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Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science. I wouldn't classify math as a science (science tries to explain how the world works). I wouldn't classify it as an art either (dictionary.com gives it as "1. Human effort to imitate, supplement, alter, or counteract the work of nature.", in which case, photography is arguably not an art). Of course, if you use another definition "5. A nonscientific branch of learning; one of the liberal arts.", it's an art because it's not scientific (but I'm sure there's more than arts and sciences, e.g. I don't think law is either). And if you use "3. High quality of conception or execution, as found in works of beauty; aesthetic value.", then pure math could be an art (since it's often pretty), but then we're left with how to place statistics (since stats isn't pretty) and, perhaps, calculus.
Most US and Canadian unis classify it as a science (though often it's in the "faculty of arts and sciences" which is more like "faculty of miscellany"), and British unis are undivided (most of them list computer science as a BSci, like Oxford, but some list them as a BA, like Cambridge). UWaterloo has a "faculty of mathematics", and you get a BMath.
Arguably, it's closer to science than art (in the sense that it mostly requires the same kind of brain as most sciences). I notice I'm rambling. -- Elektron 06:27, 2004 May 25 (UTC)
Shall we call a poll for which category we should stuff Mathematics under? -- Elektron 16:58, 2004 Jun 1 (UTC)
Gubbubu 20:26, 20 Dec 2004 (UTC)
-- Sean Kelly 20:53, 20 Dec 2004 (UTC)
I dont't think maths is fundamentally different from anything. More, maths could be considered as the modell and an ideal for all sciences. It's a superscience, and if you saw the hystory of science in the XX cent., maybe you would accept the expressions "scientifical" and "rational" became the synonyms of "deductive" and "mathematical".
But I think, this is not so right. I think, you are speaking about the written, formal mathematical proofs, but you forget, these are only the final forms or (drafting of) achievements of mathematical investigation, what is in itself likes every other sciences (see e.g. computer-helped number theory - its an experimental science).
But I think v talk not only bout đ concept of maths, but about đ concept of science. If you would be so kind to define it, maybe I could compare it with the sentences above, and my mind could conceive in wich special meaning of science math's wouldn't be a science ? Gubbubu 18:00, 21 Dec 2004 (UTC).
I think that mathematics as a body of work is not scientific. However, the practice of mathematics is in the vast majority of cases scientific: experimental examples (from special cases, enumeration, and whatnot) have been more than a little usefull throughout the development of mathematics. More importantly, a usual way of approaching a theory is "Oh, this is a nice theory, I'm going to play around with it for a bit, to gather data. When I've found it, maybe I'll find some patterns and be able to prove something". That is the crux really - it's just like the other sciences, only it has this extra step at the end of the scientific process, called proof, which is weighted with such importance that all prior steps are usually omitted (or often presented as consequences of it!). icecubex 9.14, 5 Jan 2005 (GMT)
By definition, mathematics is abstract, and science is about gathering empirical knowledge (it can refer to the process, the people, or the knowledge itself). Surely something can't be both abstract and empirical?
Why do you think science must be empirical? this is only a point of neopositivists' view.
Notice the article I linked to in the heading? That article says it is empirical. Also, the common usage (the common usage I've noticed, anyway) indicates that it is empirical. The definition at Wiktionary does not indicate that it must be empirical, but that definition seems too broad.
If you know anything about neopositivism (I don't), you can start the article! Brian j d 04:17, 2005 Mar 6 (UTC)
Why is the definition at Wiktionary too broad? Does it not imply that Wikipedia is a "science"; that accounting is a "science"? Brian j d 04:18, 2005 Mar 6 (UTC)
Some hold that since it is not empirical, it is not one of the sciences.
This implies that some take a different view, but I can find nothing in the article about any other view.
However, I found in the section "Common misconceptions" the following:
Although Einstein called it "the Queen of the Sciences", by one not-unusual definition, mathematics itself is not a science, because it is not empirical. Brian j d | Why restrict HTML? | 04:42, 2005 Apr 17 (UTC) (signature added later)
Some hold that since mathematical knowledge is not fundamentally empirical, mathematics is not itself one of the sciences, however closely allied.
This implies that some take a different view that contradicts this, but I can find nothing relevant in the article.
The following statement, that contradicts the one above, is still there:
Although Einstein called it "the Queen of the Sciences", by one not-unusual definition, mathematics itself is not a science, because it is not empirical. Brian j d | Why restrict HTML? | 04:42, 2005 Apr 17 (UTC)
Mathematics is usually regarded as an important tool for science, even though the development of mathematics is not necessarily done with science in mind
If mathematics is science, how can it be a tool for science? Brian j d | Why restrict HTML? | 04:11, 2005 Apr 23 (UTC)
The science article (for me anyway) clearly states that science is empirical. How can something be both empirical and abstract ("the science of abstraction")? Brian j d | Why restrict HTML? | 04:09, 2005 Apr 23 (UTC)
One issue here is how 'science' is defined. Most of the Anglo-Saxon world is happy with science=empirical science, but this is probably not so good in relation with usage in, say, French or German (which have more like the older idea science = any systematic knowledge). In any case a detailed argument like that might belong more in the science article. Charles Matthews 11:43, 17 Apr 2005 (UTC)
If you take a look further down Science#Mathematics and the scientific method you will see that the same controversy exists in science. I think Einstein's quote says it all: math is science. - MarSch 16:58, 23 Apr 2005 (UTC)
We shouldn't have anything in this article that indicates that it is or is not a science, since this discussion page indicates that there are people who hold both views and neither view seems to dominate. Brian j d | Why restrict HTML? | 04:08, 2005 Apr 23 (UTC)
Kindly do not comment on others' comments. Everyone has the right to an uncontested viewpoint. Let's not make this poll a springboard into fierce debate. I just wanted to record the opinions of the major editors of math-related articles, just to see where everyone stands. — Sean κ. ⇔
OK, while I think it's pretty obvious that math is not a science, I guess enough people have argued about it for long enough that I can't just walk in and say what I want. But I do think something has to be said about it, it does have to be addressed, because it's a (mis)conception that many people may have, and may come to wikipedia wanting to learn more about that idea. Maybe a heavily qualified statement like: "many people consider that math is not a science for the following reasons, while many others think math should be counted among the sciences for these reasons". Right? - Lethe | Talk 06:04, July 11, 2005 (UTC)
If mathematical knowledge exists separate from the physical world, the second sentence does not follow, unless it can be shown that knowledge of non-physical things cannot be empirical. Ontological materialists, of course, would hold that knowledge of non-physical things cannot be empirical, but fundamentally that's because they deny the existence of non-physical things, and they must therefore deny also that mathematical knowledge exists separate from the physical world. -- Trovatore 20:59, 17 July 2005 (UTC)
I've made a change to address this. I'm not entirely happy with it--the repetition of "physical world" is not perfectly euphonious, and anyway is still subject to criticism by a hypothetical ontological materialist who still thinks mathematics is an empirical science (but necessarily about the physical world, since from his perspective there's nothing but the physical world). But it's certainly better than it was. -- Trovatore 16:40, 20 July 2005 (UTC)
Here are some reasons to delete the section on "Is mathematics a science?":
Are there any reasons to keep it?
If there are no strong protests, I'll delete it. -- Ben Kovitz 06:33, 2 October 2005 (UTC)
The specific structures that are investigated by mathematicians often do have their origin in the natural sciences, most commonly in physics.
Huh? Why physics? Why only natural sciences? I've seen books that seem to give as much attention to economics as they do to physics. Brian j d | Why restrict HTML? | 07:19, 2005 Mar 6 (UTC)
I've since updated that statement to include economics, but I'm not aware of any mathematical structures that originated in economics - should it be changed back, or did I happen to get it right? Brian j d | Why restrict HTML? | 11:45, 2005 May 8 (UTC)
Practically it is possible to devide mathematics into 15 main divisions: history and foundations, number theory, arithmetic, algebra, analysis, geometry and trigonometry, combinatorics; game theory; numerical analysis; optimization; set theory; probability theory and statistics. Noteworthy is that analysis is the largest branch of mathematics.
A new organization would be very helpful and useful, especially in the advanced areas of physics such as relativity and quantum mechanics. -- Orionix 04:22, 13 Mar 2005 (UTC)
It really depends on the person and his expertise and area of interest. From the physical perspective, analysis is the central branch. For example, we have real analysis (which includes elementary calculus), numerical analysis, complex analysis, differential equations, special functions, fourier analysis, calculus of variations and functional analysis.
In algebra we have linear and multilinear algebra (also called tensor algera of vector spaces), lattice theory, groups, fields, rings, homological and universal algebra (which also connects to mathematical logic).
My favorite divisions are: analysis, algebra and geometry, set theory and logic, combinatorics etc.. -- Orionix 13:19, 23 Mar 2005 (UTC)
The fields of algebraic geometry and differential geometry are just more advanced from Euclidean geometry. Euclidean geometry forms the basics to all other non-Euclidean geometries, the most important one is Riemannian geometry.
Now most of the modern 'geometries' are basically part of abstract algebra combined with modern topics in analysis, such as fourier analysis which is widely used in astrophysics and quantum cosmology.
Algebraic geometry has its roots in analytic geometry and differential geometry has its roots in calculus. K-theory (or cohomology theory), groups shemes, lie algebras and noncommutative geometry are also active areas of research, especially in mathematical physics. -- Orionix 16:37, 23 Mar 2005 (UTC)
1. History, Foundations & Philosophy.
2. Arithmetic & Number theory.
3. Algebra & Combinatorics.
4. Geometry & Trigonometry.
5. Analysis: Calculus & Real analysis, Complex analysis, Differential equations, Theory of functions & Modern analysis. Analysis also includes Numerical analysis & Optimization. Other areas are Global analysis, Constructive analysis & Non-standard analysis.
7. Set theory & Logic
8. Modern geometries, Modern algebra and Topology
Another thing of incalculable importance in
modern physics is
group theory and the study of
lie groups,
algebraic groups and
topological groups. See also
quantum gravity and
superstring theory
[1]
In the future, mathematics will become more idealized and abstract of its subject matter. -- Orionix 01:16, 4 Apr 2005 (UTC)
After two dozen edits, I'm now fairly happy with the page, with the exception of the topics sections. Those seem to be somewhat arbitrary, and certainly not well explained. In fact, a bit old-fashioned in WP terms. What to do with those?
Charles Matthews 10:12, 4 Apr 2005 (UTC)
I see it's not possible to divide mathematics into discrete subfields, topics or branches. The boundaries are not clear. Why? Because we don't really know what math is. There are so many holes in our understanding. I think we need a theory of everything. -- Orionix 19:23, 4 Apr 2005 (UTC)
A friendlier introduction might be very desirable. Especially since some concepts are incomprehensible to non-specialists without it. How about some of the introduction from the Wikinfo article:
"1 defined by practices, not proofs 2 where mathematics comes from 3 history / origins 4 structure, space and change 5 foundations and practices "
"Mathematics (often abbreviated to math or, in British English, maths) is commonly defined as the study of patterns of structure, change, and space. It has been called the "science of measurement", measurement itself being a study of engineering (metrics) and psychology (perception).
table of contents [showhide]
In the modern view, mathematics is usually considered the investigation of axiomatically defined abstract structures using formal logic as the common or foundational framework. This was the most common view in the early 20th century and it remains common today.
However, through that century, many dissenters stated and tried to prove that this is not necessary or desirable - that social or cognitive factors specific to humans and their interactions are more basic than logic, sets or other abstractions - see philosophy of mathematics, foundations of mathematics, and Foundations and Methods references below.
In general the philosophy of mathematics one adopts has little effect on mathematical practice: mathematicians all over the world can rely on mathematics as a language even if there are arguments about the meaning or reliability of certain constructs or "words" or "phrases" used in any given "sentence". It is the practices, not the proofs, that define mathematics as a discipline, though the proofs remain persistent over time to a remarkable degree: Euclid's are still in use and are 2000 years old.
By contrast to science, politics or religion, the rationale for "why it works" has remained remarkably stable for mathematics, which is why the ability to do or check mathematical proofs is often considered to be the most basic human knowledge."
You can clarify the art vs. science thing there too. Then continue as it does:
"The specific structures investigated in mathematics often are those found useful in the natural sciences, most..."
Really? Logic, yes, but symbolic logic? Brian j d | Why restrict HTML? | 06:55, 2005 Apr 8 (UTC)
No, I don't think that's a good place to start. Much more appropriate for theoretical computer science, if you ask me. Or maybe philosophy - who knows? The emphasis on abstraction and generality is passé, also: probably went out of fashion when the new methods of knot theory came in. One might as well say mathematics is the study of equivalence relations. It's all a Procrustean bed. Tell you what, can you find a reference that defines it this way? It is not hard to find references for a generally formalist approach, I guess. If we are going to have this discussion, we need to look at sources that attempt to define mathematics, not have a fruitless discussion.
Charles Matthews 16:08, 21 Apr 2005 (UTC)
Try this: if you want to draw the line between theoretical physics and mathematical physics, I think abstraction doesn't really help. I mean, a mathematician asking about an active field like loop quantum gravity, 'how much of this is mathematics?' You find they talk about very abstract things like non-separable Hilbert spaces, and diffeomorphism groups. Some of that is mathematics by a formalist description, and some isn't. The difference is not in the use of abstraction, I say. Charles Matthews 17:09, 23 Apr 2005 (UTC)
Here's one way of making the topics section clearer. Just throwing it out there. — Sean κ. ⇔ 21:47, 23 Apr 2005 (UTC)
Ive been working on a couple templates for organizing basic math stuff. Template:numbers is being worked on now, but could use some checking/reorganizing. Working on Wikipedia:Access, Template:Access, Template:Unsolved - more offline. - SV| t 20:37, 2 May 2005 (UTC)
I've rearranged the article slightly to make the definitions easier to find, but I still don't understand what "the study of abstraction" means. It sounds like it means "study of how to remove unnecessary detail from things" but that doesn't correlate with how I've heard the word "mathematics" being used or anything else in the article (including the formal definition). Brian j d | Why restrict HTML? | 09:33, 2005 May 8 (UTC)
{{ ISBN}}
What is this section for? Brian j d | Why restrict HTML? | 03:06, 2005 May 17 (UTC)
I'm still trying to wrap my mind around this paragraph...
Unfortunately, I don't have time to go through all of MarSch's changes, but IMO a partial revert is in order. — Sean κ. ⇔ 18:17, 18 May 2005 (UTC)
Abstract algebra | Number theory | Algebraic geometry | Group theory |
File:Rubik float.png |
![]() |
![]() |
File:Rubik float.png |
Something that doesn't have to do with Rubik's cube | The elliptic curve was key in proving the most important problem in number theory for the past three centuries, Fermat's Last Theorem | Bézout's theorem, a central theorem in algebraic geometry, gives the number of interesections of two curves. | The structure of solving Rubik's Cube is an example of a problem in group theory |
I was just playing around a little more with our categories. I thought it would be fun to include Rubik's Cube as an example of a problem in group theory... anyone object? — Sean κ. ⇔ 22:22, 22 May 2005 (UTC)
Abstract algebra
The structure of solving
Rubik's Cube is an example of a problem in abstract algebra.
Number theory
The
elliptic curve was key in proving the most important problem in number theory for the past three centuries,
Fermat's Last Theorem.
Group theory
The structure of solving
Rubik's Cube is an example of a problem in group theory.
Topology
How a space or form is connected.
Category theory
The study of
morphisms and
functors.
Kevin Baas talk: new 22:42, 2005 May 27 (UTC)
how about a fifth element of the category, and a better description than "something i know nothing about"? Kevin Baas talk: new 23:40, 2005 May 27 (UTC)
Seems to have coined the phrase "queen of the sciences", but he is not a scientist and no Einstein nor Gauss either. Further he says that theology is the queen of sciences. This information does not seem relevant to me, but I also hate to see theology mentioned in this way in this article. What's your POV? -- MarSch 12:02, 8 Jun 2005 (UTC)
I think that all mathematics can be reduced to geometry. We have:
1. Precalculus: Elementary Algebra and Trigonometry ----> Linear Algebra
2. Calculus & Multivariable Calculus ----> Topology and Differential Geometry
An alternative approach is:
1. Precalculus ----> Linear & Abstract Algebra ----> Algebraic Geometry & Clifford Algebra
2. Calculus & Multivariable Calculus ----> ODEs + PDEs ----> Topology & Differential Topology
3. Real and Complex analysis ----> Functional analysis
-- Orionix 01:06, 9 Jun 2005 (UTC)
Do you have a definition of 'point'? Or would you concede that usage is pretty much the same as for 'element' of a set? Anything geometric is (nowadays, as a rule) thought of as made of something, namely an underlying set. Charles Matthews 09:49, 17 July 2005 (UTC)
hey guys just wanna give you a heads up i created the above category so, we can all get in touch with each other easier and verify articles on mathematics ^_^ Project2501a 17:42, 12 Jun 2005 (UTC)
yes, you do :) Project2501a 18:25, 12 Jun 2005 (UTC)
Hmm. Why is fluid dynamics listed in the Applied mathematics section? Seems overly specific to me. Besides, isn't it really just a subfield of mechanics, which is also listed? - dcljr ( talk) 7 July 2005 06:18 (UTC)
Agree with putting fluid dynamics back. It is a very important subject in its own right. Oleg Alexandrov 16:25, 17 July 2005 (UTC)
Yeah, it seems like every mathematics department have a few who specialize in fluid dynamics. I don't know if it, and probability theory, belong in the "applied mathematics" section, though. I think there are a lot of people who study the Navier-Stokes equations (one of the Clay millenium problems) who don't consider themselves applied mathematicians, just as most mathematical physicists don't. — Joke137 22:19, 17 July 2005 (UTC)
Actually, that wasn't quite what I was suggesting. I'm sure there are some physicists who study fluid dynamics, but I'm suggesting that what some fluid dynamicists do, just as what most probabilists and mathematical physicists do, is not really "applied" in the sense that, say, numerical analysis or financial mathematics is. — Joke137 22:55, 17 July 2005 (UTC)
Someone's changed the name back to "Why fluid dynamics?" saying that the name shouldn't be changed "mid-discussion". Firstly, there hasn't been any discussion here for a while. Secondly, how is it different to other refactoring, which seems to be acceptable? Brian j d | Why restrict HTML? | 09:29, 28 August 2005 (UTC)
What is the point of that lecture picture? Is it there because we can't find anything better? Brian j d | Why restrict HTML? | 07:28, 17 July 2005 (UTC)
The picture itself isn't that bad. The content of the blackboard could do better however. It would be great if someone provided a photo of a blackboard halfway through some extremely difficult proof; all covered with petite-sized symbols which are completely meaningless to a bystander (and most of the students as well ;-) -- Misza13 21:17:33, 2005-07-24 (UTC)
Combinatorics
This
Hasse diagram shows the
set of
partitions of a set of 4 elements.
Order theory
This
Hasse diagram shows the
partial ordering of the set of divisors of 60.
Kevin Baas talk: new 17:52, August 13, 2005 (UTC)
This section has been deleted as POV -why?
I note that the above discussion has moved into a discussion of the merits of the position taken by the deleted paragraph. I'm happy to discuss that, but I'm not sure the discussion needs to be recorded here. The relevant fact is not whether the para was right, but whether it was POV, and I think that's crystal clear. The para was virtually a formalist manifesto. It would be like putting a claim, on the God page, that theism is a "popular misconception". -- Trovatore 22:10, 13 September 2005 (UTC)
Aren't things potentially even more subtle than that? It's not clear that logic is entirely on firm footing; doesn't one already have to accept certain axioms to be able to state "P implies Q" ? So even the ability to deduce truth from axioms is provisional itself. But anyway, I don't think the original intent was to be high-brow in this way. I think the original paragraph was just trying to say that every area of mathematics articulates a certain set of relationships, and it is the articulation of these relationships that constitutes mathematical activity. Mathematics is that collection of relationships. Trying to make my last sentence more precise by talking about things like "truth" will only get you into deep doodoo. linas 00:23, 14 September 2005 (UTC)
I replaced the paragraph with this:
... but Trovatore thought even that was POV, explaining:
The first two sentences could easily be dealt with by inserting the word "physical" before "world". As for the third sentence, mathematical realism states that mathematical entities are real and not created by the human mind, but as far as I can see it does not suggest that mathematical entities have any direct connection to the physical world as we perceive it. Whether space is best described as Cartesian, hyperbolic or whatever is a question for science, not maths. – Smyth\ talk 16:58, 15 September 2005 (UTC)
We seem to be getting closer to something we can all live with. ;-) And I think that something like this is deserving of a place in the article the idea's presence in other articles non-withstanding. Paul August ☎ 19:36, 15 September 2005 (UTC)
I don't like the present definition of math. It is too abstract. I propose the following:
One can find many examples where mathematical reasoning works this way. Are there example where it is not the case? If nobody can find examples where this is not true, I'll be bold and change the definition. Vb14:13, 23 September 2005 (UTC)
I think Quantity, Structure, Space etc... are highly abstract quantities. I haven't understood what was meant till I looked at the article (where this explained by an exhaustive list of examples). This sounds like a typical math book. Vectorial analysis is the science of vectors. Consider v and u, v and u are vectors if v+u is also a vector, etc... I agree my suggestion is also abstract but one could give directly a simple example from naive basic geometry, algebra or even group theory which could be said in one line. The object could be the tympan of the Accropolys it can be modeled by three line segments in a plane, a triangle; studying the properties of abstract triangles leads us to the understanding of all object which can be assumed to be triangles. Or something alike. Oleg: Do you have examples where mathematics is not a theorem prover except in the process of defining the object and guessing the theorem? I don't but I would be interested if you could tell me one example. Vb09:44, 24 September 2005 (UTC)
Well, though esthetic plays an important role in math this not a criteria which helps defining it. The theorem prover machines are doing math even if this kind of math can be qualified as very bad one. A theorem which is proven in a very ugly style is anyway better than a conjecture and can even be very useful. Esthetic is a criteria for judging the value of a proof but not whether this is math or not. But I utterly agree on the point that math is not only a theorem prover technique! If my proposed definition was misleading on this point then I suggest you to amend it to get rid of this impression. I believe that math is much more than this because the first step in the mathematical process is abstraction which is a bit like asking the right question. This is often much more difficult than proving theorems. For example, physicists like I am, often ask questions at a much less level of abstraction than mathematicians do and therefore -from the mathematical point of view- their answer are less general and hence less useful. About not defining math or citing famous people's definitions of math: I think this is a bit like giving up the WP idea! Aren't we able to find a compromise? The present definition is just a list of things mathematicians are interested in. It is a bit as if we would define physics as the science of mechanics, heat, electromagnetism, nuclear reactions and so on. This would not sound very serious, wouldn't it? I have changed my proposal a bit
Well, I think the point was missed. Let me try again, by paraphrasing.
See what the problem is? linas 00:08, 27 September 2005 (UTC)
I agree that current definition is not only pov, but simple unscientific stupidness. What does it means maths is the study of change? So maths is dynamics? What does it mean maths is the study of patterns? What does it mean "pattern"? Fancy works? And what does it mean "structure"? And as I know, this definition can be derivated from an intuicionist mathematician (Poincaré, Weyl? I don't know), so it is the view of a little community, so-so pov. Gubbubu 06:44, 27 September 2005 (UTC)
Thank you Gubbubu for supporting my view. I am happy you used the term "unscientific stupidness" it is not politically correct but true :-) Vb 09:48, 27 September 2005 (UTC)
This is how the introduction read on 4 April:
Mathematics, often abbreviated maths (British English) or math (American English), is the investigation of axiomatically defined abstract structures using symbolic logic and mathematical notation. It is commonly defined as the study of patterns of structure, change, and space; even more informally, one might say it is the study of "figures and numbers". Some hold that since it is not empirical, it is not one of the sciences. Mathematics is widely used for the development and communication of ideas, and particularly quantitative relationships in scientific observation, reasoned analysis and prediction.
So, I preferred that. It sounds as if you might prefer that. This page is typically edited several times every day. There are more important things to fight over. Trying to get a few perfect sentences, when there are 104 mathematics pages to edit - that might be stupidity. Charles Matthews 10:33, 27 September 2005 (UTC)
Defining "mathematics" will always be problematic and controversial. That means that no matter what definition we come up with someone will always find a reason (probably valid) to complain about it. In my opinion our current "definition" is reasonably good. And in my opinion the least worst of any I've seen so far.
For comparison this is Brittannica's lead paragraph:
Mathematics: the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.
Paul August ☎ 18:53, 27 September 2005 (UTC)
I lke very much Rick's definition but since many don't like it why not simply a cut and paste of the definition at the mathematics portal:
Mathematics (Gr.: μαθηματικός (mathematikós) meaning "fond of learning") is often defined as the study of quantity, structure, change, and space. More informally, some might call it the study of "figures and numbers". In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation. In the realist view, it is the investigation of objects or concepts that exist independently of our reasoning about them. Other views are described in the philosophy of mathematics article. Due to its applicability in practically every scientific discipline, mathematics has been called "the language of science" and "the language of the universe".
Isn't that what is called NPOV? Vb 15:55, 28 September 2005 (UTC)
Try ranking, in order of what they manage to communicate about mathematics, these pages:
An exercise to try to define where effort is needed. Charles Matthews 10:20, 28 September 2005 (UTC)
Might I add also
which was COTW but hasn't still reached the FAC state but maybe is close to be. It needs thorough copyedit and systematic review Vb 16:25, 28 September 2005 (UTC)
I note that an anon editor has removed the dab to the hip-hop producer. Now, I didn't like it either. But I do wonder how people will find that article without the notice. (I also wonder why people would want to find it, but that's not my judgment to make.) -- Trovatore 18:03, 27 September 2005 (UTC)
Here's a thought: Is he really notable? Maybe we could AfD the gentleman and get rid of the problem that way. -- Trovatore 19:45, 27 September 2005 (UTC)
One can always put the more neutral {{ otheruses}} which will expand into
but I am truly not sure if it is worth putting it just for the sake of the hip hop producer. Oleg Alexandrov 00:35, 28 September 2005 (UTC)
Can someone please think of a third use, so we can do the {{otheruses]] thing? That would be much less annoying. -- Trovatore 21:00, 28 September 2005 (UTC)
I regret starting a reversion war. I usually do try out rewrites on the talk page, but the intro to math so clearly needed a rewrite, and there had already been so much discussion on the talk page with no results, that I felt the call to do something, which some people have liked (thank goodness).
As for Euclid being Eurocentric! -- Euclid wasn't a European. He lived in Africa. We don't know where he was born. There were no European mathematicians from the time of Archemedes (who was European only if you ignore the Carthegenian claim to Sicily) to the time of Fibbonacci. Most of the educated people who read Euclid before 1200 lived in Africa or the Near East (Alexandria, Bagdad, etc.) and read Euclid in Arabic. Rick Norwood 17:33, 28 September 2005 (UTC)
Paul evidently sees mathematics as a group of topics. I see mathematics as a method. There are plenty of mathematicians on both sides of this question. (It is almost as bad as the question of whether mathematics is invented or discovered -- in my intro I carefully avoided both words, in favor of "developed". But wiki is not the place to argue about this -- we want an introduction that will be helpful to non-mathematicians, which to me means mentioning both views. I'm going to try to combine the two versions and put that here, where everyone can have a try at rewriting it until we come to an acceptable compromose. Rick Norwood 18:50, 28 September 2005 (UTC)
Here is what I have come up with. My focus is on helping a non-mathematician to understand what mathematics is. Rick Norwood 19:40, 28 September 2005 (UTC)
Mathematics is often defined as the study of certain subjects, such as quantity, structure, space, and change. Others define mathematics as that body of knowledge discovered by deductive reasoning, starting from axioms and definitions, as in Euclid's Elements (circa 300 BC).
Historically, mathematics developed from counting, calculation, measurement, and geometry. Relatively few cultures have added to our mathematical knowledge. The Greek speaking Hellenic culture 4th Century BC created much of the mathematics taught in secondary school. From that time until the 13th Century, there were scattered new discoveries in mathematics in Arabia, India, China, and Japan. The renaissance of European mathematics began in the 13th Century with Fibbonacci. The base ten number system, invented in India, made mathematics much more accessable, as did the algebra notation of the European mathematician Regiomontanus. New discoveries in science and technology feuled an ever greater demand for new mathematics, and more new mathematics has been published in the last century than in all the time that went before.
The word mathematics is also used to refer to the knowledge gained by people by doing mathematics, also known as the body of mathematical knowledge. This latter meaning of mathematics includes the mathematics used to do calculations or mathematical modeling and is an indispensable tool in the natural sciences, engineering, economics, and medicine.
Mathematics is often defined as the study of certain subjects, such as quantity, structure, space, and change. Others define mathematics as that body of knowledge discovered by deductive reasoning, starting from axioms and definitions, as in Euclid's Elements (circa 300 BC).
Relatively few cultures have added to our mathematical knowledge. Particularly notable are the Greek speaking Hellenic culture 4th Century BC, the civilizations of Arabia, India, China, and Japan, and the culture of Renaissance Europe, inspired by the 13th Century work of Fibbonacci. The base ten number system, invented in India, made mathematics much more accessable, as did the algebra notation of the European mathematicians such as Regiomontanus. New discoveries in science and technology require new mathematics, and more new mathematics has been published in the last century than in all the time that went before.
The word mathematics is also used to refer to the knowledge gained by people by doing mathematics, also known as the body of mathematical knowledge. This latter meaning of mathematics includes the mathematics used to do calculations or mathematical modeling and is an indispensable tool in the natural sciences, engineering, economics, and medicine.
Rick Norwood 23:45, 28 September 2005 (UTC)
Thank you for your comments. I had Egypt in an earlier draft, but under pressure to cut the paragraph settled for a shorter list. The digit 0 has two purposes, indicating "none" and as a place holder. There were symbols for "none" and symbols for place holders before the invention of the base ten system. It is the base ten system, which combines the two uses of the zero, that was the real breakthrough. In an earlier draft, I called the system the Indo-Arabic numerals, which was what I learned in school, but the wiki article on the system is titled "Base ten" so I went with that. Of course, algebra goes back long before Regiomontanus, but the notation was cumbersome, mostly written out in words. It was the notation of Regiomontanus (and others) that streamlined algebra and made it more accessable. I don't mind taking him out, though, and putting Egypt in. Comparing this article to other articles in Wikipedia, I think a few words on history are appropriate in the introduction. In any case, I'm sure that whetever the current discussion settles on will be fine tuned many times in the future. Rick Norwood 16:40, 29 September 2005 (UTC)
So, poor old Regiomontanus is out. I'm going to go ahead and post what is left, and we can take it from there. Rick Norwood 19:55, 29 September 2005 (UTC)
So I'm still not convinced we need a history paragraph in the intro section, but I think I can live with the one that's there. I do think we need to be very disciplined about making sure it doesn't grow. I propose as a rule of thumb that a reader using 800x600 resolution and without perfect eyes (let's say he uses "large fonts" in Windows) should still be able to see the top of the TOC. This is partly as a practical matter to accomodate such a user, and partly because I just think that's about how long intros should be.
As far as history yea or nay, how about a poll? -- Trovatore 17:02, 1 October 2005 (UTC)
Keep or remove the history paragraph above the TOC?
Leaving aside the content of the introduction, how long should it ideally be? As the length of the current version may change during the poll, please compare to this version. Suggested answers: much shorter, shorter, about the same, longer, much longer, wrong question.
"Wikipedia is not the place to insist that your philosophical beliefs are the official ones" is given for reverting more than a week of work without any discussion at all.
I have reverted the reversion, returning to the version of the introduction discussed on these pages. I suggest we talk this over, instead of insisting that one point of view is objective and the other is only "philosophical beliefs".
This seems to be the crux of the matter. There are two views of mathematics.
One view, perhaps most famously enunciated in Hardy's "A Mathematician's Apology", is that mathematics is important in itself, is knowledge gained by deductive reasoning, and that the applications of mathematics are not the essential nature of mathematics. In this view, in the words of Gauss, mathemetics is the queen of the sciences.
The other view is that mathematics is "practical", that it is not the queen of anything, but rather the servent ot the sciences, and consists mainly of applications to the study of things like number, shape, and motion.
Clearly, this is not a question that Winipedia should take sides on. It should present both sides, as the version I've restored does, and not dismiss one side as mere "philosophy".
Thus my reversion.
Let's talk about it. Rick Norwood 13:28, 2 October 2005 (UTC)
First, my apologies, Rick, for reverting the change to the opening paragraph without discussion. I hadn't noticed that you'd been talking about it here for a week (my fault for not checking). Also, my apologies for saying that the previous edit insisted on one philosophical belief; that was just plain wrong of me, as the phrasing was intended to state two sides. (I was commenting only on the second one, writing the edit comment way too quickly.)
Here's why I favor reverting the change. Saying that mathematics "is often defined as the study of some subjects" and "others define it as" something else does not give an overview of the topic or describe its scope in a way useful to a likely reader. It's a listing of opposing philosophical schools of thought without first telling the reader what topic the disagreement is about. There are two problems with the second sentence: "Others define mathematics as that body of knowledge discovered by deductive reasoning, starting from axioms and definitions" states a philosophical view that I and many others think is naive, as it includes much that is outside math (economics, philosophy, physics, etc.); and "as in Euclid's Elements (circa 300 BC)" goes into detail about a subject with which the lay reader is not likely to be familiar, in a way that suggests that the reader ought to know all about Euclid's Elements before reading this article. Another problem is that the paragraph suggests that there are only two philosophical schools of thought when really there are many.
Instead of opening with what different, disagreeing unnamed groups of people define mathematics as, I think a much better opening for a lay reader would simply list some of the broadest topics of mathematics, starting with the extremely familiar (number or quantity) and including a couple that suggest the great breadth of the field, all using non-specialized language. Since we're more than a dictionary, we can do more than draw the boundaries of the subject: a good overview can say the main ways in which mathematics relates to other fields, like science and engineering, or something about how people use math or why math is so important. The rest of the article can (and does) go into more detail, briefly describing many different mathematical topics that are covered in yet more detail in other articles.
Here is how some other sources have "defined" mathematics (i.e. roughly outlined the scope of the topic):
So I propose an opening paragraph something along this line:
The vagueness suggests that this is only an outline of the scope, not a definition to end all definitions. It gives the reader the basic information he needs to know just to understand what people are talking about, whether in philosophical arguments, discussions of mathematics education from an elementary to college level, ethnographies, or anywhere else. A description of philosophical schools of thought that attempt fine-grained or counterintuitive definitions certainly belongs on Wikipedia—in the philosophy of mathematics article, where it can be assumed that the reader knows broadly what mathematics is about, since we've done such a fine job explaining that on this page.
What do you think?
— Ben Kovitz 19:12, 2 October 2005 (UTC)
The word "naive" suggests inexperienced. To call the view of Hilbert, Hardy, "Bourbaki" and others "naive" sounds like name calling. On the other hand, you are correct that non-mathematicians usually define mathematics as "that stuff I had to learn in school" about numbers, triangles and such.
The introduction you want says, in effect, "Mathematics is about certain subjects. Here is a list. Mathematics is important because it is useful."
I think Hilbert would have said something like "Mathematics is a way of thinking and is important because it is beautiful."
Let me give you an example from my own field, knot theory. In your view, knot theory is that branch of mathematics that studies the shapes of knots in loops of string, and it is important because it has practical uses such as detecting whether or not a strand of DNA has been cut. In my view, knot theory studies embeddings of n-2 dimensional objects in n dimensional space and is an extremely beautiful area of pure mathematics. I don't study it because I hope my discoveries will be useful to superstring theorists.
If it were just up to me, I would write an introduction to mathematics that went something like this.
"Mathematics is knowledge gained by deductive reasoning. It is the most certain knowledge we have, and some would say the most intellectually beautiful. It has stood the test of time, is the same in all nations, and is one of the few bodies of knowledge everyone accepts as true. Science, by contrast, uses inductive reasoning, based on careful observation, measurement, and experiment. Every branch of science and engineering uses mathematics, and new scientific discoveries require new mathematical discoveries."
But, I know I can't get away with an introduction that just gives the point of view of a pure mathematician. So, I'm willing to let the applied mathematician speak first. But I am unwilling to make mathematics sound like something people do entirely to help out scientists and engineers. Rick Norwood 20:54, 2 October 2005 (UTC)
Thank you for your thoughtful response, Rick. I don't think I succeeded in getting across my main idea, so I'd like to address a likely misunderstanding and try again.
I don't think that knot theory is important only because it's useful for string theory or that mathematics is important only because it has practical uses. Such concerns are no part of what I'm saying. I'm saying that from the standpoint of writing an encyclopedia, presenting a philosophical position is a poor introduction. Even presenting two philosophical positions is a poor introduction. A good introduction is to just outline the scope of the subject, not to make profound statements whose truth can only be appreciated by people who've spent a long time studying math. The "importance" that I spoke of was not deep, metaphysical importance where we would say that math is important because it is beautiful, but rather "encyclopedic importance": the way math connects with other subjects. In an intro, we just want to give the reader some overall idea of what this topic is about and where it sits in the big map of topics that the encyclopedia is about.
We could argue about whether the true definition of math is knowledge gained by deductive reasoning. I could point out that Spinoza famously tried to make his theory of ethics follow deductively from axioms and definitions following the pattern of Euclid, that Kant tried to build his philosophy deductively from unshakeable grounds but never confused it with mathematics, that deductive logic since Aristotle has been intended to include all subjects within its scope, not just math, that deductive reasoning predominates in law, and that Hardy and Bourbaki and the others did not actually equate mathematics with knowledge gained by deductive reasoning. You might reply that Kant allowed some non-deductive elements into his ethics, quote Hardy at greater length and prove me wrong, etc. But that would all be irrelevant, because the disagreement we are trying to resolve is not which philosophical theory is best or how to compromise between opposing philosophical theories, but what kind of introduction is good writing for a lay reader.
I think all of these topics that you bring up are great stuff for Wikipedia, and I'd love to see you write them up, just placed under some more-specialized headings or even a little later in this same article (e.g. a section on "What do professional mathematicians do?" could describe differing motivations among specialists in mathematics). For an intro to math as a whole, just outline the topic, and leave the deep explanatory theories for places where deep explanatory theories are spelled out. A good intro and survey article will provide a reader the kind of broad, shallow knowledge needed to understand what the deep explanatory theories are trying to explain. Write-ups of those theories can refer the reader here for the needed overview.
Does that sound reasonable to you—keep the intro broad, shallow, and introductory? (BTW, I thought knots could only exist in three dimensions. No?)
— Ben Kovitz 03:10, 3 October 2005 (UTC)
I understand your point. Spinoza and Kant were trying to discover a mathematics of ethics. They failed. Kant, in particular, had to admit that ethics required ideas that were, in some sense, "above" the rational. (He also, rather famously, "proved" that non-Euclidean geometry cannot exist -- just fourteen years before Lobachevski discovered non-Euclidean geometry.)
On the other hand, attempts to use the methods of mathematics in game theory and in economic theory have been remarkably successful -- more in the former case than in the latter. They are areas of mathematics that everyone agrees are mathematics, but which are not necessarily about number or shape or motion, and are only about structure in the sense that every formal study is about structure.
Most people would, I think, agree that to define science as the study of physics, chemistry, geology, astronomy, and biology would miss the point. One begins with the idea of the scientific method, and then goes on to specific subjects where that method has been successful. So, to a mathematician, it is not a philosophical question whether mathematics is primarily subject matter or method. It is a question of what mathematics really is, something that I would like to see Wikipedia enlighten people about. In other words, I want the Wikipedia introduction to mathematics to be more like the OED, and less like a paperback dictionary. The OED definition is just fine. Why don't we paraphrase that. Rick Norwood 17:36, 3 October 2005 (UTC)
Well, Rick, now I'm not sure what to do. I apologized to you on Saturday for having the impertinence to suggest that you were trying to attach the authority of Wikipedia to a philosophical belief, and now you're saying that you want to enlighten the masses about what mathematics "really" is by sticking it in the intro of Mathematics.
Regarding the OED, I'm not positive, but I think they removed the philosophical part of their definition in the current edition. I think the 1933 definition was written when Kantian philosophy was much more taken for granted, and it's rather uncharacteristic writing for the OED. In any event, pick any philosophical interpretation or definition of mathematics that you like, and it will be a matter of controversy. The other sources (and even the OED except for the Kantian part) illustrate successful ways to achieve this community's modest goal: to roughly outline the scope of mathematics, as a lead-in to a survey article that provides more details and links to more in-depth articles.
If I understand your argument correctly, it's entirely about what the true, deeply insightful interpretation of math is, and not about what makes a comprehensible, useful intro to a survey article for the general reader. You're not saying what the facts are, you're arguing for a particular interpretation of the facts. The shallow, factual overview of math is that it's about stuff like numbers, shapes, patterns, etc. That's the very basic thing that every mathematician, philosopher, anthropologist, auto mechanic, and beautician can agree on. Shallow, factual overviews are what encyclopedias provide. If you want to declare an official position on what to make of those facts, e.g. that the properties of numbers, shapes, patterns, etc. are coextensive with all conclusions arrived at by deductive reasoning, the party whose position you'll be declaring will not be Wikipedia or literate culture at large.
Now, I do think that the kinds of theories you've been arguing for ought to be described on Wikipedia, preferably along with their names and most famous advocates. Presenting these theories as theories would be factual and appropriate for Wikipedia. Presenting them in some clearly labeled place where opposing theories are listed would even be an organized and comprehensible presentation of the facts for the lay reader—exactly what a general-interest encyclopedia tries to achieve.
I've got a ton of real analysis homework to finish now, so I'll leave it to your much-respected intellect and creativity to think of a way to include a write-up of the beliefs you want the masses to hear, that accords with Wikipedia's modest goal of being an encyclopedia. There must be a lot of good ways to do that.
— Ben Kovitz 21:55, 3 October 2005 (UTC)
My inclination right now is to leave well enough alone. I think the article as it stands has a lot of good writing in it. On the other hand, I recall what happened to Faust when he said, "Enough, enough, I'm satisfied." Rick Norwood 22:40, 3 October 2005 (UTC)
Hi, Rick. I'm a little befuddled by our conversation. How about this? We could separate disagreements about content (what is factual/supported) from disagreements about writing (what is readable, organized, makes a good intro for a lay reader). My main objection has been along the latter line, but I haven't heard you address that. I just put up a new intro, which tries to incorporate some of the salient points about mathematics that you brought up, while providing a broad overview of math and the main reasons why people find it noteworthy. I hope you like it better, and I look forward to the next edit. — Ben Kovitz 05:31, 4 October 2005 (UTC)
Thanks, but too many cooks spoil the broth. I got into this, after several months of writing non-controversial math articles, because I strongly thought the introduction at the time was at odds with the rest of the article. At that time, both the intro and the history section seemed to focus on "quantity, shape, motion, and structure" which, to me, is an arbitrary list that makes no more sense than "number, lines, light, and relationships" or any other list of four things that somebody thinks of when they think mathematics. I also didn't like the style, but I figured if we could get the substance right, the style would follow. Now, I realized I opened a bag of worms, and as things stand as of this writing, I'm going to let others try to fine tune the article -- I kind of like it the way it is now (1:57 EST). Rick Norwood 17:59, 4 October 2005 (UTC)
Ok, Rick, let's let it sit for a week or so and then talk some more. I share your displeasure with the "arbitrary list" kind of (pseudo-)definition (see below). By exploring what appears to be a disagreement further, I think we'll hit on ways to make the page even better. (I believe our discussion has already led to an improved second paragraph, and we now have several topics to develop in the body of the article.) — Ben Kovitz 17:29, 6 October 2005 (UTC)
In my opinion the view of Ben Kovitz about the definition of math is not defendable. Defining math must be something like "math is the study of..." where what follows is something someone can understand and not simply a lists of abstract things : quantity, etc... But since it seems clear from the present discussion that this view is defended by many (at least within the WP editors) the NPOV attitude is to say "math is often defined by..." a list of things and "but others define math as the study of..." something (what can be deduced from axioms). This is a correct NPOV which can be of course improved but still in the objective of finding a compromise but not pushing one's own view. Vb 14:22, 4 October 2005 (UTC)
"Even so, in the past it sometimes happened that something which had supposedly been proved turned out to be false."
This sounds good -- appropriately 'umble, hat in hand, no stuck up mathematicians in hear, no suh! Further, I am certain it must be true, knowing what I know about human falibility. But I can't think of a single important example.
Newton originally got the "product rule" wrong, but he scratched it out, he didn't publish it.
I can think of lots of cases where a published proof turned out to be incorrect, from Fermat's marginal scribble to the errors in Wyles original paper, plus holes in Euclid's proofs, the incorrect proof of the Dehn Lemma, many incorrect proofs of the Four Color Theorem.
But in every case I can think of, a better proof has come along.
I'm sure one of the mathematicians on wiki will enlighten me with an example of something supposedly proved that turned out to be false. Rick Norwood 21:45, 6 October 2005 (UTC)
Is "recreation" the only way to describe mathematics engaged in the sake of beauty rather than utility? This might seem a bit like describing philosophy as "discovering and cataloging truths for beauty, without regard for practical application." Moreover, many of the Greeks (and many others, surely) saw mathematics as a kind of divine activitity, and much of the brilliance of the mathematic field comes from people striving not necessarily towards utility but the beauty and awe of knowledge...certainly not something reducable to "recreation," I venture. -- Dpr 01:50, 13 October 2005 (UTC)
I think "recreational" has strong connotations with "in your spare time" and as such has nothing to do with professional pure mathematics. Recreational mathematics might be reading a book about mathematical anecdotes. SOmething you do for (fun and not work) instead of (fun(hopefully) and work). -- MarSch 15:13, 19 October 2005 (UTC)
I have rewritten the much-disputed intro sentence as follows:
This is a loose rephrasing in modern language of the public domain 1913 Webster's Dictionary definition, which, in its original form, is as follows:
I hope this is acceptable: please feel free to improve it. -- The Anome 14:10, 18 October 2005 (UTC)
I think is better than the recent definition "maths is the study of structure, quantity and change" or what. See Wikipedia:Cite sources, Wikipedia:What is Wikipedia#Not original research Gubbubu 10:17, 19 October 2005 (UTC)
I don't like the sentences:
I would think that there are more important things to say at the beginning of the second paragraph of this very important article than emphasizing that only few cultures contributed to mathematics. Besides, these two short sentences really don't do justice to the issue I would say, and the issue itself is bigger than just mathematics, and refers to science as a whole. In short, is that text really needed there? Oleg Alexandrov ( talk) 06:38, 19 October 2005 (UTC)
I'm with Oleg on this. I really dislike those sentences, particularly their vagueness. -- MarSch 15:18, 19 October 2005 (UTC)
I think the sentence the sentences But few cultures have contributed new ideas to mathematics. There are no records of new mathematical ideas originating in Europe in the first thousand years of the Common Era, for example. should go. Paul August ☎ 18:03, 19 October 2005 (UTC)
Diophantus lived in Africa, not in Europe. His place of birth is uncertain, but is thought to be somewhere in the Near East. Rick Norwood 01:07, 20 October 2005 (UTC)
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Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science. I wouldn't classify math as a science (science tries to explain how the world works). I wouldn't classify it as an art either (dictionary.com gives it as "1. Human effort to imitate, supplement, alter, or counteract the work of nature.", in which case, photography is arguably not an art). Of course, if you use another definition "5. A nonscientific branch of learning; one of the liberal arts.", it's an art because it's not scientific (but I'm sure there's more than arts and sciences, e.g. I don't think law is either). And if you use "3. High quality of conception or execution, as found in works of beauty; aesthetic value.", then pure math could be an art (since it's often pretty), but then we're left with how to place statistics (since stats isn't pretty) and, perhaps, calculus.
Most US and Canadian unis classify it as a science (though often it's in the "faculty of arts and sciences" which is more like "faculty of miscellany"), and British unis are undivided (most of them list computer science as a BSci, like Oxford, but some list them as a BA, like Cambridge). UWaterloo has a "faculty of mathematics", and you get a BMath.
Arguably, it's closer to science than art (in the sense that it mostly requires the same kind of brain as most sciences). I notice I'm rambling. -- Elektron 06:27, 2004 May 25 (UTC)
Shall we call a poll for which category we should stuff Mathematics under? -- Elektron 16:58, 2004 Jun 1 (UTC)
Gubbubu 20:26, 20 Dec 2004 (UTC)
-- Sean Kelly 20:53, 20 Dec 2004 (UTC)
I dont't think maths is fundamentally different from anything. More, maths could be considered as the modell and an ideal for all sciences. It's a superscience, and if you saw the hystory of science in the XX cent., maybe you would accept the expressions "scientifical" and "rational" became the synonyms of "deductive" and "mathematical".
But I think, this is not so right. I think, you are speaking about the written, formal mathematical proofs, but you forget, these are only the final forms or (drafting of) achievements of mathematical investigation, what is in itself likes every other sciences (see e.g. computer-helped number theory - its an experimental science).
But I think v talk not only bout đ concept of maths, but about đ concept of science. If you would be so kind to define it, maybe I could compare it with the sentences above, and my mind could conceive in wich special meaning of science math's wouldn't be a science ? Gubbubu 18:00, 21 Dec 2004 (UTC).
I think that mathematics as a body of work is not scientific. However, the practice of mathematics is in the vast majority of cases scientific: experimental examples (from special cases, enumeration, and whatnot) have been more than a little usefull throughout the development of mathematics. More importantly, a usual way of approaching a theory is "Oh, this is a nice theory, I'm going to play around with it for a bit, to gather data. When I've found it, maybe I'll find some patterns and be able to prove something". That is the crux really - it's just like the other sciences, only it has this extra step at the end of the scientific process, called proof, which is weighted with such importance that all prior steps are usually omitted (or often presented as consequences of it!). icecubex 9.14, 5 Jan 2005 (GMT)
By definition, mathematics is abstract, and science is about gathering empirical knowledge (it can refer to the process, the people, or the knowledge itself). Surely something can't be both abstract and empirical?
Why do you think science must be empirical? this is only a point of neopositivists' view.
Notice the article I linked to in the heading? That article says it is empirical. Also, the common usage (the common usage I've noticed, anyway) indicates that it is empirical. The definition at Wiktionary does not indicate that it must be empirical, but that definition seems too broad.
If you know anything about neopositivism (I don't), you can start the article! Brian j d 04:17, 2005 Mar 6 (UTC)
Why is the definition at Wiktionary too broad? Does it not imply that Wikipedia is a "science"; that accounting is a "science"? Brian j d 04:18, 2005 Mar 6 (UTC)
Some hold that since it is not empirical, it is not one of the sciences.
This implies that some take a different view, but I can find nothing in the article about any other view.
However, I found in the section "Common misconceptions" the following:
Although Einstein called it "the Queen of the Sciences", by one not-unusual definition, mathematics itself is not a science, because it is not empirical. Brian j d | Why restrict HTML? | 04:42, 2005 Apr 17 (UTC) (signature added later)
Some hold that since mathematical knowledge is not fundamentally empirical, mathematics is not itself one of the sciences, however closely allied.
This implies that some take a different view that contradicts this, but I can find nothing relevant in the article.
The following statement, that contradicts the one above, is still there:
Although Einstein called it "the Queen of the Sciences", by one not-unusual definition, mathematics itself is not a science, because it is not empirical. Brian j d | Why restrict HTML? | 04:42, 2005 Apr 17 (UTC)
Mathematics is usually regarded as an important tool for science, even though the development of mathematics is not necessarily done with science in mind
If mathematics is science, how can it be a tool for science? Brian j d | Why restrict HTML? | 04:11, 2005 Apr 23 (UTC)
The science article (for me anyway) clearly states that science is empirical. How can something be both empirical and abstract ("the science of abstraction")? Brian j d | Why restrict HTML? | 04:09, 2005 Apr 23 (UTC)
One issue here is how 'science' is defined. Most of the Anglo-Saxon world is happy with science=empirical science, but this is probably not so good in relation with usage in, say, French or German (which have more like the older idea science = any systematic knowledge). In any case a detailed argument like that might belong more in the science article. Charles Matthews 11:43, 17 Apr 2005 (UTC)
If you take a look further down Science#Mathematics and the scientific method you will see that the same controversy exists in science. I think Einstein's quote says it all: math is science. - MarSch 16:58, 23 Apr 2005 (UTC)
We shouldn't have anything in this article that indicates that it is or is not a science, since this discussion page indicates that there are people who hold both views and neither view seems to dominate. Brian j d | Why restrict HTML? | 04:08, 2005 Apr 23 (UTC)
Kindly do not comment on others' comments. Everyone has the right to an uncontested viewpoint. Let's not make this poll a springboard into fierce debate. I just wanted to record the opinions of the major editors of math-related articles, just to see where everyone stands. — Sean κ. ⇔
OK, while I think it's pretty obvious that math is not a science, I guess enough people have argued about it for long enough that I can't just walk in and say what I want. But I do think something has to be said about it, it does have to be addressed, because it's a (mis)conception that many people may have, and may come to wikipedia wanting to learn more about that idea. Maybe a heavily qualified statement like: "many people consider that math is not a science for the following reasons, while many others think math should be counted among the sciences for these reasons". Right? - Lethe | Talk 06:04, July 11, 2005 (UTC)
If mathematical knowledge exists separate from the physical world, the second sentence does not follow, unless it can be shown that knowledge of non-physical things cannot be empirical. Ontological materialists, of course, would hold that knowledge of non-physical things cannot be empirical, but fundamentally that's because they deny the existence of non-physical things, and they must therefore deny also that mathematical knowledge exists separate from the physical world. -- Trovatore 20:59, 17 July 2005 (UTC)
I've made a change to address this. I'm not entirely happy with it--the repetition of "physical world" is not perfectly euphonious, and anyway is still subject to criticism by a hypothetical ontological materialist who still thinks mathematics is an empirical science (but necessarily about the physical world, since from his perspective there's nothing but the physical world). But it's certainly better than it was. -- Trovatore 16:40, 20 July 2005 (UTC)
Here are some reasons to delete the section on "Is mathematics a science?":
Are there any reasons to keep it?
If there are no strong protests, I'll delete it. -- Ben Kovitz 06:33, 2 October 2005 (UTC)
The specific structures that are investigated by mathematicians often do have their origin in the natural sciences, most commonly in physics.
Huh? Why physics? Why only natural sciences? I've seen books that seem to give as much attention to economics as they do to physics. Brian j d | Why restrict HTML? | 07:19, 2005 Mar 6 (UTC)
I've since updated that statement to include economics, but I'm not aware of any mathematical structures that originated in economics - should it be changed back, or did I happen to get it right? Brian j d | Why restrict HTML? | 11:45, 2005 May 8 (UTC)
Practically it is possible to devide mathematics into 15 main divisions: history and foundations, number theory, arithmetic, algebra, analysis, geometry and trigonometry, combinatorics; game theory; numerical analysis; optimization; set theory; probability theory and statistics. Noteworthy is that analysis is the largest branch of mathematics.
A new organization would be very helpful and useful, especially in the advanced areas of physics such as relativity and quantum mechanics. -- Orionix 04:22, 13 Mar 2005 (UTC)
It really depends on the person and his expertise and area of interest. From the physical perspective, analysis is the central branch. For example, we have real analysis (which includes elementary calculus), numerical analysis, complex analysis, differential equations, special functions, fourier analysis, calculus of variations and functional analysis.
In algebra we have linear and multilinear algebra (also called tensor algera of vector spaces), lattice theory, groups, fields, rings, homological and universal algebra (which also connects to mathematical logic).
My favorite divisions are: analysis, algebra and geometry, set theory and logic, combinatorics etc.. -- Orionix 13:19, 23 Mar 2005 (UTC)
The fields of algebraic geometry and differential geometry are just more advanced from Euclidean geometry. Euclidean geometry forms the basics to all other non-Euclidean geometries, the most important one is Riemannian geometry.
Now most of the modern 'geometries' are basically part of abstract algebra combined with modern topics in analysis, such as fourier analysis which is widely used in astrophysics and quantum cosmology.
Algebraic geometry has its roots in analytic geometry and differential geometry has its roots in calculus. K-theory (or cohomology theory), groups shemes, lie algebras and noncommutative geometry are also active areas of research, especially in mathematical physics. -- Orionix 16:37, 23 Mar 2005 (UTC)
1. History, Foundations & Philosophy.
2. Arithmetic & Number theory.
3. Algebra & Combinatorics.
4. Geometry & Trigonometry.
5. Analysis: Calculus & Real analysis, Complex analysis, Differential equations, Theory of functions & Modern analysis. Analysis also includes Numerical analysis & Optimization. Other areas are Global analysis, Constructive analysis & Non-standard analysis.
7. Set theory & Logic
8. Modern geometries, Modern algebra and Topology
Another thing of incalculable importance in
modern physics is
group theory and the study of
lie groups,
algebraic groups and
topological groups. See also
quantum gravity and
superstring theory
[1]
In the future, mathematics will become more idealized and abstract of its subject matter. -- Orionix 01:16, 4 Apr 2005 (UTC)
After two dozen edits, I'm now fairly happy with the page, with the exception of the topics sections. Those seem to be somewhat arbitrary, and certainly not well explained. In fact, a bit old-fashioned in WP terms. What to do with those?
Charles Matthews 10:12, 4 Apr 2005 (UTC)
I see it's not possible to divide mathematics into discrete subfields, topics or branches. The boundaries are not clear. Why? Because we don't really know what math is. There are so many holes in our understanding. I think we need a theory of everything. -- Orionix 19:23, 4 Apr 2005 (UTC)
A friendlier introduction might be very desirable. Especially since some concepts are incomprehensible to non-specialists without it. How about some of the introduction from the Wikinfo article:
"1 defined by practices, not proofs 2 where mathematics comes from 3 history / origins 4 structure, space and change 5 foundations and practices "
"Mathematics (often abbreviated to math or, in British English, maths) is commonly defined as the study of patterns of structure, change, and space. It has been called the "science of measurement", measurement itself being a study of engineering (metrics) and psychology (perception).
table of contents [showhide]
In the modern view, mathematics is usually considered the investigation of axiomatically defined abstract structures using formal logic as the common or foundational framework. This was the most common view in the early 20th century and it remains common today.
However, through that century, many dissenters stated and tried to prove that this is not necessary or desirable - that social or cognitive factors specific to humans and their interactions are more basic than logic, sets or other abstractions - see philosophy of mathematics, foundations of mathematics, and Foundations and Methods references below.
In general the philosophy of mathematics one adopts has little effect on mathematical practice: mathematicians all over the world can rely on mathematics as a language even if there are arguments about the meaning or reliability of certain constructs or "words" or "phrases" used in any given "sentence". It is the practices, not the proofs, that define mathematics as a discipline, though the proofs remain persistent over time to a remarkable degree: Euclid's are still in use and are 2000 years old.
By contrast to science, politics or religion, the rationale for "why it works" has remained remarkably stable for mathematics, which is why the ability to do or check mathematical proofs is often considered to be the most basic human knowledge."
You can clarify the art vs. science thing there too. Then continue as it does:
"The specific structures investigated in mathematics often are those found useful in the natural sciences, most..."
Really? Logic, yes, but symbolic logic? Brian j d | Why restrict HTML? | 06:55, 2005 Apr 8 (UTC)
No, I don't think that's a good place to start. Much more appropriate for theoretical computer science, if you ask me. Or maybe philosophy - who knows? The emphasis on abstraction and generality is passé, also: probably went out of fashion when the new methods of knot theory came in. One might as well say mathematics is the study of equivalence relations. It's all a Procrustean bed. Tell you what, can you find a reference that defines it this way? It is not hard to find references for a generally formalist approach, I guess. If we are going to have this discussion, we need to look at sources that attempt to define mathematics, not have a fruitless discussion.
Charles Matthews 16:08, 21 Apr 2005 (UTC)
Try this: if you want to draw the line between theoretical physics and mathematical physics, I think abstraction doesn't really help. I mean, a mathematician asking about an active field like loop quantum gravity, 'how much of this is mathematics?' You find they talk about very abstract things like non-separable Hilbert spaces, and diffeomorphism groups. Some of that is mathematics by a formalist description, and some isn't. The difference is not in the use of abstraction, I say. Charles Matthews 17:09, 23 Apr 2005 (UTC)
Here's one way of making the topics section clearer. Just throwing it out there. — Sean κ. ⇔ 21:47, 23 Apr 2005 (UTC)
Ive been working on a couple templates for organizing basic math stuff. Template:numbers is being worked on now, but could use some checking/reorganizing. Working on Wikipedia:Access, Template:Access, Template:Unsolved - more offline. - SV| t 20:37, 2 May 2005 (UTC)
I've rearranged the article slightly to make the definitions easier to find, but I still don't understand what "the study of abstraction" means. It sounds like it means "study of how to remove unnecessary detail from things" but that doesn't correlate with how I've heard the word "mathematics" being used or anything else in the article (including the formal definition). Brian j d | Why restrict HTML? | 09:33, 2005 May 8 (UTC)
{{ ISBN}}
What is this section for? Brian j d | Why restrict HTML? | 03:06, 2005 May 17 (UTC)
I'm still trying to wrap my mind around this paragraph...
Unfortunately, I don't have time to go through all of MarSch's changes, but IMO a partial revert is in order. — Sean κ. ⇔ 18:17, 18 May 2005 (UTC)
Abstract algebra | Number theory | Algebraic geometry | Group theory |
File:Rubik float.png |
![]() |
![]() |
File:Rubik float.png |
Something that doesn't have to do with Rubik's cube | The elliptic curve was key in proving the most important problem in number theory for the past three centuries, Fermat's Last Theorem | Bézout's theorem, a central theorem in algebraic geometry, gives the number of interesections of two curves. | The structure of solving Rubik's Cube is an example of a problem in group theory |
I was just playing around a little more with our categories. I thought it would be fun to include Rubik's Cube as an example of a problem in group theory... anyone object? — Sean κ. ⇔ 22:22, 22 May 2005 (UTC)
Abstract algebra
The structure of solving
Rubik's Cube is an example of a problem in abstract algebra.
Number theory
The
elliptic curve was key in proving the most important problem in number theory for the past three centuries,
Fermat's Last Theorem.
Group theory
The structure of solving
Rubik's Cube is an example of a problem in group theory.
Topology
How a space or form is connected.
Category theory
The study of
morphisms and
functors.
Kevin Baas talk: new 22:42, 2005 May 27 (UTC)
how about a fifth element of the category, and a better description than "something i know nothing about"? Kevin Baas talk: new 23:40, 2005 May 27 (UTC)
Seems to have coined the phrase "queen of the sciences", but he is not a scientist and no Einstein nor Gauss either. Further he says that theology is the queen of sciences. This information does not seem relevant to me, but I also hate to see theology mentioned in this way in this article. What's your POV? -- MarSch 12:02, 8 Jun 2005 (UTC)
I think that all mathematics can be reduced to geometry. We have:
1. Precalculus: Elementary Algebra and Trigonometry ----> Linear Algebra
2. Calculus & Multivariable Calculus ----> Topology and Differential Geometry
An alternative approach is:
1. Precalculus ----> Linear & Abstract Algebra ----> Algebraic Geometry & Clifford Algebra
2. Calculus & Multivariable Calculus ----> ODEs + PDEs ----> Topology & Differential Topology
3. Real and Complex analysis ----> Functional analysis
-- Orionix 01:06, 9 Jun 2005 (UTC)
Do you have a definition of 'point'? Or would you concede that usage is pretty much the same as for 'element' of a set? Anything geometric is (nowadays, as a rule) thought of as made of something, namely an underlying set. Charles Matthews 09:49, 17 July 2005 (UTC)
hey guys just wanna give you a heads up i created the above category so, we can all get in touch with each other easier and verify articles on mathematics ^_^ Project2501a 17:42, 12 Jun 2005 (UTC)
yes, you do :) Project2501a 18:25, 12 Jun 2005 (UTC)
Hmm. Why is fluid dynamics listed in the Applied mathematics section? Seems overly specific to me. Besides, isn't it really just a subfield of mechanics, which is also listed? - dcljr ( talk) 7 July 2005 06:18 (UTC)
Agree with putting fluid dynamics back. It is a very important subject in its own right. Oleg Alexandrov 16:25, 17 July 2005 (UTC)
Yeah, it seems like every mathematics department have a few who specialize in fluid dynamics. I don't know if it, and probability theory, belong in the "applied mathematics" section, though. I think there are a lot of people who study the Navier-Stokes equations (one of the Clay millenium problems) who don't consider themselves applied mathematicians, just as most mathematical physicists don't. — Joke137 22:19, 17 July 2005 (UTC)
Actually, that wasn't quite what I was suggesting. I'm sure there are some physicists who study fluid dynamics, but I'm suggesting that what some fluid dynamicists do, just as what most probabilists and mathematical physicists do, is not really "applied" in the sense that, say, numerical analysis or financial mathematics is. — Joke137 22:55, 17 July 2005 (UTC)
Someone's changed the name back to "Why fluid dynamics?" saying that the name shouldn't be changed "mid-discussion". Firstly, there hasn't been any discussion here for a while. Secondly, how is it different to other refactoring, which seems to be acceptable? Brian j d | Why restrict HTML? | 09:29, 28 August 2005 (UTC)
What is the point of that lecture picture? Is it there because we can't find anything better? Brian j d | Why restrict HTML? | 07:28, 17 July 2005 (UTC)
The picture itself isn't that bad. The content of the blackboard could do better however. It would be great if someone provided a photo of a blackboard halfway through some extremely difficult proof; all covered with petite-sized symbols which are completely meaningless to a bystander (and most of the students as well ;-) -- Misza13 21:17:33, 2005-07-24 (UTC)
Combinatorics
This
Hasse diagram shows the
set of
partitions of a set of 4 elements.
Order theory
This
Hasse diagram shows the
partial ordering of the set of divisors of 60.
Kevin Baas talk: new 17:52, August 13, 2005 (UTC)
This section has been deleted as POV -why?
I note that the above discussion has moved into a discussion of the merits of the position taken by the deleted paragraph. I'm happy to discuss that, but I'm not sure the discussion needs to be recorded here. The relevant fact is not whether the para was right, but whether it was POV, and I think that's crystal clear. The para was virtually a formalist manifesto. It would be like putting a claim, on the God page, that theism is a "popular misconception". -- Trovatore 22:10, 13 September 2005 (UTC)
Aren't things potentially even more subtle than that? It's not clear that logic is entirely on firm footing; doesn't one already have to accept certain axioms to be able to state "P implies Q" ? So even the ability to deduce truth from axioms is provisional itself. But anyway, I don't think the original intent was to be high-brow in this way. I think the original paragraph was just trying to say that every area of mathematics articulates a certain set of relationships, and it is the articulation of these relationships that constitutes mathematical activity. Mathematics is that collection of relationships. Trying to make my last sentence more precise by talking about things like "truth" will only get you into deep doodoo. linas 00:23, 14 September 2005 (UTC)
I replaced the paragraph with this:
... but Trovatore thought even that was POV, explaining:
The first two sentences could easily be dealt with by inserting the word "physical" before "world". As for the third sentence, mathematical realism states that mathematical entities are real and not created by the human mind, but as far as I can see it does not suggest that mathematical entities have any direct connection to the physical world as we perceive it. Whether space is best described as Cartesian, hyperbolic or whatever is a question for science, not maths. – Smyth\ talk 16:58, 15 September 2005 (UTC)
We seem to be getting closer to something we can all live with. ;-) And I think that something like this is deserving of a place in the article the idea's presence in other articles non-withstanding. Paul August ☎ 19:36, 15 September 2005 (UTC)
I don't like the present definition of math. It is too abstract. I propose the following:
One can find many examples where mathematical reasoning works this way. Are there example where it is not the case? If nobody can find examples where this is not true, I'll be bold and change the definition. Vb14:13, 23 September 2005 (UTC)
I think Quantity, Structure, Space etc... are highly abstract quantities. I haven't understood what was meant till I looked at the article (where this explained by an exhaustive list of examples). This sounds like a typical math book. Vectorial analysis is the science of vectors. Consider v and u, v and u are vectors if v+u is also a vector, etc... I agree my suggestion is also abstract but one could give directly a simple example from naive basic geometry, algebra or even group theory which could be said in one line. The object could be the tympan of the Accropolys it can be modeled by three line segments in a plane, a triangle; studying the properties of abstract triangles leads us to the understanding of all object which can be assumed to be triangles. Or something alike. Oleg: Do you have examples where mathematics is not a theorem prover except in the process of defining the object and guessing the theorem? I don't but I would be interested if you could tell me one example. Vb09:44, 24 September 2005 (UTC)
Well, though esthetic plays an important role in math this not a criteria which helps defining it. The theorem prover machines are doing math even if this kind of math can be qualified as very bad one. A theorem which is proven in a very ugly style is anyway better than a conjecture and can even be very useful. Esthetic is a criteria for judging the value of a proof but not whether this is math or not. But I utterly agree on the point that math is not only a theorem prover technique! If my proposed definition was misleading on this point then I suggest you to amend it to get rid of this impression. I believe that math is much more than this because the first step in the mathematical process is abstraction which is a bit like asking the right question. This is often much more difficult than proving theorems. For example, physicists like I am, often ask questions at a much less level of abstraction than mathematicians do and therefore -from the mathematical point of view- their answer are less general and hence less useful. About not defining math or citing famous people's definitions of math: I think this is a bit like giving up the WP idea! Aren't we able to find a compromise? The present definition is just a list of things mathematicians are interested in. It is a bit as if we would define physics as the science of mechanics, heat, electromagnetism, nuclear reactions and so on. This would not sound very serious, wouldn't it? I have changed my proposal a bit
Well, I think the point was missed. Let me try again, by paraphrasing.
See what the problem is? linas 00:08, 27 September 2005 (UTC)
I agree that current definition is not only pov, but simple unscientific stupidness. What does it means maths is the study of change? So maths is dynamics? What does it mean maths is the study of patterns? What does it mean "pattern"? Fancy works? And what does it mean "structure"? And as I know, this definition can be derivated from an intuicionist mathematician (Poincaré, Weyl? I don't know), so it is the view of a little community, so-so pov. Gubbubu 06:44, 27 September 2005 (UTC)
Thank you Gubbubu for supporting my view. I am happy you used the term "unscientific stupidness" it is not politically correct but true :-) Vb 09:48, 27 September 2005 (UTC)
This is how the introduction read on 4 April:
Mathematics, often abbreviated maths (British English) or math (American English), is the investigation of axiomatically defined abstract structures using symbolic logic and mathematical notation. It is commonly defined as the study of patterns of structure, change, and space; even more informally, one might say it is the study of "figures and numbers". Some hold that since it is not empirical, it is not one of the sciences. Mathematics is widely used for the development and communication of ideas, and particularly quantitative relationships in scientific observation, reasoned analysis and prediction.
So, I preferred that. It sounds as if you might prefer that. This page is typically edited several times every day. There are more important things to fight over. Trying to get a few perfect sentences, when there are 104 mathematics pages to edit - that might be stupidity. Charles Matthews 10:33, 27 September 2005 (UTC)
Defining "mathematics" will always be problematic and controversial. That means that no matter what definition we come up with someone will always find a reason (probably valid) to complain about it. In my opinion our current "definition" is reasonably good. And in my opinion the least worst of any I've seen so far.
For comparison this is Brittannica's lead paragraph:
Mathematics: the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.
Paul August ☎ 18:53, 27 September 2005 (UTC)
I lke very much Rick's definition but since many don't like it why not simply a cut and paste of the definition at the mathematics portal:
Mathematics (Gr.: μαθηματικός (mathematikós) meaning "fond of learning") is often defined as the study of quantity, structure, change, and space. More informally, some might call it the study of "figures and numbers". In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation. In the realist view, it is the investigation of objects or concepts that exist independently of our reasoning about them. Other views are described in the philosophy of mathematics article. Due to its applicability in practically every scientific discipline, mathematics has been called "the language of science" and "the language of the universe".
Isn't that what is called NPOV? Vb 15:55, 28 September 2005 (UTC)
Try ranking, in order of what they manage to communicate about mathematics, these pages:
An exercise to try to define where effort is needed. Charles Matthews 10:20, 28 September 2005 (UTC)
Might I add also
which was COTW but hasn't still reached the FAC state but maybe is close to be. It needs thorough copyedit and systematic review Vb 16:25, 28 September 2005 (UTC)
I note that an anon editor has removed the dab to the hip-hop producer. Now, I didn't like it either. But I do wonder how people will find that article without the notice. (I also wonder why people would want to find it, but that's not my judgment to make.) -- Trovatore 18:03, 27 September 2005 (UTC)
Here's a thought: Is he really notable? Maybe we could AfD the gentleman and get rid of the problem that way. -- Trovatore 19:45, 27 September 2005 (UTC)
One can always put the more neutral {{ otheruses}} which will expand into
but I am truly not sure if it is worth putting it just for the sake of the hip hop producer. Oleg Alexandrov 00:35, 28 September 2005 (UTC)
Can someone please think of a third use, so we can do the {{otheruses]] thing? That would be much less annoying. -- Trovatore 21:00, 28 September 2005 (UTC)
I regret starting a reversion war. I usually do try out rewrites on the talk page, but the intro to math so clearly needed a rewrite, and there had already been so much discussion on the talk page with no results, that I felt the call to do something, which some people have liked (thank goodness).
As for Euclid being Eurocentric! -- Euclid wasn't a European. He lived in Africa. We don't know where he was born. There were no European mathematicians from the time of Archemedes (who was European only if you ignore the Carthegenian claim to Sicily) to the time of Fibbonacci. Most of the educated people who read Euclid before 1200 lived in Africa or the Near East (Alexandria, Bagdad, etc.) and read Euclid in Arabic. Rick Norwood 17:33, 28 September 2005 (UTC)
Paul evidently sees mathematics as a group of topics. I see mathematics as a method. There are plenty of mathematicians on both sides of this question. (It is almost as bad as the question of whether mathematics is invented or discovered -- in my intro I carefully avoided both words, in favor of "developed". But wiki is not the place to argue about this -- we want an introduction that will be helpful to non-mathematicians, which to me means mentioning both views. I'm going to try to combine the two versions and put that here, where everyone can have a try at rewriting it until we come to an acceptable compromose. Rick Norwood 18:50, 28 September 2005 (UTC)
Here is what I have come up with. My focus is on helping a non-mathematician to understand what mathematics is. Rick Norwood 19:40, 28 September 2005 (UTC)
Mathematics is often defined as the study of certain subjects, such as quantity, structure, space, and change. Others define mathematics as that body of knowledge discovered by deductive reasoning, starting from axioms and definitions, as in Euclid's Elements (circa 300 BC).
Historically, mathematics developed from counting, calculation, measurement, and geometry. Relatively few cultures have added to our mathematical knowledge. The Greek speaking Hellenic culture 4th Century BC created much of the mathematics taught in secondary school. From that time until the 13th Century, there were scattered new discoveries in mathematics in Arabia, India, China, and Japan. The renaissance of European mathematics began in the 13th Century with Fibbonacci. The base ten number system, invented in India, made mathematics much more accessable, as did the algebra notation of the European mathematician Regiomontanus. New discoveries in science and technology feuled an ever greater demand for new mathematics, and more new mathematics has been published in the last century than in all the time that went before.
The word mathematics is also used to refer to the knowledge gained by people by doing mathematics, also known as the body of mathematical knowledge. This latter meaning of mathematics includes the mathematics used to do calculations or mathematical modeling and is an indispensable tool in the natural sciences, engineering, economics, and medicine.
Mathematics is often defined as the study of certain subjects, such as quantity, structure, space, and change. Others define mathematics as that body of knowledge discovered by deductive reasoning, starting from axioms and definitions, as in Euclid's Elements (circa 300 BC).
Relatively few cultures have added to our mathematical knowledge. Particularly notable are the Greek speaking Hellenic culture 4th Century BC, the civilizations of Arabia, India, China, and Japan, and the culture of Renaissance Europe, inspired by the 13th Century work of Fibbonacci. The base ten number system, invented in India, made mathematics much more accessable, as did the algebra notation of the European mathematicians such as Regiomontanus. New discoveries in science and technology require new mathematics, and more new mathematics has been published in the last century than in all the time that went before.
The word mathematics is also used to refer to the knowledge gained by people by doing mathematics, also known as the body of mathematical knowledge. This latter meaning of mathematics includes the mathematics used to do calculations or mathematical modeling and is an indispensable tool in the natural sciences, engineering, economics, and medicine.
Rick Norwood 23:45, 28 September 2005 (UTC)
Thank you for your comments. I had Egypt in an earlier draft, but under pressure to cut the paragraph settled for a shorter list. The digit 0 has two purposes, indicating "none" and as a place holder. There were symbols for "none" and symbols for place holders before the invention of the base ten system. It is the base ten system, which combines the two uses of the zero, that was the real breakthrough. In an earlier draft, I called the system the Indo-Arabic numerals, which was what I learned in school, but the wiki article on the system is titled "Base ten" so I went with that. Of course, algebra goes back long before Regiomontanus, but the notation was cumbersome, mostly written out in words. It was the notation of Regiomontanus (and others) that streamlined algebra and made it more accessable. I don't mind taking him out, though, and putting Egypt in. Comparing this article to other articles in Wikipedia, I think a few words on history are appropriate in the introduction. In any case, I'm sure that whetever the current discussion settles on will be fine tuned many times in the future. Rick Norwood 16:40, 29 September 2005 (UTC)
So, poor old Regiomontanus is out. I'm going to go ahead and post what is left, and we can take it from there. Rick Norwood 19:55, 29 September 2005 (UTC)
So I'm still not convinced we need a history paragraph in the intro section, but I think I can live with the one that's there. I do think we need to be very disciplined about making sure it doesn't grow. I propose as a rule of thumb that a reader using 800x600 resolution and without perfect eyes (let's say he uses "large fonts" in Windows) should still be able to see the top of the TOC. This is partly as a practical matter to accomodate such a user, and partly because I just think that's about how long intros should be.
As far as history yea or nay, how about a poll? -- Trovatore 17:02, 1 October 2005 (UTC)
Keep or remove the history paragraph above the TOC?
Leaving aside the content of the introduction, how long should it ideally be? As the length of the current version may change during the poll, please compare to this version. Suggested answers: much shorter, shorter, about the same, longer, much longer, wrong question.
"Wikipedia is not the place to insist that your philosophical beliefs are the official ones" is given for reverting more than a week of work without any discussion at all.
I have reverted the reversion, returning to the version of the introduction discussed on these pages. I suggest we talk this over, instead of insisting that one point of view is objective and the other is only "philosophical beliefs".
This seems to be the crux of the matter. There are two views of mathematics.
One view, perhaps most famously enunciated in Hardy's "A Mathematician's Apology", is that mathematics is important in itself, is knowledge gained by deductive reasoning, and that the applications of mathematics are not the essential nature of mathematics. In this view, in the words of Gauss, mathemetics is the queen of the sciences.
The other view is that mathematics is "practical", that it is not the queen of anything, but rather the servent ot the sciences, and consists mainly of applications to the study of things like number, shape, and motion.
Clearly, this is not a question that Winipedia should take sides on. It should present both sides, as the version I've restored does, and not dismiss one side as mere "philosophy".
Thus my reversion.
Let's talk about it. Rick Norwood 13:28, 2 October 2005 (UTC)
First, my apologies, Rick, for reverting the change to the opening paragraph without discussion. I hadn't noticed that you'd been talking about it here for a week (my fault for not checking). Also, my apologies for saying that the previous edit insisted on one philosophical belief; that was just plain wrong of me, as the phrasing was intended to state two sides. (I was commenting only on the second one, writing the edit comment way too quickly.)
Here's why I favor reverting the change. Saying that mathematics "is often defined as the study of some subjects" and "others define it as" something else does not give an overview of the topic or describe its scope in a way useful to a likely reader. It's a listing of opposing philosophical schools of thought without first telling the reader what topic the disagreement is about. There are two problems with the second sentence: "Others define mathematics as that body of knowledge discovered by deductive reasoning, starting from axioms and definitions" states a philosophical view that I and many others think is naive, as it includes much that is outside math (economics, philosophy, physics, etc.); and "as in Euclid's Elements (circa 300 BC)" goes into detail about a subject with which the lay reader is not likely to be familiar, in a way that suggests that the reader ought to know all about Euclid's Elements before reading this article. Another problem is that the paragraph suggests that there are only two philosophical schools of thought when really there are many.
Instead of opening with what different, disagreeing unnamed groups of people define mathematics as, I think a much better opening for a lay reader would simply list some of the broadest topics of mathematics, starting with the extremely familiar (number or quantity) and including a couple that suggest the great breadth of the field, all using non-specialized language. Since we're more than a dictionary, we can do more than draw the boundaries of the subject: a good overview can say the main ways in which mathematics relates to other fields, like science and engineering, or something about how people use math or why math is so important. The rest of the article can (and does) go into more detail, briefly describing many different mathematical topics that are covered in yet more detail in other articles.
Here is how some other sources have "defined" mathematics (i.e. roughly outlined the scope of the topic):
So I propose an opening paragraph something along this line:
The vagueness suggests that this is only an outline of the scope, not a definition to end all definitions. It gives the reader the basic information he needs to know just to understand what people are talking about, whether in philosophical arguments, discussions of mathematics education from an elementary to college level, ethnographies, or anywhere else. A description of philosophical schools of thought that attempt fine-grained or counterintuitive definitions certainly belongs on Wikipedia—in the philosophy of mathematics article, where it can be assumed that the reader knows broadly what mathematics is about, since we've done such a fine job explaining that on this page.
What do you think?
— Ben Kovitz 19:12, 2 October 2005 (UTC)
The word "naive" suggests inexperienced. To call the view of Hilbert, Hardy, "Bourbaki" and others "naive" sounds like name calling. On the other hand, you are correct that non-mathematicians usually define mathematics as "that stuff I had to learn in school" about numbers, triangles and such.
The introduction you want says, in effect, "Mathematics is about certain subjects. Here is a list. Mathematics is important because it is useful."
I think Hilbert would have said something like "Mathematics is a way of thinking and is important because it is beautiful."
Let me give you an example from my own field, knot theory. In your view, knot theory is that branch of mathematics that studies the shapes of knots in loops of string, and it is important because it has practical uses such as detecting whether or not a strand of DNA has been cut. In my view, knot theory studies embeddings of n-2 dimensional objects in n dimensional space and is an extremely beautiful area of pure mathematics. I don't study it because I hope my discoveries will be useful to superstring theorists.
If it were just up to me, I would write an introduction to mathematics that went something like this.
"Mathematics is knowledge gained by deductive reasoning. It is the most certain knowledge we have, and some would say the most intellectually beautiful. It has stood the test of time, is the same in all nations, and is one of the few bodies of knowledge everyone accepts as true. Science, by contrast, uses inductive reasoning, based on careful observation, measurement, and experiment. Every branch of science and engineering uses mathematics, and new scientific discoveries require new mathematical discoveries."
But, I know I can't get away with an introduction that just gives the point of view of a pure mathematician. So, I'm willing to let the applied mathematician speak first. But I am unwilling to make mathematics sound like something people do entirely to help out scientists and engineers. Rick Norwood 20:54, 2 October 2005 (UTC)
Thank you for your thoughtful response, Rick. I don't think I succeeded in getting across my main idea, so I'd like to address a likely misunderstanding and try again.
I don't think that knot theory is important only because it's useful for string theory or that mathematics is important only because it has practical uses. Such concerns are no part of what I'm saying. I'm saying that from the standpoint of writing an encyclopedia, presenting a philosophical position is a poor introduction. Even presenting two philosophical positions is a poor introduction. A good introduction is to just outline the scope of the subject, not to make profound statements whose truth can only be appreciated by people who've spent a long time studying math. The "importance" that I spoke of was not deep, metaphysical importance where we would say that math is important because it is beautiful, but rather "encyclopedic importance": the way math connects with other subjects. In an intro, we just want to give the reader some overall idea of what this topic is about and where it sits in the big map of topics that the encyclopedia is about.
We could argue about whether the true definition of math is knowledge gained by deductive reasoning. I could point out that Spinoza famously tried to make his theory of ethics follow deductively from axioms and definitions following the pattern of Euclid, that Kant tried to build his philosophy deductively from unshakeable grounds but never confused it with mathematics, that deductive logic since Aristotle has been intended to include all subjects within its scope, not just math, that deductive reasoning predominates in law, and that Hardy and Bourbaki and the others did not actually equate mathematics with knowledge gained by deductive reasoning. You might reply that Kant allowed some non-deductive elements into his ethics, quote Hardy at greater length and prove me wrong, etc. But that would all be irrelevant, because the disagreement we are trying to resolve is not which philosophical theory is best or how to compromise between opposing philosophical theories, but what kind of introduction is good writing for a lay reader.
I think all of these topics that you bring up are great stuff for Wikipedia, and I'd love to see you write them up, just placed under some more-specialized headings or even a little later in this same article (e.g. a section on "What do professional mathematicians do?" could describe differing motivations among specialists in mathematics). For an intro to math as a whole, just outline the topic, and leave the deep explanatory theories for places where deep explanatory theories are spelled out. A good intro and survey article will provide a reader the kind of broad, shallow knowledge needed to understand what the deep explanatory theories are trying to explain. Write-ups of those theories can refer the reader here for the needed overview.
Does that sound reasonable to you—keep the intro broad, shallow, and introductory? (BTW, I thought knots could only exist in three dimensions. No?)
— Ben Kovitz 03:10, 3 October 2005 (UTC)
I understand your point. Spinoza and Kant were trying to discover a mathematics of ethics. They failed. Kant, in particular, had to admit that ethics required ideas that were, in some sense, "above" the rational. (He also, rather famously, "proved" that non-Euclidean geometry cannot exist -- just fourteen years before Lobachevski discovered non-Euclidean geometry.)
On the other hand, attempts to use the methods of mathematics in game theory and in economic theory have been remarkably successful -- more in the former case than in the latter. They are areas of mathematics that everyone agrees are mathematics, but which are not necessarily about number or shape or motion, and are only about structure in the sense that every formal study is about structure.
Most people would, I think, agree that to define science as the study of physics, chemistry, geology, astronomy, and biology would miss the point. One begins with the idea of the scientific method, and then goes on to specific subjects where that method has been successful. So, to a mathematician, it is not a philosophical question whether mathematics is primarily subject matter or method. It is a question of what mathematics really is, something that I would like to see Wikipedia enlighten people about. In other words, I want the Wikipedia introduction to mathematics to be more like the OED, and less like a paperback dictionary. The OED definition is just fine. Why don't we paraphrase that. Rick Norwood 17:36, 3 October 2005 (UTC)
Well, Rick, now I'm not sure what to do. I apologized to you on Saturday for having the impertinence to suggest that you were trying to attach the authority of Wikipedia to a philosophical belief, and now you're saying that you want to enlighten the masses about what mathematics "really" is by sticking it in the intro of Mathematics.
Regarding the OED, I'm not positive, but I think they removed the philosophical part of their definition in the current edition. I think the 1933 definition was written when Kantian philosophy was much more taken for granted, and it's rather uncharacteristic writing for the OED. In any event, pick any philosophical interpretation or definition of mathematics that you like, and it will be a matter of controversy. The other sources (and even the OED except for the Kantian part) illustrate successful ways to achieve this community's modest goal: to roughly outline the scope of mathematics, as a lead-in to a survey article that provides more details and links to more in-depth articles.
If I understand your argument correctly, it's entirely about what the true, deeply insightful interpretation of math is, and not about what makes a comprehensible, useful intro to a survey article for the general reader. You're not saying what the facts are, you're arguing for a particular interpretation of the facts. The shallow, factual overview of math is that it's about stuff like numbers, shapes, patterns, etc. That's the very basic thing that every mathematician, philosopher, anthropologist, auto mechanic, and beautician can agree on. Shallow, factual overviews are what encyclopedias provide. If you want to declare an official position on what to make of those facts, e.g. that the properties of numbers, shapes, patterns, etc. are coextensive with all conclusions arrived at by deductive reasoning, the party whose position you'll be declaring will not be Wikipedia or literate culture at large.
Now, I do think that the kinds of theories you've been arguing for ought to be described on Wikipedia, preferably along with their names and most famous advocates. Presenting these theories as theories would be factual and appropriate for Wikipedia. Presenting them in some clearly labeled place where opposing theories are listed would even be an organized and comprehensible presentation of the facts for the lay reader—exactly what a general-interest encyclopedia tries to achieve.
I've got a ton of real analysis homework to finish now, so I'll leave it to your much-respected intellect and creativity to think of a way to include a write-up of the beliefs you want the masses to hear, that accords with Wikipedia's modest goal of being an encyclopedia. There must be a lot of good ways to do that.
— Ben Kovitz 21:55, 3 October 2005 (UTC)
My inclination right now is to leave well enough alone. I think the article as it stands has a lot of good writing in it. On the other hand, I recall what happened to Faust when he said, "Enough, enough, I'm satisfied." Rick Norwood 22:40, 3 October 2005 (UTC)
Hi, Rick. I'm a little befuddled by our conversation. How about this? We could separate disagreements about content (what is factual/supported) from disagreements about writing (what is readable, organized, makes a good intro for a lay reader). My main objection has been along the latter line, but I haven't heard you address that. I just put up a new intro, which tries to incorporate some of the salient points about mathematics that you brought up, while providing a broad overview of math and the main reasons why people find it noteworthy. I hope you like it better, and I look forward to the next edit. — Ben Kovitz 05:31, 4 October 2005 (UTC)
Thanks, but too many cooks spoil the broth. I got into this, after several months of writing non-controversial math articles, because I strongly thought the introduction at the time was at odds with the rest of the article. At that time, both the intro and the history section seemed to focus on "quantity, shape, motion, and structure" which, to me, is an arbitrary list that makes no more sense than "number, lines, light, and relationships" or any other list of four things that somebody thinks of when they think mathematics. I also didn't like the style, but I figured if we could get the substance right, the style would follow. Now, I realized I opened a bag of worms, and as things stand as of this writing, I'm going to let others try to fine tune the article -- I kind of like it the way it is now (1:57 EST). Rick Norwood 17:59, 4 October 2005 (UTC)
Ok, Rick, let's let it sit for a week or so and then talk some more. I share your displeasure with the "arbitrary list" kind of (pseudo-)definition (see below). By exploring what appears to be a disagreement further, I think we'll hit on ways to make the page even better. (I believe our discussion has already led to an improved second paragraph, and we now have several topics to develop in the body of the article.) — Ben Kovitz 17:29, 6 October 2005 (UTC)
In my opinion the view of Ben Kovitz about the definition of math is not defendable. Defining math must be something like "math is the study of..." where what follows is something someone can understand and not simply a lists of abstract things : quantity, etc... But since it seems clear from the present discussion that this view is defended by many (at least within the WP editors) the NPOV attitude is to say "math is often defined by..." a list of things and "but others define math as the study of..." something (what can be deduced from axioms). This is a correct NPOV which can be of course improved but still in the objective of finding a compromise but not pushing one's own view. Vb 14:22, 4 October 2005 (UTC)
"Even so, in the past it sometimes happened that something which had supposedly been proved turned out to be false."
This sounds good -- appropriately 'umble, hat in hand, no stuck up mathematicians in hear, no suh! Further, I am certain it must be true, knowing what I know about human falibility. But I can't think of a single important example.
Newton originally got the "product rule" wrong, but he scratched it out, he didn't publish it.
I can think of lots of cases where a published proof turned out to be incorrect, from Fermat's marginal scribble to the errors in Wyles original paper, plus holes in Euclid's proofs, the incorrect proof of the Dehn Lemma, many incorrect proofs of the Four Color Theorem.
But in every case I can think of, a better proof has come along.
I'm sure one of the mathematicians on wiki will enlighten me with an example of something supposedly proved that turned out to be false. Rick Norwood 21:45, 6 October 2005 (UTC)
Is "recreation" the only way to describe mathematics engaged in the sake of beauty rather than utility? This might seem a bit like describing philosophy as "discovering and cataloging truths for beauty, without regard for practical application." Moreover, many of the Greeks (and many others, surely) saw mathematics as a kind of divine activitity, and much of the brilliance of the mathematic field comes from people striving not necessarily towards utility but the beauty and awe of knowledge...certainly not something reducable to "recreation," I venture. -- Dpr 01:50, 13 October 2005 (UTC)
I think "recreational" has strong connotations with "in your spare time" and as such has nothing to do with professional pure mathematics. Recreational mathematics might be reading a book about mathematical anecdotes. SOmething you do for (fun and not work) instead of (fun(hopefully) and work). -- MarSch 15:13, 19 October 2005 (UTC)
I have rewritten the much-disputed intro sentence as follows:
This is a loose rephrasing in modern language of the public domain 1913 Webster's Dictionary definition, which, in its original form, is as follows:
I hope this is acceptable: please feel free to improve it. -- The Anome 14:10, 18 October 2005 (UTC)
I think is better than the recent definition "maths is the study of structure, quantity and change" or what. See Wikipedia:Cite sources, Wikipedia:What is Wikipedia#Not original research Gubbubu 10:17, 19 October 2005 (UTC)
I don't like the sentences:
I would think that there are more important things to say at the beginning of the second paragraph of this very important article than emphasizing that only few cultures contributed to mathematics. Besides, these two short sentences really don't do justice to the issue I would say, and the issue itself is bigger than just mathematics, and refers to science as a whole. In short, is that text really needed there? Oleg Alexandrov ( talk) 06:38, 19 October 2005 (UTC)
I'm with Oleg on this. I really dislike those sentences, particularly their vagueness. -- MarSch 15:18, 19 October 2005 (UTC)
I think the sentence the sentences But few cultures have contributed new ideas to mathematics. There are no records of new mathematical ideas originating in Europe in the first thousand years of the Common Era, for example. should go. Paul August ☎ 18:03, 19 October 2005 (UTC)
Diophantus lived in Africa, not in Europe. His place of birth is uncertain, but is thought to be somewhere in the Near East. Rick Norwood 01:07, 20 October 2005 (UTC)