Talk:Mathematics/Archive 11

GA Re-Review and In-line citations
''Note: This article has a small number of in-line citations for an article of its size and subject content. Currently it would not pass criteria 2b.'' Members of the WikiProject Good articles are in the process of doing a re-review of current Good Article listings to ensure compliance with the standards of the Good Article Criteria. (Discussion of the changes and re-review can be found here). A significant change to the GA criteria is the mandatory use of some sort of in-line citation (In accordance to WP:CITE) to be used in order for an article to pass the verification and reference criteria. It is recommended that the article's editors take a look at the inclusion of in-line citations as well as how the article stacks up against the rest of the Good Article criteria. GA reviewers will give you at least a week's time from the date of this notice to work on the in-line citations before doing a full re-review and deciding if the article still merits being considered a Good Article or would need to be de-listed. If you have any questions, please don't hesitate to contact us on the Good Article project talk page or you may contact me personally. On behalf of the Good Articles Project, I want to thank you for all the time and effort that you have put into working on this article and improving the overall quality of the Wikipedia project. Agne 05:52, 26 September 2006 (UTC)
 * The requirements of GA are now converging rapidly on those for FA, thus making the whole idea worthless. Also no one cares about GAs. Soo 12:57, 26 September 2006 (UTC)
 * This change in criteria have caused some upset, some discussion of relavance of inline cites in mathematics can be found at Wikipedia talk:WikiProject Good articles. On the Template:Grading scheme GA is seen as a step towards FA. I see article grading as an important process for the mathematics articles, as it helps us get an overview of where there are weeknesses in our coverage. Work on grading mathematics can be found at WikiProject Mathematics/Wikipedia 1.0. --Salix alba (talk) 13:28, 26 September 2006 (UTC)
 * I offer myself as a counterexample to Soo's "nobody cares" assertion. I'll get to work on in-line citations right away. Rick Norwood 13:30, 26 September 2006 (UTC)
 * I also am happy to help. And as it happens, I like in-line citations :-) Stephen B Streater 19:07, 26 September 2006 (UTC)
 * If it matters, I disagree that this article doesn't have enough inline citations, it looks pretty good reference-wise to me. (And I mean good as in, you know, Good good.) Homestarmy 19:24, 26 September 2006 (UTC)

I have made a request regarding this issue here. --ScienceApologist 21:03, 26 September 2006 (UTC)


 * Nice to hear that we have some citation volunteers. As well as mathematics Georg Cantor, David Hilbert, Alan Turing, Calculus, Euclidean geometry, Homotopy groups of spheres, Nash equilibrium, Ordinal number, Polar coordinate system, Pythagorean theorem, Rubik's Cube, (in fact all the maths GA's) received the same message. --Salix alba (talk) 00:55, 27 September 2006 (UTC)

If mathematics is a discipline, does it hurt much?
The word discipline in the definition of mathematics invokes evokes some very negative connotations that reinforce the perception that mathematics is not fun.

The word discipline has, among others, these senses (from various dictionaries):


 * 1 : PUNISHMENT
 * 3a. Control obtained by enforcing compliance or order.
 * c. A state of order based on submission to rules and authority: a teacher who demanded discipline in the classroom.
 * 4. Punishment intended to correct or train.
 * 3. punishment inflicted by way of correction and training.
 * 8. an instrument of punishment, esp. a whip or scourge, used in the practice of self-mortification or as an instrument of chastisement in certain religious communities.
 * 5: the act of punishing; "the offenders deserved the harsh discipline they received"

And if the above is not enough, there is BDSM.

--Jtir 14:57, 30 September 2006 (UTC)

Definitions of mathematics from several sources

 * Here are definitions of mathematics from several sources:


 * "Yet as a science in the modern sense mathematics only emerges later, on Greek soil, in the fifth and fourth centuries B.C.", Richard Courant and Herbert Robbins, What is Mathematics?, 1941, p. xv.
 * Originally, the collective name for geometry, arithmetic, and certain physical sciences (as astronomy and optics) involving geometrical reasoning.
 * In modern use applied,
 * (a) in a strict sense,
 * to the abstract science
 * which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra; and
 * (b) in a wider sense, so as to include
 * those branches of physical or other research which consist in
 * the application of this abstract science to concrete data.
 * When the word is used in its wider sense,
 * the abstract science is distinguished as pure mathematics, and
 * its concrete applications (e.g. in astronomy, various branches of physics, the theory of probabilities)
 * as applied or mixed mathematics.
 * Oxford English Dictionary, 2nd Ed.
 * the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations Merriam-Webster Online Dictionary.
 * the branch of science concerned with number, quantity, and space, either as abstract ideas (pure mathematics) or as applied to physics, engineering, and other subjects (applied mathematics). Compact Oxford English Dictionary.
 * a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement WordNet 2.0, Princeton University, 2003.
 * Mathematics is a science. Encyclopaedia Britannica, 1911.
 * the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. Random House Unabridged Dictionary, 2006.
 * The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. American Heritage Dictionary, 4th Ed., 2000.


 * I prefer study, because it embraces industrial, hobbyist, and academic mathematics.
 * And it is easier to spell.
 * --Jtir 16:21, 30 September 2006 (UTC)

Further comments
Discipline (definition 2): a branch of knowledge, typically one studied in higher education : sociology is a fairly new discipline. Stephen B Streater 17:23, 30 September 2006 (UTC)


 * To answer the question in the subject header -- yes, it can hurt a lot, especially if you love it. Barbie took heat from feminists for saying "math is hard", but she spoke only the truth, and no one knows it better than mathematicians. "Discipline" is an excellent word because it makes mathematicians "disciples", a word that conveys the sense of sacrifice and spirituality involved in the undertaking.
 * The other side of the coin is that being a mathematician is the best job on the planet, when things are going well. You might have a point in that the word "discpline" doesn't get that fact across too well. --Trovatore 21:15, 30 September 2006 (UTC)


 * Calculus hurts so much that most calculus students wind up in L'Hospital. Rick Norwood 12:45, 1 October 2006 (UTC)
 *  L'Hospital --Jtir 17:35, 1 October 2006 (UTC)
 * Quite good! LOL capitalist 03:08, 2 October 2006 (UTC)

Is mathematics a science?
This is, of course, a subject that keeps coming up. I think the current opening paragraph captures what the quoted definitions have in common, and the question of whether math is a science is too controversial for us to use the word "science" in the opening paragraph.

The quoted definitions take one (or both) of two approaches. One approach is to give some kind of list of the topics studied by mathematics. This is, I think, how non-mathematicians think of mathematics. Note that astronomy was once considered mathematics but is now considered a science. Any such list is necessarily incomplete, but our list seems to me as good as any, and is reflected throughout the article as well as in other articles. I opposed the "list" approach, but when I was outvoted, I helped to make the list consistent throughout.

It seems clear to me, as a professional mathematician, that the other approach is correct, and that mathematics is that body of knowledge discovered by deductive reasoning. Science, in contrast, is that body of knowledge discovered by inductive reasoning based on careful observation and measurement. But, again, I was outvoted, and am happy that deductive reasoning at least gets a mention in the opening paragraph. Rick Norwood 13:17, 2 October 2006 (UTC)


 * I haven't observed that it is possible to induce "quantity" without observation. Even marks on a page, or symbols in a human mind is observation of quantity. Eric Norby 02:32, 29 January 2007 (UTC)


 * Your distinction makes sense to me and is essentially what I was thinking until I did the research. I gave a nod toward your distinction in my rewrite of the definition by quoting the OED, saying abstract science.  Did you notice that the science article considers mathematics a science?  It reads in part "Mathematics has both similarities and differences compared to other fields of science." The article then goes on to clarify the distinction.  The editors did a nice job.  Something similar could be done with the mathematics article. --Jtir 14:27, 2 October 2006 (UTC)


 * It clearly depends on what is meant by "science", and is discussed in the article. As Rick says, this is controversial and potentially confusing enough to avoid going one way or the other in the introduction. A quote suggesting that at some point maths emerged as a science is definitely not enough to justify the statement "Mathematics is the abstract science..." in the very first sentence. JPD (talk) 15:10, 2 October 2006 (UTC)


 * As can easily be verified, the word "science" has multiple meanings. One meaning excludes maths. I think that more restrictive meaning is the one that is prevalent by now in the US. The you get definitions as in the American Heritage Dictionary: "The observation, identification, description, experimental investigation, and theoretical explanation of phenomena". In that view, science is what is produced by the "scientific method". It is unwise to call maths a science in the lead given the fact that a large part of our readership has been spoonfed a definition of science according to which maths is not a science. --Lambiam Talk  18:08, 2 October 2006 (UTC)


 * I also think that science should not be in the intro. Seeing abstract science, most readers will be wondering what that is - it is much more obscure than Mathematics itself. Stephen B Streater 18:46, 2 October 2006 (UTC)


 * I wikified science, but there are no suitable links for abstract. NB, those are the exact words the OED uses. --Jtir 19:27, 2 October 2006 (UTC)
 * "I think that more restrictive meaning is the one that is prevalent by now in the US." "the fact that a large part of our readership has been spoonfed a definition of science according to which maths is not a science."
 * If there are verifiable sources for these suppositions, maybe something about popular conceptions of mathematics could be worked into the article.
 * Laudably, the AHD is self-consistent on the matter. It is one dictionary that does not define math as any kind of science.
 * I'm not sure what the RHUD has in mind:
 * science - a branch of knowledge or study dealing with a body of facts or truths systematically arranged and showing the operation of general laws: the mathematical sciences . RHUD
 * mathematics - the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. RHUD
 * I like those defs. --Jtir 19:24, 2 October 2006 (UTC)
 * I'm still wary of a definition which is immediately so obscure that it needs two wikilinks. Stephen B Streater 19:32, 2 October 2006 (UTC)


 * As others have said, this is an issue that does not need to be treated in the lead section; we can just choose a neutral word, like "study" or "discipline", and defer the question to later in the article. My main concern is that that later section should not take a position on whether mathematics is a science, or an empirical science, or a natural science, or whether it makes falsifiable assertions or uses the scientific method. It can discuss all those questions, in the sense of attributing various views to various thinkers, but should not pretend that there is a noncontroversial answer to any of them. --Trovatore 19:35, 2 October 2006 (UTC)


 * Well said. I would be OK with branch of knowledge per the RHUD or study per the AHD.  discipline has too many negative connotations (with BDSM being the most extreme) and further, the posited link from discipline to Academic discipline is POV because it excludes industrial and amateur mathematicians. --Jtir 19:52, 2 October 2006 (UTC)


 * I used to prefer discipline, but will defer to your superior knowledge of BDSM ;-) Your latest two proposal are sourced, which is a plus. I opposed study in our earlier discussion, because it makes it sound too much like work. Stephen B Streater 20:04, 2 October 2006 (UTC)


 *  I am now favoring branch of knowledge, because it pleasantly suggests growth and because knowledge can be easily interpreted without a dictionary as stuff that people know. So the lead to the lead would read:
 * "Mathematics is a branch of knowledge that ..." (RHUD uses "a").
 * This start can accommodate any number of topics for the following lead paragraphs, or indeed, the entire article. --Jtir 20:41, 2 October 2006 (UTC)
 * This meets my constraints, and allows us to add another cite. That must be at least six since our ultimatum ;-) But I am always open to improvements on the definition. Stephen B Streater 21:00, 2 October 2006 (UTC)
 * Thanks. The exemplary citing in 0.999... does set the bar high. BTW, I just noticed that the RHUD def using branch of knowledge is science and not mathematics. --Jtir 21:29, 2 October 2006 (UTC)

I prefer discipline and I'm opposed to science, simply because it's not agreed this is the case (evidenced in the monster of a footnote you have had to include, an entire section in this article and the science article), not because I have an opinion either way. Is branch of knowledge really better than discipline?

I think most people refer to these .. things .. as an academic discipline as opposed to a branch of knowledge (google agrees with me when I googled those two terms), though either seem to be in use so I'm not totally opposed to the latter being used here. Not that my own limited experience counts for much, but I've found that branch of knowledge tends to be reserved for more specialised topics within a discipline.

I don't think it's possible to mix up academic discipline and BDSM given the context of the article. If you're really concerned about this, a simple wikilink as I've done here and the addition of the word academic would more than suffice. Most words have more than one meaning, we'd never be able to avoid them all, but at least this way we remove the possible ambiguity and stick to the apparent norm for describing subjects/fields of study/etc while the academic discipline article goes into more detail if necessary for the user, including using the term branch of knowledge. darkliight[&pi;alk] 06:47, 3 October 2006 (UTC)


 * "...I googled those two terms..." Good idea. Could you post the details? --Jtir 11:36, 3 October 2006 (UTC)
 * Results 1 - 10 of about 168,000 for "branch of knowledge" and Results 1 - 10 of about 1,440,000 for "academic discipline"? It can just give a rough idea of the prevelence of the term. The actual results can give a rough idea of its use, and I think that helps here too. darkliight[&pi;alk] 12:38, 3 October 2006 (UTC)
 * That's excellent and thanks for including the links. I get slightly different numbers and they change dramatically when I do reloads, but the conclusion is still the same. BTW, I usually exclude -wikipedia when doing searches like yours because it is cloned so much. --Jtir 18:00, 3 October 2006 (UTC)


 * "Branch of knowledge" or even "field of study" would be ok, but I fail to see how any negative connotations of discipline are actually relevant here. Even "academic discipline" indicates only that it is studied in an academic manner, and does not imply that mathematicians must be academics. JPD (talk) 14:15, 3 October 2006 (UTC)

need a quote and page number from Jourdain
I left in the Jourdain footnote while editing the lead, but it is not at all clear what it is sourcing.

Can someone add a quote and page number? --Jtir 14:00, 2 October 2006 (UTC)

moved almost all of the inline references to a separate References section
I have moved almost all of the inline references to a separate References section, fixed a few templates, and added one new reference on the Fields Medal. For most of the ones that I moved, the inline citations are now either an author's name, or a quote with the author's name. This approach is nice because it puts the references in one place and it removes clutter from the body of the article (esp. clutter due to the templates, which, although they are very flexible, take up a lot of space). It is also easy to cite a work multiple times. The order and naming of the end sections follow Guide_to_layout. The Earliest Uses of Various Mathematical Symbols is a fantastic reference, which could probably be cited more than once in the article, but I haven't decided what "author" I should attribute it to. --Jtir 21:12, 2 October 2006 (UTC)
 * Thanks - I was lucky to find this reference. It looks like a real gold mine, and well sourced too. I'm not sure yet about the Notes/References change. It does make the article easier to edit, but makes it a three step process to go to the reference (instead of two), so makes it more awkward for the reader. The refs I have added (8, 14 and 15) are easier this way. Perhaps it all depends on how many references we add. If we really pack the article (as we're supposed to now), it could be quite hard to edit the old way, but then the notes will be more than a page long, so it could be quite awkward to find the relevant reference as the note and reference won't both be visible at the same time. I'd be interested to know what other people think. Stephen B Streater 21:34, 2 October 2006 (UTC)


 * Oh, those are the three I couldn't decide about. :-)
 * Maybe those are OK there because they are actual links.
 * Ideally, software would automatically create links between each footnote and its reference. The tags support a sort of labeling that allows a long inline reference to be given a shorter name that can be used for subsequent citations.  But it does not allow a reference to be cited several times with different page numbers .  The 0.999... article uses the same reference more than once with different pages numbers. --Jtir 22:09, 2 October 2006 (UTC)

copyedit of lead
The new lead sentence contains all the mathematical concepts that I could identify in the first and third sentences of the original, adds some others (notably theorem and proof), and organizes them in a logical sequence.

The second sentence distinguishes between pure and applied -- it needs to be extended.

The third sentence concerning the evolution of math in the new lead is not changed from the original. It needs a brief transition from the definition of applied.

I used bulleted clauses after realizing the sentence would be unreadable otherwise. The new structure is highly adaptable and can be extended ad infinitum, if needed. The phrase relating mathematics to knowledge and science is also flexible and it could even be removed without modifying the rest of the definition. I didn't wikify it because it is too hard to edit complex sentences that are fully wikified and I expect there will changes. :-)

--Jtir 17:39, 3 October 2006 (UTC)


 * Sorry, but the new first sentence is incomprehensible. Fredrik Johansson 17:41, 3 October 2006 (UTC)
 * Thanks for your comment. If you are referring to this part: "Mathematics, as a branch of knowledge sometimes regarded as a science, ...", I agree and will try to do something about. --Jtir 19:15, 3 October 2006 (UTC)
 * ISTM, that methods of doing proofs, and mathematical notation and terminology are also mathematical knowledge. --Jtir 20:50, 3 October 2006 (UTC)


 * "''...abstracts axioms and definitions from such concepts as quantity, structure, space, and change,.."
 * Mathematics also abstracts from existing mathematics (generalization). --Jtir 21:07, 3 October 2006 (UTC)

I'm very impressed with this article, especially the illustrations. But the first sentence is really awkward. Of the definitions cited above, I'm most impressed with Britannica ("Science of structure, order, and relation that has evolved from counting, measuring, and describing the shapes of objects.") Yes, I have read the above discussion and I have already learned that it is difficult to give good definitions on wikipedia, where you have to accommodate every minority view. That is a pity, because this article is really beautiful. (well done!) --Vesal 21:02, 3 October 2006 (UTC)

organization of talk page...
Since I'm new to wikipedia I was wondering if there is any reason why the talk pages are so badly organized? I believe it is okay to create subpages of the talk page, so is it okay if I create a FAQ based on previous discussion and create subpages /lead and /Popper on math (to begin with) and gather all previous discussion from the archive on those pages? --Vesal 21:20, 3 October 2006 (UTC)
 * I looked around a bit, and found totally different layouts on talk pages. It seems doing a FAQ is not a bad idea (WP:RTP), the example given is Talk:Mother Teresa/FAQ. Otherwise, I also like the Talk:Abortion style. --Vesal 21:41, 3 October 2006 (UTC)

So anyway, this is what I will do, I will create the subpages, and since it is your discussion that I will be summarizing, I would ask you guys to look them over. If you find them helpful, we could put them on more prominent position. I will begin with the /FAQ and then the topical subpages: --Vesal 21:50, 3 October 2006 (UTC)
 * /First paragraph
 * /Popper
 * I was thinking along similar lines. I have collected numerous quotes and definitions that are getting buried in the main page.  It would be more convenient to reference them with a topical name.  Indeed it seems to me that any project that needs to collect evidence from research and organize it for convenient access would benefit from your approach. I have just finished typing in two more quotes from Courant & Robbins' classic "What is Mathematics?" that I was just about to put on the main page. Where would you suggest putting them? --Jtir 22:04, 3 October 2006 (UTC)
 * Some projects have a /Sources subpage, maybe that is a good place to just collect and classify sourced statements. As I understand, we are quite free to structure the talk pages the way we want. I will focus on summarizing what has currently been said in this discussion pages, because I need to read them anyway (damn, you guys write a lot :), this will take some time. --Vesal 22:30, 3 October 2006 (UTC)
 * /sources must be sourced. I completely agree.  ISTM that some subpages might be subdivided by editor. This would allow each editor to develop, for example, a "personal" list of The Important Theorems and Conjectures with sourcing and possibly with annotation. --Jtir 23:16, 3 October 2006 (UTC)
 * Yes I have even seen people develop articles in their User namespace, but I've always wondered how such stuff is merged. By the way, I started on the /Sources page, set up very strict rules, but the idea is to collect and classify not so much discuss them there... I still added some discussion space at the bottom. --Vesal 23:38, 3 October 2006 (UTC)

I started the FAQ, see what you think. Since this is a core article of wikipeda, there might be future drives to improve quality and I think a FAQ is useful to avoid having the same "Is math a science debates" all over again. I personally would state in the lead that math is a science, but the purpose of the FAQ is to give the answer quickly to people like me. --Vesal 15:54, 4 October 2006 (UTC)


 * Looks fantastic! Were you going to create a link to the FAQ on the Talk page? Also, I find all those boxes at the top distracting and would prefer to see the TOC first. --Jtir 17:00, 4 October 2006 (UTC)
 * Yes, but I don't know where to put the link, I was thinking about having it at the bottom of the archive box... But, about those boxes, I think we could add another box (the skip to TOC box) to make it even more annoying for you :) --Vesal 17:06, 4 October 2006 (UTC)


 * There is a distinction between the citation style the reader sees and what editors use to implement it. Readers see a slightly modified form of Harvard referencing. The editors implement them with the so-called "Cite.php footnote system", which employs the inline tags. Further, the tag goes in the Notes section, not the References section.  Basically, the names don't match the uses. --Jtir 17:19, 4 October 2006 (UTC)
 * Ok, well I was primarily targetting the FAQ to would-be editors of this article. Other than that, I will fix the mismatches, unless you already have... --Vesal 17:39, 4 October 2006 (UTC)


 * The forward link to the TOC works for me. How about putting the FAQ link at the top of the topical subpage list -- it's a topic too. :-) --Jtir 17:27, 4 October 2006 (UTC)
 * Now, why didn't I think of that????? Instead, I created a general topics section and it's getting to big, I'm thinking of moving the precedents in discussions section into the FAQ, because it makes more sense to first summarize it and then give the links. As a new user I'm too inclined to click on those precedent discussion links anyway, a FAQ however, might be something to look at. --Vesal 17:39, 4 October 2006 (UTC)


 * OK, The FAQ! link works fine. Regular editors probably won't need it too often. --Jtir 17:45, 4 October 2006 (UTC)

What is the point of this, if you are not going to go back through the archives to find all the discussions about the intro, etc? If the subpage is only going to contain the recent discussion, it may as well be on this page. JPD (talk) 10:26, 5 October 2006 (UTC)
 * Wait, it will take some time to do that, and I don't know exactly what the best solution is. But I think a FAQ with a short answer and then links to the topics in the archive is helpful. --Vesal 22:54, 5 October 2006 (UTC)
 * Well, anyway, I removed the link to the Lead paragraph subpage. I moved the dictionary defs to the Conceptions page for reference. I sort of lost motivation (or gained some perspective)... I no longer think the lead paragraph needs focus or discussion. Well, when there are no more mathematics related stubs, say in 10 years, I will turn back to it :) --Vesal 18:04, 9 October 2006 (UTC)

Mathematics as Science, Philosophy of Mathematics
Maybe this should be renamed or incorporated into a Philosophy of Mathematics section... Mathematics as Science is quite POV (actually my own POV), but there are a also a few more things that could be incorporated: Very much in line with the todo list. --Vesal 23:27, 4 October 2006 (UTC)
 * Relationship between mathematics and physical reality... This is a very important issue and should not be tossed into the misconception section.
 * Psychology of mathematics, i.e Numerical cognition.

I wouldn't object to this section being incorporated into a philosophy of mathematics section, but I don't think that it is POV use something like "mathematics as science" as a heading for a section discussing the different views on whether/how maths is (a) science. On that topic, now that Poppers views on the matter have been clarified, the mention of Lakatos' work doesn't quite fit in. I am not sure whether it should be removed or reworded. I am also not sure why the Einstein footnote is needed. As the text suggests, the Einstein quote supports the view that pure maths is not science if you take a partiuclar view of what science is. The text doesn't suggest any more than this, and the footnote doesnt' seem to clarify much anyway. JPD (talk) 10:45, 5 October 2006 (UTC)


 * The subpage Talk:Mathematics/Conceptions of mathematics is intended to collect all of the dueling quotes. ATM it is stubs. --Jtir 12:19, 5 October 2006 (UTC)


 * This section should be moved into a separate article. The mathematics article is an introduction to mathematics for the rest of the encyclopedia, not a treatise. A phrase like "experimentally falsifiable" cannot possibly be explained in an introduction. Popper and Lakatos are far too advanced for an introductory article. There are already articles on Philosophy of mathematics and the Philosophy of science, where Popper and Lakatos could be discussed at the required technical level. --Jtir 12:48, 5 October 2006 (UTC)
 * Well, the nice thing with naming it Phiosophy of Mathematics, is that we could have a main article link to that page and briefly summarize the issues. However, issues about philosophy of math and psychology of mathematics, certainly need to be on this page! --Vesal 21:55, 5 October 2006 (UTC)

It is not entirely clear what the new paragraph means in its assertion that mathematics is not "reducible" to logic. I suppose it has something to do with undecidable questions such as the continuum hypothesis. I know there are some people who claim that we can replace logic by either experiment (if a computer can't find a counterexample to the Riemann Hypothesis, then it must be true) or intuition (the Riemann Hypothesis must be true because it is beautiful) but as several people have observed, comments on the subject in this article should be a) basic and b) referenced. Rick Norwood 13:37, 5 October 2006 (UTC)
 * Yes, the math is not redicible to logic part is quite bad, but it is directly from Popper, see /Sources and the most important limitative results that he referse to are Tarski's indefinability theorem and Gödel's incompleteness theorem I believe... --Vesal 21:09, 5 October 2006 (UTC)

About the footnote on Einstein... Yes, it is very badly written, I tried quite hard, but it is very difficult to explain a long essay in a few words with my bad english. The problem is that the quotation is out of context. When Einstein is arguing that a pure branch of mathematics like geometry really depends on the physical world, I find it unethical to cite him as saying quite the opposite. Disclaimer! I might be wrong about what Einstein is arguing for in the essay, but I did read all of it, and it sure seemed the point was that pure math is a science. In fact, very similar to the views of Popper. --Vesal 21:24, 5 October 2006 (UTC)
 * Good thing I put a disclaimer, I think I am wrong... Well, I'm certainly wrong about the views being similar to that of Popper, since Einstein's essay is from 1921. But it is a complex issue. Currently, we can leave it be, as I have changed the footnote already. On the other hand, we have now three references to the unreasonable effectiveness. --Vesal 22:22, 5 October 2006 (UTC)

three more Talk subpages
I have created three more Talk subpages and put links to them in the Talk page infobox: ]] (see also Mathematical joke, Mathematical game) They have already been set up with a brief descriptive header and populated with a least one entry. You may use that entry as a prototype for more entries.
 * 1) Notable mathematicians (There is no article by this name.)
 * 2) Conceptions of mathematics (This is where the dueling quotes go. There is no article by this name.)
 * 3) [[Talk:Mathematics/Jokes, anecdotes, puzzles, games, riddles|Jokes, anecdotes, puzzles, games, riddles

The objective is to collect material that can be used to broaden the scope of the article. It is not to redo the work of wikipedians, which is what I was doing until Vesal set me straight. :-)

Would a subpage called "Mathematical Creativity, Intuition, and Aesthetics" be of interest? (I could probably find some quotes by Poincaré, Polya, or Einstein in this area.) --Jtir 18:12, 5 October 2006 (UTC)


 * The article is long enough already... but we could use the new material to make new articles which are linked to from the main article. If you have too many sub pages (which may be more than none), you may find that most of them have no readers! Stephen B Streater 19:28, 5 October 2006 (UTC)
 * Hehe, I think you are right, we have now more subpages than even the most controversal topics like abortion, well maybe Jesus has more, but we'll get there... :P --Vesal 22:39, 5 October 2006 (UTC)
 * And where would the "Mathematics in the Bible" section go? Perhaps a page to discuss it ;-) Stephen B Streater 11:17, 6 October 2006 (UTC)
 * Yes, yes... maybe it can be cohosted with Talk:Book_of_Numbers. --Vesal 12:26, 6 October 2006 (UTC)
 * I like the idea that the more important/commonly discussed topics could be highlighted and added to in the archive box, but I think it should be just kept as an archive box. That is, I think all discussion should still take place on this page. We don't get that much discussion on this page as it is, and spreading what little there is across several pages is going to just make it harder to keep track of. darkliight[&pi;alk] 12:42, 6 October 2006 (UTC)

Some of the subpages are not for discussion, they are for collecting and organizing information. Editors may add information to them or correct them, but there is no discussion on them. Here are two hints: --Jtir 15:13, 6 October 2006 (UTC)
 * You can add subpages to your watchlist.
 * You can create links to subpages and sections within subpages. To illustrate: I have typed in some quotes and saved them in the Conceptions of mathematics subpage.


 * I reorganized the Talk infobox to more clearly distinguish the information subpages from the discussion subpages. I turned the Sources subpage into information only. --Jtir 15:55, 6 October 2006 (UTC)


 * What I have in mind for the information subpages is similar to what is being done in Talk:Jesus/Cited Authors Bios, where there is this comment:
 * "... it provides a handy reference for us as we develop this page.". --Jtir 16:58, 6 October 2006 (UTC)

Perspectives on mathematics
Maybe the argumentative parts of "mathematics as science" could be entirely ditched in favor of a portrait gallery of famous mathematicians, philosophers and artists with a quotations on mathematics. In particular on what they think mathematics is. So the idea is a slight extreme of what has already be suggested, that we don't define mathematics. But what do you think about the idea of basically only pictures and quotations instead of "Some mathematicians think...". A good place to collect them is of course the /Conceptions_of_mathematics page, even if we don't do pictures and quotes, it would still be very nice to have opinions from maybe Hardy on math for math's on sake etc, maybe like "A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." stolen from the G. H. Hardy page. --Vesal 23:05, 5 October 2006 (UTC)
 * Bravo! That's a great idea. Replace the philosophy of math stuff with something non-technical people might enjoy reading. Mathematicians say and do plenty of interesting things. Maybe there is even an astronaut-mathematician. :-) And what was the theorem on the blackboard in the movie Good Will Hunting?  Sarah Flannery's family had a chalkboard in their kitchen where her mathematician father posed puzzles for his children. She's great story teller  -- Gorenstein's chauffeur. And a 12 year old boy (Patrick Bossert) wrote a book on solving Rubik's Cube. --Jtir 23:55, 5 October 2006 (UTC)


 * Just found a whole page of math quotes (which would need to be sourced and fact checked, of course)..
 * Politics is for the moment. An equation is for eternity. (Albert Einstein) --Jtir 00:06, 6 October 2006 (UTC)


 * Yes, maybe the common misconceptions section could be spiced up with "Math in Popular Culture" material or rather common conceptions of mathematics, but I guess screencaps from movies like Good Will Hunting or a Beautiful Mind, are copyrighted? --Vesal 01:12, 6 October 2006 (UTC)


 * If we're going to ditch a section to make room for a section of quotations, I think it's "Misconceptions" that should go, not "Science". "Science" is, I think, an admirably balanced section, whereas "Misconceptions" section are POV bait by their very nature. --Trovatore 01:17, 6 October 2006 (UTC)


 * I agree that the Maths as Science section is worth keeping, at the very least as part of a philosophy of mathematics section. Wikipedia isn't the place for collections of quotes, and I'm not that keen on the "popular culture" sections in most articles. Introductory articles (actually overview articles) aren't meant to completely avoid issues due to their technical nature, but cover them in summary style. JPD (talk) 09:26, 6 October 2006 (UTC)
 * Don't worry, I have very much respect for the current content, any major "ditching" will be propsed by first showing the alternative on one of the many subpages here. This is a class A article and we will be very conservative in changes... Now the quotes gallery as I envision would be quite small on screen estate somewhat like the illustration in the Fields of mathematics section, and it should reflect the different views on mathematics in an effectful way. So we have first two philosophers one probably Popper claiming math is like physical science the other offering a philosophical platonist view, then there are two mathematicians on pure vs. impure, like Hardy saying he has never done anything of real use, and somebody claiming "Math is of value inasmuch as it contributes to the growth of the US economy" :) Anyway, the point is, we need sources for citation anyway! Take the following section: While some in applied mathematics feel that they are scientists, those in pure mathematics often feel that they are working in an area more akin to logic and that they are, hence, fundamentally philosophers. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. I agree wikipedia is not wikiquote, but it shoudln't be an opinion poll in prose either. --Vesal 10:58, 6 October 2006 (UTC)
 * Obviously finding sources is more important than talking about the format, but since you've brought up the question, I do think prose backed up by sources would be better than a quote gallery. JPD (talk) 11:22, 6 October 2006 (UTC)
 * That's fine, I prefer more specific prose, but I will look for the material and leave the wording/formatting to you people. I will look for mathematicians as philosophers first... does anyone, preferrably from the 20th century, come to mind? It would be excellent to find some who does abstract algebra or general topology and considers that to be philosophical. --Vesal 12:22, 6 October 2006 (UTC)
 * Agreed with JPD. I also think that section is getting a bit long. Why do we need an entire paragraph on awards in mathematics, in this section? And just out of curiosity, since when is the Nobel Prize a science award? I think it might be best to try and avoid a 'this person said that, but this other person said this' format and instead offer a general overview of the differing views, with references to works/people that hold that view. I agree with Vesal too that maybe this section should be given a proper overview in philosophy of mathematics and just briefly discussed here, with a link back to the philosophy of mathematics article as the 'main' article. darkliight[&pi;alk] 12:59, 6 October 2006 (UTC)
 * I will now look at the philosophy of mathematics page first, see what can be done about it. And then maybe move some stuff back here. --Vesal 18:04, 9 October 2006 (UTC)


 * Independently of the above, I still think we ought to dump "misconceptions". Just cut it out and throw it away; redirect output to /dev/null. It has less than zero value for the article. --Trovatore 17:27, 6 October 2006 (UTC)

sample gallery in sandbox
I've written a sample gallery into the sandbox. The Picture_tutorial illustrates some other options. --Jtir 14:34, 6 October 2006 (UTC)
 * Here is another sample. This one has a vertical format and is implemented with an infobox. Infoboxes are much more flexible than tags. The Numerical analysis article uses infoboxes. The Fields of mathematics section would benefit from vertical layout of the images. The images could also be annotated more fully. --Jtir 19:13, 6 October 2006 (UTC)

the movie "Good Will Hunting"
The movie Good Will Hunting is an example of mathematics in popular culture. I was thinking of developing an infobox featuring it. The use of low-resolution images is OK, according to the copyright boxes for these images.
 * [[Image:NonFreeImageRemoved.svg|50px]] [[Image:NonFreeImageRemoved.svg|100px]]

My reasons are:
 * The main character is a mathematical prodigy (Will Hunting).
 * It features a professor who is a Fields Medalist and combinatorialist.
 * Will solves a graduate level problem overnight.
 * The theorem on the blackboard in the hall is Parseval's theorem from Fourier analysis, a second year problem in algebraic graph theory.

NB: The Good Will Hunting article refers to the problem as an equation: "...wondering who could have solved the equation." At another point the article reads: "...astonished assistant staring at the correctly solved theorem." The Parseval's theorem article does not mention the movie.

Any comments or suggestions? --Jtir 20:34, 6 October 2006 (UTC)
 * I don't think you can use fair use images in templates, it counts as decoration. You can only claim fair use if the picture is directly illustrating a particular point in the text. Soo 23:23, 6 October 2006 (UTC)
 * I don't think showing a picture of two actors really does alot for the article either. darkliight[&pi;alk] 00:12, 7 October 2006 (UTC)
 * I thought the character of Will Hunting didn't accurately portray a typical Mathematical prodigy. I wouldn't mention it in this article at all. —The preceding unsigned comment was added by 143.210.181.32 (talk • contribs) 12:50, October 22, 2006 (UTC).
 * I concur. --Lambiam Talk  17:15, 22 October 2006 (UTC)

Mathematics is also a thing that each and every single one of us do every single day. For example, when you are drinking water, your arm makes an angle. When a person lays down on a bed, this person makes a straight line. And if you pay attention of what a person is doing, he or she would see that Mathematic in there. Franck Jr. Colas, 18:26, May 14, 2007.

The mathematics article currently uses styles in six places
For example:
 * {| style="border:1px solid #ddd; text-align:center; margin: auto;" cellspacing="15"


 * [[Image:Pythagorean.svg|96px]] || [[Image:Taylorsine.svg|96px]] || [[Image:Osculating_circle.svg|96px]] || [[Image:Torus.png|96px]] || [[Image:Koch curve.png|96px]]
 * Geometry || Trigonometry || Differential geometry || Topology || Fractal geometry
 * }
 * }

which looks like:
 * {| style="border:1px solid #ddd; text-align:center; margin: auto;" cellspacing="15"


 * [[Image:Pythagorean.svg|96px]] || [[Image:Taylorsine.svg|96px]] || [[Image:Osculating_circle.svg|96px]] || [[Image:Torus.png|96px]] || [[Image:Koch curve.png|96px]]
 * Geometry || Trigonometry || Differential geometry || Topology || Fractal geometry
 * }
 * }

BTW, WP:MOS is a guideline. --Jtir 16:18, 7 October 2006 (UTC)
 * That is because it is more like a navigation box than an image with a caption. The images with captions will use the image: syntax in this article. — Mets 501  (talk) 16:27, 7 October 2006 (UTC)


 * User:Mets501 wrote in this summary: "revert, basically all images on Wikipedia use the image: syntax, so this article will as well".
 * Is this a policy then? Today's FA is doing basically the same thing --Jtir 16:50, 7 October 2006 (UTC)

{| cellpadding=3px cellspacing=0px class="wikitable" style="float:right; border:1px solid; margin-left: 1em" !style="background:#f90; border-bottom:1px solid" align=center colspan=2|Adi Shankara
 * colspan=2 align=center style="margin: 10px; border-top:1px solid"|[[Image:Adi Shankara.jpg|center|200px]]
 * align=center style="border-top:1px solid"|Dates:||style="border-top:1px solid"|c. 788 to 820 AD  There is some debate regarding this issue.
 * align=center style="border-top:1px solid"|Dates:||style="border-top:1px solid"|c. 788 to 820 AD  There is some debate regarding this issue.

What you've pasted is part of an infobox, not an image with a caption. You'll notice that all the images throughout that FA use the Image: syntax. darkliight[&pi;alk] 17:02, 7 October 2006 (UTC)


 * I don't see any distinction between that infobox and a picture with a wiki-linked caption. For the reader there is no distinction -- it's a picture with some links to click on. However, I do see a distinction in how they are implemented. Infoboxes are better because they are far more flexible and powerful. What am I missing? --Jtir 17:45, 7 October 2006 (UTC)

Why infoboxes are better than inline images.
Text alignment can be controlled. I have centered some captions and left-aligned others. The alignment may need some tweaking. Further, they eliminate clutter -- namely the small resize icon that indicates the image is a thumbnail. In some cases the result has been a smaller box, which saves screen space. Also they support titles above the boxes if editors feel those would be desirable.

Another advantage to infoboxes is that images and captions can be stacked more densely, as this sandbox example shows.

User:Fredrik raised these issues (which I am paraphrasing):
 * 1) Using infoboxes is inconsistent.
 * 2) Infoboxes are complex.
 * 3) They interfere with custom style sheets.


 * Actually, I was increasing consistency.


 * I agree that they are more complex (I know, because I just got through trying to get them to display correctly and then tediously editing six of them without breaking anything.), but they also are more capable.


 * I don't know if the last is true. However, most readers of the encyclopedia probably don't know how to edit custom stylesheets, so they deserve to see an attractive layout without them.

The six blocks of images illustrating fields of mathematics waste screen space at each side and have poor integration with the article. What does a picture of Rubik's Cube captioned Abstract algebra tell a reader? Further, it is far more conventional to put that info along the sides. Now they divide the article.

I converted the panel illustrating number systems into a vertical infobox and it looks fantastic. Before, it was hard to see where one system ended and the next began.
 * Now they are easy to see.

I rv'ed it when I realized that doing the same thing with the remaining panels would end up with panels running far past the text they are supposed to illustrate. I like the images, but there are too many of them for side panels. I am now thinking they might be better collected in a separate gallery section with more informative captions.

--Jtir 17:35, 7 October 2006 (UTC)
 * Image syntax is easy to understand, and very logical. Making all images into a table is just pointless.  And like above, even you say they look alike, and the image syntax has the advantage of working with custom style sheets.  Can you find any other articles that use this infobox structure instead of the image syntax? I can't. And the resize icon is not just clutter, it's there for a purpose: it's saying that you can click on the image to see it larger. — Mets 501  (talk) 18:23, 7 October 2006 (UTC)


 * Every image is already clickable, so what does that icon add? The Lord of the Rings does not have that icon and clicking on the image brings up a larger image. It is redundant and therefore clutter. Or am I missing something? --Jtir 18:59, 7 October 2006 (UTC)


 * "Image syntax is easy to understand, and very logical.": Now I realize what I have been missing. This is all about keeping things simple for the editors . That is certainly desirable, since editors of any skill level may edit the WP. It would make things even simpler if there were no style boxes at all. :-) And I also just realized the examples I have been citing are automatically generated from templates. Editors can see the complexity after expansion, but they don't have to know how to create it in detail.  However they still have to edit after expansion, as this edit of the FA shows. --Jtir 19:30, 7 October 2006 (UTC)


 * I'm sure simplicity for editors is an important consideration, but there is also the issue of having a consistent style. The infobox style is for infoboxes, and the image style is for images. Mixing them around goes against the point of having the styles in the first place. It is true that the displays in the fields of mathematics section are an anomaly - they are not simply images or infoboxes. The best description of them probably is navigation boxes with images. At any rate, they shouldn't be used as a model for other images in the article, and if there are problems with the image style, then the style should be discussed, not simply discarded in this article. JPD (talk) 14:44, 9 October 2006 (UTC)

Possible Additions:
Under "things math is not", I'd like to see a comparison with algorithms. Is the difference the capacity of algorithms to save state?

A high school student (many I'm sure) recently asked, "What's this stuff good for". How about a page on the applications of high school math?

Will Brown 12 October 2006

There's not much wrt history, specifically contribution of eg al gorizme or whatever his name was, that is the arab contribution... Also deals with categorising, rather than actually talking about the content, simple things like place value and the representation '0'... is this on purpose? I am sure you guys have talked about this... I was just curious --Fidocancan 12:25, 29 October 2006 (UTC)


 * I can't remember any specific discussions about that; more on whether maths was to be called a "science" or a "discipline" :-). The problem is that there is so much to say that you must make a selection, and even then can only touch on the topics. What we have tried to do is identify the major topics and refer to other articles dealing with the topic in more detail. So the section on History refers to the article History of mathematics, and as you will see that is by itself so rich that it is essentially impossible to give a balanced summary in the present article that would specifically name al-Khwarizmi. History of mathematics further refers to History of mathematical notation which refers to History of the Hindu-Arabic numeral system. Important as this has been to the development of mathematics, many other things not mentioned directly in the text are equally important to the whole subject. We are open to all suggestions for improvements, though. --Lambiam Talk  15:26, 29 October 2006 (UTC)

Discrete Mathematics
The page classifies discrete mathematics as a relatively new discipline. In so far as it is concerned with computer science this is true but discrete mathematics itself (contrasted to more continuous settings like topology or calculus) is hardly new. Some of the earlier purely combinatorial works are attributed to Bhaskara (1114- circa 1185) and Gerson (1288-1344). Combinatorics falls neatly into the category of discrete mathematics so these counter examples serve to debunk the idea that discrete mathematics is new. May of the principal results are in fact significantly older than calculus. [ED Forgot Spell check] —The preceding unsigned comment was added by 68.48.143.71 (talk • contribs) 01:57, November 1, 2006 (UTC).

in the Discrete Math section, the word 'soluable' is used where it seems the word 'solvable' was meant. Pulseczar 03:43, 25 January 2007 (UTC)
 * Or perhaps soluble (http://www.thefreedictionary.com/soluble) 213.143.18.224 15:06, 23 April 2007 (UTC)

"academic discipline"?
I know this was discussed above but unfortunately I wasn't around to participate. It is very misleading to describe mathematics as " the academic discipline that …". Mathematics is not just an academic discipline. Academics is in no way a defining property of mathematics. Mathematics and mathematicians, existed before the existence of formal academic institutions, exist now outside of them, and would continue to exist without them. For now I've returned the lead to the original "discipline". "Body of knowledge" would also be acceptable to me. Even science used in the broad sense would be better than "academic discipline". Paul August &#9742; 06:03, 2 November 2006 (UTC)
 * I haven't read back over the discussion, but I think it was added because of a concern that people might confuse it with ... bondage. While I don't mind the sentence without the word academic in it, I never thought it was a problem with it in there either. Now I'm a bit curious, what does academic really mean in this context? darkliight[&pi;alk] 08:53, 2 November 2006 (UTC)
 * See Academic discipline. The fact that maths is an academic discipline now does not imply it has always been so – although the notion of it being academic has a long background. The discussion about "discipline" is still on this page. Personally I prefer "study", which is the noun that was used until early June. Objection: it excludes the use of mathematics. OK, make it "branch of knowledge". No, too static. And so on and on. This may be one of these cases where for every choice there is a majority who prefers another choice. --Lambiam Talk  09:42, 2 November 2006 (UTC)
 * Hmm. I always felt that an academic discipline was something that could be studied at a college or university level (as Academic discipline suggests), but that just because something could be studied outside of those environments it should not be excluded as an academic discipline. I'm not arguing one way or the other here, I'm just trying to get a good sense of the what the words really mean. darkliight[&pi;alk] 10:03, 2 November 2006 (UTC)

Mathematics is certainly an academic discipline, but it is clearly more than that. The implication of the former lead was that mathematics was restricted to being an academic discipline, which is just not true. Paul August &#9742; 18:42, 2 November 2006 (UTC)
 * Well, if one won't do, let's try three. Of the three I put in, I like Peirce best; but I'm a Hilbertian. Septentrionalis 19:27, 2 November 2006 (UTC)

What mathematics is not
I see an anon has taken the plunge and removed the controversial "What mathematics is not", as was discussed on here recently. Before some reactionary leaps to revert the change, I'll point out that I completely agree with this deletion. How does everyone else feel? Soo 22:57, 15 November 2006 (UTC)
 * I agree, and have said so repeatedly. I wouldn't mind getting rid of the "misconceptions" section along with it. -- Trovatore 22:59, 15 November 2006 (UTC)


 * I disagree because many people who are not familiar with what mathematics is really all about would be MOST helped by the delineations between mathematics and accountancy, mathematics and numerology, etc. That's my objection to the "not useful" argument.  As for the "pretentious" argument, I'm unclear about what the anon editor thinks this section is "pretending" to do.  capitalist 04:34, 16 November 2006 (UTC)

Actually, a lot of this article is horribly pretentious and not useful, nor understandable by a person interested in looking at the various fields of mathematics. The subject is hard as it is, why deter people from it by just stating that everyone who studies it has their head stuck up their asses? BTW, I was the one who removed it, and I think the "misconceptions" part should be taken too. —The preceding unsigned comment was added by 216.165.37.30 (talk • contribs) 03:16, November 16, 2006 (UTC).


 * There are many things mathematics is not: arm wrestling, butterfly collecting, change ringing, .... We can't possibly list everything mathematics is not, and the idea (as I understood it) was to take away some common misconceptions. So we need at most one section, say "misconceptions", rather than "is not" + "misconceptions". I'm not sorry to see "is not" go, because it wasn't good. "Misconceptions" is also not very good, and, I'm afraid, in fact OR, so I won't mourn its demise either. I can see the potential value of a well-written section, but this is not it. --Lambiam Talk  07:31, 16 November 2006 (UTC)


 * I agree that there should only be one section and that it should be titled "Misconceptions". I never liked the "What Mathematics is not" title because it seemed kind of in the reader's face.  I would like to see the deleted information on accountancy, theorom solving and numerology appear in the "Misconceptions" section however.  capitalist 05:14, 18 November 2006 (UTC)


 * I can agree in principle that a well-written "Misconceptions" section could, in theory, have (extremely modest but) positive value to the article, taken in isolation. But trying to keep it well-written would be a continuing annoyance that I think would tip it over into the negative. The things are pure POV bait, not worth the trouble. Deep-six it now and leave it out. --Trovatore 05:24, 18 November 2006 (UTC)


 * Actually, I take the first part of it back. I can't imagine such a section being well-written. I think they're not just POV bait; they're pretty much ineherently POV. Salt and burn its bones. --Trovatore 05:37, 18 November 2006 (UTC)


 * Smells like consensus... Soo 22:35, 20 November 2006 (UTC)


 * Bah! I'm going to lunch. ;-)   capitalist 03:57, 22 November 2006 (UTC)

Featuredness
The drive to get this article to featured status seems to have stalled in recent months. I will try to push things along over the next few months but I don't have enough knowledge of the subject (or time) to do everything myself. Anyone willing to get one of Wikipedia's most important (and most read) articles to Featuredhood, say "aye". Soo 22:35, 20 November 2006 (UTC)
 * I'll do my best to help, but it'll be easier if someone could tell us what to do. I think it's a great article, but it's kind of difficult for a new-comer to determine its needs? Snailwalker | talk 13:23, 15 December 2006 (UTC)
 * The to-do list at the top would be a pretty good place to start. The hard bit is sourcing everything. Soo 18:46, 15 December 2006 (UTC)

Multi-dimensional Math
Multi-dimensional Math

While this is part of THE LIGHT; The Rainbow of Truth collection of research ideas attributed to the philosophy of Thinking in Colour, it is also the logical development of Alfred North Whitehead and his observation that "There are no whole truths; all truths are half-truths. It is trying to treat them as whole truths that plays the devil".

In the realm of Mathematics we appear focused in the world of concrete reality, whereby in the world of abstracts and mult-dimensional representations we have different answers. American school redefines mathematical paradigm

The simple reality that in the absolute concrete world, 1+1 = 11, and in the pure abstract world, 1+1=1, case example, the cities of Port Arthur and Fort William were united and formed the city of Thunder Bay, Ontario. Here the addition of concepts remains the same, only that the concept is larger.

The addition of variables can have different values if the value of each integer has sub-values. One bag of marbles plus one bad of marbles equals 200 marbles. The complexity of the variables has changed in this particular example; multi-dimensional math.

--Son of Maryann Rosso and Arthur Natale Squitti 22:54, 29 December 2006 (UTC)
 * Heehee, top trolling sir. Soo 23:54, 29 December 2006 (UTC)

Commercial / Business Math
I can't find a topic for an important branch of lower mathematics: that which is often called commercial math or business math. It is a practical subject, emphasizing simple arithmetic, percentages, and fractions, but also covering things such as banking transactions (writing checks, for example), purchase orders and invoices, consumer and business loans, etc. All of these things have a mathematical component, or at least a computational one, and they are very widely taught in commercial courses around the world. Does this subject have an article? If not, should it? What should it be called? Lou Sander 20:18, 13 January 2007 (UTC)
 * You may want to post this at Wikipedia talk:WikiProject Mathematics where most math discussion takes place. Oleg Alexandrov (talk) 21:24, 13 January 2007 (UTC)
 * It's called arithmetic. Rick Norwood 13:41, 30 January 2007 (UTC)
 * Oh, bravo! :o)  capitalist 04:16, 3 February 2007 (UTC)

locked?
Why is this article partially locked? Charles (Kznf) 21:35, 29 January 2007 (UTC)

Probably because someone, likely a high school student or University student was vandalizing the page. I'm not sure, but it seems like the kind of thing that would happen... Trocisp 17:44, 30 January 2007 (UTC)

Link to the mathematics portal?
Why isn't there a link to the mathematics portal? There's a link to the chem portal under Chemistry =o. It'd also be nice to have a table on the right hand side with all the different disciplines in math, much like the way the one under political science. Talcite 02:43, 5 February 2007 (UTC)

Oh wait, its under the picture, never mind =P. Talcite 02:44, 5 February 2007 (UTC)

Notation, language, and rigor
The article asserts that fallible intuitions have led to mistaken "theorems" often in the history of Mathematics. This may be, but the link provided does not seem to support the assertion very well. Most of the examples at the other end of the link are simple algebraic errors, not mistakes of the kind Newton, Descartes or Euler might have stumbled into for lack of sufficient rigor. I have not edited the article, but if you want to edit it in the matter, feel free. Tbtkorg 17:11, 28 February 2007 (UTC)


 * I think you are right, but I'm not sure what to do about it except to remove the sentence entirely. All of the mathematical mistakes that come to mind, "proofs" of the parallel postulate, the Dehn lemma, "proofs" of Fermat's last theorem, and so on are mistaken proofs rather than mistaken theorems.  Can anyone think of a mathematical assertion that was a mistake -- in the sense that, for example, phrenology or N-rays were scientific mistakes? Rick Norwood 19:42, 28 February 2007 (UTC)


 * I think the parallel postulate is an OK example. Although no proof was ever generally accepted, plenty of people claim to have proved it from the Euclidean axioms, whereas we now know it to be independent of them. EdwardLockhart 09:07, 17 March 2007 (UTC)

This is not quite what I'm looking for, since most people would naively agree that in the plane given a point and a line there is one and only one parallel to the line through the point. This can't be proven, but it is generally accepted. I'm looking for a statement that was asserted by mathematicians, and given a falacious proof, which later turned out to be mathematically (rather than metamathematically) incorrect. Of course, there have been lots of "proofs" that, for example, pi is rational, but no real mathematician accepts these. Can the record of mathematics in establishing truth really be that good? Rick Norwood 12:55, 17 March 2007 (UTC)


 * Yes, Rick, that is right. That is also what I was looking for.  There may well exist somewhere an example of a false result stumbled into, and accepted, through lack of rigor, but I doubt very much that this happens often.  To call a theorem false because someone has derived it outside the strict limits of some formal algebra is a tautology, implicitly restricting the meaning of the verb to derive, the noun theorem, or both.  "To derive" does not mean "to derive rigorously," else there would be no need for the adverb rigorously.  (One could call a nonrigorously proved theorem a "conjecture," but a conjecture is usually thought of a claim whose truth remains unknown, rather than merely one whose known truth has never been cast in the terms of some formal algebra.)  Edward's example is a good one, but it does not quite seem to address Rick's point in my view.  Tbtkorg 13:04, 21 March 2007 (UTC)


 * I remember a book mentioning one that I think fits the spirit of the statement, but I can't remember quite what the theorem was. It was geometric, and visual, involving I believe the drawing of graphs on 3D shapes, and the counterexample was something along the lines of a cube with a spherical hole cut out of it. According to the book (I'll see if I can find it again), this followed far enough after the proof and was enough of a shock that nobody would trust visual (rather than analytic) proofs for quite awhile afterwards. It's possible that mistakes of that type just aren't very common, though. Maybe it would be better to focus the sentence on the eternal war against new mistakes, rather than on mistakes from the past. Black Carrot 08:39, 2 June 2007 (UTC)


 * Come to think of it, isn't there something called "naive set theory", a widely accepted precursor to modern set theory, that turned out to have some gaping logical holes in it? Black Carrot 08:41, 2 June 2007 (UTC)
 * This is a can of worms, and a problem I've been meaning for some time to address in several of our articles. There's a popular notion that Cantor's non-axiomatic (or "naive") set theory was responsible for the antinomies such as Russell's paradox, and that the solution was to put it on an axiomatic basis. In actual fact, what Russell's paradox directly refuted was a particular axiomatic system, due to Frege, that formally encoded what was the real problem (the conflation of the extensional and intensional notion of set).
 * Whether Cantor's system, as he understood it (say, after coming to terms with Cantor's paradox) was actually vulnerable to the antinomies is a point of dispute. Hao Wang thought not; others have disagreed.
 * In any case, in my opinion, the idea that formal axiomatizations such as ZFC are the solution to the antinomies is a red herring. The real solution was to re-conceive the notion of set, not as "extension of a definable property", but as "object that appears somewhere in the cumulative hierarchy". Cantor took at least some steps in this direction; the notion at the time was called limitation of size. --Trovatore 08:57, 2 June 2007 (UTC)


 * A clearer case was Alfred Bray Kempe's universally-accepted but fallacious 1879 proof of the Four-Colour Map Theorem. When Percy John Heawood exposed the error, Kempe credited him, thanked him, and retracted it. Tilsit 09:45, 15 June 2007 (UTC)

Creation of a Mathematical Formula Page
I believe it would be a good idea to add a page with formulas used in mathematics. They could be grouped into categories of different areas of math with explanations and examples of the equation. This would be of great help for many students.--Trd89 23:16, 2 March 2007 (UTC)
 * This question is better suited for Wikipedia talk:WikiProject Mathematics. — M ETS 501 (talk) 23:37, 2 March 2007 (UTC)

Maths, math
The discussion of this trivial point is now far too long and disrupts the flow. Prior to making it longer, it was irritating because it sort of sounded like "those dumb North Americans don't know that it's called maths". I say dump the whole thing; it's not worth the trouble (and wouldn't be, even if it were almost no trouble). --Trovatore 21:54, 6 March 2007 (UTC)


 * I agree. We have the redirect pages for math and maths. This should be enough. This article is about mathematics, not about common short forms/abbreviations for the term mathematics in every English dialect. Ocolon 22:03, 6 March 2007 (UTC)

I disagree. For most terms in wiki - we do add the [edit] Etymology of the word, as well as possible variants.Sardonicone 04:59, 7 March 2007 (UTC)
 * It's just way too much text for its usefulness (which is essentially zero, as everyone knows the words). --Trovatore 05:05, 7 March 2007 (UTC)

If it's fairly close to standard practice with other articles, I fail to see how it really affects this one as well.

However if there's a concensus to remove it, I'll refrain from adding it back in.Sardonicone 05:07, 7 March 2007 (UTC)
 * My biggest objection is really that it's an invitation to a completely unnecessary Yank-Brit war (whose form goes first?), or at least to annoyance on one or the other part even if we don't war about it. Given that it's so completely useless, I think we should avoid even a tiny temptation in that direction. --Trovatore 05:10, 7 March 2007 (UTC)

Being from North America, it was not my intent to start up any sort of Dialect war. I can see how that could happen though, which is why I'll be glad to concede that point pending a sort of consensus on the issue.

Apologies if I offended anyone.Sardonicone 05:20, 7 March 2007 (UTC)

The brief note on math/maths stood without bothering anybody for years. If either side is offended because the other side comes first, I suggest that they have way too much spare time on their hands. Rick Norwood 19:37, 7 March 2007 (UTC)


 * No, it hasn't been there for "years"; it was added sometime last summer or fall. And it's bothered me for that long; I just haven't been irritated enough to do anything about it, until recently when it started getting long. Personally I find "maths" grating, and am more than willing to forgo "math" too, to keep "maths" out of the first sentence. No doubt there are others who feel the same way with the roles of the two terms reversed. --Trovatore 19:46, 7 March 2007 (UTC)

I've made it shorter. Is that better? I grew up with "math" -- "The guy who taught us math, who never took a bath, acquired a certain measure of renown..." -- but I find "maths" charming, like schoolgirls in neckties. Rick Norwood 20:11, 7 March 2007 (UTC)


 * It's less offensive than it was. I still think it's unnecessary and would be better removed entirely. --Trovatore 20:18, 7 March 2007 (UTC)

Typo in picture description
"Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens." By by? This needs correcting. Bye :)


 * I corrected it. Thanks a lot. :) — Ocolon 15:20, 12 March 2007 (UTC)

references problem
This article contains many sections without any references. I tagged those sections with unreferenced--Sefringle 03:45, 23 March 2007 (UTC)


 * Just because sections do not contain inline references, it does not mean they are not referenced. Many mathematical topics, particularly basic and general ones, such as this article, have little or no inline references, as they are not suited for this purpose.  See the ~20 references at the bottom of the page; all of them were used to write the entire article. — M ETS 501 (talk) 04:16, 23 March 2007 (UTC)

According to wiki policy, if something is so well known that it doesn't "need" a reference, then it is also so well known that it is easy to provide a reference, so provide one. Why not? I'll start. Rick Norwood 12:58, 23 March 2007 (UTC)

I took care of two. Are there others? Rick Norwood 13:32, 23 March 2007 (UTC)


 * No-one has suggested here that the requested references are not needed because something is well known. The statement was instead: Yes, there are references, only, they are not inline. They are in the References section. Wikipedia has no requirement that references be inline. See also Scientific citation guidelines. --Lambiam Talk  19:42, 23 March 2007 (UTC)

But inline references are handy, and answered a question that somebody went to the trouble to ask. Rick Norwood 00:29, 24 March 2007 (UTC)

Also note the comment in the discussion of whether or not this should be a Featured Article where someone faults the paucity of inline references. Rick Norwood 21:08, 29 March 2007 (UTC)

Simple expression of complex concept
The expression beneath the image under the heading Notation, language, and rigor doesn't seem particularly simple. These things are relative, of course, but a Mandelbrot or Julia fractal would give a better illustration. Pavium 01:27, 7 April 2007 (UTC)


 * While the expression is fairly simple as they come, I don't understand what the concept is supposed to be that is being described here. For simplicity leading to complexity, a puffer train in Conway's Game of Life is illustrative – but I'm against animated images unless strictly needed, and the concept is not that complex. What are we trying to say here anyway? Is the sentence actually true? --Lambiam Talk  06:37, 7 April 2007 (UTC)

Marsch made the edit "fixme: explain what that equation has to do with the picture". I undid the edit. The picture is clearly a plot of the function z(x,y) with the color determined by the z value at every point (x,y). Eighty 05:58, 25 April 2007 (UTC)


 * The caption should say this. Rick Norwood 13:13, 25 April 2007 (UTC)

General Issue
I'm not sure if this is where the following belongs. I rather suspect that it is not. But I don't know where it actually does belong.

Wikipedia is suppossed to be a general encyclopedia. Why then, do so many articles on mathematics related subjects require a strong background in mathematics in order to understand them? Is it truly impossible to write an article on a mathematics subject that is comprehensible to the general public -and- useful to someone who requires more in depth information - particularly is it truly impossible given that we can benefit from Wikipedia not being paper?-198.97.67.56 16:10, 20 April 2007 (UTC)


 * I, and a number of others, have been working for some time to make math articles more accessible. Try the calculus article and let us know what you think.  On the other hand, there are some math topics that are not accessible without prerequisites.  For example, to understand finite simple group you must first understand group (mathematics).  In those articles, the prerequisites should be stated in the introductory paragraph.


 * If you have specific examples, report them here and I'll be glad to try to improve them. Rick Norwood 12:32, 21 April 2007 (UTC)


 * You bring up a good point regarding prerequisites. I've searched in the past for some kind of reference that shows the hierachical structure of present mathematical knowledge.  I'm familiar with the Mathematics Subject Classification system but that doesn't show a hierarchy;  there's no way to tell that you need to know arithmetic before algebra, algebra and trigonometry before calculus and so forth.  Creating such a hierarchy would be orignal research, but surely there must be some standard reference that universities use to figure out what is a prerequisite of what.  Such a hierarchy would be a great tool in organizing the mathematics articles on Wikipedia, especially for calling out prerequisites at the beginning of articles (which is a great practice btw).   Does anyone know of any such reference that we could somehow add to this article or to WikiProject Mathematics?  capitalist 04:43, 22 April 2007 (UTC)


 * I have a rough "hierarchy" that I teach in my courses, but that would, I suppose, be OR since the only reference I could give would be the prerequisites listed in university catalogs. I've run into the problem before, that common professional knowledge often never appears in books, because "everybody already knows that".  I've published several articles in Mathematics Teacher, but I've also had articles rejected because "everybody knows that", even though I know many examples of math teachers who do not "know that".


 * Here is my own tree of mathematical prerequisites:


 * arithmetic:::::geometry:::::::::probability
 * algebra::::::::trigonometry:::::statistics
 * analytic geometry
 * linear algebra:::::calculus:::::calculus based statistics
 * logic and set theory:::::linear programming, numerical methods
 * abstract algebra:::real and complex analysis:::::topology

Rick Norwood 15:16, 22 April 2007 (UTC)

Is 2 a complex number?
Under title Fields of mathematics - Quantity, it is written that number 2 is a complex number. Is this a mistake, or is my basic knowledge of mathematic so weak?--Yovi 15:41, 12 May 2007 (UTC)
 * 2 is indeed a complex number (think of it as 2 + 0i if you like). The natural numbers are integers are quotients are real numbers are complex numbers, or speaking a little more mathematically, the natural numbers are a proper subset of the integers etc. So although 2 may be a natural number, this does not preclude it from being a complex number also. As far as I can recall, this image was made to emphasize this fact. Hope that helps, darkliight[&pi;alk] 14:28, 12 May 2007 (UTC)


 * In fact, each of the articles Integer, Rational number, Real number and Complex number explains how the concept is an extension of the previous concept (for integer: natural number), and how the conceptually simpler numbers are embedded in each extension. In particular, Complex number states this in the opening paragraph, as follows: "Real numbers may be considered to be complex numbers with an imaginary part of zero; that is, the real number a is equivalent to the complex number a+0i." --Lambiam Talk  14:38, 12 May 2007 (UTC)


 * But doesn't, by definition, the complex number need it's real part and it's imaginary part of the complex number? So that number 2 can not be a complex number, but 2 - 0i is a complex number.--Yovi 15:41, 12 May 2007 (UTC)


 * If you use the normal rules of algebra, then 0 × x = 0 and x + 0 = x for all x. So then 2 + 0 × i = 2 + 0 = 2. The real answer is that the complex numbers contain an embedding of the real numbers, given by the injection mapping the real number x to the complex number (x, 0), usually written as x+0i. However, by convention, the real line is identified with the algebraically indistinguishable clone found embedded in the complex plane. Computer scientists might call this a "coercion" (implicit type conversion), but mathematicians tend not to think in terms of typed objects.  --Lambiam Talk  16:50, 12 May 2007 (UTC)


 * If I can locate the number somewhere on the complex plane then it's a complex number. The fact that it happens to lie on one of the axes doesn't make it any less so.  I may be making a simplistic interpretation of Lambiam's point above, but I think the distinction that you're making between "2" and "2+0i" is a typographical one as opposed to a mathematical one.  capitalist 02:42, 13 May 2007 (UTC)


 * I think one needs to state, if it should matter, whether 2 is being viewed as an element of R or of C. If we're in R, then there is no such thing as sqrt(-(2)), while in C there is. After all, 2 is 1x1 matrix too, if one chooses to view it that way; you could interpret 2+2 as adding the doubling operator to itself to get the quadrupling operator. By default, I would view 2 as an element of C. But to simply say "2 is a complex number" overlooks the importance of context, to my mind. In fact, 2 is a numeral and it could well be acting as the "ten" of the base 3 system Z3 for all I know. Context matters--one must state with what kinds of objects one is working. JJL 03:20, 13 May 2007 (UTC)
 * Shouldn't we then change the article, so that it would be clearer, "with what kind of objects one is working"? I find that number 2 very confusing. Perhaps we could change it to $$2 + 0i = 2, etc.$$?--Yovi 21:08, 13 May 2007 (UTC)
 * No, I wouldn't do that. Its the same thing as having 2 in Natural Numbers and Integers - Integers is a superset of natural numbers just as complex numbers are a superset of reals. (John User:Jwy talk) 21:30, 13 May 2007 (UTC)
 * And the only reason the naturals are a subset of the integers is that we choose to identify them with the non-negative integers. Quoting from Integer: "Taking 0 to be a natural number, the natural numbers may be considered to be integers by the embedding that maps n to [(n, 0)], where [(a,b)] denotes the equivalence class having (a, b) as a member." Even if you balk at identifying the real number 2 with the complex number 2, it is true that among the inhabitants of the complex domain there is one that, among possible other names, also goes by the moniker "2". The caption of the box in the article sets the context that makes it explicit that here by "2" the complex number of that name is meant. --Lambiam Talk  22:13, 13 May 2007 (UTC)
 * It doesn't really matter whether we are choosing to identify natural numbers with subsets of larger sets, or constructing the larger sets from the natural numbers, or whatever. Put simply, the article doesn't say "2 is a complex number", it gives 2 as one of the examples of complex numbers, precisely to make the point that Yovi finds confusing - the fact that the complex numbers include numbers with a zero imaginary part that does not need to be written (or a zero real part, for that matter). JPD (talk) 14:09, 14 May 2007 (UTC)