Talk:Mathematics/Archive 12

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Archive 5

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Archive 10

Pictures

This is just a suggestion but could we try to find better pictures for differentail equations (and possibly differential geometry) in "Fields in mathematics" since these two don't look as asthetically pleasing as the others. Algebra man 13:18, 20 June 2007 (UTC)

Cubes

"Mathematics has since the time of ancient Greece, been a set of variances of agreed scale to calculate the number of singular item of use to the commercial environment, even the Romans had no use for 0, nothing or not worth anything. Why then is there a need today for any other type of mathematics, well no longer are we limited by the use of the human brain for our ability to recall knowledge, we have at our control the ability to program a system of electrical impulses to enable a Knowledge retrieval and calculation system called a computer via a system of knowledge retrieval information pods called the internet, to store unique pieces of Knowledge relating to unique areas at unique times. It is this unique singular ability that will take us to another system of calculation, “compound mathematics”. This is the ability to use the unique singular knowledge about any unique area of space at any unique singularity of time, to justify the next event in that unique area of space in the next singularity of time. It is the interaction of the compound effect of the changes caused to these unique areas of space by other unique areas of space over a series of singularities of time that along side the basic principal of physics will become a fresh approach to Calculate the evolvement of knowledge. So how do I justify this claim, for many years I have been trying to justify the ability, to fit a cube in a square. Infact my entire life has been trying to achieve this impossibility. Yet, Albert Einstein tells us in one of the greatest scientific human achievements of all time “The splitting of the atom” that Energy is Equal to the mass of light squared."

Perhaps if I was an academic person I could understand the basis of this equation better, but I am not an academic. I consider myself a practical person and earn my living not from theoretical theories but my ability to turn ideas into actual practical useful objects. It is this practical perspective of the laws of nature that have given me so much trouble with Mr Einstein Equation. A mass by dictionary definition must be, Lump – a body of matter that forms a whole but none definable shape. Collection – a collection of many individual parts. Great unspecified Quantity – a large but unspecified number or Quantity. Physics physical quantity – the property of an object that is a measure of it inertia, the amount of matter it contains, and its influence in a gravitational field. Symbol m. The problem I have is how you can put a mass in a square. A dictionary definition of a mathematical square “the produce of multiplying a number or term by itself”.

In the practical world I live in any number multiplied by itself must produce an area, and because it has no third dimension. This area cannot even have a surface, because to create a surface you require three dimensions. How does a non-academic practical person, tell a world of academic physics masters. That the most famous equation in the World, E=Mc2 is fine if you are looking to justify the “current situation”. But this must lock you into the present situation where you have only one fixed set of variance. Somewhere between a super nova spewing massive amounts of matter from a magician’s hat, to a matter gobbling black hole putting it back into the magician’s hat, but unable to explain “WHY”. Hence a singularity. “ONE EVENT”. So how can we take the singularity of this one event to another horizon. simply by taking energy to the point its velocity is equal to the mass of light cubed. you will not only discover your missing "dark matter" but find a path through the wormhole of your current impass. To understand more go to Spacetime @ talk page of cubedmass. --Thor 14:54, 1 July 2007 (UTC)

First, please add new material at the bottom of the page. That is where people look for it. I've moved your comments here.

I think most of your problems are linguistic rather than scientific or mathematical. For example, in E = mc^2, the "square" has nothing to do with the geometric figure called a "square" but rather uses the second definition of the word, a number multiplied by itself. You seem to think that the only situation in which we might multiply numbers is to find area, but that is not the case. For example, if I multiply the number of hours I work each day by the number of days I work each week, I arrive at the number of hours I work each week. But there is no area involved -- no-one is claiming that hours can be an area. Rick Norwood 15:09, 1 July 2007 (UTC)

I thank you for your help and comment I also understand that in the singular how you have arrived at the number of hour worked in a given period of time. but to work that number of hours in that period of time energy must have been transfered from your effort to your product.

This must as you have stated occured over time and the energy you have used to create your product will have to be replaced or you cannot work further periods of time. My though is how does this energy transfer through time. in your use of singular mathmatics you have explained how you have used unique amount of energy in a unique period of time to create a unique product in a unique area of space. this i see as singular or squared area. What i want to create is a interaction between the singularity of a unique energy used at a unique space and time and its effect on the total amount of available energy over the total amount of available space in the next unique moment in time. Because energy can,t be lost it can only be changed. i know this cannot be done using singular or squared energy to mass ratio because although this justifies the current unique spacetime it can,t take you to the next unique spacetime because it has no means of interacting with other units of unique spacetime. to do this you need a compound or cubed energy to mass ratio it is that will create the interaction to create the next unique unit of spacetime.--Thor 18:14, 1 July 2007 (UTC)

The energy expended in answering your question comes ultimately from the source Einstein described in his famous equation. In the sun, gravity causes fusion which releases energy which reaches the earth in the form of sunlight. The sunlight falling on plants is converted into chemical energy by photosynthesis. When I eat those plants, the chemical energy is converted into a form that can be used by my muscles to allow me to type on a keyboard and answer your questions.

The energy to mass ratio is not squared. Rather it is itself a square, the square of c. This is, however, a purely arithmetic square, not a geometric square.

You seem to assume that the time dimension is quantized rather than continuous. This is still an open question. As you observe, the assumption that time is quantized leads to serious problems. It may be that time is continuous. Rick Norwood 22:45, 1 July 2007 (UTC)

again i must thank you for your comments. the main problem, i have no academic knowledge (although i'm tryin to learn/understand it). so my ability to comunicate in the type of academic language curently understood will not be possiable. However i hope you will hummer my quantized rather than constinuous vision of time, because that is what it is. because i'm not academic i perceive in my practical Knowledge as a quantified chain of events not a continous chain of events. Einstiens relativity along with newtons "equal and opposite" and the many other rules or laws of physics have built up into our knowledge of the subject. you will find in general that the rules or laws that our individual knowledge of physics understands is the rules and laws as individuals we will agree with or disagree with. This in turn will devide us into groups that agree or disagree with therectical propositions. but every individual or group will base there view on the level of knowledge at there disposal. I will know try to my comunicate my views,Sorry i have to go back latter--Thor 16:09, 2 July 2007 (UTC)

The purpose of this page is to discuss improvements to our encyclopedia article on Mathematics. Wikipedia is not a forum for general discussion. If you have specific questions on scientific topics, you can ask such questions at the Science section of our reference desk. Thank you. --Lambiam ^Talk 08:35, 3 July 2007 (UTC)

Many thanks for your comments,I honestly wish my ability to comunicate was better but I understand better than most in "conceptual fact" but my beleif is still nobody will ever understand the dicipline of the ability to use mathematics with your academic perception of only one understanding of 0 plus the ability to use + or - to take towards or away from the concept of answer you seek but I wish you well and will continue to follow. cubedmass 13.07 07

It appears Math and physics in fact are unified should we mention it?

Alright, here we go... Light and energy transmit patterns that we can measure (detect with our organs).

To measure is to calculate (create) distinctions, to be distinct then, is to be NOT EQUAL, to any other distinct-pattern (red, is not equal to blue). Now create is a verb, and therefore a function. Our eyes and ears gather information, distinct visual forms and detectable forms of energy through our ears, neurons, synapses etc. from the patterns of energy in the environment.

Therefore our organs measure (enumerate) the forms of energy(light, etc) in the environment. This would mean that math is actually the study of patterns of light and energy. An abstract concept, actually exists as data in our minds, as a form of stored energy. Therefore an abstract concept is stored as and is a form of stored energy.

Agree, disagree? If so explain and demonstrate how this is wrong. BeExcellent2every1 (talk) 06:58, 20 November 2007 (UTC)

BeExcellent2every1 (talk) 06:58, 20 November 2007 (UTC)

It's just wrong. It's against Wikipedia:No original research, also. Charles Matthews (talk) 08:28, 20 November 2007 (UTC)

You haven't explained how it is wrong. I just said that if you're going to make a CLAIM that you KNOW the STATEMENT is WRONG, then what do you KNOW about the STATEMENT that is WRONG. How do you know it is wrong? You must be able to point out the flaw for your claim to be valid, if you say the definitions are invalid, then which word, and which definition. Where exactly, which words, are wrong? so I can find which definition is incorrect. Otherwise you're claim is invalid, you must show your work, which words or statements are incorrect, otherwise you are making no intelligible claim.BeExcellent2every1 (talk) 08:58, 20 November 2007 (UTC)

Even if you were correct, this talk page is the wrong venue for such ideas. In fact, there really isn't any place (on wikipedia) where such discussion is appropriate. Such speculation violates our most basic policies, as Charles pointed out. —Cronholm¹⁴⁴ 09:45, 20 November 2007 (UTC)

I understand no-original research, I was just wondering if anyone had discovered that little fact. Apparently not. "Even if you were correct" the fact that you said this, means you dont know. So please accept that you don't know, and your claim to my first paragaphs incorrectness is invalid.BeExcellent2every1 (talk) 10:08, 20 November 2007 (UTC)

Cronholm¹⁴⁴ did not claim that you were incorrect. Within the realm of Wikipedia, the verifiability requirement means the burden of evidence (not proof, but evidence) falls on the person wanting to include the information. So if you'd like to discuss some sources, (for example, whether or not a source does a good job explaining the idea, or whether or not a source would be considered reliable) this is the place to do that, but it is not the place to discuss whether the idea is correct. --Smiller933 (talk) 15:00, 20 November 2007 (UTC)

First Sentence

I believe there should be an "is" after "change" in the first sentence. ". . .change, and is also. . ." Otherwise, "the academic discipline" is also one of the subjects of "Mathematics" along with quantity, structure, space, and change. I'm not an English teacher, so someone else will have to make this change.

The present first sentence is short for:

"Mathematics is the body of knowledge centered on concepts such as quantity, structure, space, and change,

and

mathematics is also the academic discipline that studies them."

In an English sentence like that, you can leave out the repetition of "mathematics is" after "and" without change of meaning. Although potentially ambiguous, I think the risk of an actual misinterpretation is small. --Lambiam ^Talk 10:15, 6 July 2007 (UTC)

you're right, their should be a "is" after the change. repeating the word mathematics is redundant. Also there shouldn't be an "and" before change. the sentence should be, "Mathematics is the body of knowledge centered on concepts such as quantity, structure, space, change, and is also the academic discipline that studies them." Lou Dofoye (talk) 03:00, 7 February 2008 (UTC)Lou Dofoye

I respectfully disagree. By the same token, one should also change

Arthur had smuggled in beer, nuts, and crisps, and hidden the contraband in the closet

into

Arthur had smuggled in beer, nuts, crisps, and had hidden the contraband in the closet.

That is not an improvement, however. --Lambiam 06:29, 7 February 2008 (UTC)

It is fine because the "and change" shows the termination of the list of "concepts". However, I think the following sentence is more accurate and succinct:

"Mathematics is the science of quantity, structure, space, and change."

Calling something a "science" automatically implies that it is both a body of knowledge and an academic discipline.To alleviate confusion with natural and social sciences, you could also write:

"Mathematics is the formal science of quantity, structure, space, and change."

Check the articles on science and formal science for more information. Aetherealize (talk) 09:33, 20 February 2008 (UTC)

Does anyone have a problem with the above as the first sentence, or think that the current one is better? Aetherealize (talk) 23:53, 25 February 2008 (UTC)

Just calling it "a science" is clearly not acceptable, as that term is largely identified today with "natural science". A problem with using "formal science" is that this is not a well-known and generally understood concept. Our article Formal science does not do a good job of explaining the notion either; formulations like "A formal science is a theoretical study" and "The formal sciences are built up of theoretical symbols and rules" make me shudder. While the present first sentence is decidedly inelegant, I think its message is clearer. --Lambiam 11:29, 26 February 2008 (UTC)

Mathematics concept collage

Recently the image Mathematics concept collage was added to the article. Personally I'm less than enthusiastic about the image, because it reflects and propagates the misconception that Mathematics = lots of formulas. I don't see any added value. But perhaps others are happy with it and I'm just alone in my grumpiness. So I don't want to remove it straightaway without inviting some opinions. --Lambiam ^Talk 14:26, 7 July 2007 (UTC)

I don't mind the image. If several people dislike it, it should go. I moved it a little further down the page. Rick Norwood 14:28, 7 July 2007 (UTC)

I don't like it much. The images in the navigation boxes further down do a much better job of expressing the richness of mathematics, in my view. It might be more appropriate for an article on high school mathematics, or perhaps mathematics education. -- Avenue 15:27, 7 July 2007 (UTC)

I agree with Avenue. It doesn't seem to take a broad enough view of mathematics. JPD (talk) 18:51, 8 July 2007 (UTC)

I also dislike the image, for similar reasons to Lambiam, Avenue, and JPD. EdwardLockhart 08:06, 9 July 2007 (UTC)

I dislike it too. I've removed it. --Zundark 08:18, 9 July 2007 (UTC)

Good. Paul August ☎ 22:56, 9 July 2007 (UTC)

Lambiam, I felt I should point out that you sort of used a formula improperly in your argument that mathematics is more than just formulas. (". . . the misconception that Mathematics = lots of formulas.") However, I agree with your main point. Also, the picture is ugly. - Chris Klein —Preceding unsigned comment added by 131.215.172.198 (talk) 08:40, 7 November 2007 (UTC)

Images for applied mathematics

Mathematical physics

Analytical mechanics

Mathematical fluid dynamics

Mathematical economics

Financial mathematics

I liked the idea of illustrating different areas of mathematics with images and thought I'd extend it to the section about applied maths. Above, I give an incomplete suggestion and I need you to fill in the gaps. Does protein folding fall under mathematical biology? This could be an alternative for fluid mechanics. Finally, the table is too wide and won't break automatically, so someone who knows how needs to fix it. —Bromskloss 15:54, 30 May 2007 (UTC)

Nice work, Bromskloss. I've gone ahead and added a shortened version of the the above to the article. Paul August ☎ 13:48, 20 June 2007 (UTC)

I replaced the ugly Image:Trapetsmetoden.png (see right) with Image:Finite element solution.svg (see right, below). The latter is my own picture, and better illustration of numerical analysis can be found perhaps, but I'd argue it beats the original picture. Oleg Alexandrov (talk) 03:54, 16 July 2007 (UTC)

Ok. Actually, I don't think the original one was that bad. Handwriting is kind of cosy :-) and you can hardly see it in the scaled-down image anyway. However, the main reason for the original image was that it is analogous to the one used for calculus. —Bromskloss 14:55, 22 July 2007 (UTC)

I call for others to share their opinion on this. I think we are better off with the original image (numerical integration) than the current (finite element), for three reasons:

More fundamental to numerical analysis
Analogous to calculus illustration
The current one is very similar to the optimization illustration

—Bromskloss 05:17, 10 August 2007 (UTC)

.

Then how about using the "new numerical integration" picture on the right, instead? My only concern is that the hand-drawn picture is ugly. Oleg Alexandrov (talk) 15:33, 10 August 2007 (UTC)

I think it is best to view the three images in context. So we have:

Original:


Mathematical physics	Mathematical fluid dynamics	Numerical analysis	Optimization	Probability	Statistics	Financial mathematics	Game theory

Current:


Mathematical physics	Mathematical fluid dynamics	Numerical analysis	Optimization	Probability	Statistics	Financial mathematics	Game theory

Proposed:


Mathematical physics	Mathematical fluid dynamics	Numerical analysis	Optimization	Probability	Statistics	Financial mathematics	Game theory

So far I like the original best. (Oleg: there is just no accounting for taste ;-) I think the handwritten drawing looks informal but not ugly. For me the "beauty" of a diagram is one that communicates well — so I hope all of the blackboard drawings I've done were beautiful, though handwritten. It also helps to remind (or even inform!) that mathematics (for a little while longer at least) is done by people not machines. Paul August ☎ 16:42, 10 August 2007 (UTC)

I also prefer "original". I can see the charm of hand-drawn diagrams but that's not the most important for me. The "current" diagram does not yell out numerical analysis to me. I don't like the "proposed" diagram at this size; I'm not sure what the problem is but the red and blue curves are hard to distinguish and the dash lines look odd when shrunk so much.

I notice only now that there is also a "game theory" image. That disappears behind the right edge of the window on my laptop (screen size 1024 x 768). Should we distribute the images over two rows for poor people like me with small screens? -- Jitse Niesen (talk) 04:23, 11 August 2007 (UTC)

Apart from the charm of diagrams and such clumsily drawn on napkins (but how did the stars get there? were they pasted on?), can someone explain how the "original" image illustrates the "Trapezium rule" other than in a quite confusing way, what with the steps up and down? Or if it does not illustrate that rule, then what does it represent? Maybe we should also consider other candidates, for example an illustration of finding zeros using the regula falsi or Newton's method. --Lambiam 05:12, 11 August 2007 (UTC)

Indeed, is the hand-drawn picture an accurate representation of the trapezium rule? Oleg Alexandrov (talk) 06:04, 11 August 2007 (UTC)

On the right are some other numerical pictures, following Lambiam's suggestion. They won't scale well either to the small size.

If agreed that the illustration of the trapezoidal rule without the staircase which Jitse says has the curves too close (I agree) is a better picture, I can make a version of it to look better small (disclaimer: I made the original, although I hope I am not driven by selfish interests here :) If the hand-drawn picture is agreed to be a better illustration, it can be recreated in SVG easily. Oleg Alexandrov (talk) 06:25, 11 August 2007 (UTC)

For an image illustrating the trapezoidal rule I'd recommend to have equidistant "sampling". The following function has been engineered to have a few relatively but not excessively large deviations from the linear approximations when sampled at integral x, for x = 0 to 5:

0.05x⁶ − 0.762x⁵ + 4.229x⁴ − 10.188x³ + 9.116x² + .638x + 0.66.

--Lambiam 20:30, 11 August 2007 (UTC)

Nice job, that should be easy enough to plot. What we did not agree on is whether there should be a staircase or trapezoids when discretizing (I'd argue for the latter). Oleg Alexandrov (talk) 00:34, 12 August 2007 (UTC)

If it was called the staircase rule I'd probably argue for staircases, but it ain't. --Lambiam 02:21, 12 August 2007 (UTC)

But it doesn't resemble the non-numerical image, and it would be nice if it did. —Bromskloss 07:29, 20 August 2007 (UTC)

(de-indenting) I think the hand-drawn picture is a fair representation of the trapezium rule. It it based on the fact that the trapezium has the same area as the hexagon. My hand-drawing-with-ASCII-symbols skills are not up to the challenge, but perhaps the following will clarify what I mean.

               _
    /|        | |
   / |        | |
  /  |  =    _| |   
 |   |      |   | 
 |___|      |___|

However, I didn't recognize the picture as illustrating the trapezium rule. I thought it was approximation by a piecewise constant function, as in nearest neighbor interpolation, or perhaps integration by the midpoint rule. There is nothing in the picture that says it's the trapezium rule; where did you get that idea? -- Jitse Niesen (talk) 02:44, 12 August 2007 (UTC)

I don't know. :) I guess from comparison with the other picture. The point is still the same, to we want trapeziums or a staircase? In the meantime I tweaked the trapezium rule picture to make it look better small, see to the right (96px, as in the gallery above). Again, the ugly picture can me made svg too, the issue is of staircase vs trapezium. Oleg Alexandrov (talk) 03:30, 12 August 2007 (UTC)

The name of the image is Image:Trapetsmetoden.png, and trapetsmetoden is Swedish for "the trapezium method". The article on the Swedish Wikipedia actually contains ASCII art, with a staircase, of arguably greater sophistication than displayed by Jitse. --Lambiam 06:12, 12 August 2007 (UTC)

Apparently the discussion here died out. I replaced my finite element picture with the trapezoidal rule seen on the right. Again, if people prefer the straircase figure, let me know, and I can recreate it in SVG. Oleg Alexandrov (talk) 01:55, 18 August 2007 (UTC)

More images

Purely a question of aesthetics. Preferences? Suggestions? --Cronholm¹⁴⁴ 03:01, 19 August 2007 (UTC)

I prefer the left one. —Bromskloss 15:11, 11 September 2007 (UTC)

The left one makes the overlap more obvious — the gradients are only distracting. ⇌Elektron 16:53, 11 September 2007 (UTC)

I think the one on the right is prettier, while the one on the left might be clearer. For a survey article, I favor using the prettier one (which is indeed in use now). Suggested improvement: don't lighten the border around the left circle in the overlap area. Right now, both images look like two colored lenses, with the right lens above the left one. Removing the visual cue for Z-order will help lead the viewer to see this as a symmetrical relationship. --Ben Kovitz 14:53, 17 October 2007 (UTC)

GA status reviewed

This article has been reviewed as part of Wikipedia:WikiProject Good articles/Project quality task force. I believe the article currently meets the criteria and should remain listed as a Good article. The article history has been updated to reflect this review. Regards, OhanaUnited^{Talk page} 21:44, 8 September 2007 (UTC)

Abstract Algebra image

Why is the abstract algebra image a Rubik's Cube? —Preceding unsigned comment added by 151.63.84.14 (talk) 17:34, 17 September 2007 (UTC)

Abstract algebra studies algebraic structures, such as groups, and Rubik's Cube provides a tangible representation of a group: Rubik's Cube group. --Lambiam 19:45, 17 September 2007 (UTC)

Maths vs. Math

The current introduction reads "Mathematics (colloquially, maths or math)" and I highly disagree with the placement of "maths" in front of "math". Math is the better spelling, and should come first. I believe this point is extremely important and worthy of debate. Please change this or argue about why it should not be changed. —Preceding unsigned comment added by 129.173.121.142 (talk) 20:52, 18 September 2007 (UTC)

If you're that worried about which of the two comes first, change the order. I can't see that it actually matters. --Mark H Wilkinson ^{(t, c)} 21:38, 18 September 2007 (UTC)

I don't care which comes first, but you are on very shaky ground if you argue that one form is "better" than the other - they are just different conventions. I believe that "maths" is the norm for English speakers outside of North America. Certainly "maths" is the norm in England - saying "math" here would make you sound very ... um ... American. Gandalf61 06:40, 19 September 2007 (UTC)

As a maths graduate from the UK, I would suggest people used mathematics wherever possible to avoid this kind of argument. My own opinion is that maths is the correct spelling, but then I'm biased. SpaceLem 17:41, 8 November 2007 (UTC)

I still say we should simply remove the parenthetical remark about the colloquial names. What does it add to the article? What information does it convey that the reader doesn't already know? Just stick to "mathematics", which is the appropriate form for the formal register used in an encyclopedia article. --Trovatore 15:15, 19 September 2007 (UTC)

For one thing, Maths and Math both redirect to Mathematics, so the parenthetical remark offers an immediate explanation to the reader redirected here of the why of that redirection. I would not assume that all readers already know that these forms are common colloquial abbreviations of the word "mathematics". --Lambiam 17:14, 19 September 2007 (UTC)

1AT. If it is a redirect, it is not needed, b/c obvious. Wikipedia:What Wikipedia is not#Wikipedia is not a dictionary, so the inclusion of the terms is unnecessary and to that degree arguably unencyclopedic~~, if they were colloquialisms[]~~.

2AT. None of my 3 standard dictionaries list then as a colloquialisms. Rather, they are alternative labels.

3AT. Should the article be suggesting in the lead sentence that the bolded terms math & maths should be used as equivalent? Only if it is meant to suggest that they are equivalent terms, say in the U.S. and in Great Britain. I doubt that there is one standard encyclopedic source that so indicates for either, much less both. In that respect, it does not satisfy WP:VERIFY policy. Rather, it is an invitation to misuse or misunderstanding as to regional differences, b/c of American & British parochialism.

4AT. If it is to be mentioned at all, the British-U.S. differences should be noted so as not to mislead, as it is in section 1. Why not keep the first sentence clean and simple and save the parochialism for sect. 1? --Thomasmeeks 02:07, 17 October 2007 (UTC)

Its confusing to pop to this page without the lead explaining why. Its NOT obvious to someone from the opposite side of the Atlantic.
I'm not sure calling them alternate labels means much. People will enter math or maths wanting the information in this article.
math & maths ARE used as equivalents to mathematics. Entering "math" or "maths" in m-w.com and you get: "mathematics."
What are people going to be mislead about? In any event, I think any possibility of misleading are minimal compared to the confusion someone coming on the page by surprise.

I think it useful to have math and maths in the lead to explain why the user ended up on the page. If they want to know the etymology, it can be found below. I plan to switch the lead back. (John User:Jwy talk) 02:55, 17 October 2007 (UTC)

I think it's really unlikely that there are very many users who type either of those terms into the search box and are surprised where they wind up (likewise for internal links). I don't question the accuracy, which was what you referred to in your edit summary, just the value. It's aesthetically hideous, so for it to be there there ought to be a compelling reason, and I just don't see it. --Trovatore 03:22, 17 October 2007 (UTC)

The phrase "(colloquially, maths or math)" strikes me as pedantic. I don't think a native English speaker would be confused by the redirect. However, a great many Americans don't know that "maths" is the British version of "math". So, even though I don't like it, I think it's OK to keep the phrase. Also, I just checked refrigerator and indeed it mentions "fridge" in a parenthetical comment in the lead. Television includes lots of colloquialisms. So I think we are actually being consistent with the rest of Wikipedia here, without becoming a dictionary. --Ben Kovitz 14:44, 17 October 2007 (UTC)

Well, I withdraw my point (2AT) above. My standard dictionaries don't seem to mention any colloquialisms.

1BT. Still, "(colloquially, maths or math)" is unnecessary there. In the Lead, the phrase lacks the context provided in sect. 1 on regional differences, thus arguably violating WP guideline to WP:LEAD#establish context. Including that context in the Lead would bog the Lead down. A N. American might wrongly conclude they the terms were equivalent in use there, when they are not. That is misleading. Similarly for a Brit. Inclusion there is unnecessary except on the assumptions the few readers unfamiliar with the connection of mathematics to math(s) would not be able to figure it out from the Redirect. So, not only is its presence there arguably pedantic but patronizing and misleading. Finally, mathematics is a scholarly subject. Shouldn't the lead indicate that, rather than not try to bulwark against extremely rare contingencies by writing down to its audience?

On Jwy's points, I regret that I was unclear, concerning which let me try to do better:

1CT. Assuming away common knowledge is arguably a formula for overwriting.

2CT. Point (2) above neglects the misleading incompleteness of the objectionable phrase. Anyone who ends up at the article has the Disambiguation at the top & sect. 1 to help out. I believe that the hypothetical person of concern is a virtual singularity.

3CT. The objectionable terms are not equivalent within each geographic region, the failure of which to mention is misleading.

4CT. See (4CT) immediately above. --Thomasmeeks 20:20, 17 October 2007 (UTC)

Thomas, it sounds like we agree in our distaste for the phrase. Let's see if I can answer your objections, though, since, after reading all of the above and thinking about it, I came to the opposite conclusion.

Is there a concise way to say that "maths" is British and "math" is common in the United States? I agree that we are perilously close to overwriting here, so an especially concise wording would be best. Or, in favor of keeping the phrase and not explaining the regions in the lead: Is it not fairly common on Wikipedia for introductions to mention alternate terms in use in different geographical regions, without spelling out these regions? For examples, see Dosa, Hing. Hero sandwich locates the main phrase to New York but doesn't tell which regions use the alternate terms. This seems encyclopedic to me without stepping into dictionary-land or regional-glossary-land.
Indeed, assuming away common knowledge is terrible, but I'm quite sure that a majority of Americans don't know that mathematics is called "maths" elsewhere. (I didn't know that hing is called "giant fennel" until a moment ago.;))
Isn't it commonplace and proper on Wikipedia to list common informal synonyms in the lead, especially when it's likely that many people haven't heard them? (Refrigerator, Television, etc. You can probably find more.) This combined with the previous point is what swung me to favor keeping the phrase.
Regarding establishing context, I believe that principle refers to the conceptual context of the whole topic, not the geographic context of the term, e.g. "abstract algebra" in the article Group (mathematics).

--Ben Kovitz 21:38, 17 October 2007 (UTC)

Just because there are lots of other articles that do it doesn't necessarily mean it's a good thing to do. But to be honest, I don't care as much if refrigerator is written in a distracted and distracting style. Who cares? It's just about refrigerators. I don't think we should have that tone here.

It may be that there are lots of Americans who don't know the word "maths", but I don't think we should be particularly at pains to teach it to them in this article, and certainly not in the lead. It's not a mathematical issue. --Trovatore 22:15, 17 October 2007 (UTC)

1DT. Ben Kovitz's 1sr point above is to recomment a concise statement of U.S./British use of math/maths in the Lead rather than simply deferring it to section 1, presumably b/c different colloquialisms in the 2 different national areas are too interesting and important to defer and are worth repeating. This despite the presumable minuscule fraction of potential readers unaware that math or maths refers to mathematics. If that is not overwriting, it would be hard to cite an example. Asserted analogies cited are Dosa, Hing. Hero sandwich Refrigerator, Television, "etc.," but significantly not scholarly subjects (for one good reason, they do not appear in WP or other broad scholarly-discipline articles). The lack of parallel is unpersuasive.

2DT. Is it so important nations learn each others colloquialisms in the Lead?

3DT. On WP:LEAD#Establish_context, the link recommends "supplying the set of circumstances or facts that surround it," the U.S./British use of math/maths in the Lead would do that but at the cost of overwriting noted in (1A). The current "(colloquially, maths or math)" lacks context. Examples that cite synonyms within a country are not parallel, nor are examples offered for the purpose of providing colorful alternatives. Why create a problem in the Lead that sect. 1 avoids?--Thomasmeeks 04:18, 18 October 2007 (UTC)

I'm not convinced:

The phrase is short and to the point, perhaps misleading - but misleading only in its brevity. It IS the lead, the full explanation follows for those who might get it wrong. I agree adding a fuller explanation in the lead would make it awkward (although I'd be happy to be proved wrong).
I still think it helps establish a navigation context for a user who was redirected.
In addition, including the phrase as is allows a reasonably smooth way of providing the two (very common) alternatives terms early so they may appear in bold, consistent with WP:LEAD#Bold title.
While I agree we can assume common knowledge, cross-Atlantic linguistic awareness cannot be assumed (especially my fellow Americans' awareness of Britishisms).
And finally, we must remember the target audience of this article is not just academics.

I think these items are enough to justify a slight awkwardness. (John User:Jwy talk) 07:22, 18 October 2007 (UTC)

John & others, I have relabelled my points above for ease of reference & Strikethrough where appropriate. Let me try to address the above points. Essentially all differences come down to predicted effects on the readers, negative and positive (including future reviewers of this article).

1ET. "(colloquially, maths or math)": Short, to the point, and misleading -- not only in its brevity but its placement -- in the lead. Additionally it is likely to be offputting to readers on each side of the ponds (Atlantic & Pacific), b/c it seems to patronize & b/c it informs without informing sufficiently to avoid the suggestion that math & maths are to be used equivalently in each country, when most readers will know that for their country the terms should not be so used. What they might well infer is sloppy editing (discussed in more detail at 1AT, 3AT, 4AT). That is offputting and should not happen in the lead, which should rather "invite a reading of the full article." Why read further if the 1st line seems amateurish (even if it is less so than it seems)?

2ET. Confusing & patronizing readers and overwriting come closer to insult than help, despite good intentions, esp. given the Redirect note that would appear at the top and the Disambiguation note there, both of which provide "navigation context" without introducing misleading context (on which see 3AT).

3ET. "Math(s)" as "smooth transition" in the lead? Not if the above 2 points apply. WP:LEAD#Bold title does say the bolded term "may include variations." "May" is not "must," esp. for colloquialisms otherwise absent from enclopedia articles on scholarly subjects. Their use in the article lead suggests an attempt to appeal by "dumbing down," which readers are unlikely to appreciate and could well misunderstand.

4ET. "Cross-Atlantic linguistic awareness" not only cannot be assumed. It should not be assumed. That is the argument for omitting "math(s)" in the lead as likely to cause confusion and misunderstanding, as discussed in detail at (3AT, 1BT), and deferring it to sect. 1 where it does not do all that bad stuff.

5ET. The audience for the article may include quite young people. World Book is written for them. It is noted for good writing and does not include "math(s)" in the corresponding article. More immediately, the lead should be written to entice and inform, not drive away by overwriting, misleading, or confusing in a way made unnecessary by coverage in section 1. --Thomasmeeks 16:29, 18 October 2007 (UTC)

Responding to Jwy: I certainly don't assume that everyone will know both "math" and "maths"; what I can't figure out is why we should care whether they know them. That isn't a mathematical issue, and this isn't a linguistics article. It would be different if there were two different full, formal names for the subject; then we would have to treat the issue up front, because we would have to choose one of them for the title of the article. But these are essentially just two different abbreviations; if we stick to the correct formal name it's the same in all dialects. --Trovatore 16:34, 18 October 2007 (UTC)

It's teapot tempest time in Wikipedia. Rick Norwood 17:22, 18 October 2007 (UTC)

Seemingly, yes, but if this page can take care of some "little" things by appealing to such norms as provided in Wiki policies & guidelines and appeals to good sense in their apps. for making cumulative progress on the article. On the other hand, if discussions get stalled in editors talking past each other, progress on the article may get stalled as well. I believe that the above very long discussion illustrates both possibilities. It also illustrates how hard it is, even with the best of intentions to make positions as clear and compelling as may be necessary to make progress & the necessity of close reasoning for bridging differences. Fortunately there are ways of settling disputed points, starting with this page. It should not be a matter of picking sides but working toward the common end of article improvement. If anyone (else) wants to add to the discussion, please consider doing so. There is always the WP:RfC option, but resolving the matter on this page seems preferable. --Thomasmeeks 17:31, 19 October 2007 (UTC)

I couldn't care less about "policies" in this connection. I'm not interested in wonking a micro-issue like this. I simply state my opinion that the parenthetical is ugly, and my reasons for believing that it's unnecessary. If people agree with me, we can get rid of it; if not, I'm not going to fight about it. --Trovatore 17:48, 19 October 2007 (UTC)

I agree - it's ugly and unnecessary. EdwardLockhart 11:18, 20 October 2007 (UTC)

It's harmless -- well, mostly harmless. The usage is ubiquitous. Do you say, "I am teaching a mathematics class." or "I'm teaching math."? Rick Norwood 13:34, 21 October 2007 (UTC)

There is no doubt that it's ubiquitous; that's not the point. It's not formal writing. We're not chatting with our friends here; we're writing encyclopedia articles, and a certain tone is expected. If we were to actually use the word "math" or "maths" in the article, it would detract from the appearance of seriousness.

Of course the current parenthetical remark isn't use, it's mention, so I don't object to it on those grounds. But I still think it's ugly and unnecessary. It isn't "harmless" because it's ugly; this is an aesthetic judgement on my part with which you may agree or disagree. It's unnecessary because it's not relevant to the subject matter. --Trovatore 18:45, 21 October 2007 (UTC)

In the UK, you would say "I'm a maths teacher", or "I teach maths". SpaceLem 17:52, 8 November 2007 (UTC)

Usage is always relevant. Darn, I swore I wasn't going to get drawn in to this discussion. "The guy who taught us math, who never took a bath, acquired a certain measure of renown..." Rick Norwood 21:19, 22 October 2007 (UTC)

Hmm? No, usage is not always relevant. This article is not about English usage; it's about mathematics. --Trovatore 21:21, 22 October 2007 (UTC)

One can agree that usage is always relevant in a dictionary, but see above (1A & following, after 17:14, 19 September 2007 Edit) that Wikipedia:What Wikipedia is not#Wikipedia is not a dictionary as to its relevance in the present discussion. Thomasmeeks 12:52, 25 October 2007 (UTC)

On the Wikipedia:Redirect page, under the heading What needs to be done on pages that are targets of redirects?, we find:

Normally, we try to make sure that all "inbound redirects" are mentioned in the first couple of paragraphs of the article.

Mentioning "math" and "maths" in the lead section is simply a way of complying with this rule. Is there a method of gauging the injury to some editors' esthetic sensibilities so as to ascertain that the amount of pain experienced outweighs the utility of doing here what we "normally" try to do? --Lambiam 16:29, 25 October 2007 (UTC)

Frankly I think that recommendation should be changed. There are a great many reasons for making a redirect that do not merit mentioning the term in the lead. --Trovatore 18:47, 25 October 2007 (UTC)

Still, an interesting question, even if unanswerable in any conclusive sense. But it is only also fair to note the "common sense" and "occasional exceptions" links at the top of that (and every) Guideline p. and the use of "normally" in the above quotation. Virtually every point above arguing that the phrase in question should go gives reasons as why guideline should not apply here. The paragraph immediately above the one quoted refers to Wikipedia:Guide to writing better articles#Principle of least astonishment, which arguably is violated in parochial pairing in "math" and "maths" (on which, see (1ET-4ET) above). I believe that one piece of evidence can suggest that the phrase "(colloquially, maths or math)" in the 1st sent. of the article is not useful. A Google search of "math" puts the article at #16 on hit list. Is inclusion of "math" in the 1st sent. of marginal help? Arguably not for 3 reasons: (a) virtually every searcher already knows the math-mathematics link, (b) it will be close to instantly figured by others, or (c), in the absence of (1) or (2), there is the Disambig for "math" above. Use of the term "colloguially" seems much more likely to confuse than the failure of (a-c). That leaves "maths." If its inclusion were useful, one might expect that Mathematics to be near the top as to Google hits for "maths" (like a bear finding honey). It is not only not near the top. I could not find it in the top 300 Google hits for "maths" & stopped searching beyond that. --Thomasmeeks 19:19, 25 October 2007 (UTC)

The purpose of Wikipedia is to present information about topics in a way useful to the audience, not just to follow rules and not just to "be formal." In my opinion, having math and maths mentioned early in the article - and in bold - is important to give the user context should they enter "math" or "maths." Their ubiquitous use also makes them appropriate content for the article. The aesthetics are no worse, in my opinion, than birth-death years or pronunciation guides. And information trumps aesthetics. Hopefully, we find both, but I'm not creative enough to think of something better than what is there. Those are my thoughts from the north side of the teacup. (John User:Jwy talk) 00:31, 26 October 2007 (UTC)

I just don't see the scenario as likely that a user enters "math" or "maths" into the search box, and is then surprised when an article called "Mathematics" pops up. --Trovatore 00:56, 26 October 2007 (UTC)

Many arguments above against including the questioned "math(s)" phrase above (such as (1ET-4ET) or the 19:19, 25 October 2007 Edit) are also arguments that the phrase is misleading and unnecessary, and therefore a distraction. On the contrary, I believe inclusion of the phrase itself introduces unnecessary context problems. --Thomasmeeks 22:10, 27 October 2007 (UTC)

The current Edit of Mathematics has this at the top:

This effectively says to any interested or (hypothetically) disoriented reader that the quoted terms are being used interchangeably & makes the phrase in the first line the article "(colloquially, maths or math)" even more redundant, which makes removal of the phrase that much easier. Full discussion is instead taken up in section 1. --Thomasmeeks 18:11, 28 October 2007 (UTC)

Beauty is in the eye of the beholder. Your objection to "math" and "maths" baffles me. More information is better than less, and the mention in the article is about 1/1000 as long as the discussion about it here. Rick Norwood 12:20, 29 October 2007 (UTC)

That's why I did not refer to "beauty," although it is relevant to refer to properties associated with beauty that can be expected to result in a more beautiful Edit. If the "more information" provided is unnecessary, distracts from the article content, & misleads (as my numbered points above have argued above repeatedly), that does not make it better. The question is not whether the information should be included at all but whether it should be deferred to section 1 where the objections to its misleading & distracting incompleteness are removed. --Thomasmeeks —Preceding comment was added at 12:56, 29 October 2007 (UTC) (Proofing edit Thomasmeeks 13:19, 29 October 2007 (UTC))

More information is certainly not always preferable to less. Rick, be serious; that remark is absurd on its face. Yes, beauty is subjective and I make no bones about my objection to the parenthetical being subjective. But subjectively, I really do think it's hideous and ought to go. I won't act unilaterally on that perception, but I will state it, so that if enough other people agree, then we can get rid of it. --Trovatore 17:10, 29 October 2007 (UTC)

Disambiguation text at top of article: Edit conflict

This is a subsection, since apparently one of the disputants thinks that it is related to the "Maths vs. Math" section immediately above. The following Disambiguation text at tht top of the article was restored:

Edit summary of orginal Edit; Disambig at top: "For other meanings of 'mathematics" or "math'," --> "For different uses of similar terms,. . ".: more concise, accurate ("uses" vs. "meanings"), less distracting

The revert-Edit that it replaced was:

Edit summary; Undid revision by ... Please stop this nonsense

The last comment seems to violate Wikipedia:Talk page guidelines#Behavior that is unacceptable, Wikipedia:Neutral point of view, and Wikipedia:Assume good faith guidelines. There is nothing in the Edit summary to support its assertion, and the the Edit summary of what was reverted was ignored without good reason. --Thomasmeeks 03:40, 30 October 2007 (UTC)

I'm not opposed to the disambig text being changed, but I don't think that "For different uses of similar terms" is very clear. Ben 04:33, 30 October 2007 (UTC)

On further thinking, is it possible to combine the the disambig text with the "maths and math" text problem discussed above? Is something like:

possible? Ben 04:38, 30 October 2007 (UTC)

I was indeed annoyed, after such earlier edits as these: [1], [2], and [3]. They introduce pointless complications; in particular, there is no point in disambiguating maths because it is not used with a secondary meaning in Wikipedia articles, so what are the other uses? Also, having more than one wikilink in a line on a dab page is in contravention of the guidelines. I see no plausible argument for replacing the standard wording for dablinks, used everywhere else, by an ad-hoc wording that is in my opinion rather unclear (what are "similar terms"?). In the BOLD–revert–discuss paradigm, if your bold edit gets reverted you don't reapply it, but discuss first. What particularly irked me was this comment made shortly after the edit creating the situation referred to, which suggests to me that there is an agenda behind the otherwise pointless edits, to wit to gain an advantage in the teapot debate. --Lambiam 11:00, 30 October 2007 (UTC)

0T. I do appreciate the gentle opening and close of Ben's Edit summary for the 2nd revert of my Edit and invitation for further discussion ("Revert for now. ... Can we wait and see what happens on the talk page?"). On the middle portion, see below.

1T. A general comment first. Things go better all around with a mutual, thorough, & unsparing application of the principle of charity. "Unsparing," b/c the principle can be used to peel away charitable (roughly, truth-maximizing) interpretations to reveal remaining points at issue. It includes a willingness to abandon (a) defective formulations (possible, yea, even in my case) when they are shown to be such and (b) interpretations of an Edit that fail to comprehend in what sense(s) the editor might have a valid point.

2T. The Edit of Lambiam ("L." for short) above begins by expressing annoyance resulting at 3 earlier Edits of someone else (me), 2 of them earlier or later reverted by 1 person (L.), and 2 of them on a disambiguation page, the same Edit but with a more complete Edit summary in the 2nd case. I did see a problem with the remaining Edit (pointed out by L.) & tried to avoid the repeating the mistake. But the earlier Edits do not exculpate multiple violations of WP guidelines (referred to at the top) in the "Edit summary" reverting a later Edit by that same person (me). Toward the end above, L. again expresses annoyance at an earlier Edit summary of mine, which names no one but refers to what I identified as a problem in the Edit it replaced. I regret that L. is also annoyed at that. Like L., I do have an agenda: to improve the article.

3T. Lambian asserts that "having more than one wikilink in a line on a [Disambig note] is in contravention" of WP guidelines, for which the link is Wikipedia:Disambiguation. If there were such a guideline, it would be absurd. (Why cavil at "1 line" but not 1 line + 1 word on another line?) There is no such guideline to my knowledge.

4T. Lambiam's Edit above has a link to Wikipedia:Disambiguation. But the link in no way supports a "standard" wording that does not concisely or precisely describe the Disambiguation.

5T. The greater conciseness of the 1st Disamb above is of course why "more concise" was used in the accompanying top "Edit summary" above.

6T. The imprecision of the 2nd Disambig above (referred in the 1st "Edit summary" at the top) is in its use of 'meaning' which suggests a "definition," the first dictionary-definition of 'meaning'. But the other terms on the Mathematics (disambiguation) are

Mathematics (producer), a hip-hop producer.
Mathematics (album), an album by the band The Servant.
Mathematics (song), a song by Mos Def
Mathematics Magazine, a publication of the Mathematical Association of America.

These are better described, not as "meanings," but as "uses" as multiple similar examples on Wikipedia:Disambiguation#Disambiguation links illustrate. Similarly for Math (disambiguation). That is why the assertion was made for the first "Edit summary" above that it was more precise.

7T. "Mathematics" and "math" are "similar terms," including spelling and a common dictionary reference for the 2 terms. Within each of the Disambig pages, there are also multiple similar terms or articles describing different uses.

8T. Ben's suggested alternative I believe comes closer to being a disambiguation itself, rather than referring as briefly as convenient to a Disambig page.

9T. I believe that the Edit summary at the top stands up well against all of the criticisms above raised against the corresponding Edit. --Thomasmeeks 21:28, 31 October 2007 (UTC) (2 minor typos fixed Thomasmeeks 19:33, 1 November 2007 (UTC))

I do not think that "maths" is a colloquialism. For example, in schools in Britain one will find the word "maths" used extensively on formal documents such as timetables, school reports et cetera. It is by far the most common term found in the media. Finally it is used in formal publications by the government, such as [4]. It is an abbreviation, but used in Britain as a slightly less formal synonym. I can not comment on the usage of "maths" in the rest of the world (it use is not confined to Britain) or of that of "math". Thehalfone 06:04, 6 November 2007 (UTC)

Probably "abbreviation" is a better word than "colloquialism" for the American use of "math", as well. I still don't see that it belongs in the lead. --Trovatore 08:08, 6 November 2007 (UTC)

I take it that the above 2 comments are related to Ben's suggestion above for this subsection referring to "colloquial" in the Disambig. It is both a colloquialism and an abbreviation (called a "shortened form" in my unabridged dictionary). Should either be mentioned? Arguably neither should, because it just adds unnecessary detail ("clutter") to the Disambig. --Thomasmeeks 14:24, 6 November 2007 (UTC) (minor typo fixed. Thomasmeeks 21:08, 6 November 2007 (UTC))

The above proposed Edit (1st indent at the top), has been reverted once apiece by 2 different editors, who suggested that it was "unclear." I have used plain words to express a plain meaning in the Edit. I have shown above the relevant sense in which the Edit is plain. One editor called the words "ad hoc." I have shown in what relevant senses the wording is standard, less misleading, and more informative than the alternative. I believe that these advantages trump the "ad hoc" label.

Concerning the above discussion, if one has nothing further to add, there is no reason merely to reiterate points already made. There may be of value, however, in attempting to defeat an argument against one's own argument, which would not be reiteration. Otherwise a non-response is open to the inference that no defense is possible, which is a questionble type of "consensus" solution. An expressed plurality opposing an Edit is especially vulnerable to challenge if the supporting argument is defective.

Comments are welcome concerning the proposed Edit, whether favorable or not to the position of this writer, --Thomasmeeks —Preceding comment was added at 18:08, 8 November 2007 (UTC)

For other uses, see Mathematics (disambiguation) and Math (disambiguation) would work fine for me. (John User:Jwy talk) 21:08, 8 November 2007 (UTC)

The "standard" way to handle this would be to use:

{{redirect|Math}}

{{otherusesof}}

resulting in:

"Math" redirects here. For other uses, see Math (disambiguation).

For other uses of "Mathematics", see Mathematics (disambiguation).

This, I feel, is perfectly clear. This could be combined on one line, for example in the form as suggested several times above, by John/Jwy, or – less concisely, but perhaps clearer – in the form:

"Math" redirects here. For other uses of "Math" or "Mathematics", see Math (disambiguation) and Mathematics (disambiguation).

An explanation of the alternative formulation involving the phrase "similar terms" is provided above. Unfortunately, we cannot add this explanation to the line or lines in the article. I still feel that (without accompanying explanation) this is less clear than basically any other formulation that has been considered. --Lambiam 21:37, 8 November 2007 (UTC)

John's proposed (3rd) alterative above has the advantage of brevity but the disadvamtage of possibly raising the distracting question of "Other uses of what?" The proposed Edit at the top avoids raising this question in clear enough terms.

Lambiam's 1st ("standard") alterative is not currently used for the article, presumably b/c what is there is better.

Lambiam's 2nd alternative above is an improvement over the current Edit by replacing "meanings" with "uses," as suggested at (6T) above. But it has the problem of telling those who search for "math" that they have beem redirected to "Mathematics," which they cam see for themselves and of which they are so informed by the little math#note right under the article title just for them. There is no reason to think that anyone else would care to be so informed. So, the first sentence ("Math" redirects here.) is doubly unnecessary, except to explain the second sentence. The proposed Edit does the same thing in one line and one sentence. (7T) above explicitly addresses the assertion that the phrase "similar terms" in unclear, the first sentence of which points out what should make assertion of unclarity unpersuasive. For that reason, I simply do not believe most people would find that phrase at all unclear in context. I realize that the following objection could be made: "If is it clear, why did you have to point to those things that make it clear?" My answer is, "because those things seem to have been overlooked in the assertion of unclarity." My prediction is that most people would regard the proposed Edit at the top as clear and favor it for its brevity. And, to the contrary, for those opposed. Additional comments are welcome, pro or con. --Thomasmeeks 02:26, 9 November 2007 (UTC)

From responses above, there seems to be a recognition of problems with the Current Disambig (CD) at the top of the article:

,

starting with use of the word "meanings," The phrase "For other meanings of ,,, math" misleadingly suggests that "Math" is the title of the ariicle. The question is what to do about it. The lack of concisensss is a distractng drawback of (CD). There would surely be wide agreement that a good reason for a Disbambig is to inform readers of where to find links distinguishing "different uses of similar" terms. The proposed Edit for example tells everyone who searched for "mathematics" or "math" where to look if they were suprised or disappointed to end up at the "Mathematics" article. So does (CD) but more verbosely and misleadingly.

A personal note: why spend all this anergy on such a small matter? I'm referring not merely to myself but everyone else who has read or contributed to this subsection and section that precedes it. Part of the answer may be that the subject (math) is considered important enough to make the lead as good as it can be. First impressions, for good or ill, can make a difference. That's worth discussion if it thare is a prospect for improving the article. It would be nice if there were a template that solved every problem beforehand. In the absence of that, reasoned discussion of relevant alterives might be second-best. What I have found offputting about (CD) is its awkwardness and length in trying to explain not one but 2 terms and Disambig links. It is not spam, but it may result in a similar reaction. Comments welcome. --Thomasmeeks 17:36, 10 November 2007 (UTC) (Minor typos fixed. Thomasmeeks 21:43, 10 November 2007 (UTC))

Request for comment on Disambiguation text for "Mathematics"

This section is a request for comment on the current Disambiguation text (labelled CD) below) at the top of Mathematics:

CD: For other meanings of "mathematics" or "math", see Mathematics (disambiguation) and Math (disambiguation).}}

Proposals have been made in the previous subsection (Talk:Mathematics#Disambiguation text at top of article: Edit conflict) with accompanying comments, pro and con. The proposals are labelled for convenience below.

D1: For different uses of similar terms, see Mathematics (disambiguation) and Math (disambiguation).

D2: Maths and math are colloquialisms of mathematics and redirect here. For other meanings of "mathematics" or "math", see Mathematics (disambiguation) and Math (disambiguation).

D3: For other uses, see Mathematics (disambiguation) and Math (disambiguation)

D4: "Math" redirects here. For other uses of "Math" or "Mathematics", see Math (disambiguation) and Mathematics (disambiguation).

D5: "Math" redirects here. For other uses, see Math (disambiguation). (together two lines)

For other uses of "Mathematics", see Mathematics (disambiguation).

Issues about the proposals relate to clarity, accuracy and conciseness. There seems to be an impasse in discussion of the previous section, which additional comments below might alleviate. Please indicate (and sign) which of the alternatives below is top-ranked in your judgment, with or without reasons. (Ties are permitted for equally top-ranked akternatives under "Other".) This "straw poll" will end in a week.

Top-ranked:

CD:

Minor Support: I have no real objections to this version, other than the note in my D4 support. Ben 00:01, 21 November 2007 (UTC)

D1:

Unsurprisingly perhaps, in light of discussion in the preceding section, I believe that this one is least objectionable. --Thomasmeeks 13:48, 14 November 2007 (UTC) (Minor typos fixed above, Thomasmeeks 15:36, 14 November 2007 (UTC))

Per comments under "Other" below, I accept that D2-D4 would be an improvement over CD. --Thomasmeeks (talk) 15:02, 17 November 2007 (UTC)

Oppose: This is just not clear. The idea here is to help lost or confused readers. If some lost or confused reader ends up here then I can not possibly fathom how this message would help them back on their way, other than providing the disambiguation links. Since every other disambiguation message has those links, I can't see a reason to support this. Ben 00:01, 21 November 2007 (UTC)

Relative to the "Top-ranked:" heading, the above comment is misplaced and possibly prejudicial. I believe that D1 speaks for itself to most readers. A full response was provided in the previous subsection. See also P.S. near bottom below. --Thomasmeeks (talk) 14:40, 21 November 2007 (UTC)

D2:

Neutral: I only floated the idea of moving the colloquialism stuff into the disambiguation as an alternative to removing it completely, per the discussion going on above this one. I have no strong feelings for it. Ben 00:01, 21 November 2007 (UTC)

D3:

Oppose: This is just too short, maybe even an ambiguous disambiguation. :) Ben 00:01, 21 November 2007 (UTC)

D4:

Support: I prefer this over CD since it's apparently convention to note a redirect in the disambiguation. A small caveat, I think that and should be changed to or (or vice versa), and I think uses should be changed to meanings as in CD. Ben 00:01, 21 November 2007 (UTC)

D5:

Minor support: I believe this is how the Wikipedia guidelines specify this should be done (though I could be confused!), so I wouldn't oppose it - but spreading this over two lines? Ugh. Ben 00:01, 21 November 2007 (UTC)

Other (please indicate)):

I like CD just fine, and find this whole fuss a waste of time. Rick Norwood 16:45, 13 November 2007 (UTC)

D1 is confusing, and D2 is an unnecessary and avoidable complication, compared to CD. The others are fine. --Lambiam 21:15, 14 November 2007 (UTC)

This discussion is a waste of time. The only improvement would be a technical solution assuring that only the relevant disambiguation page is offered for "Mathematics" or "Math", and none is offered for "Maths". For instance, whenever an article "X (Disambiguation)" exists, the server could add the disambiguation link automatically to the article "X". This could be decided before doing a redirect. This technical solution may not be worth implementing. In this case/in the meantime all proposed solutions are fine. --Hans Adler 21:25, 14 November 2007 (UTC)

Responding to RfC. Keep it simple since readers are only interested in locating the disamb. link , so: D1.Labongo 15:10, 16 November 2007 (UTC)

D1 is the simplest except it ought to say math(s) to cover both terms as 'math' is not used in many countries. Fainites ^barley 19:36, 16 November 2007 (UTC)

- The point is not that some countries use math, some maths. The point of a disambiguation page is that math has other meanings and maths apparently has only one meaning. The is no page maths (disambiguation).

Rick Norwood (talk) 16:06, 20 November 2007 (UTC)

As to the last sentence of the introduction for this subsection (above the "Top-ranked:" line), a week has elapsed for this "straw poll with comments" in an effort to break the apparent impasse of the preceding subsection at Talk:Mathematics#Disambiguation text at top of article: Edit conflict and locate a possible consensus as to proposed Disambigs. The last Edit by a "new discussant" (for this subsection) was 4 days ago. A ranking of the current and proposed Disambig texts from highest to lowest consensus (as nearly as I can determine is from the above 6 "votes" cast) is:

~~D1 preferred by 3 & accepted by 1 as good as other Ds~~

~~D3: accepted by 1 as good as other Ds & by 1 as good as D2-D5~~

~~D4: accepted by 1 as good as other Ds & by 1 as good as D2-D5~~

~~D5: accepted by 1 as good as other Ds & by 1 as good as D2-D5~~

~~D2: accepted by 1 as good as other Ds~~

~~CD: preferred by 1.~~

Currently D1 has the widest acceptance with a "majority" of "votes" in a large field of proposed alternatives. Moreover, "voters" have had a chance to inspect detailed discussion of the preceding subsection. Prudence might be advisable at this point.

I propose that if the above top-ranking of D1 is maintained for another week, it be used as a substitute for CD in the article. In the meanwhile, additional comments and/or "votes" on (de)merits of the alternatives are welcome. (Editors new to this subsection might wish to consult the preceding subsection for more detailed discussion.)

It is appropriate to recognize that not everyone might agree with D1 to replace CD (not to mention that the result could overturned). But there might still be acceptance of the process in this section for resolving the impasse of the preceding subsection. If there are no further comments in that time, I'd further propose that this section be deleted as suggested in Wikipedia:Requests for comment#Example use of RFCxxx Template (to be reposted if anyone sees fit now or later). --Thomasmeeks (talk) 17:56, 20 November 2007 (UTC) (See discussion below.)

I don't quite recognize the votes in the above tally. Maybe I have a different interpretation of "as good as". You don't seem to take account of objections, in which you might include the editor who wrote in an edit summary: "The new version doesn't make sense".[5] I must say it is slightly curious that the two outside commenters prefer D1 because it "is the simplest", while clearly D3 is simpler. I don't know what this says about how serious we should take their opinions. ~~Also, these two users appear to be both editing pages as disparate as Lavvu and Talk:OPV AIDS hypothesis, and somehow manage never to be editing at the same time; perhaps they share their computer.~~ --Lambiam 20:15, 20 November 2007 (UTC)

Thank you, Lambiam. Let me attempt to respond. This subsection is a request for comment to locate where the largest consensus was. Anyone was free to comment or not in this subsection, including those from who commented in the previous subsection. I counted above only those who chose to respond in this subsection. I do not think that all earlier comments of someone necessarily carry over to later discussion, much less a following subsection. (That user of course is free to determine whether or not to comment here.) The discussion of the previous subsection was extended. Nothing in the above tally was assumed either way as to how a user who responded in an earlier subsection would respond if that user did not in fact respond in this subsection. It appeared to me that Rick Norwood preferred CD, that Hans Adler ranked D1-D4 equally over CD, and that Lambiam ranked D3-D5 over CD & D1. Lambiam, I believe, raises interesting prima facie points about the 2 "votes" for D1, in light of which I'd say, "Let the voting/comments continue above or below and I'll strike out the provvisional result above. I hope that there would be continued effort to find s consensus. An attempt to improve the article by transparently fair means remains the only objective here. (talk) 23:11, 20 November 2007 (UTC)

P.S. I must say, I continue to be astonished at the reassertions that those entering "Math" would find D1 about "Math" and "Mathematics" unclear (& presumably without motivation to go to the respective Disambiguation pages), with or without the extended discussion of the previous section. --Thomasmeeks (talk) 00:50, 21 November 2007 (UTC)

P.P.S. There is this to be said for D1 being "simple" (but not too much so). It is more concise than any alternative except D3. D3 is shorter but question-begging: "other uses" in D3 wrongly suggests that the term "math" that follows has already been used in the article, so it is not simplest in the relevant sense of avoiding unnecessary quetions. Hence, there is some validity in both comments of the "outsiders" as to simplicity of D1. (Incidentallyy if we are going to start not counting "votes" that we think don't quite compute, I don't think that that should stop with the "outsiders.") --Thomasmeeks (talk) 21:56, 29 November 2007 (UTC)

I understand why math redirects here, and I still find D1 unclear. I imagine a confused reader would be worse off. What are these similar terms you mention? Why would I care about other uses of them? Ben 07:58, 21 November 2007 (UTC)

So, "math" redirecting to Mathematics is clear but D1 is unclear? Again, I'm astonished at such an assertion. The "similar terms" are "mathematics" & "math," mentioned in an obvious context in D1 (& commented on in (7T) of the previous subsection). Again, I believe that few readers would be puzzled, despite contrary expressions above. I do agree that anyone who accepts the above argument would likely instead "vote for" some other alternative. --Thomasmeeks (talk) 14:40, 21 November 2007 (UTC)

Additional comments and/or "votes" for at least 1 of CD and D1-D5 above as indicated or here:

Example of complex number (Fields of Mathematics: Quantity)

The last example of a complex number given in this section is 2*e^(i(4*pi/3)). But doesn't this evaluate to 2 [e^(i*pi)=-1 so 2*e^(i(4*pi/3))=2*(e^(i*pi))^(4/3)=2*(-1)^(4/3)=2*1=2]? I'm not sure if the expression itself is still considered complex or what or whether or not this is worth changing. -- 86.134.205.5 22:17, 27 October 2007 (UTC)

In the complex domain you cannot use e^ab = (e^a)^b without restrictions, otherwise we would get this:

e^x = (e^2πi)^x/(2πi) = 1^x/(2πi) = 1.

To interpret something of the form e^ix, just use Euler's formula. --Lambiam 22:34, 27 October 2007 (UTC)

Oh yeah. Thanks for the insight. --86.134.205.5 22:42, 27 October 2007 (UTC)

I dislike that formula being used as an example of a complex number. It is complex, but it's not in the spirit of the other examples. You need to use Euler's formula to expand it, and the leading 2 is extraneous. None of other examples invite manipulation, nor do they have "extras" - there is no 3π in the real numbers category, for example. Aetherealize (talk) 09:51, 20 February 2008 (UTC)

The example on the main page has changed to include an i, seems okay by me. I also suggest we retain the 2 as it makes it clear (as we see below) reals, integers and natural numbers are a subset. (John User:Jwy talk) 19:23, 20 February 2008 (UTC)

The example is the same now as it was at 15:56, September 2, 2006. There has been no recent change. See Talk:Mathematics/Archive 10#Suggested Improvement for the discussion leading up to the present version. --Lambiam 20:08, 21 February 2008 (UTC)

As it currently stands, it looks as though it is giving 2 as an example of a complex number, i'm almost certain that it isn't a complex number. Cmdr Clarke 23:18, 11 November 2007 (UTC)

You're welcome to check out the complex numbers article for the details, but briefly, complex numbers are of the form a + bi, where a and b are real numbers, and i is a number that squares to give -1. There is nothing wrong with, or stopping us from choosing, the real numbers a = 2 and b = 0. So we can see that the real numbers are in fact a proper subset of the complex numbers. Hope that helps. Ben 23:49, 11 November 2007 (UTC)

Math isnt all correct but it is supposed to be all correct

Most math problems are corect but there are some math equasions that are incorrect.Math is suposevly all are corect but this is not true (as it states in the last sentence) —The preceding unsigned comment was added by 24.217.141.75 (talk • contribs) 03:31, November 14, 2007 – Please sign your posts!

It would help us understand your point if you give an example. Rick Norwood 13:31, 14 November 2007 (UTC)

If we're going to do this, let's move the new vote to the bottom of the page.

C1 is "unclear" because of the word "similar". Disambiguation pages deal with identical words, not similar words. What are some of these words "similar" to mathematics and math?

While there are clearly too few votes to be meaningful, I continue to support the status quo.

Rick Norwood (talk) 13:46, 21 November 2007 (UTC)

On the 1st point, see (7T) at Talk:Mathematics#Disambiguation text at top of article: Edit conflict. Disambiguation pages deal with identical terms and in some cases similar terms, e.g. Mathematics (disambiguation) and Math (disambiguation). So can disambiguation texts. On the title of this seciton, undiscussed in the text that follows, let's let the RfC remain as long as there is active comment, or objection, or the 30 days indicated at as indicated in Wikipedia:Requests for comment#Example use of RFCxxx Template. --Thomasmeeks (talk) 15:03, 21 November 2007 (UTC)

We can be quite certain that the user who ends up on this page looking for "similar" terms is in fact not looking for terms that are just similar, but for other meanings of one of the terms mathematics or math themselves. Now, although in mathematical jargon identical triangles are similar, in everyday use of English the terms identical and similar are mutually exclusive: if it is identical, it is not similar. Just search for "not similar" in any of [6], [7], [8], [9], and [10], to give but a few examples. This makes the use of "similar" here unnecessarily confusing. --Lambiam 18:12, 21 November 2007 (UTC)

This has become an absurd waste of time for everyone concerned. Rick Norwood (talk) 13:57, 22 November 2007 (UTC)

I don't see a compelling reason to change anything. Anyone encountering this message for the first time that is remotely curious will click one of the links to see what possible other meanings/uses/similarities there could be. They will then say "oh" and go about their business. Unfortunately, we end up advertising a bunch of marginal articles, but it can't be helped. --agr (talk) 16:00, 22 November 2007 (UTC)

Well I do believe that it can be helped -- by making the Disambig as short as reasonnably possible, as discussed above at Talk:Mathematics#Request for comment on Disambiguation text for "Mathematics". --Thomasmeeks (talk) 17:11, 23 November 2007 (UTC)

What is mathematics a branch of?

Is there a larger category to which mathematics belongs?

Is it a branch of science?

Is it a branch of philosophy?

The Transhumanist (talk) 19:20, 25 November 2007 (UTC)

Views vary. It is a formal science, but whether or not formal sciences are science is disputed. —Ruud 20:01, 25 November 2007 (UTC)

Assuming that by "science" you mean natural science then the answers are clearly "no" and "no". There are branches of mathematics that overlap with branches of natural science, and there are other branches that overlap with philosophy, but there are also parts of mathematics that have no connection with either natural science or philosophy. Therefore mathematics is not a branch of either natural science or of philosophy. Gandalf61 (talk) 20:18, 25 November 2007 (UTC)

Mathematics is a part of collated human knowledge and is very a important tool when used to calculate third party effect within other branch's of human knowledge. My personal opinion is

mathematic along with language ( as a means of communication both spoken and written) is part of the root and trunk of the tree of human knowledge from which the branches grow.86.146.123.97 (talk) 19:13, 29 November 2007 (UTC)

Singular noun?

The article currently describes "mathematics" as a singular noun. But surely it is uncountable isn't it? (Therefore neither singular nor plural) - ^Estoy_Aquí^{(t • c • e)} 17:20, 9 December 2007 (UTC)

It's a "mass noun construed as singular". By contrast "people" is a mass noun construed as plural. I think the text is there to warn non-native-English speakers not to use it with plural verb forms (which would not seem unreasonable a priori; I think etymologically it is plural). --Trovatore (talk) 17:54, 10 December 2007 (UTC)

I think it is more accurate to classify "people" as a count noun that, despite its singular form, acts as a plural of the word "person". --Lambiam 19:49, 10 December 2007 (UTC)

Well, you can make that argument. To my mind the only true plural of "person" is "persons"; "people" is often used in that role but is actually a separate word. But anyway, if you don't like the example, substitute "cattle". --Trovatore (talk) 19:59, 10 December 2007 (UTC)

But I can't quite let the "people" thing go, because someone whose first language is a neo-Latin language might be reading this, and they tend to get confused on this point. So: "people" used to mean something like "nation" is indeed a count noun, but "people" in the sense of "persons" is a mass noun construed as plural, like "cattle". --Trovatore (talk) 20:05, 10 December 2007 (UTC)

You can't say: "A Wagogo family had on the average three cattle"; you have to use something like "three heads of cattle". You can't count cattle but only their heads. But you can say, at least in English as I know it: "Three people are vying for the position": you can count people, that is, "people" is noun that can be modified by a numeral – except the numeral "one". If two of these contenders drop out of the race, you can only say that "one person" is left. So "person" functions as the singular of "people" here. While you can say: "Three persons are vying for the position", that is not truly idiomatic English. See also Wikipedia:Reference desk archive/Language/2006 October 2#People v. Persons and Wikipedia:Reference desk/Archives/Language/2007 October 7#People, Persons. --Lambiam 21:44, 10 December 2007 (UTC)

The only true plural of "person" is "persons". "People" has extra baggage. It's true that it's not completely parallel with "cattle" because of the "*three cattle" problem, though the idiomatic expression is "three head of cattle", not "three heads". --Trovatore (talk) 22:57, 10 December 2007 (UTC)

Yes I'm aware it is treated as singular, but the point I was making is that it isn't (because it is uncountable). But anyway, I see that it's gone now. - ^Estoy_Aquí^{(t • c • e)} 09:54, 15 December 2007 (UTC)

Fields of mathematics

What is the goal of the "Fields of mathematics" section? To give the reader (another) overview of the main themes of math, or to introduce her to the research subdivisions? I suggest we try for the latter, because it is not done anywhere else in the article and it would dovetail nicely with the later discussion of the misconception that no new math is being done. The goals of the section (which could be renamed "Mathematics research" or so) could be:

subsections corresponding to the major subfields, with the adult terminology (e.g. Algebra, Geometry, Analysis, Logic, etc.), as used in the American Mathematical Society's Mathematics Subject Classification, or its annual survey of new Ph.D.s granted, or some other math organization's system
gentle tie-in to how these build off and connect the historical themes of math
less technical jargon (fiber bundle comes half a paragraph after Pythagorean theorem)
major subfields identified in a sane way (group theory is a subfield of abstract algebra, not a peer of it)
major outstanding problems clearly identified.

Joshua R. Davis (talk) 21:40, 12 December 2007 (UTC)

Math as Science Section

I do not mean to bring up this dispute again, but "those in pure mathematics often feel that they are working in an area more akin to logic and that they are, hence, fundamentally philosophers.", seems odd to me. Pure mathematics is not fundamentally Philosophy. No Philosophy department offers courses in pure mathematics, nor do any math text books mention philosophy. The philosophy of mathematics on the other hand is philosophy, but it is not pure mathematics. Please realize that things like Homological Algebra are not applied in any sense, they are not immediate abstractions of anything physical, they are not philosophical, and they are not considered primarily for aesthetics. Hence, such branches of mathematics can only be called pure mathematics; and it is absurd to suppose that anyone working in such fields would think otherwise.Phoenix1177 (talk) 11:08, 1 January 2008 (UTC)

As no reply has been made to the above, I am going to remove the offending sentence. If anyone objects to this please add the sentence back and we can discuss here.Phoenix1177 (talk) 04:18, 2 January 2008 (UTC)

http://www.st-andrews.ac.uk/~slr/teach07-8.html#3 appears to be a course in pure Mathematics offered by a Philosophy department. EdwardLockhart (talk) 13:16, 2 January 2008 (UTC)

That is a course in Logic, which can be mathematical. However, all of pure mathematics is not a course in logic; especially, courses mentioning modal logic. Furthermore, branches of philosophy interact with one another, the same holds for mathematics. I can see no way in which homological algebra is relevant to epistemology; I can see how it is relevant to topology and sheafs and so on. Homological algebra is not the only theory for which this holds; I can think of no theorem in group theory, differential geometry, complex analysis, etc that has any bearing on the JTB theory of knowledge or on any other such matter. In fact, none of the above branches even has a bearing on the veracity of modal realism, which is involved with modal logic. Lastlty, philosophers attempt to generate theories correspondant to reality, thus why they are debatable; the point of epistemology is to characterize actual knowledge, not to formally define an interesting concept and call it knowledge. Mathematics on the other hand does not have a serious debate over which axioms to employ; at least not since the modern times and non-euclidean geometry. Mathematics does not care if the axioms used correspond to reality, only if the results are useful/interesting; philosophers do care if their axioms correspond, indeed, it is their goal.Phoenix1177 (talk) 06:50, 3 January 2008 (UTC)

In (Vaughan Pratt (1995). "Rational mechanics and natural mathematics". In TAPSOFT’95, LNCS 915, pp. 108–122) the author proposes Chu spaces as a solution to Descartes’ problem of mind-body interaction. The idea is that the points of a Chu space are interpreted as "body", whereas its dual points are "mind". The matrix of the Chu space then serves as the mechanism of their interaction. I'm not making this up. --Lambiam 11:15, 3 January 2008 (UTC)

I never heard of him before, I'll have to look at his work sometime. Nonetheless, this example only serves my point; he is applying an interpertation to mathematical results in order to do philosophy. This is, although not so seemingly strangely, what science does to math aswell; it adds an interpertation to it. My point is this, to make mathematics philosophically useful, you must add an interpertation. We do not have to add an interpertation to results in one area of math to be able to apply it to another. For clarity: the above bit about Chu spaces is an interpertation, we could not simply say something like, "...in every hausdorff space compact sets are closed, it follows that the Gettier problems are resolvable by adding an additional contstraint on knowledge as JTB rather than taking knowledge to be fundamental...". Of course, when it comes to things like the foundations of mathematics, things can get a little blurred; but this is a far cry from saying that pure mathematics is fundamentally philosophy; any more so than saying that a chef is fundamentally a chemist.Phoenix1177 (talk) 11:42, 3 January 2008 (UTC)

But note that Pratt proposed Chu spaces, not as a model of the problem, but as a solution. My pineal gland is still reverberating from the shock. All I can say is: Monsieur, (a+bⁿ)/n = x, donc Dieu existe – répondez! --Lambiam 13:51, 3 January 2008 (UTC)

Whatever Pratt did, does not change the issue at hand. My point was not that someone would never philosophize mathematically, but that pure mathematics is not fundamentally philosophy. It is no more true that theoretical physics is fundamentally mathematics, that writing fictional novels is fundamentally linguistics(your citing Pratt, would be like me citing Tolkien here), that philosophy is fundamentally writing fiction(although Ayn Rany used fiction), that writing novels is fundamentally mathematics(Von Neumann did this once), or any other concoction. If we submit that pure math is philosophy, we might aswell just say X is fundamentally Y, irregardless of X and Y because someone will be able to find some instance of it or a case where the lines blur.Phoenix1177 (talk) 02:13, 4 January 2008 (UTC)

By the way, I love the quote from Euler above; and I did not look at Pratt's work, so I was speculating; I apologize for that bit, but my argument still stands.Phoenix1177 (talk) 02:13, 4 January 2008 (UTC)

(reset) The contentious issue is whether contemporary mathematicians consider math a science, right? Can we find reliable sources that make definitive statements one way or the other? (The paragraph as it stands is loaded with weasel words.) Joshua R. Davis (talk) 04:01, 4 January 2008 (UTC)

Actually, the issue was if pure mathematics was fundamentally philosophy; which it said in the article before removal. However, the mathematics as science view is something I also object to. However, I did not bring this up, as it appears that the current section is a comprimise based on past arguments. I will bring up my objections now, since you have brought the matter up.

The first paragraph of the section states that mathematics, pure, is not a science in the empirical sense. Which is good, in fact this is a common misconception and I'm happy that it is cleared up in the article. However, the sentence, " If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science." is unusual in that Wikipedia's science article does consider science to be about the physical world, although, the last paragraph of its article seems to contradict this view nonetheless.

Also, "...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." has little to do with mathematics being science. Observe, that it talks about mathematics connection to science, which seems to implicitly deny mathematics actually being science; I would not talk about quantum mechanics connections' to science, but of quantum mechanics as science and its connections to other parts of science. This is not all, the sentence is actually talking about mathematicians thinking that the connection between math and science should not be downplayed; the article presents this asif mathematicians were saying that math and science being the same thing should not be downplayed, which is misleading as the sentence only implies that an important connection between them exists. There are important connections for between food and parties, and these have inspired many types of foods; still I would not say that this is an argument for a chef being a host.

The next sentence seems to imply that mathematics is either science or art; or some synthesis of the two. It is none of these. Art's primary aim is aesthetics, mathematics is not. Science aims for results about the natural world, math does not. Both may inspire mathematics, and mathematics be viewed in relation to both; then again, so can cooking.

Lastly, the paragraph on awards does not belong in the section. The first sentence talks about equivalent awards in science, but then says that the Fields medal is misleadingly considered to be the Nobel prize of mathematics. While these two sentences are not contradictory, they do seem to clash. Furthermore, awards deserve their own section, and we could equally talk about the equivalent awards in acting or literature; thus, showing that science and math have awards does not relate to their relation as fields. This paragraph should be moved and the mention of science in it should be taken out.

If no one objects, or if some agree, then I will change some of these things in the next few days; I do not have time right now.Phoenix1177 (talk) 04:48, 4 January 2008 (UTC)

Let me rephrase. All of the talk here, while insightful and interesting, doesn't make an encyclopedia. Opinions should come from reliable sources. Can we find published, citable quotes by Atiyah, Serre, Smale, or others? The current article sorely lacks these, so I don't object to your removing some claims, but ideally we would be disciplined in adding new claims. Joshua R. Davis (talk) 14:11, 4 January 2008 (UTC)

Adding my two pennies: much of the on one hand... on the other hand... of this section is not about the epistemological position of mathematics per se – although confusingly presented as such – but rather about the meaning of the term science. People who understand the term science as a notion comprising all systematic accumulation of knowledge are more likely to consider mathematics a science than those who understand the concept of science as being the systematic effort to understand how nature works, with observable physical evidence as the basis of that understanding, acquired through observation of phenomena, and/or through experimentation that tries to simulate events under controlled conditions. --Lambiam 15:05, 4 January 2008 (UTC)

Nevertheless mathematics answers to the latter description just fine, provided that you don't require the objects being studied to themselves be physical. --Trovatore (talk) 17:21, 4 January 2008 (UTC)

That just underscores Lambiam's well put point: in order to decide if mathematics is a science or not, you need to define science. A definition that includes "observable physical evidence as the basis of that understanding" would seem to me to indicate the "objects" have to be "physical." If we do include mathematics as a science, it has to have an "asterisk" that explains it is "detached" from the physical world in a way distinct from other sciences. In most cases, it is not an "experimental" science, one might say. (John User:Jwy talk) 22:25, 4 January 2008 (UTC)

The evidence can be physical, without the objects being physical. And I think it is an experimental science, at bottom, though perhaps many sorts of mathematicians will often not experience it that way in practice. --Trovatore (talk) 22:35, 4 January 2008 (UTC)

WARNING: I enjoy this discussion such that I will probably get far away from the topic at hand. I think there have been books written pro and con what you say in the first sentence. And I don't think modern mathematics has much experiment involved. How do you prove with a physical experiment some hypothesis about non-Euclidean geometry? But the main point is there is something very different about mathematics and the other things we call science (whether we call math a science or not) and it has to do with its looser connection with physical reality. I don't believe I am capable of expressing it in a way appropriate for the article. . . (John User:Jwy talk) 23:54, 4 January 2008 (UTC)

Non-Euclidean geometry is not a particularly good example. Some of the clearest examples are actually large cardinal axioms, though they're a little bit complicated for quick discussion. We could take a mini analogue of LCAs -- the existence of the powerset of the naturals. Does there exist, Platonistically, a completed totality containing all sets of natural numbers?

Well, if such a thing does exist, then we can predict that lots of theories are consistent, because they have P(ω) as a model. And in turn the consistency of a theory has consequences we can observe in the physical world, such as certain computer programs not halting. To be sure, the same consequences follow from ontologically more parsimonious hypotheses -- but these hypotheses, though they assume the existence of fewer entia, are nevertheless more complicated and less motivated. Without the existence hypothesis, the latter ones appear to be "brute facts" without explanation.

Of course this discussion quickly gets very involved and very controversial; just scratching the surface here. --Trovatore (talk) 00:28, 5 January 2008 (UTC)

While interesting, what you said doesn't support the claim of mathematics being an empirical science with nonphysical objects. Most of this argument seems to be: let me show the case where it is most favorable, followed by sweeping hypothesis. If mathematics is indeed as you say, then please give me an example of an experiment that will provide physical evidence for some result from algebraic topology; how about Van Kampen's theorem.

I agree that we need more sourced material, but I didn't come here to provide it; only to remove a line about pure math being philosophy. Lastly, I don't thik the average person uses science to mean the systematic accumulation of knowledge, but uses it to refer to natural science. Although, the mentioning of this difference in usage and saying that math belongs to the first type is not a bad thing; mainly because some people tend to think mathematics is a natural science(I know many people who think that mathematics is either physics or accounting, sadly).Phoenix1177 (talk) 05:54, 5 January 2008 (UTC)

What I said does indeed support my claim. I don't claim that it has to convince you; it's an explanation of how I and some others see it. Some philosophers of mathematics who have made similar arguments (though I don't know if they'd sign on to this specific one) are Quine, Putnam, and Lakatos.

Not all of mathematics has to be empirical. Not all of physical science is empirical either; much of it is the elaboration of consequences of that which we believe to be the underlying reality, but which we can never observe phenomenally. I don't know much about algebraic topology so I can't help you there -- I'm a set theorist and naturally my examples are going to come from set theory.

As for the text you removed, frankly it was not that bad and something along those lines perhaps should be restored, though admittedly it should be sourced. It didn't say that mathematics is a philosophy, just that many pure mathematicians consider what they are doing to be part of philosophy. And it's absolutely true that many of them so consider it, or at least consider it to be on the borderline. Whether they're right or not is not the point, from the standpoint of this article. But I think they are right, and I don't think there's any contradiction between that and saying that they're doing empirical science. There are no exact demarcations between any of these things; it is quite possible for an activity to be simultaneously philosophy and science. --Trovatore (talk) 07:22, 5 January 2008 (UTC)

I removed one sentence. The sentence said that pure mathematics is considered to be fundamentally philosophy. I have never heard of anyone calling topology a branch of philosophy. I do believe that set theory and philosophy might blur, but people working on the Riemann Hypothesis would most likely not call themselves philosophers, nor consider themselves as such; or do you not include analytic number theory and complex analysis in pure mathematics?

Also, you claimed that mathematics can be viewed as an empircal science. I did not ask you to prove it to me, I do expect you to back up the claim if you make it, however. There are thousands of experiments performed in science, I know of no experiment ever done that provided physical evidence for a theorem of high mathematics.

I do not recall saying that there is a contradiction between mathematics being philosophy and science at the sametime, I said that mathematics is neither of these things. In fact, I don't see why there is such a need to classify mathematics as being some other field; mathematics is its own. It would make more sense to point out the several connections between mathematics and philosophy, science, art, etc. and let the reader draw their own conclusion.Phoenix1177 (talk) 08:06, 5 January 2008 (UTC)

I did back it up. I gave specific examples of the sorts of experiments that provide physical evidence for axioms of mathematics (axioms of course are also theorems, but what I'm interested in finding evidence for is more the axioms, though in some cases they might not seem so axiomatic in character). You are not required to be convinced, but don't claim I didn't back it up.

Whether you (or I) think that mathematics is or is not philosophy, or is or is not science, is quite beside the point here, though. You think it's neither, I think it's both, but neither of those things matters. What matters is presenting the views of informed currents of opinion.

Oh, and by the way, no, the sentence you removed did not say that pure mathematics is considered to be fundamentally philosophy. If that's how you read it then you read it wrong. --Trovatore (talk) 08:31, 5 January 2008 (UTC)

I agree with Trovatore. The sentence said made the point that pure mathematicians (it really should have been some pure mathematicians) consider themselves "fundamentally philosophers" because their work is more in line with logic than sciences. This is quite different to saying they think pure maths is a branch of philosophy, but at any rate is a claim about how mathematics is viewed, not what it actually is. JPD (talk) 12:41, 5 January 2008 (UTC)

What is mathematics?

JPD wrote: "... is a claim about how mathematics is viewed, not what it actually is."

Since mathematics is an abstraction, not a physical object, what is "is" and "how it is viewed" have the same meaning. Abstractions exist only to the extent there is a consensus about them.

The problem with mathematics seems to be that non-mathematicians almost always get it wrong, so mathematics "is" one thing to non-mathematicians (usually something like "really hard arithmetic") and "is" something else to mathematicians (often something like "axioms, definitions, theorems, and proofs"). Mathematics differs from philosophy because mathematical reasoning has led to a large, useful, and generally accepted body of knowledge, while philosophical reasoning has led to many "schools" of philosophy but little in the way of philosophical "truths" that all philosophers agree on. Until the twin primes conjecture is accepted on the basis of a very large number of experiments, mathematics is not a science.

Because Wikipedia relies on published sources, this article has to go with the best definition non-mathematicians have come up with, which is usually something like "the science of number and shape". But we also mention the view of pure mathematicians. By its very nature, this disjunction will not be resolved, unless we can convince everyone in the world to become a mathematician, which would mean the end of civilization as we know it. Rick Norwood (talk) 14:31, 5 January 2008 (UTC)

I thought we were talking about the section that discusses views of mathematics as science, in particular the paragraph about the views of mathematicians on the subject. The issue isn't a definition of mathematics, but how to convey the different views, even among mathematicians. In this context, it doesn't matter what what "is" makes any sense or whether we agree with the view being described - the question is simply whether we are accurately describing a common view. JPD (talk) 15:04, 5 January 2008 (UTC)

If I ask someone what they are and they say they are fundamentally a philosopher; I would assume that they do philosophy. If you tell me most pure mathematicians feel they are fundamentally philosophers, then I assume that they feel that pure mathematics is philosophy. I suppose you could also take it to mean that their methods are fundamentally philsophical; but I don't see how this applies to mathematics in general; my point is that if I misread it, I'm sure others did. Regardless, I'm sure some mathematicians view themselves as fundamentally mathematicians, and my whole point of contention was that the article does not mention this; I read it as portraying mathematics as being some cross between philosphy, science, and art.

Also, to Trovatore: You gave one example. One example does not establish that , "...And I think it is an experimental science". I can point to a large number of physical experiments in physics, eventhough all of physics may not be experimental. You even say non-euclidean geometry is not a good example, why is it not a good example? Suppose I just make up some axioms that are rules governing symbols, arbitrarily; it may not be interesting, but I'm sure I could make it so that it wasn't experimentaly supportable in a physical way. Perhaps, mathematics evolved the way it did because we think the way we do; being such it seems to correspond to the physical world. We could generate equally complex theories from a perspective that could not even correspond via analogy, we wouldn't because they would it would be of no interest, but we could do so vallidly.

Also, I mentioned earlier about moving the awards from that section. We could equally say that the awards in science have an equivalent in acting. All awards are given based upon merit, the paragraph has nothing to do with mathematics being science; only that mathematics has awards. As such, it doesn't belong.Phoenix1177 (talk) 20:02, 5 January 2008 (UTC)

The point of the awards paragraph is to give more evidence that math is regarded as something outside science (in that the sciences have Nobel prizes and math doesn't). This is the intent of the opening sentence. I do agree, however, that the second half of the paragraph, beginning with Hilbert's problems, is irrelevant. Joshua R. Davis (talk) 22:04, 5 January 2008 (UTC)

History of Mathematics

Earliest evidence on Mathematics is found in Africa (South Africa, Congo). But there is no mention of these countries. I don't think it is fair. —Preceding unsigned comment added by Observer8 (talk • contribs) 16:15, 8 January 2008 (UTC)

Well, this is Wikipedia, the free encyclopedia that anyone can edit. That's not an invitation to add just anything, of course, but if you have reliable sources to which you can attribute your claim, I think it would certainly be worth a mention. Do be a little careful if your sources are in some way polemical or see themselves as trailblazing -- if the claims are not accepted by the history-of-math or anthropological communities at large, they can still be mentioned, but should not be presented as accepted fact. --Trovatore (talk) 19:08, 8 January 2008 (UTC)

Please find reliable sources. If you can find a reliable source which suggest that the earliest evidence on Mathematics is found in Africa, it should be mentioned in the article. Masterpiece2000 (talk) 02:48, 16 January 2008 (UTC)

Accuracy dispute?

Why is this page in the category Accuracy disputes? I can’t seem to find what, if anything is disputed, nor what is causing it to appear in this category. GromXXVII (talk) 23:03, 18 January 2008 (UTC)

When Estoy Aquí added the {{dubious}} tag in this edit the page was automatically added to the Accuracy disputes category. Ben (talk) 01:06, 19 January 2008 (UTC)

Since the user who added it has failed to give any indication why the statement is disputed in over a month, I have removed the tag. --Lambiam 06:02, 19 January 2008 (UTC)

Protection

Surely it's time for this article to be unprotected? 86.27.59.185 (talk) 23:42, 30 January 2008 (UTC)

The "protected" tag may have to stay. Large numbers of bored high school students want to add to the article, "My math teecher suks." Now, they can do that during class, using their cell phones.Rick Norwood (talk) 14:16, 31 January 2008 (UTC)

Bored with maths - I can't believe that! Anyway, we should unprotect it and see if this really does happen. By default Wikipedia articles shouldn't be protected. If a big problem emerges then it can soon be re-protected. 86.27.59.185 (talk) 21:38, 2 February 2008 (UTC)

I agree. Last anon edit was back on Nov 2nd. That's long enough. Unprotect it and reprotect only if/when persistent vandalism (re-)occurs. - dcljr (talk) 19:24, 4 February 2008 (UTC)

Could a passing Admin please unprotect. 81.76.82.232 (talk) 17:41, 6 February 2008 (UTC)

Peirce's quotation needs to have the period within the quotation marks.

—Preceding unsigned heading added by 76.22.155.72 (talk • contribs) 09:28, February 1, 2008 (UTC)

Not really. There are two competing conventions for the placement of punctuation marks – inside or outside – at the end of a quotation: "American style" or typesetter's quotation and "British style" or logical quotation. The Mathematics article is rather inconsistent in this respect, but the Manual of Style prescribes the use of logical quotation, which for the Peirce quote means: outside. --Lambiam 20:02, 1 February 2008 (UTC)

In the Mathematics article, it makes sense to standardise on the "logical quotation". Do do otherwise would be "odd." Stephen B Streater (talk) 09:40, 5 February 2008 (UTC)

Overview look

Angeliccare (talk) 10:28, 8 June 2008 (UTC): Any ideas how to modify the following text so it could be added to the article?

Mathematics in it's full glory, in limit - is a represenation of human mind: the whole mind: including thinking.

However mathematics does not (even in the full glory) include many things:

the life: life produces the need of thinking. What is brain and thinking without the need itself for the things to be thought of?
Mathematics is only relevant and pertinent in the context of life.
the names: how names becomes valuable and meaningful? They represent something more-in-depth than mind, the names are something that can be said via mathematics but is not included into.

Between these 2 - the life and the names lies the whole mathematics.

Personally, I wouldn't put any version of that into the Mathematics article. I don't know where it comes from, but it sounds like a variant on the embodied mind theory of mathematics. Most mathematicians would prefer a more objective and more down-to-earth definition of their subject. I don't think this minority view is sufficiently notable to be mentioned in the subject's top-level article. Gandalf61 (talk) 11:32, 8 June 2008 (UTC)

Angeliccare (talk) 11:50, 8 June 2008 (UTC): Every definition can be described as a projection of the term to it's maximum glory. Such projections, when collected together can in total give the full overview for the subject. Definitions are given when Life demands something strict. But how can you give something strict without completeness? Downed-to-earth is good and practical but loses the fullness.

Sorry, Angeliccare, but I agree with Gandalf61. This does not belong in this article. Rick Norwood (talk) 13:44, 8 June 2008 (UTC)

Hermeneutics meets math. Take a page from aerospace engineering and simplify the entire concept into component equations. There is a point where cognitive overreach turns airy.

Question

I need someone to explain something for me, because I could not find it in any article on wikipedia.

What is meant by the term "subleading order"?

For example,

"Show P(N)=1/ln(N). Assume N is large and ignore terms in your answer that are of subleading order in N."

Gagueci (talk) 17:12, 11 June 2008 (UTC)

This page is for discussing improvements to the article mathematics, not for asking questions about mathematics. But we do have a nice place for just that: Please take it to WP:RD/MATH. --Trovatore (talk) 18:40, 11 June 2008 (UTC)

Factual Inaccuracy

The following is factually inaccurate:

"However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently.""

1. this "work" 1930s is undoubtedly Godel's work on axiomatic systems and the discover of a Godelian assertion. Godels work does not in fact imply that mathematics does not reduce to logic because mathematics is only logic. All mathematics is only logic. This is not a matter of opinion. It is fact. All mathematical study consists of forming a set of axioms and definitions and using logic to connect the definitions using the axioms. All Godel did with the incompleteness theorems is demonstrate that there are statements that are not provable using logic.

2. Popper's quote does not imply that mathematics does not reduce to logic. Popper may be remarking on the fact that the study starts with conjecture and then proceeds to look for proof. This is indeed the case in both mathematics and natural science.

3. If, however, Popper is suggesting that mathematics is not pure logic, he is wrong. Just because he is respected doesn't mean he isn't extremely wrong. Mathematics is only logic. Mathematics is in no way a science that uses observation or measurement in any way in order to provide proof of an assertion.

I will let the author change it so that the flow of the paragraph can be maintained. The point is, Godel's work is celebrated as a breakthrough in logic, not a demonstration that mathematics is not purely logical. Mathematics is defined for all practical purposes as "logical evaluation of what follows from assumptions", so how can that not be logic?--Gtg207u (talk) 06:06, 10 February 2008 (UTC)

Well, the important thing about Gödel's incompleteness theorems is that they show that in any (moderately powerful) formalisation of mathematics there are mathematical statements expressed within that formalisation that are not only not provable starting from the axioms of that formalisation but are also self evidently true. So to see that these statements are true, we must be using reasoning that cannot be captured within our chosen formalisation. I don't think this immediately leads to the conclusion that mathematics is a science - although mathematicians do hold a range of different opinions on that topic as well, some of which are discussed in the Mathematics as science section of the article. I am very dubious about any attempt to define all of mathematics theory and practice within a single phrase or sentence - that is why the article is so long (and this talk page and its archives are much longer).

Anyway, there is no single author of this article - like every article in Wikipedia, it is a piece of collaborative writing - see Wikipedia:About for more information on how Wikipedia works. So you don't have to wait for the "author" to come by and fix things. If you think you can improve on this part of the article then dive in and change it - or, if you prefer, propose a new version here on the talk page first. Gandalf61 (talk) 09:32, 10 February 2008 (UTC)

There are different views on what constitutes mathematics; see Foundations of mathematics and Philosophy of mathematics. The point of view that all mathematics is (reducible to) logic is certainly not universally shared, and some would, rather conversely, maintain that logic (inasmuch as it can be made rigorous) is a form of mathematics. By Gödel's results we know for a fact that there is no single formalization of mathematics that is sound and whose theorems encompass all mathematical statements that are provable. And we definitely have no general method for determining whether a proposed formalization of a fragment of mathematics (like for example ZFC) is sound, nor is there any basis in logic for preferring AC over (for example) AD. Therefore it is too bold to label the contended statement "factually inaccurate". Without looking it up, I don't know if the rendering of Popper's conclusion is adequate, but if it is: this is Popper's conclusion, not Wikipedia's. --Lambiam 18:40, 10 February 2008 (UTC)

CRITICISM OF THE ABOVE ARGUMENT:

When mathematicians say that "math is not reducible to logic" they are alluding to Godel incompleteness. If Godel Incompleteness were not true, then we could aspire to a day when all mathematic truths were deducible from a finite set of axioms. Then mathematics would become essentially a branch of logic and mathematicians could be replaced by computers. But, by Godel incompleteness, such a system will never be constructed and therefore we cannot even aspire to this. This is all we mean when we say "math is not just logic." The comments above have naively interpreted "math is not logic" to mean that mathematics does not employ the tools and methods of logical analysis. In this case the criticism is basically correct. But, by this definition, the creative process of constructing new axioms is "logic" and that is not the usual employment of the term. Creating new axioms is not a purely logical procedure (if by logical procedure we mean step by step deduction), it requires creativity and intuition. Add to this the fact that there will never be a perfect set of axioms, and you have shown that mathematics will never become logical deduction. Imagine if Godel proved arithmetic to be complete. Then Fermat's last theorem probably could have been proven by a computer much sooner then it was. It is a simple arithmetical statement, easily expressible as a logical formula in a first order logic.

-Barry Barrett B.S. in Mathematics University of Rhode Island —Preceding unsigned comment added by 68.226.94.121 (talk) 08:55, 18 May 2008 (UTC)

It can be reduced to apparent logic. If mathematics could be reduced to whole logic, which at this time is beyond human cognition, humans would have qualitative control over every aspect of being. There remains no Unified Theory. —Preceding unsigned comment added by 74.13.63.120 (talk) 15:42, 16 August 2008 (UTC)

algebra

(a+b) —Preceding unsigned comment added by 203.126.166.172 (talk) 08:30, 3 May 2008 (UTC)

Under the entry «Mathematics (disambiguation)» is given the correct definition of the term 'mathematics':

Mathematics is the body of knowledge justified by deductive reasoning about abstract structures, starting from axioms and definitions.

I want to add here that the 'abstract structures' are created by humans and can not be indefinite.

Under the entry «Mathematics» one can read: «...mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects.»

Yes, it evolved from theoretical physics in particular but the modern trend not to separate theoretical physics and mathematics (and call the whole thing mathematics) is an abomination. Mathematicians are studying indefinite objects (which is o.k. in theoretical physics but not in mathematics). Geometry that is taught in schools is not mathematics – it is theoretical physics. —Preceding unsigned comment added by Oldsmobill (talk • contribs) 13:07, 15 May 2008 (UTC)

All of these statements are controversial and could elicit much discussion, but this page is for discussing improvements to the Mathematics article, not for discussing mathematics. If you would like these views to be represented in the article (or in Philosophy of mathematics?) then you should find reputable, verifiable sources that express the views, and cite those sources. Joshua R. Davis (talk) 13:58, 15 May 2008 (UTC)

A little history:

When this article was in its formative stages, there was a big controversy about the definition of mathematics. Is mathematics that body of knowledge that arrises from deductive reasoning, or is mathematics the study of shapes and numbers? Both sides were sure they were right, but the dictionary overruled the pure mathematicians, and the dictionary says shapes and numbers. Further compromises added other subject areas and a nod to the pure mathematicians in the last sentence of the first paragraph. None of us who took part in that long, long battle wants to reopen the question now, since the end result is apt to be the same. Rick Norwood (talk) 12:47, 16 May 2008 (UTC)

There's a startling aversion to accepting that indefinite conclusions eventually have definite values. This is why it took so long to solve Fermat's last theory. There is value in the philosophical question of a man's grasp exceeding his reach. As for theoretical physics, mathematics is the Rosetta Stone between sciences, as well as the intrepretive language and atomic structure of being. We cannot stop grasping further than we can reach, just because we don't have a number for it yet. —Preceding unsigned comment added by 74.13.63.120 (talk) 15:58, 16 August 2008 (UTC)

Wrong impression

Dear all,

I am sick and tired (just to exaggerate) of people who think that mathematics is only about numbers. Any mathematician who reads this will understand what I am trying to say. Mathematics is such a diverse field and in my opinion this should be mentioned as early as possible in the article. Just to make my point clear, it is virtually impossible for anyone in the current day to learn all of mathematics.

Also, I think that the article conveys the impression that mathematics is about numbers from the start. For instance, the article claims that mathematicians seek patterns. In topology for instance, I have never even encountered a problem that requires one to find patterns. This statement is only true in the most obscure sense and therefore it should be made more precise.

I hope that you agree with me; if not, please give your opinion on the matter.

Topology Expert (talk) 06:48, 19 August 2008 (UTC)

I think the article makes it very clear that mathematics is not arithmetic with its emphasis on "patterns". I'm not sure why you think otherwise.

For example, take the bridges of konigsberg problem. Some arrangements of bridges and islands permit a tour, some do not. It is the essence of mathematical thinking to look for a pattern in this - to try to see what the arrangements which permit a tour have in common, which they do not share with the others. EdwardLockhart (talk) 08:23, 19 August 2008 (UTC)

There is more to topology than the bridges of Konigsberg problem; one of the other common misconceptions about mathematics is that topology deals with shapes. It does, but topology is much more abstract than that. Perhaps when one views the fundamental group of the circle; the intuitive idea behind this is that the number of turns in a given loop determines its uniqueness (uniqueness in this sense means homotopic to no other loop with a different number of turns). This suggests that the given fundamental group is isomorphic to the integers. If you are suggesting that this is why mathematics has a link with patterns then I agree. However, now that you have found this link between the fundamental group of the circle and the integers, you must actually prove your claim (i.e construct an isomorphism between these two mathematical objects). This is an example in which one can believe that mathematics is about patterns. However, the 'patterns part' of the problem accounts for only 20% of the thinking.

I can even construct other such examples where one does not even encounter a pattern. For instance (a typically easy problem), how would one prove that every locally compact separable metric space is sigma compact? There is more to this then just finding patterns. One would use the local compactness of the space (choose a compact set for each point in the space that also contains a neighbourhood of the point in question). Then one must reduce this collection to a countable number. One may note that if the space is countable this is trivial and then notice that countable spaces are Lindelof. Since the metric space is separable, it must be Lindelof (which one should prove), and the result follows.

I am not particularly a fan of the bridges of Konigsberg problem. It gives the wrong impression of topology and really, finding an arrangement that permits a tour is just plain luck; proving that there exists no arrangement for a particular network involves more thinking. Surely you do not claim that topology is centered on this problem?

Topology Expert (talk) 03:24, 20 August 2008 (UTC)

I didn't pick the Konigsberg problem because of any supposed link to topology, but as an example of mathematics as a search for (not necessarily numerical) patterns. It has a personal resonance for me because the first explicitly mathematical thinking I remember doing as a child was to try to work out which house-like join-the-dots shapes could be drawn without retracing a line - essentially the same problem. For what it's worth, I would consider it to lie in the domain of combinatorics.

I would consider many mathematical theorems to be statements of a certain kind of regularity or pattern. To take your example, one might wonder whether separable metric spaces are sigma compact, perhaps initially supposing that all of them are, before coming up with a counterexample or two. Then observe that one needs some sort of local compactness property, convince oneself that this is sufficient, and finally procede to state and prove a theorem. This is very much a search for a pattern.

You are certainly right; this is how I would also approach the problem. I can now see you logic in why mathematics is related to patterns and I have also found many examples to convince myself. However, ignorant people who think mathematicians deal with numbers should actually learn that mathematics is a lot more diverse. My original request was to somehow emphasise in the lede paragraph that mathematics is a diverse field and perhaps list some branches of mathematics. In my opinion, this should be emphasised throughout the article. I agree with you regarding the claim that mathematics is, in a way, related to patterns but the average reader may interpret this in the wrong way and conclude that mathematics is about numbers. My intention is to do something about this. Do you have any suggestions?

Topology Expert (talk) 11:24, 23 August 2008 (UTC)

Lastly, you seem to think that "patterns" means something necessarily numerical. I'm not sure why this is, but I don't think it's the common understanding of the term. The article I think rightly emphasises "patterns" to counteract the (otherwise likely) view that mathematics is all about computation, or symbol manipulation, or formal proof. EdwardLockhart (talk) 09:44, 20 August 2008 (UTC)

MMPR VS PRDT episode 1/8 "1st Fight"

dMMPR VS PRDT episode 1/8 "1st Fight" —Preceding unsigned comment added by 71.190.84.21 (talk) 20:09, 23 August 2008 (UTC)

Type of science

maths is a type of science —Preceding unsigned comment added by 84.69.67.194 (talk) 08:50, 14 September 2008 (UTC)

Indeed it is - it is a formal science, rather than a natural science. See the Mathematics as science section of the article for more details. Gandalf61 (talk) 10:26, 14 September 2008 (UTC)

Oh, that formal science article needs work. To call mathematics a "formal science" and then say that formal sciences are "built up of formal symbols and rules" is unacceptably POV. To a realist, the formal symbols only denote and describe the underlying mathematical reality, and the rules are valid in it; they're not what mathematics is "made of". --Trovatore (talk) 18:30, 14 September 2008 (UTC)

Hear, hear. While the practice of mathematics may not be bound by the physical reality in the sense of computation and measurement, it is naive to think that mathematical thinking is detached from reality in some meaningful wider sense. Certainly we can do away with the parallel postulate, skip euclidean geometry and replace it with general relativity, extoll the genius of the uncertainty principle, but thinking that one can push symbols in a way independent of our perceptions is a variety of fundamentalist belief (i.e. POV as above). In fact the corresponding section of mathematics should be rewritten or deleted. Katzmik (talk) 16:42, 17 September 2008 (UTC)

Link to wilbourhall.org

I run a website, Wilbourhall.org that distributes PDF files of many important ancient and medieval mathematical texts in Greek, Latin, Arabic and Sanskrit, along with translations for most of them. As I explain on the website, I of the things I try to do is to repair scans of these texts from Google books, the Digital Library of India and elsewhere by replacing missing pages with my own scans, digital photographs etc. For example, Google books has many versions of Heiberg's Greek edition of Euclid's Elements available for download, but the vast majority of them are missing anywhere from several to (in one case) several hundred pages. The "repaired" version of Euclid is available on wilbourhall.org and is hopefully complete. I named the site after Wilbour Hall at Brown University, former home to the History of Mathematics Department, where I had the pleasure of studying with Dr. David Pingree. In the year and few months the site has been operating, it has distributed tens of thousands of these texts worldwide. Please take a look at the site and let me know if you think it would be appropriate to have a link to it from this page. (I completely understand if you think it would be more suitable for other, more historically-oriented articles on mathematics). Thank you for your time. BillLoney (talk) 04:31, 22 September 2008 (UTC)

I've two thoughts on this. One, the site does seem to offer useful information, and is relatively free of advertising. However, I'm honestly not impressed by the message you've posted at the top of your page.

(Text of Wikipedia criticism from "Wilbourhall" site removed to facilitate discussion)

Why would you trash Wikipedia's "History of math" article on your site, then come here and ask to post a link? --Ckatz^chat _spy 04:57, 22 September 2008 (UTC)

You're right. Its gone. Apologies. Nevermind. —Preceding unsigned comment added by BillLoney (talk • contribs) 05:55, 22 September 2008 (UTC)

They were stupid remarks I wrote several days ago when I was very upset. I thought I had removed them, when in fact I had only commented them out. I have yet to learn the value of "restraint of tongue and pen", quite obviously. I did not mean to re-open an issue that was resolved. It was completely inadvertent. Apologies while I go crawl under a rock and hide. —Preceding unsigned comment added by BillLoney (talk • contribs) 06:05, 22 September 2008 (UTC)

I feel absolutely horrible. I honestly thought I had removed those stupid words. I have removed everything from the site except the links to the PDFs. I think it would be nice if these PDFs were available for distribution. I apologize again and again for my sheer stupidity. I am really, really, really not cut out for this. Again I am sorry for any offense. —Preceding unsigned comment added by BillLoney (talk • contribs) 06:14, 22 September 2008 (UTC)

Please. No one from Wikipedia ever contact me again under any circumstances. I really can not take any more of this. Apologies again to anyone and everyone. You win. —Preceding unsigned comment added by 68.195.75.223 (talk) 06:24, 22 September 2008 (UTC) Please remove the remarks you quoted above from my webpage. Distateful as you may find them I OWN THE COPYRIGHT. YOU COPIED IT AND POSTED IT WITHOUT MY PERMISSION. PLEASE REMOVE THEM AND ALL REFERENCE TO THEM. PLEASE COMPLY WITH WITH THE LAW REGARDING COPYRIGHT INFRINGEMENT. AS IT SAYS "Content that violates any copyright will be deleted." PLEASE DO SO IMMEDIATELY. —Preceding unsigned comment added by BillLoney (talk • contribs) 15:43, 22 September 2008 (UTC)

FYI, there was absolutely no question of copyright infringement, especially given that the quote of your tirade against Wikipedia was part of a discussion with you, regarding your efforts to list your site on Wikipedia. In addition, the text was not taken from "source code"; it was clearly visible to anyone who visited the site (at least with Firefox; can't speak for other browsers.). You claim to have commented it out, but obviously had not done so when I visited the site. --Ckatz^chat _spy 08:00, 23 September 2008 (UTC)

Bill, please restore your website first and then I will restore the links. We can't have the link to your website unless you restore it first. Khoi khoi 21:03, 23 September 2008 (UTC)

Agreed. Bill, your site appears to contain useful information, and no-one wants you to close it down. There is no "war" against you; it would be great if you would return here so that we can find a way to incorporate a valuable resource. --Ckatz^chat _spy 21:58, 23 September 2008 (UTC)

It is online again. He does seem might pissed of with wikipedia though. --Salix (talk): 08:14, 27 September 2008 (UTC)

Yes, he certainly does. Quite the imagination as well. --Ckatz^chat _spy 08:25, 27 September 2008 (UTC)

Making mathematics articles more accessible to a general readership

Please visit ~~Wikipedia:Village pump (proposals)#Easy as pi?~~(this archive) to see a discussion about making mathematics articles more accessible to a general readership.

-- Wavelength (talk) 17:37, 1 September 2008 (UTC)

I'm afraid I have insufficient interest to wade through that very long discussion thread. Is there an actual proposal in there somewhere ? If there is, perhaps you could summarise your proposal in two or three sentences. In a nutshell, can you please make "making mathematics articles more accessible" more accessible. Gandalf61 (talk) 10:34, 14 September 2008 (UTC)

According to my interpretation of your request, and according to my view of the discussion, the best summary is found where I listed several options under Wikipedia:Village pump (proposals)#WHICH problem??? by a distance of about three screen-heights.

Please specify the size and resolution of the screen you are talking about. Also, which browser were you using? Did it have any "sidebars" reducing the width of the displayed page, what font size was it using and did your browser window fill the screen? —Preceding unsigned comment added by 86.139.109.135 (talk) 11:33, 11 November 2008 (UTC)

At this stage of the discussion, I do not have a preference from among the various options, which include the following.

linking to articles in Wikibooks
linking to articles in Wikiversity
linking to other articles in Wikipedia
linking to a very large prerequisite chart of articles (and/or to one of a number of smaller prerequisite charts of articles)
having a feature similar to the one currently used with articles about cities (where a mouse over a globe icon, in the upper right corner of the page, displays the expression "show location on an interactive map"), but with an interactive prerequisite chart instead of an interactive geographical map

Any one or combination of those options is acceptable to me.

The proposal (initiated by another editor) is to make mathematics articles more accessible to a general readership, and the options are proposals on how to fulfill that proposal.

-- Wavelength (talk) 03:56, 16 September 2008 (UTC)

It is worth pointing out that the original poster's complaint was focused on equations — that there was notation, such as lim, that he could not decipher. Of course explaining notation such as variable names is essential, but the original poster wanted explanations even of standard notation, since he did not know it. I'm not sure how we can explain "lim" everywhere it arises without destroying the text. The Village pump discussion suggests annotated equations; they are too cluttered, I think.

I dislike the "prerequisite chart" idea; it will have to be so large as to be unnavigable and difficult to maintain. I like the "linking to other articles in Wikipedia" idea, because that is what we already do. When you read a math article and come across a term you don't know, it should be linked, and you should follow the link to learn about it. (Of course, you might not return for some months...) The prerequisite chart is already encoded into the structure of wikilinks among the articles, or should be.

Many of the math articles (including ones I've written) could be made more accessible by adding more "soft text" about the history, goals, and applications of the idea in question. I strongly support that. But we do need to remember that Wikipedia is not a textbook. It is not designed to lead a beginner from no-knowledge to full-knowledge. Mgnbar (talk) 12:19, 16 September 2008 (UTC)

According to the past discussions listed under Wikipedia:Village pump (proposals)#Subsection 5, editors of Wikipedia mathematics articles have disagreed for years on whether and how to make them more accessible to a general readership. Is this problem (this lack of consensus) destined to continue ad infinitum or does it have a solution?

-- Wavelength (talk) 16:19, 17 September 2008 (UTC)

Inaccessibility is a problem in math and physics articles, but are these subjects truly qualitatively different from other subjects, so that we must create a special system for accessibility? I don't think so. We just need to keep doing more of what is done throughout Wikipedia — making sure opening paragraphs are not just jargon, linking to prerequisites, giving plenty of history, motivation, and applications, etc.

Perhaps we are not doing these things as fast as we'd like — because there are not enough editors? One problem I've seen is that mathematically unsophisticated editors are sometimes intimidated (even abused) by the math-savvy around here, and leave. These laypeople, even if they can't write about math with encyclopedic precision, can help us write articles that are actually useful to laypeople. Mgnbar (talk) 21:02, 17 September 2008 (UTC)

My interest in the accessibility of Wikipedia articles (without limitation as to subject) predates my involvement in the current discussion at Wikipedia:Village pump (proposals), a discussion which began on 22 July, 2008. Because the focus of the discussion is on mathematics articles, the focus of my involvement in it is likewise. If certain suggestions are followed first on mathematics articles, they can be followed later on other articles.

When I mentioned prerequisite charts, I was referring to prerequisite charts of any kind(s), although I was thinking mostly of prerequisite flowcharts. Incidentally, a prerequisite chart can provide a convenient overview that would be difficult to picture mentally from merely examining links to prerequisite articles.

Here is a suggestion which I made on 9 August 2005.

Temporary link: User talk:Wavelength#Subject (difficulty) level
Permanent link: User talk:Wavelength (subsection 4.2) "Subject (difficulty) level"

Here is a suggestion which I made on 15 January 2008.

Temporary link: User talk:Wavelength#Suggestion: readability test(s) for Wikipedia articles
Permanent link: User talk:Wavelength (section 36) "Suggestion: readability test(s) for Wikipedia articles"

Here is a permanent link to the current discussion:

Wikipedia:Village pump (proposals) (first section) "Easy as pi?"

When I noticed the discussion two months ago, I postponed some major plans and I added my comments, hoping that I could make a positive difference toward a consensus. Maybe someone else can help toward consensus. My involvement in the discussion has almost finished. Here are three related links.

-- Wavelength (talk) 00:48, 26 September 2008 (UTC)

It occurs to me that Wikipedia: Wikiproject Mathematics would be a better forum for this discussion. Maybe that's why others are not commenting here. On the other hand, if your goal is to institute changes throughout Wikipedia, then going back to the Village pump might be best. To respond to one point that you made: I know that you were suggesting prerequisite flowcharts. I don't think the idea is inherently bad; I just think that the flowchart for mathematics would be impossibly large. Mgnbar (talk) 12:43, 26 September 2008 (UTC)

Thank you both for your comments. I have added sub-subheadings, including some which indicate the presence of proposals. Some sub-subsections are still long, because I decided not to split any post into more than one sub-subsection. Generally, the very long posts were made my me, and I probably would have separated them into smaller posts at that time, if I had anticipated that I would be adding sub-subheadings. I named one subsection "Subsection 0" for consistency with the other numbered subsections. There is already a link to "Subsection 5" from this section of this page; otherwise, I would probably rename the numbered subsections by increasing each number by one.

-- Wavelength (talk) 01:06, 28 September 2008 (UTC)

It has been archived at Wikipedia:Village pump (proposals)/Archive 35#Easy as pi?.

-- Wavelength (talk) 15:14, 28 September 2008 (UTC)

Mathematics vs. Maths

Here thar be trolls

The following discussion has been closed. Please do not modify it.

Maths is a much more accurate term as opposed to Mathematics. I think we should replace the Mathematics title with Maths. --MeatJustice (talk) 19:50, 23 October 2008 (UTC)

Everyone please take a look at this editor's brief history before considering how to respond (or, probably better, not). --Trovatore (talk) 20:01, 23 October 2008 (UTC)

Everyone please take a look at the pompousness of this editor before considering his condescension and inability to accept other people's ideas. --MeatJustice (talk) 00:19, 28 October 2008 (UTC)

I have the impression "math" is used in the US while "maths" is used in English. Both are very informal. There does not seem to be any compelling reason to change the current title which is hardly too long, so I concur with Trovatore. Katzmik (talk) 09:32, 28 October 2008 (UTC)

I would at least suggest making a reference to the usage of "maths" in the start of the article, so as not to confuse anyone. Also, the English invented the English language so they are correct. --MeatJustice (talk) 15:15, 3 November 2008 (UTC)

If there are no objections, I will change the article's title to Maths. I am willing, however, to hear more arguments for keeping it as is. --MeatJustice (talk) 23:19, 6 November 2008 (UTC)

Please quit trying to be provocative. This is a serious article; people who make frivolous edits here are not going to get along well. --Trovatore (talk) 23:26, 6 November 2008 (UTC)

Both Maths and Math are abbreviations of mathematics. As a general rule Wikipedia avoids abbreviations in article titles. Further choosing either Math or Maths would be wrong for either the UK or the US audiences. I agree that some mention of the abbreviations should be made high up in the article. However, renaming the article would be poor style and exclude half our audience. --Salix (talk): 00:12, 7 November 2008 (UTC)

I do not think any mention of the short forms needs to be made, high in the article, or for that matter anywhere in the article. There is no risk of confusion, and no important information about mathematics is conveyed by discussing them. This is not a dictionary entry. --Trovatore (talk) 00:25, 7 November 2008 (UTC)

Can I say that I agree with Salix alba and "Finder"?-Jorgen W (talk) 01:38, 7 November 2008 (UTC)

(OT linguistic digression: trovare is "to find", but trovatore does not mean "finder"; it means troubadour. Easy mistake though!) --Trovatore (talk) 01:47, 7 November 2008 (UTC)

(A wanted mistake, I studied some italian literature and french, too). Please, Trovator, give a look to my comment in the previous paragraph. Jorgen W (talk) 02:09, 7 November 2008 (UTC)

I don’t think any modifications are necessary. The article name should not be an abbreviation. And there is even the disambiguation statement at the top of the article noting that maths redirects here. Also note that neither of two premier English schools call their math department a “maths department”: at Oxford they call it “Mathematical Institute”, and at Cambridge they call it “Department of Mathematics”. GromXXVII (talk) 12:13, 7 November 2008 (UTC)

Look, let's quit discussing this as though it were a serious, good-faith proposal, which it was obviously not. "Assume good faith" can't mean "be a bloody idiot about it". The ludicrous suggestion was put forth by an editor whose other edits were redirects from implausible misspellings of meat. --Trovatore (talk) 20:13, 7 November 2008 (UTC)

Quite frankly, your referring to my edits, all of which have been attempts to help this encyclopedia, as ludicrous and in bad taste is insulting. Most people wouldn't take the time out of their day to even consider discussing ways to improve Wikipedia. Maybe you should treat everyone with a little more respect, even if they don't have the same number of edits as you. --MeatJustice (talk) 21:02, 7 November 2008 (UTC)

I am willing to accept that we probably shouldn't change the title, however some mention of various colloquial terms should be included. --MeatJustice (talk) 02:22, 10 November 2008 (UTC)

Hmm. How about a section called Etymology? Seems like a great idea to me. (John User:Jwy talk) 02:33, 10 November 2008 (UTC)

It seems that the article has been protected without any edits having been made. I apologize to anyone who may have been offended by frank discussion, and hope that we can resolve the issue and keep Wikipedia open. --MeatJustice (talk) 00:28, 11 November 2008 (UTC)

The edit protection is there as the article gets a high amount of vandalism, being first target for school kids who just had a hard maths leason. The move protection is there as there has been a prolific page move vandle, admins tend to remove all traces of this vandles work. Nothing to do with the current discussion. --Salix (talk): 00:42, 11 November 2008 (UTC)

Thanks for clearing that up, I didn't know. --MeatJustice (talk) 01:29, 11 November 2008 (UTC)

I added a clarification to where the term maths is used with regards to various regions, I hope that ends this debate. --MeatJustice (talk) 20:23, 11 November 2008 (UTC)

Your edit reduced the accuracy of the article, and has been reverted. Your claim that "Maths" is used only in England and India is unsupportable; "math" is the North American English term, and "maths" is the term favoured in international English. Per WP:ENGVAR and project-wide convention, we mention both variants of terms like this, but we do not prefer one over the other; we use U.S. or international English consistently in an article, and we do not engage in edit wars to "correct" the usage to one or the other variant of English. Please scrutinise the introductory notes at Carburetor and Engine knocking for examples of how it's done. —Scheinwerfermann (talk) 21:12, 11 November 2008 (UTC)

Please -- there is no "international English". There is Commonwealth English, but it is not correctly called "international English". For example American forms tend to be favored in East Asia, exclusive of former British colonies. --Trovatore (talk) 21:24, 11 November 2008 (UTC)

You completely misunderstood my edit, and I have to side with Trovatore on this one. This didn't favor either of the forms, it simply added a clarification as to where one was used. Thanks for your valuable input, though. --MeatJustice (talk) 21:38, 11 November 2008 (UTC)

I think the consensus here is to rename the article to "Maths". This will be kept open for discussion. --MeatJustice (talk) 23:11, 3 December 2008 (UTC)

WP:IDHT. Ben (talk) 05:43, 4 December 2008 (UTC)

Vectors generalized to vecor spaces?

An important concept here is that of vectors, generalized to vector spaces,

What does that mean? Vector spaces aren't generalized vectors. You can't just string words mathematicians use and call yourself a mathematician!! —Preceding unsigned comment added by 141.211.62.162 (talk) 14:22, 9 December 2008 (UTC)

If you know what a vector space is, it is obvious that what is meant is the generalization from vectors to elements of a vector space; so need to be insulting. Still, it should be changed. Phoenix1177 (talk) 10:58, 27 January 2009 (UTC)

Mathematics as a science in the lead

The lead seems to imply that Mathematics is definitely a science, whereas the Mathematics#Mathematics as science goes into a lot of depth and contains many essentially contrary views, including one by Einstein. I don't think that the lead is handling this correctly right now. I also wonder at the quote in the second sentence being disconnected from the following paragraph- it seems to me that those should be in one paragraph, probably the quote should be moved down.- (User) Wolfkeeper (Talk) 18:50, 30 January 2009 (UTC)

I thought this spelled out the semantic confusion pretty clearly:

Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".^[1] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date.

Actually, I am not aware that the older meaning of "science" has passed out of use. I believe that it's still the primary sense of the word. A quick look at http://dictionary.reference.com/browse/science reveals:

"1. 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." (Random House, 2006)

"1. Knowledge; knowledge of principles and causes; ascertained truth of facts. ... 2. Accumulated and established knowledge, which has been systematized and formulated with reference to the discovery of general truths or the operation of general laws; knowledge classified and made available in work, life, or the search for truth; comprehensive, profound, or philosophical knowledge." ("Webster's" Revised Unabridged, 1998)

Some of the definitions do limit themselves to the natural sciences, though, like the American Heritage definition. If you'd like to try wording things so the reader does not get confused by the semantic confusion around the word "science", please have a go! --Ben Kovitz (talk) 01:57, 1 February 2009 (UTC)

Early Zero

The Olmecs in Mexico developed the Zero before the Indus Valley civilzation (India). There should be a mention of them and of course the Mayas. —Preceding unsigned comment added by 128.196.165.102 (talk • contribs) 22:24, June 13, 2008 (UTC)

No, it shouldn't. This is not an article on the invention of zero or even the history of mathematics. What you are proposing is that this article on mathematics, the subject, become an exhaustive listing of everyone who ever invented anything in mathematics, and that's not the focus. --C S (talk) 19:40, 10 March 2009 (UTC)

Definition

Wanted to change the definition to make it grammatically clear (a "body of knowledge" can't "study" anything). I was thinking something like `Mathematics is the body of knowledge and academic discipline arising from the study of such concepts as quantity, structure, space, and change.' but I can't change it. Not sure about "arise" either, hopefully another author will think of something better. —Preceding unsigned comment added by 150.203.114.44 (talk) 15:45, 8 October 2008 (UTC)

OK, you do sort of have a point here, I think. Be aware that much blood has been spilled over this sentence; changes to it have to be made delicately. Maybe something like mathematics is the body of knowledge comprising the study...? --Trovatore (talk) 19:15, 8 October 2008 (UTC)

I feel very uncomfortable with the words "body of knowledge", though I am sure that the idea behind is good; in my opinion mathematics is "techniques, methods, models" in relation to the real world (including human mind) and its (scientific) description.Jorgen W (talk) 01:35, 7 November 2008 (UTC)

I don't completely understand. You don't think the knowledge that we've discovered using those techniques, etc, is part of mathematics? Or you don't think it constitutes knowledge at all? --Trovatore (talk) 02:55, 7 November 2008 (UTC)

I didn't say that the knowledge using those techniques isn't part of maths: on the contrary, I'm just saying that mathematics is also techniques, methods, models, arguably (a naive point of view), maths is "motion-movement", transformation of natural phenomena into abstract and wider structures; the term body of knowledge sounds a bit more "static" or just symbolic (sort of meta-language used as a tool by natural scientists). Furthermore, let's compare the heading of Physics and Science. :) --Jorgen W (talk) 01:18, 9 November 2008 (UTC)

Right, it's those things as well, I agree. I sort of prefer the version that said mathematics was a discipline. But unfortunately some people read that word and automatically think bondage and ... or some such thing. Someone asked the question once, if mathematics is a discipline does it hurt much?. We obviously can't use science because a significant point of view denies that it is a science (though personally I say it is). So it's tricky. All in all I'd be happy to go back to discipline; what do others think? --Trovatore (talk) 02:04, 9 November 2008 (UTC)

Oh, wait a minute; it says now that it's a discipline, and a supporting body of knowledge, and it's said that for a while. I hadn't bothered to check. So now I'm really confused as to what it is you don't like -- your techniques etc are subsumed in discipline, and you agreed that the "body of knowledge" was also part of mathematics. --Trovatore (talk) 10:12, 10 November 2008 (UTC)

Shouldn't this, somewhere in the lead, call Mathematics a hard science and an exact science. If we can't attribute those characteristics to math, then what? :) Student7 (talk) 02:10, 28 January 2009 (UTC)

Both of those names have just a tinge of a dig taken at the social sciences. I don't see that that's helpful here. --Trovatore (talk) 02:15, 28 January 2009 (UTC)

It just seems to me that in mathematics, there is a certain rigor lacking in other fields. Even in astronomy theories have changed. "We see so and so, therefore x must be true." That is, observed data changes the theory which is sometimes "just a theory" in the lay use of the word, rather than being driven by underlying principles.

It's much worse in evolution, where digging up a new relic (or a new discovery in biology) causes scientists to say, "Since animal x had characteristic y, therefore it must be a survival technique and they had it for reason z." There are new reasons z each year! So forget the alleged "dig" against the social sciences, the physical sciences don't do that well either IMO! There should be something in philosophy that describes this and which should maybe be here. Student7 (talk) 14:12, 24 March 2009 (UTC)

Jorgen, I share your opinion about math. However, our opinions about math, no matter how well-reasoned or even true, are not relevant on Wikipedia. All we do here is summarize information from reliable, published sources: see WP:OR. Most other reference works define mathematics as something like the study of quantity, form, structure, pattern, etc. Philosophically inclined folks like to try to invent a super-rigorous definition that aims to capture all of that, usually under something like "drawing necessary conclusions". Posting that sort of definition would take a side in a debate (see WP:NPOV). Let's just do a good job summarizing the main, non-side-taking definitions. (And, hey, please argue for your definition in scholarly forums! I'd like to see that POV much more widely recognized.) --Ben Kovitz (talk) 04:24, 29 January 2009 (UTC)

Here are some definitions:

from the 1933 OED: "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."

from the 2000 American Heritage Dictionary: "The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols."

from Random House: "the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically."

from one of those Webster's dictionaries: "That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations."

from WordNet: "a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement."

from the American Heritage New Dictionary of Cultural Literacy: "The study of numbers, equations, functions, and geometric shapes (see geometry) and their relationships."

Since we're writing an encyclopedia, we can have a more in-depth definition. We can surely do better than any of the above. But we shouldn't violate the basic idea: math is the science that studies certain things: quantity, structure, pattern, relation, <argue for stuff here>, etc. (Replace "science that studies" with "study of" if you're worried about people who construe the word "science" narrowly; but I think that's unnecessary.)

--Ben Kovitz (talk) 04:43, 29 January 2009 (UTC)

I am astonished at the lack of rigour in most of the above defintions of mathematics. I hope that you see that something along the lines of "The application of logic to axiomatically defined systems" is much more appropriate.

Maths Graduate, UK —Preceding unsigned comment added by 212.2.4.82 (talk) 11:29, 18 February 2009 (UTC)

I am not sure I agree with you. First, which logic, constructive mathematics formally uses a different logic than other parts of mathematics. Second we usually are only interested in axioms that represent some system we feel is meaningful. With slight alterations you can make the axioms for a group completely independent. Meaning you could negate any one of the axioms and still have a consistent system of axioms. But no one studies such systems. Mathematics is notoriously hard to define. Though I once heard the following definition attributed to Erdös "Mathematics is what mathematicians do." Not apropriate for the article but nice because it is such a mathematical way to look at the problem. Thenub314 (talk) 19:12, 18 February 2009 (UTC)

I'm astonished by how many people learn some mathematics and think they can go around telling people mathematics is and is not such-and-such. Plenty of mathematicians "do mathematics" without working in axiomatically defined systems, and certainly mathematics involves considerably more than just "the application of logic". --C S (talk) 19:45, 10 March 2009 (UTC)

By not specifying a variety of logic, have I not included all of them? I agree that we are usually only interested in meaningful axioms but that does not mean that the study of less meaningful axioms is not maths. The Erdös quote made me smile. :) I would genuinely like to know what mathematics doesn't have its foundation in axioms. —Preceding unsigned comment added by 93.96.239.83 (talk) 22:09, 6 April 2009 (UTC)

It's not Wikipedia's place to settle controversies about the definition of mathematics. However, your insights and research might do a lot to improve definitions of mathematics. Hint, hint. :) --Ben Kovitz (talk) 15:46, 22 February 2009 (UTC)

move structure below space and change?

I think the "structure" section, being a more advanced topic would work better under the "space" and "change" section. I want to hear other's thoughts and opinions before making the change. Kevin Baas^talk 17:26, 13 February 2009 (UTC)

I think it's a good idea. Abstract algebra is abstracted from more-"concrete" ideas, so I think a lay reader will have an easier time following the sequence that you propose. Your sequence might even provide a nice way to work in a reference to the Erlangen program. --Ben Kovitz (talk) 03:56, 14 February 2009 (UTC)

Here there be tygers! The "quantity, structure, space, and change" rubric is the result of a long hard fight between the "mathematics is a subject" contingent and the "mathematics is a method" contingent. (I want the article to say "Mathematics is that body of knowledge discovered by deduction, just as science is that body of knowledge discovered by induction," but I lost the fight and the current lede was a compromise.) The "quantity, structure, space, and change" definition now appears not just here but across the mathematics portal. If you change it here, you should change it everywhere, and be prepared to fight every inch of the way. I would suggest not even starting unless you have at least six months with nothing better to do. Rick Norwood (talk) 17:25, 14 February 2009 (UTC)

Have you read the proposed idea carefully? It's not proposing to change the definition, it's proposing to change the order of some of the sections later in the article. --Ben Kovitz (talk) 18:09, 14 February 2009 (UTC)

I understood that. But if you change the order of the subsections, shouldn't you change the order in the lede to reflect that? And if you change the order in the lede, shouldn't you change the order in all the other articles that list those topics in that order?

The order does not seem that important to me, because I don't believe that definition of mathematics for a minute. (Where does game theory fit? How about probability and statistics? How about mathematical economics, which John Nash won a Nobel prize for?) But that definition follows a number of standard dictionary definitions, or is, rather, an expansion of them. Most dictionaries limit mathematics to the study of numbers and shapes.)

But if you do start to make structural changes in the article, just be aware that this article is closely watched. Rick Norwood (talk) 21:55, 14 February 2009 (UTC)

Here goes... Kevin Baas^talk 16:47, 17 February 2009 (UTC)

btw, i visualize mathematics. thats what i see when i do it: quantity, space, and change. that's how math is done. that's what it's used to describe. etc. etc. etc. space + change = structure (hence analytic geometry, etc.) (perhaps you can add in quantity to count sets and all that.) but there is no way to derive change from quantity, space, and structure. you take out the change and you break it - there are things it can't do anymore. (like adaptive control, feedback, nonlinear dynamics, etc. you know, the really cool stuff.) so unless you seek to regulate calculus or violate set theory in your basic definition of mathematics (oh, the irony!), i would say the foundations of mathematics, if one is to name just three, are quantity, space, and change. just my opinion. and frankly, i'd hate to see how critics of that see mathematics - it'd be like a world without color. Frozen in time, as it were, in a neat little box. Kevin Baas^talk 17:07, 17 February 2009 (UTC)

While it has nothing to do with the page, I see mathematics as only dealing with structure, and I don't find it to be a world without colour; nor do I follow how not viewing quantity, space, and change as foundational would violate anything, Category Theory is foundtional and is entirely structural. Maybe I'm missing the point of what you're saying. At any rate, if the order is being changed to facilitate simplicity, then your suggestion makes sense; if it is being changed to better illuminate the foundational philosophy of mathematics, then I whole heartedly disagree with you. Phoenix1177 (talk) 06:04, 18 February 2009 (UTC)

I see how in a sense mathematics is purely structural: it's a set of symbols manipulated by a set of rules. Such that one can view any mathematical statement (read:string of symbols) as a point that is connected to other statements by lines represented the different rules by which you can manipulate the statement. Thus, mathematics is one giant web and in that sense - purely structural. However, the usefulness of viewing math in this way, save proving the inherent incompleteness of it (ahem) is fairly limited. Math is used to describe the world which is not "purely structural" (though i suppose this depends on your definition of "structure" which is a whole philosophical/semantic discussion which would probably go nowhere anyways). It is probabilistic and differential. And although math - as manipulation of symbols; as something "structural" can solve many differential problems, there are many that it simply _cannot_. As godel's theorems proofs, there are some points that are unreachable from other points. And even beyond that, there are some (in fact, many) diff eq problems that are simply not analytic. On a different vein, numbers themselves don't lend themselves to a purely structural perspective. There are Real numbers that are simply not computable, yet when a mathemetician or engineer envisions their problems i'm sure they see _continuous_ motion. and Hava Siegelmann has shown that a machine can be built that operates on un-computable numbers - is this machine therefore not mathematical? etc. etc. etc. point is that there are differential situations that are spatial-mathematical and cannot be discretized. you take out change - differential systems w/a time differential that can't be factored out, and you take out entire branches of mathematics some of which (e.g. chaos theory), at least conceptually - and certainly in the real world - transcend the limits of computability; of mathematics as structure. and "structure" by definition doesn't include time. nor does space or quantity. so unless you include change, your set is incomplete.

but in any case, it reads simpler, yes, that's why. but have you ever thought about WHY it reads simpler? the mind develops concepts from the bottom up - it cant start with the top. it needs the pieces to put together and build up to it. the more you put something in that order the easier it will be to read and vice versa. so, if, as you acknowledged, the present order is easier to read, then as far as the brain's grasping of concepts is concerned, "change" is more foundational than "structure" in mathematics.

i could go on (and the above-cited Erlangen_program is yet another good example) but i hope that is enough for you to understand my reasoning. Let me reiterate that i'm quite aware of things like whitehead's principa mathemetica which shows that all mathematical proofs (all of "mathematics" in a way) can be broken down into a small set of logical axioms. In a way that is "foundational" but I don't suppose that's the way math is taught in schools or how it developed epistemologically (and category theory was certainly a later development). Altogether it seems quite possible that we have different definitions of "foundational". For instance, what you might call "foundational" I might call "abstract", which to me is pretty much the _opposite_ of "foundational". In any case, I hope you see my meaning clearer now. Kevin Baas^talk 18:32, 19 February 2009 (UTC)

Are you guys arguing about what is the correct definition of mathematics? Instead of doing that, how'd you like to help improve definitions of mathematics? --Ben Kovitz (talk) 15:40, 22 February 2009 (UTC)

I have thought about why it is simpler, I think it is because the way most people think, not how mathematics is. At any rate, I do not view mathematics as just a bunch of symbol manipulation, that is not what I mean by structure; for example, you say that people think of the continuum when dealing with the reals, I think of a topology with nice algebraic and order structures. At any rate, there is no "right" or "standard" way of looking at mathematics, there is mathematics and the perspective of the mathematician.

Also, the Principia does not show that all mathematics can be broken down into some axioms; I can take any list of axioms and axiom schemas and call it a system, even if they're not all interesting, they are no less mathematical. I've always read "foundations" to mean that it involves a system off of which modern mathemaical knowledge can be based; and you're right that we don't teach this way in schools, reason being is that you can't really grasp things like Category Theory until you have a firm grasp on all the variety of structure that it is abstracting, this has nothing to do with what mathematics is, only how people learn.

Finally, Ben Kovitz is right, we should stop arguing(me especially, if argument you call this) and do something useful :) Phoenix1177 (talk) 12:11, 27 February 2009 (UTC)

awkward wording

"The first abstraction was probably that of numbers: the realization that two apples and two oranges (for example) have something in common was a breakthrough in human thought." seems like a suspiciously worded claim

better as, e.g., "The first abstraction was probably that of numbers: that is, the realization that two apples and two oranges (for example) have something in common." --Rainjacket (talk) 01:25, 24 February 2009 (UTC)

It is not only suspicious but wrong! Lots of animals from rats up can be shown to have an abstract concept of number. For example a rat trained to press a level after a particular number of sounds will also press it after the same number of flashes of light. I changed the text with a reference. McKay (talk) 12:15, 24 February 2009 (UTC)

Also note the article in the April 2009 Science News, which shows that chickens can count (but not, presumably, before they hatch). It's called "Counting Chicks", and is about a study of baby chicks that shows that they can not only count, they can do simple arithmetic! Rick Norwood (talk) 18:50, 24 April 2009 (UTC)

I need an answer to my question plz , wikipedia sent me here plz !!

your macro lens permits you to make close ups at a reproduction ratio of 1:3.If you are taking a close up of a flower that is 3/8 across , how wide will the flowers image be on film? —Preceding unsigned comment added by 75.152.125.141 (talk) 21:06, 6 May 2009 (UTC)

Wikipedia:Reference desk. Hut 8.5 06:33, 7 May 2009 (UTC)

multiplication Kevin Baas^talk 16:44, 7 May 2009 (UTC)

when is a circle said to be a quad —Preceding unsigned comment added by 217.117.2.100 (talk) 17:51, 16 May 2009 (UTC)

plane shapes

when a degree is divided into sixty equal parts,it is called —Preceding unsigned comment added by 217.117.2.100 (talk) 18:00, 16 May 2009 (UTC)

See degree Rich Farmbrough, 15:26 24 May 2009 (UTC).

Applied math

There is a great heterogeneity in the topics listed under applied math, and one of them seems misplaced to me.

I think most probabilisty students would agree that the area is more suited to the "Change" section. Even applied probability is too theoretical for most statisticians/numerical analysts/physicists and such.

--Lucas Gallindo (talk) 14:18, 19 May 2009 (UTC)

Lead.

I am a bit confused by the phrase "patterns and other quantitative dimensions", mostly because I have no idea what a "quantitative dimension" is, and I honestly have no idea what this is supposed to mean. My best guess is that dimension is being used as a synonym for aspect. I was wondering how people felt about changing this sentence. I might suggest:

Mathematicians seek out patterns and other quantitative aspects of the entities they study, whether these entities are numbers, spaces, natural sciences, computers, or abstract concepts.

I like to critique my own work, so I would say the above sentence fails in that much of what some mathematics is could be thought of as qualitative and not quantitative. But adding the term qualitative then makes it sound as if mathematicians do everything under the sun. Also, the list of "entities" they study is necessarily eclectic, but I don't know particularly how to improve it. Does anyone object to me making this change?

As a last comment. One of the citations we give points to [11], which on my browser loads a blank page, are other people encountering this? Thenub314 (talk) 10:34, 11 June 2009 (UTC)

i hope you could help me

hello my name is arman jabari i come from iran and i am 18 years old i affirmed pythagores but i don't know i affirm it in a new way or someone have affirmed it from this way before could you guide me which way i have to know it —Preceding unsigned comment added by 217.219.195.15 (talk) 07:09, 14 June 2009 (UTC)

???

hello i affirmed pythagores but i don't know i affirm it in a new way or someone have affirmed it from this way before could you guide me which way i have to know it my email is: arman.jabari@yahoo.com —Preceding unsigned comment added by 217.219.195.15 (talk) 07:26, 14 June 2009 (UTC)

Einstein quote: not worth of an encyclopedia

Despite all his accomplishments, Einstein is not a mathemathecian or a philosopher of mathematics. Yet, we find one of his quote in the introduction of this article, which discusses about what are mathematics. To have a quote of Einstein in this particular article is a false appeal to an expert testimony, something an encyclopedia like wikipedia should avoid. 142.85.5.20 (talk) 01:34, 3 July 2009 (UTC)

Although not a mathematician, Einstein was certainly a scientist, and made extensive use of mathematics in his work, so his views on the relationship between mathematics and science are relevant in the "Mathematics as science" section of the article. Other viewpoints are also presented. Wikipedia does not say that Einstein was right; it only quotes him as representative of a significant point of view. Gandalf61 (talk) 10:42, 3 July 2009 (UTC)

Not a mathematician? Who sez? Einstein was often referred to as a mathematician in his lifetime. Someone like Einstein is just as much a mathematician as Edward Witten. I notice the article on Stephen Hawking, Lucasian Professor of Mathematics, has resolved the discipline issue by referring to him as physicist and "applied mathematician" in the infobox (Hawking, interestingly, is more often described as a mathematician in British publications (usually in conjunction with "physicist") than American, where he is almost always described as a physicist, although even publications like the New York Times will occasionally refer to him as a mathematician) --C S (talk) 18:34, 8 July 2009 (UTC)

Japanese interlink

Japanese interlink is not in alphabetic order: currently it lies between ne and no. Please change this. 82.52.179.192 (talk) 09:16, 8 July 2009 (UTC)

Mathematics is the study of space?

My revision of the first paragraph was reverted, so I'm bringing it here for discussion. If anyone wants to point me to prior conversations I should be aware of, I'll try to bring myself up to speed. My proposed revision:

Mathematics is the science of applying mathematical techniques to the study of quantity, structure, space, and change.

Rationale: First, as the "tics" suffix of the word "Mathematics" indicates, it is a practice and methodology rather than a study, and indeed, it is. Second, although mathematical techniques are fantastic for exploring concepts such as stquantity, structure, space, or change, it is a mistake to conflate the tool with the concept itself. If we're going to say that mathematics is the study of everything to which mathematical techniques can be applied, then we just have to say that mathematics is the study of everything (like the physicists do

D) —Preceding unsigned comment added by Pooryorick (talk • contribs) 08:02, 12 July 2009 (UTC)

You should not use a form of a word (mathematical) in the definition of a word (mathematics). Rick Norwood (talk) 12:27, 12 July 2009 (UTC)

Hmm, this is actually done all the time, and is in fact often the only sane approach to defining words that derive from adjectives, as "mathematics" does. For example, "heuristics" is defined as "the study or practice of heuristic procedure". Such definitions follow the pattern "of or relating to...". "heuristics" also illustrates what I am saying about words that end in "tics", which are defined as the "art", "science", "practice", "process", and, secondarily, the study of the practice itself. The dead giveaway is the "s" at the end, which differentiates the latin suffix "ic" (pertaining to the nature of) and "ics" (used to form names of arts and sciences). Pooryorick (talk) 16:46, 12 July 2009 (UTC)

But mathematical redirects here. Such circularity is to be avoided if at all possible. I certainly prefer the current form to your suggestion. Calling math. a 'science' is also apt to be contentious, though I myself can go either way on it. JJL (talk) 19:41, 12 July 2009 (UTC)

OK, a few points in no particular order:

Relating to Pooryorick's proposal:
- As noted above, I don't think we can say in the opening sentence that mathematics is a "science". I personally have defended the claim that math is a science, and indeed an empirical science, but we can't pick one side of such a vexed question, certainly not in the lede. Farther down, we present pro and con views, as is appropriate.
- The apparent circularity doesn't necessarily have to be a real circularity (I'll defend mathematics is the study of mathematical objects), but it's bad stylistically even if not logically.
As to Pooryorick's criticisms of the existing text:
- "study" includes "practice". The practice is how you study it.
- I really don't think we can make any clear deductions from the suffix of the word. Possibly such inferences might have had force at some time in the past, but we're talking about mathematics as understood now.
Here's perhaps the most important point: We give a "definition of mathematics" sort of pro forma, because it's expected. The fact is that there is no satisfactory definition of mathematics that exactly delineates what is mathematics from what is not. So whatever we put here is going to be kind of bogus. But it should be bogus in as inclusive a way as possible — we should not exclude from mathematics things that any significant number of mathematicians consider to be mathematics.

Summarizing: The existing opening sentence would not have been my personal ideal, but I see no strong grounds to change it at this time. --Trovatore (talk) 20:53, 12 July 2009 (UTC)

One last note: If Math was a separate page, it would neatly resolve the issue of whether mathematics is a science or not All th e philosophical discussion and debate about the nature of math itself could go there, and this article could, by definition, happily focus on the art of doing math. 206.53.79.172 (talk) 14:45, 13 July 2009 (UTC)

First two sentences

{{editsemiprotected}}

The first two sentences are incorrect in my opinion. And incorrect in such way that I am motivated to beg you all to please change it! To express my concerns:

The first sentence attempts to give a definition of mathematics: "Mathematics is the science and study of quantity, structure, space, and change." If this is the definition of mathematics, then what am I doing when I say Zermelo's theorem is equivalent to Zorn's Lemma. Study of structure? Architecture. Study of space? Feng shui. Study of change? "i ching" maybe? My point is that each item on its own is offensively vague. To put three vague characterizations in one definition, compels me to write this note.

Here are two much better ways to define mathematics. The first way is to define it by enumerating its fields, and then defining each of those fields. Math is largely divided into five fields: Geometry, Algebra, Analysis, Number Theory, Combinatorics. Geometry encompasses the study of Euclidian Geometry, Topology, Differential Geometry, etc. Algebra is the study of algebraic structures such as groups, rings, fields, and algebras. Please note that the statement "math is the study of structures" is offensively vague, in my opinion, while the completely different statement "math includes the study of algebraic structures" is fine. Analysis incorporates such familiar things as calculus, differential equations. Number Theory is at its heart just arithmetic, of course built into a magnificently rich theory. Combinatorics can in broad sense incorporate set theory and in a sense shares (with analysis) the theory of probability.

The second way to define mathematics would be with a much broader statement. Ideally we there would be a razor thin definition such as the first sentence I quoted above, but the problem is to get razor thin makes it wrong. Therefore we are forced to define it broadly. Some such definition as "Math is the practice of determining the consequences of axioms" would be one general way to go. Another would be " Math is the study of numbers and their relationships" I suppose.

I do not purport to be an expert on writing encyclopedias. I do purport to recognize something that must absolutely be changed. Please let us change the first two sentences of this article. Thank you

Request --> Please change "Mathematics is the science and study of quantity, structure, space, and change." to read: "Mathematics is the study of axioms and their consequences, of numbers and their relationships; of Geometry, Algebra, Analysis, Number Theory, and Combinatorics." Thyg (talk) 01:53, 19 September 2009 (UTC)

Oppose. I won't repeat myself at length as to why; look through the archives. As to the specific proposal, it overemphasizes axiomatics in the first half, and is overly detailed in the second. --Trovatore (talk) 02:13, 19 September 2009 (UTC)

Comment I have cancelled out this request-to-edit-semiprotected for now, as this obviously needs discussion; if a consensus to change it can be reached, please use an {{editsemiprotected}} then. Yar! Chzz ► 02:41, 19 September 2009 (UTC)

Agreed & I see I should have read further. Kind of a wiki contribution novice though I absolutely love using it. The core of my objection is that "structure, space, change" is vague. How about "algebraic structures, topological spaces, infinitesimal rate of change" turns something false or at best vague, into something acceptable. PS I would hate to offend anyone, I agree this is my opinion only, I just read somewhere that it's ok to be bold with change suggestions??!! PS I could get some preeminient math professors to provide definitions, which I feel would be a better source than a laymans dictionary. Would that help engender a change? And maybe I am misunderstanding: do the majority of people out there like it the way it is right now?

Hi, Thyg. Vague, in this case, is good. It's deliberately vague. Anything not vague would choose one side over another in the wars over conceptions of mathematics (Platonist/formalist, pure/applied, a-priori/empirical, etc).

As to whether a majority like it, I seriously doubt it. I don't like it. I don't know that anyone likes it. If we could, the best solution would be to drop the "definition" part entirely. But unfortunately we have to have it; it's expected, and I think this expectation is even codified somewhere. So we have this thing, and it's more or less acceptable to most people, which is probably the best we can do. Probably there are other formulations that would satisfy a similar proportion of editors, but to change it would just re-open the wars, and no one really wants that. So my preference is to leave it alone, not because it's ideal, but because there's no clear reason to think the pain it would cause would actually improve things. --Trovatore (talk) 07:48, 19 September 2009 (UTC)

GA Reassessment

This discussion is transcluded from Talk:Mathematics/GA1. The edit link for this section can be used to add comments to the reassessment.

Most of this article is unreferenced. Please reference the paragraphs that don't have any inline citations. Gary King (talk) 06:48, 27 July 2009 (UTC)

I'm delisting this article since these issues were not resolved. Gary King (talk) 03:29, 3 August 2009 (UTC)

You really think this is appropriate? The idea is to improve articles for readers, not to reference paragraphs where there is no particular advantage in so doing. The article offers 29 inline references. I think you should be more specific; I think you should make a case that something important here has gone unreferenced; and you haven't said anything helpful about what you take "these issues" to be. And you do seem keen to move ahead to a delisting of what is by any standards one of the most prominent articles in the encyclopedia on the minimum scale as far as time and engagement is concerned, at a period when many people may be more concerned about vacations than dealing with opinionated critics. Charles Matthews (talk) 11:24, 3 August 2009 (UTC)

Please see WP:CITE and the Wikipedia:Good article criteria. There is plenty of information that is uncited when it should be, including most of Etymology, History, Mathematics as science, Fields of mathematics, and Common misconceptions. The advantage in referencing information is so that it can be verified by readers when they wish to do so. Just because the article has 29 inline references does not mean that this is "enough"; for a 5 kb article, sure it might be sufficient, but it's probably not likely enough for a 100 kb article. It all depends on the specific article. And, why would I be keen in delisting articles? I don't gain anything from it; what I typically do is make a list of articles that I have put up for reassessment, and then just return to them when I have time any time after the standard seven days to make a decision; if there is a discussion going on, then I am willing to let it go on for a much longer period of time—feel free to check my hundreds of reviews, I have never had a problem with delisting an article too early when someone was working on it. Ultimately, good article status is a concern left to editors, not readers; it exists to help editors better improve articles, which is what I try to do when I perform a review. Delisting an article does not change the article at all, from a reader's perspective.

After doing some research, it looks like this article never went through a good article review to begin with. Here is where the article was listed as a good article, sans review. A lack of inline citations was then later brought up at the FAC by several editors. The article Force is a pretty good example of a well-written, well-referenced article in the science field. Gary King (talk) 15:23, 3 August 2009 (UTC)

Clearly, this conversation has failed to even consider the relevant citation guideline: WP:SCG. This is unfortunate; a "review" by a single reviewer, without addressing our policy or whether the assertions in question are challenged or likely to be challenged, which is the standard set in actual policy, does not add credibility to GA. Septentrionalis PMAnderson 18:08, 3 August 2009 (UTC)

Please just have a look at some of the better-referenced articles, such as those in Wikipedia:WikiProject Mathematics/Wikipedia 1.0/FA-Class mathematics articles, for an idea of what I am looking for in terms of referencing. Gary King (talk) 18:24, 3 August 2009 (UTC)

Really? I looked at Group (mathematics) and see about the same density of citation (including the silly reference to a 1908 edition of Galois for the story of his life. This source is neither contemporary nor current, and in any case unnecessary; none of the aspects likely to be challenged is mentioned). Please specify what statements seem to you challengable and unsourced - as policy requires; this vague hand-waving is no service to anybody. Septentrionalis PMAnderson 18:39, 3 August 2009 (UTC)

Really? Compare Group_(mathematics)#History to Mathematics#History, for instance. If the Sevryuk citation is used for the entire section, then okay, at the very least please copy the citation to the end of each paragraph in the section so that this is clear (it also makes it easier to determine what reference is used for that information if the text is ever changed/moved around). "Etymology" is almost entirely unreferenced, save for the quoted text; surely this information is not common knowledge. Gary King (talk) 18:45, 3 August 2009 (UTC)

Having consulted the OED, I conclude that you have not; whether or not the etymology of mathematics is common knowledge, it (including the reference to mathematiques) is to be found there.

On the other hand, the outline of the evolution of mathematics in the section on History (on that level of generality) is common knowledge, and to be found in any of the sources in the notes. I would think two claims likely to be challenged - and they are the two that have notes.

The idea of repeating a footnote at the end of successive paragraphs is appalling. That's dreadful style, and insisting on it does harm to Wikipedia. I thought that GA could be left to twiddle with its stars and give them out to each other while the rest of us got on with writing an encyclopedia; as Lotte Lenya sang: guess not. Septentrionalis PMAnderson 19:12, 3 August 2009 (UTC)

I don't understand. If you don't care about GA status, then why put so much effort into it? Ultimately, as I said earlier, this article was promoted without a review in the first place, so delisting it essentially brings it back to the status that it would have had, anyway. Gary King (talk) 19:18, 3 August 2009 (UTC)

Because I am disappointed. Good Articles could be a useful process if it were conducted as a light process, without a self-important "review" ("I approve this" = GA; "well, actually, it could use work" = not GA; "there, it's better" = GA). This is how it was originally designed.

It could also be a useful process if an intelligent reviewer read the article, (preferably with some knowledge of the subject matter - although a detailed response by someone who didn't could also be very valuable) and saw what could actually be useful to it.

This, however, is neither: I see no sign that Gary King read this article before ten minutes ago; I don't include counting footnotes or other purely mechanical tests - especially when the test is not based on guidelines or on common sense. Septentrionalis PMAnderson 19:31, 3 August 2009 (UTC)

If you disagree with my review, you can always renominate the article at WP:GAN or bring it to WP:GAR for a reassessment, similar to Wikipedia:Good article reassessment/Special relativity/1. Gary King (talk) 19:34, 3 August 2009 (UTC)

No, I think you are being given a chance to read the section on "Summary style" in WP:SCG where this precise article is named. Let's look

Many articles on broad subjects, such as Albert Einstein, special relativity, big bang, and, indeed, physics and mathematics, have a series of sub-articles. In this case, the summary style may be used, in which a broad overview is given in the main article, and details can be found in subarticles. For citations, the summary style article says:

There is no need to repeat all the references for the subtopics in the main "Summary style" article, unless they are required to support a specific point.'

Therefore I think by asking for inline citations for each para, you are either disregarding this guideline where in terms the point at issue is dealt with, or disqualifiying yourself as a competent reviewer by lack of knowledge of the most relevant material. Please come back with a more considered approach to this article, and the task of assessing it. You are playing one-club golf with a prominent article, and you can be expected to put up a better argument than that this is a fait accompli. Where it says 'specific points', I believe that means you should be conducting this review by means of specific points, where we could have a reasonable discussion on the appropriate level of referencing for a "broad subject". You are not supposed to subvert the spirit of guidelines with such direct application. Charles Matthews (talk) 20:37, 3 August 2009 (UTC)

Okay, let's start with the "Mathematics as science" section, particularly the last paragraph. I don't think that most of that information is common knowledge, and so I think that they need inline citations. In addition, the page that you linked to mentions this also:

When adding material to a section in the summary style, however, it is important to ensure that the material is present in the sub-article with a reference. This also imposes additional burden in maintaining Wikipedia articles, as it is important to ensure that the broad article and its sub-articles remain consistent.

Some of the information found in this article's summary-style sections are not found anywhere in their respective main articles, or they are not referenced in both this article and their main article. One example is "In the 18th century, Euler was responsible for many of the notations in use today."; actually, the entire Mathematical notation article only has one reference, and that is only for a single statement. Also, weasel words should be avoided, such as "Many mathematicians feel that...", per WP:CITE#CHALLENGED. (This happens several times in the article.) Gary King (talk) 21:20, 3 August 2009 (UTC)

That is common knowledge - for citations on the subject, see Leonhard_Euler#Mathematical_notation, which is noticeably incomplete. (Summary style does include linked articles, but feel free to add any section header you think actually helpful to readers.) Septentrionalis PMAnderson 00:35, 4 August 2009 (UTC)

Please leave any article on my watchlist alone. Septentrionalis PMAnderson 19:39, 3 August 2009 (UTC)

Numbers can be used for many things, such as change, space, velocity, and infinitely many other concepts

I believe that the difference between pure mathematics and applied mathematics should be discussed prominently in the first paragraph, since it is crucial to establishing the definition of what the article is talking about. In my opinion, units can be attached to number to lend different meanings in different concepts; for example, if we are talking about physics, a unit such as the Newton, the standard unit of force, can be attached to a number. In this case, we are discussing applied mathematics, or pure mathematics AND a unit or units.

If you are talking about two apples, or two people, you are using applied mathematics, since you are using the "apple" as a unit, and also the word "people." Pure mathematics lacks units.

Therefor, the first sentence should be revised. Applied mathematics deals with change, structure, and all that sort of thing; however, that is not part of the inherent nature of mathematics in the pure sense. This difference should be elucidated carefully. —Preceding unsigned comment added by Onefive15 (talk • contribs) 17:24, 1 August 2009

Please take a look back in the archives before proposing any change to the opening sentence. It is the outcome of long and tedious discussions.

Fundamentally there is no agreement in the community (either the community of mathematical WP editors, or the mathematical community at large) as to what demarcates mathematics from non-mathematics. And in some sense we don't really need a definition; it's not as though mathematics is likely to be an unfamiliar word to anyone who comes here. But unfortunately we have to have something that looks like a definition, because this is the expected form. I think it's even codified somewhere in broader WP policies/guidelines.

So we give this not-terribly-meaningful collection of generalities, reminiscent of a traditional definition, sort of as a placeholder. It doesn't say much, but at least it doesn't attempt to put mathematics into a contentious ideological box.

A side comment: You seem to be focused on mathematics as the study of numbers. There's no precise definition of either mathematics or number, but nevertheless it is clear that mathematics studies much more than just numbers. Sets, functions, algebraic structures, topological spaces with or without additional structure; none of these are really well described by the word number. --Trovatore (talk) 21:46, 1 August 2009 (UTC)

It might be just as well to move Peirce's definition up higher. That's as close to consensus as anything is likely to come. (A minor suggestion.) Septentrionalis PMAnderson 00:57, 4 August 2009 (UTC)

Also, the line between pure and applied mathematics is not as clear as all that; the seven bridges of Koenigsberg was written as applied mathematics, but is close to the core of topology, which is as pure as math comes - unless Trovatore wants to plug category theory. ;-> Septentrionalis PMAnderson 00:57, 4 August 2009 (UTC)

I understand that mathematics is difficult to define, and that the opening sentence tries to present a sensible summery of it that is generally correct.

However, I think we can improve. Saying that most people who will visit this web page already have some idea of what mathematics is is a cop-out: the best Wikipedia entries make their contents clear to the unfamiliar novice.

Therefore, I propose opening the first sentence to revision. One simple way to define pure mathematics is when we are talking about quantity only, or quantitys of quantitys. If we are talking about quantitys of non-quantitys, e.g. quantitys of change, distance, or money, that is applied mathematics. Simple.

Not so simple. Abstract Algebra. No numbers. Applied math involves math problems resulting from "real world" problems, including physics, chemistry, etc. Pure math involves problems about abstract mathematical structures and relationships - whether or not they relate to "real world" problems. I know. I haven't define math here. Its hard. (John User:Jwy talk) 21:20, 7 August 2009 (UTC)

I say again, this has already been discussed at great length, and all of these suggestions have been made many times before. Rick Norwood (talk) 13:53, 8 August 2009 (UTC)

"Quantity, space, change, structure", the first sentence, and the "fields of mathematics" section

(1) After some thought, I believe the first sentence is not an accurate description of what math is. One might be able to say (in a later sentence) "Most areas of mathematics can be sorted into one or more of the following four general concepts: quantity, space, change, structure". But math is more about how one approaches the study of something, and what one considers a satisfactory answer to a question (or what questions are even allowed). See for example the first paragraph of Science. Science is rightly not described as the study of Physics, chemistry, biology, psychology, etc, because science is about method. Similarly, mathematics is something of the sort. I don't know how to describe it, but it's likely there's references that can address this.

(2) Whether or not you agree with point (1), I'd suggest that the "Fields of mathematics" section be revamped. Though one may argue that "quantity, space, change, structure" is better for laymen than "arithmetic, geometry, analysis, algebra", I think that the fields of mathematics should be subdivided between the branches of mathematics, which are arithmetic, geometry, analysis, algebra, foundations, etc. Because most of the more advanced fields bleed into more than one branch, I'm also not sure that the current organization of this section is adequate. Though I'm not sure what would be a better method. Also, there should be a discussion of what criteria to use to decide which fields of mathematics to include in this section. RobHar (talk) 15:39, 16 August 2009 (UTC)

As an addendum to point (2), the classification is really off. I think it may be because the author(s) attempted to fit facts into a narrative based on "quantity, space, change, structure". Narratives are dangerous. I just moved Vector and Tensor calculus from structure (because somebody found that they allowed "structure" to expand into a the fourth fundamental area of "change"), though they clearly belong in "change" or "space". I placed them in space because "calculus of manifolds" was already mentioned there, but the graphic for "Vector calculus" appears in the "change" section. Similarly, topological groups are a matter of algebra and analysis, not geometry, though it currently occurs in "space". Algebraic geometry is not about space or quantity, it is about structure. This section is a mess. RobHar (talk) 15:54, 16 August 2009 (UTC)

You're right. The first sentence is not an accurate description of mathematics. It is, however, what standard reference works say mathematics is and Wikipedia is not the place to correct standard reference works. The first sentence was constructed using a large number of references, including the Oxford English Dictionary, and any change is unlikely to be acceptable unless you can find a source that is generally considered more authoritative than the OED. Rick Norwood (talk) 20:41, 16 August 2009 (UTC)

Were any secondary sources (rather than tertiary sources) used to construct the lead sentence? Is there a good way to search the talk page archive of this article? Google did not work. RobHar (talk) 21:19, 16 August 2009 (UTC)

A large number of sources of all kinds were used. The definitions of mathematics by non-mathematicians tended to be given more emphasis that I thought best, the definitions of mathematics by mathematicians are relegated to a later sentence.

I pushed for "Mathematics is that body of knowledge arrived at by deduction from axioms." But it didn't fly. Rick Norwood (talk) 12:50, 25 August 2009 (UTC)

Hey, Rick, if you can find a secondary source for it, how about adding your preferred definition to Definitions of mathematics? It could use some good work. —Ben Kovitz (talk) 16:07, 25 August 2009 (UTC)

IMO, the first two sentences are good enough and should be left alone. Clearly, however, many are dissatisfied with it, and understandably so. The main problem, as noted earlier, is the impression given in the first sentence that math is an arbitrary collection of four domains. While most who come to this article will already have an idea of what math is, those who come to read the first paragraph are probably looking to satisfy their confusion over what exactly math is. The Internet is full of people asking, "So what is math really?" I don't think it is such a bad idea to desire to provide a better idea of what unifies math in the opening sentence of this Wikipedia article.

If you want to go to primary sources, then in my experience, mathematicians tend to give one of three answers:

1. Math is the study of patterns, or pattern and structure. This is the definition I prefer, and it really stands alone, though for this article it would probably be best to integrate that with the listing of the four domains for the opening sentence, something like "Mathematics is the study of pattern, especially with respect to quantity, structure, space, and change."

2. Math is the study of abstractions. This one has a lot going for it. It is the definition Wolfram Mathematics goes with. (Yes I know Wolfram is not a good reference, but it is a popular one.)

3. Math is the study of axioms and theorems. This is definitely the worst of the three, for reasons already discussed above. Courant & Robbins in their classic work, What is Mathematics? caution in their introduction against this kind of definition. It is also problematic historically. Other than geometry, math was not really axiomatized until the 19th century. What were non-geometer mathematicians doing until the 19th century if not mathematics? What do mathematicians do today when they first explore a new concept? Axiomatization is undoubtedly the most important and powerful thing to happen to math in the last two hundred years, but it is hardly a proper definition.--seberle (talk) 18:59, 27 September 2009 (UTC)

Common Misconceptions Section

The Common Misconceptions section is poorly written, and portrays a certain sense of bias. Therefore, I think it should be deleted. Anyone have any objections? --Trehansiddharth (talk) 21:26, 22 October 2009 (UTC)

I'd be in favour of its deletion, although it does contain some useful information (e.g. that mathematics is not closed) which might not be obvious to all readers. Maybe it can be merged into a broad part of the article. calr (talk) 21:45, 22 October 2009 (UTC)

The three main ideas I found within the section are:

(1) Mathematics is not a closed intellectual system

(2) Not numerology and is not restricted to arithmatic

(3) Why misconceptions occur

Psuedomathematics can be added to the "See Also" section, ideas (1) and (2) can be added to the introduction, and idea (3) can go in the "Notation, Language and Rigor" section. --Trehansiddharth (talk) 00:31, 24 October 2009 (UTC)

This strikes me as a very good outline. That section has been an irritant for a long time and I'd love to see it dumped; you seem to have a good plan for saving the salvageable. --Trovatore (talk) 02:10, 27 October 2009 (UTC)

Psuedomathematics portion is especially bad. I think someone just made that up off the top of their head. The author seems to define it in terms of any challenge to famous questions. Andy (talk) 23:04, 29 October 2009 (UTC)

Apparently, I can't edit the page because it's semi-protected. Does anyone else have the power to do it?--Trehansiddharth (talk) 18:53, 15 November 2009 (UTC)

See Wikipedia:Semi-protection_policy#Semi-protection. You can request changes here or wait a while. . . (John User:Jwy talk) 19:00, 15 November 2009 (UTC)

Actually, you probably can edit the article now (see WP:autoconfirmed). Paul August ☎ 19:05, 15 November 2009 (UTC)

I edited the page and commented out the Common Misconceptions section. But there's a part in the third paragraph of the Notation, Language, and Rigor section that says "Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics", and I think it needs improvement.--Trehansiddharth (talk) 01:28, 17 November 2009 (UTC)

2 a complex number?

You have 2 listed as a complex number in the image. —

2 is a complex number. It can be written in the form 2 + 0i. Rick Norwood (talk) 22:02, 5 November 2009 (UTC)

Each set in the image is a subset of the sets to the right. Each set contains all the numbers in the sets to the left. By including 2, this makes this inclusion more clear. --seberle (talk) 16:32, 6 November 2009 (UTC)

Citation needed?

Hi folks, the following has a {{fact}} tag...

Further steps needed writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.[citation needed]

But what needs sourcing? I'm a bit confused... - Tbsdy lives (formerly Ta bu shi da yu) ^talk 06:29, 12 September 2009 (UTC)

Part of the problem here is that the sentence is written rather badly, so it's hard to say exactly what needs sourcing (or what the sentence means at all, I'm afraid!) Whatever the main point is (that further steps in mathematical development required writing or other notation systems?), there needs to be a citation to a reliable source that stated this is the case. It isn't enough just that a point makes sense or is logical; it must be sourced. Ideally you'd source both the main point, AND a source stating that the Inca used quipu to store data. (I didn't add the tag btw, just my thoughts). 121.98.130.90 (talk) 02:43, 27 November 2009 (UTC)

"Mathematics is a language"

Yes, mathematics is a language. but mathematics is not just a language. I would like to see the following added:

"Mathematics is a language, method, and body of knowledge ... "

Comments? Rick Norwood (talk) 12:56, 28 October 2009 (UTC)

I have reverted the change entirely. Do we have to go here again? If so, discuss first. --Trovatore (talk) 12:59, 28 October 2009 (UTC)

Mathematics has a language, just as every field does. But that's as much as you can say. Medical researchers use language not familiar to lay people; but that doesn't mean medicine is a language. Astronomers use language not familiar to lay people, but that doesn't mean astronomy is a language. Michael Hardy (talk) 02:53, 29 October 2009 (UTC)

Michael Hardy is correct. Similarly, math has methods and perhaps contains a body of knowledge, but these are insufficient for a definition. I believe the claim that mathematics is a language has been historically supported by some who adopt a formalist view of mathematics, but this is a minority philosophical view and should not be used here. Rick Norwood, you might want to expand on this in the Definitions of mathematics article where different philosophical views are discussed. --seberle (talk) 15:20, 30 October 2009 (UTC)

The philosophy is interesting, but I want this article to be mainstream. "Mathematics is a science and study..." is better than "Mathematics is a language..." but not much better, because science is inductive and mathematics is deductive. A large number of examples can show that water flows downhill, but no number of examples can prove the twin primes conjecture. Rick Norwood (talk) 15:30, 30 October 2009 (UTC)

Actually, I agree with you. I am not happy about the use of the word "science" in the opening sentence. I think "study" would be sufficient. But this has apparently been debated quite a bit in archived discussions of this page (though I don't see a clear consensus on this point—did I miss it?) and apparently this is the wording that was decided. Fortunately, there is a satisfying discussion of whether math is a "science" later in the article. --seberle (talk) 16:38, 30 October 2009 (UTC)

I don't believe there was ever a real consensus for science. I think someone slipped that in at a point when people were tired of arguing about it. While I personally am fine with science, I think it would probably be better to put that back to the status quo ante, as it's a controversial point, and also because it's not the outcome of the long painful argument. --Trovatore (talk) 19:39, 30 October 2009 (UTC)

Yes I think Trovatore is correct about the status of "science". I think it should probably be removed. Paul August ☎ 19:44, 30 October 2009 (UTC)

I do not like including "science" for stylistic reasons. As the "Mathematics as science" section makes clear (I think), math is not a science in the sense of a natural science or experimental science, but it is a science in the original sense of a "field of knowledge" or study. But this means that "science and study" is redundant (and possibly confusing). Omitting the word science in the opening sentence does not mean Wikipedia is decreeing that math is not a science; it is just putting off the debate until another section. (A second reason for omitting science in the first sentence is that this is a debated term and should not be included in the opening definition.) If the decision is made to retain "science", please consider rewording in a way that does not sound redundant, such as "Mathematics is the science that studies ..." --seberle (talk) 15:39, 2 November 2009 (UTC)

Well, no, the section presents differing viewpoints on the question of whether mathematics is a science in those senses. As I have expressed elsewhere, my position is that mathematics is indeed an experimental science. But the point being controversial, the first sentence should probably not appear to take a position on it. --Trovatore (talk) 20:56, 2 November 2009 (UTC)

Mathematics is certainly not a science in my view. Firstly, mathematics is an axiomatic theory; that is, a mathematican defines truth, and deduces further truth. In science, the "axioms" may change as new theories develop since the axioms must usually be verified by experiment in science. Secondly, mathematics is not done solely for science (lay readers may interpret "mathematics is a science" in this way, as many are already under this misconception). Branches of mathematics such as general topology or number theory are not done for science only. Therefore, I think that we should not make links to science until later in the article. --PS T 00:59, 3 November 2009 (UTC)

This is not really the right forum for discussing whether math is a science, but just to clarify my position: as a mathematical realist I consider mathematics to be about objects that exist independently of our reasoning about them, and therefore the truths about them do not derive from the axioms; rather the axioms must be chosen so that they are true. Figuring out which ones are correct (for example, large cardinal axioms) as opposed to the ones that are wrong (for example, the axiom of constructibility) is a scientific, and partly empirical, endeavor, and is definitely part of mathematics. --Trovatore (talk) 01:13, 3 November 2009 (UTC)

I think a strong argument could be made that math is an experimental science, but I did not see this in the "Mathematics as science" section. I am not at ease with the purely axiomatic view of math. It would be good to hear more about this, but Trovatore's link to "mathematical realist" does not go anywhere. I am glad for the revert to "study" for now, if only for reasons of style and clarity. But it would be good to understand this further, perhaps with some links to other references? Or is everyone tired of this debate? --seberle (talk) 03:08, 3 November 2009 (UTC)

Oh, I can argue about it endlessly — this just isn't really the right place. --Trovatore (talk) 03:10, 3 November 2009 (UTC)

The introduction should use a neutral word. I think "Mathematics is the study of..." is fairly neutral. Let's leave it at that unless there is a consensus on some other word. Rick Norwood (talk) 16:15, 3 November 2009 (UTC)

I agree completely. The lower part of the article can discuss the "science" term at length, as it should be. — Carl (CBM · talk) 21:38, 3 November 2009 (UTC)

"strict syntax"?

Under "Notation, language, rigor", there's a statement "modern mathematical notation has a strict syntax". I'd like to see a citation to back this up, or indeed a link to a wp page describing "the" strict syntax, especially since there's a link for musical notation. In my experience there are a variety of different notations used in math, often varying within fields, between authors in the same field, and sometimes even on the same page of exposition. Gwideman (talk) 14:42, 11 December 2009 (UTC)

I think the idea being expressed here is not that there is one and only one strict syntax for all of mathematics but rather that the syntax, whatever it may be, is strict, and you cannot, for example, write x+1^n when you mean (x+1)^n. It could probably be expressed more clearly. Rick Norwood (talk) 20:09, 11 December 2009 (UTC)

Perhaps "precise" would be a better word than "strict"?Paul August ☎ 20:15, 11 December 2009 (UTC)

I agree that the objective is to convey an idea precisely from author to reader. And agreed that within a subset of notation, the writer has to write it in accordance with that idea, and the reader has to read it the same way. However there are all sorts of ambiguities: Should one write d/dx, or use prime, or a dot? If one reads x', does that mean derivative of x, or a second variable that may or may not be related to an x somewhere else? What does a caret ("hat") mean? Are square brackets an array subscript or an operator of some kind? Does the author's choice of beginning- and end-of-alphabet letters (a, b, c vs x, y, z) distinguish coefficients from variables, or not? What about i, j, k: Variables? Subscript? Sqrt(-1)? Much relies on context and upon the reader being familiar with the author's mathematical "culture". Which is to say, the notation not strict or precise according to some consistent standard, but relies on the reader divining which convention the author is using, or sometimes even what convention the author has invented on the spot.

Anyhow, I'm certainly not qualified to characterize this -- I was hoping someone would point to a reference or two on the subject! Gwideman (talk) 02:22, 12 December 2009 (UTC)

Calculations, measurement, arithmetic; not part of mathematics?

The following sentence is pasted directly from the lead: "Although incorrectly considered part of mathematics by many, calculations and measurement are features of accountancy and arithmetic."

I am not an authority on mathematics, so perhaps I fall under the category of the many who make incorrect assumptions according to that statement, but the arithmetic lead on this same wikipedia mentions that subject specifically as being "the oldest and most elementary branch of mathematics".

There is definitely a contradiction here.

Measurement too might be argued to be a branch of mathematics (i.e. geometry ("earth-measuring")); Euclid's Elements, which is a treatise on geometry, is specifically mentioned in the lead as an example of mathematics. Zalmoxe (talk) 16:12, 24 December 2009 (UTC)

There have been several changes for the worse in the lede, including the sentence you mention and another sentence claiming that numerology is mathematics. I've tried to restore things to the stable lede that has existed for a long time. Rick Norwood (talk) 13:47, 6 January 2010 (UTC)

Thank you Rick. This new old lede [interesting ortographic revival there btw, I wasn't aware of it until now] seems more consistent. Zalmoxe (talk) 05:11, 18 January 2010 (UTC)

Does mathematics study the shapes and motions of physical objects or of abstractions?

To say that mathematics studies physical objects is misleading. Of course, mathematics is applied to the physics of motion, but the mathematics is first developed with reference to abstract shapes, such as triangles, before considering questions such as the irregularities, imperfections, and discontinuities of any physical triangle. Mathematics first considers ideal motion, usually of a point mass not subject to friction, air-resistance, or uncertainty, before considering all of the messy reality of the physics of actual motion in the real world.

Which should the lede state, abstract objects or physical objects?

Rick Norwood (talk) 14:45, 6 January 2010 (UTC)

I agree that saying that mathematics "studies studies physical objects is misleading". But the article does not say that. What it says is that "mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects" — that's quite a different thing. The idea in this sentence is to give a sense of where mathematics came from, not what it is. Paul August ☎ 17:05, 7 January 2010 (UTC)

Good point! Rick Norwood (talk) 12:55, 8 January 2010 (UTC)

Citation for maths vs. math

I am not a registered user, but can somebody find a citation for maths vs. math. I'm an American having an argument with a British friend over whether mathematics is plural or singular. I say it is singular, therefore mathematics should be shortened to math. But he insists that mathematics refers to a diversity of strands and is therefore plurally maths. —Preceding unsigned comment added by 137.205.222.238 (talk) 13:08, 13 January 2010 (UTC)

As well argue over the spelling of "color"/"colour". They do it one way in the US and a different way in England, and there is no right or wrong. The main reason to use "math" is that it gave Tom Lehrer a rhyme. "The guy who taught us math, who never took a bath, acquired a certain measure of renown..." Rick Norwood (talk) 15:06, 13 January 2010 (UTC)

I've read a topic on this, and the whole math/maths thing can be summed up to a) the Germans and/or the French as American math developed from German mathematik and French mathematiques (s is silent) b) the fact that even in britain "Maths IS my favorite subject" Math is while a plural noun an uncountable noun and treated as singular, because you'd never say "Maths ARE my favorite subject" as subject would have to plural as well and Mathematics is a subject and not subjects.96.3.141.210 (talk) 19:08, 17 January 2010 (UTC)

And what about "physics", which is grammatically similar but is "la physique" in French (also "mécanique" is considered a distinct subject here, but I doubt whether that explains the singular). Too bad nobody abbreviates it to "phys", and even then, this wouldn't help... Marc van Leeuwen (talk) 10:44, 25 January 2010 (UTC)

Throw physic to the dogs; — I'll none of it. GDallimore (Talk) 01:16, 4 June 2010 (UTC)

Awards and Prizes in Mathematics

This should include the William Lowell Putnam Mathematical Competition for college undergraduates in the US and Canada. —Preceding unsigned comment added by 166.82.218.97 (talk • contribs)

Debate about whether numbers exist naturally

To those of you who can't understand or refuse to believe that mathematics is not a human invention but a Universal property: Where did the capacity for humans to think mathematically come from? Did humans invent the brain mechanisms that recognize mathematical and logical truth? Obviously, no. Does human mathematical thought require mathematical truth to exist as a prerequisite? Obviously, yes; besides the fact that our brains operate according to the laws of physics which are themselves embodiments of mathematical truth, there would be no way to reach mathematical conclusions without mathematical brains. Seven is not a prime number because people decided it should be divisible only by itself and 1, humans recognize it as prime because it is logically found to be divisible by itself and 1. Some might try to argue that curiosities like this are consequences of the base-10 system of numbers, but no matter what system is used, the primes are still prime, the squares are still square, pi is still pi, and so on. A musical major triad sounds the way it does not because human ingenuity invented a pleasing harmony, but because the sound waves' frequencies mathematically correspond in whole number ratios, which our naturally logical brains recognize as pure (5:4 between third and root, 6:5 between fifth and third, and 3:2 between the fifth and the root). The examples are endless because everything that exists arises from the foundation of cosmic logic, undying truth. -Mcgriggin —Preceding unsigned comment added by 66.32.130.224 (talk • contribs) 20:32, 1 July 2010

This is not the place to argue/discuss mathematics/philosophy - that is original research. It is a place to report what reliable sources have to say about the topic. --John (User:Jwy/talk) 18:51, 2 July 2010 (UTC)

Sorry if this subject has already been discussed (I tried to check) but I find the second paragraph of the lede utter nonsense:

There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".[5] Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]

Maybe my problem is that I misinterpret "exist naturally"; I can't think of any other meaning than "exist in nature", and in that case I find it hard to imagine any serious debate about the question: numbers and other mathematical abstractions do not exist in nature. I will admit the existence of a black hole at the other end of the galaxy, but not that of a complex number (or natural number for that matter, say 0:-) in my back yard. This is not to say that mathematical abstractions are (uniquely) human inventions, I would expect any extraterrestrial civilisation to come up with the same, or very similar, abstractions.

The two citations do not express opposing views either. Mathematics draws necessary conclusions, but those conclusions apply to the object of mathematics, that is to abstractions. Such conclusions only apply to reality insofar as reality is willing to abide by the laws of mathematical abstractions; since this is not certain, neither is the application of conclusions to reality, which is what Einstein appears to say. Marc van Leeuwen (talk) 10:23, 25 January 2010 (UTC)

You need to inform yourself that although one can follow and believe some philosophical line there are others. Take a look at Philosophy of mathematics, in particular to the section Mathematical realism. franklin 11:31, 25 January 2010 (UTC)
Also what one can understand by "exists", or "nature", or "naturally", or even "exists naturally" have many many different possible answers. That paragraph is pointing to the problem of the ontological existence of mathematical objects. franklin 11:58, 25 January 2010 (UTC)

I'm not taking any particular philosophical position, I'm just saying this paragraph is not making any clear sense. If "existing naturally" is a reference to one or various ontological positions, this should be made clear. I would have less difficulty with "There is debate over the kind of existence, if any, that can be attributed to mathematical abstractions such as numbers", as it more clearly indicates that the discussion is about "existence", rather than about numbers themselves. Also I do not believe that the two citations belong to the ontological debate you refer to. Peirce does not refer to existence or reality at all, and Einstein most probably (as a physicist) is talking about physical reality, not some kind of Platonic reality. So I'm just saying this paragraph is lousy. Marc van Leeuwen (talk) 12:56, 25 January 2010 (UTC)

Well, you starting saying that numbers do not exist in nature and that you expect extraterrestrials to come out with the same abstractions (which is a valid but still very particular philosophical position) as your point for labeling the paragraph as utter nonsense but let's forget that you did. About other ways of writing this paragraph, which is one in the lede of a very general article, there are probably hundreds of possibilities. If you bring some alternative it can be seen if it is a better choice. I don't find the current version a bad one as holes can always be found in three lines about a topic that takes volumes to explain. The two quotations, although there are probably several alternatives, refer to problems linked to the ontology of mathematical abstractions. Peirce's key word is "necessary" and the level in which it is necessary, and Einsteins' to to a possibly antagonistic position again depending what we take as the meaning of "reality". This topics we very hot at Einsteins time and environment. He and Godel probably talked about these things a lot. I wouldn't dare to narrow what he really meant with "reality" there. Again, several options for quotations are available. If you bring some options it can be seen whether they are more suitable than the current ones. One thing that can help is a link inside that paragraph to a more specific article explaining all this. An option is philosophy of mathematics. franklin 13:59, 25 January 2010 (UTC)

At worst, numbers exist naturally as singularly probable singularities in nonlinear dynamical systems that exist naturally. thus, to say that they don't exist naturally would be like saying that the non-trivial zeros of the Riemann zeta-function don't exist. Kevin Baas^talk 16:21, 26 January 2010 (UTC)

No better explanation exists. franklin 17:30, 26 January 2010 (UTC)

Not so, I came up with a clearer one: Consider the vibration of a string. Because of certain mathematical relationships, the harmonics will always be at 2x,3x,4x,5x, etc. of the base frequency. Any thing not a natural number multiple would only dampen or eliminate the vibration. Since "1" is defined simply as "There is a number one.", to say that "natural numbers exist naturally" one only needs to show that there's some kind of physical law that strongly prefers natural number multiples. And that is precisely what we just showed. As they say in the industry: Q.E.D., b@%&@! Kevin Baas^talk 16:10, 29 January 2010 (UTC)

I was being ironic, sorry. You need to read a little more about this topic. I guess you can start reading philosophy of mathematics and then whatever related book you can find. franklin 17:51, 29 January 2010 (UTC)

Actually for any physical string (and any finite amplitude of vibration) the harmonics will not be exactly as 2x, 3x etc, because of parameters like stiffness of the string that are ignored in the mathematical model of the string. Does that show that natural numbers are not naturally exact integers? No, it just shows that this particular physical problem does not have the exact properties of the mathematical model.

I.e. regardless of the correctness or incorrectness of the diff eq. describing the model, there are more concurrent diff. processes at work. That doesn't diminish my point.

My point was that, so long as there is positive and negative feedback, periodic relationships will inexorably emerge from natural phenomena. And what can "numbers exist naturally" mean if not precisely that? But I digress --- just wanted to make sure my meaning was clear. Kevin Baas^talk

But enough of this; in spite of my initial somewhat provocative language, I did want to pose a serious question, not evoke a philosophical debate. The first sentence of the paragraph is not clearly formulated; at best it indicates a somewhat esoteric philosophical debate that I think does not deserve to be mentioned in the lede of an article on mathematics. The citation by Peirce is not about ontological questions, but an introduction to broadening the sense of the term "mathematics" to more than purely quantitative questions (notably he mentions quaternions as not being covered by that); while understandable in the late 19-th century context, such broadening is no longer relevant since it has been completely integrated into mathematics already. Einstein's quote may be pertinent, but is more about the role of mathematics in the sciences than about the philosophy of mathematics itself. Altogether, the paragraph seems less than helpful to readers who want to learn about mathematics. Marc van Leeuwen (talk) 16:51, 29 January 2010 (UTC)

Well, about the form of the paragraph I agree about the possibility of improving it. About the pertinence of having such an information in the lede i think it is more than indispensable. This is the main article about a science or a discipline ( or whatever mathematics is). The first thing you do when you start to study a science is to define its subject of study. In the case of the other sciences, although it is equally difficult to define the subject for most of the people is clear (at least intuitively) what they are. Take for example Biology, it studies living beings. Although it is very hard to define (appropriately) what is a living being it is mush more clear (at least intuitively) what they are. In the case of mathematics, even from a materialist perspective the nature of the subject matter of the science (if it is a science) is less clear. I guess I don't have to point out what is the importance of defining the subject matter of a science but let me just say that that defines its purpose and applicability which then determines its evolution (for example if it develops more as a game like chess, or to things with direct application or both). For example compare the volumes of publications in Physics and Mathematics that are about situations that do not correspond to reality. This example is interesting since these two sciences are very similar in many of the techniques they use. In Physics you always have the oracle of "reality" that is the ultimate verification. As well as in Mathematics it is valid the question about what is the nature of this "real world", and whether it even exists. Now in mathematics the objects have a different nature. There is also no such "oracle" validating the results or even the axioms. that is why the question becomes even more important in this case. franklin 17:51, 29 January 2010 (UTC)

Newton's picture

The use of Newton is needlessly anglophilic. —Preceding unsigned comment added by 129.120.193.30 (talk) 23:08, 16 February 2010 (UTC)

Isaac Newton is almost universally acknowledged as one of the greatest mathematicians of all time. To omit him would be anglophobic. Rick Norwood (talk) 13:43, 23 March 2010 (UTC)

^ Waltershausen

[1] Waltershausen

[1]