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Mathematics was one of the Mathematics good articles, but it has been removed from the list. There are suggestions below for improving the article to meet the good article criteria. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.

Article milestones Date Process Result January 22, 2006 Good article nominee Listed May 19, 2006 Peer review Reviewed April 3, 2007 Featured article candidate Not promoted September 8, 2007 Good article reassessment Kept August 3, 2009 Good article reassessment Delisted August 26, 2009 Good article reassessment Not listed This article was on the Article Collaboration and Improvement Drive for the week of May 23, 2006. Current status: Delisted good article

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The solution to whether to put "maths" or "math" first is just not to mention them at all. --Trovatore (talk) 15:03, 13 November 2017 (UTC)[reply]

Agree. Paul August ☎ 15:45, 13 November 2017 (UTC)[reply]

17×19607843=333333331, not 33333331. 80.98.179.160 (talk) 15:51, 24 November 2017 (UTC)[reply]

I removed it. I did not check that error, but the strange formatting of the numbers made it hard to follow, the writing otherwise needed cleaning up, but most importantly it simply did not belong in the lead as an obscure, unsourced example, breaking the flow of the existing text.--JohnBlackburne^words_deeds 15:56, 24 November 2017 (UTC)[reply]

Were there not eight 3s ? The example is famous, and a good example of why proof is needed. The formatting could be solved afterwards, I didn't want to turn everything upside down. Boeing720 (talk) 22:14, 24 November 2017 (UTC)[reply]

This is "the first" math article, or where to begin possibly. Just a lot of philosophy isn't sufficient. Interesting examples really is a good way in order to reach readers. Yes axioms are mentioned later, but that part could well have been deleted instead. Axioms are fundamental. Everything else must be derived (either from axioms or what's already proven). And at a certain level, there are assumptions. But just using words isn't helpful for our readers. I strongly believe examples are required, and as interesting as possible. And interesting at a level that reaches at least readers who have studied math at secondary school level. Lots of our "deeper" math related articles are simply just understandable to those who have studied mathematics for years at university level. But with examples would they become "within reach" for so many more. Assumptions belongs mainly at the very highest level, like that estimation of how often a prime occurs (don't remember it all), but it was an assumption which later was proven to be either far to high or far to low, regarding very high such numbers. But anyways, in this article I think that 31, 331, 3331 etc series really illustrates the importance of proof, even if assumptions also later can be proven to be correct. I assume, by the way, that most of our math-related articles just will be for the extremely few (also very few among math educated people, like engineers) if we avoid examples in hard numbers. All formulas are available at other sites, so why use Wikipedia (for math) if we don't (or won't even) offer something more ? (I have nothing against philosophy but that subjest can't be the main issue in this article) Boeing720 (talk) 22:09, 24 November 2017 (UTC)[reply]

Boeing720, I agree with you that examples accessible to a lay reader are most helpful, and that most philosophy and graduate-level mathematics are off-putting to a lay reader in a general survey article like this. The article has improved enormously over the years, but it could still use much improvement. It's a very difficult job, though, partly because mathematics is such a vast topic, it's hard to do justice to it in a brief survey, and partly because mathematics is itself difficult to explain clearly in a way that's easy to understand. I can tell you how to do it, though. You probably already know, since you've been on Wikipedia a long time, but I'll mention it for anyone else reading. The thing to do is get a few books that are general surveys of mathematics for a lay reader, find their main points and examples, and summarize them in the article. There are actually quite a few of these general-survey books published. This is a lot of work, of course, but this kind of work is the most fulfilling part of editing Wikipedia. —Ben Kovitz (talk) 18:04, 2 February 2018 (UTC)[reply]

Shall we add a section about mathematics education? Benjamin (talk) 02:48, 20 December 2017 (UTC)[reply]

Why? I see no similar sections in any of top pages devoted to disciplines (Biology, Chemistry, Physics, Sociology, etc.). As a math educator myself, I am not adverse to talking about math education, but I do not see such a discussion as adding anything to the topic of mathematics. --Bill Cherowitzo (talk) 04:48, 20 December 2017 (UTC)[reply]

I thought it would be relevant. Benjamin (talk) 06:59, 20 December 2017 (UTC)[reply]

@Purgy Purgatorio: would you please explain why you rolled back these edits? They're small, mostly copyedits, but I'm puzzled by the rollback. Specifically, here's what you rolled back that I find puzzling: (1) deletion of modern Greek from the etymology, where it's not relevant; (2) fixing the use of St. Augustine's "warning" as an example of a mistranslation (the mistranslations, not the original text, are the mistranslations); (3) the ordinary phrase "takes a singular verb" instead of the puzzling plural "singular verb forms". —Ben Kovitz (talk) 09:00, 17 January 2018 (UTC)[reply]

I tried to paraphrase (3) and (1) in my edit summary, by using more than a "singular verb" in "singular verb forms", and by the use of "etymologists" to refer to the linguistically interesting remarks in the alluded section, which I restored. I think that the given information is sourced, is interesting, and belongs to the topic of this article. My intentions regarding (2) were to respect the weight of St. Augustine as a philosopher and the correspondingly widespread "mistranslation" by restoring the attribute "notorious". In my perception, "condemnation of mathematicians" is a notorious "misinterpretation" of St. Augustine's statement(s) caused by too literal a translation of "condemnatio mathematici".

I am myself puzzled about the possible intentions of your edits, and believe my partial rollback is reasoned and reasonable, too. Purgy (talk) 09:37, 17 January 2018 (UTC)[reply]

@Purgy Purgatorio: Thanks for explaining, Purgy. Let's take these one at a time, starting with (2). I agree that it would be nicer to retain the word "notorious", but I couldn't think of a graceful way to do that. The current version has two errors: it says that St. Augustine's warning was itself a mistranslation; and it calls mathematici a "notion", making it appear that a concept rather than a word is at issue. How about I fix these errors and leave it to you to find a way to weave in the word "notorious" without reintroducing these errors? (If I think of a graceful way to include it myself, I'll put it in.) —Ben Kovitz (talk) 10:01, 17 January 2018 (UTC)[reply]

@BenKovitz:I apologize for not having answered your comment, but I understood your edits of the article as implementing your intentions, with which I did not want to interfere any further. Yes, I consider the notion, addressed by St. Augustine with the word "mathematici", as a concept, and therefore the word's translation to "mathematicians" is correct in its literal meaning, but is wrong in rendering the intended (by St. Augustine) notion, aka concept (Mit Worten läßt's sich trefflich streiten, mit Worten ein Gebild bereiten. Faust).

In a similar vein, I perceive a fundamental difference between "singular verbs", possibly denoting "always only one verb", and "singular verb forms", being explicit that all employed verbs are to be used in their singular form (see my earlier edit summary for multiple verbs, all in plural forms).

I stated already that I consider the content, which you removed and I restored, as "sourced, interesting, and belonging to the topic of this article". However, again, I will not interfere with your intentions beyond what I did already. Simply let me know, if you think I could answer further questions. Purgy (talk) 08:48, 26 January 2018 (UTC)[reply]

Thanks for your answer, Purgy. Unfortunately, I don't understand most of it. I hope I understood the part where you gave me the green light to go ahead with my copyedits. I'll do that next. Regarding "takes a singular verb" vs. "takes singular verb forms", please see this Ngram and this sampling of books written in English. A Ngram by itself can't settle such a matter, of course; it must be combined with experience and common sense. Hopefully sampling actual usage will help with that, hence the book search. Anyway, the phrase "takes a singular verb" is customary in English to name the kind of grammatical agreement that we're talking about; it does not imply that "mathematics" can't be the subject of a sentence with more than one verb. —Ben Kovitz (talk) 17:01, 31 January 2018 (UTC)[reply]

It's really funny how much dissent influences (factual or claimed) non-understanding of a statement, especially when it is formulated in a borderline style. But, main thing, you are right, I do not care (beyond what I did) about phrases being logically worse but more wide-spread, nor do I mind deleting content, which is "undisputed, interesting, and belonging to the topic of this article", just because some editor dislikes it; and the same holds for changing interpretations of some "transcribed" opinion of an ancient Doctor of the Church about mathematicians (in whatever meaning). So yes, go on, improve the article to your measures (and degrade it to mine). Purgy (talk) 09:36, 1 February 2018 (UTC)[reply]

In my last two (reverted) edits I did not intend to simply dispute B. Oakley in general, but to demonstrate that the ideas, which are excerpted in the article from the two references before my edits, (1) the claim about natural language, "where people can often equate a word (such as cow) with the physical object" and (2) the idiosyncrasy that in math a "single symbol can encode a number of different operations or ideas", are not undisputed in the scientific world. BTW, her claim of "multiplication being repeated addition" is also not accepted in the erudite math world.

I gave two sources and one obvious referral to the mentioned "cow", establishing the inherent incapacity of "natural languages" to unambigously identify physical objects, and cited from the "Begriffschrift" by G. Frege, more than two centuries ago, that any intuitive understanding has to be secured by strict formalism, thereby excluding any "encoding" of different concepts in one notion.

Imho, the given refs suffice to render the transcribed ideas of Oakley in "stark contrast" to other, relevant reliable sources. I am not after calling Oakley "generally refuted". Purgy (talk) 16:53, 31 January 2018 (UTC)[reply]

In the article, there are three ideas attributed to Oakley:

1. Mathematical notation is more abstract than natural language.

2. Mathematical notation is more encrypted than natural language.

3. The greater abstractness and encryptedness are the reason why beginners often find mathematical notation daunting.

The first two seem self evidently true. The third seems highly plausible. I don't see how any of the three ideas "are in stark contrast not only to the rigor, which, in generality, all mathematicians strive for, but also to the inherent ambiguities of the natural languages." Paul August ☎ 18:45, 31 January 2018 (UTC)[reply]

I explicitly referred to the disputed ideas, subsumed under your wishy-washy points (encrypted! in natural language), and do it again:

"... natural language, where people can often equate a word (such as cow) with the physical object it corresponds to, ..."

"... meaning a single symbol can encode a number of different operations or ideas ..."

"By encryptedness, I mean that one symbol can stand for a number of different operations or ideas, just as the multiplication sign symbolizes repeated addition."

The first two claims ARE in "stark contrast" to the sources I cited, the last repeats the opinion of the second, and adds elementary level teacher's misunderstanding. Purgy (talk) 19:12, 31 January 2018 (UTC)[reply]

Purgy, I checked the sources that you provided, and they did not contrast Oakley's ideas about mathematical notation with mathematical rigor. On Wikipedia, we just summarize the authoritative sources about each article's topic; please see WP:V. The articles on requirements engineering were not even about mathematics as such. The Mathematics article is a general survey of mathematics; it should be a summary of authoritative surveys of mathematics. —Ben Kovitz (talk) 20:15, 31 January 2018 (UTC)[reply]

By the way, I'm not sure that we should even be mentioning Barbara Oakley by name here. The salient points are just that modern mathematical notation is more rigorous and abstract than ordinary language, and that this rigor and abstractness often presents difficulties for beginners. The claim that the "encryptedness" of mathematical notation is greater than that of ordinary words sounds dubious, and the word choice is strange; normally we would say that mathematical notation is "equivocal" or "ambiguous". I haven't checked the sources about this, though. Are facts about Barbara Oakley (as opposed to mathematical notation) salient here? And does the claim that mathematical notation is more equivocal than ordinary words fairly represent scholarly consensus? —Ben Kovitz (talk) 20:15, 31 January 2018 (UTC)[reply]

I hadn't actually read what Oakley meant by "encryptedness". I agree that "encryptedness" is a strange choice for meaning that "one symbol can stand for a number of different operations or ideas"; "ambiguous" would be better. And thinking about it with that meaning in mind, her claim 2 does now seem questionable. Paul August ☎ 20:43, 31 January 2018 (UTC)[reply]

When Oakley says mathematical symbols can stand for a number of different operations or ideas, what she really means is that mathematical symbols can encode or symbolize a number of different operations or ideas. She isn't claiming mathematical symbols are more ambiguous than ordinary words (although I can see how the sentence quoted in the article, without the context of the surrounding paragraph, gives that impression). To illustrate the encryptedness of mathematical symbols, she gives the example of multiplication (e.g., the succinct 5 × 5 encoding the more complicated 5 + 5 + 5 + 5 + 5). This is not the best example, as Purgy pointed out, because multiplication can only represent repeated addition in a rather limited context; explaining the multiplication of fractions or negative numbers as repeated addition would be rather tricky, for instance. A much better example of how mathematical symbols are more highly encrypted than regular words is the prime symbol (') used as a differential operator for single variable functions, as described by User:Wcherowi in this discussion I had with him. A couple other examples of encrypted mathematical symbols I can think of off the top of my head are 5! encoding 5 × 4 × 3 × 2 × 1, and 2⁶ encoding 2 × 2 × 2 × 2 × 2 × 2. Basically, Oakley is saying that mathematical symbols can have a number of different operations or ideas packed into them, and that this helps explain why mathematical language is more difficult to process than natural language. Lord Bolingbroke (talk) 00:13, 1 February 2018 (UTC)[reply]

It does help to have the full context from the source available. Paul August ☎ 01:10, 1 February 2018 (UTC)[reply]

Do you think the current wording of the article regarding the encryptedness of mathematical symbols needs to be clarified at all? If people such as you and Mr. Kovitz are being mislead, I don't know if the average reader will fair much better. Lord Bolingbroke (talk) 01:12, 1 February 2018 (UTC)[reply]

First, an apology having misspelled the name above.
As already stated, I do not care very much about the content of this article, only if I get -by chance- confronted with it. So I also will not mind any further, if you leave untouched the obviously inconsistent formulations by Oakley (ambiguity in math notation, superiority of natural language to avoid ambiguity) as a leading idea in the disputed paragraph. Oakley is certainly on to something with the formalism and abstractness of math, making math hard to comprehend for beginners, but the current three sentences and the given citations cannot express the reasons in a concise (mathematical???) way. Partly, this is induced by the unlucky term "encryption", which she has a hard time herself to interpret, and ends in an inappropriate claim of ambiguity. Maybe she should have used "compressed" instead of "encrypted". Concise, strict, rigorous, abstract, compressed, formal, even unintuitive, all being not very sexy attributes, are acceptable reasons for beginner's difficulties, but encrypted and less attached to a well defined meaning are not.

If you mind comparing these three sentences to the content of the following paragraphs in this section, you will see a demonstration that it is not math to be accused of equivocation, but exactly Oakley's c-o-w. These paragraphs give a way more consistent description of reasons for perceiving math as difficult than Oakley's idea of "encryptedness", which, imho, just serves to appeal to the mystic impression math has in the general public ("I'm a rational person, I've never been good at math, and PROUD of it").

Considering that the citations of Oakley start with unfounded conjectures (no human evolution), continue with banalities (math being abstract), and end in obvious wrongness (well formed math notation is ambiguous), I suggest to remove wholesale the last three sentences of the first paragraph, starting with the editorializing "According to Barbara Oakley ..." Purgy (talk) 10:50, 1 February 2018 (UTC)[reply]

BTW, on 22. Oct. 2017 I left the following on Lord Bolingbroke's TP as an additional reply to his attempt to "further illustrate" Oakley's ideas:

While stalking the talk page of Wcherowi I noticed your intent to illustrate the claim of B. Oakly that "mathematical symbols were more highly encrypted than regular words". By referring to the mentioned "cow" I claim that this notion is by far "more highly encrypted" than the notion of "addition". Comparing the plethora of properties involved in identifying something as "cow" (this is a cow, and that also, yes, this too, ...) to the few -admittedly- abstract axioms making up the "addition", makes it obvious to me that Oakly is carried away by the rigorous abstractness of this notion, and involves a new term, encryption, which is imho not appropriate. It is not true that "cow" would not exist as a fully abstract notion, as the cow-in-itself; and it exists in an enormous variety as concrete animal, chewing grass on a meadow, within a, possibly big, herd, made up of, all of them, cows. On the other hand, the "addition" is not only an abstract notion, but has real manifestations, too: adding a dash of salt to some soup "adds" the number representing the mass of the dash to the number representing the mass of the soup (both in the same units, neglecting mass deficits of chemical reactions), putting together a number of fruits from one basket to a number of fruits from another basket adds these numbers to the total number of fruits, and infinitely many other instantiations. The essence of mathematical addition is to embody the addition-in-itself, disregarding any concrete realisation. Maybe, we cannot phrase (axiomatize) a "cow-in-itself", but math works on phrasing "addition-in-itself". Imho, with respect to addition, Oakly is right, when talking about "abstraction", wrong, when denying "physical analogs", and quite meaningless, when mentioning "encryption". Therefore, I object to any illustration of this thought.

Just to make interests more clear. Purgy (talk) 11:29, 1 February 2018 (UTC)[reply]

@Lord Bolingbroke: In answer to the question you asked above: "Do you think the current wording of the article regarding the encryptedness of mathematical symbols needs to be clarified at all?" Yes I think so. Paul August ☎ 12:00, 1 February 2018 (UTC)[reply]

I agree with Purgy's suggestion about removing the sentences about Barbara Oakley opinions, which are taken from A Mind For Numbers. If we quote or paraphrase Oakley's opinions in her own language this is confusing; if we try to interpret them this is speculation; and these opinions are based mostly on her own personal experiences anyway. Her book is interesting, but should not be given undue weight in this article. Gandalf61 (talk) 15:20, 1 February 2018 (UTC)[reply]

I also concur that the article's explanation of "encryptedness" is ambiguous to the point of being misleading (misleading one to think that it means ambiguity!). Possibly the whole passage on Barbara Oakley's ideas should be removed. I'm not at all sure that it reflects scholarly consensus, but I haven't looked into it enough yet to know. I'm happy to read some of Oakley's book and search for other sources on the same topic if no one else gets to it before I do. I probably won't have time until the middle of next week. —Ben Kovitz (talk) 13:08, 2 February 2018 (UTC)[reply]

@Purgy Purgatorio: I tried to clarify this in my previous message, and I'll try to do so again now. Oakley doesn't claim that math notation is especially ambiguous. I'm not sure where in the text you find her making claims that "well formed math notation is ambiguous" or about the "superiority of natural language to avoid ambiguity". As I mentioned above, the wording "one symbol can stand for a number of different operations or ideas" does make it sound like she's saying math notation is ambiguous; considered in context, however, it is obvious she's saying math notation is compressed (i.e., that multiple operations or ideas are embedded in just a few symbols). For this reason, I agree with you that the term "compressed" would probably be a better term than "encrypted" to get Oakley's message across. I'm open to removing the in-text attribution of Oakley and to rewording the article to clarify what she's talking about (perhaps by adding an example of compressed math notation such as exponents or factorials as I discussed above); I just hope we can do so without attributing positions to Oakley that she's not actually taking. Lord Bolingbroke (talk) 06:55, 3 February 2018 (UTC)[reply]

@BenKovitz: I don't think the whole passage on Oakley's ideas should be removed. I'm inclined to agree with Mr. August that two of the three ideas attributed to Oakley (that mathematical notation is more abstract and more compressed than natural language) are pretty much common sense. Of course, "common sense" shouldn't be the criterion for the inclusion of content in Wikipedia, given how nebulous and subjective it is. I'm confident, however, that sources can be found for these claims (Oakley herself cites a few, as I recall). I think sources could also be found supporting Oakley's claim that the abstractness and compressedness (?) of mathematical notation may be why it more difficult to process than natural language.

At this point, it seems like there's something close enough to consensus that I'd like to offer a few concrete suggestions:

• Remove the in-text attribution of Barbara Oakley.
• Remove the term "encrypted" from the passage, along with Oakley's citation supporting it. (This specific citation seems to be the one that's causing the most controversy.) Instead, use the term "compressed" to describe what Oakley is trying to communicate, and possibly include an example of what this term means.
• Keep the rest of the wording the same for now, but search for more sources to see whether there is any disagreement among scholars about the claims Oakley is making. If there is disagreement, this discussion can be restarted to see how we should proceed.

Any thoughts? Lord Bolingbroke (talk) 07:51, 3 February 2018 (UTC)[reply]

Well, I received your ping (not required in a discussion), read your remarks and acted according to the statements as of then. I did not know then about your addiction of keeping popular rubbish like "encryptedness" (= exclude access to non-in-the know) of math within the article, and also not of your attempts of making new suggestions to save a (recently used) "bee in your bonnet". Imho, it does not help to defend Oakley by simply substituting in your suggestion my coinage of "compressed" for her -by prevailing opinion- misleading formulation. As of now I stop at my sole discretion commenting on this article until further notion. Purgy (talk) 08:42, 3 February 2018 (UTC)[reply]

Purgy, I'm actually not that attached to this specific source by Oakley. As I've come to see through this discussion, the wording she uses is far from ideal. I do think the basic ideas she's trying to communicate are important though. What do you think of this suggestion: I will remove all references to Oakley from the article (citations, in-text mentions, the whole caboodle), but take a shot at rewording the material attributed to her in order to clarify what she's trying to say. As I stated earlier, I'm confident other sources can be found to substantiate the claims she's making (or something close to them). And really, it's these claims that I'm interested in seeing in the article, not Oakley's specific description of them. Do you mind if I take a shot at this? If you and other editors sill want to remove the new wording in light of future sources or further discussion, I won't object too vehemently. It is, after all, just three sentences that we've been discussing this whole time. Lord Bolingbroke (talk) 09:31, 3 February 2018 (UTC)[reply]

For what it's worth, the wording of the article before I added the material sourced from Oakley's book went like this: "Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is compressed: a few symbols contain a great deal of information." This second sentence on how mathematical symbols are compressed is basically the same thing Oakley was trying to communicate about mathematical symbols being encrypted; it's just slightly different terminology. In the reword I'm thinking of, I'd have a sentence along these lines, and then add another sentence about how mathematical notation is more abstract than natural language. I think I would leave out the claim that these things are what make mathematical notation difficult for beginners, at least for now. Again, I'm open to having this new wording changed per sources and discussion. Do you have any objections? Lord Bolingbroke (talk) 09:47, 3 February 2018 (UTC)[reply]

I substantially support all of Lord Bolingbroke's suggestions. Paul August ☎ 13:00, 3 February 2018 (UTC)[reply]

Lord Bolingbroke, here are my thoughts. (1) Remove in-text attribution of Oakley: yes. (2) Remove "encrypted": yes. (3) Regarding the word "compressed": How about "terse"? That's ordinary English for the same concept. I don't think there's any need to invent jargon here. (4) More sources: yes, another source or two would probably help a lot. (5) I'm actually not convinced that mathematical notation is more abstract than natural language. Is "+" more abstract than "plus"? At least, the notion of 'abstraction' probably needs to be broken down to indicate what sort of abstractness the source is talking about. (6) Even if we haven't settled every detail, go ahead and edit. Sources and further editing by other editors to fine-tune wording or anything else should improve this part of the page pretty quickly, without extended discussion. If some new problem comes up, we can discuss it then. —Ben Kovitz (talk) 14:58, 3 February 2018 (UTC)[reply]

I concur with Ben Kovitz' points. Schmandt-Besserat, cited above, points out that the use of clay tokens (the precursors of writing) of ownership pre-dates numerals, which predate writing. The clay containers for these tokens of ownership were marked by embossing symbols for the tokens, which in fact are the direct precursors of writing on clay tablets. Thus the symbols, far from encrypting the number of sheep in a flock, aided the ownership of, and trade in, wealth. --Ancheta Wis   (talk | contribs) 15:17, 3 February 2018 (UTC)[reply]

"Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of topics such as quantity (numbers),[1] structure,[2] space,[1] and change.[3][4][5] There are many views among mathematicians and philosophers as to the exact scope and definition of mathematics.[6][7]"

Somethings missing in the basic introduction - that math is the study of portion and proportion, and done so systematically such as to provide a basis for science (and engineering). That a good portion of math is portions, which is to say holistic study - the study of whole forms and how they are divided. Math is not measurement itself, but it uses measurements to build formula upon. -Inowen (talk) 22:14, 1 February 2018 (UTC)[reply]

The introduction to this article is contentious, perhaps because the article attracts many editors with differing backgrounds. For you to convince your fellow editors of your viewpoint, you will surely need Wikipedia:Reliable sources. Then the discussion can begin. Mgnbar (talk) 02:40, 2 February 2018 (UTC)[reply]

"Ratio and proportion are fundamental to mathematics and important in many other fields of knowledge. Many phenomena can be expressed as some proportional relationship between specific variables, often leading to some new, unique entity." [1] -Inowen (talk) 04:15, 2 February 2018 (UTC)[reply]

Aren't "ratio" and "proportion" already subsumed in "quantity"?

The important thing to keep in mind is that the first sentence of this article is intentionally vague. We will never be able to give a complete, defensible demarcation between what is mathematics and what is not. But pro forma we have to put something that looks like a definition at the start of the article; it's part of the style, and it's what readers expect.

We're not going to be able to convey an awful lot of actual information there. I think it is useful that we mention "structure", because a lot of less mathematically inclined readers probably don't think of math as being about structure, and that might spur productive thought patterns for reading the rest of the article. But other than that, we mostly want to get the "definition" out of the way without doing any harm. Possible ways to do harm would be to make it seem more precise than it is, to limit it more than it should be limited, or to drag it on too long. Adding "ratio" and "proportion" would seem fine as regards the first two criteria, but the danger is in letting it drag on too long, especially if it sets a precedent for people to add other things that might occur to them. --Trovatore (talk) 05:26, 2 February 2018 (UTC)[reply]

Indeed, "There are many views among mathematicians and philosophers as to.." Portion and proportion could be mentioned briefly (with due weight) in Section "Definitions of mathematics", not in the lead. Boris Tsirelson (talk) 06:52, 2 February 2018 (UTC)[reply]

I support the views of Trovatore and Tsirel (about vagueness, inclusion of "portion, ..." in quantity, importance of structure, mentioning "portion, ..." later on), but oppose to the claim that math builds formulae upon measurement. Rather, math is eager to develop formulae, capable of predicting measurement, possibly inspired by existing measurements. Furthermore, this were only a description of the applied part of math. Maybe, it is true that the mentioning of "math being essential in many fields" is a somewhat meager denotation of the central and crucial role of math in all natural science. I am unsure, if the importance of the accentuated notions "portion, ..." matches the importance of other mathematically refined notions, which are shaping our lives, not covered by elementary math. Purgy (talk) 09:08, 2 February 2018 (UTC)[reply]

"Math builds formulae upon measurement"? Is it written in the article? I fail to find it there. Boris Tsirelson (talk) 17:02, 2 February 2018 (UTC)[reply]

Sorry for invasion: (math) "uses measurements to build formula upon" by Inowen above. Purgy (talk) 18:00, 2 February 2018 (UTC)[reply]

Tsirel, the lead says "mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects." I'm not sure if Purgy is arguing against that proposition, but it's well established fact. —Ben Kovitz (talk) 17:22, 2 February 2018 (UTC)[reply]

Sorry for invasion: You are mixing up different statements, and insinuate alien claims! Purgy (talk) 18:00, 2 February 2018 (UTC)[reply]

Thanks for the clarification, Purgy. I found "[math] uses measurements to build formula upon" too vague and ambiguous to agree or disagree with. I couldn't tell if it was a reference to the sentence in the lead, or what. I couldn't tell what you meant by "math builds formulae upon measurement", either. (No insinuations intended.) —Ben Kovitz (talk) 18:11, 2 February 2018 (UTC)[reply]

Take a look at Definitions of mathematics for a survey of some of the many competing definitions. —Ben Kovitz (talk) 12:51, 2 February 2018 (UTC)[reply]

Inowen seems to have a bee in his bonnet about "portion and proportion". The book he cites above does not even use the phrase. Of course, "ratio and proportion" are important topics in mathematics, but they are hardly the most important topics, and "uses measurements to build formula upon" is, as BenKovitz says, too vague and ambiguous to agree or disagree with. Rick Norwood (talk) 21:05, 2 February 2018 (UTC)[reply]

Please change "Mathematics [...] is the study of topics such as quantity (numbers) [...]" to "Mathematics [...] is the study of quantity (numbers) [...]", removing the "topics such as". The current wording including "such as" is not mentioned by the cited sources and is editorializing. 58.167.81.248 (talk) 02:37, 3 February 2018 (UTC)[reply]

The sources do not agree on any exact list. See this page, especially this section. Some wording like "such as" is needed to avoid saying that mathematics is limited to exactly those four topics. The list is imprecise; "such as" or some equivalent is needed to indicate that. I would favor "such topics as" over "topics such as", since the latter suggests that mathematics is a grab bag of topics, whereas the former suggests a little better that there is some kind of unity to the topics of mathematics even if people can't agree on what it is or put their finger on it precisely. That is indeed what the sources indicate. —Ben Kovitz (talk) 03:42, 3 February 2018 (UTC)[reply]

Well put. Lord Bolingbroke (talk) 07:55, 3 February 2018 (UTC)[reply]

I suppose your suggestion is a little better. The current wording doesn’t particularly define anything except to say "these four things are a part of mathematics, but there are many other things that aren’t listed here". 1.129.109.81 (talk) 08:43, 4 February 2018 (UTC)[reply]

I worked on this article extensively when the "quantity, structure, space, and change" formulation was agreed upon. I didn't like it, preferring the definition "mathematics is that body of knowledge discovered by pure reason", but I lost the argument. Dictionaries tend to agree with the "quantity, structure, space, and change" formulation, or some subset thereof. For example, Merriam-Webster says "the science of numbers and their properties, operations, and relations and with shapes in space and their structure and measurement".

The "quantity, structure, space, and change" formulation was a compromise which most of the large number of people working on the article at the time could accept. It is now incorporated in a great many Wikipedia articles, and to change it would require changes in all of those articles as well. Since many other sources pick up material from Wikipedia, a Google search of "quantity, structure, space, and change" (in quotes) returns 212,000 hits, and a book published by Springer in 2013 is titled "Mathematics, The Study of Quantity, Structure, Space, and Change". So, like it or not, trying to change that definition now would prove extremely difficult. Rick Norwood (talk) 12:38, 4 February 2018 (UTC)[reply]

I remember the many long discussions involving that formulation. Many editors for whom I have great respect participated. A lot of careful thought when into those discussions. In my opinion, the formulation that we arrived at is a good one. This is born out, to some extant, by its apparent adoption elsewhere. (And, as an aside, I think it's something we can all feel proud of.) Paul August ☎ 17:45, 4 February 2018 (UTC)[reply]

It's a shame that those four broad subtopics of mathematics have been taken as an authoritative definition. While I think it's a good compromise for a general reference work such as Wikipedia, no one should find it satisfying as a true definition of the topic. The sources certainly do not support that. The sources support that no definition yet proposed is satisfactory. That's why we use an ostensive definition rather than the usual definition by genus and differentia. The "such as" ought to indicate that it's only a rough, ostensive definition, but evidently that's not enough to convey that the definition is nonstandard and unsatisfactory. I recall that we used to have clear wording about this: not the mealy-mouthed "many views", but a plain and direct statement that there is no consensus about the definition. I'll see if I can find it and put it back in, or write something straightforward. Feel free to have a go at it yourself, of course, if you get an idea first. —Ben Kovitz (talk) 14:34, 11 February 2018 (UTC)[reply]

Sorry, it's not a shame, but rather speculating with the existence of some authoritative definition of mathematics is a shame. Perhaps, may I point you to the linguistic side of this problem, via Wittgenstein (... thereof we must remain silent), or to the more formal aspect of incompleteness of anything sufficiently strong to fix arithmetic, via Gödel? Clear wordings, outside of formal models just appear as such to the uninitiated, and those looking for being deceived. Sorry, I felt provoked; nevertheless, happy editing. Purgy (talk) 15:00, 11 February 2018 (UTC)[reply]

Actually, I think a sound definition is possible. Mathematics is that body of knowledge arrived at by pure reason. Science is that body of knowledge arrived at by reason, observation, and experiment. So, definition is possible, just not practical in the context of Wikipedia. Rick Norwood (talk) 12:13, 12 February 2018 (UTC)[reply]

This isn't the place to argue about it, but as long as you've brought it up, I'm going to say I disagree with that definition. It both includes things it shouldn't be included (such as natural theology) and excludes things that ought to be included (such as experimental mathematics). --Trovatore (talk) 19:56, 12 February 2018 (UTC)[reply]

@User:Lord Bolingbroke, rather than Oakley, might I suggest Schmandt-Besserat. See Denise Schmandt-Besserat's From Accounting to Writing which shows that clay tokens were a concrete representation of things such as chattel, in the fertile crescent, where clay was abundant. Thus these tokens encoded the notion of 'the size of my flock of sheep', and enumeration of 'kinds of things', before development of the general concept of 'number'. Hence according to Schmandt-Besserat, the notion of number appeared thousands of years before writing. These abstractions evolved gradually through usage over thousands of years. But after their formulation these notions spread rapidly via trading across long distances. --Ancheta Wis   (talk | contribs) 15:40, 3 February 2018 (UTC)[reply]

Ancheta Wis, I took a quick look at "From Accounting to Writing". Its main point seems to be that phonetic writing grew out of inscribed symbols for accounting, the impetus for the shift being a religious ritual that required the names of the dead to be spoken aloud. It doesn't seem to be a source primarily about mathematics, and I don't think that it makes the points that number concepts predate writing or that the earliest number-words referred to specific types of things counted rather than fully abstract quantities (though I could have missed something). If that's right, it would be WP:SYNTHESIS for us to pull those conclusions from this source. However, I think these points are made elsewhere in sources that make broad surveys of mathematics—the ideal kind of source for this article. It's been a while since I looked at it, but I think the first chapter of Number Words and Number Symbols: A Cultural History of Numbers by Karl Menninger discusses these same facts explicitly. That book is widely considered a classic, so it's probably one of the best sources we could go to for this article. By WP:BALASPS, the fact that these facts are in such a source, even given some prominence, suggests that they probably belong in this article.

One slight basis for doubt, though. I'm not sure that number-words began tied to specific kinds of objects being counted in all languages. Menninger's book was written in 1934, and IIRC it reasoned on the basis of the grammar of some American Indian languages. It would be nice to check against recent scholarship (even if it's not published in a book that makes a broad survey of mathematics, of course).

—Ben Kovitz (talk) 17:00, 3 February 2018 (UTC)[reply]

Ben Kovitz, Schmandt-Besserat wrote a lot. What I recited came from a Scientific American article "The Earliest Precursor of Writing" by Denise Schmandt-Besserat. Scientific American. June 1977, Vol. 238, No. 6, p. 50-58. which did not tie the clay tokens (10,000 years ago) to rites (5,000 years ago), but to accounting. In her web page Denise Schmandt-Besserat 'The Evolution of Writing' I refer you to section 1: "1. Tokens as Precursor of Writing". --Ancheta Wis   (talk | contribs) 19:45, 3 February 2018 (UTC)[reply]

It seems to me that linear algebra is missing from the different links to mathematical topics. — Preceding unsigned comment added by 97.102.177.26 (talk) 13:02, 8 February 2018 (UTC)[reply]

We typically add new comments to the bottom of our talk pages (unlike many other sites on the web), so I have taken the liberty of repositioning yours here. As to your comment, look at the section titled Algebra under Pure Mathematics. --Bill Cherowitzo (talk) 20:32, 8 February 2018 (UTC)[reply]