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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)
"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)
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)
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
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)
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.
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
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:
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:
Check the articles on science and formal science for more information. Aetherealize ( talk) 09:33, 20 February 2008 (UTC)
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 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)
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I think it is best to view the three images in context. So we have:
Original:
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Mathematical physics | Mathematical fluid dynamics | Numerical analysis | Optimization | Probability | Statistics | Financial mathematics | Game theory |
Current:
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Mathematical physics | Mathematical fluid dynamics | Numerical analysis | Optimization | Probability | Statistics | Financial mathematics | Game theory |
Proposed:
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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)
(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)
Purely a question of aesthetics. Preferences? Suggestions? -- Cronholm 144 03:01, 19 August 2007 (UTC)
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)
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)
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)
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)
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)
On Jwy's points, I regret that I was unclear, concerning which let me try to do better:
I'm not convinced:
I think these items are enough to justify a slight awkwardness. (John User:Jwy talk) 07:22, 18 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)
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)
On the Wikipedia:Redirect page, under the heading What needs to be done on pages that are targets of redirects?, we find:
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)
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)
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)
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:
The revert-Edit that it replaced was:
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 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)
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)
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))
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)
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))
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)
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)
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.)
Additional comments and/or "votes" for at least 1 of CD and D1-D5 above as indicated or here:
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)
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)
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!
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)
This has become an absurd waste of time for everyone concerned. Rick Norwood ( talk) 13:57, 22 November 2007 (UTC)
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)
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)
The article currently describes "mathematics" as a singular noun. But surely it is uncountable isn't it? (Therefore neither singular nor plural) - EstoyAquí( t • c • e) 17:20, 9 December 2007 (UTC)
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:
Joshua R. Davis ( talk) 21:40, 12 December 2007 (UTC)
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)
(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)
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)
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)
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)
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)
Surely it's time for this article to be unprotected? 86.27.59.185 ( talk) 23:42, 30 January 2008 (UTC)
—Preceding unsigned heading added by 76.22.155.72 ( talk • contribs) 09:28, February 1, 2008 (UTC)
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:
Mathematics is only relevant and pertinent in the context of life.
Between these 2 - the life and the names lies the whole mathematics.
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.
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)
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)
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)
(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)
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)
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)
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)
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)
dMMPR VS PRDT episode 1/8 "1st Fight" —Preceding unsigned comment added by 71.190.84.21 ( talk) 20:09, 23 August 2008 (UTC)
maths is a type of science —Preceding unsigned comment added by 84.69.67.194 ( talk) 08:50, 14 September 2008 (UTC)
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)
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)(Text of Wikipedia criticism from "Wilbourhall" site removed to facilitate discussion)
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)
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)
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.
Here thar be trolls |
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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)
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)
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)
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)
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) |
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)
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)
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)
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)
Here are some definitions:
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)
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)
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)
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)
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)
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)
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)
"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)
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)
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)
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)
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)
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)
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:
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)
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)
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)
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)
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
OK, a few points in no particular order:
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)
{{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)
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?
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)
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)
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
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)
Please leave any article on my watchlist alone. Septentrionalis PMAnderson 19:39, 3 August 2009 (UTC)
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
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.
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)
(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)
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)
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)
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)
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 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)
Hi folks, the following has a {{ fact}} tag...
But what needs sourcing? I'm a bit confused... - Tbsdy lives (formerly Ta bu shi da yu) talk 06:29, 12 September 2009 (UTC)
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)
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 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)
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)
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)
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)
Good point! Rick Norwood ( talk) 12:55, 8 January 2010 (UTC)
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)
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)
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
Sorry if this subject has already been discussed (I tried to check) but I find the second paragraph of the lede utter nonsense:
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)
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)
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.
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)
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)
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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)
"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)
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)
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
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)
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.
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
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:
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:
Check the articles on science and formal science for more information. Aetherealize ( talk) 09:33, 20 February 2008 (UTC)
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 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)
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I think it is best to view the three images in context. So we have:
Original:
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Mathematical physics | Mathematical fluid dynamics | Numerical analysis | Optimization | Probability | Statistics | Financial mathematics | Game theory |
Current:
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Mathematical physics | Mathematical fluid dynamics | Numerical analysis | Optimization | Probability | Statistics | Financial mathematics | Game theory |
Proposed:
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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)
(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)
Purely a question of aesthetics. Preferences? Suggestions? -- Cronholm 144 03:01, 19 August 2007 (UTC)
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)
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)
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)
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)
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)
On Jwy's points, I regret that I was unclear, concerning which let me try to do better:
I'm not convinced:
I think these items are enough to justify a slight awkwardness. (John User:Jwy talk) 07:22, 18 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)
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)
On the Wikipedia:Redirect page, under the heading What needs to be done on pages that are targets of redirects?, we find:
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)
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)
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)
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:
The revert-Edit that it replaced was:
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 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)
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)
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))
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)
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))
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)
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)
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.)
Additional comments and/or "votes" for at least 1 of CD and D1-D5 above as indicated or here:
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)
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)
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!
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)
This has become an absurd waste of time for everyone concerned. Rick Norwood ( talk) 13:57, 22 November 2007 (UTC)
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)
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)
The article currently describes "mathematics" as a singular noun. But surely it is uncountable isn't it? (Therefore neither singular nor plural) - EstoyAquí( t • c • e) 17:20, 9 December 2007 (UTC)
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:
Joshua R. Davis ( talk) 21:40, 12 December 2007 (UTC)
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)
(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)
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)
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)
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)
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)
Surely it's time for this article to be unprotected? 86.27.59.185 ( talk) 23:42, 30 January 2008 (UTC)
—Preceding unsigned heading added by 76.22.155.72 ( talk • contribs) 09:28, February 1, 2008 (UTC)
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:
Mathematics is only relevant and pertinent in the context of life.
Between these 2 - the life and the names lies the whole mathematics.
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.
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)
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)
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)
(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)
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)
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)
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)
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)
dMMPR VS PRDT episode 1/8 "1st Fight" —Preceding unsigned comment added by 71.190.84.21 ( talk) 20:09, 23 August 2008 (UTC)
maths is a type of science —Preceding unsigned comment added by 84.69.67.194 ( talk) 08:50, 14 September 2008 (UTC)
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)
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)(Text of Wikipedia criticism from "Wilbourhall" site removed to facilitate discussion)
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)
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)
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.
Here thar be trolls |
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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)
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)
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)
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)
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) |
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)
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)
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)
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)
Here are some definitions:
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)
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)
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)
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)
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)
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)
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)
"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)
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)
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)
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)
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)
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)
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:
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)
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)
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)
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)
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
OK, a few points in no particular order:
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)
{{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)
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?
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)
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)
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
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)
Please leave any article on my watchlist alone. Septentrionalis PMAnderson 19:39, 3 August 2009 (UTC)
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
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.
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)
(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)
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)
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)
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)
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 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)
Hi folks, the following has a {{ fact}} tag...
But what needs sourcing? I'm a bit confused... - Tbsdy lives (formerly Ta bu shi da yu) talk 06:29, 12 September 2009 (UTC)
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)
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 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)
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)
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)
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)
Good point! Rick Norwood ( talk) 12:55, 8 January 2010 (UTC)
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)
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)
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
Sorry if this subject has already been discussed (I tried to check) but I find the second paragraph of the lede utter nonsense:
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)
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)
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.
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)
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)