Talk:History of mathematics

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Contents

[edit] last section

The section about 21st centuary isn't that very objective.

[edit] older comments

I'm developing some material from zero divided by zero for inclusion here. But its taking some time to work up so I'm putting it here temporarily while it is worked on. Feel free to contribute Barnaby dawson 10:13, 22 Sep 2004 (UTC)

Could somebody please clean up the first lines of 'complex numbers'? They sound rather trivial or non-encyclopedic. Radiant! 22:09, 12 Feb 2005 (UTC)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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.

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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

[edit] 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)


[edit] 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)


[edit] 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)

[edit] M a s edit

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

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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.

[edit] 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)

[edit] 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)

[edit] 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 histriography. I don't know any books on the histriography of math. Someone should write one. Rick Norwood 13:50, 27 November 2006 (UTC)


[edit] Spherical trigonometry

I disagree with the following sentence "spherical trigonometry was largely developed by the Persian mathematician Nasir al-Din Tusi (Nasireddin) in the 13th century." Other mathematicians wrote about spherical trigonometry before him, including Ptolemy in the 100s AD in his book the Almagest long before the 1200s. NikolaiLobachevsky 21:08:49 12/26/2006 (UTC)


[edit] 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)


[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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,[3] 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)

[edit] 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)

[edit] 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)

[edit] 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)[4]

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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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.

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

[edit] 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)

[edit] 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

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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

[edit] 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 ture: the actual command was books held by nongovernmental-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)

[edit] 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)