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== Awards and Prizes in Mathematics == |
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This should include the [[William Lowell Putnam Mathematical Competition]] for college undergraduates in the US and Canada. <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:166.82.218.97|166.82.218.97]] ([[User talk:166.82.218.97|talk]] • [[Special:Contributions/166.82.218.97|contribs]]) </span></small><!-- Template:Unsigned --> |
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== Math Causes Cancer!!! == |
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== |
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Hi folks, the following has a {{tl|fact}} tag... |
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:Further steps needed writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.[citation needed] |
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But what needs sourcing? I'm a bit confused... - [[User:Tbsdy lives|Tbsdy lives]] (formerly [[User:Ta bu shi da yu|Ta bu shi da yu]]) <sup>[[User talk:Tbsdy lives|talk]]</sup> 06:29, 12 September 2009 (UTC) |
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:Part of the problem here is that the sentence is written rather badly, so it's hard to say exactly what needs sourcing (or what the sentence means at all, I'm afraid!) Whatever the main point is (that further steps in mathematical development required writing or other notation systems?), there needs to be a citation to a reliable source that stated this is the case. It isn't enough just that a point makes sense or is logical; it must be sourced. Ideally you'd source both the main point, AND a source stating that the Inca used quipu to store data. (I didn't add the tag btw, just my thoughts). [[Special:Contributions/121.98.130.90|121.98.130.90]] ([[User talk:121.98.130.90|talk]]) 02:43, 27 November 2009 (UTC) |
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== "Mathematics is a language" == |
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Yes, mathematics is a language. but mathematics is not just a language. I would like to see the following added: |
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"Mathematics is a language, method, and body of knowledge ... " |
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Comments? [[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 12:56, 28 October 2009 (UTC) |
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:I have reverted the change entirely. Do we have to go here again? If so, discuss ''first''. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 12:59, 28 October 2009 (UTC) |
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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. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 02:53, 29 October 2009 (UTC) |
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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. --[[User:Seberle|seberle]] ([[User talk:Seberle|talk]]) 15:20, 30 October 2009 (UTC) |
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:The philosophy is interesting, but I want this article to be mainstream. "Mathematics is a science and study..." is better than "Mathematics is a language..." but not much better, because science is inductive and mathematics is deductive. A large number of examples can show that water flows downhill, but no number of examples can prove the twin primes conjecture. [[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 15:30, 30 October 2009 (UTC) |
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::Actually, I agree with you. I am not happy about the use of the word "science" in the opening sentence. I think "study" would be sufficient. But this has apparently been debated quite a bit in archived discussions of this page (though I don't see a clear consensus on this point—did I miss it?) and apparently this is the wording that was decided. Fortunately, there is a satisfying discussion of whether math is a "science" [[Mathematics#Mathematics as science|later]] in the article. --[[User:Seberle|seberle]] ([[User talk:Seberle|talk]]) 16:38, 30 October 2009 (UTC) |
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:::I don't believe there was ever a real consensus for ''science''. I think someone slipped that in at a point when people were tired of arguing about it. While I personally am fine with ''science'', I think it would probably be better to put that back to the status quo ante, as it's a controversial point, and also because it's not the outcome of the long painful argument. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:39, 30 October 2009 (UTC) |
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::::Yes I think Trovatore is correct about the status of "science". I think it should probably be removed. [[User:Paul August|Paul August]] [[User_talk:Paul August|☎]] 19:44, 30 October 2009 (UTC) |
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::::I do not like including "science" for stylistic reasons. As the "[[Mathematics#Mathematics as science|Mathematics as science]]" section makes clear (I think), math is not a science in the sense of a natural science or experimental science, but it ''is'' a science in the original sense of a "field of knowledge" or study. But this means that "science and study" is redundant (and possibly confusing). Omitting the word science in the opening sentence does not mean Wikipedia is decreeing that math is not a science; it is just putting off the debate until another section. (A second reason for omitting science in the first sentence is that this is a debated term and should not be included in the opening definition.) If the decision is made to retain "science", please consider rewording in a way that does not sound redundant, such as "Mathematics is the science that studies ..." --[[User:Seberle|seberle]] ([[User talk:Seberle|talk]]) 15:39, 2 November 2009 (UTC) |
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:::::Well, no, the section presents differing viewpoints on the question of whether mathematics is a science in those senses. As I have expressed elsewhere, my position is that mathematics is indeed an experimental science. But the point being controversial, the first sentence should probably not appear to take a position on it. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:56, 2 November 2009 (UTC) |
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::::::Mathematics is certainly ''not'' a science in my view. Firstly, mathematics is an axiomatic theory; that is, a mathematican defines truth, and deduces ''further truth''. In science, the "axioms" may change as new theories develop since the axioms must usually be verified by experiment in science. Secondly, mathematics is not done solely for science (lay readers may interpret "mathematics is a science" in this way, as many are already under this misconception). Branches of mathematics such as general topology or number theory are not done for science only. Therefore, I think that we should not make links to science until later in the article. --[[User:Point-set topologist|<font color="#000000">PS</font>]][[User talk:Point-set topologist|<font color="#000000">T</font>]] 00:59, 3 November 2009 (UTC) |
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:::::::This is not really the right forum for discussing whether math is a science, but just to clarify my position: as a [[mathematical realist]] I consider mathematics to be about objects that exist independently of our reasoning about them, and therefore the truths about them do not derive from the axioms; rather the axioms must be chosen so that they are true. Figuring out which ones are correct (for example, [[large cardinal axiom]]s) as opposed to the ones that are wrong (for example, the [[axiom of constructibility]]) is a scientific, and partly empirical, endeavor, and is definitely part of mathematics. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 01:13, 3 November 2009 (UTC) |
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::::::::I think a strong argument could be made that math is an experimental science, but I did not see this in the "Mathematics as science" section. I am not at ease with the purely axiomatic view of math. It would be good to hear more about this, but Trovatore's link to "mathematical realist" does not go anywhere. I am glad for the revert to "study" for now, if only for reasons of style and clarity. But it would be good to understand this further, perhaps with some links to other references? Or is everyone tired of this debate? --[[User:Seberle|seberle]] ([[User talk:Seberle|talk]]) 03:08, 3 November 2009 (UTC) |
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:::::::::Oh, I can argue about it endlessly — this just isn't really the right place. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 03:10, 3 November 2009 (UTC) |
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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. [[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 16:15, 3 November 2009 (UTC) |
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:I agree completely. The lower part of the article can discuss the "science" term at length, as it should be. — Carl <small>([[User:CBM|CBM]] · [[User talk:CBM|talk]])</small> 21:38, 3 November 2009 (UTC) |
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== "strict syntax"? == |
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Under "Notation, language, rigor", there's a statement "modern mathematical notation has a strict syntax". |
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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. |
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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. [[User:Gwideman|Gwideman]] ([[User talk:Gwideman|talk]]) 14:42, 11 December 2009 (UTC) |
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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. [[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 20:09, 11 December 2009 (UTC) |
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:Perhaps "precise" would be a better word than "strict"?[[User:Paul August|Paul August]] [[User_talk:Paul August|☎]] 20:15, 11 December 2009 (UTC) |
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::I agree that the objective is to convey an idea precisely from author to reader. And agreed that within a subset of notation, the writer has to write it in accordance with that idea, and the reader has to read it the same way. However there are all sorts of ambiguities: Should one write d/dx, or use prime, or a dot? If one reads x', does that mean derivative of x, or a second variable that may or may not be related to an x somewhere else? What does a caret ("hat") mean? Are square brackets an array subscript or an operator of some kind? Does the author's choice of beginning- and end-of-alphabet letters (a, b, c vs x, y, z) distinguish coefficients from variables, or not? What about i, j, k: Variables? Subscript? Sqrt(-1)? Much relies on context and upon the reader being familiar with the author's mathematical "culture". Which is to say, the notation not strict or precise according to some consistent standard, but relies on the reader divining which convention the author is using, or sometimes even what convention the author has invented on the spot. |
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::Anyhow, I'm certainly not qualified to characterize this -- I was hoping someone would point to a reference or two on the subject! [[User:Gwideman|Gwideman]] ([[User talk:Gwideman|talk]]) 02:22, 12 December 2009 (UTC) |
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== Calculations, measurement, arithmetic; not part of mathematics? == |
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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''." |
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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". |
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There is definitely a contradiction here. |
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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. [[User:Zalmoxe|Zalmoxe]] ([[User talk:Zalmoxe|talk]]) 16:12, 24 December 2009 (UTC) |
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::There have been several changes for the worse in the lede, including the sentence you mention and another sentence claiming that numerology is mathematics. I've tried to restore things to the stable lede that has existed for a long time. [[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 13:47, 6 January 2010 (UTC) |
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:::Thank you Rick. This new old lede [interesting ortographic revival there btw, I wasn't aware of it until now] seems more consistent. [[User:Zalmoxe|Zalmoxe]] ([[User talk:Zalmoxe|talk]]) 05:11, 18 January 2010 (UTC) |
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== Does mathematics study the shapes and motions of physical objects or of abstractions? == |
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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. |
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Which should the lede state, abstract objects or physical objects? |
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[[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 14:45, 6 January 2010 (UTC) |
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:I agree that saying that mathematics "studies studies physical objects is misleading". But the article does not say that. What it says is that "mathematics '''evolved from''' counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects" — that's quite a different thing. The idea in this sentence is to give a sense of where mathematics came from, not what it is. [[User:Paul August|Paul August]] [[User_talk:Paul August|☎]] 17:05, 7 January 2010 (UTC) |
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Good point! [[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 12:55, 8 January 2010 (UTC) |
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== Citation for maths vs. math == |
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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. <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/137.205.222.238|137.205.222.238]] ([[User talk:137.205.222.238|talk]]) 13:08, 13 January 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot--> |
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:As well argue over the spelling of "color"/"colour". They do it one way in the US and a different way in England, and there is no right or wrong. The main reason to use "math" is that it gave [[Tom Lehrer]] a rhyme. "The guy who taught us math, who never took a bath, acquired a certain measure of renown..." [[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 15:06, 13 January 2010 (UTC) |
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::I've read a topic on this, and the whole math/maths thing can be summed up to a) the Germans and/or the French as American math developed from German mathematik and French mathematiques (s is silent) b) the fact that even in britain "Maths IS my favorite subject" Math is while a plural noun an uncountable noun and treated as singular, because you'd never say "Maths ARE my favorite subject" as subject would have to plural as well and Mathematics is a subject and not subjects.[[Special:Contributions/96.3.141.210|96.3.141.210]] ([[User talk:96.3.141.210|talk]]) 19:08, 17 January 2010 (UTC) |
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:::And what about "physics", which is grammatically similar but is "la physique" in French (also "mécanique" is considered a distinct subject here, but I doubt whether that explains the singular). Too bad nobody abbreviates it to "phys", and even then, this wouldn't help... [[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen|talk]]) 10:44, 25 January 2010 (UTC) |
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== Debate about whether numbers exist naturally == |
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Sorry if this subject has already been discussed (I tried to check) but I find the second paragraph of the lede utter nonsense: |
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:There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".[5] Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6] |
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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. |
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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. [[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen|talk]]) 10:23, 25 January 2010 (UTC) |
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* You need to inform yourself that although one can follow and believe some philosophical line there are others. Take a look at [[Philosophy of mathematics]], in particular to the section '''Mathematical realism'''. <small><span style="border:1px solid black;padding:0.5px;">[[User:franklin.vp|<font style="color:black;background:white;font-family:sans-serif;" >''' franklin '''</font>]]</span></small> 11:31, 25 January 2010 (UTC) |
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* Also what one can understand by "exists", or "nature", or "naturally", or even "exists naturally" have many many different possible answers. That paragraph is pointing to the problem of the ontological existence of mathematical objects. <small><span style="border:1px solid black;padding:0.5px;">[[User:franklin.vp|<font style="color:black;background:white;font-family:sans-serif;" >''' franklin '''</font>]]</span></small> 11:58, 25 January 2010 (UTC) |
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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. [[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen|talk]]) 12:56, 25 January 2010 (UTC) |
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* Well, you starting saying that numbers do not exist in nature and that you expect extraterrestrials to come out with the same abstractions (which is a valid but still very particular philosophical position) as your point for labeling the paragraph as utter nonsense but let's forget that you did. About other ways of writing this paragraph, which is one in the lede of a very general article, there are probably hundreds of possibilities. If you bring some alternative it can be seen if it is a better choice. I don't find the current version a bad one as holes can always be found in three lines about a topic that takes volumes to explain. The two quotations, although there are probably several alternatives, refer to problems linked to the ontology of mathematical abstractions. Peirce's key word is "necessary" and the level in which it is necessary, and Einsteins' to to a possibly antagonistic position again depending what we take as the meaning of "reality". This topics we very hot at Einsteins time and environment. He and Godel probably talked about these things a lot. I wouldn't dare to narrow what he really meant with "reality" there. Again, several options for quotations are available. If you bring some options it can be seen whether they are more suitable than the current ones. One thing that can help is a link inside that paragraph to a more specific article explaining all this. An option is [[philosophy of mathematics]]. <small><span style="border:1px solid black;padding:0.5px;">[[User:franklin.vp|<font style="color:black;background:white;font-family:sans-serif;" >''' franklin '''</font>]]</span></small> 13:59, 25 January 2010 (UTC) |
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:At worst, numbers exist naturally as singularly probable singularities in nonlinear [[dynamical system]]s that exist naturally. thus, to say that they don't exist naturally would be like saying that the non-trivial zeros of the [[Riemann zeta-function]] don't exist. [[User:Kevin_Baas|Kevin Baas]]<sup>[[User_talk:Kevin_Baas|talk]]</sup> 16:21, 26 January 2010 (UTC) |
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:* No better explanation exists. <small><span style="border:1px solid black;padding:0.5px;">[[User:franklin.vp|<font style="color:black;background:white;font-family:sans-serif;" >''' franklin '''</font>]]</span></small> 17:30, 26 January 2010 (UTC) |
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::Not so, I came up with a clearer one: Consider the vibration of a string. Because of certain mathematical relationships, the harmonics will always be at 2x,3x,4x,5x, etc. of the base frequency. Any thing not a natural number multiple would only dampen or eliminate the vibration. Since "1" is defined simply as "There is a number one.", to say that "natural numbers exist naturally" one only needs to show that there's some kind of physical law that strongly prefers natural number multiples. And that is precisely what we just showed. As they say in the industry: Q.E.D., b@%&@! [[User:Kevin_Baas|Kevin Baas]]<sup>[[User_talk:Kevin_Baas|talk]]</sup> 16:10, 29 January 2010 (UTC) |
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::* I was being ironic, sorry. You need to read a little more about this topic. I guess you can start reading [[philosophy of mathematics]] and then whatever related book you can find. <small><span style="border:1px solid black;padding:0.5px;">[[User:franklin.vp|<font style="color:black;background:white;font-family:sans-serif;" >''' franklin '''</font>]]</span></small> 17:51, 29 January 2010 (UTC) |
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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. |
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:I.e. regardless of the correctness or incorrectness of the diff eq. describing the model, there are more concurrent diff. processes at work. That doesn't diminish my point. |
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:My point was that, so long as there is positive and negative [[feedback]], periodic relationships will inexorably emerge from natural phenomena. And what can "numbers exist naturally" mean if not precisely that? But I digress --- just wanted to make sure my meaning was clear. [[User:Kevin_Baas|Kevin Baas]]<sup>[[User_talk:Kevin_Baas|talk]]</sup> |
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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 [[quaternion]]s 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. [[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen|talk]]) 16:51, 29 January 2010 (UTC) |
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* Well, about the form of the paragraph I agree about the possibility of improving it. About the pertinence of having such an information in the lede i think it is more than indispensable. This is the main article about a science or a discipline ( or whatever mathematics is). The first thing you do when you start to study a science is to define its subject of study. In the case of the other sciences, although it is equally difficult to define the subject for most of the people is clear (at least intuitively) what they are. Take for example Biology, it studies living beings. Although it is very hard to define (appropriately) what is a living being it is mush more clear (at least intuitively) what they are. In the case of mathematics, even from a materialist perspective the nature of the subject matter of the science (if it is a science) is less clear. I guess I don't have to point out what is the importance of defining the subject matter of a science but let me just say that that defines its purpose and applicability which then determines its evolution (for example if it develops more as a game like chess, or to things with direct application or both). For example compare the volumes of publications in Physics and Mathematics that are about situations that do not correspond to reality. This example is interesting since these two sciences are very similar in many of the techniques they use. In Physics you always have the oracle of "reality" that is the ultimate verification. As well as in Mathematics it is valid the question about what is the nature of this "real world", and whether it even exists. Now in mathematics the objects have a different nature. There is also no such "oracle" validating the results or even the axioms. that is why the question becomes even more important in this case. <small><span style="border:1px solid black;padding:0.5px;">[[User:franklin.vp|<font style="color:black;background:white;font-family:sans-serif;" >''' franklin '''</font>]]</span></small> 17:51, 29 January 2010 (UTC) |
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