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The discrete mathematics paragraph of this article was misleading. All the areas mentioned include both discrete and continuous mathematics.
Information theory isn't only discrete mathematics, information theory also applies to analog signals. See Differential entropy.
Analog signal processing is also a form of computation.
Theoretical computer science considers both discrete and continuous computational processes, and both discrete and continuous input/output:
Including the question of P!=NP over Real numbers
For now, I've added the above information to the article, but the whole section on discrete mathematics should be deleted, and its content redistributed to elsewhere in the article. Bethnim ( talk) 13:37, 23 March 2010 (UTC)
User Keifer Wolfowitz seems to be claiming completely without any references of [or? Kiefer.Wolfowitz ( talk) 19:57, 23 March 2010 (UTC) ]
proof that Maths is not part of logic. Note that logic includes the study of inconsistent systems and many other things, and it does not seem that there is any part of mathematics that is not part of logic in the broad sense. But there are clearly parts of logic which are not usually considered mathematical.- Wolfkeeper 19:43, 23 March 2010 (UTC)
He also seems to be maintaining a claim that maths is actually not logical in the article; with an apparent oblique reference to Gödel's incompleteness; this is frankly a bizarre non sequitor and completely unreferenced.- Wolfkeeper 19:43, 23 March 2010 (UTC)
There's no need for personal attacks or displays of emotion in our discussion about logic - it's illogical, Captain ;-) What we should do is, as was pointed out above, report what good sources have claimed, after reaching consensus here. These ideas were not born fully formed, and it is not surprising that there is some inconsistency in their use. But the subject is big, and it is important to keep it concisely worded. Stephen B Streater ( talk) 20:37, 23 March 2010 (UTC)
http://www.mathscentre.ac.uk/students.php http://www.mathtutor.ac.uk/ I would add them to the article but it is locked. Please could someone add it when it is unlocked, as I may not return here for years. Thanks 89.240.44.159 ( talk) 12:42, 19 April 2010 (UTC)
In my humble opinion, more attention should be paid to the definition of mathematics. The first line of the article - as far as I know - is not a recognised definition, more a sketchy impression what mathematics is roughly like.
In a true "mathematical" spirit, the question must be answered first whether the concept "mathematics" can be defined at all. A mathematician told me that it is fundamentally impossible to give such a definition, at least to mathematicians (but I did not fully understand his reasoning).
Is mathematics perhaps just a term culturally attached to a rather arbitrary choice of some forms of logical reasoning? Perhaps a definition can only be given if a purpose is decided first (e.g. there are "ontological" and "teleological" definitions: the former try to grasp the essence of a concept, the latter are a choice that can be practical or unpractical, but is never correct or incorrect).
On reason to be strict in the definition of mathematics is the continuing debate about the patentability of mathematical algorithms. Some argue that all algorithms are inherently mathematical, I am inclined to believe that none of them is: only the proof of an algorithm is - perhaps - mathematics. Is all mathematics inherently so fundamental knowledge that it should never be withdrawn from the public domain? But many mechanical inventions basically are geometric, and geometry is a branch of mathematics.
Is there a fundamental difference between philosophy and mathematics? Or is it just cultural: mathematicans prove, philosophers cite. Both are disciplines that do not depend on observations (which is afaik a reason not to consider them "science" at all, in some perceptions - which must be a deception for a PhD in maths!). Maths often deal with quantities, but e.g. boolean algebra doesn't. Math's use shorthands - formulas - but the Pythagoras theorem in plain language is still mathematics.
Who responds to the challenge? Rbakels ( talk) 19:54, 12 August 2010 (UTC)
The BBC programme In Our Time presented by Melvyn Bragg has an episode which may be about this subject (if not moving this note to the appropriate talk page earns cookies). You can add it to "External links" by pasting * {{In Our Time|Mathematics|p00545hk}}. Rich Farmbrough, 03:17, 16 September 2010 (UTC).
An image displaying the infinity symbol eight times seems unnecessary to me, and I think that removing it would greatly improve the article's aesthetic quality. —Preceding unsigned comment added by 131.104.240.36 ( talk) 10:28, 20 October 2010 (UTC)
That doesn't really work either, as it doesn't illustrate the parallel postulate, or at least not without far more explanation (and it's not controversial in modern mathematics). But it's on history, not mathematical theory, so detailed explanation is just distracting. What about something from commons:Category:History_of_mathematics ? A few things stand out as historic and interesting to me, such as file:Yanghui_triangle.PNG, or one of these commons:Category:Geometria by Augustin Hirschvogel.-- JohnBlackburne words deeds 22:48, 20 October 2010 (UTC)
Though I have nothing but respect for Euclid, I'm not sure that Raphael's painting of him is the best lead image for this article. It seems to me that something more lively or visually interesting might be preferable, similar to the approach used by the biology and chemistry articles.
I would suggest a pair of images, similar to the layout on the chemistry article. My pick would be the two images on the right. The first image illustrates geometry (specifically Desargues' theorem), and was obtained from Portal:Mathematics/Featured picture archive. The second is a picture of the Mandelbrot set, and is a mathematics-related featured picture on Wikipedia (see Wikipedia:Featured pictures/Sciences/Mathematics). Jim.belk ( talk) 00:16, 16 November 2010 (UTC)
I propose that we add www.onlinemathcircle.com to the external links section. A good explanation of what it is would be: "A Community Dedicated to Making Mathematics More Open" —Preceding unsigned comment added by Shrig94 ( talk • contribs) 02:14, 21 November 2010 (UTC)
It seems inappropriate to me to advertise prizes in this article. The article is about mathematics, not mathematicians. There are no such sections in the articles about politics or economics, for instance. The insertion appears to have been placed arbitrarily into a section in which it does not belong, in any case. 131.111.184.95 ( talk) 08:37, 15 December 2010 (UTC)
Let's agree to disagree. Rick Norwood ( talk) 14:19, 17 December 2010 (UTC)
In reference to the disputed claim that
I have to say that I agree that this assertion should not appear without qualification, because not everyone agrees (for example Torkel Franzen did not agree). However it is a serious and widely held point of view, likely the majority view, and ought to be represented. I suspect that Wolfkeeper and Kevin Bass don't understand the argument for how Goedel's theorems might be said to refute logicism, and this is a bit subtle and I'm not going to go into it right now. It is strictly speaking beside the point anyway for purposes of editing the encyclopedia — the challenge is to source the statement, and to correctly attribute it, not to a single thinker because it's a widely held view, but to a current of mathematical thought. -- Trovatore ( talk) 21:24, 23 March 2010 (UTC)
Nearly all mathematical concepts are now defined formally in terms of sets and set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces are all defined as sets having various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of relations is entirely grounded in set theory.
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.
Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes derivations of more than 10,000 theorems starting from the ZFC axioms and using first order logic.
(unindent)they seem rather straightforward to me, but yes, yes, i shall digress. the issue is the phrase "...important work in mathematical logic showed that mathematics cannot be reduced to logic." , with which i still take issue with, if for slightly altered reasons. by "reduced to logic" i read something like "proofs of theorems be expressible in a reduced formal grammar", and I maintain that they can, e.g. a turing machine or a ZFS+AOC system. and i contend that the subtler point that you need a few axioms beyond the basic and,or, in, not operations, which are noentheless expressible with those operations, and that some egregious purists might balk at that is a bit trivial and a bit to fine to be stated so boldly, apart from the phrasing as worded being -- as we have just witnessed -- misleading. Kevin Baas talk 21:02, 24 March 2010 (UTC)
There's a lot of nonsense said above (on both sides of the argument, if there is one) that I will not comment about. However there is a simple issue that needs no technical arguments. The current version, which Trovatore keeps reverting to, starts: "Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper." I think all mathematicians would agree that mathematics is not experimentally falsifiable; it is hard to imagine how an experiment could falsify mathematics. It is quite conceivable that mathematics, or some part of it, will one day be found to be inconsistent (remember Russell's paradox?), but if it happens, it will have nothing to do with experiment. The universe has been found to not be a Euclidean space, but that does not affect the work of Euclid (which is not entirely rigorous anyway, but that is another issue) any more than it affects hyperbolic geometry (which does not model the universe either). I'm unsure what Popper actually though about mathematics (his WP article does not mention mathematics as subject at all), but I think falsifiability can only be taken to characterize empirical science, which mathematics simply isn't. As an aside, it would be more interesting to know if many philosophers believe that philosophical theories are experimentally falsifiable. But I digress.
Next sentence: "However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, 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.'". OK, so (if I understand this correctly) Popper believes (around 1995, presumably a bit before his death in 1994) that mathematical hypotheses are conjectures that could be experimentally falsified, which just shows he doesn't understand what a hypothesis is in mathematics. But in any case there is no relation with Gödel's findings of the 1930's; I think this sentence makes a completely unjustified link between them and Popper's quote (in which "even recently" is unlikely to mean "before 1930"). And I would like to know what kind of non-logical element "many mathematicians" would like to invoke to resolve statements that according to the incompleteness theorem cannot be decided by logic alone.
So to get to the point I really wanted to make, there are two issues mixed up in this sentence which in fact are totally unrelated: (1) the question of whether mathematical axioms are assumptions about reality that could be experimentally falsified, and (2) the question whether in principle all mathematical statements can be proved or disproved in an appropriate formal logical system. Gödel's incompleteness theorem shows that (under mild assumptions on what "mathematical statement" and "formal system" mean) the answer to (2) has to be "no". So there will be in every theory some statements that cannot be decided from the axioms of the theory by pure logic. But that is miles away from anything involved in question (1). First of all such a statement is not an axiom of the theory, or a hypothesis of any particular theorem, which are simply assumed to be true in order for the theory/theorem to be applicable; it is a statement whose truth or falsehood one might think to be deducible from the (assumed) truth of the axioms, but in fact is not. And second, axioms have long since ended being considered to be "self-evident truths", they are just starting points of a theory that implicitly determine what the theory is about; they are meaningless in reality, or in any other theory. Take the axioms of your favorite theory: groups, probability, topology, mathematical analysis, (and yes) geometry, set theory, even mathematical logic itself. Which means that the answer to (1) is also "no", but with no relation whatsoever to Gödel's work.
And logicism in all this? It certainly never assumed that mathematics needs no axioms. It is inconceivable to base say geometry on pure logic without some axioms telling what geometric notions like "point" and "line" mean. It also does not affirm that all mathematical statements can be decided by pure logic (from a given set of axioms). What it probably does affirm (I'm in no way an expert on this) is that apart from the rules of logic and the axioms, mathematics needs no vague kind or reasoning based on things that are "obvious" without being able to be formalized. It definitely rejects the idea that there are mathematical statements whose truth is open to experimental verification or falsification, but that does not distinguish it from other schools of thought (Popper notwithstanding). Marc van Leeuwen ( talk) 14:03, 10 January 2011 (UTC)
Not to anyone particular, but to the attitude at the heart of this whole fight: please do not use philosophical waffle without being clear about the mathematics first, or you will be part of the grand scheme to reduce philosophy to random bits of pretentious babble about other subjects as spouted by those who have never actually studied the other subjects for their own sake.
Dudes. Why is there a bunch of infinity signs in this article? They serve no purpose. I suggest we remove it, —Preceding unsigned comment added by 134.173.58.98 ( talk) 04:07, 27 January 2011 (UTC)
Some races, talk mathematics structure within their plain language, and have no written symbols to prove it. One example might be that according to a person who lives where there is no winter, he/she might only know snow and ice, but to a person who lives where it is winter for half the year, there are numerous kinds of snow definitions. There is granular, powdered, drifting, hard packed, just right to make an igloo, too hard even to leave tracks and I can go on and on, and another person will understand just what I am describing in exact detail. —Preceding unsigned comment added by 65.181.32.135 ( talk) 16:58, 9 February 2011 (UTC)
This is a terrible article starting with a shit definition of maths justified by poor sources, and suitable only for children. How is this definition any different for an equivalent definition of physics by replacing the word mathematics for the word physics? Quantity, change and space, are physical concepts, mathematics deals with concepts that are patently not physical. That leaves structure. To say that algebraic topology, finite geometry, graph theory etc deals with structure is uninformative and misleading
The following are examples of awful sentences:
Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[8]
Mathematics arises from many different kinds of problems.
Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area.
Number theory also holds two problems widely considered to be unsolved: the twin prime conjecture and Goldbach's conjecture.
Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set.
The study of space originates with geometry – in particular, Euclidean geometry.
Who the fuck writes this shit? Do they read what they wrote?
The only half-decent math entries on wikipedia, tend to be those that are too technical for idiots to fake, and even then they tend to be verbose, repetitious and awkwardly phrased. —Preceding unsigned comment added by 86.27.195.112 ( talk) 13:42, 20 February 2011 (UTC)
"....establish truth by rigorous deduction from appropriately chosen axioms and definitions" This is not uncontroversial. It seems like what mathematicians wnat to believe about themselves, more than something factual. Also, why can't I edit this article? Sincerely, Mythirdself. —Preceding unsigned comment added by Mythirdself ( talk • contribs) 19:10, 28 April 2011 (UTC)
The reference for the sentence I criticized isn't even comprehensible. It's (I assume) an author: "^ Jourdain". What's the work, page number, etc.? —Preceding unsigned comment added by Mythirdself ( talk • contribs) 23:31, 29 April 2011 (UTC)
I removed this quote from the lede, because of undue weight:
Arnold wrote that mathematics is a branch of physics, among other entertaining absurdities. Kiefer. Wolfowitz 08:51, 2 May 2011 (UTC)
I removed the following section, which seems to be far below the rest of this article in quality, and which also seems to give undue weight to speculations:
Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper. [1] However, in the 1930s Gödel's incompleteness theoremsconvinced many mathematicians who? that mathematics cannot be reduced to logic alone, 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." [2] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.
An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [3] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method. citation needed
The opinions of mathematicians on this matter are varied. Many mathematicians who? feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others who? feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics. citation needed
IMHO, the article needs a discussion of the unity of mathematics, how the same objects appear in apparently disparate fields of inquiry, such as the role of groups in complex analysis (homotopy or homology), geometry, and polynomial equations. (Peirce referred to this as surprising to find the same _____ in an African jungle and the Alaskan Klondike!)
Kiefer.
Wolfowitz 12:26, 1 May 2011 (UTC)
I moved that section to the bottom, because imho it still reads like an essay, rather than an encyclopedia article, and its weight may be undue. Editor Trovatore disagrees, so this is worth a discussion. Kiefer. Wolfowitz 20:24, 26 May 2011 (UTC)
From the start, I think the link to "space" in the first line:
Is probably meant to link to the http://en.wikipedia.org/wiki/Space_(mathematics) page instead, though neither of them come anywhere close to being explanations of "Space" that belongs in the definition of "Mathematics"...
Neither space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. [6]
Nor http://en.wikipedia.org/wiki/Space_(mathematics) In mathematics, a space is a set with some added structure.
Seem to be relevant descriptions for the more abstract concept "space" of which Mathematics studies.
The intro / declaration from the Mathematics portal seems to be a more fundamental description: Mathematics, from the Greek: μαθηματικά or mathēmatiká, is the study of patterns. Such patterns include quantities (numbers) and their operations, interrelations, combinations and abstractions; and of space configurations and their structure, measurement, transformations, and generalizations. Mathematics evolved through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the systematic study of positions, shapes and motions of abstract objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.
There's a typo in the Applied Mathematics section. Looks like someone only half pasted a quote: 'formulation and study of mathematical models. — Preceding unsigned comment added by 94.195.50.242 ( talk) 10:30, 30 May 2011 (UTC)
Let's try to work on a version of the lede here instead of getting into an edit war. As I mentioned, I object to tracing the axiomatic method back to Euclid in describing modern mathematics, because this was simply not the case before Peano, Hilbert, and Co, and we don't really know what Euclid and his contemporaries did. Describing mathematics flat out as formal derivation of theorems from axioms is philosophically naive. Tkuvho ( talk) 10:42, 2 May 2011 (UTC)
I still have philosophical problems with this: "Since the pioneering work of Giuseppe Peano, David Hilbert, and others on axiomatic systems in the late 1800s, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions." The first sentence seems more applicable to logic than math. The second is a tautology: is the definition of "a good model" that which "provides insight and predictions?" Isn't insight subjective? — Preceding unsigned comment added by Mythirdself ( talk • contribs) 18:25, 26 May 2011 (UTC)
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I'm not sure quoting a famous and idiosyncratic figure like Quinne is a good idea. He's brilliant, but he has his own very Quinnian views. Something blander (hey, it's an encyclopedia not a revolution) might be more appropriate. Mythirdself ( talk) 18:37, 2 June 2011 (UTC)
Mathematics undergoes specialization, yet continuity within mathematics is maintained by the discovery and elaboration of structures/theories whose powerful abstractions provide fresh insight. This quote form Charles Sanders Peirce is probably too long for inclusion:
The host of men who achieve the bulk of each year's new discoveries are
mostly confined to narrow ranges. For that reason you would expect the arbitrary hypotheses of the different mathematicians to shoot out in every direction into the boundless void of arbitrariness. But you do not find any such thing. On the contrary, what you find is that men working in fields as remote from one another as the African diamond fields are from the Klondike reproduce the same forms of novel hypothesis. Riemann had apparently never heard of his contemporary Listing. The latter was a naturalistic geometer, occupied with the shapes of leaves and birds' nests, while the former was working upon analytical functions. And yet that which seems the most arbitrary in the ideas created by the two men are one and the same form. This phenomenon is not an isolated one; it characterizes the mathematics of our times, as is, indeed, well known. All this crowd of creators of forms for which the real world affords no parallel, each man arbitrarily following his own sweet will, are, as we now begin to discern, gradually uncovering one great cosmos of forms, a world of potential being.
Charles Sanders Peirce (``CP 1.646)
Hi, Perhaps the user has not understood the comments made to 'change the syntax'. The [ | reference book]'s [ | introduction], [ | Development of Philosophy], etc. are indeed exercises in glorifying Greek Mathematics. How is this a reliable source is just one question, the point here is 'The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.' while at other places like India [ | you may find the same], another source. The book therefore can only be interpreted as a reference to beginning of the systematic study of mathematics in its own right in the Ancient Greece. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 17:00, 6 June 2011 (UTC)
Hi, I would like to know why are the changes reverted after deleting 4 sources in the guise of statement that one source is unreliable without giving any proofs. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 07:10, 6 June 2011 (UTC)
I think you are confusing mathematics with the various rules-of-thumb used to build structures. Standard reference works agree that Greek mathematics began at an earlier date than Indian mathematics. That in no way minimizes the many important contributions of Indian mathematics. Rick Norwood ( talk) 14:19, 7 June 2011 (UTC)
I thought discussions on Wikipedia would be at a higher level than the usual Internet fare. Especially for the article on mathematics. Does being educated and sharing a good cause reduce the incidence of flame-wars? — Preceding unsigned comment added by Mythirdself ( talk • contribs) 01:45, 9 June 2011 (UTC)
Just found a page on Ruler on Wikipedia for information. I am sure it applies to mathematics to an extent. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 19:52, 16 July 2011 (UTC)
I think the Etymology from Semitic Arabic is the important thing. Because it tells the history of Mathematic just from the etymology and begin crediting some specific mathematic such as Al-Jabar, Arithmatic, Logarithma (read Loharitema), and Algorithm. It opens mind that the Greek doesn't invent this specific mathematic since The Arab invent the specific system number which lead to many specific mathematic knowledge such as Al-Jabar, Arithmatic, Logarithma, and Algorithm. See all the Al- infront of words, explain they come from Arab with Arabic names and Arabic system number. So be honest and let's tell the truth to the world this mathematic come from Arab. The Arab own the system number and put God name in Arab on mathematic knowledge. For the editor Please be honest because this is an important encyclopedy. I hope the etymolog from Semitic Arab soon move to the main page. And Please Complete the article about Al-jabar, Arithmatic, Log, Logarithma and Algorithma it self. — Preceding unsigned comment added by 202.152.202.248 ( talk) 04:06, 28 July 2011 (UTC)
Now that the lead sentence has (weak) citations, which is great, there is still the issue that "quantity, structure, space, change" are listed as "fields of mathematics". Mathematicians simply do not talk like this. Maybe philosophers of math talk like this? Even if so, it needs citation, and equal weight should be given to how mathematicians talk about math. Most fields of math are about specific problems (or classes of problems, or sets of axioms) that draw on more than one of these four "ingredients".
Further, if the graphics are supposed to suggest that group theory is a subdiscipline of "structure" (let's accept that it is, for now), then they also suggest that "complex numbers" is a subdiscipline of "quantity". Is "complex numbers" supposed to be complex analysis? Or is that a subdiscipline of "change"? I know that this task is difficult, but the current solution is not good. We could do better by simply citing from the American Mathematical Society's (or another comparable organization's) classifications and copying text from the appropriate Wikipedia articles. We should also describe historical views on the organization of mathematics (e.g. as indistinguishable from physics or natural philosophy?). The current solution seems neither contemporary nor historical. Mgnbar ( talk) 15:59, 22 July 2011 (UTC)
Field | 2010 | 2009 |
---|---|---|
Algebra, Number Theory | 230 | 223 |
Real, Complex, Functional, and Harmonic Analysis | 102 | 98 |
Geometry, Topology | 149 | 139 |
Discrete Math, Combinatorics, Logic, Computer Science | 116 | 141 |
Probability | 73 | 82 |
Statistics, Biostatistics | 495 | 483 |
Applied Math | 229 | 169 |
Numerical Analysis, Approximations | 88 | 93 |
Nonlinear Optimization, Control | 27 | 24 |
Differential, Integral, and Difference Equations | 102 | 117 |
Math Education | 15 | 14 |
Other | 6 | 22 |
Total | 1632 | 1605 |
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The word "mathematics" (jovo or java in modern language) comes from the semitic hebrew Arabic (Arabic: محمد Muḥammad, pronounced [mʊˈħæmmæd], which means The Praised Man or The Honourable Man, The Man was the Mesenger of the religion of Islam(born 570, Mecca, Arabia [now in Saudi Arabia]—died June 8, 632, Medina) [Britannica Group 1] [1] , and is considered by Muslims to be a messenger and prophet of God, the last law-bearer in a series of Islamic prophets, and, by most Muslims,the last prophet of God as taught by the Quran. This Original word is Mohammadika which means Your Praised Man or Your Honourable Man. Muhammad in Al-Qur'an, had been told by God to read (in Al-Alaq first verse) and to seek refuge to Lord (keeper) of Calculation (Robbilfalaq) (it means Allah) in Al-Falaq first verse.
The word mathematicis irrelevant with máthēma comes from μανθάνω (manthano) in ancient Greek and from μαθαίνω (mathaino) in modern Greek, both of which mean to learn Since The Arabian originally invent the zero system number and the Decimal system number (read Al-Qur'an see the system number for Al-Qur'an Juz) which mean the Greek (Pythagoras and Archimedes)and The Roman didn't know the Arabic system number until Muhammad spread Islam. The word Mathematic become Jovo or Java in modern time it came from the original word in Arabic محمد. From Al-Qur'an the Arabian muslim originally know all human Calculation is relative than God calculation so in the mathematic knowledge they use Al-Asmaul-Husna(99 names of the only one God of Islam) [2] [3] [4] [5] as the name of this knowledge such Ar-Rahim ( Arithmatic, see rythm from "rhyme" and rhyme from rahim), Allah ( log), Allah Ar-Rahim ( logarithm), Al-Gofur Ar-Rahim (AlGorithm, some say from Al-Khawarijm) and Al-Jabar (Al-Gebra). Since this knowledge come from Arab soon people call it "mathematics" refers to Muhammad.
110.137.147.244 ( talk) 14:18, 27 July 2011 (UTC)
I have been musing on the definition of mathenatics as "the study of quantity, structure, space, and change". It strikes me that I could take my camera, get out there and contrast a single human being with a crowd, compare the structure of a leaf with a network of roads, picture the clever use of a tiny volume in a yacht and the humbling vastness of the mountains, and document the changes of seasons. This would provide me material for a show "the study of quantity, structure, space, and change", and we would not recognise any of it as mathematics.
Rather than characterise mathematics by WHAT it studies (even though the list is very compact and comprehensive), I would attempt to characterise, in one sentence, HOW it does it. So my two cents: "Mathematics is the art of rigorous abstract thinking"
Does it make sense?
Obviously the proposed definition touches on many elements already well discussed in the article.
Philippe Maincon ( talk) 17:24, 15 July 2011 (UTC)
Charles Sanders Peirce's New Elements of Mathematics has an thorough and stimulating discussion of previous definitions of mathematics. Kiefer. Wolfowitz 10:49, 29 July 2011 (UTC)
Looking over the above exchanges, I find that I have left some things unexplained that I now believe I know how to say more clearly, and on a related note, that I also do not like the first paragraph as it stands. (Well, I never did really, I just thought it was "least bad", but I now find that it has evolved in a way that I think is suboptimal, and I have a candidate point in the past where I think the text was better, that we should consider a starting point.)
What is a definition? In mathematics itself, our definitions, say of a "ring" just for example, provide precise demarcations. They divide things into rings and non-rings, with nothing in between (ignoring quibbles about whether you require a unit). It's not unnatural that mathematicians would like to be able to provide that sort of a definition for mathematics itself.
Perhaps such a project is possible; I am skeptical, but for the sake of argument suppose that it is possible to define "mathematics" in such a way that it precisely demarcates that which is mathematics from that which is not mathematics. We are left with the problem that no such definition is agreed among mathematicians or philosophers of mathematics. It is not Wikipedia's function to pick one from among them. We simply may not do that; it is a blatant violation of WP:NPOV.
To preserve neutrality, we could futz around with competing definitions and say who uses them. But I hope everyone agrees that the lead is not the place for that. The article is about mathematics, not about how to define the term mathematics, and more than two or three sentences is too much to spend on the definition in the lead section.
But luckily, we don't have to. All we have to do is recognize that the sort of "definition" required in the lead paragraph is not a demarcation at all. The first sentence of dog does not give you the information required to divide all objects into dog and non-dog, and it can't be expected to. Rather, it gives you enough information to identify what the article is talking about, and possibly tells you something you might not have known about what is included. (Are dingoes dogs? Does mathematics deal with things other than numbers?)
That's why an NPOV definition in the lead will necessarily have a "shopping-cart" aspect to it. It can't demarcate what is mathematics from what is not, but it can say some things that we all agree are mathematics, including some that some readers may not have realized are mathematics. -- Trovatore ( talk) 07:50, 1 August 2011 (UTC)
Here are the first three sentences as they stand: (I'll leave the refs but won't put a reflist):
Here is the version I like better, from 1 February 2009:
Here's the main reason I like it better: The current version pretends to be a demarcation. It lists four (vague) things; those are mathematics, nothing else is. As I argue above, that cannot possibly meet NPOV. The Feb 2009 version does not; rather, it lists some of the things that mathematics studies, without claiming to exhaust the subject.
That's not to say it can't be improved. I would change "the academic discipline" to "an academic discipline", making the non-demarcative nature more explicit. Also I would probably incorporate some of the language of the current version as well; I have no detailed proposal at this time for how to do that. But I hope my main point is clear. -- Trovatore ( talk) 07:50, 1 August 2011 (UTC)
{{edit semi-protected}} After " Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[10]" add "At times it is difficult to know where pure mathematics ends and applied mathematics begins." Knwlgc ( talk) 05:12, 13 September 2011 (UTC)
Procedural note: I'm removing this 'edit semi-prot request' for now, pending consensus Chzz ► 01:37, 17 September 2011 (UTC)
My request still exists, it has not gone away. It was assumed when I sent in my edit request that there was not consensus (otherwise this article would not need semi-protection), how has the state of my request changed since its conception? I should add that an admission of willingness to discard my edit request, is not a request to discard my edit request (had it been then I would have discarded it). Knwlgc ( talk) 03:08, 17 September 2011 (UTC)
The work in the Centre for Experimental and Constructive Mathematics is more than twenty years ahead of Mathematics as an international discipline. This has been inside information which might now be public.
It seems as though there could have been some confusion about what Applied Mathematics was. Generally, Applied Mathematics has been a variety of subsets of ad hoc amalgamations of Theoretical Physics, Statistics, Computing Science, Mathematics, Engineering Science, and almost anything else. The common denominator clarifying what Applied Mathematics has been was tricky to find. If/when ad hoc disciplines (an oxymoron) try existing primarily as theoretical constructions built for the purpose of trying to get money any old way, the result is internal organizational inefficiency. It would be unfair and dreamy of Mathematics, as the international discipline this is, to ask other organizations to have internal cohesion due to our recent update of the definition of if, without first demonstrating what we mean by internal cohesion.
(Instantiations and examples differ slightly: instantiations are generalizable and examples may have generalizable properties and features. However, part of this work includes teaching mathematics to seven billion people, thus for now I preferentially use the word example.)
An example of confusion arising from lack of internal organizational cohesion due to presence of ad hoc discipline, is the 50% vote of support Jonathan and Peter Borwein received from participating voters (abstention rate unknown) for establishing the Centre for Experimental and Constructive Mathematics at Simon Fraser University, a tie which was broken in preference of establishing Experimental and Constructive Mathematics by someone in senior administration circa 1992, and the upshot of which includes the Organic Mathematics Project which singularly redefined Mathematics online education and collaboration; correction of Aristotle; redefinition of the word if; free demonstrative and instructional tutoring services for the world's central banks with respect to the additive and multiplicative identities; open questions including where Mathematics proofs come from, ownership of intellectual property in collaborative processes, how Mathematics and Mathematically informed disciplines develop communities; and the intellectual property ownership question: who owns Mayer Amschel Rothschild's intellectual property conceived circa 1794, still in circulation, and which I previously cited in my work as osmosis.
Generally and obviously, a discipline is not yet qualified to offer its services to customers until after the discipline has demonstrated the same expertise internally. Having organizations' internal and external services match works. The Centre for Experimental and Constructive Mathematics and our network is perfect for driving optimization by osmosis and naturally occurring, real selection processes. Therefore handling the question << what is Mathematics >> is part of this constructive instruction. This exemplifies what Applied Mathematics really is, both in this self-referencing demonstration and explanatory definition update, and in real world ubiquitous application across all disciplines everywhere; therefore we acknowledge Mathematics was previously domesticated partly under Philosophy and partly under Science, and might be correctly understood as a profession under the Institute for Electric and Electronic Engineers, who as an organization has the highest standards in ethics and professional conduct. This Applied Mathematics includes Information Theory and Computer Architecture. Having the discipline Mathematics perfectly located under the IEEE solves all problems related to franchising Mathematics other than my unique personal problem if The Rothschild Family prefers to take me to court for accidental intellectual property theft.
References:
http://www.cecm.sfu.ca/organics/project/
http://www.ieee.org/index.html
Founder by Amos Elon, ISBN 0 670 86857 4
JenniferProkhorov ( talk) 19:23, 19 October 2011 (UTC)
Does mathematics really belong to the mathematical sciences? Mathematical sciences says
Mathematics isn't primarily mathematical in nature. It is COMPLETELY mathematical in nature.
I don't know of a better place to discuss this than the wiki page for Mathematics, although the question pertains to any page under the section of Mathematics.
Can we implement mathematical symbols that are ALSO hyperlinks to wiki pages for each symbol?
I do not think all mathematical symbols have wiki pages, but I don't see why not.
At least they could link to a relevant page in which the symbol is heavily used.
I believe this would make it significantly easier to learn mathematics from wikipedia. — Preceding unsigned comment added by 140.247.59.253 ( talk) 23:45, 27 July 2011 (UTC)
The idea is not just to have a list of symbols, but to use hyperlink versions of the symbols on any or all wiki pages within Mathematics. Like many special terms, the first usage of a symbol on any wiki page could be a hyperlink version of the symbol. Much in the same way that unique terms can be clicked on to bring the wiki-reader to the definition of that term, so too could she more quickly learn about the mathematical symbols that crop up in whatever section of mathematics she is currently browsing. — Preceding unsigned comment added by 140.247.59.84 ( talk) 15:29, 9 August 2011 (UTC)
I'm sure we've had this discussion before as I remember thinking it a very bad idea at the time, for various reasons.
If a symbol really needs explaining then add a sentence, e.g. from Euler's formula:
This is far clearer than linking any symbol.-- JohnBlackburne words deeds 02:34, 30 November 2011 (UTC)
Why do we have long paragraph of quotations about mathematics in the lead? Shouldn't that go in Wikiquote? Kaldari ( talk) 22:18, 4 December 2011 (UTC)
How about adding "Mathematosis" to the "See Also" section? 164.107.189.191 ( talk) 14:37, 6 December 2011 (UTC)
It's nice to see this right in the first paragraph of a significant article: "Galileo Galilei (1564-1942) said" I never knew the man lived to be almost 400. Good job, Wikipedia. And the article is locked so I can't even fix this boneheaded error. Ugh. — Preceding unsigned comment added by 131.193.127.17 ( talk) 16:03, 6 December 2011 (UTC)
hi.
I would replace
"However, mathematical proofs are less formal and painstaking than proofs in mathematical logic"
with
"Mathematical proofs are written in a formal language provided/analysed by mathematical logic".
The main reason for this exchange is that mathematical logic is itself a part of math! Therefore, the above statement means something like "trains are faster than TGVs". It is just nonsense. Another reason is that proofs in e.g. algebra are just as formal and painstaking as proofs in mathematical logic...
best regards a phd-student in math — Preceding unsigned comment added by 138.246.2.177 ( talk) 17:48, 20 December 2011 (UTC)
Mathematical proofs really are less formal and painstaking than proofs in mathematical logic. Open any math book to a proof. I'll pick one at random off the shelf behind me, and open it to a random page. "Proof: Recall that a subspace Y of L is said to be convex if for every pair of points a, b of Y with a < b, the entire interval [a, b] of points of L lies in Y." I think this is fairly typical of how a mathematical proof is written. Now, compare with a proof in mathematical logic:
More formal. More painstaking.
Mathematicians usually assume that the kinds of proofs we do in our work could, if necessary, be reduced to mathematical logic, but we never, in practice, do that.
Rick Norwood ( talk) 19:13, 20 December 2011 (UTC)
Mathematical logic usually means mathematical logic, as in Hamilton's Logic for Mathematicians or Manin's A Course in Mathematical Logic. The other topics you mention are in Foundations, rather than in Mathematical Logic. I agree that the proofs in essentially all areas of mathematics except formal mathematical logic are in the metalanguage rather than in the object language. Rick Norwood ( talk) 19:53, 20 December 2011 (UTC)
03-XX Mathematical logic and foundations 03-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 03-01 Instructional exposition (textbooks, tutorial papers, etc.) 03-02 Research exposition (monographs, survey articles) 03-03 Historical (must also be assigned at least one classification number from Section 01) 03-04 Explicit machine computation and programs (not the theory of computation or programming) 03-06 Proceedings, conferences, collections, etc. 03Axx Philosophical aspects of logic and foundations 03Bxx General logic 03Cxx Model theory 03Dxx Computability and recursion theory 03Exx Set theory 03Fxx Proof theory and constructive mathematics 03Gxx Algebraic logic 03Hxx Nonstandard models [See also 03C62]
previous unsigned comment by User:Rick Norwood 21:25, 20 December 2011 (UTC)
thanks for that, rick. see? "Proof theory and constructive mathematics" one section. notice the absence of a separate, independant section on "Proof theory and constructive mathematics for mathematical logic, specifically, which is for some reason different". Kevin Baas talk 21:27, 20 December 2011 (UTC)
I reverted your change because, though you say one thing above, the change you made says another. We need to either omit this entirely or find a way of saying it that is both intelligible to the layperson and mathematically accurate. Rick Norwood ( talk) 14:14, 21 December 2011 (UTC)
In the mathematical sense of the phrase "formal language", it is not possible to be "more formal" or "less formal". A "formal language" is one where the proofs depend only on the form the symbols take, and not on the meaning of the symbols. If you prefer "formal grammar", that is also used in the same sense. I'm primarily following Hamilton's Logic for Mathematicians.
But, back to the point of this discussion. I agree the disputed sentence should either be improved or, if nobody can come up with a good way to improve it, removed.
Rick Norwood ( talk) 16:29, 21 December 2011 (UTC)
A new suggestion:
"For convenience, most proofs are written in a metalanguage and, therefore, have to deal with the insufficiencies of each metalanguage. Nonetheless, mathematical proofs should (!) be written in such a way that a mathematician could translate them into a more formal language with an unambiguous grammar. This more formal proof could then even be checked by an computer. Usually, one of these more formal languages is taught at the beginning of mathematical logic"
btw.: less attacking and more suggestions and we could have closed this secition yesterday....!!! — Preceding unsigned comment added by 138.246.2.177 ( talk) 17:14, 21 December 2011 (UTC)
I actuallyLOVE the lead, and appreciate the approach of quoting a few important folks as they described mathematics. However, I found the context lacking, especially for Galileo's quote. It is too often that the poor metaphors of natural law and language are used to describe mathematics. Such conceptions are invaluable to its history and this article, but there is a responsibility to more properly contextualize this paragraph I'm question. I believe the final sentence, a quote by Einsteiniis intended to achieve this effect, but j would rather see a punchline less punched if it meant clarity that could prevent further propagation of naive interpretations, despite also being valuable for other reasons. — Preceding unsigned comment added by 67.161.64.224 ( talk) 07:43, 6 January 2012 (UTC)
The choice of entries in the See also section looks rather arbitrary to me and is, in my opinion, quite uneven. Iatromathematicians is an extreme obscure topic. Why is Self-similarity on this list? Any suggestions for a criterion to decide what should be on this list? -- Lambiam 12:42, 12 January 2012 (UTC)
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Please transpose the words space and quantity. It was recently found that the "all pages lead to philosophy" loop was broken by an edit to the mathematics page and the problem would be easily fixed by transposing these two nouns. menaing of the page would not be changed and many meme-ers would be made very happy. Thank you.
66.99.120.222 ( talk) 15:24, 20 January 2012 (UTC)
Sorry, to edit articles to cause them to lead to philosophy violates the idea that all articles naturally lead to philosophy. Also, the current word order is used throughout the article and in several other articles. Rick Norwood ( talk) 16:02, 20 January 2012 (UTC)
.................................................................................................. — Preceding unsigned comment added by 205.125.65.84 ( talk) 15:46, 26 January 2012 (UTC)
MATH is defined as the expression of logic through the use of quantity.
It is often believed to be a determining science - which means it has causal properties. This is not true. A symbol cannot have any bearing on physical or metaphysical phenomena. At its best, it expresses logic or the scale and change of entities. At its worst...well let's not talk about it...:))))
LOGIC can be expressed in different forms - Math, Words, Pictures — Preceding unsigned comment added by 117.207.152.172 ( talk) 13:15, 17 March 2012 (UTC)
The first sentence doesn't agree with the citation. The reference says that mathematics is primarily about abstraction, not simply that it is about space, quantity etc. Removing the part about abstraction significantly changes the expression and the meaning, it does not encompass the same range. Teapeat ( talk) 03:47, 19 March 2012 (UTC) For example boolean logic is not really, in any normal sense, to do with space, quantity, structure or change, but is normally considered to be part of Mathematics. It is however an abstraction of logic. Teapeat ( talk) 03:47, 19 March 2012 (UTC)
Given this, I am considering the verification of the reference given to have failed, and the removal of that part to be original research. Teapeat ( talk) 03:47, 19 March 2012 (UTC)
The cited source is not reliable. It's a web page about a university math/comp-sci department. It's probably just copying us, and it appears to have misattributed the famous Galileo quotation to Newton. The virtues of our definition are that it accords well with the kind of definition offered in other reference works, and it reflects the body of the article that we actually have. Please see Definitions of mathematics. I would favor removing the citation and leaving our definition unsourced, but wording it in such a way that it does not come across as the final word on the subject. How about we simply mention the lack of any consensus on the definition of mathematics, and link to Definitions of mathematics? — Ben Kovitz ( talk) 10:02, 11 June 2012 (UTC)
I just made some changes: a new section on definitions, and a rewording of our opening sentence so it addresses the lack of consensus on a definition of mathematics, but is still informative and reflects the body of the article. I removed the old source that was probably quoting us, and added some new sources that I think are pretty solid, with the exception of the two I found for calculus (as the study of change). It would be better to have a single good source for that. I found a pretty amusing source on the inadequacy of any definition to cover all of mathematics: a published critique of this very article in a general survey of mathematics.
There are still problems with the lead. The lead is now five paragraphs, which is too long. Much of that bulk says rather doubtful things that are not covered in the article, no doubt the residue of long-past soapboxing. I think that by merely summarizing the body of the article, the lead could easily be brought back into shape. But, I'm done for the day. — Ben Kovitz ( talk) 23:22, 16 June 2012 (UTC)
MATH: Mental Abuse To Humans — Preceding unsigned comment added by Ilovenickelback17 ( talk • contribs) 00:53, 29 March 2012 (UTC)
Maurice Carbonaro ( talk) 08:00, 20 July 2012 (UTC)
Math seems to be going on in multiple different directions. But if mathematicians could define a goal that all of their research amounts to (like why we're trying to solve the Riemann hypothesis and other unsolved problems, and/or what they have in common), that would be great to include here. (Hint: It's definitely not 42 or NaN.) 68.173.113.106 ( talk) 21:48, 6 March 2012 (UTC)
What I mean is, some people want to investigate topology, others want to solve P versus NP, still others want to do advanced complex analysis. Personally, I'm looking into mathematical finance (even though I'm just a kid). So what do all of these approaches have in common? If at all, why is it important? (Added to 68.173.113.106 ( talk)'s previous comment on 21:53, 6 March 2012 (UTC))
Do we have to find an ultimate goal? Perhaps a better posed question is: Can we find traits common to all branches of mathematics? Quantity, logic, and intuition are all partially correct but not generic enough (e.g., algebra intentionally leaves quantity unspecified). How about "Mathematics seeks to construct abstract entities and discover their properties through logic based on a limited set of axioms." Of course, in trying to be generic I have introduced some terminology, and others are welcome to improve upon this. — Preceding unsigned comment added by 71.236.24.128 ( talk) 22:01, 18 August 2012 (UTC)
I think math as a hobby could become a useful addition to the article if someone worked hard enough on writing it. Marvin Ray Burns ( talk) 02:16, 19 May 2012 (UTC)
There are good examples of math as a hobby, but none of the three names you mention seems to fit well under that heading. Rick Norwood ( talk) 11:54, 10 September 2012 (UTC)
A C.S. Peirce scholar brought this line in the article to my attention:
Two examples of logicist definitions are "Mathematics is the science that draws necessary conclusions" (Benjamin Peirce)[24] and "All Mathematics is Symbolic Logic" (Bertrand Russell).[25]
Russell's quote is quite logicist, but Benjamin Peirce's quote doesn't seem so, since logicism usually means the idea that much or all of mathematics is reducible to logic, not merely the idea that mathematical conclusions are logical or deductive. Have anti-logicists generally held that mathematical conclusions are not generally deductive?
Now, the B. Peirce quote is from "Linear Associative Algebra" in which he goes on to say, beginning lower on the same page (the article's first page),
Mathematics, under this definition, belongs to every enquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by observation. In its pure and simple form, the syllogism cannot be directly compared with all experience, or it would not have required an Aristotle to discover it. It must be transmuted into the all the possible shapes in which reason loves to clothe itself. The transmmutation is the mathematical process in the establishment of the law. [....]
That really doesn't sound like logicism at all. It's more to say the same as his son C.S. Peirce said, that the mathematician aids the logician, not vice versa. (I'm not arguing about whether they were right, just that that was their view.) Anyway, I suggest that Benjamin Peirce's definition of maths not be characterized as "logicist." The Tetrast ( talk) 01:00, 7 September 2012 (UTC).
If nobody comments during the coming five days or so, I'll go ahead and make the change. The Tetrast ( talk) 02:56, 8 September 2012 (UTC)An example of a logicist definition is "All Mathematics is Symbolic Logic" (Bertrand Russell).[25].
"In Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic."[25] Related but more subtle views were put forward by Benjamin Peirce, who wrote that mathematics is "the science that draws necessary conclusions" [24], and his son Charles, who wrote that logic is "the science of drawing necessary conclusions".
We have mathematics subject since we are in elementary school, but not physics or chemistry until high school 180.194.246.163 ( talk) 10:14, 12 November 2012 (UTC)
As a graduate student in group theory, I do find it a bit alarming that this picture of a rubix cube has become the cornerstone of Wikipedia's imagery on groups. Yes, the rubix cube does form a group under composition of turns, but is it a key, or even interesting example of hows a group can function? I motion that it is changed to something more relevant to the field itself (e.g. an illustration of a dihedron group might be nice as it is the lodestone of much of most introductory texts). If I am alone on this issue, I will simply retract my argument, but I do find it a tad annoying. — Preceding unsigned comment added by 76.126.169.150 ( talk) 03:46, 13 January 2013 (UTC)
Hello, i have a problem with my homework. and i was wondering if you can help me with it. — Preceding unsigned comment added by Mickenson45 ( talk • contribs) 00:41, 6 March 2013 (UTC)
"Mathematics as profession" does not at all treat the right topic. "Renowned Math prizes" would be a more appropriate title. — Preceding unsigned comment added by 85.224.152.237 ( talk) 20:20, 6 April 2013 (UTC)
I agree with the fact that the content of "Mathematics as profession" has nothing to do with it's title. "Math Prizes" or "Math Awards" would a more appropriate title for that content. Anyway, would be a good thing to have a section titled "Mathematics as profession" or something similar to deal with the professional jobs where mathematicians apply their knowledge. MickMurillo ( talk) 21:21, 3 May 2013 (UTC)
The philosophy of mathematics, the branch of philosophy that studies mathematical assumptions, foundations, and implications, is one of the biggest branches of philosophy in the world. This branch never stops growing, from Thales, a great mathematician from Ancient Greece, to V. N. Bhat, a small mathematician from India, they have all added what little they could. Every idea and theory has helped this art grow. Every year mathematicians discover something new that helps us understand mathematics a little better. The mathematicians don't MAKE or ADD to the subject, they DISCOVER something that is already there. This is the crux of the matter, you cannot make something in math, everything is just there. There can, however, be another way to interpret the art. The Arabic numerals, which were actually made in India but were carried to England by Arabic Traders, are known as the language of math. However, if you lived in Babylon in prehistoric times, you would have a whole different way to express mathematics. The Arabic numerals aren't math, they are just a way to interpret the art. However that means that mathematics cannot be defined because there is no definite way to express it. From the plastic ladybugs used by teachers in 2nd grade to help you add and subtract, to the Arabic numerals used around the world, they are just different languages used to express math, just like all the different languages in the world that are used to express people. Math is a never ending problem, that can be used to solve problems. Just like you can never count to infinity using a language of math, you can never be finished with math. There will always be a new theory to try or a new method that explains another one of life's great questions.
-- Professor Captiosus — Preceding
unsigned comment added by
Professor Captiosus (
talk •
contribs) 17:29, 20 January 2013 (UTC)
Given the warning in the article pseudo-code about changing the opening, I decided to bring my proposal here, since the way it currently reads is awkward, in my opinion. I propose:
Mathematics (from
Greek μάθημα máthēma, “knowledge, study, learning”) is the study of abstract objects and the
logical relationships among such objects. Mathematics encompasses topics including
quantity,
[14]
structure,
[15]
space,
[14]and
change,
[16]
[17]
[18] although it has no generally accepted
definition.
[19]
[20]
...for the following reasons: 1) Mathematical entities, as represented by symbolic notation, are abstract. They are well defined for the purposes of an axiomatic system, and the rules and operations between mathematical objects are logical in character. I'm sure there is no objection here. 2) The topics listed are not well defined, and while math is used to study things like structure and change, it is NOT the study of naturally occurring structure or change, but the study of abstract representations of such. That is to say, the quantum zeno effect negates the direct correspondence with reality of infinitesimal calculus. Topology deals with abstract surfaces ect. 3) The "and more" is amateurish and doesn't do anything to inform a reader about what mathematics actually is. 4) I don't even know what is meant by "the abstract study of subjects" - the verb study is surely only undertaken by a physical human or physical computer, an abstract object cannot "study" anything as far as I know. The word "subjects" is too ambiguous and probably incorrect. In colloquial parlance "subjects" can mean topics of learning in school or whatnot, but "fields" or "disciplines" works better if I understand the connotation correctly. In any event, Math is the the discipline that makes use of well-defined abstract objects and manipulates them by logical rules and operations. One might even include "rigorous" before "study" in my proposed intro, but it's not particularly important. What I see as important is to do away with "abstract study" because nobody even knows what abstract study is. Either everything that could conceivably studied is abstract or nothing is. My point is that I could see dog feces on my shoe and look at it carefully and in my brain associate dog feces with my previous understanding of dog feces, and the structure of the smear on my shoe could be "structure", and by the current lead paragraph I would be doing math. To me, a definition that excludes nothing is a poor definition. -
Fcb981(
talk:
contribs) 00:56, 29 August 2012 (UTC)
{{
cite journal}}
: Unknown parameter |month=
ignored (
help). For this reason, it's not appropriate for us to give a proper definition that draws a clear boundary between what is and is not mathematics; we settle for a rough distinction that leaves the boundary indeterminate.Wrong link in lead paragraph - the space link should go to Space (mathematics) not Space. — Preceding unsigned comment added by 173.79.197.180 ( talk) 02:02, 27 February 2013 (UTC)
It is just a minor change but i cant touch it.
On top section last paragraph,it said, "Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind."
It is ambiguous. A mathematician can engage in "pure mathematics" without mathematics and any application in mind. The sentence isn't exactly false but ambiguous. If i can edit i would just delete ", or mathematics for its own sake, ". If you think we must mention something like "pure mathematics often has mathematics in mind", then try split the sentence in better shape.
Also,this wiki article is trivial and important for all, extra care on wordings/semantics must be given, so it doesn't spread any misleading information. 14.198.221.131 ( talk) 16:34, 23 December 2012 (UTC)
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The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.
Mathematics…is simply the study of abstract structures, or formal patterns of connectedness.
Calculus is the study of change—how things change, and how quickly they change.
{{
cite book}}
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link)
The mathematical study of change, motion, growth or decay is calculus.
Mura
was invoked but never defined (see the
help page).Runge
was invoked but never defined (see the
help page).![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 10 | Archive 11 | Archive 12 | Archive 13 | Archive 14 | Archive 15 |
The discrete mathematics paragraph of this article was misleading. All the areas mentioned include both discrete and continuous mathematics.
Information theory isn't only discrete mathematics, information theory also applies to analog signals. See Differential entropy.
Analog signal processing is also a form of computation.
Theoretical computer science considers both discrete and continuous computational processes, and both discrete and continuous input/output:
Including the question of P!=NP over Real numbers
For now, I've added the above information to the article, but the whole section on discrete mathematics should be deleted, and its content redistributed to elsewhere in the article. Bethnim ( talk) 13:37, 23 March 2010 (UTC)
User Keifer Wolfowitz seems to be claiming completely without any references of [or? Kiefer.Wolfowitz ( talk) 19:57, 23 March 2010 (UTC) ]
proof that Maths is not part of logic. Note that logic includes the study of inconsistent systems and many other things, and it does not seem that there is any part of mathematics that is not part of logic in the broad sense. But there are clearly parts of logic which are not usually considered mathematical.- Wolfkeeper 19:43, 23 March 2010 (UTC)
He also seems to be maintaining a claim that maths is actually not logical in the article; with an apparent oblique reference to Gödel's incompleteness; this is frankly a bizarre non sequitor and completely unreferenced.- Wolfkeeper 19:43, 23 March 2010 (UTC)
There's no need for personal attacks or displays of emotion in our discussion about logic - it's illogical, Captain ;-) What we should do is, as was pointed out above, report what good sources have claimed, after reaching consensus here. These ideas were not born fully formed, and it is not surprising that there is some inconsistency in their use. But the subject is big, and it is important to keep it concisely worded. Stephen B Streater ( talk) 20:37, 23 March 2010 (UTC)
http://www.mathscentre.ac.uk/students.php http://www.mathtutor.ac.uk/ I would add them to the article but it is locked. Please could someone add it when it is unlocked, as I may not return here for years. Thanks 89.240.44.159 ( talk) 12:42, 19 April 2010 (UTC)
In my humble opinion, more attention should be paid to the definition of mathematics. The first line of the article - as far as I know - is not a recognised definition, more a sketchy impression what mathematics is roughly like.
In a true "mathematical" spirit, the question must be answered first whether the concept "mathematics" can be defined at all. A mathematician told me that it is fundamentally impossible to give such a definition, at least to mathematicians (but I did not fully understand his reasoning).
Is mathematics perhaps just a term culturally attached to a rather arbitrary choice of some forms of logical reasoning? Perhaps a definition can only be given if a purpose is decided first (e.g. there are "ontological" and "teleological" definitions: the former try to grasp the essence of a concept, the latter are a choice that can be practical or unpractical, but is never correct or incorrect).
On reason to be strict in the definition of mathematics is the continuing debate about the patentability of mathematical algorithms. Some argue that all algorithms are inherently mathematical, I am inclined to believe that none of them is: only the proof of an algorithm is - perhaps - mathematics. Is all mathematics inherently so fundamental knowledge that it should never be withdrawn from the public domain? But many mechanical inventions basically are geometric, and geometry is a branch of mathematics.
Is there a fundamental difference between philosophy and mathematics? Or is it just cultural: mathematicans prove, philosophers cite. Both are disciplines that do not depend on observations (which is afaik a reason not to consider them "science" at all, in some perceptions - which must be a deception for a PhD in maths!). Maths often deal with quantities, but e.g. boolean algebra doesn't. Math's use shorthands - formulas - but the Pythagoras theorem in plain language is still mathematics.
Who responds to the challenge? Rbakels ( talk) 19:54, 12 August 2010 (UTC)
The BBC programme In Our Time presented by Melvyn Bragg has an episode which may be about this subject (if not moving this note to the appropriate talk page earns cookies). You can add it to "External links" by pasting * {{In Our Time|Mathematics|p00545hk}}. Rich Farmbrough, 03:17, 16 September 2010 (UTC).
An image displaying the infinity symbol eight times seems unnecessary to me, and I think that removing it would greatly improve the article's aesthetic quality. —Preceding unsigned comment added by 131.104.240.36 ( talk) 10:28, 20 October 2010 (UTC)
That doesn't really work either, as it doesn't illustrate the parallel postulate, or at least not without far more explanation (and it's not controversial in modern mathematics). But it's on history, not mathematical theory, so detailed explanation is just distracting. What about something from commons:Category:History_of_mathematics ? A few things stand out as historic and interesting to me, such as file:Yanghui_triangle.PNG, or one of these commons:Category:Geometria by Augustin Hirschvogel.-- JohnBlackburne words deeds 22:48, 20 October 2010 (UTC)
Though I have nothing but respect for Euclid, I'm not sure that Raphael's painting of him is the best lead image for this article. It seems to me that something more lively or visually interesting might be preferable, similar to the approach used by the biology and chemistry articles.
I would suggest a pair of images, similar to the layout on the chemistry article. My pick would be the two images on the right. The first image illustrates geometry (specifically Desargues' theorem), and was obtained from Portal:Mathematics/Featured picture archive. The second is a picture of the Mandelbrot set, and is a mathematics-related featured picture on Wikipedia (see Wikipedia:Featured pictures/Sciences/Mathematics). Jim.belk ( talk) 00:16, 16 November 2010 (UTC)
I propose that we add www.onlinemathcircle.com to the external links section. A good explanation of what it is would be: "A Community Dedicated to Making Mathematics More Open" —Preceding unsigned comment added by Shrig94 ( talk • contribs) 02:14, 21 November 2010 (UTC)
It seems inappropriate to me to advertise prizes in this article. The article is about mathematics, not mathematicians. There are no such sections in the articles about politics or economics, for instance. The insertion appears to have been placed arbitrarily into a section in which it does not belong, in any case. 131.111.184.95 ( talk) 08:37, 15 December 2010 (UTC)
Let's agree to disagree. Rick Norwood ( talk) 14:19, 17 December 2010 (UTC)
In reference to the disputed claim that
I have to say that I agree that this assertion should not appear without qualification, because not everyone agrees (for example Torkel Franzen did not agree). However it is a serious and widely held point of view, likely the majority view, and ought to be represented. I suspect that Wolfkeeper and Kevin Bass don't understand the argument for how Goedel's theorems might be said to refute logicism, and this is a bit subtle and I'm not going to go into it right now. It is strictly speaking beside the point anyway for purposes of editing the encyclopedia — the challenge is to source the statement, and to correctly attribute it, not to a single thinker because it's a widely held view, but to a current of mathematical thought. -- Trovatore ( talk) 21:24, 23 March 2010 (UTC)
Nearly all mathematical concepts are now defined formally in terms of sets and set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces are all defined as sets having various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of relations is entirely grounded in set theory.
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.
Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes derivations of more than 10,000 theorems starting from the ZFC axioms and using first order logic.
(unindent)they seem rather straightforward to me, but yes, yes, i shall digress. the issue is the phrase "...important work in mathematical logic showed that mathematics cannot be reduced to logic." , with which i still take issue with, if for slightly altered reasons. by "reduced to logic" i read something like "proofs of theorems be expressible in a reduced formal grammar", and I maintain that they can, e.g. a turing machine or a ZFS+AOC system. and i contend that the subtler point that you need a few axioms beyond the basic and,or, in, not operations, which are noentheless expressible with those operations, and that some egregious purists might balk at that is a bit trivial and a bit to fine to be stated so boldly, apart from the phrasing as worded being -- as we have just witnessed -- misleading. Kevin Baas talk 21:02, 24 March 2010 (UTC)
There's a lot of nonsense said above (on both sides of the argument, if there is one) that I will not comment about. However there is a simple issue that needs no technical arguments. The current version, which Trovatore keeps reverting to, starts: "Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper." I think all mathematicians would agree that mathematics is not experimentally falsifiable; it is hard to imagine how an experiment could falsify mathematics. It is quite conceivable that mathematics, or some part of it, will one day be found to be inconsistent (remember Russell's paradox?), but if it happens, it will have nothing to do with experiment. The universe has been found to not be a Euclidean space, but that does not affect the work of Euclid (which is not entirely rigorous anyway, but that is another issue) any more than it affects hyperbolic geometry (which does not model the universe either). I'm unsure what Popper actually though about mathematics (his WP article does not mention mathematics as subject at all), but I think falsifiability can only be taken to characterize empirical science, which mathematics simply isn't. As an aside, it would be more interesting to know if many philosophers believe that philosophical theories are experimentally falsifiable. But I digress.
Next sentence: "However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, 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.'". OK, so (if I understand this correctly) Popper believes (around 1995, presumably a bit before his death in 1994) that mathematical hypotheses are conjectures that could be experimentally falsified, which just shows he doesn't understand what a hypothesis is in mathematics. But in any case there is no relation with Gödel's findings of the 1930's; I think this sentence makes a completely unjustified link between them and Popper's quote (in which "even recently" is unlikely to mean "before 1930"). And I would like to know what kind of non-logical element "many mathematicians" would like to invoke to resolve statements that according to the incompleteness theorem cannot be decided by logic alone.
So to get to the point I really wanted to make, there are two issues mixed up in this sentence which in fact are totally unrelated: (1) the question of whether mathematical axioms are assumptions about reality that could be experimentally falsified, and (2) the question whether in principle all mathematical statements can be proved or disproved in an appropriate formal logical system. Gödel's incompleteness theorem shows that (under mild assumptions on what "mathematical statement" and "formal system" mean) the answer to (2) has to be "no". So there will be in every theory some statements that cannot be decided from the axioms of the theory by pure logic. But that is miles away from anything involved in question (1). First of all such a statement is not an axiom of the theory, or a hypothesis of any particular theorem, which are simply assumed to be true in order for the theory/theorem to be applicable; it is a statement whose truth or falsehood one might think to be deducible from the (assumed) truth of the axioms, but in fact is not. And second, axioms have long since ended being considered to be "self-evident truths", they are just starting points of a theory that implicitly determine what the theory is about; they are meaningless in reality, or in any other theory. Take the axioms of your favorite theory: groups, probability, topology, mathematical analysis, (and yes) geometry, set theory, even mathematical logic itself. Which means that the answer to (1) is also "no", but with no relation whatsoever to Gödel's work.
And logicism in all this? It certainly never assumed that mathematics needs no axioms. It is inconceivable to base say geometry on pure logic without some axioms telling what geometric notions like "point" and "line" mean. It also does not affirm that all mathematical statements can be decided by pure logic (from a given set of axioms). What it probably does affirm (I'm in no way an expert on this) is that apart from the rules of logic and the axioms, mathematics needs no vague kind or reasoning based on things that are "obvious" without being able to be formalized. It definitely rejects the idea that there are mathematical statements whose truth is open to experimental verification or falsification, but that does not distinguish it from other schools of thought (Popper notwithstanding). Marc van Leeuwen ( talk) 14:03, 10 January 2011 (UTC)
Not to anyone particular, but to the attitude at the heart of this whole fight: please do not use philosophical waffle without being clear about the mathematics first, or you will be part of the grand scheme to reduce philosophy to random bits of pretentious babble about other subjects as spouted by those who have never actually studied the other subjects for their own sake.
Dudes. Why is there a bunch of infinity signs in this article? They serve no purpose. I suggest we remove it, —Preceding unsigned comment added by 134.173.58.98 ( talk) 04:07, 27 January 2011 (UTC)
Some races, talk mathematics structure within their plain language, and have no written symbols to prove it. One example might be that according to a person who lives where there is no winter, he/she might only know snow and ice, but to a person who lives where it is winter for half the year, there are numerous kinds of snow definitions. There is granular, powdered, drifting, hard packed, just right to make an igloo, too hard even to leave tracks and I can go on and on, and another person will understand just what I am describing in exact detail. —Preceding unsigned comment added by 65.181.32.135 ( talk) 16:58, 9 February 2011 (UTC)
This is a terrible article starting with a shit definition of maths justified by poor sources, and suitable only for children. How is this definition any different for an equivalent definition of physics by replacing the word mathematics for the word physics? Quantity, change and space, are physical concepts, mathematics deals with concepts that are patently not physical. That leaves structure. To say that algebraic topology, finite geometry, graph theory etc deals with structure is uninformative and misleading
The following are examples of awful sentences:
Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[8]
Mathematics arises from many different kinds of problems.
Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area.
Number theory also holds two problems widely considered to be unsolved: the twin prime conjecture and Goldbach's conjecture.
Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set.
The study of space originates with geometry – in particular, Euclidean geometry.
Who the fuck writes this shit? Do they read what they wrote?
The only half-decent math entries on wikipedia, tend to be those that are too technical for idiots to fake, and even then they tend to be verbose, repetitious and awkwardly phrased. —Preceding unsigned comment added by 86.27.195.112 ( talk) 13:42, 20 February 2011 (UTC)
"....establish truth by rigorous deduction from appropriately chosen axioms and definitions" This is not uncontroversial. It seems like what mathematicians wnat to believe about themselves, more than something factual. Also, why can't I edit this article? Sincerely, Mythirdself. —Preceding unsigned comment added by Mythirdself ( talk • contribs) 19:10, 28 April 2011 (UTC)
The reference for the sentence I criticized isn't even comprehensible. It's (I assume) an author: "^ Jourdain". What's the work, page number, etc.? —Preceding unsigned comment added by Mythirdself ( talk • contribs) 23:31, 29 April 2011 (UTC)
I removed this quote from the lede, because of undue weight:
Arnold wrote that mathematics is a branch of physics, among other entertaining absurdities. Kiefer. Wolfowitz 08:51, 2 May 2011 (UTC)
I removed the following section, which seems to be far below the rest of this article in quality, and which also seems to give undue weight to speculations:
Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper. [1] However, in the 1930s Gödel's incompleteness theoremsconvinced many mathematicians who? that mathematics cannot be reduced to logic alone, 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." [2] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.
An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [3] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method. citation needed
The opinions of mathematicians on this matter are varied. Many mathematicians who? feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others who? feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics. citation needed
IMHO, the article needs a discussion of the unity of mathematics, how the same objects appear in apparently disparate fields of inquiry, such as the role of groups in complex analysis (homotopy or homology), geometry, and polynomial equations. (Peirce referred to this as surprising to find the same _____ in an African jungle and the Alaskan Klondike!)
Kiefer.
Wolfowitz 12:26, 1 May 2011 (UTC)
I moved that section to the bottom, because imho it still reads like an essay, rather than an encyclopedia article, and its weight may be undue. Editor Trovatore disagrees, so this is worth a discussion. Kiefer. Wolfowitz 20:24, 26 May 2011 (UTC)
From the start, I think the link to "space" in the first line:
Is probably meant to link to the http://en.wikipedia.org/wiki/Space_(mathematics) page instead, though neither of them come anywhere close to being explanations of "Space" that belongs in the definition of "Mathematics"...
Neither space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. [6]
Nor http://en.wikipedia.org/wiki/Space_(mathematics) In mathematics, a space is a set with some added structure.
Seem to be relevant descriptions for the more abstract concept "space" of which Mathematics studies.
The intro / declaration from the Mathematics portal seems to be a more fundamental description: Mathematics, from the Greek: μαθηματικά or mathēmatiká, is the study of patterns. Such patterns include quantities (numbers) and their operations, interrelations, combinations and abstractions; and of space configurations and their structure, measurement, transformations, and generalizations. Mathematics evolved through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the systematic study of positions, shapes and motions of abstract objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.
There's a typo in the Applied Mathematics section. Looks like someone only half pasted a quote: 'formulation and study of mathematical models. — Preceding unsigned comment added by 94.195.50.242 ( talk) 10:30, 30 May 2011 (UTC)
Let's try to work on a version of the lede here instead of getting into an edit war. As I mentioned, I object to tracing the axiomatic method back to Euclid in describing modern mathematics, because this was simply not the case before Peano, Hilbert, and Co, and we don't really know what Euclid and his contemporaries did. Describing mathematics flat out as formal derivation of theorems from axioms is philosophically naive. Tkuvho ( talk) 10:42, 2 May 2011 (UTC)
I still have philosophical problems with this: "Since the pioneering work of Giuseppe Peano, David Hilbert, and others on axiomatic systems in the late 1800s, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions." The first sentence seems more applicable to logic than math. The second is a tautology: is the definition of "a good model" that which "provides insight and predictions?" Isn't insight subjective? — Preceding unsigned comment added by Mythirdself ( talk • contribs) 18:25, 26 May 2011 (UTC)
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I'm not sure quoting a famous and idiosyncratic figure like Quinne is a good idea. He's brilliant, but he has his own very Quinnian views. Something blander (hey, it's an encyclopedia not a revolution) might be more appropriate. Mythirdself ( talk) 18:37, 2 June 2011 (UTC)
Mathematics undergoes specialization, yet continuity within mathematics is maintained by the discovery and elaboration of structures/theories whose powerful abstractions provide fresh insight. This quote form Charles Sanders Peirce is probably too long for inclusion:
The host of men who achieve the bulk of each year's new discoveries are
mostly confined to narrow ranges. For that reason you would expect the arbitrary hypotheses of the different mathematicians to shoot out in every direction into the boundless void of arbitrariness. But you do not find any such thing. On the contrary, what you find is that men working in fields as remote from one another as the African diamond fields are from the Klondike reproduce the same forms of novel hypothesis. Riemann had apparently never heard of his contemporary Listing. The latter was a naturalistic geometer, occupied with the shapes of leaves and birds' nests, while the former was working upon analytical functions. And yet that which seems the most arbitrary in the ideas created by the two men are one and the same form. This phenomenon is not an isolated one; it characterizes the mathematics of our times, as is, indeed, well known. All this crowd of creators of forms for which the real world affords no parallel, each man arbitrarily following his own sweet will, are, as we now begin to discern, gradually uncovering one great cosmos of forms, a world of potential being.
Charles Sanders Peirce (``CP 1.646)
Hi, Perhaps the user has not understood the comments made to 'change the syntax'. The [ | reference book]'s [ | introduction], [ | Development of Philosophy], etc. are indeed exercises in glorifying Greek Mathematics. How is this a reliable source is just one question, the point here is 'The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.' while at other places like India [ | you may find the same], another source. The book therefore can only be interpreted as a reference to beginning of the systematic study of mathematics in its own right in the Ancient Greece. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 17:00, 6 June 2011 (UTC)
Hi, I would like to know why are the changes reverted after deleting 4 sources in the guise of statement that one source is unreliable without giving any proofs. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 07:10, 6 June 2011 (UTC)
I think you are confusing mathematics with the various rules-of-thumb used to build structures. Standard reference works agree that Greek mathematics began at an earlier date than Indian mathematics. That in no way minimizes the many important contributions of Indian mathematics. Rick Norwood ( talk) 14:19, 7 June 2011 (UTC)
I thought discussions on Wikipedia would be at a higher level than the usual Internet fare. Especially for the article on mathematics. Does being educated and sharing a good cause reduce the incidence of flame-wars? — Preceding unsigned comment added by Mythirdself ( talk • contribs) 01:45, 9 June 2011 (UTC)
Just found a page on Ruler on Wikipedia for information. I am sure it applies to mathematics to an extent. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 19:52, 16 July 2011 (UTC)
I think the Etymology from Semitic Arabic is the important thing. Because it tells the history of Mathematic just from the etymology and begin crediting some specific mathematic such as Al-Jabar, Arithmatic, Logarithma (read Loharitema), and Algorithm. It opens mind that the Greek doesn't invent this specific mathematic since The Arab invent the specific system number which lead to many specific mathematic knowledge such as Al-Jabar, Arithmatic, Logarithma, and Algorithm. See all the Al- infront of words, explain they come from Arab with Arabic names and Arabic system number. So be honest and let's tell the truth to the world this mathematic come from Arab. The Arab own the system number and put God name in Arab on mathematic knowledge. For the editor Please be honest because this is an important encyclopedy. I hope the etymolog from Semitic Arab soon move to the main page. And Please Complete the article about Al-jabar, Arithmatic, Log, Logarithma and Algorithma it self. — Preceding unsigned comment added by 202.152.202.248 ( talk) 04:06, 28 July 2011 (UTC)
Now that the lead sentence has (weak) citations, which is great, there is still the issue that "quantity, structure, space, change" are listed as "fields of mathematics". Mathematicians simply do not talk like this. Maybe philosophers of math talk like this? Even if so, it needs citation, and equal weight should be given to how mathematicians talk about math. Most fields of math are about specific problems (or classes of problems, or sets of axioms) that draw on more than one of these four "ingredients".
Further, if the graphics are supposed to suggest that group theory is a subdiscipline of "structure" (let's accept that it is, for now), then they also suggest that "complex numbers" is a subdiscipline of "quantity". Is "complex numbers" supposed to be complex analysis? Or is that a subdiscipline of "change"? I know that this task is difficult, but the current solution is not good. We could do better by simply citing from the American Mathematical Society's (or another comparable organization's) classifications and copying text from the appropriate Wikipedia articles. We should also describe historical views on the organization of mathematics (e.g. as indistinguishable from physics or natural philosophy?). The current solution seems neither contemporary nor historical. Mgnbar ( talk) 15:59, 22 July 2011 (UTC)
Field | 2010 | 2009 |
---|---|---|
Algebra, Number Theory | 230 | 223 |
Real, Complex, Functional, and Harmonic Analysis | 102 | 98 |
Geometry, Topology | 149 | 139 |
Discrete Math, Combinatorics, Logic, Computer Science | 116 | 141 |
Probability | 73 | 82 |
Statistics, Biostatistics | 495 | 483 |
Applied Math | 229 | 169 |
Numerical Analysis, Approximations | 88 | 93 |
Nonlinear Optimization, Control | 27 | 24 |
Differential, Integral, and Difference Equations | 102 | 117 |
Math Education | 15 | 14 |
Other | 6 | 22 |
Total | 1632 | 1605 |
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The word "mathematics" (jovo or java in modern language) comes from the semitic hebrew Arabic (Arabic: محمد Muḥammad, pronounced [mʊˈħæmmæd], which means The Praised Man or The Honourable Man, The Man was the Mesenger of the religion of Islam(born 570, Mecca, Arabia [now in Saudi Arabia]—died June 8, 632, Medina) [Britannica Group 1] [1] , and is considered by Muslims to be a messenger and prophet of God, the last law-bearer in a series of Islamic prophets, and, by most Muslims,the last prophet of God as taught by the Quran. This Original word is Mohammadika which means Your Praised Man or Your Honourable Man. Muhammad in Al-Qur'an, had been told by God to read (in Al-Alaq first verse) and to seek refuge to Lord (keeper) of Calculation (Robbilfalaq) (it means Allah) in Al-Falaq first verse.
The word mathematicis irrelevant with máthēma comes from μανθάνω (manthano) in ancient Greek and from μαθαίνω (mathaino) in modern Greek, both of which mean to learn Since The Arabian originally invent the zero system number and the Decimal system number (read Al-Qur'an see the system number for Al-Qur'an Juz) which mean the Greek (Pythagoras and Archimedes)and The Roman didn't know the Arabic system number until Muhammad spread Islam. The word Mathematic become Jovo or Java in modern time it came from the original word in Arabic محمد. From Al-Qur'an the Arabian muslim originally know all human Calculation is relative than God calculation so in the mathematic knowledge they use Al-Asmaul-Husna(99 names of the only one God of Islam) [2] [3] [4] [5] as the name of this knowledge such Ar-Rahim ( Arithmatic, see rythm from "rhyme" and rhyme from rahim), Allah ( log), Allah Ar-Rahim ( logarithm), Al-Gofur Ar-Rahim (AlGorithm, some say from Al-Khawarijm) and Al-Jabar (Al-Gebra). Since this knowledge come from Arab soon people call it "mathematics" refers to Muhammad.
110.137.147.244 ( talk) 14:18, 27 July 2011 (UTC)
I have been musing on the definition of mathenatics as "the study of quantity, structure, space, and change". It strikes me that I could take my camera, get out there and contrast a single human being with a crowd, compare the structure of a leaf with a network of roads, picture the clever use of a tiny volume in a yacht and the humbling vastness of the mountains, and document the changes of seasons. This would provide me material for a show "the study of quantity, structure, space, and change", and we would not recognise any of it as mathematics.
Rather than characterise mathematics by WHAT it studies (even though the list is very compact and comprehensive), I would attempt to characterise, in one sentence, HOW it does it. So my two cents: "Mathematics is the art of rigorous abstract thinking"
Does it make sense?
Obviously the proposed definition touches on many elements already well discussed in the article.
Philippe Maincon ( talk) 17:24, 15 July 2011 (UTC)
Charles Sanders Peirce's New Elements of Mathematics has an thorough and stimulating discussion of previous definitions of mathematics. Kiefer. Wolfowitz 10:49, 29 July 2011 (UTC)
Looking over the above exchanges, I find that I have left some things unexplained that I now believe I know how to say more clearly, and on a related note, that I also do not like the first paragraph as it stands. (Well, I never did really, I just thought it was "least bad", but I now find that it has evolved in a way that I think is suboptimal, and I have a candidate point in the past where I think the text was better, that we should consider a starting point.)
What is a definition? In mathematics itself, our definitions, say of a "ring" just for example, provide precise demarcations. They divide things into rings and non-rings, with nothing in between (ignoring quibbles about whether you require a unit). It's not unnatural that mathematicians would like to be able to provide that sort of a definition for mathematics itself.
Perhaps such a project is possible; I am skeptical, but for the sake of argument suppose that it is possible to define "mathematics" in such a way that it precisely demarcates that which is mathematics from that which is not mathematics. We are left with the problem that no such definition is agreed among mathematicians or philosophers of mathematics. It is not Wikipedia's function to pick one from among them. We simply may not do that; it is a blatant violation of WP:NPOV.
To preserve neutrality, we could futz around with competing definitions and say who uses them. But I hope everyone agrees that the lead is not the place for that. The article is about mathematics, not about how to define the term mathematics, and more than two or three sentences is too much to spend on the definition in the lead section.
But luckily, we don't have to. All we have to do is recognize that the sort of "definition" required in the lead paragraph is not a demarcation at all. The first sentence of dog does not give you the information required to divide all objects into dog and non-dog, and it can't be expected to. Rather, it gives you enough information to identify what the article is talking about, and possibly tells you something you might not have known about what is included. (Are dingoes dogs? Does mathematics deal with things other than numbers?)
That's why an NPOV definition in the lead will necessarily have a "shopping-cart" aspect to it. It can't demarcate what is mathematics from what is not, but it can say some things that we all agree are mathematics, including some that some readers may not have realized are mathematics. -- Trovatore ( talk) 07:50, 1 August 2011 (UTC)
Here are the first three sentences as they stand: (I'll leave the refs but won't put a reflist):
Here is the version I like better, from 1 February 2009:
Here's the main reason I like it better: The current version pretends to be a demarcation. It lists four (vague) things; those are mathematics, nothing else is. As I argue above, that cannot possibly meet NPOV. The Feb 2009 version does not; rather, it lists some of the things that mathematics studies, without claiming to exhaust the subject.
That's not to say it can't be improved. I would change "the academic discipline" to "an academic discipline", making the non-demarcative nature more explicit. Also I would probably incorporate some of the language of the current version as well; I have no detailed proposal at this time for how to do that. But I hope my main point is clear. -- Trovatore ( talk) 07:50, 1 August 2011 (UTC)
{{edit semi-protected}} After " Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[10]" add "At times it is difficult to know where pure mathematics ends and applied mathematics begins." Knwlgc ( talk) 05:12, 13 September 2011 (UTC)
Procedural note: I'm removing this 'edit semi-prot request' for now, pending consensus Chzz ► 01:37, 17 September 2011 (UTC)
My request still exists, it has not gone away. It was assumed when I sent in my edit request that there was not consensus (otherwise this article would not need semi-protection), how has the state of my request changed since its conception? I should add that an admission of willingness to discard my edit request, is not a request to discard my edit request (had it been then I would have discarded it). Knwlgc ( talk) 03:08, 17 September 2011 (UTC)
The work in the Centre for Experimental and Constructive Mathematics is more than twenty years ahead of Mathematics as an international discipline. This has been inside information which might now be public.
It seems as though there could have been some confusion about what Applied Mathematics was. Generally, Applied Mathematics has been a variety of subsets of ad hoc amalgamations of Theoretical Physics, Statistics, Computing Science, Mathematics, Engineering Science, and almost anything else. The common denominator clarifying what Applied Mathematics has been was tricky to find. If/when ad hoc disciplines (an oxymoron) try existing primarily as theoretical constructions built for the purpose of trying to get money any old way, the result is internal organizational inefficiency. It would be unfair and dreamy of Mathematics, as the international discipline this is, to ask other organizations to have internal cohesion due to our recent update of the definition of if, without first demonstrating what we mean by internal cohesion.
(Instantiations and examples differ slightly: instantiations are generalizable and examples may have generalizable properties and features. However, part of this work includes teaching mathematics to seven billion people, thus for now I preferentially use the word example.)
An example of confusion arising from lack of internal organizational cohesion due to presence of ad hoc discipline, is the 50% vote of support Jonathan and Peter Borwein received from participating voters (abstention rate unknown) for establishing the Centre for Experimental and Constructive Mathematics at Simon Fraser University, a tie which was broken in preference of establishing Experimental and Constructive Mathematics by someone in senior administration circa 1992, and the upshot of which includes the Organic Mathematics Project which singularly redefined Mathematics online education and collaboration; correction of Aristotle; redefinition of the word if; free demonstrative and instructional tutoring services for the world's central banks with respect to the additive and multiplicative identities; open questions including where Mathematics proofs come from, ownership of intellectual property in collaborative processes, how Mathematics and Mathematically informed disciplines develop communities; and the intellectual property ownership question: who owns Mayer Amschel Rothschild's intellectual property conceived circa 1794, still in circulation, and which I previously cited in my work as osmosis.
Generally and obviously, a discipline is not yet qualified to offer its services to customers until after the discipline has demonstrated the same expertise internally. Having organizations' internal and external services match works. The Centre for Experimental and Constructive Mathematics and our network is perfect for driving optimization by osmosis and naturally occurring, real selection processes. Therefore handling the question << what is Mathematics >> is part of this constructive instruction. This exemplifies what Applied Mathematics really is, both in this self-referencing demonstration and explanatory definition update, and in real world ubiquitous application across all disciplines everywhere; therefore we acknowledge Mathematics was previously domesticated partly under Philosophy and partly under Science, and might be correctly understood as a profession under the Institute for Electric and Electronic Engineers, who as an organization has the highest standards in ethics and professional conduct. This Applied Mathematics includes Information Theory and Computer Architecture. Having the discipline Mathematics perfectly located under the IEEE solves all problems related to franchising Mathematics other than my unique personal problem if The Rothschild Family prefers to take me to court for accidental intellectual property theft.
References:
http://www.cecm.sfu.ca/organics/project/
http://www.ieee.org/index.html
Founder by Amos Elon, ISBN 0 670 86857 4
JenniferProkhorov ( talk) 19:23, 19 October 2011 (UTC)
Does mathematics really belong to the mathematical sciences? Mathematical sciences says
Mathematics isn't primarily mathematical in nature. It is COMPLETELY mathematical in nature.
I don't know of a better place to discuss this than the wiki page for Mathematics, although the question pertains to any page under the section of Mathematics.
Can we implement mathematical symbols that are ALSO hyperlinks to wiki pages for each symbol?
I do not think all mathematical symbols have wiki pages, but I don't see why not.
At least they could link to a relevant page in which the symbol is heavily used.
I believe this would make it significantly easier to learn mathematics from wikipedia. — Preceding unsigned comment added by 140.247.59.253 ( talk) 23:45, 27 July 2011 (UTC)
The idea is not just to have a list of symbols, but to use hyperlink versions of the symbols on any or all wiki pages within Mathematics. Like many special terms, the first usage of a symbol on any wiki page could be a hyperlink version of the symbol. Much in the same way that unique terms can be clicked on to bring the wiki-reader to the definition of that term, so too could she more quickly learn about the mathematical symbols that crop up in whatever section of mathematics she is currently browsing. — Preceding unsigned comment added by 140.247.59.84 ( talk) 15:29, 9 August 2011 (UTC)
I'm sure we've had this discussion before as I remember thinking it a very bad idea at the time, for various reasons.
If a symbol really needs explaining then add a sentence, e.g. from Euler's formula:
This is far clearer than linking any symbol.-- JohnBlackburne words deeds 02:34, 30 November 2011 (UTC)
Why do we have long paragraph of quotations about mathematics in the lead? Shouldn't that go in Wikiquote? Kaldari ( talk) 22:18, 4 December 2011 (UTC)
How about adding "Mathematosis" to the "See Also" section? 164.107.189.191 ( talk) 14:37, 6 December 2011 (UTC)
It's nice to see this right in the first paragraph of a significant article: "Galileo Galilei (1564-1942) said" I never knew the man lived to be almost 400. Good job, Wikipedia. And the article is locked so I can't even fix this boneheaded error. Ugh. — Preceding unsigned comment added by 131.193.127.17 ( talk) 16:03, 6 December 2011 (UTC)
hi.
I would replace
"However, mathematical proofs are less formal and painstaking than proofs in mathematical logic"
with
"Mathematical proofs are written in a formal language provided/analysed by mathematical logic".
The main reason for this exchange is that mathematical logic is itself a part of math! Therefore, the above statement means something like "trains are faster than TGVs". It is just nonsense. Another reason is that proofs in e.g. algebra are just as formal and painstaking as proofs in mathematical logic...
best regards a phd-student in math — Preceding unsigned comment added by 138.246.2.177 ( talk) 17:48, 20 December 2011 (UTC)
Mathematical proofs really are less formal and painstaking than proofs in mathematical logic. Open any math book to a proof. I'll pick one at random off the shelf behind me, and open it to a random page. "Proof: Recall that a subspace Y of L is said to be convex if for every pair of points a, b of Y with a < b, the entire interval [a, b] of points of L lies in Y." I think this is fairly typical of how a mathematical proof is written. Now, compare with a proof in mathematical logic:
More formal. More painstaking.
Mathematicians usually assume that the kinds of proofs we do in our work could, if necessary, be reduced to mathematical logic, but we never, in practice, do that.
Rick Norwood ( talk) 19:13, 20 December 2011 (UTC)
Mathematical logic usually means mathematical logic, as in Hamilton's Logic for Mathematicians or Manin's A Course in Mathematical Logic. The other topics you mention are in Foundations, rather than in Mathematical Logic. I agree that the proofs in essentially all areas of mathematics except formal mathematical logic are in the metalanguage rather than in the object language. Rick Norwood ( talk) 19:53, 20 December 2011 (UTC)
03-XX Mathematical logic and foundations 03-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 03-01 Instructional exposition (textbooks, tutorial papers, etc.) 03-02 Research exposition (monographs, survey articles) 03-03 Historical (must also be assigned at least one classification number from Section 01) 03-04 Explicit machine computation and programs (not the theory of computation or programming) 03-06 Proceedings, conferences, collections, etc. 03Axx Philosophical aspects of logic and foundations 03Bxx General logic 03Cxx Model theory 03Dxx Computability and recursion theory 03Exx Set theory 03Fxx Proof theory and constructive mathematics 03Gxx Algebraic logic 03Hxx Nonstandard models [See also 03C62]
previous unsigned comment by User:Rick Norwood 21:25, 20 December 2011 (UTC)
thanks for that, rick. see? "Proof theory and constructive mathematics" one section. notice the absence of a separate, independant section on "Proof theory and constructive mathematics for mathematical logic, specifically, which is for some reason different". Kevin Baas talk 21:27, 20 December 2011 (UTC)
I reverted your change because, though you say one thing above, the change you made says another. We need to either omit this entirely or find a way of saying it that is both intelligible to the layperson and mathematically accurate. Rick Norwood ( talk) 14:14, 21 December 2011 (UTC)
In the mathematical sense of the phrase "formal language", it is not possible to be "more formal" or "less formal". A "formal language" is one where the proofs depend only on the form the symbols take, and not on the meaning of the symbols. If you prefer "formal grammar", that is also used in the same sense. I'm primarily following Hamilton's Logic for Mathematicians.
But, back to the point of this discussion. I agree the disputed sentence should either be improved or, if nobody can come up with a good way to improve it, removed.
Rick Norwood ( talk) 16:29, 21 December 2011 (UTC)
A new suggestion:
"For convenience, most proofs are written in a metalanguage and, therefore, have to deal with the insufficiencies of each metalanguage. Nonetheless, mathematical proofs should (!) be written in such a way that a mathematician could translate them into a more formal language with an unambiguous grammar. This more formal proof could then even be checked by an computer. Usually, one of these more formal languages is taught at the beginning of mathematical logic"
btw.: less attacking and more suggestions and we could have closed this secition yesterday....!!! — Preceding unsigned comment added by 138.246.2.177 ( talk) 17:14, 21 December 2011 (UTC)
I actuallyLOVE the lead, and appreciate the approach of quoting a few important folks as they described mathematics. However, I found the context lacking, especially for Galileo's quote. It is too often that the poor metaphors of natural law and language are used to describe mathematics. Such conceptions are invaluable to its history and this article, but there is a responsibility to more properly contextualize this paragraph I'm question. I believe the final sentence, a quote by Einsteiniis intended to achieve this effect, but j would rather see a punchline less punched if it meant clarity that could prevent further propagation of naive interpretations, despite also being valuable for other reasons. — Preceding unsigned comment added by 67.161.64.224 ( talk) 07:43, 6 January 2012 (UTC)
The choice of entries in the See also section looks rather arbitrary to me and is, in my opinion, quite uneven. Iatromathematicians is an extreme obscure topic. Why is Self-similarity on this list? Any suggestions for a criterion to decide what should be on this list? -- Lambiam 12:42, 12 January 2012 (UTC)
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Please transpose the words space and quantity. It was recently found that the "all pages lead to philosophy" loop was broken by an edit to the mathematics page and the problem would be easily fixed by transposing these two nouns. menaing of the page would not be changed and many meme-ers would be made very happy. Thank you.
66.99.120.222 ( talk) 15:24, 20 January 2012 (UTC)
Sorry, to edit articles to cause them to lead to philosophy violates the idea that all articles naturally lead to philosophy. Also, the current word order is used throughout the article and in several other articles. Rick Norwood ( talk) 16:02, 20 January 2012 (UTC)
.................................................................................................. — Preceding unsigned comment added by 205.125.65.84 ( talk) 15:46, 26 January 2012 (UTC)
MATH is defined as the expression of logic through the use of quantity.
It is often believed to be a determining science - which means it has causal properties. This is not true. A symbol cannot have any bearing on physical or metaphysical phenomena. At its best, it expresses logic or the scale and change of entities. At its worst...well let's not talk about it...:))))
LOGIC can be expressed in different forms - Math, Words, Pictures — Preceding unsigned comment added by 117.207.152.172 ( talk) 13:15, 17 March 2012 (UTC)
The first sentence doesn't agree with the citation. The reference says that mathematics is primarily about abstraction, not simply that it is about space, quantity etc. Removing the part about abstraction significantly changes the expression and the meaning, it does not encompass the same range. Teapeat ( talk) 03:47, 19 March 2012 (UTC) For example boolean logic is not really, in any normal sense, to do with space, quantity, structure or change, but is normally considered to be part of Mathematics. It is however an abstraction of logic. Teapeat ( talk) 03:47, 19 March 2012 (UTC)
Given this, I am considering the verification of the reference given to have failed, and the removal of that part to be original research. Teapeat ( talk) 03:47, 19 March 2012 (UTC)
The cited source is not reliable. It's a web page about a university math/comp-sci department. It's probably just copying us, and it appears to have misattributed the famous Galileo quotation to Newton. The virtues of our definition are that it accords well with the kind of definition offered in other reference works, and it reflects the body of the article that we actually have. Please see Definitions of mathematics. I would favor removing the citation and leaving our definition unsourced, but wording it in such a way that it does not come across as the final word on the subject. How about we simply mention the lack of any consensus on the definition of mathematics, and link to Definitions of mathematics? — Ben Kovitz ( talk) 10:02, 11 June 2012 (UTC)
I just made some changes: a new section on definitions, and a rewording of our opening sentence so it addresses the lack of consensus on a definition of mathematics, but is still informative and reflects the body of the article. I removed the old source that was probably quoting us, and added some new sources that I think are pretty solid, with the exception of the two I found for calculus (as the study of change). It would be better to have a single good source for that. I found a pretty amusing source on the inadequacy of any definition to cover all of mathematics: a published critique of this very article in a general survey of mathematics.
There are still problems with the lead. The lead is now five paragraphs, which is too long. Much of that bulk says rather doubtful things that are not covered in the article, no doubt the residue of long-past soapboxing. I think that by merely summarizing the body of the article, the lead could easily be brought back into shape. But, I'm done for the day. — Ben Kovitz ( talk) 23:22, 16 June 2012 (UTC)
MATH: Mental Abuse To Humans — Preceding unsigned comment added by Ilovenickelback17 ( talk • contribs) 00:53, 29 March 2012 (UTC)
Maurice Carbonaro ( talk) 08:00, 20 July 2012 (UTC)
Math seems to be going on in multiple different directions. But if mathematicians could define a goal that all of their research amounts to (like why we're trying to solve the Riemann hypothesis and other unsolved problems, and/or what they have in common), that would be great to include here. (Hint: It's definitely not 42 or NaN.) 68.173.113.106 ( talk) 21:48, 6 March 2012 (UTC)
What I mean is, some people want to investigate topology, others want to solve P versus NP, still others want to do advanced complex analysis. Personally, I'm looking into mathematical finance (even though I'm just a kid). So what do all of these approaches have in common? If at all, why is it important? (Added to 68.173.113.106 ( talk)'s previous comment on 21:53, 6 March 2012 (UTC))
Do we have to find an ultimate goal? Perhaps a better posed question is: Can we find traits common to all branches of mathematics? Quantity, logic, and intuition are all partially correct but not generic enough (e.g., algebra intentionally leaves quantity unspecified). How about "Mathematics seeks to construct abstract entities and discover their properties through logic based on a limited set of axioms." Of course, in trying to be generic I have introduced some terminology, and others are welcome to improve upon this. — Preceding unsigned comment added by 71.236.24.128 ( talk) 22:01, 18 August 2012 (UTC)
I think math as a hobby could become a useful addition to the article if someone worked hard enough on writing it. Marvin Ray Burns ( talk) 02:16, 19 May 2012 (UTC)
There are good examples of math as a hobby, but none of the three names you mention seems to fit well under that heading. Rick Norwood ( talk) 11:54, 10 September 2012 (UTC)
A C.S. Peirce scholar brought this line in the article to my attention:
Two examples of logicist definitions are "Mathematics is the science that draws necessary conclusions" (Benjamin Peirce)[24] and "All Mathematics is Symbolic Logic" (Bertrand Russell).[25]
Russell's quote is quite logicist, but Benjamin Peirce's quote doesn't seem so, since logicism usually means the idea that much or all of mathematics is reducible to logic, not merely the idea that mathematical conclusions are logical or deductive. Have anti-logicists generally held that mathematical conclusions are not generally deductive?
Now, the B. Peirce quote is from "Linear Associative Algebra" in which he goes on to say, beginning lower on the same page (the article's first page),
Mathematics, under this definition, belongs to every enquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by observation. In its pure and simple form, the syllogism cannot be directly compared with all experience, or it would not have required an Aristotle to discover it. It must be transmuted into the all the possible shapes in which reason loves to clothe itself. The transmmutation is the mathematical process in the establishment of the law. [....]
That really doesn't sound like logicism at all. It's more to say the same as his son C.S. Peirce said, that the mathematician aids the logician, not vice versa. (I'm not arguing about whether they were right, just that that was their view.) Anyway, I suggest that Benjamin Peirce's definition of maths not be characterized as "logicist." The Tetrast ( talk) 01:00, 7 September 2012 (UTC).
If nobody comments during the coming five days or so, I'll go ahead and make the change. The Tetrast ( talk) 02:56, 8 September 2012 (UTC)An example of a logicist definition is "All Mathematics is Symbolic Logic" (Bertrand Russell).[25].
"In Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic."[25] Related but more subtle views were put forward by Benjamin Peirce, who wrote that mathematics is "the science that draws necessary conclusions" [24], and his son Charles, who wrote that logic is "the science of drawing necessary conclusions".
We have mathematics subject since we are in elementary school, but not physics or chemistry until high school 180.194.246.163 ( talk) 10:14, 12 November 2012 (UTC)
As a graduate student in group theory, I do find it a bit alarming that this picture of a rubix cube has become the cornerstone of Wikipedia's imagery on groups. Yes, the rubix cube does form a group under composition of turns, but is it a key, or even interesting example of hows a group can function? I motion that it is changed to something more relevant to the field itself (e.g. an illustration of a dihedron group might be nice as it is the lodestone of much of most introductory texts). If I am alone on this issue, I will simply retract my argument, but I do find it a tad annoying. — Preceding unsigned comment added by 76.126.169.150 ( talk) 03:46, 13 January 2013 (UTC)
Hello, i have a problem with my homework. and i was wondering if you can help me with it. — Preceding unsigned comment added by Mickenson45 ( talk • contribs) 00:41, 6 March 2013 (UTC)
"Mathematics as profession" does not at all treat the right topic. "Renowned Math prizes" would be a more appropriate title. — Preceding unsigned comment added by 85.224.152.237 ( talk) 20:20, 6 April 2013 (UTC)
I agree with the fact that the content of "Mathematics as profession" has nothing to do with it's title. "Math Prizes" or "Math Awards" would a more appropriate title for that content. Anyway, would be a good thing to have a section titled "Mathematics as profession" or something similar to deal with the professional jobs where mathematicians apply their knowledge. MickMurillo ( talk) 21:21, 3 May 2013 (UTC)
The philosophy of mathematics, the branch of philosophy that studies mathematical assumptions, foundations, and implications, is one of the biggest branches of philosophy in the world. This branch never stops growing, from Thales, a great mathematician from Ancient Greece, to V. N. Bhat, a small mathematician from India, they have all added what little they could. Every idea and theory has helped this art grow. Every year mathematicians discover something new that helps us understand mathematics a little better. The mathematicians don't MAKE or ADD to the subject, they DISCOVER something that is already there. This is the crux of the matter, you cannot make something in math, everything is just there. There can, however, be another way to interpret the art. The Arabic numerals, which were actually made in India but were carried to England by Arabic Traders, are known as the language of math. However, if you lived in Babylon in prehistoric times, you would have a whole different way to express mathematics. The Arabic numerals aren't math, they are just a way to interpret the art. However that means that mathematics cannot be defined because there is no definite way to express it. From the plastic ladybugs used by teachers in 2nd grade to help you add and subtract, to the Arabic numerals used around the world, they are just different languages used to express math, just like all the different languages in the world that are used to express people. Math is a never ending problem, that can be used to solve problems. Just like you can never count to infinity using a language of math, you can never be finished with math. There will always be a new theory to try or a new method that explains another one of life's great questions.
-- Professor Captiosus — Preceding
unsigned comment added by
Professor Captiosus (
talk •
contribs) 17:29, 20 January 2013 (UTC)
Given the warning in the article pseudo-code about changing the opening, I decided to bring my proposal here, since the way it currently reads is awkward, in my opinion. I propose:
Mathematics (from
Greek μάθημα máthēma, “knowledge, study, learning”) is the study of abstract objects and the
logical relationships among such objects. Mathematics encompasses topics including
quantity,
[14]
structure,
[15]
space,
[14]and
change,
[16]
[17]
[18] although it has no generally accepted
definition.
[19]
[20]
...for the following reasons: 1) Mathematical entities, as represented by symbolic notation, are abstract. They are well defined for the purposes of an axiomatic system, and the rules and operations between mathematical objects are logical in character. I'm sure there is no objection here. 2) The topics listed are not well defined, and while math is used to study things like structure and change, it is NOT the study of naturally occurring structure or change, but the study of abstract representations of such. That is to say, the quantum zeno effect negates the direct correspondence with reality of infinitesimal calculus. Topology deals with abstract surfaces ect. 3) The "and more" is amateurish and doesn't do anything to inform a reader about what mathematics actually is. 4) I don't even know what is meant by "the abstract study of subjects" - the verb study is surely only undertaken by a physical human or physical computer, an abstract object cannot "study" anything as far as I know. The word "subjects" is too ambiguous and probably incorrect. In colloquial parlance "subjects" can mean topics of learning in school or whatnot, but "fields" or "disciplines" works better if I understand the connotation correctly. In any event, Math is the the discipline that makes use of well-defined abstract objects and manipulates them by logical rules and operations. One might even include "rigorous" before "study" in my proposed intro, but it's not particularly important. What I see as important is to do away with "abstract study" because nobody even knows what abstract study is. Either everything that could conceivably studied is abstract or nothing is. My point is that I could see dog feces on my shoe and look at it carefully and in my brain associate dog feces with my previous understanding of dog feces, and the structure of the smear on my shoe could be "structure", and by the current lead paragraph I would be doing math. To me, a definition that excludes nothing is a poor definition. -
Fcb981(
talk:
contribs) 00:56, 29 August 2012 (UTC)
{{
cite journal}}
: Unknown parameter |month=
ignored (
help). For this reason, it's not appropriate for us to give a proper definition that draws a clear boundary between what is and is not mathematics; we settle for a rough distinction that leaves the boundary indeterminate.Wrong link in lead paragraph - the space link should go to Space (mathematics) not Space. — Preceding unsigned comment added by 173.79.197.180 ( talk) 02:02, 27 February 2013 (UTC)
It is just a minor change but i cant touch it.
On top section last paragraph,it said, "Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind."
It is ambiguous. A mathematician can engage in "pure mathematics" without mathematics and any application in mind. The sentence isn't exactly false but ambiguous. If i can edit i would just delete ", or mathematics for its own sake, ". If you think we must mention something like "pure mathematics often has mathematics in mind", then try split the sentence in better shape.
Also,this wiki article is trivial and important for all, extra care on wordings/semantics must be given, so it doesn't spread any misleading information. 14.198.221.131 ( talk) 16:34, 23 December 2012 (UTC)
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The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.
Mathematics…is simply the study of abstract structures, or formal patterns of connectedness.
Calculus is the study of change—how things change, and how quickly they change.
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The mathematical study of change, motion, growth or decay is calculus.
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