mathematics; the scientific revolution; the modern age.

I am going to take a shot at the introduction, and then work slowly, one section at at time, at a rate of about one section per week. Rick Norwood 17:07, 21 December 2005 (UTC)

A few words on the new introduction I just posted. I've focused here on the times and places in which new mathematics has been discovered for the following reason -- to begin to list famous mathematicians or to explain mathematics more complicated that the Pythagorean Theorem would make the introduction too long and too hard to read. In the sections following the summary, I believe only the most important mathematicians and mathematical discoveries should be mentioned, e.g. Euclid but not Heron, the infinity of primes but not the quadriture of the lune. Finally, each section in this article should reference another article which contains a fuller account. Rick Norwood 17:54, 21 December 2005 (UTC)

Hardly a comprehensive history of mathematics

A history of mathematics which makes no significant mention of differential calculus, set theory, or metamathematics? No mention of Leibniz, Euler, Gauss, Cantor, or Godel? This article is a hodgepodge mess of trivia on various different ancient cultures' mathematics (even this is not fleshed out, and consists mostly of links to other articles), with some stuff on complex numbers slapped on; it needs a lot of work to even approach what its title suggests. -69.249.40.134 06:06, 7 January 2006 (UTC)

If you had read the paragraph directly above yours, you would have noticed that this is a work in progress, and that I am working slowly and deliberately, to leave time for comments on each section before moving on to the next section. Why don't you help? Rick Norwood 14:46, 7 January 2006 (UTC)

(I am the author of the anonymous comment above). Sorry, I apologize if my tone was too snippy. The current state of the article had aggravated me into quickly posting my disappointed impression, and I did indeed miss your comment. But, of course, the Wikipedia spirit is to provide help where it is needed, and so I will indeed try to help you fix up this article. You propose a slow and deliberate pace of change, though; I think an initial bold, almost complete restructuring of the article would be worthwhile instead. We could at least begin by modifying the sections to be approximately those you suggested. -Chinju 00:29, 8 January 2006 (UTC)

new section on Arabic mathematics

I've written a section on Arabic mathematics. Additions and suggestions are welcome. Next: Italy and the Renaissance. Rick Norwood 01:57, 8 January 2006 (UTC)

Still to do

The section on complex numbers needs to be rewritten due to changes upstream, and the sections on the 19th, 20th, and 21st centuries must be written. Rick Norwood 22:35, 22 January 2006 (UTC)

Manifold

Hey, if there are any experts reading this talk page, it would be great to see the Manifold#History section fleshed out. Thanks. –Joke 03:16, 2 February 2006 (UTC)

More on Africa and Mayan math please?

I'm a math teacher to be and will be working in a district where most students are hispanic and african american. I'm trying to find any non-egyptian african math/science/astronomy, since it is hotly debated whether or not the egyptians were "black". Also, more on the math of the mayans. But otherwise, great start! I'll be coming back frequently.

Let me recommend The Cartoon History of the Universe volume three, which has a great deal on the history and technology of sub-Saharan Africa. Rick Norwood 13:12, 19 February 2006 (UTC)

new material added

A lot of interesting new material has been added to this article. I would like to suggest that those making future additions keep in mind the Wikipedia "ideal article length" and carefully consider what is important enough to go into the general overview, and what should go in one of the more specialized articles. Rick Norwood 14:00, 21 February 2006 (UTC)

Kelvin Case rewrite

Good work, Kelvin. I do have one question.

"(also, the paper itself upon which the Indian mathematics is written is ultimately of Egyptian origin, yielding to another possible source of learning.)" Paper or papyrus?

Rick Norwood 14:36, 26 February 2006 (UTC)

Minor point from the Classical Indian Section

The cosmological time cycles explained in the text, which was copied from an earlier work, corresponds an average sidereal year of 365.2563627 days, which is only 1.4 seconds longer than the modern value of 365.2563627 days — Preceding unsigned comment added by 195.60.5.35 (talk • contribs) 14:56, 4 April 2006 (UTC)

indian math: sidereal year

The indian value of the sidereal year appears equal to the modern one, although it is said they differ by 1.4 s Gakrivas 11:57, 10 April 2006 (UTC)

The Indian measurement seems remarkable, but the modern one is accurate to many more decimal places than are indicated here. Rick Norwood 13:12, 10 April 2006 (UTC)

Complex numbers

Imaginary numbers were used by Cardan in his treatment of cubic equations, in the 15thC. Euler was the first to treat them with respect, the first to use i, etc, but he wasn't the first to ask what the square root of minus one is.

This is an Ok article, it's getting better I think, but it does sound as if it's written with High School teachers in mind. Being culturally sensitive is great, but 17th - 21st c. maths is pretty important to the way math is taught and done and appreciated today. --M a s 01:20, 3 May 2006 (UTC)

M a s edit

Good edit, M a s. Rick Norwood 15:57, 3 May 2006 (UTC)

Platonism? Disovered or Invented

I'm an avowed Platonist, and I do think that mathematical ideas are discovered, but there does seem to be an inconsistent use of "discovered" vs. "invented" in this page. I think this might invite controversy. Any suggestions on consistency? Does anybody like "developed" for every instance of "discovered" and "invented," except for notational references (pi, e, etc?) Thanks! --M a s 02:00, 3 May 2006 (UTC)

I just did some (hopefully micro) surgery ala the above. I left the archeoligical stuff as "discoveries," and some of the terminology as "inventions," but I changed most everything else to development / proof / etc. Development I think is a little weaker word than either discovered or invented, and when something is significant or when priority is important, I tried to emphasize that. I'm not happy with all of it, but pls. comment if something is glaring. Thanks! --M a s 16:39, 3 May 2006 (UTC)

Possibly useful quotes

Here are some possibly useful/interesting/helpful quotes from: The Mathematical Universe / An Alphabetical Journey Through the Great Proofs, Problems, and Personalities by William Dunham, 1994 John Wiley & Sons, Inc. ISBN 0471536563

• "Egyptian mathematics can be traced back at least 4,000 years"
  • "Scholars have deciphered papyrus rolls predating 1500 B.C., some of which are indisputably mathematical." p. 179
• "Ahmes papyrus from about 1650 B.C. and named for the scribe who wrote it. This 18 foot-long document was purchased in Egypt in 1858 and now resides somewhere in the British Museum."
  • "the 64th problem of the Ahmes papyrus is: / Divide 10 hekats of barley among 10 men so that the common difference is 1/8 a hekat of barley."
  • "Ahmes gave the solution correctly, stating that the first individual should get 1/4 + 1/8 + 1/16 hekats of barley" p. 180
• "the sum of fractions each with numerator of 1. These are called unite fractions, and the Egyptians used them almost exclusively."
  • "the Egyptian notational scheme: to represent a reciprocal, they used a symbol - looking something like a floating cigar - atop an integer. ... all fractions had to be assembled from unit fractions, with the lone exception of 2/3, which had it's own unique symbol." p. 181
• problem 50: / A circular field has diamater 9 khet. What is its area? / According to the scribe, the answer is obtained by subtracting one-ninth of the diameter and then squaring the result."
  • "an estimate of [pi] can be unearthed from this solution."
  • "[analysis gives a value of pi=3.1605] which is often cited as the Egyptian approximation of [pi]" p. 182

Good stuff 12. Are they in the Rhind Papyrus article? --M a s 23:03, 9 May 2006 (UTC)

Medieval quadrivium

As a medievalist, I'm a bit concerned that the chronological layout gives the impression there was no mathematical activity in Western Europe between the end of Greek and Hellenistic mathematics (AD 200) and the beginning of the European Renaissance[s] (AD 1200). I'm not claiming great mathematical breakthroughs in the Middle Ages, but mathematics was used and understood -- although in ways which may not conform to the expectations of modern mathematicians.

I'm also surprised that there's no mention of the quadrivium, which provided one of the basic frameworks within which mathematics was taught and understood during the Middle Ages. --SteveMcCluskey 22:30, 6 June 2006 (UTC)

These are good points, but a complete history of mathematics would fill volumes. I'm not aware of any mathematics in the Middle Ages in Western Europe as important as the least of the mathematical discoveries that are included in what is necessarily a survey article. I would be glad to hear about them if there were any. And I do thing there should be a mention of the trivium and quadrivium. Maybe you would like to add them. By the way, did you know that the trivium and quadrivium have been revived as part of the home schooling movement in the American South? Rick Norwood 22:35, 6 June 2006 (UTC)

Science, math, and technology definitions

First of all, mathematics is not a science. Science is empirical, mathematics is theoretical. Second, the examples listed of older "sciences" list technologies. I don't think these were ever considered "science". Ancient "science" was called natural philosophy, and I don't think the creation of useful things was considered part of that endeavor. Moreover, all of these technologies were invented in the pre-historic period, so we have no writings to give us any hint of what field they might have been considered part of, if any. -- Beland 17:49, 26 Feb 2005 (UTC)

If mathematics is not a science, perhaps you should inform the Mathematical Sciences Research Institute, at [1]. Perhaps this might help: [2]. Although your definition is a valid one, it is not as common as other definitions in modern usage. Remember that science -- the accumulation of knowledge -- is as much (1) observation as it is (2) experimentation. What's important is that it is verifiable by others in the scientific community, and mathematics certainly fits this description, as any other mathematical scientist may tell you. --a scientist

I wish to make another rebuttal of "mathematics is not a science". I find this a very troubling statement as mathematics is very much 'inspired' by (physical) reality. I believe it is (still) debatable as to whether mathematics (as we know it, at least) is empirical, like science. --GrimRC

This is a subject much debated by mathematicians. Some would say that mathematics is deductive, science inductive. That mathematics tends to be right the first time, while science tends to approach the right answer by successive approximations (e.g. Newton to Einstein). And finally that mathematics is abstract while science is concrete. Others argue the other side of the question with equal passion. Rick Norwood 21:35, 14 May 2006 (UTC)

The Historian of Science, Marshall Clagett, once defined science this way:

Science comprises, first, the orderly and systematic comprehension, description and/or explanation of natural phenomena and, secondly, the [mathematical and logical] tools necessary for the undertaking.

His definition fits mathematics in, but only as a tool to assist those sciences which deal directly with natural phenomena. --SteveMcCluskey 01:23, 8 June 2006 (UTC)

Egyptian mathematics

I'd appreciate it if someone could have a look at the recent edits there. Tom Harrison ^Talk 03:22, 11 June 2006 (UTC)

I presume you mean the recent deletions of everything about Egyptian math (twice); Babylonian math (twice); and Early math, Chinese, Greek and Hellenistic, Persian and Islamic, and European Renaissance mathematics.

Two things of note:

• Both deletions were done from IP addresses in the range 59.176...
• Both deletions left Indian mathematics unscathed.

It seems we may have an edit vandal with an agenda. --SteveMcCluskey 12:57, 12 June 2006 (UTC)

Oops; I think I may have looked at the wrong article. I just saw the edits in Egyptian mathematics. My comments about the edits in History of mathematics still stand. --SteveMcCluskey 13:14, 12 June 2006 (UTC)

Sidereal Year

The modern value of the sidereal year in the discussion of Indian Mathematics was recently changed from 365.2563627 days to 365.25636305 days, without any citation of a source for this value.

I found the following authoritative data for the modern value at the website of the National Physical Laboratory (UK).

The following values of the lengths of the year, month and day are expressed in units of 1 day of 86 400 SI seconds; T is measured in Julian centuries from 2000.0.

• • 1 tropical year (equinox to equinox) = 365d.242 193 − 0d.000 0061 T
  • 1 sidereal year (fixed star to fixed star) = 365d.256 360 + 0d.000 0001 T

Computing the value of the sidereal year for AD 400, we get

365.256 360 - 0.000 0016 = 365.256 358 4 days

This differs from both the values given in the article.

There is a further problem with this value in that it uses the modern definition of the "day", in terms of SI seconds which are defined in terms of the constant vibrations of an atomic clock. The astronomical value given in the Surya Siddhanta measures the year in terms of the changing length which reflects the slowly changing rate of rotation of the Earth.

Does anyone have other reliable sources for the length of the sidereal year? --SteveMcCluskey 14:23, 24 June 2006 (UTC)

Squaring the Circle

It menions in the section on ancient indian mathematics that The Sulba Sutras gave the method for squaring the circle. Clearly this can't be the case since it has been proven no such method can exist. — Preceding unsigned comment added by 58.160.189.201 (talk • contribs) 03:00, 17 September 2006 (UTC)

More Pictures...

I think the article would be better if more pictures could be provided. By that, I don't mean picture of mathematician, maybe a front cover of a historical manuscript at the top would be nice, I'll look around and see if I can find one, but I hope this article can be better illustrated. Angrynight 04:58, 12 October 2006 (UTC)

Okay, I've added a few pics. Please enhance! Angrynight 05:18, 12 October 2006 (UTC)

Where's Leibniz?

Although Newton is mentioned as having discovered calculus, Leibniz isn't even mentioned. I think the article would be more complete if Leibniz were mentioned. --Propower 05:06, 15 November 2006 (UTC)

Good point. Leibniz added. Rick Norwood 14:04, 15 November 2006 (UTC)

Nothing on "History of Mathematics"

Hi, I came to this page because I wanted to read about the History of Mathematics - not the history of Mathematics. If you see what I mean. Most maths degrees offer a module in History of Mathematics and I would appreciate a little section in the ways that mathematicians and historians discover the mathematical past - what they think is significant, any big discoveries, and the reason for being interested in the first place. wikitruth_lover42

What you are interested in is called historiography. I don't know any books on the historiography of math. Someone should write one. Rick Norwood 13:50, 27 November 2006 (UTC)

last section

The section about 21st centuary isn't that very objective.— Preceding unsigned comment added by Andreas2001 (talk • contribs) 19:56, 27 November 2006 (UTC)

Apology

I made an edit to the article to remove what I thought was a misleading statement, that few cultures ever make new developments in mathematics. But I accidentally caused something to go wrong with the page in the process. Sorry. NikolaiLobachevsky 12/20/2007 16:45:26 (UTC)

Let me suggest you go a little slower. You have a lot to contribute to the article, but some of it seems to be hastily written. Rick Norwood 17:01, 20 January 2007 (UTC)

I've done some rewrites of your rewrites. I hope you will not mind a couple of suggestions. You cannot rewrite part of a section without considering the flow of the entire section, or the section becomes unbalanced and hard to read -- lumpy. Also, we cannot cram everything into this short article. This article just touches on the high points, links provided additional information. Rick Norwood 16:54, 27 January 2007 (UTC)

Wikiproject

Since mathematics is not one of the natural sciences is it appropriate to include the history of mathematics article in the history of science wikiproject. I think this article should be switched to the scope of wikiproject mathematics. Prb4 12:50:37 2/11/2007 (UTC)

Articles can be and frequently are within the scope of multiple WikiProjects. It doesn't imply any sort of ownership or authority, just that WikiProject members are likely to interested in the article and have relevant expertise.--ragesoss 22:04, 11 February 2007 (UTC)

rewrite

Someone who knows almost nothing of mathematics or history has done a major rewrite since I last visited this page. (For example, they do not know that Pythagoras lived in Italy!) I'm trying to repair some of the damage, but much more work remains to be done. Rick Norwood 13:05, 15 March 2007 (UTC)

Wisdom of the ancients

A lot of nonsense has been added to this article about supposed ancient wisdom. mostly without references. For some time now, citations have been requested, and none have been forthcoming. If references are not added soon, I plan to remove all assertions about ancient mathematics that are not either well known or referenced. Rick Norwood 13:13, 15 March 2007 (UTC)

Pythagoras

I've removed a lot of silly stuff from the paragraph on Pythagoras, e.g. that irrational numbers were discovered by division! Note that we have a long article on Pythagoras. This is not the place for stories. Rick Norwood 13:12, 2 April 2007 (UTC)

European periodization:

I see some problems with the periodization of the European sections of this article. At present it goes:

6 Greek and Hellenistic mathematics (c. 550 BC—AD 300)

...

10 European Renaissance mathematics (c. 1200—1600)

This leaves a 900-year gap between AD 300 and c. 1200, and it makes the period 1200-1400 part of the Renaissance. Most historians of science would see the Renaissance as beginning at 1350 at the earliest, and some would put it a century later. I consider that Nicole Oresme (d. 1382) who incorporated mathematics in the Aristotelian philosophy he taught at the University of Paris clearly belongs in the Middle Ages.

I suggest the following division:

6 Greek and Hellenistic mathematics (c. 550 BC—AD 300)

...

10 Medieval European mathematics (c. 300-1400)

11 European Renaissance mathematics (c. 1400—1600)

I plan to make the division and then to flesh out the medieval period. Could someone comment before I start this revision. --SteveMcCluskey 16:11, 2 April 2007 (UTC)

The problem is that what a professor of The History of Mathematics means by that phrase -- in other words, current academic practice -- is not A History of How People Used Arithmetic to do Business. That would be an interesting article, but that is not this article. In most reference books on the subject, the last person in the Greek tradition to do anything that a mathematician would recognize as mathematics was Pappus, who lived toward the end of the 3rd century. Living mathematics then passed from Europe to the Moslem cultures of Asia and Africa. The next person in Europe to do anything that a modern mathematician would consider real mathematics was Fibonacci, who was born in about 1175. Between 1175 and 1400, there were at most three mathematicians, Fibonacci, Nemorarius, and Oresme. Roger Bacon admired mathematics, but didn't actually do any. Oresme died in 1382.

In all books on the History of Mathematics that I am familiar with, the division is as it currently appears in this article. If you want to call the period "Early Modern Mathematics" rather than "Renaissance Mathematics" I have no objection. but if you want to insert a new period, "Medieval Mathematics", then you need to cite a reference. Rick Norwood 22:19, 2 April 2007 (UTC)

Rick,

You misunderstand me; I'm not at all interested in the history of business arithmetic. I'm interested in the kind of theoretical discussions which are found in Buridan, Oresme, Swineshead, Bradwardine, and the Oxford calculatores.

Rather than argue about abstractions, I'll put together something on one of my talk pages so you can see what I'm thinking about.

Your suggestion of Early Modern Mathematics deserves some thought, but the Early Modern Period is generally considered to be the Late Renaissance, while the present "Renaissance" period goes in the other direction into the Late Middle Ages. --SteveMcCluskey 23:09, 2 April 2007 (UTC)

Sorry to have misunderstood you. Maybe you can come up with some interesting stuff -- though where Buridan is concerned I don't think his proof that 3 = 1 (in the case of the trinity) is really mathematics. His work in logic does not really bear on mathematical logic. I like his ass, though. I don't know anything at all about Swineshead or Bradwardine. Maybe there is an interesting paragraph there after all, not covered in the standard texts. As for the current section, how about "pre-Reanassance mathematics" as a title? Rick Norwood 12:59, 3 April 2007 (UTC)

Defining History of mathematics

Since its been added, I've been disturbed by the restrictive definition of the History of Mathematics at the head of this article. A discussion on the Historia Matematica Mailing list defines the History of Mathematics in a much more expansive way. One revealing comment says a lot:

With the advent of a professional history of science, a new and more sophisticated historiography has arisen and is being put into practice in the history of mathematics.

This historiography measures events of the past against the standards of their time, not against the mathematical practices of today. The focus is on understanding the thought of the period, independent of whether it is right or wrong by today's account. The historiography is more philosophically sensitive in its understanding of the nature of mathematical truth and rigor, and it recognizes that these concepts have not remained invariant over time. This new historiography requires an investigation of a richer body of published and unpublished sources. It does not focus so exclusively on the great mathematicians of an era, but considers the work produced the journeymen of mathematics and related scientific disciplines. It also investigates the social roots: the research programs of institutions and nations; the impact of mathematical

patronage; professionalization through societies, journals, education, and employment; and how these and other social factors shape the form and content of mathematical ideas. (Kitcher, P. and W. Aspray (1988). History and Philosophy of Modern Mathematics. Minneapolis, University of Minnesota Press, cited by Roy Weintraub)[3]

I'd like some comments before I remove the recently added definition. --SteveMcCluskey 17:04, 5 April 2007 (UTC) (A historian of science exploring the wilds of the history of mathematics)

Please note that the interesting quote above is about the historiography of mathematics. The field currently called "History of Mathematics" is designed for the training of mathematicians, and as such it is mainly concerned with what works, however interesting an insight into the culture of the time failures may be. The difference is primarily one of emphasis. Are we talking about the HISTORY of mathematics or the history of MATHEMATICS.

Similar conflicts occur in every field. A person interested in the HISTORY of music may find a chapter devoted to the exceptionally poor quality of the lute players of Provonce of great interest, but a person interested in the history of MUSIC is going to be much more interested in a chapter on Bach.

Encyclopedia articles should reflect current practice, not try to change current practice, no matter how strongly you think the change is a good one. Every year at least a hundred mathematicians teach and thousands of math majors take a course in the history of MATHEMATICS. You tell me how many historians teach and how many history majors take a course in the HISTORY of mathematics. As the term is in fact currently used, "History of Mathematics" means "history of MATHEMATICS" and if you want to change that, you need to change it in academia before you get to change it in Wikipedia.

Having said that, I find your suggested rewrite of the material on the Middle Ages interesting. Why not post it and see what happens. As with all Wikipedia articles, I reserve the right to edit it -- as do lots and lots of other Wikipedians.

Rick Norwood 19:02, 5 April 2007 (UTC)

Historiography is the theoretical discussion of the writing of history. Essays about historiography of mathematics are about how the history of mathematics should be written. As the passage I provided explicitly says, it's about how this historiography "is being put into practice in the history of mathematics." That discussion is about current academic practice.

I'll run the material on the Middle Ages up the flagpole and see who salutes it -- and who shoots at it. The bit on Fibonacci and the section on Early Modern math still needs work; anyone familiar with those issues is more than welcome to contribute. --SteveMcCluskey 21:54, 5 April 2007 (UTC)

Mathematics had become an international endeavour

Does anyone know what this passage in the section on the 17th century "Science and mathematics had become an international endeavor, which would soon spread over the entire world.", which has been around for over a year,[4] is intended to mean. As a medievalist, I would insist that mathematics and science were international activities from the 8th to the 14th centuries; The textbooks of Bede (an Englishman) were read and copied in the Carolingian court (a cosmopolitan center if ever there was one); the medieval universities of the 13th and 14th centuries drew scholars from throughout Europe, who studied from texts originally written in Greek and Arabic.

It should be clarified to spell out what is meant, otherwise I'd suggest deleting it as empty puffery. --SteveMcCluskey 20:27, 28 March 2007 (UTC) (who sounds a bit too grumpy).

The problem, of course, is that Bede did not do any mathematics (beyond arithmetic). Neither did the Carolingians. The first European mathematician following the Dark Ages was Fibonacci, in the 13th Century. During the 13th century, most "mathematics" consisted of a painful rediscovery of a few of the ideas known to the Greeks, and well known in the Arabic world. The next century, the 14th, was most famous for the Black Death, and there was very little mathematics done of any kind. If you really want to push it, you might claim that mathematics became international in the 15th century. Regiomontanus traveled extensively, as did Pacioli. But most of their work had to do with mathematical notation rather than with mathematics per se. Certainly, they laid vital groundwork, making it possible for the first time since the Greeks for European mathematicians in different cities to read each others work. Aside from the all important development of a common notation, however, the main thrust of mathematics in the 15th century was the rediscovery of the rules of arithmetic. Numerology flourished.

In the 16th century, some original mathematics (unknown to the ancients and also unknown in Asia and Africa) appeared in Europe. Viete in France may have known about the work of Tartaglia in Italy. But this is nothing compared to the explosion of new discoveries in the 17th century. Now, instead of one mathematician in one country hearing about the discoveries of one mathematician in another country, every mathematician in Europe was in touch with every other mathematician in Europe. When Galileo was placed under house arrest, they knew about it in Amsterdam in a matter of weeks, and scientists and philosophers from all over Europe came to visit. Nothing like this happened in the 16th century. The expansion of mathematical and scientific communication was unprecedented. Rick Norwood 21:44, 28 March 2007 (UTC)

Rick. I think our differences say a lot about history written by mathematicians and history written by historians. Let me make a few short points:

• The history of mathematics is not just about new discoveries, it's about how mathematics was used, studied, or taught in various times and places. If you limit a histor to the history of mathematical progress, you ignore the important question of how mathematics was practiced in a relatively static environment. Not all history is a history of progress.
• In the medieval universities
  • texts on algorism became part of the teaching of mathematics.
  • original insights into the application of mathematics to the study of motion were developed at Paris and Oxford and studied at Vienna and Erfurt (among other places).

these need to be considered.

• Your comment that numerology flourished is at best an oversimplification, and in many cases is flat out false. Bede used mathematical symbolism extensively in his scriptural commentaries, but strictly avoided it when he was writing about astronomical calculations.

Once we accept the full range of the history of mathematics, my question about the need for clarity in the internationalization of mathematics in the 17th century remains valid. --SteveMcCluskey 14:46, 2 April 2007 (UTC)

Our differences also say a lot about the way mathematicians use the word "mathematics" and the way non-mathematicians use the word. For example, to a non-mathematician I suspect that 2 + 2 = 4 is "mathematics", while to a mathematician, 2 + 2 = 4 is arithmetic, and mathematics is, for example, Fermat's Last Theorem. In other words, to a mathematician, mathematics means "new discoveries in mathematics" and the "history of mathematics" is a history of those new discoveries. I'm sure a very interesting book could be written about the ways arithmetic was used in the past, but it would be a history of arithmetic, not a history of mathematics. I think this article should make that clear, and I'll think about ways that can be done.

I am interested in your comment on original insights into the application of mathematics to the study of motion. Can you cite some names and dates? Usually Galileo is credited with the first insights to go beyond the errors of Aristotle.

Numerology flourished. This does not mean that numerology was universal, but that numerology was widespread. See, for example, the writings of Stifel, Napier, Father Bongus, and many others.

I'll try to make it clear that the comment in the article refers to new discoveries in mathematics. Rick Norwood 15:05, 2 April 2007 (UTC)

Rick,

The version you're insisting on, is not just a matter of phraesology. It makes the demonstrably false claim that before the 17th c. mathematics was not an international endeavor. During the middle ages almost all scholarly activities, from mathematics to theology, were fully international. The text quoted in your note to support this says nothing about international activities or about the state of mathematics in the 17th c.

Ranking my preferences regarding this I would list them.

1. Something like my version (I'm open to edits there).
2. Delete the sentence (it really isn't necessary or useful).
3. Retain your version (if relevant sources can be found to support the new internationalization of mathematics).

--SteveMcCluskey 23:00, 2 April 2007 (UTC)

Sorry to butt in here as a newby but Rick, Steve is right and you are wrong. European mathematics was very much an international endeavour through most of the Middle Ages and the Renaissance. The only real change in that in the 17th C. is that non European countries were added to the intellectual exchange, mainly through the Jesuit missions to China, India, South America etc.

Rick your definition of the differences between arithmetic and mathematics is not correct! Mathematics is the whole discipline, Arithmetic is a branch of that discipline and Fermat's Last Theorem is a theorem of Number Theory, which is a branch of Arithmetic.

Have made some minor changes to Renaissance Maths hope they meet approval!Thony C. 11:55, 7 April 2007 (UTC)

Mathematics in the middle ages

A few comments on the new section.

As requested, I'll add a bit on what Fibonacci did, but probably not until Monday.

The new section mentions a "complete translation" of Euclid's Geometry. Does this mean a complete translation of Elements or just a translation of the books in Elements on the subject of geometry?

I think the business about motion is too specialized for this article, and should have its own article with a link from this page.

The point about the book on arithmetic was that it was printed rather than copies. I'll try to make that clearer.

Rick Norwood 13:01, 6 April 2007 (UTC)

Yes the Euclid translation is indeed complete. I have modified and extended the comment on printed mathematical books. At some point I will do a whole article on this theme or at least one on the early printing of science. As well as an article on Ratdolt! Thony C. 12:02, 7 April 2007 (UTC)

Rick,

Thanks for undertaking to clear up the Fibonacci section.

As to removing the "business about motion" as too specialized for an article on the history of mathematics, by the same argument the "business about motion" in Kepler and Newton would be inappropriate. Both are good examples of new developments in mathematics that were driven by attempts to solve physical problems. As I see it, that's one of the central themes of European mathematics and its evolution should be clearly displayed in this article. --SteveMcCluskey 13:28, 7 April 2007 (UTC)

Bourbaki

I am not one of those who believe dullness is an important quality in an encyclopedia. There is something of a tradition of describing Bourbaki as a "nonexistant French mathematician", and an explanation is only a click away. Rick Norwood 13:05, 25 April 2007 (UTC)

While mathematicans who know of Boubaki may appreciate this in-joke, it did make me smile when I read it, would the lay man who has not encountered the group before understand and apreciate it? I've not encountered this tradition, only one google hit for the term. As for being only one click away I see hyperlinks as being an optional thing which people can click if they want to find out more, not something which a reader is required to read if they want to understand the paragraph. --Salix alba (talk) 14:31, 25 April 2007 (UTC)

Rigveda

A few months ago a change asserted that Rigveda was relevant to the history, yet there is no mention of it on that article and I have had no response regarding its omission on Talk:Rigveda#mathematics. Is there some truth to the anons edit? John Vandenberg 01:55, 3 May 2007 (UTC)

The burden of proof lies with the poster. If no reference is provided, the citation should be removed. Rick Norwood 13:16, 3 May 2007 (UTC)

Hindu-Arabic Numerals

A caption to a figure in the article makes the unsourced claim that "The Arabic numerals were developed in the Maghreb from the characters of the Arabic sentence: my goal is calculation وهدَفي حسابْ in 792."

This disagrees with everything I've read about the origins of Arabic numerals and the following Wikipedia articles: Arabic numerals, Hindu-Arabic numeral system, Eastern Arabic numerals and History of the Hindu-Arabic numeral system. Since Wikipedia is not a Reliable Source, further history is available at the The MacTutor History of Mathematics archive at the University of St. Andrews.

• Indian numerals by J J O'Connor and E F Robertson
• Arabic numerals by J J O'Connor and E F Robertson
• The Arabic numeral system by: J J O'Connor and E F Robertson

I am removing the figure and its caption. --SteveMcCluskey 00:11, 4 June 2007 (UTC)

E8

I don't see why the result on E8 is so important. The Green-Tao theorem or the solution of the Poincare conjecture should be there instead. Kope 12:39, 21 July 2007 (UTC)

Put them in. Obviously, the section on the 21st Century is a work in progress. Rick Norwood 12:56, 21 July 2007 (UTC)

CH passage

I've removed this sentence from the "Complex numbers" section:

Interestingly the independence of the continuum hypothesis can be seen as an inability to prove whether or not certain real numbers should be thought to exist.

First of all, it's not about complex numbers. Secondly, it's pretty meaningless. Thirdly, to the extent a meaning can be imputed to it, it's not clearly correct. --Trovatore 23:03, 4 October 2005 (UTC)

The comment makes sense if they meant the axiom of choice. —Preceding unsigned comment added by 99.233.27.82 (talk) 20:01, 5 October 2007 (UTC)

How about them Greeks

Most people who write about the history of mathematics give special credit to the Hellenistic mathematicians, such as Euclid, Eratosthenes, and Archemedes. This material, all of it referenced, was recently deleted by someone claiming that it represented a Eurocentric POV. My own view is that to fail to mention the importance of Greek mathematicians would distort the history of mathematics, and so I reverted the deletion. Comments? Rick Norwood 13:10, 27 June 2007 (UTC)

Nobody is deprecating importance of Greek mathematicians. But the general hellenistic bias in western history texts on science and mathematics is well known, with the deleted statement being a good example. The deleted statement was a blatant POV in claiming 'Greek Mathematics' to be 'better' and more sophisticated than other ancient systems, hence I'm deleting it again.199.43.48.131 19:11, 27 June 2007 (UTC)

The statement that Greek mathematics is more sophisticated than other ancient systems is referenced. This article and many others in Wikipedia go into considerable detail about the "other ancient systems" you mention. None of these "other ancient systems" has anything that even comes close to the pattern of axiom, definition, theorem and proof found in Euclid's Elements, the mathematical method in universal use today.

In any case, if you want to contest the primacy of Greek mathematics, you need to quote an authority to back up your claim. Rick Norwood 14:57, 28 June 2007 (UTC)

So, your argument is that the relatively limited Greek mathematical achievemnts (compared to its contemporary Indian achievemnts) are superior just because modern mathematical approach has historic ties (but is very different) to greek epistemological approach. The reason for that has nothing to do with greek mathematical superiority, but with modern mathematics developing mainly in Europe. Indian mathmatics has its own, if somewhat different mathematical methods rooted in its various schools of darshan. Although, Indian and Greek logical methods are different with Indians recognising the boundary between the deductive and the inductive to be blur (which might have enabled them in developing far more abstract concepts then the simplistic greek geometrical formulations). As for the question of providing references. Sir, where is your reference for the claim that Archimidies was 'Some say the greatest of Greek mathematicians, if not of all time..'? Btw, I did add a reference in my last edit, which you removed while demanding 'refence challenging greek primacy'. Also, no modern scholarly work on history of mathematics will make such a blatant claim of 'Greek Primacy'. This ehnocentric (and perhaps racist) view of history has been repeatedly (and succesfully challanged) over last half century, i will add several references of that. I would expect any serious student of history of mathematics to be well aware of that, unless your source of history is still colonial historains (like in the reference mentioning greek superiority in current article).199.43.32.85 19:11, 5 July 2007 (UTC)

No, my argument is that if you want to claim that Indian achievements are both contemporary with and superior to Greek achievements, you need to cite an authority as your source. You ask for references for the claims the article makes for Greek mathematics. The article has footnotes. Those are the references for the statements in the article. Rick Norwood 14:15, 6 July 2007 (UTC)

I strongly suppoert Rick and strongly oppose 199.43.32.85. For example, what periods is 199.43.32.85 comparing when he talks about Indian mathematics and Greek/Western mathematics? In the 17th and 18th century, which is the period I'm most interested in, as far as I know Western mathematicians were studying MUCH more abstract ideas than Indian mathematicians. And I would like at least one good example for the comment about blurring the line between deductive and inductive. 99.233.27.82 20:10, 5 October 2007 (UTC) Jordan

Removal of introductory part

I removed what I deemed to be an ill-placed sentence on a definition of mathematics. Maths doesn't need to be defined in this article. Hope this is ok. MP ^{(talk•contribs)} 10:47, 15 October 2007 (UTC)

Milogardner's edit

I was forced to revert Milogardner's interesting edit due to incomprehensible sentences such as "Five vulgar fraction conversion date to methods to 1650 BCE, or earlier, the 1800 BCE Egyptian Mathematical Leather Roll". If I could figure out what Milogardner was trying to say, I'd be glad to help include the information. Rick Norwood 12:56, 19 October 2007 (UTC)

Dates

Please be wary of antiquity frenzy. The individual sections should address a reasonable time-span for the respective traditions, not some fancy date in prehistory due to the excavation of some Bronze Age notches that look like numerals. The Early and Middle Bronze Ages should be treated under "early mathematics". It is only from about 1800 BC that we have a recognizable Near Eastern mathematical tradition, from ca. 900 BC in India, and from about 200 BC in China. Yes, there are numerals on tortoise shells from 1300 BC China, and regularly spaced notches from 2500 BC India, but that's not "Chinese mathematics" or "Indian mathematics" in any meaningful historical sense. dab (𒁳) 17:08, 22 June 2007 (UTC)

I'm not expert enough to correct these, but:

http://en.wikipedia.org/wiki/Indian_mathematics dates the Vedic period to 1500 BCE - 400 BCE

http://en.wikipedia.org/wiki/History_of_mathematics#Ancient_Indian_mathematics_.28c._900_BC.E2.80.94AD_200.29 claims that Vedic mathematics begins in the 9th century BC

http://en.wikipedia.org/wiki/Timeline_of_mathematics dates a work of "Vedic India" to 1800 BC

There may be others... Scorwin (talk) 17:06, 19 November 2007 (UTC)

Arabic mathematics

I have a problem with the following paragraph in the article.

"The first known proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.[16] The historian of mathematics, F. Woepcke,[17] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, and using the method of induction, he developed a method for determining the general formula for the sum of any integral powers, which was fundamental to the development of integral calculus."

This paragraph seems to confuse the use of the word "integral" in the calculus sense with the use of the word "ingegral" meaning, "the property of being an integer'. I don't know enough about Arabic mathematics to correct this. It may be that the final phrase should be deleted.

-- Rick Norwood (talk) 20:26, 16 November 2007 (UTC)

Rick, the name of this section should really change. You've named it "Islamic and Arabic mathematics," but virtually every mathematician you cite is Persian. Not only were they Persian, but most of them existed under Persian regimes: for example, Omar Khayyam lived during a time where that particular section of Iran (Nishapur) was ruled by the Samanids, not the Abbasids. (Who, coincidentally, were also heavily Persian-dominated.) Many of the treatises of the time were written in both Persian and Arabic, and as time passed Persian works began to eclipse those written in Arabic. (Eventually, Persian became the lingua franca for the Islamic world, as the Ottoman Empire adopted it for nearly all of their internal proceedings.) That brief history lesson aside, dubbing the section "Arabic" mathematics when the contributors are chiefly Persian is misleading. I suggest we change it to "Persian and Islamic mathematics." (Because not all of the Persian mathematicians of the time were necessarily Muslim--Zoroastrianism remained prevalent until the 11th century.) Let me know what you think. Spectheintro (talk) 19:37, 27 December 2007 (UTC)spectheintro

A statement on History of China

"In China, in 212 BC, the Emperor Qin Shi Huang (Shi Huang-ti) commanded that all books outside of Qin state to be burned."

this is not quite true: the actual command was books held by non-governmental individuals should be burned, while the government still had its own storage of books, which, unfortunately, burned in another war. —Preceding unsigned comment added by 136.152.181.228 (talk) 23:45, 25 April 2008 (UTC)

Eighteenth century

This section is not up to the standard of the other sections. (And the text there presently is largely not about the eighteenth century). Was there some better text there that has been deleted?

Anyway, the section needs a complete rewrite. m.e. (talk) 08:14, 3 May 2008 (UTC)

Speculative History

Why are we including speculation as history fact? There is absolutely nothing indicating the popularity of hypotheses concerning the markings on these ancient bone artifacts and other ancient lunar calendars. Even with the hypotheses themselves, they aren't supported by anything other than speculation.

Based on what I have stated below, I'm finding it hard to believe that these hypotheses are being supported by more than a small minority of experts and that they're uncontested, but this isn't my area of expertise, so someone else would be better at finding sources that argue against them. This page summarizes some criticism from critics, but I couldn't find who the critics are.

Considering that most histories of mathematics aren't including them, it is an NPOV violation to include them. Is there any evidence that this is not a fringe viewpoint?

The lunar calendars are literally nothing more than lunar calendars; the inference that they were used by women to track cycles is based exclusively on the fact that lunar cycles were recorded.

None of the rows on the bones are consistent with each other beyond the sum total of all the tally marks. One supposedly is primes, another is supposedly counting by 10 +/- 1, and yet another is supposedly a random sequence of addition, multiplication and division. There's no rhyme or reason to it.

-Nathan J. Yoder (talk) 09:28, 21 August 2008 (UTC)

I tend to agree. People add their current enthusiasms to articles. In this case, I don't know as much about the situation as you do. Why don't you edit accordingly. Rick Norwood (talk) 12:32, 21 August 2008 (UTC)

I only know what I learned recently; I'm hardly an expert. I created this section so that people who know more than me can voice objections, if any. If there aren't any, I'll revise the section after a while. -Nathan J. Yoder (talk) 02:55, 23 August 2008 (UTC)

21th century?

What about the development of mathematic in 21th century? Newone (talk) 04:32, 11 November 2008 (UTC)

fifth century

There was a mathematician who lived in the fifth century-but his name is unknown. We don't know much about this mathematician either. —Preceding unsigned comment added by 99.245.115.135 (talk) 18:37, 10 April 2009 (UTC)

Misleading/mistranslated Augustine Quote.

The "Medieval European mathematics (c. 500–1400)" section had a quote from Augustine condemning "mathematicians" that was used to support the notion that early Medieval churchmen condemned mathematics as evil:

"The good Christian should beware the mathematician and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of hell."

This quote is often used in this way, but this is based on a modern misunderstanding of the meaning of the Latin word "mathematici", which means astrologers and numerologists and not mathematicians in the modern sense. This can be seen if the quote is looked at in the context of the rest of Augustine's text. By including it in this article this mistranslated quote gives a completely false impression of the early Medieval attitude to mathematics - the Church actually had no problem with mathematics at all. Wikipedia should not be perpectuating this kind of myth. I have removed the quote as a result. TimONeill (talk) 01:04, 16 October 2009 (UTC)

Recent changes

I have taken the liberty to make some changes, particularly to the Greek and Chinese mathematics sections. I have replaced some things that I didn't think were significant advances in mathematics (e.g. the motto of Plato's academy or the mention of the I Ching) with some that were (e.g. the Pythagorean school and the rod numeral system). I have replaced a couple of dubious cites in the Chinese section and added some more, but it could use a lot more. I am also debating whether to move the sentence about the book burning to the beginning of the section or leave it where it is. Any suggestions are welcome. Athenean (talk) 00:46, 21 February 2010 (UTC)

Future of mathematics

The "future of mathematics" section has incredibly poor quality. I keep trying to delete it, but it seems that it must come to a vote; what do you think?18.111.19.44 (talk) 18:33, 29 April 2010 (UTC)

I agree that the section as it stands is piss poor but instead of deleting it why don't you simply re-write so that its good?Thony C. (talk) 21:47, 29 April 2010 (UTC)

I'm not at all sure that predicting the future is a reasonable thing for this article to do. Rick Norwood (talk) 13:24, 30 April 2010 (UTC)

The books "A history of mathematics" by CB Boyer, UC Merzbach, and "Mathematical Thought from ancient to Modern Times" by Morris Kline, finish up with saying a few words about the future, so I think it is quite reasonable that this article should do the same. The last section of Boyer is called "Future Outlook" and it gives a couple of quotes by André Weil. Kline's book doesn't say much about it, but the point is that it does end by making a remark about the future. The section acts as a lead in to the main article on Future of mathematics, and it is appropriate for this article to end with such a section as it is a natural sequel to this article that readers might want to read next, and it reflects the same practice that history books follow in writing their final pages. Bethnim (talk) 15:18, 30 April 2010 (UTC)

Chinese Place Value System

I have reverted the edit from Arjun024 because the Chinese did have a place value number system which is describe on Wikipedia, I have supplied a link and two references. I also removed the Ifrah quote that he had added to the references because it contradicts what Ifrah himself wrote in his Universal History of Numbers where he gives detailed descriptions of the Babylonian, Chinese and Mayan place value systems as well as the Indian one. If somebody can replace my Ifrah reference for the Chinese system with the equivalent section from the English translation (I only own the German one) I would be very grateful.Thony C. (talk) 17:20, 13 May 2010 (UTC)

mean value theorem in Kerala?

The claim that they proved the mean value theorem in Kerala seems wild. To state the theorem in the first place, one needs the concept of the derivative. Does anyone have any information about this? Tkuvho (talk) 04:13, 20 August 2010 (UTC) Will deal with what Kerala did or did not achieve later today. I had to read up on it myself first last night! Bhaskara II did have somethimg approximating to the mean value theorem as he did have something approximating to a derivative.Thony C. (talk) 06:11, 20 August 2010 (UTC)

POV-pushing

This [5] edit by Arjun024 is unacceptable. Most scholarly sources on the subject chronologically place Greek mathematics before both Chinese and Indian mathematics, for several good reasons. First and foremost, the golden age of Greek mathematics occurred centuries before the golden age of the other two civilizations. This does not in any way diminish the accomplishments of other civilizations, it's just a historical fact, just like Egypt and Mesoptamia preceded Greece. The oldest Greek mathematical documents also predate any written documents from India and China (The Bakshali manuscript is only from the 7th century AD). Yet here we have an editor arrogantly dismissing the sum of the literature on the subject with "it's not about how western sources arrange things chronologically, it's about how I do it." In other words, to hell with the expert sources, they are automatically dismissed as "western". This is NOT how we do things here. We follow scholarly consensus, which is clear to anyone who has opened a book on the subject. This [6] is also problematic. Indian Cultural Heritage For Tourism Perspective? I don't think so. Seems like what we have here is a case of "antiquity frenzy", i.e. "we were the first". The Indus Valley Civilzation, fascinating as it was, did not leave behind mathematical treatises, unlike Egypt and Mesopotamia. Athenean (talk) 03:23, 8 July 2010 (UTC)

You have presented the facts completely correctly and have my blessing to revert Arjun24's editsThony C. (talk) 06:16, 8 July 2010 (UTC)

Indian Cultural Heritage Perspective For Tourism is not an acceptable source. Pride of India may be, but it is out of print, and I've been unable to locate a copy or find a review in a major journal. On the one hand, a great deal of modern work on Indian mathematics has been done in recent years, and should be reflected in the article. On the other hand, extraordinary claims require extraordinary proof, and should not be reflected in the article until more widely accepted in academic circles. Rick Norwood (talk) 12:27, 8 July 2010 (UTC)

The claims of Indian priority in the invention of calculus seem to be mostly based on the online item "Science and technology in free India". Government of Kerala — Kerala Call, September 2004. It appears that the most one can argue reasonably is a power series expansion for the arctangent function, exploited to determine pi with substantial accuracy. The claim that this book is the first book in differential calculus is exaggerated. Tkuvho (talk) 16:01, 16 August 2010 (UTC)

Hi Rick. Is Plofker one of the "supporters"? I was unable to find a reference to his book. Where are these allegedly exaggerated claims found? If the only place is the booklet from the government of Kerala, I think we should implement a wholesale deletion of these claims, rather than express scepticism. Tkuvho (talk) 15:28, 17 August 2010 (UTC)

I have Plofker at hand in my office too, and access to a university library with a wide variety of books published in the Indian subcontinent. I too notice that there are quite a few historical claims in articles about India that cannot be verified by indisputable sources, by which I mean that there are plenty of Indian scholars who make much more cautious statements, and none who show rigorous sourcing for some of the kinds of statements I currently see in Wikipedia articles. There does seem to have been considerable POV-pushing in various Wikipedia articles on these points. It seems to be without dispute that the Kerala school of mathematicians made and published some early findings (during approximately the Renaissance period of European history) that anticipated some findings anywhere else in the world, but there is little evidence any of those findings were transmitted outside India, or even to other parts of India. -- WeijiBaikeBianji (talk) 15:42, 17 August 2010 (UTC)

Could you please mention a few examples published by a reputable historian? There is nothing wrong a priori with the fact that a given finding was not transmitted. For example, Bolzano made some remarkable discoveries in analysis as early as 1817, which were not "discovered" by his European colleagues until nearly half a century later. Nonetheless, his results are significant and should be mentioned. If it is true that they invented infinitesimal calculus in Kerala 500 years ago, I will be the first to add this to the article and defend its inclusion vigorously. I suspect such claims are merely periphero-centric (mirror image of euro-centric). Tkuvho (talk) 15:47, 17 August 2010 (UTC)

I'll try to look up what Plofker (the only source I have at hand) says later. The short answer is that my recollection, not aided by sources as I type this, is that the Kerala school investigated Taylor sequences, but didn't develop a full-fledged theory of calculus (that is, a branch of mathematics based on the fundamental theorem of calculus). I could be mistaken in my recollection, but I invite other eidtors to check the sources meanwhile. Yes, I would agree that a lone discovery that doesn't get transmitted to other mathematicians but that can be reliably sourced might still belong in this article, depending on space and emphasis considerations. -- WeijiBaikeBianji (talk) 23:28, 19 August 2010 (UTC)

Does Plofker really say that in Kerala they proved the mean value theorem, as the article currently claims? Tkuvho (talk) 04:09, 20 August 2010 (UTC)

I have removed some of the wilder and ridiculous claims from this section and added references for reliable modern professional sources. On which subject Howard Eves is not a reliable source for the history of Indian mathematics. It is a college textbook written in the 1950s by a mathematician who was not a historian; he deals with the whole of Chinese, Indian and Islamic mathematics in 26 pages! Alone the quote brought by Rick in his footnote shows that Eves doesn't know what he's talking about. Thony C. (talk) 19:56, 19 August 2010 (UTC)

Thanks for the edits. There are some other Western mathematicians who are nonhistorians who make some amazing claims about mathematics around the world, and we have to be careful about sourcing this article. -- WeijiBaikeBianji (talk) 23:28, 19 August 2010 (UTC)

Thank you for providing a better source than Eves. Rick Norwood (talk) 12:11, 20 August 2010 (UTC)

errors in Kerala third order Taylor approximations to sine

The article may be of interest: Plofker, Kim: The "error" in the Indian "Taylor series approximation" to the sine. Historia Math. 28 (2001), no. 4, 283--295. Tkuvho (talk) 15:54, 17 August 2010 (UTC)

Thanks for pointing this out. Especially illuminating is Plofker's comment:

"Sankara’s description of a clever extension of this procedure up to a sixth order approximation makes it clear that the discrepancies between these rules and Taylor polynomials represent not so much "errors" as an entirely different approach to the problem." (p. 293)

--SteveMcCluskey (talk) 15:35, 20 August 2010 (UTC)

Bhaskara II on infinitesimals

Hi Thony, The latest is a very interesting edit. Who was it precisely that wrote about Bhaskara on infinitesimals, and what did Bhaskara say about them exactly? Tkuvho (talk) 14:24, 20 August 2010 (UTC)

The MacTutor article on Bhaskara II has the following comment:

In dealing with numbers Bhaskaracharya, like Brahmagupta before him, handled efficiently arithmetic involving negative numbers. He is sound in addition, subtraction and multiplication involving zero but realised that there were problems with Brahmagupta's ideas of dividing by zero. Madhukar Mallayya in [14] argues that the zero used by Bhaskaracharya in his rule (a.0)/0 = a, given in Lilavati, is equivalent to the modern concept of a non-zero "infinitesimal". Although this claim is not without foundation, perhaps it is seeing ideas beyond what Bhaskaracharya intended.

Mallaya is: V Madhukar Mallayya, Arithmetic operation of division with special reference to Bhaskara II's Lilavati and its commentaries, Indian J. Hist. Sci. 32 (4) (1997), 315-324.

Takao Hayashi (source see footnotes in article) writes: Bhaskara II also used a kind of integration, the summation of infinitesimal parts, to compute the volume and the surface of a sphere in his astronomical work Goladhayaya ('Chapter on the Sphere') (Sengupta 1932) Sengupta, P. C. 1932, "Infinitesimal calculus in Indian mathematics - Its origin and development", Journal of the DEpartment of Letters, University of Calcutta, 22, 1- 17.

Joseph write: In computing the instantaneous motion of a planet, the time interval between successive positions of the planet was no greater than a truti, or 1/33 750 of a second, and his measure of velocity was expressed in this 'infinitesimal' unit of time.

Oppossed to this Plofker writes: It has been noted* that this and related statements reveal similarities between Bhaskara's ideas of motion and concepts in differential calculus. (In fact, perhaps these ratios of small quantities are what he was referring to in his commentary on Lilavati 47 when he spoke of calculations with factors of 0/0 being "useful in astronomy.") This analogy should not be streched too far: for one thing, Bhaskara is dealing with particular increments of particular trigonometric quantities, not with general functions or rates of change in the abstract. But it does bring out the conceptual boldness of the idea of an instantaneous speed, and of its derivative by ratios of small increments.

• For example, in [Rao2004], pp. 162 - 163.

[Rao2004]=Rao, S. Balachandra. Indian Mathematics and Astronomy: Some Landmarks, 3rd ed. Bangalore: Bhavan's Gandhi Centre, 2004.Thony C. (talk) 15:20, 21 August 2010 (UTC)

A question

I noticed that some things in the India section are sourced to this [7]. What kind of source is westgatehouse.com? Who is Dwight Johnson? What kind of editorial oversight is there? I notice there is no bibliography. I accordingly removed the claim about the sidereal year known accurately to a zillion decimal places, though i kept the rest about the sines and cosines being present in the Surya Siddantha. I need to consult my sources on this, but it seems less of a REDFLAG than the sidereal year claim. If it is decided that the source is problematic, it shouldn't be too difficult to source the claim about the trigonometric functions. Athenean (talk) 21:35, 28 August 2010 (UTC)

India section

I merged the "ancient" and "classical" Indian sections into one. I frankly don't see why they need to be split up into two sections. I also replaced some of the more outlandish claims about the Indus Valley Civilization with a sourced passage from Boyer and Merzbach. --Athenean (talk) 05:33, 20 October 2009 (UTC)

Good idea. I was about to do the same myself the other day but didn't. Fowler&fowler«Talk» 08:45, 20 October 2009 (UTC) This section needs to be rewritten as it contains generalisation that distort and exaggerate the facts. I suggest a rewrite based on Plofker.Thony C. (talk) 09:05, 20 October 2009 (UTC)

I've done a little work on the India section. Much more remains to be done. I've fixed some grammatical errors. Since the text sometimes uses the "Oxford comma" and sometimes omits it, to support uniformity of style I've added a comma before "and" in conjunctions of three or more terms. I've also removed some claims that are mathematically impossible, but I have left claims that, while unsupported, are merely unlikely. Anyone here read Sanskrit? Rick Norwood (talk) 13:15, 20 October 2009 (UTC)

A bit, but English sources are usually easier to find. :) The impossible claim I presume is "squaring the circle" (turns out it was a poor approximation); which are the unlikely ones? Shreevatsa (talk) 16:39, 20 October 2009 (UTC)

I've written large parts (the non-listy ones) of the Indian mathematics article. The best thing to do here would be to summarize that article. I haven't got around to the classical mathematics section there, but will do so now that there is some interest. I have all the sources, including Plofker 2009, and Katz 2007 and all the papers. Please see the references in Indian mathematics. I can probably also add references for the Chinese math section. No need for Sanskrit sources; they would be primary ones anyway. Fowler&fowler«Talk» 17:23, 20 October 2009 (UTC)

Shreevatsa: Your edit much improves mine. Rick Norwood (talk) 18:39, 20 October 2009 (UTC)

How about sourcing some parts of the section to Boyer & Merzbach in addition to Katz and Plofker? --Athenean (talk) 01:41, 21 October 2009 (UTC)

__________________________

Are you sure on this sentence ? The reference(66) (67) can be to use in no appropriate manner

Progress in mathematics along with other fields of science stagnated in India with the establishment of Muslim rule in India.[66][67]

Without a good mathematics is possible build Taj Mahal ?..... the mathematics of Moghul is consider India or Islamic mathematics ? word "India" is consider geographic or cultural ? This sentence must be precised. About word "science" there is a epistemological question. I am not sure that this word can be used outside western culture before the industrial revolution colonialism. --84.222.77.75 (talk) 20:41, 22 December 2010 (UTC)

Other sentence but in islamic section. This sentence has non sense.

During the time of the Ottoman Empire from the 15th century, the development of Islamic mathematics became stagnant.

Became stagnant because is impossible to define an "islamic mathematics" into Ottoman Empire. Ottoman Empire lasted from XIV to XX century. In XIX century the Empire had steam boats, steel guns, gas lamps and Constantinople was a big cosmopolitan town as Paris, Wien, London, Moscow etc. After italian renaissance, is not easier to divide islamic mathematics from western mathematics into Ottoman Empire because the cultural intelectual relations was as today. Books printed in Venice during XV century, were read in Alessandria, Constantinople, Tunis ecc., in every part of mediterranean. As today in the globalize world is impossible to divide chinese mathematics from western mathematics. Is a nonsense. I would consider Ottoman Empire as European state, it was integred in "the game of European powers". You rebembered "the patiente of Europe".

--84.222.77.75 (talk) 22:37, 22 December 2010 (UTC)

Andriolo

Prehistory section

I expanded this section, re-arranged, and added some references. Hope that is fine with everyone. If this article is to achieve GA status, it will need some expansion and especially more sources. --Athenean (talk) 07:03, 20 October 2009 (UTC)

Athenean: You are doing good work, but it is a truism that scholars should not assume that ancient hunter gatherers resembled modern hunter gatherers. Rick Norwood (talk) 12:49, 20 October 2009 (UTC)

Thank you for your comment, Rick. I see your point, however, I would like to point out somewhere that mathematical ability would not have been without survival value to hunter gatherers. Mathematical ability, like all human traits is subject to natural selection and would not have evolved in humans if it didn't confer a survival advantage. Perhaps I shouldn't say that it was of survival value to early hunter gatherers as opposed to current hunter-gatherers, so what if I just removed the "early"? --Athenean (talk) 01:39, 21 October 2009 (UTC)

While the "one, two, many" mathematics of modern hunter-gatherers in, for example, Borneo, is interesting, I don't think it played any part in the history of mathematics. For the prehistory of mathematics, I think it is best to rely on physical evidence (as the article now does) rather that speculate, however plausible those speculations may be. Rick Norwood (talk) 12:17, 21 October 2009 (UTC)

Number systems are the foundation of all mathematics and it can be shown liguistically that almost all natural number systems started out as one, two, many (the word three in almost all languages is derived from a word meaning many) so this is more than speculation and a valid claim for the prehistory section of an article on the history of mathematics. (for details see Karl Menninger "Number Words and Number Symbols")Thony C. (talk) 13:21, 21 October 2009 (UTC)

Sounds good to me. It was the story about counting antelope that seemed to me to cross the line. If you add it to the article, be sure to include the reference. I teach History of Math, and I had not run across three = many before, though I had run across seven = several. Rick Norwood (talk) 14:39, 21 October 2009 (UTC)

Is it possible to get some examples of these languages not having words for numbers higher than two? This seems very strange to me, I know how to count in several languages and I can't think of one where the word for three is even remotely similar to the one for many. —Preceding unsigned comment added by 196.2.126.171 (talk) 12:29, 24 December 2010 (UTC)

Modern math revolution started with Italian mathematician of XVI century, they are not mentioned at all

Scipione del Ferro, Cardano, Tartaglia discovered general solution to cubic and quartic equations, it was a major achievement since the quadratic solution was already known to Babylonian. Cardano's "Ars magna" is the first modern math book, without Bombelli's imaginary number discovery we wouldn't be able to operate electricity or telecomunication devices. All these men marked the start of west supremacy on math which last till today. It was one of the most revolutionary period in the history of math. I think they not only deserve a mention (which is not present) but a whole paragraph. Magnagr (talk) 09:26, 6 November 2010 (UTC)

Math is as music, exist in the man mind (perhaps in the Λóγοσ mind), and the man mind discovered its when he has the need.

"Philosophy is written in this grand book which stands continually open to our eyes (I mean the universe), but it can not be understood without first learns to comprehend the language and know the characters. It is written in mathematical language, and its characters are triangles, circles and other geometric figures" I have traslated a sentence of a big italian that is Galileo Galilei, reported on italian item of wiki on Galileo.

I agree with you that this article still needed of job. But I don't to think, perhaps i have, misinterpret, that exist an italian mathematics. The scientific culture in western europe was unitary. The student goes in England in France in Italy in Germany ecc. They studied in the international universities in latin and followed the "important" professors. The corrispondences between professors (famous are the letters between Galileo to Kepler) were as XX century. After the discovered, the telescope in few months invaded europe.

--84.222.77.75 (talk) 23:30, 22 December 2010 (UTC)

Andriolo

I see that they are cited in the article, perhaps for Tartaglia and Bombelli some more lines...

Andriolo--84.222.74.135 (talk) 18:02, 24 December 2010 (UTC)

The decimal system.

"The contribution of Indian mathematics was also of great importance in the early Middle Ages, giving rise to the decimal system of mathematics that was later adopted in Europe and that is used in present day.(reference)"The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." - Pierre Simon Laplace http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html"

The above was deleted. The only problem I can see is the word "early". Please explain why the decimal system is not important enough to mention. Rick Norwood (talk) 13:21, 14 October 2009 (UTC)

I agree totally with Rick and don't understand why this line was deleted. I also find the use of the Laplace quote as a footnote perfectly in order. If you require other sources for the importance of the place value decimal system in the history of mathematics I can supply a whole truck load! Thony C. (talk) 14:08, 14 October 2009 (UTC)

I removed it because it is an oversimplification, and it is not properly sourced. The story of the decimal system is far more complex and nuanced than that. It is true that India played a major role, but the Babylonians, for example had a similar placeholding system, even though it was sexagesimal rather than decimal, and Chinese mathematics also had a way of expressing numbers as the sum of powers of 10. Laplace is moreover somewhat of a primary source. I also note that the editor who added this passage has a history of boosterism, so I think I can be forgiven for not giving him the benefit of the doubt. Now, if one of you guys could provide a contemporary, secondary source that says something to the effect of the Indian contribution is of major significance because it developed the decimal placeholding system, I would gladly re-instate it. --Athenean (talk) 21:13, 14 October 2009 (UTC)

The idea of the same symbol representing both a number and certain multiples does, as you note, go back to the Babylonians, and such a number system, called a based number system, was discovered independently by several cultures, including the Chinese. The Indian contribution was the use of the zero as a place holder. This idea spread to the Arab empire and from there to the rest of the world, and its use is now universal, except for archaic numbers such as the Roman Numerals. Here is another reference, from Howard Eves' "An Introduction to the History of Mathematics", "The earliest preserved examples of our present number symbols are found on some stone columns erected in India about 250 B.C. by King Asoka." Rick Norwood (talk) 14:01, 15 October 2009 (UTC)

While I agree with what you are saying, the source you have presented says something very different from the sentence I removed. Your source talks about the number symbols, and I do not dispute that. The sentence I removed said something completely different. Moreover, although the current use of zero as a placeholder undoubtedly comes from Indian Mathematics, the use of zero as a placeholder is also found in Mayan mathematics (which are missing from this article). So it's like I was saying: The development of the modern numeral system is a far more complex story than the rather blunt sentence that I removed would have us believe. I suppose we could introduce a more suitably worded sentence to that effect in the lead though. On another note, while a tremendous amount of good work has been done in this article, there are a number of improvements I would suggest:

• The Indian sections should be merged into one (seems odd to have two).
• A Japanese and Mayan mathematics sections would be nice. These are notable mathematical schools.
• The Islamic mathematics section should be somewhat pruned to be of similar length to the others.
• More on the development of mathametical rigour and the Hindu-Arabic numeral system in use today.

It is my goal to eventual to goal to raise this article to GA status. It is a pity that such an important article is B class. I have wanted to do this for a long time, but simply didn't have the time. I plan to devote to it some more time now though. Let me know of any suggestions you might have for improving the article. Regards, --Athenean (talk) 23:50, 15 October 2009 (UTC)

For a general history of mathematics, and especially the lead, it seems appropriate to mention the developments that led to modern mathematics, whether or not earlier/related developments happened elsewhere. The sentence that was removed was "The contribution of Indian mathematics was also of great importance in the early Middle Ages, giving rise to the decimal system of mathematics that was later adopted in Europe and that is used in present day", which seems to be mostly accurate and close to what most history books say (but it omits the Islamic contribution). The Chinese and Mayan developments, though we should certainly mention them, are not direct ancestors of the Hindu-Arabic numeral system in use today. Shreevatsa (talk) 03:39, 16 October 2009 (UTC)

I agree that a sentence regarding the development of the Hindu-Arabic numeral system in the lead is in order, given it's importance. What I object to is the way it was done previously. I removed that sentence because it was PEACOCKy and an oversimplification. That the Hindu-Arabic numeral system is of fundamental importance is understood by all, there is no need to say that is is important. Thus, I would propose something along the lines of The Hindu-Arabic numeral system, in use throughout the world today, evolved over the course of the first millenium AD in India and was transmitted to the west via Islamic mathematics. Better to have a sentence that conveys more information to our readers and shows how the system developed rather than telling them it is important (i.e. show, don't tell). --Athenean (talk) 04:36, 16 October 2009 (UTC)

I agree; your sentence is better. Shreevatsa (talk) 13:57, 16 October 2009 (UTC)

I also agree that your sentence is an improvement on what was there before and had you added it whilst removing the previous sentence then I would not have commented. I will now however comment on a false statement from both Rick and yourself in the discussion concerning the role of zero. In the Babylonian and Mayan number systems the zero is indeed just a placeholder but in the Indian number system, as presented by Brahmagupta in the 6th century CE, zero is a number, the difference is subtle but highly significant and is what distinguishes the Indian from all other early number sytems. Thony C. (talk) 15:08, 16 October 2009 (UTC)

"Athenian"s removal is ridiculous and unwarranted. If the said user disagreed with the wording, then it could be ammended. However, givin the users name as well as the POV style edititing removals, it is safe to say he is simply trolling. Clearly it is ludicrous not to mention that the decimal system that we posess, derives from India. It is the first recorded system that included zero as a number, hence its importance. I have altered the wording slightly in line with the suggestions.

KnowledgeAndVision (talk) 14:00, 18 October 2009 (UTC)

Givin (sic) the users name as well as POV style editing...Excuse me? Is there a problem here? The fact the numeral system evolved in India is clearly mentioned; I don't understand what your problem is. Incidentally, the only trolling and POV editing here is from you. As you can see, a number of users think my wording is preferrable to your uninformative PEACOCKy wording. Looking at you contribs log, I see you have a history of POV pushing and personal attacks. Further personal attacks and trolling will not be tolerated. The next step is reporting you for disruption. --Athenean (talk) 19:33, 18 October 2009 (UTC)

Let's avoid ad hominem attacks, please. Incidentally, the sentence previous to the one in question says "The Greek and Hellenistic contribution [...] is widely considered the most important for greatly refining… and expanding…" (emphasis mine). This seems against the spirit of "Show, don't tell", too (and might have been what prompted the "was also of great importance" sentence about Indian mathematics in the first place). How about changing it to "Greek mathematics greatly refined… and expanded…"? Shreevatsa (talk) 20:11, 18 October 2009 (UTC)

You're probably right. How about Greek mathematics greatly refined and expanded the scope of methamatics, particularly through the introduction of rigor, and was thus the first to treat mathematics as a science. (sourced to Heath). --Athenean (talk) 21:00, 18 October 2009 (UTC)

Enough with the rather petty and childish attacks. Kindly "Athenian", refrain from this type of behaviour. It is not beneficial to Wikipedia. I wil only say that "Athenian" should follow his/her own advise. I am unaware of any 'history of boosterism', it would seem the personal attacks have been made only by yourself. I also do regard total removals of cited sections such as when you removed the Indian contribution paragraph as blatant trolling. You are correct, that type of behaviour will not be tolerated. So don't do it. The part about Greek contribution being 'the most important' has also now been ammended, as it is a clear POV statement.

KnowledgeAndVision (talk) 08:59, 19 October 2009 (UTC)

I have phrased the paragraph to be more inclusive, as the contribution of Indian mathematics was not limited to just the decimal system. The wording now reflects this.

KnowledgeAndVision (talk) 09:23, 19 October 2009 (UTC)

Would you care to explain which other contributions were made by Indian mathematics that are important enough to be mentioned in a three line lead in to an article that covers about 1000 years of maths history? Secondly your amendments are simply a return to boosterism so I suggest you rephrase your latest contribution to a more moderate and therefore more suitable form of expression.Thony C. (talk) 10:03, 19 October 2009 (UTC)

The leading section, currently 16 lines, is certainly where the Indian, Greek and Islamic mathematical contributions must be mentioned. To say that the Indian contribution is greatly significant is not in the slightest bit controversial nor is it POV. I suggest you try to explain how you think that statement is in anyway 'boosterism'. It is a fact frequently mentioned by historians. If you think it should be worded differently then kindly put forward your alternate suggestions. The Indian contribution to Algebra and calculus, amongst other areas, are two other extremely significant contributions that rank alongside the decimal system. Therfore, the wording should refelct that the Indian contribution as a whole has been extremely significant in the history of mathematics, the most notable contrbibution being the decimal system.

KnowledgeAndVision (talk) 12:33, 19 October 2009 (UTC)

The Indian contribution to the development of algebra consists of the rules for the arithmetical operations which I have now included in the text. Whilst it is true that mediaeval mathematicians in India made important discoveries in the theory of infinite series, earlier than their European counterparts, that are important in the prehistory of calculus these discoveries were made in isolation and there is no evidence, to date, that they played a role in the general development of mathematics. Therfore their presence in an article on the history of Indian mathematics is wholly justified but in a general survey outlining the historical development of mathematics questionable. Thony C. (talk) 14:00, 19 October 2009 (UTC)

There is ample evidence, that the Indian contributions towards calculus and algebra were very significant in the development of mathematics. Your statement that 'these develepents took place in isolation' is nonsensical, and there is no basis to support it. All of the evidence that we have in fact indicates Islamic mathematicians were very much influenced by these developments, and they helped to form the bedrock of the Islamic mathematical developments that were later transmitted to the West and elsewhere. Therefore, it is fair to say that the decimal system was a significant part of the wider Indian contribution to mathematics.

KnowledgeAndVision (talk) 17:31, 21 October 2009 (UTC)

Instead of making empty unsupported claims why don't you give some reliable sources to back up what you are saying? The algebra that came into Europe from the Arabs is based mostly on Greek and Babylonian sources. The unique Indian contribution was the decimal number system and its rules of operation. There is no Indian calculus! The Kerela mathematicians in the middle ages did some very impressive work in the theory of infinite series that would lay the foundations of integral calculus but did not take this further step. Also to date there is no evidence that their work was ever known outside of Kerela.Thony C. (talk) 13:22, 22 October 2009 (UTC)

Not quite. Islamic mathematics was much more heavily influenced by Greek and Hellenistic mathematics (and by the way, Indian mathematics also shows Greek and Hellenistic influence), and in the history of mathematics books I've read The roots of calculus and analytical geometry are Greek rather than Indian (Boyer and Merzbach, p. 224). --Athenean (talk) 19:04, 21 October 2009 (UTC)

You make the unsupported claim that Algebra that came with Islamic mathematicians was based mainly on Greek sources. Where is your basis for making this statement? Why don't you provide some sources? The Greeks didn't even have a decimal system to work with. Their contributions were necessarily limited compared with Indian developments in the field of algebra. Where the Greeks really excelled, was trigonometry. Islamic mathematics was based on Indian developments primarily, the Greek influence was also there but to a lesser extent.

KnowledgeAndVision (talk) 17:42, 22 October 2009 (UTC)

Try any standard work on the history of mathematics e.g. Boyer, "A History of Mathematics", Grattan-Guinness, "The Rainbow of Mathematics", Gericke, "Mathematik in Antike, Orient und Abendland" or the chapter "Pure Mathematics in Islamic Civilisation" by J. P. Hogendijk in "Companion Encyclopedia of the History & Philosophy of the Mathematical Sciences" ed. Grattan-Guinness all of which emphasise the origines of Islamic algebra in the geometric-algebra of the Greeks. The latter being something you apparently have never heard of. You also have apparently never heard of Diophantus in your dismissal of Greek algebra. If you wish to edit an article on the history of mathematics you should at least take the trouble to inform yourself on the subject. Thony C. (talk) 12:27, 23 October 2009 (UTC)

I've replaced the peacocky text with more neutral and accurate text, cited to modern sources. Clearly Laplace is not a modern source. Fowler&fowler«Talk» 12:41, 19 October 2009 (UTC)

Replaced Plofker with Kaplan as reference because Kaplan is a book specifically about the genesis, evolution and transmission of the place value decimal sytem with zero whereas Plofker is a complete history of mathematics in India, as such I find that Kaplan is a more appropriate source for this statement.Thony C. (talk) 13:26, 19 October 2009 (UTC)

That's fine, although I wouldn't include mention of the rules of arithmetic as a uniquely Indian achievement. The Chinese place value system (using counting rods) circa 500 BCE had rules for multiplication and division as well. Katz's book has an extensive discussion of this. Kaplan, by the way, is generally not considered an academic source. However, if you want to use to source the shorter statement that the decimal numeral system in use today evolved in Indian mathematics, that would be fine. Fowler&fowler«Talk» 21:26, 19 October 2009 (UTC)

I did some further reading on the decimal placeholding system, and the picture that emerges is far more complex than I first thought. From Boyer & Merzbach, Ch. 12:

• From the Brahmi ciphered numerals to our present-day notation for integers two steps are needed. The first is a recognition that, through the use of the positional principle the ciphers of the first 9 units can serve as ciphers for the corresponding multiples of ten...It appears from extant evidence that the change took place in India, but the source of the inspiration for the change is uncertain. Possibly the so-called Hindu numerals were the result of internal development alone; perhaps they developed first along the interface between India and Persia, where remembrance of the Babylonian positional notation nay have led to modification of the Brahmi system. It is possible that the newer system arose along the eastern interface with China where the pseudopositional rod numerals may have suggested the reduction to nine ciphers. There is also a theory that this reduction may first have been made in Alexandria within the Greek alphabetic system and that subsequently the idea spread to India.
• The history of mathematics holds many anomalies, not least that the...earliest undoubted occurence of zero in India is in an inscription from 876...more than two centuries after the first reference to the other nine numerals...It is quite possible that zero originated in the Greek world, perhaps at Alexandria, and that it was transmitted to India after the decimal positional system had been established there.
• The new numeration...is mereley a new combination of three basic principles, all of ancient origin: (1) a decimal base (2) a positional system and (3) a ciphered form for each of the ten numerals. Not one of these three was due originally to the Hindus, but it presumably is due to them that the three were first linked to form the modern system of numeration.
• Nevertheless, in modern mathematics there are at least two reminders that mathematics owes its development to India as well as many other lands. The trigonometry of the sine function came presumably from India; our own system of numeration for integers is appropriately called the Hindu-Arabic system to indicate its probable origin in India and its transmission through Arabia. (emphasis mine) --Athenean (talk) 05:10, 20 October 2009 (UTC)

Yes, please see the section Indian_mathematics#Numerals_and_the_Decimal_Number_System for an accurate description (according to the latest understanding) of the Indian contribution. Please also see the lead section for the sine definition. That's why I prefer to simply say that the decimal system in use today was first recorded in Indian mathematics. That way, we don't have to worry about exactly where and how it evolved. Fowler&fowler«Talk» 08:43, 20 October 2009 (UTC)

I am well aware of the fact that Kaplan is a so called popular book but it does however fulfil the criteria for a Wiki reliable sorce. I suggested it because it deals with all of the points raised by Athenean in his last comment above. On the question of the arithmetical operations I think that the main purpose of a general article on the history of mathematics should be as follows. There is a universally acknowledge body of mathematical knowledge that is taught in schools and universities throughout the world. This article should outline the origins and sketch the historical lines of transmission of the principle elements of that body of knowledge. Now whilst I would not argue against the fact that not only the Chinese but also the Babylonians, the Egyptians and also other culture had arithmetical operation it is an established fat that the arithmetical operators and the rules for their use were introduced into mainstream mathematics (i.e. that body of mathematical knowledge) through the works of Indian mathematicians most notably Brahmagupta. To quote Plofker, whom you seem to be rather fond of, "Brahmagupta is the first known mathematical author to classify arithmetic topics [operations and procedures] in this way..." It is of course perfectly correct to also mention that other cultures also indepently developed arithmetical operations but the recognition for having introduced that knowledge into the universal body of mathematical knowledge goes, without question, to the Indians.Thony C. (talk) 09:43, 20 October 2009 (UTC)

Sounds good to me. We speak the same language. I'm happy to help you with the article if you are planning a GA run. Regards, Fowler&fowler«Talk» 09:49, 20 October 2009 (UTC)

I think it would be hard to show that Brahmagupta "introduced that knowledge into the universal body of mathematical knowledge". Rather, from what I've read, it seems that Brahmagupta's work was largely forgotten, and only rediscovered after the decimal system was already well established. (Every civilization has rules of arithmetic. Mathematically, there is no difference between 2*3 = 6 and II*III=VI.) Brahmagupta set out rules for the decimal system in a systematic way. This was a great achievement. But the transmission of the decimal system, through Arabic and Jewish sources was probably piecemeal. I'm not downplaying what Brahmagupta did. I'm just saying that Regiomontanus, for example, probably never heard of it, but got his ideas from Arab mathmaticians, who in turn also probably never heard of Brahmagupta. To credit Brahmagupta with introducing these ideas into the universal body of mathematical knowledge, you need to show a line of descent. Rick Norwood (talk) 12:41, 24 October 2009 (UTC)

"Brahmagupta holds a special place in the history of mathematics. As we shall see in the next chapter, it was partly through a translation of his Brahma Sputa Siddhanta that the >Arabs, and then the West, became aware of Indian astronomy and mathematics. This was to have momentous consequences for the development of the two subjects." G. C. Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, Penguiguin Books, London, 1992, p. 267.

It is a well established fact that the Islamic mathematicians acquired their knowledge of the decimal system and the arithmetical operations from Indian sources which they themselves freely acknowledge. It is also a well known fact that this knowledge came into Europe through al-Kwarizmi who actually calls them Indian numbers. Regiomotanus got his knowledge of the decimal system from European sources because by the time he was active the decimal system had been being taught in Europe in the Universities as Algorism and in the Reckoning Schools of the North Italien and Southern German traders for two centuries and was as a result very widespread and totally accepted. Thony C. (talk) 09:34, 25 October 2009 (UTC)

Thanks for providing an excellent reference. Rick Norwood (talk) 13:45, 25 October 2009 (UTC)

I think that we confuse the positional system with the decimal system.

About the origin of the decimal system, if we count the fingers of both hands we can to understand that it is not exclusive to one culture. However, the decimal system in Europe is fully used only with the Napoleon metric standard. But for the money and the time, the agrimensura the system remain no-decimal and for agrimensura remain non standard with differences between cities and villages.

In ancient times the Aegean system and the Attic system were decimal but non positional. A strange positional systems was present in Ionian or Milesian system that make the count easier. The The Hellenistic astronomers used a sexagesimal positional numbering system including a special symbol for zero. The greek zero is used today in sets algebra as empty.  used also from Byzantines. Simple cardinal number count methods (roman), were teachers in some italian public schools until recently to learn the use of Abaco.

In the West the Indian-Arabic method has been adopted by Italian merchants of Italian city-states for standardized reasons to resolve questions about mesures, payments, taxes in the North Africa and Central Asia emporias. At the time when the italian merchants adopted arabic numerals, the economies in the Islamic world were the richest. Otherwise it could to be possible to use in the positional manner the Milesian system but at the time the Greek world was not more at the center of the world.

Andriolo --84.222.75.217 (talk) 09:35, 4 February 2011 (UTC)

P.S: Today we must learn the Chinese....... if we want to do business with China.

In the “accounts books” of Italian merchants of the XIV century, the Arabic positional system never completely replace the Roman cardinal system. There were lot of merchants that preferred the Roman system. Only recently, after the Scientific Revolution and with Illuminism the use of Arabic numerals became prevailing. Soon if I will have a time I will provide a reference on this. --Andriolo (talk) 21:08, 7 February 2011 (UTC)

"Father of..."

In order to try and forestall and edit war over al Khwarizmi I am in total agreement with the removal of the expression "father of" not only here but everywhere where it occurs in connection with the history of science. It is a meaningless and usually incorrect piece of rubbish. I can list at least a dozen fathers of the computer and half a dozen fathers of optics without trying. In terms of the history of algebra and the relative status of al Khwarizmi and Diophantus the discussion completely ignores the achievements of the Old Babylonians who were developing surprisingly advanced algebraic concepts a couple of thousand years before al Khwarizmi started crapping in his nappy and those of the Indian mathematicians from whom he borrowed a lot of his work.Thony C. (talk) 10:41, 18 February 2011 (UTC)

If you want to be taken seriously, adopt a more serious tone. Rick Norwood (talk) 19:00, 18 February 2011 (UTC)

Pardon? Was that intended to be a constructive contribution? William M. Connolley (talk) 19:13, 18 February 2011 (UTC)

Rick, I'm well aware of the fact that you regard this page as your private property but I certainly don't need instructions from you or anybody else for that matter on how I should or should not conduct myself and if you don't like it that's your problem.Thony C. (talk) 19:34, 18 February 2011 (UTC)

There are "only" three Fathers of algebra : Diophantus, Al-Khwarizmi and Evariste Galois (father of modern algebra) and one mother : Emmy Noether [8]. --El Caro (talk) 19:43, 18 February 2011 (UTC)

And what is with Cardano, Bombelli and Viète not to mention Harriot and Descartes Thony C. (talk) 20:37, 18 February 2011 (UTC)

I agree that the whole "Father of X" is kinda silly and unnecessary, not to mention a little arbitrary and old fashioned. Athenean (talk) 22:08, 18 February 2011 (UTC)

Fictitious portraits of mathematicians

I propose that we do away with fictitious portraits of mathematicians whom no one had any idea what they looked like (Pythagoras, Archimedes, Al-Khwarizmi, etc), and replace them with illustrations of important mathematical discoveries from that era. For example, a while back I removed an image of a rather tacky-looking statue ofAryabhatta (no one has any idea what he actually looked like, the statue was an entirely fictitious image). A good replacement would be an image of the Hindu numerals, arguably the most significant mathematical discovery to come out of India. Similarly, I was thinking of replacing the portrait of Pythagoras with an illustration of the theorem that bears his name, and that of Archimedes with an image of his use of the method of exhaustion to estimate the value of Pi. Athenean (talk) 02:50, 21 May 2011 (UTC)

Great idea, go ahead Thony C. (talk) 06:03, 21 May 2011 (UTC)

Great, I'm glad you agree. Athenean (talk) 18:04, 21 May 2011 (UTC)

Great idea. We should go through all the history articles and remove the vaguely obscene indian statues, the remainders of Jaggedalia, as well as remaining references to treatises by one Ian Pearce, which unfortunately continue to pollute the mathematics pages. Tkuvho (talk) 20:53, 21 May 2011 (UTC)

Yes, definitely. That one really tacky one of Aryabhatta especially. I will begin with cleaning up this article, then move on to specific sub-articles (geometry, trigonometry, etc.).

There is no problem with referencing Ian Pearce if one actually quotes what he wrote and doesn't quote mine or distort him.Thony C. (talk) 08:20, 22 May 2011 (UTC)

You seem to have missed the discussion at WPM. The consensus in the community is that Ian Pearce is unreliable. His pieces are student projects and not the main MacTutor historical pages. To give you an example, he claims in one of his pieces that Bhaskara had as good an understanding of calculus as Newton. This appears to be unsourced. At any rate it impressed few people at WPM. Tkuvho (talk) 09:38, 22 May 2011 (UTC)

I withdraw my previous comment. I had in earlier edits followed dubious claims about Indian maths in the Wiki articles back to Pearce as supposed source only to discover that Pearce had not made the claimed statement; I had never bothered to read the whole article as I own all the original texts he references. However you are quite correct he does indeed make the mind boggling stupid claim about Bhaskara that you paraphrase (by the way without any source!) and as such completely disqualifies himself as source for anything.Thony C. (talk) 18:43, 22 May 2011 (UTC)

OK, that was the consensus at WPM as well. Even if that particular sentence is removed, his article at MacTutor is not part of their official biography database, and appears at a different web address. It's a student project, and certainly can't be used as a reliable source. A quick web search reveals that in the intervening decade, he has not published anything, which tells us something about what the historical community thinks of his competence as a historian. Tkuvho (talk) 04:34, 23 May 2011 (UTC)

Number of Erdos publications

The article currently mentions (in the 20th century mathematics section) that Erdos published more papers than any mathematician in history. This is 'technically' not true, in that Euler actually proved more theorems and generated more mathematical literature than Erdos did. However, Euler lived in a time when peer-reviewed publications did not exist, so maybe attributing the most number of publications in that sense to Erdos is okay, although I think this might be somewhat misleading. The statement that I have heard to be unequivocally true is that Erdos had more coauthors than any other mathematician in history. Any thoughts on whether the statement should be changed from its current form? SnehaNar (talk) 09:57, 4 June 2011 (UTC)

Scientific Revolution

What on earth is a heading like "Scientific Revolution" doing in an article on mathematics (Boyer or no Boyer)? AWhiteC (talk) 23:22, 7 June 2011 (UTC)

We could go with "Early Modern Mathematics" or something like that. Athenean (talk) 00:13, 8 June 2011 (UTC)

From context, it appears to be short for "Mathematics during the Scientific Revolution". I have changed the sub-head accordingly. Gandalf61 (talk) 10:39, 8 June 2011 (UTC)

I see two problems with the suggested change 1) It is currently disputed in the history of science whether there ever was a scientific revolution 2) Even if one ignores this dispute the conventional theory says that the scientific revolution started in the 16th century whereas the section so labeled begins in the 17th century. A similar problem exists with the label 'early modern' as the Early Modern Period is considered by many historians, including myself, to begin in 1400.Thony C. (talk) 11:07, 8 June 2011 (UTC)

If we accept the Scientific Revolution categorisation as at all useful, then it follows the Renaissance and the dividing line is normally taken to be the publication of De revolutionibus in 1543 (see History of science in the Renaissance). So, as you say, the final parts of the "Renaissance mathematics" section are misplaced, and the "Scientific Revolution" sub-head should actually appear 2 or 3 paragraphs further up. Or you could drop the whole Renaissance/Scientific Revolution division and just use the century divisions. I don't have strong views either way. Gandalf61 (talk) 12:29, 8 June 2011 (UTC)

Recent undo.

I just reverted a rather large addition, so I thought I should explain. A quick google test showed that the addition seemed to originate at this blog. Thenub314 (talk) 16:31, 17 June 2011 (UTC)

Oops, my bad. Saw a large removal by an IP and thought it was vandalism (happens frequently here). Athenean (talk) 16:46, 17 June 2011 (UTC)

No prob., perhaps you could be good enough to look into the large removal at Decimal for me? I have an appointment to keep and can't check it out now. Thenub314 (talk) 16:57, 17 June 2011 (UTC)

African Mathematics section

I hate to do this, but upon closer inspection, there are several issues with this section, detailed below.

• The ancient Nubians used a trigonometric methodology similar to the Egyptians. - Actually the only mention of the Nubians I see in the source is "Hence we can say that Armenia, Syria, and Nubia best preserved the original Greek pattern". Not only does the source say something completely different from what is claimed in the article, but it only mentions preservation, not any original contribution.

• Τhe second sentence belongs more to History of Astronomy, moreover the second source used in it refers to Neolithic peoples, not Meroe.

• The Axumites (fl. 100-900 AD) were certainly not "among the first to study mathematical astronomy", as they were preceded by several thousand years by the Egyptians, Babylonians, and other. This sentence is peacockery.

• That their astronomical knowledge derives from the book of Enoch and their astronomical calendar was based on the Egyptian one is again more relevant to History of Astronomy than here. I would be happy to move the content to that article.

• The only information that is truly about mathematics is that regarding the binary system, but the sources used are dubious (geez.org, science20.com).

• Regarding fractals, I am sorry, but there is nothing to corroborate the claim about "Advanced knowledge of fractal geometry". As far as I know, "advanced knowledge" of fractals did not occur till relatively late in the history of mathematics, in the late 19th century. I am moreover unable to verify the source as no page number is given.

• The last paragraph really belongs to Medieval Islamic mathematics. I mean, Ibn al Yasamin is only "African" in a very narrow geographical sense. He was a product of the medieval Islamic civilization, as were many other North African philosphers and mathematicians.

In light of this, I propose that poorly-sourced material be removed, and material that belongs better in other articles be moved to those articles. Athenean (talk) 23:12, 1 September 2011 (UTC)

Certainly the paragraph makes use of a creative interpretation of the sources cited. I could only verify to some extent the second sentence, but, as you say, this belongs more to the field of astronomy. Removing poorly-sourced material may be well amount to removing the paragraph given that the references curiously die out altogether wherever the stronger claims are put forth. Gun Powder Ma (talk) 18:34, 2 September 2011 (UTC)

I concur with both of you. I found the section very suspect but knowing nothing about so-called "African" mathematics I have refrained from commenting till nowThony C. (talk) 10:14, 3 September 2011 (UTC)

Hello, I wrote part of that section--namely the material on Axum and Ibn al Yasamin. I'd like to address some of the issues that Athenean raised.

First, the act of measuring time via sundials and triangles (this is the "trigonometric methodology" referenced in the first paragraph) was indeed developed in Egypt. Nevertheless, Athenian is correct in stating that Nubia, along with Armenia and Syria, merely preserved that practice. As did Greece.

Second, the claim that the Axumites were among the first ancient peoples to study mathematical astronomy is not negated by the precedence of Egyptian and Mesopotamian astronomy. The fact of the matter is, relative to most people, in most parts of the world, in the years between 100 and 900 AD, the Axumites' knowledge of Astronomy (and hence, mathematics) was quite profound, if not entirely "original". And still other things mentioned would undoubtedly constitute original advances (i.e., the Axumite numerals and system of multiplication). Of course, I acknowledge that neither of these things played a role in shaping mathematics in Europe (and thus, played no role in shaping modern mathematics). So, your deleting the section is understandable.

Regarding Ibn al Yasamin: He was African in more than a narrow geographical sense. The following is a direct quote from the Encyclopedia of Islamic Science and Scientists, pg. 454. "As his name indicates, he was originally from a Berber tribe from the Maghreb (North Africa), and, according to Ibn Sa'id, he was black like his mother."

North African/Maghrebi is not synonymous with "non-black", if that's what you meant by "African in a very narrow geographical sense". Vilehumanbeing (talk) 15:30, 20 September 2011 (UTC)

What I meant was that culturally (and therefore mathematically) he was part of medieval Islamic civilzation, not of a sub-Saharan African civilzation (like, say, the Great Zimbabwe or Benin). It's culture and mathematical tradition that matter, not biological race. So if he is to be included in the article at all, which I don't think he should be, as he is a relatively minor figure, it would be in the Islamic section, not an African section. Athenean (talk) 21:44, 20 September 2011 (UTC)

I understand that culture and mathematical tradition is the topic at hand. And I agree that al Yasamin's contributions to mathematics are dwarfed by what is cited in the Islamic section. If the goal of this article is to exlplain the origins of fundamental and well known results, I see no harm in not including al Yasamin. In retrospect, I'd say that the material posted under the African section was better suited for a survey of mathematics from around the world. In the future, less known traditions from sub-saharan Africa (and the Maghreb) will be reserved for such an article. I understand that "The History of Mathematics" deals exclusively with topics that are paramount to the development of modern mathematics. That's perfectly fine. Vilehumanbeing (talk) 21:40, 20 September 2011 (UTC)

The last Greek mathematician

I'm removing the word "last" for Archimedes. Apollonius of Perga came after Archimedes and was no minor figure.

Perga is in Asia Minor, not in Europe. Rick Norwood 22:37, 22 January 2006 (UTC)

From that section - "Some say the greatest of Greek mathematicians was Archimedes 287 BC - 212 BC of Syracuse. At the age of 75, while drawing mathematical formulas in the dust, he was run through with a spear by a Roman soldier. The Romans had absolutely no interest in mathematics." The Romans had absolutely no interest in mathematics? Surely this isn't true. I'm not very knowledgable in math history, so I am hesistant to take out that sentence, but it seems to be there more for humor than for fact. Riddlefox 18:34, 21 February 2006 (UTC)

I have never heard of one single Roman mathematician, nor any non-Roman mathematician who (during the days of the Roman Empire) wrote in Latin rather than Greek. Nor is one mentioned in any of the math history books I've read. If you can find one, please let me know. Rick Norwood 19:36, 21 February 2006 (UTC)

I dunno about famous mathematicians, but the Romans had to be interested in at least applied math to build the Colusseum, all the roads, and keep their legionaires paid and supplied. For instance, there's Vitruvius, who was the original architect. I don't know too much about him, but I think there's some stuff in there about math.

I think a better wording of the sentence would be "Romans were more interested in pragmatic application of mathematics, rather than exploring the theory of math." Something along those lines, at least. When you combine the "romans had no interest in math" along with the previous sentence about running Achimedes through, it sounds like that the Romans killed him out of spite and disdain for math than anything else.

Oh yeah, what about Roman numerals? :) Riddlefox 02:02, 24 February 2006 (UTC)

I've made it: "The Romans had absolutely no interest in pure mathematics."

I don't think Wikipedia can speak for every single Roman that lived prior to the fall of the Roman Empire. If by Roman one means someone who wrote in Latin from ~27BCE to ~473CE, then relative to the Greeks the statement is on the right path. But http://www-groups.dcs.st-and.ac.uk/~history/ shows a lot of exceptions. Thanks, --M a s 00:45, 3 May 2006 (UTC)

Why is there no mention of Hypatia of Alexandria in this article, nor mention made of the shift in mathematical study from Alexandria to Persia as a result of her murder? As a Greek and a women she deserves a place in this section? Thanks - Chris, 10May2012 — Preceding unsigned comment added by 155.91.28.231 (talk) 21:29, 10 May 2012 (UTC)

Good point. I've added her to the article. She wasn't a major contributor to mathematics, but as the first woman mathematician (as far as we know) she certainly deserves mention.Rick Norwood (talk) 12:39, 11 May 2012 (UTC)

"Islamic mathematics?" - Excuse me?!

There is and never was such a thing as Islamic mathematics. I don't know who the author of this article is, but seems to me this is yet another wikipedia biased article written by narrow minded incompetent pov-pushing people. Islamic mathematics as the author calls it - wherever he might have found the term doesn't matter - is nothing else than "Persian mathematics". 95% of all scientist and mathematicians who were muslim in name of belief were Persian. Not Arabs, but Persian. Their vast contributions and development of mathematics had nothing to do with Islam nor arabic culture. Only thing arabic about their works was that most of them were forced to write their books in arabic. As always i won't bother even touching the article to correct it, even though i have strict reliable sources to confirm my claims, such as "Encyclopedia Iranica". Because i know it wouldn't lead to any change, cause of the enormous biased system of wikipedia. There goes my two cents, like it or not, that i leave to you. 82.209.153.239 (talk) 18:44, 27 November 2011 (UTC)

I'm glad you got that off your chest! By the way, what nationality are you? AWhiteC (talk) 20:05, 28 November 2011 (UTC)

But he has a point. Why should Persian mathematics always be subsumed under the umbrella term "Islamic" which itself is very problematic (there is no claim to Christian, Confucian, Buddhist, Zoroastrian or Hindu mathematics, is there?). The ultimate source for this approbiation, however, is no Wiki bias, but to a large extent the Islamic Republic of Iran itself whose state ideology has placed from the beginning its Islamic religion resolutely above its Iranian identity and history. WP, in a way, just mirrors this official view as far as the Iranian contribution to science is concerned. Sadly so. Gun Powder Ma (talk) 09:15, 6 June 2012 (UTC)

Dubious timeline

The timeline in the lede is dubious. I agree with the IP editor who removed it. Though ostensibly sourced, it is giving undue weight to the source it is based on. The dates given are at odds with what we know on the history of mathematics. The end date of 260 AD for Egyptian and Babylonian mathematics is arbitrary. The start date of Chinese mathematics of 1030 BC contradicts what we know about Chinese mathematics, the oldest extant mathematical texts dating from the 4-5th centuries BC. Ditto the 1200 AD end date for Arabian mathematics, since there are many significant mathematical manuscripts in Arabic dating from the 14th and 15th centuries. As far as I know there is no such thing as "Dark Ages mathematics", and the periods "Period of Transmission", "Modern First Half", and "Modern Second Half" are completely arbitrary. Athenean (talk) 10:04, 9 June 2012 (UTC)

Correction in the India Section

The oldest civilization in historical India is actually the Mehrgarh civilization found on the Kachi Plain in Balochistan. It's also the oldest archaeological remnant of an urbanized settlement. — Preceding unsigned comment added by 142.59.203.143 (talk) 16:15, 14 January 2013 (UTC)

Prehistoric mathematics

"The origin of mathematical thought lie in the concepts of number, magnitude, and form"

Form then links to modular form which is "In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory."

Prehistoric mathematicians did not use complex numbers, no? — Preceding unsigned comment added by 64.189.242.76 (talk) 03:23, 14 September 2013 (UTC)

Maya numeration system

do not figure in article, neither the quipus of ancient peruviens.--Tarpuq (talk) 20:26, 8 January 2014 (UTC)

Difficult to read

As someone who does not know the subject matter I found this article very difficult to follow. Rather than a history of math it reads more like an article on nationalistic contributions to math. I would just ask, other than context, what does Greek/Chinese/Babylonian/etc have to do with mathematics? I get what context it brings, but contex should be secondary, not the core division/perspective. 64.64.178.234 (talk) 22:47, 21 August 2014 (UTC)

I'm not sure what you are asking. The history of mathematics, like most history, is chronological, and the different countries were where, in the ancient world, mathematics began. Until the modern era, it was rare for more than one country to contribute to mathematics in the same century. Rick Norwood (talk) 12:17, 22 August 2014 (UTC)

References

I have been doing some work with the references. Over time they have grown sloppy and have been subject to some vandalism. There are a couple of long quotes that have been put into the notes that do not have full citations (names and dates but no book titles or other bibliographic info). I've decided to pull these out and put them here so that if this information can be found we can decide whether or not to replace them. Bill Cherowitzo (talk) 04:32, 9 January 2015 (UTC)

Pingree 1992, p. 562 Quote: "One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution."

(Bressoud 2002, p. 12) Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."

Plofker 2001, p. 293 Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that “the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)” [Joseph 1991, 300], or that “we may consider Madhava to have been the founder of mathematical analysis” (Joseph 1991, 293), or that Bhaskara II may claim to be “the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus” (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian “discovery of the principle of the differential calculus” somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential “principle” was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here"

Bill. I traced down the references and restored the citations to the article. This should provide a starting point for further editing. SteveMcCluskey (talk) 15:43, 9 January 2015 (UTC)

Steve. Thanks, I appreciate the effort and knew that it would be easier for someone else to do. I do have a concern about these and other lengthy quotations in the footnotes of this article. It appears to me that some kind of debate (or perhaps just one side of one) is being carried out in the footnotes. I understand that there is some controversy here with some probably overzealous comments made and quoted and then countered by other quotes, and I don't mean to shy away from reporting on such controversy; but I wonder whether this is the appropriate article to do this in and whether or not the "footnote wars" are the right way to handle it. Bill Cherowitzo (talk) 18:22, 9 January 2015 (UTC)

You're welcome. I agree that the footnote wars do present too many lengthy quotations. When they were added they were probably felt necessary to counter claims by Indian nationalist historians. I suggest trimming the quotation from Bressoud's citation, which is almost a tertiary source. Plofker's comment represents the most recent serious historical study so I'd keep it. The quotation from Katz is excessively long and jumps between Islam and India so really isn't on target for this discussion. I like Pingree's as an analysis by a distiguished scholar, rejecting claims for Indian calculus in the context of a discussion that Asian mathematics was not adequately appreciated by "Hellenophilic" historians. I'll strip the quotations from Bressoud and Katz and leave the remainder. SteveMcCluskey (talk) 20:23, 9 January 2015 (UTC)

Good work. If we could now trim some of those "Arabian" quotations I think we will be in pretty good shape. Bill Cherowitzo (talk) 22:18, 9 January 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:History of mathematics/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Looks good for everything upto the 17th century, after that things become very brief, sub articles needed to expand on later centuries in greater detail. --Salix alba (talk) 15:44, 22 April 2007 (UTC) Consider mathematics A-Class review. Geometry guy 21:47, 9 June 2007 (UTC)

Last edited at 21:47, 9 June 2007 (UTC). Substituted at 19:34, 1 May 2016 (UTC)

start date of indian mathematics per boyer's page 208

hi,

it says in the wiki text that history of indian mathematics starts around 2600-1900BC, however the cited source (p 208 boyer edition 1) states:

"Chronology in ancient cultures of the Far East is scarcely reliable when orthodox Hindu tradition boasts of important astronomical work more than 2,000,000 years ago and when calculation leads to billions of days from the beginning of the life of Brahman to about A.D. 400. References to arithmetic and geometric series in Vedic literature that purport to go back to 2000 B.C. may be more reliable, but there are no contemporary documents from India to confirm this".

thus i'm wondering if even the dates 2600BC-1900BC are inappropriate? given that the edit i had to revert tried to extend this from 3000 BC, and the cited text says 2000 BC, i'm having a hard time understanding why it's 2600 BC to 1900 BC.

any clarification would be great. — Preceding unsigned comment added by 174.3.213.121 (talk) 00:18, 29 March 2015 (UTC)

Influence

Boyer does not present any evidence for "All of these results are present in Babylonian mathematics, indicating Mesopotamian influence" and "showing strong Hellenistic influence". He may be correct that these results were there earlier, but "influence" is speculation. Evidence in the form of direct or indirect contact between the Indians, Babylonians and Greeks is lacking. Including material like this is giving undue weight to a badly written source.

JS (talk) 21:47, 6 April 2015 (UTC)

Boyer is a reliable source (not "badly written") and I take him at his word. You have nothing to base your claim that this is "speculation". His wording is not speculative. Athenean (talk) 22:49, 6 April 2015 (UTC)

very much agree with athenean. if further reverts occur i'll page in an editor. -- 174.3.213.121, 20:53, 12 April 2015‎

Article Neutrality, Eurocentricity and Context

Don't you think we should reduce the eurocentric skew from these articles ?

Statements like

• The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics"
• Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in mathematical proof

are unclear and false.

Firstly, Mathematics does not become a 'subject in its own right' by someone coining the term for it in a english(greek/latin) related language. Every language had a term for the field Mathematics.

Secondly, Mathematical rigor and deductive reasoning were evident in textual material ranging from the Egyptian and Babylonians in the West to the Indian and Chinese texts in the East.

I believe adding context to such statements will reduce friction and add accuracy to such topics. — Preceding unsigned comment added by Vanya (talk • contribs) 04:19, 13 September 2015 (UTC)

unclear and false. Firstly, Mathematics does not become a 'subject in its own right' by someone coining the term for it - you've failed to understand the sentence you've quoted. The qualifier ", who coined the term "mathematics" merely states that William M. Connolley (talk) 06:36, 16 September 2015 (UTC)

You do notice that the sentence starts with a statement about where the study of mathematics in its own right began and ends with a statement about how the word mathematics originates from a Greek word meaning "subject of instruction". I see the qualifier bit now but if this is not the meaning attempted the two statements may be better of in different sentences.

No comments on the Eurocentric leaning of at least the intro ?

I did a revision which added details of other cultures and reduced the content on the greek in that specific para above. It was reverted because "too much detail for an intro". The word count difference between revisions was ~3. The reverter did not consult the talk page before reverting. Of course it seems like too much detail if everyone starts talking about what every 'other' culture was up to in the intro. But who decides how much space which culture gets ? Vanya (talk) 16:25, 17 September 2015 (UTC)

That "The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans" is sourced to Carl Boyer, History of Mathematics, page 48. It is not "eurocentrism" as you falsely claim, and by changing the wording you are altering the meaning of a sourced statement, which constitutes source falsification. As for the rest of your edit, we cannot cram every mathematical advance of every civilization in the lede. That you are intent on doing so constitutes POV-pushing. Athenean (talk) 20:22, 25 September 2015 (UTC)

Boyer's statement can be challenged. According to one source, Old Babylonian scribal schools (between 1900 and 1600 BC) also studied mathematics as a subject.[9] Quote: "Mathematics as a subject was born in the scribal traditions and the scribal schools." (p. 7) "The mathematical activity of the Babylonian scribes seems to have arisen from the everyday necessities of running a central government. Then, in the context of the scribal schools, people became interested in the subject for its own sake, pushing the problems and techniques beyond what was strictly practical." (p. 10) "A social change at the end of the Old Babylonian period seems to have brought this fertile time for mathematics to an end. [...] Scribal arts became family traditions, and scribes no longer specialized in mathematics. As a result, we see tablets in which mathematics is mixed with several other subjects. Mathematics loses its separate identity, and most of the enthusiasm and creativity disappear." (p. 12). Wiqi⁽⁵⁵⁾ 02:19, 26 September 2015 (UTC)

Very interesting but it appears this didn't last very long, thus there is no continuity in the study of mathematics as subject on its own before the Pythagoreans. Athenean (talk) 03:02, 26 September 2015 (UTC)

Not to mention that the source "A gentle introduction for teachers and other" isn't quite of the same caliber as Boyer. Athenean (talk) 03:06, 26 September 2015 (UTC)

Boyer's statement (as rendered here anyway) doesn't imply continuity. Hence it's more accurate to say that "The study of mathematics as a subject in its own right began in the Old Babylonian period". The source making this point[10] is definitely reliable and more recent (2004) compared to Boyer (~1976). Wiqi⁽⁵⁵⁾ 04:58, 26 September 2015 (UTC)

Nonsense. By your reasoning, it would also be accurate to say that the study of mathematics as a subject in its own right ended in the Old Babylonian period. Mathematics as a subject in its own right has been studied continuously since the Pythagoreans, and this is explicitly backed by Boyer. Furthermore there is no break in the mathematical tradition since the Pythagoreans started studying mathematics for its own sake and the present day. Are you suggesting there is? If so, that would be original research. By the way, Boyer's book was revised in 1991, and a few years' difference means nothing regarding a source's reliablity, so much for that card. Athenean (talk) 05:10, 26 September 2015 (UTC)

Could you quote Boyer's full statement from the 1991 edition? We only lack evidence that mathematics continued as a subject for the Babylonians after the Old Babylonian period (mostly because their older tablets are better preserved). Nevertheless, it is still notable to point out that mathematics as a subject began in that period and within a scribal tradition. Both these facts are notable and supported by reliable sources. If you think the continuity claim is important you can mention that too (although more clearly). But it's obvious that this concept precedes the Pythagoreans by more than a 1000 years. Wiqi⁽⁵⁵⁾ 06:37, 26 September 2015 (UTC)

...it is evident the Pythagoreans played an important role - possibly the crucial role - in the history of mathematics. In Egypt and Mesopotamia the elements of arithmetic and geometry were primarily exercises in the application of numerical procedures to specific problems, whether concerned with beer or pyramids or the inheritance of land. There had been little in the way of intellectual structure and perhaps nothing resembling philosophical discussion of principles. Thales is generally regarded as having made a beginning in this direction, although tradition supports the view of Eudemus and Proclus that the new emphasis in mathematics was due primarily to the Pythagoreans. With them mathematics was more closely related to a love of wisdom than to the exigencies of practical life, and it has had that tendency ever since.

As you can see, Boyer is quite explicit. As for the Babylonian claim, your own source explicitly states that they lost interest in the study of mathematics on its own, so it's not a question of a lack of evidence. It would be misleading to our readers to say that the study of mathematics on its own began with the Babylonians, unless we mentioned that this was a brief blip that died out several hundred years later (and even then "nothing resembling philosophical discussion of principles"). That, however, would be far too much detail for the lede. Athenean (talk) 07:20, 26 September 2015 (UTC)

The part you quoted doesn't mention anything about "mathematics as a subject". I guess it's reasonable to assume that Boyer never made any claims about mathematics as a subject, contrary to what's in the article. Furthermore, sources that explicitly mention "mathematics as a subject" point out that this development happened in the Old Babylonian Period. In short, the article misrepresents Boyer and ignores more relevant sources. Wiqi⁽⁵⁵⁾ 18:34, 26 September 2015 (UTC)

I had a feeling the BS was about to start. "I guess it's reasonable to assume that Boyer never made any claims about mathematics as a subject"? What does that even mean? Then as what? A hobby? This is sophistry, and pretty banal at that. What the lede currently states faithfully summarizes the spirit of what Boyer wrote. I have very little patience for this kind of nonsense, and even less time for it. Athenean (talk) 19:16, 26 September 2015 (UTC)

To spell it out for you, Boyer never made this claim: "The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans". That statement is just poor quality original research. Reliable sources (e.g., [11]) state that the study of mathematics as a subject began in the Old Babylonian Period. Calling this "sophistry" is just being disingenuous. Wiqi⁽⁵⁵⁾ 20:33, 26 September 2015 (UTC)

If you cannot or don't want to comprehend what Boyer wrote, that's not my problem. He did not use those exact words, but the current wording of the lede captures the spirit of what he wrote perfectly. And if he had used those exact words, you would doubtless play the copyvio card. Like I said earlier, I have no time for sophistry and word games. So long. Athenean (talk) 21:35, 26 September 2015 (UTC)

Well, everyone can see that the above quote by Boyer says nothing about mathematics as a subject. Also, recent scholarship tend to reject the Pythagoreans' "important role" theory: "The majority opinion, however, now seems to be that the Pythagoreans did not play a unique role in the development of the mathematical sciences".[12] Wiqi⁽⁵⁵⁾ 11:52, 2 October 2015 (UTC)

(outdent) How do you know that one source is the majority opinion? William M. Connolley (talk) 15:28, 2 October 2015 (UTC)

"Majority opinion" means his own opinion. He is playing word games and refusing to comprehend what the source (Boyer) wrote. Clear case of WP:IDONTLIKEIT. Athenean (talk) 20:53, 2 October 2015 (UTC)

@Athenean: Sorry for the delay in a reply.

• In continuation of @Wiqi55:'s points. Your only refutation to his citations is that it looses continuity. The current statement on the wiki article has no reference to continuity. I don't know why you would bring that up as a refutation to his statement.
• I have cited two statements from the introduction as examples of eurocentricity in it. How is attributing one of the statements in the introduction to its author a refutation of the eurocentricity of the introduction or even that statement ?

You need to be clearer about what you want to articulate. Correct me if I'm wrong but it seems like what you actually want is to get a higher cumulative 'citation worthiness' for alternate viewpoints to edit or remove such a statement. Is that correct ?

• you make the point about 'POV-pushing', do you get that limiting what is allowed to be written is also 'POV-pushing' ? Controlling what is to be allowed in an article has to be a form of 'POV-pushing' on non-science topics. The best we can do is keep the article neutral. I was attempting this by increasing the cultural diversity of history of mathematics being discussed. Mathematics is not just or even pre-dominantly a european-centered topic.

Can we discuss this like civilized people who don't assume the person on the other side of every connection is a low IQ spammer here to waste your time ? We can discuss the Boyer point but all the rest of the edit did was increase the diversity and still kept the conciseness of the introduction. I don't see the problem with that ?

Vanya (talk) 20:56, 9 November 2015 (UTC)

I just came across this discussion so let me toss in my two cents worth. Boyer's discussion, quoted above, deals with the question of when mathematics changed from a practical discipline to a concern with abstract proof. Boyer advances the commonly held view that proof was a product of Greek philosophical thought and can be read as saying that without proof there is no mathematics.

More recent studies have questioned the importance of abstract proof and whether it is exclusively a Greek phenomenon. The Historian of Greek science, G. E. R. Lloyd, has in many places traced the Greek concern with proof to the agonistic (i.e., argumentative) nature of Greek society; at times he suggests that claims of proof were used as social claims for the superiority of one school of thought (e.g., the philosophers against the sophists). Interestingly, his co-authored comparison of Greek and Chinese science (Lloyd and Sivin, The Way and the Word, 2002) while pointing out the Greek emphasis on proof, points out the different Chinese approach to demonstrations: "Rather than formally proving that these methods will always work with the pertinent problem type, the compilers make the point with examples" (p. 230).

As I understand recent thinking in the history of mathematics, there is a tendency to move away from the notion that demonstrative proof, in the manner of Euclid, is the sole method of doing mathematics.

In conclusion, I suggest that the passage in the lede be changed to read: "The study of mathematics as a demonstrative discipline begins in the 6th century BC with the Pythagoreans,…"

--SteveMcCluskey (talk) 17:43, 10 November 2015 (UTC)

I disagree. Studying a subject in its own right and logical proofs are two different things. Biologist study biology for its own sake, without there being proofs in biology. Furthermore, Boyer nowhere links the studied mathematics as a subject for its own sake and the use of proofs. He mentions "philosophical principles", "intellectual structures" and "love of wisdom", but that is not the same thing as the concept of a proof. People like Thales and Pythagoras lived centuries before the formal Euclidean proof evolved. I also worry that almost all of our readers have absolutely no idea what "demonstrative discipline" means. Athenean (talk) 03:07, 11 November 2015 (UTC)

"I also worry that almost all of our readers have absolutely no idea what "demonstrative discipline" means" I think you are truly scraping the bottom of the barrel using that claim as an argument. If a reader doesn't understand the expression 'demonstrative discipline' they certainly won't understand a lot of other terms used in the article. This is an article about the history of mathematics, an academic subject, and not a first reading book for second graders. Thony C. (talk) 06:49, 11 November 2015 (UTC)

Not all of our readers are academics. In fact the vast majority are not. Wikipedia is meant to be accessible to the general public, it's not meant to be an elite academic publication venue. In any case, that is a secondary point. My main point is that the development of the mathematical proof and the study of mathematics as a subject in its own right are two entirely separate developments, separated by a couple of centuries. Athenean (talk) 07:54, 11 November 2015 (UTC)

The "majority opinion" is an assessment made by the source I quoted, not my own. Here is the quote again: "The majority opinion, however, now seems to be that the Pythagoreans did not play a unique role in the development of the mathematical sciences".[13] This quote is seemingly at odds with the weight given to the Pythagoreans in the lede of this article. It may also suggest that using more recent scholarship than Boyer should be preferred. Wiqi⁽⁵⁵⁾ 05:02, 8 December 2015 (UTC)

Confusion in the "Babylonian mathematics" section

In the beginning of the fourth paragraph, it is stated that "unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system". However, the last sentence in the same paragraph instead states "thus the Babylonians came close but did not develop a true place value system". Both sentences cite the same source, and seems to have been added in this diff. So which of the statements is true?

Edit: I found some references that may shed some light. Quoting Positional notation#History, first paragraph:

"For example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. [...] Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number."

Boyer, C.B. (1991) [1989], A History of Mathematics (2nd ed.), New York: Wiley, ISBN 0-471-54397-7 is cited multiple times on this page, and on page 30 states:

"The Babylonian zero symbol apparently did not end all ambiguity, for the sign seems to have been used for intermediate empty positions only. There are no extant tablets in which the zero sign appears in a terminal position. This means that the Babylonians in antiquity never achieved an absolute positional system."

Due to these sources, I'm inclined to believe that the last sentence in the "Babylonian mathematics" section is the factual statement, and that the sentence "unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system" should be removed. Thoughts?

Axel Andersson (talk) 18:56, 24 March 2017 (UTC)

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new findings, ancient Babylonian trigonometry on tablet Plimpton 322

A new study published in 2017 found that Plimpton 322 might be the first use of advanced mathematics and trignometry in the ancient world!!! and also might be more accurate than even present-day mathematicians!!!! this information needs to be included in this entry!!! --Sm8900 (talk) 16:00, 25 August 2017 (UTC)

There is a case for its mention on the Plimpton page, but certainly not here. This is an interpretation by one pair of authors. It's not backed by any other work and not an accepted interpretation. One of the authors, Wildberger, is known for his distinctly non-mainstream theory of Rational trigonometry which the paper looks like another version of.--Salix alba (talk): 16:06, 25 August 2017 (UTC)

Languages

This phrase, "The idea of the 'number' concept evolving gradually over time is supported by the existence of languages" found in the article, I found is too general and does not offer insightful facts as to which languages exactly contributed to the "idea of the number concept." As a reader it would be more fascinating to know exactly which languages were the inspiration for this concept. I do not have the right sources at the moment to edit this myself so if anyone is willing and able to obtain more information on this topic, that would be wonderful. — Preceding unsigned comment added by Af1317a (talk • contribs) 04:50, 30 August 2017 (UTC)

You are misreading the statement. The statement only refers to the evidence that the number concept evolved over time that is given by the fact that some early languages did not have words for numbers greater than two but later ones did. One could learn more about this from the reference given. It would be pretty far afield to go into specific languages since they all have some form of number concept, be it very limited or more advanced. There is a book by Dantzig, called Number / The Language of Science ISBN 0-02-906990-4 that deals with this in greater detail.--Bill Cherowitzo (talk) 18:29, 30 August 2017 (UTC)

Mathematical reasoning

Be careful with history. This article in mathematica [14] states there is only three. — Preceding unsigned comment added by 159.100.67.82 (talk) 10:03, 25 August 2018 (UTC)

Article Preview

When this page is linked from another article (such as https://en.wikipedia.org/wiki/Contributions_of_Leonhard_Euler_to_mathematics), and you put your mouse over without clicking, you see the preview "today id dfnvklbkjkjbklj pofdjif,awno;n;nm". I don't know how to fix it, so please do if you know how :) 95.165.7.165 (talk) 20:25, 21 October 2019 (UTC)

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Observation of Patterns In Nature led to Origin of Math & Science

When did math originate? In connection with science and the observation of patterns in the heavens: the lunar cycle, solar year, movement of the planets. 2601:589:4800:9090:F05C:A40C:B4A:8885 (talk) 15:22, 17 August 2020 (UTC)

"Mathematikhistoriker" listed at Redirects for discussion

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User:1234qwer1234qwer4 (talk) 01:50, 12 February 2022 (UTC)

Mathematics in Japan during Edo period

Let's talk about mathematics during Edo period of Japan Obiwana (talk) 07:46, 3 December 2022 (UTC)

I see you're doing exactly that, adding material to the article! Largoplazo (talk) 12:18, 3 December 2022 (UTC)

I returned the beginning of the section "Medieval Europe" to how it was. I think you changed it by mistake. Check it if you want. 142.247.107.229 (talk) 20:21, 28 January 2023 (UTC)

Regarding its origin

The origin of Mathematics is from India. Here is the proof their was a Maths book named Lilavati writen by (Maharishi Bhaskra acharya.Book contains arthmatic, algebra and first degree intermainate equations. Which is further read by alot of people and whoever read they tried to give the knowledge and didn’t told the reference from they got it and took the wnole credit. when Arabic invaders came ti India they get the knowledge and spread it by stating it as there own knowledge. However Indians have each proof of it in there old books. 2605:8D80:641:12A:65E0:E836:D03B:D00D (talk) 10:46, 14 May 2023 (UTC)

Bhāskarāchārya was born in 1115 CE. He wrote Lilavati in 1150 CE. At that time, mathematics had already been a field of study for thousands of years. Euclid, for example. preceded him by 14 centuries, and Pythagoras by 17. Largoplazo (talk) 11:08, 14 May 2023 (UTC)