Talk:Mathematics

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)

Mathematics is not a science, it is the one of the furthest things from a science you can get. Science is empirical, i.e. it is based on observations. Mathematics is not, it is based on reason.--ShaunMacPherson 04:00, 21 Jun 2004 (UTC)

This isn't true enough, observations in maths are so important, see computer algebra or computer-helped number theory. We don't state anything we can't observe nowhere and by no means. The problem is not that whether maths is a science, but that is it an empirical "natural" science (?).Gubbubu 09:28, 2 Sep 2004 (UTC)

Agreed, Mathematics, like science, begins with conjecture and theory, and is proven to be absolute based on observation. --Will2k 14:57, Sep 2, 2004 (UTC)

In science, we do not "prove (things) to be absolute". We assume things to be true if they have been well-tested, but we recognise that they may not be true.
In mathematics however, we are able to prove things to be absolute.
Brianjd 07:23, Sep 12, 2004 (UTC)

That has got to be the most false statement I have ever heard! Believe me, I am a scientist. Science is constantly striving to find absolute truth. It is the development of technology that allows us to determine that what was proposed as a scientific theory is, in fact, truth. The "assumption" you describe is applied to the scientific theory which is part of the entire process known as science. Every branch of science will not quit until the truth is proven. --Will2k 21:15, Sep 12, 2004 (UTC)

Every statement was true. Science looks at the consequences of something and tries to formulate rules for the world. Mathematics defines rules for a world and tries to figure out the consequences. No (natural) science can "prove" anything, but for the most part, we take Newtonian mechanics as 'good enough'. Of course, for statistics, evidence is 'good enough', but not for pure math. Of course, it depends on how you define 'science', but if anyone can find any other 'science' that isn't empirical, feel free to let me know (boolean algebra counts as math). Elektron 18:56, 2004 Nov 1 (UTC)

Shall we call a poll for which category we should stuff Mathematics under? --Elektron 16:58, 2004 Jun 1 (UTC)

The problem of where to put mathematics is one common to many universities as well. Usually it is grouped with the science since mathematics is usually a required course for the sciences more then other subjects. Recently though at the university I goto they put the math department with the humanities, i guess since it also has much to do with philosophy (i.e. non empirical seeking of knowledge).

Maybe just have a catagory of mathematics, it could be that it is different enough to warrent its own catagory ;). --ShaunMacPherson 04:00, 21 Jun 2004 (UTC)

I agree.
Brianjd 07:23, Sep 12, 2004 (UTC)

Dec 2004 – Feb 2005

Well, could you give me resources who said M. is not a science? I think this is a so curious oppinion, not debated by fews, but reserved by fews.
Could you explain to me circumstantially why it wouldn't be a science? How would you define science, then?

Gubbubu 20:26, 20 Dec 2004 (UTC)

I began to answer this up top (old posting of Not a science?) Maybe we should bring the old discussion down here.

Note that I believe m. is not a science even if we remove empirical from the definition of science. I don't think it even comes up in the discussion.

I'm arguing that at the very least, m. is fundamentally different in the way it proves things from every other modern science (yes, including natural and social sciences). I made the point up top. --Sean Kelly 20:41, 20 Dec 2004 (UTC)

Oh, but I also think the point is moot. The average person who reads this article believes (incorrectly perhaps) that science is the study of the natural world. When we say "m. is not a science," we are eliminating the misconception that m. somehow derives from the natural world.

--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".

Most theory of maths are really derived from the real world, this is detectable well by investigations on the history of this science. Empirism, hypothe is so important
it is not concerned finding evidence .. hah? 90% of mathematical works is finding an evidence (called "proof").
a proof begins by assuming the hypothesis (?) Where's the difference?

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).

finally, mathematics is not an extension of phisycs, but physics is an extension of maths. Nowadays mathematicians find out a lot of theory, and astronomers, cosmologists etc. engage these. Gubbubu

I would agree that mathematics is a superscience (metascience), but I don't think that makes it a science.

If you dont't think, this will drive you to question in the future that 90% of sciences are really sciences. Other sciences probably will be "overmathematized" so in the future.

Surely we agree that there are subsets of mathematics, such as computer science, that are science. Can we agree that there are subsets of mathematics, say "pure mathematics", which have no relation to any other science?

No. On the 1 hand I think there is no pure mathematics in the classic meaning of this expression. On the other hand I don't think the expression "science" can be defined exactly. As I remember, someone "proved" about Dillinger or Capone he was one of the greatests scientists in the world - his methods to prepare for robberies and carried them out completely fitted the positivists' definitions of sciences.

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 -- I would say that proofs are mathematical investigation. The fact that we humans need to use computers and computation to help us construct our proofs should not take away from the purity of the proof itself. If the universe was destroyed tomorrow, the Axiom of Choice would still be an independent axiom, and Gödel's incompleteness theorem would still be true.

I'm starting to understand why U say maths is not science. But I debate in that axioms and other mathematical objects has independent existence from us (despite that I'm a light formalist-structuralist and maybe a halfplatonist). There are a lot of warming signals to support my debates, e.g. Reuben Hersh's books (despite that he wrote a lot of neomarxistic stupidness in his new bokks as some answers to this problem, his questions and problems are good), then the intuicionism and the discoveries in non-classical logics; then Darwin's theory on evolution and so on). Gödel's theorem, like whole maths and the whole science, is founded on a lot of methaphisical assumptions. Maybe it isn't true at all. Gubbubu 17:49, 21 Dec 2004 (UTC)

I'm sorry, I shouldn't have used the word "true" to describe Gödle's incompleteness theorem. Yes, it's based on metaphysical assumptions, but that's not the point. The point is that it's valid no matter what. Mathematicians are fine with making valid statements like, "If 0=2, then pi=3". Sure, the premise and conclusion are false, but it's mathematically valid. --Sean Kelly 18:50, 21 Dec 2004 (UTC)

Yes. But it's a historical accident, caused by formal (extensional) definition of implication/causality. Intensional logic is in its primitive, initial stadium but it exists (Church, Lewis). Formal and extensional implication don't has to be the one and only method of mathematical thinking, what's more, I think we even don't use it in practice, working on a long proof (intuition etc.).

These things constitute pure mathematics—they are part of the foundations of set theory and complexity theory. By what definition are they considered science? -- Sean Kelly 00:58, 21 Dec 2004 (UTC)

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).

For a definition of science I would direct one to the article on the Scientific method, say, or Descartes' Discourse on Method. But that's still beside the point. The fact of the matter is that the average user will consider science to be the study of the natural world, and mathematics to be the study of numbers. On this intro page, we should not get into a heated debate over whether these are the correct definitions. The point of the statement "Mathematics is not science" is to eliminate the misconception that mathematics is the study of computation, or a tool to describe the universe. Do you disagree with this?

Now from this discussion I can see we have a huge cultural difference. I studied mathematics and never touched a calculator. You, I'm assuming, think mathematics consists of intuition and computation followed by proof. I think that mathematics would exist in the same state it is in now whether or not we existed. You, perhaps, think that mathematics is a human construction based out of the need to solve problems in the sciences. We could argue our points indefinitely, but I'd rather concentrate on writing for the average reader, who is unaware of any distinctions. --Sean Kelly 21:55, 21 Dec 2004 (UTC)

I can't believe for an avarage user maths is not a science. I think this belief must be quite curious and not too widespead. In most European countries it is absolutely accepted as a science. A lot of similarity (the institute system, the methods etc.) must put us on guard saying not to be a science.
It is not sure the study of numbers is not the study of the natural world. More, great scientists (like Galilei) said the book of nature is written on the language of mathematics. I think maths is so related to other (natural) sciences than we can't say it wouldn't be a science.
I think if we can't define something, we must'nt. We must not treat people as idiots, cause they are not; and admit if we didn't know something. As a matter of fact, I think the whole article is quite inexact, naive, and misleading. For example, what does it mean "maths" is the sutdy of patterns? What patterns? Of fancy works? Of Picasso's paintings? And so on. I think english editors threw out the formalists' maths definition, but couldn't give something better (i wrote on this topic above on this page). I think we shouldn't aggravate this situatinon writing more POV sentences "maths is not a science". It's not a shame to show some question is undecidable, but editors of this article seems to be not agreeing with me. Gubbubu 17:52, 22 Dec 2004 (UTC)

So, I can accept something like this: "Someone say, maths is not a science, cause ... ", but I think the judgement "Math is is not a science", this way, without anything else, is so-so POV. Gubbubu

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)

I agree with everyone that mathematics can be scientific or artistic, but I don't agree that it is a science or an art. I think that you can stretch the definitions of science and mathematics so that one seems to be an instance of the other, but you can do this with any subject. For example, I can say that a gourmet cook has a theory about a dish, then plays around with the ingredients for a while, gathers data, and then comes up with a recipe. Cooking is like a science---I can even be poetic and say that it is metaphorically a science---but it is not a science. Golf is like a science, I have a theory about the hit I'm going to make, I can take some practice strokes... I think you see where I'm going. IMHO, the word "science" is reserved for a few specific subjects, and mathematics is not one of them. --Sean Kelly 03:40, 8 Feb 2005 (UTC)

Yes, cooking and golf is not a science. But mathematics is not cooking and not golf, it's more serious. Gubbubu 18:37, 8 Feb 2005 (UTC)

Mar 2005 "Not a science, by definition"

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)

Mathematics is not a science. Mathematics is mathematics, it is ubiquitous, it is not a subcategory of something greater. We don't have to define it in terms of something else. --Tothebarricades.tk 05:46, 6 Mar 2005 (UTC)

Calling it a science no more "defines it in terms of something else" than calling it one of the academic disciplines "defines it in terms of something else". Michael Hardy 02:03, 20 Mar 2005 (UTC)

April 2005 "Science or not?"

The article contradicts itself

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:

How about replacing the sentence "Some hold ... allied" with "It is debatable that mathematics is not itself one of the sciences, however closely allied, since mathematical knowledge is not fundamentally empirical." But I don't see how the "Although Einstein ... empirical" sentence contradicts "Some hold ... allied". Could you please explain? -- Jitse Niesen 11:22, 17 Apr 2005 (UTC)

The "Some hold...allied" sentence implies that mathematics may be a science. The "Although Einstein...empirical" sentence states that mathematics is definitely not a science. Brian j d | Why restrict HTML? | 03:58, 2005 Apr 23 (UTC)

In my reading, the "Although Einstein ... empirical" sentence says that mathematics is not a science, if a certain definition is used. The sentence is badly formulated IMHO, but I think the "one not-unusual definition" refers to defining science as empirical. If another definition for science is used, then mathematics may be a science. Jitse Niesen 19:36, 23 Apr 2005 (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)

Mathematics is not science. re: french usage, science as systematic knowledge. one would then have to call religion science. science, in regards "scientific" refers to empirical knowledge that is falsifiable, universally verifiable, etc. for instance, an analysis of a survey would have to include the raw data from the survey in order to be "scientific". Does math have to include the raw data from the survey? This question is nonsensical, as it is with religion and other systematic knowledges that are not "scientific". Yes, math is a systematic knowledge. No, there is nothing scientific about math. Kevin Baas^talk 04:47, 2005 Apr 23 (UTC)

The mathematics article and the Science article contradict each other

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)

To think in abstractions is the strategy of simplification of detail. You can select an empirical detail from a mass of other empirical detail, thus simplifying or abstracting. No contradiction. Ancheta Wis 08:27, 23 Apr 2005 (UTC)

What is science?

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)

Such an argument does not seem to exist. We have the science article that says that science is empirical and a comment on the talk page that says that mathematics is not a science. The mathematics article and the science article should not contradict each other. Brian j d | Why restrict HTML? | 06:11, 2005 Apr 23 (UTC)

When I say "Such an argument does not seem to exist.", I mean that there seems to be no mention on the talk page except for the one in my previous comment. Brian j d | Why restrict HTML? | 06:28, 2005 Apr 23 (UTC)

See note above. Ancheta Wis 08:28, 23 Apr 2005 (UTC) I came to this page to answer a point which Brianjd raised in the mathematical notation article, but the link failed to resolve to a specific point. Care to repeat it, so I can attempt to respond? Ancheta Wis 08:35, 23 Apr 2005 (UTC)

I agree that the word "science" is commonly used in the less specific sense, including in many dictionary definitions of "mathematics". However, that is not what science and the related articles cover, and I have added a note to science's introduction to that effect. – Smyth\^talk 10:42, 23 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)

Neutral?

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)

Let's review the changes you just made:

You placed a NPOV warning on top of the page, which IMO is a misunderstanding of NPOV. No editor has brought any bias to this article, and the question of whether mathematics is a science is a minor one which has no bearing on the vast majority of this article's content.
You moved all of the categories and language links to the top of the article for some reason.
You put mathematics into the "Science" category despite your own apparent views that it is not a science.
Your argument for NPOV is that the article 'indicates that [Mathematics] is or is not a science,' which it barely does in at most two sentences. I hardly think this justifies calling the entire article biased.

Based on this, I'm reverting all of your changes, and I hope you will not revert back without addressing the points I just made. —Sean κ. ⇔ 06:07, 23 Apr 2005 (UTC)

"Minor question" or not, it is part of the article.
It seems more logical to have them at the top. I haven't seen any policy or guideline on this.
If some people think it's a science, doesn't it belong in the "Science" category, no matter what the editors think about it?
There's nothing in the NPOV tag that indicates that "the entire article (is) biased". Brian j d | Why restrict HTML? | 06:19, 2005 Apr 23 (UTC)

and 4. The NPOV warning is very prominent and will tend to call the accuracy of the whole article into question. If there is only one point in dispute, there are smaller per-section templates that should be used.
If 99.99% of articles doing it the other way isn't a guideline, I don't know what is. As they're considerably less prominent (and important) than the article itself, they should obviously not appear before the article body.
No, we should resolve the dispute first.

– Smyth\^talk 09:53, 23 Apr 2005 (UTC)

Poll

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. ⇔

Is Mathematics a science?

No. Though m. is like a science, and metaphorically a science. --Sean Kelly
No Charles Matthews
No Gandalf61
No Kevin Baas Science is making statistical inferences from empirical phenomena, and is always falsifiable. Mathematics is a tool for navigating spatial relationships that may exist as theoretical models of empirical phenomena, and is not falsifiable. Science can exist without math, and math without science.
Yes. 1) Einstein knew it. 2) Computer science which is really a subfield of mathematics is a science. Mathematics may not be a _natural_ science, but it is a science. Every statement in mathematics is also falsifiable, simply by giving a counterexample. Sometimes it happens there are no counterexamples. If not does this mean that when, someday in the future, physicists find the ultimate theory which describes everything correctly and thus has no counterexamples, that that theory is not a scientific theory? -MarSch 17:46, 23 Apr 2005 (UTC)

1) Einstein may have been using the more general meaning of "science". For all I know, that was a perfectly common usage of the word at the time, and perhaps he was speaking in German. 2) "Computer science", despite its name, is either mathematics or engineering.

And mathematics does not proceed by the balancing of evidence for and against a theory. Fermat's last theorem was not proved by accumulating endless lists of numbers that were consistent with it, but by a literally unfalsifiable argument that no inconsistent numbers could ever exist. Science does not deal in the unfalsifiable. – Smyth\^talk 18:20, 23 Apr 2005 (UTC)

Yes! It is the first science. All others grew out of dependence on it. Without mathematics, there would be no real science. EDN. [Left by anon User:12.72.229.2 ].
No. This is more about the definition of science. I think science is empirical, and mathematics not, so mathematics is not a science. By the way, the Oxford English Dictionary says (science, meaning 5b): "In modern use, often treated as synonymous with ‘Natural and Physical Science’, and thus restricted to those branches of study that relate to the phenomena of the material universe and their laws, sometimes with implied exclusion of pure mathematics. This is now the dominant sense in ordinary use." On the other hand, mathematics (in modern use) is defined as "the science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis; mathematical operations or calculations." -- Jitse Niesen 19:53, 23 Apr 2005 (UTC)
No. It doesn't follow the scientific method. Theories are not really stablished, nor demonstrated: either they're relevant or not. Really, I don't even think mathematics makes sense as a subject; rather, we spend time mathematizing ideas, first through abstraction, then classification, then ..., etc, etc, etc. -- irrªtiºnal 20:03, 23 Apr 2005 (UTC)
No, but studying and understanding why people produce symbols and responses for math can be scientific. Maybe math should be a subcategory of psych disorders. GabrielAPetrie 19:12, 5 May 2005 (UTC)
No, clearly not, for all the reasons given during the discussion. Math is a part of the noosphere. linas 04:28, 13 Jun 2005 (UTC)
Yes, certainly. My God, what a stupidness. We will bring a democratic judgement about that what would be a science? When will we take a voting about that is 2+2=5 or not? Well, I can't imagine why the hell Jitse Niesen voted with no, when his resources says yes. Mathematics is a science, naturally. I think, you must rather correct the "scientific method" article, it semms to be stupidness if it says m. is not science. Maybe I will do that in august. Gubbubu 09:36, 19 Jun 2005 (UTC) (mathematician)

Is Mathematics empirical?

No. Empiricism means applying logical induction to the real world. Mathematicians use logical induction to come up with conjectures, but pure mathematics is independent of physics. --Sean Kelly
Sometimes it is, actually. Charles Matthews
Empirical methods used to form hypotheses but logic used to prove theorems - so results are not contingent, and mathematics is not just empirical investigations. Gandalf61
No Kevin Baas^talk "Numbers exist independantly of the things they number." 'Nough said.
I guess you mean "Does mathematics include taking measurements in the real world?" and then my answer would certainly have to be No. Instead from intuition you may suspect that a certain statement holds. This is your hypothesis. Then you can proceed to investigate and try to find a proof or a counterexample. These are your measurements. In this way you could say that mathematics is empirical. -MarSch 17:46, 23 Apr 2005 (UTC)
Yes always! Its basic roots are from observations. The roots can't be pruned. EDN. [Left by anon User:12.72.229.2 ].
No, at least not fundamentally. -- Jitse Niesen 19:53, 23 Apr 2005 (UTC)
I shall follow the current trend and simply say that it's a matter of taste... and it's blasphemously irrelevant. I would like to quote perhaps the one line which hinted me I wanted to be a mathematician, by Richard Courant, since we are all being intransigent preacher of our own faith: For scholars and laymen alike it is not philosophy but active experience in mathematics itself that alone can answer the question: What is mathematics? -- Wether uncontested opinions are enforced or not, this is not really just a poll, people is arguing and need to argue other opinions... else, please change the title of this section to something like Discussion... and then create an actual poll.
No. Mathematics reifies what is being observed and distances the observer. GabrielAPetrie 19:12, 5 May 2005 (UTC)
No. Err, sort of ... Math does depend on the mathematician making observations and noticing correlations. The act of creating mathematics requires empiricism, but the final product is not empirical. linas 04:35, 13 Jun 2005 (UTC)
Sometimes yes, sometimes no. Gubbubu

Is Mathematics an art?

No. Though m. is artist and beautiful, and metaphorically an art. --Sean Kelly
No Charles Matthews
No Kevin Baas^talk If math was an art, it would be taught in art school. Math does not express human emotion or irrational impressions in a relatively unrestrained form. To summarize, there are a few basic subjects taught in middle school: art, science, math, history, english. give or take a few, like phy. ed., etc. in any case, point made. is art a language? is history a science?
Unsure. Mathematics requires great skill and ingenuity, which would also make engineering an art. Yes art is a language and music is a language, even mathematics is a language. History is surely a science. It is even certainly empirical. How else do we know about our evolution or dinosaurs? "Hey, when I dig here I get a lot of bones and broken crockery. I wonder how it got here. Hmmm" -MarSch 17:46, 23 Apr 2005 (UTC)
Yes! Scientists are artists. Einstein ran his hand up and down looking at a fountain to freeze one drop and said "Never forget, this is science">>> PLAY! EDN. [Left by anon User:12.72.229.2 ].
No. Again, this is more about the definition of art. -- Jitse Niesen 19:53, 23 Apr 2005 (UTC)
Yes: it is an anxiety-produced symbolic representation of reality (as is art); it is crafted (and perceived) arbitrarily by some and institutionally by others, both of whom claim 'authority' (as is art); and provides the same distancing from and irrelevance to reality that art does. And just as art cannot be proven to be necessary, no-one has ever proven the necessity of the mathematical model. GabrielAPetrie 17:41, 4 May 2005 (UTC)
No. Lets not confuse the process of making math with the final result. To create math, one must be inspired, visionary, struck by lightening, a little bit crazy, etc. Ditto for the creation of great art or of great science. Possibly even politics or marketing. Can we say "marketing is an art"? "politics is an art"? Creation requires inspiration; however, just because its inspired does not make it "art". linas 04:45, 13 Jun 2005 (UTC)
Yes it's an art and it's a sience. On the one hand, every science can be an art. On the other hand, maths is the scientific art and the artificial science. Gubbubu

Origin in natural sciences?

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)

Think calculus and Isaac Newton. Ancheta Wis 08:58, 23 Apr 2005 (UTC)

The derivative (one could say that originated from the problem of finding the tangent line)? The integral (that originated from the need to find areas)? Brian j d | Why restrict HTML? | 11:45, 2005 May 8 (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)

How should the "Topics" section be structured?

Orionix's proposal

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)

Though I feel that the current scheme is a bit whimsical, I'm not sure I would agree with your divisions. I feel most of these can be combined or split depending on who is reading them. History and foundations doesn't seem to be a division of mathematics, rather a division of history about mathematics. I would ask why arithmetic split off from number theory, and why geometry and trig split off from analysis. And why is analysis the largest division? Why not group theory or algebra? I personally find the most useful scheme of divisions to be: algebra, analysis, logic, and other. --Sean Kelly 07:38, 15 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)

Geometry is a tough one, though. Classical geometry is clearly a unique field, but it's also very old, and I'm not sure how many geometers are left. Modern geometry seems to be its bastard child. Once we abandon the preoccupation with Euclidean geometry, then the study of spaces in general falls under analysis and topology. Algebraic geometry is really just utilizing what we learned back in classical geometry to solve tough problems in algebra. So we can't say that geometry is analysis, can't say it's algebra. Thoughts? --Sean Kelly

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)

Orionix's other proposal

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.

6. Probability and statistics

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] (http://superstringtheory.com/math/index.html)

In the future, mathematics will become more idealized and abstract of its subject matter. -- Orionix 01:16, 4 Apr 2005 (UTC)

It is really difficult to partition mathematics. Instead all these proposals are coverings. Thus it does no harm to provide several different "orthogonal" coverings. -MarSch 17:53, 23 Apr 2005 (UTC)

More discussion

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)

Here are some suggestions to bridge the period till we discover a theory of everything. I agree that the "topics" section, and also the subsequent "tools" section, is not very informative, and I think they should ideally be removed. Thanks to Charles' excellent work, most of the "topics" section (specifically, all except Theorem & Conjectures and World of Mathematicians) can in my opinion already be deleted. We might need a section on the activity of doing mathematical research, with links as Fields medal and Fermat's last theorem. I think there should also be a bit more on applications. But it is very hard to write an article on the whole of mathematics (which is of course why I am only making suggestions instead of editing the page directly!) -- Jitse Niesen 12:02, 5 Apr 2005 (UTC)

I think we should move the topics section to something like list of mathematical topics, perhaps mathematical topics or fields of mathematics. Tools section can be deleted. -MarSch 18:02, 23 Apr 2005 (UTC)

Friendler introduction?

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 (http://wikinfo.org/wiki.php?title=mathematics|nabbing):

"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..."

Symbolic logic?

Really? Logic, yes, but symbolic logic? Brian j d | Why restrict HTML? | 06:55, 2005 Apr 8 (UTC)

Your removal of mathematical notation there hasn't helped. A page of algebra, or Euclidean geometry, is typically representing what? Actually, it is probably best seen as an abbreviated, somewhat informal way of writing down mathematical arguments which in their full-on, unabbreviated form would be full of symbolic logic notations. I think your edit is a bit perverse, therefore. Most people are much more familiar with mathematical notations than with the logical notations (which were only made explicit in current form about a century ago). So I am going to put this back. Charles Matthews 09:13, 8 Apr 2005 (UTC)

How is mathematical notation an "abbreviated, somewhat informal way of writing"? Can you add some explanation to the article? Brian j d | Why restrict HTML? | 04:38, 2005 Apr 17 (UTC)

There is a famous joke (and not really a joke) about '2' being an abbreviation for about 10000 symbols of a correct definition starting with set theory. Even 2x + y + z abbreviates by leaving out parentheses as compared with ((2×x) + y) + z. I'd agree, having looked at it, that the notation article needs work. Charles Matthews 11:48, 17 Apr 2005 (UTC)

Abstraction

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)

I think Charles has hit upon the right way out of this definitional quagmire we seem to find ourselves in. That is, rather than rely upon our own understanding of what mathematics is, we should instead examine what other respected sources say, and synthesize a definition from them. What we have been doing up to now, is probably akin to "original research". Paul August ☎ 16:34, Apr 21, 2005 (UTC)

computer science is just one branch of mathematics. But I think we can all agree that mathematics is inspired by reality. Then to get from reality to mathematics we do abstraction. Of course there is more to it than that, since this also what physics does. Then perhaps the distinction between the two is that mathematics investigates the abstractions while physics investigates reality.

Now to your suggestions. No, you cannot say that mathematics is the study of equivalence relations, might as well go all the way to sets. That may be the incarnation of a lot of mathematics, but it is merely one axiom system. It doesn't describe the spirit of mathematics. I would say philosophy is no science, but perhaps we should say something about logic, you will find little of that in philosophy. "The emphasis on abstraction and generality is passé", then what is the emphasis on now? No seriously, everything in mathematics is an abstraction, like sets for example and hypersets another. Whether an abstraction is general or not wholly depends on the abstraction. What new methods of knot theory BTW?

Anyway it is probably a good idea to find some sources that try to define mathematics. -MarSch 12:27, 22 Apr 2005 (UTC)

I think he means that it is now socially acceptable to study the digits of pi, as opposed to trying to find a theory of everything. linas 05:13, 13 Jun 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)

Yes, this is much the same as what I said. For some things we are still looking for the right abstractions. Thus these non-existent abstractions do not yet form part of mathematical knowledge, but the search for them is mathematics. What do you think about: theoretical physics tries to explain experimental data by a physical model of the universe, and mathematical physics tries to cast these models into mathematical theories. -MarSch 18:16, 23 Apr 2005 (UTC)

Ahem, most physicists know that there is a very very clear difference between theoretical physics and mathematical physics. The former attempts to explain nature, often making use of dubious, unjustified, hand-waving arguments. Physicists see this as "the good kind of physics, that which must be encouraged in students". The other thing, "mathematical physics", is an attempt to take things that are "clearly" understood by physicists (such as quantization, brimming with misunderstandings on the talk page), and put it on a rigorous mathematical footing, with theorems and proofs. Most physicists treat this thing, "mathematical physics", as a bit of a leper, a disease that must be avoided, a waste of time of time and energy; for it fails to advance the cause of physics. Most physicists and even many/most theoretical physicists, haven't a clue what math is. linas 05:13, 13 Jun 2005 (UTC)

(See, for example, Talk:Color charge esp. near the remark "its not a theory of anything". Bambiah is a physicist, and he is virulently rejecting any math (however correct and provable that math may be) that isn't firmly anchored in experiment. The topic under discussion is quite abstract.)linas 05:24, 13 Jun 2005 (UTC)

I think he is just saying that the math belongs in a general article about Field theories and not in the article about the particular field theory QCD. --MarSch 14:59, 13 Jun 2005 (UTC)

Topics redesign

Here's one way of making the topics section clearer. Just throwing it out there. —Sean κ. ⇔ 21:47, 23 Apr 2005 (UTC)

Quantity

<math>1, 2, \ldots<math>	<math>0, 1, -1, \ldots<math>	<math>\frac{1}{2}, \frac{2}{3}, 0.125,\ldots<math>	<math>\pi, e, \sqrt{2},\ldots<math>	<math>i, 3i+2, e^{i\pi/3},\ldots<math>
Natural number	Integers	Rational numbers	Real numbers	Complex numbers

Change

Ways to express and handle change in mathematical functions, and changes between numbers.

<math>36 \div 9 = 4<math>	<math>\int_0^1 x^2\,dx<math>	<math>\oint_{\ell} f(x,y)\,dy<math>	<math>\int 1_S\,d\mu=\mu(S)<math>	<math>\frac{d^2}{dx^2} y = \frac{d}{dx} y + c<math>
Arithmetic	Calculus	Vector calculus	Analysis	Differential equations

Spatial relations

A more visual approach to mathematics.


Topology	Geometry	Trigonometry

Comments on redesign

I like the concept very much. :) Kevin Baas^talk 19:01, 2005 Apr 24 (UTC)

I like the concept with the pictures. Although the grid on the torus reminds me some of differential geometry. First though we should be clear on what divisions we want to make. -MarSch 14:51, 2 May 2005 (UTC)

Appropriate pictures are good! Why do we have, for probability, under discrete mathematics, some normal pdfs (we should have discrete pdfs)? Brian j d | Why restrict HTML? | 10:12, 2005 May 8 (UTC)

(I'm new here, so i hope im doing this correct) I was just going to comment upon the same as the previous user. The descrete part of probability theory is just a fraction of probability theory. Probability theory is a subgenre of measure-theory or analysis and not only descrite mathematics. I think those pictures look great, so I dont want to try to change anything though. Steffen Grønneberg 12:21, 12 June 2005 (CET)

I agree, especially since the picture of the normal distribution is very much not discrete. So, I removed probability theory from the discrete mathematics section. It is still listed under applied mathematics. -- Jitse Niesen 11:17, 12 Jun 2005 (UTC)

Templates

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)

Are you aware of the discussion of the similar Template:Calculus at Wikipedia talk:WikiProject Mathematics#Template:Calculus -- is that needed?, where some people question its usefulness? I think that the numbers template also takes a lot of room and the reason why it is created is not quite clear to me. But perhaps it is time to discuss the wider issue on how the reader is expected to navigate through all the maths pages. -- Jitse Niesen 21:15, 2 May 2005 (UTC)

Dodgy definition

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)

Definition from Abstraction (mathematics): Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications.. In my view, defining mathematics as "the study of abstraction" is valid, but rather ... errm ... abstract, and so not very helpful in isolation. Gandalf61 13:27, May 8, 2005 (UTC)

It is relevant in that mathematicians tend to be Platonists, and Platonism accentuates the abstraction from the real world to an ideal level. It's not much good as a definition; it doesn't have much explanatory value, and it is quite possible to be a kind of nominalist mathematician who rejects all that (Haskell Curry comes to mind). It is just not a lot of fun thinking of mathematical reasoning that way. Charles Matthews 13:35, 8 May 2005 (UTC)

ISBN

Template:Tl

What is this section for? Brian j d | Why restrict HTML? | 03:06, 2005 May 17 (UTC)

Wading through the recent changes

I'm still trying to wrap my mind around this paragraph...

Since the result of mathematics inspired by mathematics is often pure mathematics and thus has no applications outside of mathematics yet, the only value it has is in its aesthetics. Surprisingly often, it has happened that pure mathematics, which was considered only of interest to mathematics, has become applied mathematics because of some new insight, as if it anticipated later needs.

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)

I understand that you find my formulation suboptimal. Template:Tl. Perhaps you should take a look at what was in the old version before saying you want to revert. -MarSch 12:35, 19 May 2005 (UTC)

While the formulation may indeed well be "suboptimal" in many places, it is not worse than before. However, I do not understand why the section Overview of fields of mathematics was moved to history of mathematics. In fact, I found this the best section in the whole article. As the heading already indicates, it is not about the history, though it does sometimes mention the fields in historical order because that happens to be the most logical order. Therefore, I undid the move. -- Jitse Niesen 23:42, 9 Jun 2005 (UTC)

Fine by me. I didn't move it, but merged it, since it was duplicated at history of. --MarSch 09:19, 10 Jun 2005 (UTC)

More categorization

Abstract algebra	Number theory	Algebraic geometry	Group theory
Missing image Rubik_float.png	Missing image Elliptic_curve_simple.png	Missing image Bezout_simple.png	Missing image 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)

not I --MarSch 11:22, 25 May 2005 (UTC)

not I. I like the small text underneath. We need more graphics for the "structure" category before we can make a visual depiction group for it. Kevin Baas^{talk: new (http://en.wikipedia.org/w/index.php?title=User_talk:Kevin_baas&action=edit&section=new)} 22:23, 2005 May 27 (UTC)

Kevin Baas^{talk: new (http://en.wikipedia.org/w/index.php?title=User_talk:Kevin_baas&action=edit&section=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 (http://en.wikipedia.org/w/index.php?title=User_talk:Kevin_baas&action=edit&section=new)} 23:40, 2005 May 27 (UTC)

Thomas Aquinas

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)

NPOV, but seems to be irrelevant indeed. Both the person and what he said. --R.Koot 00:02, 9 Jun 2005 (UTC)
Agree that it is irrelevant. —Sean κ. + 00:55, 9 Jun 2005 (UTC)

Mathematics and Geometry

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)

[[Category:Wikipedian mathematicians]]

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)

Hey, that sounds as if I need to say Wikipedia:WikiProject Mathematics, we really need that template for establishing territory...--MarSch 17:54, 12 Jun 2005 (UTC)

Cool, I'll put myself in it right away --CircleSquarer56

Me too! —CuBeDubler

Me three! ~ ~ Aangle Trisecter

yes, you do :) Project2501a 18:25, 12 Jun 2005 (UTC)

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