# Talk:Mathematics/Archive 5

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## Is mathematics a science?

### May–Nov 2004

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! Brianjd 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"? Brianjd 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?"

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. Brianjd | 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. Brianjd | Why restrict HTML? | 04:42, 2005 Apr 17 (UTC)

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. Brianjd | 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? Brianjd | 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 Baastalk 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")? Brianjd | 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. Brianjd | 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. Brianjd | 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. Brianjd | Why restrict HTML? | 04:08, 2005 Apr 23 (UTC)

Let's review the changes you just made:
1. 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.
2. You moved all of the categories and language links to the top of the article for some reason.
3. You put mathematics into the "Science" category despite your own apparent views that it is not a science.
4. 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)
1. "Minor question" or not, it is part of the article.
2. It seems more logical to have them at the top. I haven't seen any policy or guideline on this.
3. If some people think it's a science, doesn't it belong in the "Science" category, no matter what the editors think about it?
4. There's nothing in the NPOV tag that indicates that "the entire article (is) biased". Brianjd | Why restrict HTML? | 06:19, 2005 Apr 23 (UTC)
1. 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.
2. 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.
3. 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?

• Yes! If you define science as the study of phenomenon and the development of tools to do so. It is not religious or artistic to develop the tools of science. So it is a scientific exploration to develop those tools. Those tools are science. They are part of the body of knowledge known as science. EDN.[Left by anon User:12.72.229.2 ].
• 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)
• Yes. In fact it's an empirical science, at its edges, such as the study of large cardinals. These are not posited because their existence is considered a priori obvious, but because of the way their consequences cohere. They are also falsifiable in Popper's sense. --Trovatore 20:07, 17 July 2005 (UTC)
Oh, you get bonus points for that! That's strong enough to maybe even change my vote. linas 22:52, 22 July 2005 (UTC)
• No. -Lethe | Talk 00:59, July 23, 2005 (UTC)
• No. For me (but not for everyone), and as far as school subjects are concerned, science is about concrete things. Mathematics is not concerned with concrete things. Brianjd | Why restrict HTML? | 14:52, 23 July 2005 (UTC)
• Yes and No: it's a semantic question. The word "science" stands for a number of different things, and math fits some and doesn't fit others. Traditionally, a science is knowledge of "what is" as opposed to art, which is knowledge of "how to" (as in "the industrial arts"). By that definition, math straddles science and art. More recently, some people have tried to emphasize a basis for knowledge in experimental test as an important distinction, and have tried to limit science to just that kind of knowledge. By this definition, it's arguable whether math is a science is not, but most would say it doesn't meet that criterion. I propose that we simply remove the Queen of the Sciences section. If there is interest, we can make a page describing the philosophical schools of thought about this question. It's not Wikipedia's place to make philosophical rulings. --Ben Kovitz 05:02, 31 July 2005 (UTC)
• No.: mathematics is an extension of natural language, with increased precison. it is a tool for decription, measures and communication, and for the development of ideas about quantitative relationships.
• Yes, according to the Oxford English Dictionary: "the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations..." (part of a much longer definition). That probably won't satisfy anyone, because the debate turns on which sense of the word "science" you have in mind, and many people are unaware that the word has other senses than the one they're using. --Ben Kovitz 06:33, 2 October 2005 (UTC)
• Yes, It is the science of mathematics. Science is a way of thinking and vialualising/understanding the world. 68.55.145.124 01:50, 20 October 2005 (UTC)

#### 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 Baastalk "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
• Yes. See above remarks (in #Is Mathematics a science?) on large cardinals. Really this is true even much lower down. For example the platonic existence of the set of all natural numbers is a falsifiable hypothesis; it would be refuted by the discovery of a contradiction from the Peano Axioms. --Trovatore 20:11, 17 July 2005 (UTC)
• Yes. Not only I agree with above (choice of axioms), but choice of definitions is matter of what is useful and which ones lead to shortest proofs. Samohyl Jan 19:22, 18 July 2005 (UTC)

#### Is Mathematics an art?

• No. Though m. is artist and beautiful, and metaphorically an art. --Sean Kelly
• No Charles Matthews
• No Kevin Baastalk 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
• Could be. If defining "science" is always risky (because you might exclude things you later see you don't want to), defining "art", exclusionarily, is positively contrary to the spirit of art. And there's a nice artistic little antinomy somewhere in there. --Trovatore 20:15, 17 July 2005 (UTC)
• No and Yes: it's a semantic question. Traditionally an "art" is a set of methods or know-how for doing things, as in "the industrial arts", including engineering. Methods of calculation are a pretty important part of math, indeed the main focus for non-mathematicians, so math is an art in this sense. People sometimes use "art" to mean something that doesn't have a completely systematic method, like the fine arts. In this sense, calculation methods are not art, but proof-writing is. But math is primarily the objects and relationships that we learn about when we learn math: numbers, sets, geometric relationships, patterns, etc. This makes math not an art but a science as traditionally defined (but please see my comments above about math as science). To paraphrase linas, let's not confuse the process of discovering math with what it is that we discover. --Ben Kovitz 06:04, 2 October 2005 (UTC)

### latecomer to the debate

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)

The best we have so far is kind of a compromise in the subheading titled "Queen of the Sciences":
Albert Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions, a phrase first used by Carl Friedrich Gauss. If one considers science to be strictly empirical, then mathematics itself is not a science. That is, mathematical knowledge exists separate from the physical world.
Mathematics shares much in common with the sciences. Experimentation plays a large role in the formulation of reasonable conjectures, and therefore is not by any means excluded from use by research mathematicians. However, theorems are only accepted if proofs have been found for them.
Actually, a lot of people hold the opinion that there shouldn't be a "what mathematics is not" section, based on the arguments above. I'm beginning to concur. —Sean κ. + 06:11, 11 July 2005 (UTC)

### "Queen of science" section problematic

Albert Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions, a phrase first used by Carl Friedrich Gauss. If one considers science to be strictly empirical, then mathematics itself is not a science. That is, mathematical knowledge exists separate from the physical world.

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)

### Proposal: delete this section

Here are some reasons to delete the section on "Is mathematics a science?":

• It starts semantic arguments (see above).
• It's about the philosophy of mathematics. Pretty much any answer you give will make one controversial claim or another.
• It distracts from presenting an overview of the subject suitable for a non-expert.
• The article is too long anyway, and there are many more-important things that should be kept instead.

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)

Someone will re-open the question. Charles Matthews 12:50, 2 October 2005 (UTC)

## 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. Brianjd | 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)? Brianjd | 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? Brianjd | 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.

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.

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)

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:

"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).

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? Brianjd | 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? Brianjd | 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

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

### Change

Ways to express and handle change in mathematical functions, and changes between numbers.
 ${\displaystyle 36\div 9=4}$ ${\displaystyle \int _{0}^{1}x^{2}\,dx}$ ${\displaystyle \oint _{\ell }f(x,y)\,dy}$ ${\displaystyle \int 1_{S}\,d\mu =\mu (S)}$ ${\displaystyle {\frac {d^{2}}{dx^{2}}}y={\frac {d}{dx}}y+c}$ Arithmetic Calculus Vector calculus Analysis Differential equations

### Spatial relations

A more visual approach to mathematics.

I like the concept very much. :) Kevin Baastalk 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)? Brianjd | 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). Brianjd | 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

{{ISBN}}

What is this section for? Brianjd | 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. {{sofixit}}. 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 128px 128px 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 Baastalk: new 22:23, 2005 May 27 (UTC)

Kevin Baastalk: 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 Baastalk: 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)

Where does set theory of any kind fit into this? Brianjd | Why restrict HTML? | 09:33, 17 July 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)

## [[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)

## Fluid mechanics

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)

I agree. I have removed it. Brianjd | Why restrict HTML? | 07:27, 17 July 2005 (UTC)
It should go back, I think. It is a subfield of continuum mechanics, perhaps - but mechanics tout court more likely implies just the mechanics of rigid bodies, plus perhaps celestial mechanics. Anyway, why be cheese-paring about links that help navigate to big areas of the site? Charles Matthews 08:13, 17 July 2005 (UTC)
1. One category path on Wikipedia is: Mechanics > Classical mechanics > Continuum mechanics > Fluid mechanics (aka fluid dynamics). So, fluid dynamics is under mechanics.
2. I think that this section should be an overview of where mathematics can be applied that is as simple as possible. I think we can make a much more exhaustive list but 1) why should we? and 2) why put it there? Brianjd | Why restrict HTML? | 09:18, 17 July 2005 (UTC)
I agree with Charles. When I read "mechanics" by itself, I indeed think of mechanics of rigid bodies. Furthermore, fluid dynamics is a huge subject within applied maths (at least as the term applied mathematics is used in Britain), definitely bigger and more important than cartography. -- Jitse Niesen (talk) 14:08, 17 July 2005 (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)

A good point indeed is that fluid people might not consider themselves mathematicians, rather physicists. Either way, I did not include fluid dynamics in the list of mathematics categories so far. I don't know if I should. Oleg Alexandrov 22:33, 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)

Well, the ones funded by the oil industry probably are applied people. It depends where you are, I think. It's basically a physics/engineering/numerical analysis area with inescapable amounts of heavy mathematics; anyone theorising abstractly about turbulence could claim to be doing fluids. 'Just a subfield of mechanics' seems rather a bald description. Charles Matthews 09:52, 18 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? Brianjd | Why restrict HTML? | 09:29, 28 August 2005 (UTC)

IMO changing section headers is bad form because it breaks links to the sections found, for example, in the page history. These get broken when discussion is archived, but that usually happens much later. (And while there hadn't been discussion of this topic "for a while", it is still the third-to-last section on this page... it's not like it's ancient.) - dcljr (talk) 10:51, 28 August 2005 (UTC) ("someone")

## Lecture picture

What is the point of that lecture picture? Is it there because we can't find anything better? Brianjd | 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)

The picture seems to be good but inappropriate. This article is about mathematics itself, and even if it did have a section on mathematics lectures, the picture doesn't belong in the "history of mathematics" section. Brianjd | Why restrict HTML? | 03:08, 25 July 2005 (UTC)
I don't imagine you'd object to including a picture, in the Chemistry article, of workers holding test tubes with colorful liquids in them over Bunsen burners--that's the practice of chemistry. How would you rather show the practice of mathematics? Some guy sitting in a chair staring into the middle distance? I'm not really attached to the picture, but I think on balance it's probably better than nothing. --Trovatore 04:46, 25 July 2005 (UTC)
We can show pictures like those on other mathematics pages, inside and outside of Wikipedia, to help understand mathematics (mostly graphs, I suppose). Brianjd | Why restrict HTML? | 05:34, 26 July 2005 (UTC)
But that doesn't show people doing mathematics. I suppose the lecture doesn't either, but it's closer. Better perhaps would be a picture of a collaboration, two people working on a problem at a chalkboard, but it would look staged (because it almost certainly would be). --Trovatore 05:50, 26 July 2005 (UTC)
Why do we want to show people doing mathematics? Brianjd | Why restrict HTML? | 02:56, 27 July 2005 (UTC)
The other sciences do it. --Trovatore 03:03, 27 July 2005 (UTC)
Firstly, as far as I'm concerned, mathematics isn't a science. Secondly, what the other sciences show is something that's unique to a school subject called "science", while this lecture picture shows something that's common to many (all?) subjects. Brianjd | Why restrict HTML? | 04:34, 31 July 2005 (UTC)
I moved the picture to Mathematics education. Judging from the above discussion, I think no one would oppose. If we want to put a picture related to history of mathematics, some historical note or sketch might be appreciate; but certainly not a contemporary lecture on linear algebra. -- Taku 04:52, July 31, 2005 (UTC)
Picture related to the history of mathematics? Snapshot of Fermat's marginal note. :-) Less spectacular but more relevant: snapshots of original writing by some famous mathematician on some pivotal topic in mathematics. JRM · Talk 17:59, 13 August 2005 (UTC)

## candiates for category pics

Kevin Baastalk: new 17:52, August 13, 2005 (UTC)

## M is not absolute truth

This section has been deleted as POV -why?

Mathematics is not about absolute truth. As is well known in the case of geometry, (see Non-Euclidean geometry) it is not important to know whether points and lines exist as absolute truths, but it only matters to know what relations there are between points and lines - namely, whether two points determine a unique line, whether two lines determine a unique point, and so on. The truth of mathematics is limited to the axioms and postulates that we feed into the system and nothing more nor less can be read from its theorems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
I think that this is absolute(ly) truth (right :-); and that this section is an intelligent part of the article. Gubbubu 06:41, 12 September 2005 (UTC)
It is POV because it presents a subjective opinion that is not universlaly accepted, without at the same time presenting alternative views. To be NPOV it could say Some mathematicians think that mathematics is not about absolute truth ... Other mathematicians think that mathematical entities have an existence that is independent of the human mind ... - but these issues are already deailt with (in considerable detail) in the philosophy of mathematics article. Gandalf61 13:45, September 12, 2005 (UTC)
But even if math is about real objects, it's still not "absolute truth" because it relies on unproven (unprovable) assumptions about those objects. - dcljr (talk) 22:51, 12 September 2005 (UTC)
What exactly does "not absolute truth" mean? The most straightforward interpretation of "not absolutely true" is "a little bit false". Do you think mathematics is a little bit false? From your phrasing I'd guess not; you just think that our knowledge of it is a little bit uncertain.
I actually agree with you on that point; I don't buy the notion of the apodeictic certainty of mathematics, and in fact consider mathematics an empirical science, subject to doubt like the other sciences. But I wouldn't put that assertion in the article, because it would be POV. --Trovatore 23:05, 12 September 2005 (UTC)
There is another possibility. Instead of negating the word "truth" you can change the word "absolute". Mathematics is conditional truth: true given the assumptions you've made. - dcljr (talk) 21:51, 13 September 2005 (UTC)
But if we're talking about real objects, then those assumptions are either true or false of the real objects being discussed. If they're in fact true, then the truth of the conclusions is no longer conditional. --Trovatore 21:59, 13 September 2005 (UTC)
Yes but determining the truth or falsity of those assumption concerning real objects lies outside of mathematics. However, that mathematics provides conditional truths, seems to me to be a kind of absolute truth. Paul August 22:29, 13 September 2005 (UTC)

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)

I agree as written it was not completely NPOV, but perhaps we should fix it rather than delete it? Paul August 22:29, 13 September 2005 (UTC)
The first time I deleted it I put in the edit summary that it could be useful if made NPOV. But on reflection I don't think that anymore; as Gandalf61 pointed out, this is all dealt with in philosophy of mathematics, and anyway if it weren't, that would be the place to do it. I can't think of anything related to this topic that could be called a "common misconception" in an NPOV way. --Trovatore 22:33, 13 September 2005 (UTC)
Maybe the real problem was the reference to absolute truth. Otherwise the original paragraph seemed okay (although in need of a rewrite) up to the phrase "nothing more nor less can be read from its theorems". That seemed a bit harsh. (dcljr -- continued below)
Well, no, I don't agree that it was OK up to that point. "it is not important to know whether points and lines exist as absolute truths". Who says? Skipping over the fact that no one claims lines are truths (that would be part of your rewrite), I think it would be very interesting to know whether points and lines exist as real objects. It's true that you don't need to know that in order to prove theorems about them, but I don't think mathematics is reducible to formal theorems. --Trovatore 23:59, 13 September 2005 (UTC)
As for salvaging something from all this for the article, how about mentioning how mathematical truth is conditional, always depending on underlying assumptions which could be called into question by new results. Seems like this would go nicely after the sentence "There is no shortage of open problems." (Unless it's already in the article -- I haven't read the whole thing in a while.) - dcljr (talk) 23:37, 13 September 2005 (UTC)
The claim "mathematical truth is conditional" is clearly POV. Maybe what you really mean is that mathematical knowledge is conditional, or perhaps a better word would be "provisional". I actually agree with that. Nevertheless that would also be POV--many claim that mathematical truths are logically necessary and can be known with justified certainty. --Trovatore 23:59, 13 September 2005 (UTC)
And what was one hair is now two... This is getting ridiculous. I give up. - dcljr (talk) 16:48, 14 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:

Mathematics does not produce absolute facts about the world. In the case of geometry, for example (see Non-Euclidean geometry), it is not relevant to mathematics to know whether points and lines exist in any real sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our real-world experience, they are not dependent on it. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

... but Trovatore thought even that was POV, explaining:

According to the realist view of mathematics (sometimes called "Platonist", but this term is contentious), mathematical objects are real things that exist independently of our reasoning about them. Thus they are part of the "world", as I understand that term. Then mathematical reasoning--starting from assumptions that are in fact true of the objects they purport to speak of--can indeed produce absolute facts about the world.

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)

Hadn't occurred to me to add "physical". That's certainly getting much closer. I can still imagine someone quibbling it, but I probably wouldn't, myself. Of course "exist in any real sense" would also have to change to "exist in any physical sense". --Trovatore 17:13, 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)

It's OK with me, given the changes discussed above. --Trovatore 23:59, 15 September 2005 (UTC)

## Definition of Math

I don't like the present definition of math. It is too abstract. I propose the following:

Mathematics is the science of abstraction, that is of selecting some relevant properties of some objects, and, in a second step, of establishing the properties (theorems) of all objects which are qualified by the same properties.

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)

That's less abstract? Actually, the definition has been discussed many times by now. Some of us feel it would be better to cite definitions from other sources, rather than post new, home-made ideas. Charles Matthews 16:18, 23 September 2005 (UTC)
I like the present definition better. And I agree with Charles. Paul August 17:44, 23 September 2005 (UTC)
I find the proposed introduction highly abstract. all properties, and objects qualified with the same properties. Also, this makes math look like some formal theorem prover. That's of course a big part of what math is, but that's more as a means rather than a primary purpose as that paragraph would imply. Oleg Alexandrov 18:50, 23 September 2005 (UTC)
Right. My feeling about defining math is that, to the extent possible, I'd really prefer not to. Of course we have to say something; the readers expect to see it in that particular place, but it should be as non-prescriptive and non-exclusionary as reasonably achievable. --Trovatore 23:41, 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)

Vb, Trying to argue that math is about theorem proving is like trying to argue that literature is about the construction of grammatically correct sentances. (Mathematicians know that theorem provers are isomorphic to semi-Thue systems which are isomorphic to context-sensitive grammars which are isomorphic to Turing machines. We can make computers prove theorems; but the result is like gramatically correct monkeys at typewriters, compared to Shakespeare.). Charles is right, Oleg is right. This discussion would be beneficial if you provided references to famous personages attempting to describe mathematics, rather than arguing with mathematicians about what math is. linas 15:34, 24 September 2005 (UTC)
(It does beg the question: we have Markov chains that can write passable thrillers and steamy romances; can similar generators of mathematical theorms say anything passably interesting?) linas 19:35, 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

Mathematics is the science of abstraction and deductive reasoning, that is of selecting some relevant properties of some objects, and, in a second step, of establishing the properties (theorems) of all objects which are qualified by the same properties. Vb 07:48, 26 September 2005 (UTC)
Well, doesn't that just leave out too much? (I am well aware that pure mathematics has sometimes been defined along such lines.) Charles Matthews 08:42, 26 September 2005 (UTC)
Not defining math or citing famous definitions is giving up on the WP idea? How ya figure? NPOV is a big part of the WP idea, and any definition like yours is necessarily going to be POV.
I think the current definition is good precisely because it's obviously not to be taken too seriously; pro forma everyone expects a definition as the first sentence, so we get the silly thing out of the way and go on to more important topics. But if the article really has to spend screen ink on this--I hope it doesn't--then it will necessarily have to give various competing definitions from different philosophical schools. --Trovatore 15:56, 26 September 2005 (UTC)

Well, I think the point was missed. Let me try again, by paraphrasing.

Literature is the art of abstraction and non-linear reasoning, that is of selecting some relevant properties of some objects, and, in a second step, of establishing the emotional relationships of all objects which are qualified by the same emotions (felt by the reader).

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)

Don't use terms like stupidness. The current definition is not the only possible approach. But it aims to be helpful, and not to close down (to 'formal' mathematics) prematurely. Charles Matthews 07:15, 27 September 2005 (UTC)
I believe the word for pattern in Hungarian is minta. A pattern is that which is recognized by pattern recognition (alakfelismerés ?). --noösfractal 07:42, 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)

Maybe really not unscientific stupidness, but in a philosophically view, it is too weak, too primitive. Gubbubu
Don't use terms like stupidness. The current definition is not the only possible approach. But it aims to be helpful, and not to close down (to 'formal' mathematics) prematurely. Charles Matthews 07:15, 27 September 2005 (UTC)
Yes, I agree that current definition is not the only approach - so pov. So we should talk about some other approaches in the head of the article, and then make a new article like "Maths philosophy#Possible Definitions of Mathematics" or like this. I think we should admit that we can't define maths exactly, but this is a living scientific problem with lots of arguments and works (like Courant-Robbins:What is math., Hersh: The nature of math., Lakatos: Proves and refutations etc.). I know what pattern means in Hungarian (as you see, I asked you mean fancy works or the science of bathroom tiles, you know that was ironic sarcasm). So I think this definition is too general and not too exact, then the formalist definition is better (despite it is not wholly true, too). Gubbubu 10:24, 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)

I think this is a formalist view, and we should talk about this. But this section is only a point of view, too. The truth is that - imnsho - we can't define maths, and there isn't a full definition for that what mathematics would be. But i think this section is even better then the recent article. Gubbubu 17:47, 27 September 2005 (UTC)
I still say we should talk about it as little as possible here. It's a question for the philosophy of math article. How about something along the lines of
Mathematics is a discipline with no single agreed definition, but whose objects of study include this and that and the other thing, and that uses methods such as foo and bar.
where we keep the lists of objects and methods short, but sufficiently vague as not to leave anything out. --Trovatore 17:55, 27 September 2005 (UTC)
I would object strongly to that one as written, because of the inclusion of axiomatics as part of the definition. I consider axiomatics a method; mathematics does not all start with axioms. --Trovatore 16:06, 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)

IMHO, the definition of the 4th of April is far better than the current one. I agree axiomatics should not be there but if one skip this, one still obtains something nice. I don't like Brittanica's definition either. I think it is not better than the one we have now. Vb09:58, 28 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)

I don't like Rick's definition because:
1. The stuff it cuts out is important.
2. Some of the stuff it adds in is eurocentric e.g. The most famous ancient mathematical text is Euclid's Elements, which was studied by every educated person from the time it was written, circa 2000 B.C., until roughly 1960.
Gandalf61 16:19, 28 September 2005 (UTC)
It must stay simple at least in the first sentence, I mean understandable to anybody who thinks he knows what math is (without being a mathematician). If you think it is eurcontric then change it into One of the most famous ancient mathematical text is Euclid's Elements, which was studied by every educated person in Europe from the time it was written, circa 2000 B.C., until roughly 1960 and add some link to some Indian or Chinese mathematician if you know any. Vb 17:02, 28 September 2005 (UTC)
I suggest moving the discussion to the bottom of the page, since this subhead has gotten very long. Rick Norwood 18:42, 28 September 2005 (UTC)

## Time out

Try ranking, in order of what they manage to communicate about mathematics, these pages:

(A) axiomatic system (B) Bourbaki, (C) Category:Mathematics, (D) formalism, (E) history of mathematics (F) mathematics, (G) philosophy of mathematics.

An exercise to try to define where effort is needed. Charles Matthews 10:20, 28 September 2005 (UTC)

Eigenvalue, eigenvector, and eigenspace

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)

## Mathematics (producer)

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)

Well, you can alwas try. I suspect that the AfD will fail; however, this raises an issue about the bar for "notability" in WP. Recently, CH made a case for what constitutes a "notable" mathematician; I suspect the bar he set is higher than it is in other areas of WP. (The point being that there are others who are more notable than this producer, and yet they've already been ruled as "not notable enough"). linas 22:38, 27 September 2005 (UTC)

One can always put the more neutral {{otheruses}} which will expand into

For other uses, see mathematics (disambiaguation)

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)

As a general rule, if there is only one other use of a term, even if it's vastly less common than the main use, then a direct link should appear from the top of the main use's page. If this producer is noteworthy enough to have an article, then the link must be restored. – Smyth\talk 16:42, 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)

## new intro

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. This was the method of Euclid (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, which spread across the Middle East, North Africa, and the Mediterranean, created much of the mathematics taught in primary and secondary school, but the Greek numbers were cumbersome and hard to use compared with the base ten numbers we use today. After the time of Archimedes (287 BC - 212 BC) no new mathematics was reported in the West for more than a thousand years. In that period, new mathematical discoveries were made in China, Japan, India, North Africa, and the Middle East. Renewed interest in mathematics in the West started with Fibbonacci (1170 - 1250), and led to a rennaissance in mathematics. In particular, the importation of base ten numbers from India by way of Arabia made mathematics much easier, as did the algebraic notation developed by mathematicians such as 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 of history before 1900.
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.
The first paragraph is fine, except for the pronoun "this" without a clear referent. The rest of it is just way way way too long-winded. It might fit elsewhere in the article, but doesn't belong above the TOC. Fold it into the history secton, maybe. --Trovatore 19:54, 28 September 2005 (UTC)
Hi Rick. Thanks for bringing your ideas here to the talk page. One thing it would be good to keep in mind is that, collectively, we have been discussing the first few sentences of this article for years. I won't ask you to read through all the voluminous talk page archives (see above - although it wouldn't hurt ;-) but let me assure you that a lot of smart and knowledgeable people (present company excluded of course ;-) professional mathematicians and othwerwise, have spent a lot of time and "ink" trying to figure out what those few sentences should be. (it might also be an instructive excersize going through the page history to see how the"definition" has changed over time). Paul August 20:43, 28 September 2005 (UTC)

## Second try at a compromise, based on apt criticism

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.

Better, but I still don't see why we need any significant historical discussion above the TOC. How about just paras 1 and 3, and work para 2 into the history section?. --Trovatore 20:44, 28 September 2005 (UTC)
Agree with Trovatore's reversion and reasons. Also, why is every newcomer mathematician's first itch to scratch is putting his/her own ideas into this article? Make an account baby, sign up in the math wikiproject, and help with less glamorous tasks than putting your stamp on the main math article. Oleg Alexandrov 22:04, 28 September 2005 (UTC)
I did sign up with the math wikiproject months ago, and I've been working my way through maths7 -- I can't think of a less glamorous task than that. And I am not the only one who saw the old intro as stodgy and not giving an accurate picture of mathematics. You make it sound as if I had just tossed in ideas off the top of my head, instead of thinking about and working on a revised introduction for a substantial amount of time -- not to mention thinking about and writing about what mathematics is for my entire professional life.
So then the anon contributor was you. Got it. Oleg Alexandrov 00:44, 29 September 2005 (UTC)
Let me explain why I think some history is necessary. Most people think that mathematics is just something mathematicians make up -- they have no idea what a mathematical proof is -- never seen one, never even heard of one. Further, most people think mathematics was all made up a long time ago. They think mathematicians just repeat this stuff to the next generation. They are surprised and largely disbelieving when I tell them that new mathematics is discovered all the time and that I've made a few very modest contributions myself. Third, most people think mathematics is completely useless. I remember a graduation address by a writer for the New Yorker magazine who told the students, to cheers, "There's no such thing as algebra in the real world."
That said, I'll try to cut further the history section. Rick Norwood 23:37, 28 September 2005 (UTC)

## Once more unto the breach, dear friends.

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)

1. The list of civilisations should include ancient Egypt - see Egyptian mathematics.
2. Specifically, it was the invention of the digit zero in India around the year 300 that simplified arithmetic calculations. Mentioning base ten is potentially confusing, as people were counting in powers of ten and using base ten notations in many cultures long before this.
3. Algebra goes back a long way before Regiomontanus - there is a short history in the algebra article.
4. IMO, the middle paragraph belongs in the History section, not in the introduction.
Gandalf61 15:32, 29 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)

While there's some merit in your points about popular misperceptions of mathematics, if you want to convince naive readers that math wasn't "all made up a long time ago", I hardly think talking about Regiomontanus is going to help. One problem with counting on future fine-tunings is that tendentially those make things longer, not shorter; another is that if the material doesn't belong in the intro then neither does a fine-tuned version of it. --Trovatore 17:02, 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)

## History in the intro section

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)

### Poll-history in intro

Keep or remove the history paragraph above the TOC?

• weak remove --Trovatore 17:02, 1 October 2005 (UTC)
• Remove - it's not the history of mathematics article. Charles Matthews 20:24, 1 October 2005 (UTC)
• keep - but do not allow to grow to more than the five lines it currently occupies. It is consistant with other wikipedia articles in having a small about of history in the introduction. Rick Norwood 22:50, 1 October 2005 (UTC)
• Remove - either move it down to the History section or remove it completely. Gandalf61 14:07, 2 October 2005 (UTC)
• remove - Mentioning a little history in the intro could be OK, but the present history is not a good intro: listing cultures that contributed to math, how much math was published in the 20 century, etc. makes the opening incoherent in my opinion. There is already a section in the article on the history of math, and it comes first. And there is an entire article on history of mathematics. --Ben Kovitz 17:08, 2 October 2005 (UTC)
• I don't favor polls, but until I have the time and inclination to write more, I'll say remove for now. Paul August 04:42, 3 October 2005 (UTC)

### Poll-length of intro

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.

• shorter It isn't massively too long at the moment, but I worry that the length may start creeping up. ---Trovatore 19:25, 1 October 2005 (UTC)
• about the same -- it currently meets the test of the TOC showing without scrolling down, at least on my browser, and I think that is a good test. Rick Norwood 22:51, 1 October 2005 (UTC)
• wrong question—It depends on what the intro says and what's in the rest of the article. —Ben Kovitz 20:47, 2 October 2005 (UTC)
• As long as it needs to be — and not a word longer ;-) Paul August 04:43, 3 October 2005 (UTC)
• Well, so what I'm really trying to get at with this question is, at what length does there need to be a serious justification for making it longer? Surely we agree that it's not like the article as a whole--those can go as long as they need to to cover the topic, more or less, though eventually you start to think about writing more than one article (or a book). Introductions aren't like that, right? They should have a more-or-less maximum length, independent of the complexity of the topic, do we agree on that? --Trovatore 04:47, 3 October 2005 (UTC)
No, I don't believe in having a length rule like that. Let's just keep searching for ways to make the article better, without predefining what we're going to find. --Ben Kovitz 12:14, 3 October 2005 (UTC)
• Shorter, i.e. in line with the policy that says three paras maximum, news-style. (Wikipedia:Manual of Style says The appropriate lead length depends on the length of the article, but should be no longer than three paragraphs in any case.) Not an essay. Charles Matthews 09:33, 3 October 2005 (UTC)
Listen to Charles, In my experience, he knows (generally) whereof he speaks. If we must have a rule of thumb about length, then the above seems about as good as any. Paul August 17:59, 3 October 2005 (UTC)

## reversion of changes without discussion

"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):

• The American Heritage Dictionary (1976): "The study of number, form, arrangement, and associated relationships, using rigorously defined literal, numerical, and operational symbols."
• The American Heritage Dictionary (2000): "The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols."
• Concise Columbia Encyclopedia (1983): "Deductive study of numbers, geometry, and various abstract constructs" followed by an overview of the main branches of math and then a short history.
• WordNet 2.0 (2000): "a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement."
• The OED (1933): "[pure mathematics:] the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra; [applied mathematics:] ...those branches of physical or other research which consist in the application of this abstract science to concrete data."

So I propose an opening paragraph something along this line:

Mathematics is the study of such topics as quantity, shape, pattern, and structure. The natural sciences, engineering, economics, and medicine depend heavily on mathematics to provide descriptive vocabulary for quantitative relationships within those fields and to provide dependable theorems and methods of calculation.

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)

I note that someone else has reverted to the earlier version. I think the OED version, which is quoted above, captures the idea of mathematics very well, and that the current introduction reflects that. There does, however, seem to be a feeling that the history doesn't work, and I tend to agree -- I thought it worked better when it was longer, but now it is probably not worth saving. I'm going to try to come up with a one sentence explanation of where math comes from, which I do think needs to be in the introduction, since most people do not know. Rick Norwood 00:59, 3 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)

Knot theory does address higher dimensions. In general, I wish people would micro-edit what is there, rather than propose radical change each time. Charles Matthews 09:52, 3 October 2005 (UTC)
I would like to echo what Charles (who is by the way one of the most experienced wiki-mathamaticians I know) has said above. And by way of offering some support for Charles' plea for conservative edits, for those of you who may have missed it, I will quote my self from above:
One thing it would be good to keep in mind is that, collectively, we have been discussing the first few sentences of this article for years. I won't ask you to read through all the voluminous talk page archives (see above - although it wouldn't hurt ;-) but let me assure you that a lot of smart and knowledgeable people (present company excluded of course ;-) professional mathematicians and otherwise, have spent a lot of time and "ink" trying to figure out what those few sentences should be. (it might also be an instructive exercise going through the page history to see how the "definition" has changed over time).
Paul August 18:41, 3 October 2005 (UTC)
Thanks for the warning, Paul. This is indeed potentially a major time-sink! I thought the previous (quasi-)definition was a bit clumsily worded, but OK: it was the kind of approach that emerges out of a long period of refinement by many people, and it's probably no coincidence that it was similar in spirit but a bit more refined than the other major sources.
Regarding radical edits vs. micro-edits, I also wince when losing a good bit of writing when someone walks in and tries something radical, but I think part of the glory of Wikipedia is that we stay open to bold, radical edits even when they temporarily worsen the page. Here on the discussion page, we can hash it out with the new author (as we have been the last week or so). If we are really good Wikipedians, we'll find some way to incorporate what the new author has to contribute, without losing the refinement that took so long to achieve. Who knows, we might get Rick Norwood to write us a nice, new, well-researched section on what professional mathematicians do and why they do it!
On this page we have a bit of a problem in that roughly half the article is inappropriate for an encyclopedia: four of the sections read more like personal essays declaring the author's deep beliefs about math than factual summaries ("Inspiration...", "Notation...", "Is it a science?", and "Common misconceptions"). Perhaps this has set a precedent, inviting authors to use the page for advocacy. If we do some work establishing a factual tone throughout, we might arrive at something more stable and that doesn't beckon for radical edits as strongly.
Ben Kovitz 22:20, 3 October 2005 (UTC)
Inspiration and Notation both need some editorial attention. This kind of contribution is easily defended, though. General WP policy is to aim for a degree of completeness; so addressing issues that round out popular ideas is basically good. Charles Matthews 11:29, 4 October 2005 (UTC)

## What math "really" is.

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)

No, I think it's a thoroughly philosophical question. To some mathematicians, tendentially realists, mathematics is the study of mathematical objects, by whatever (valid) method. If you could find out whether the continuum hypothesis were true, not by any traditional mathematical methodology, but simply by asking God, that would still be mathematics. To prescribe a methodology, particularly the methodology of proof, is unnecessarily restrictive (for example, it would exclude the extraordinarily convincing arguments in favor of Goldbach's conjecture from the realm of mathematical knowledge). --Trovatore 17:43, 3 October 2005 (UTC)
Now, you could say that this simply shifts the question slightly to "what is a mathematical object?", and you'd be right, of course. I don't propose any final definition of that. I think the best we can do is give examples and say, "get the idea?". --Trovatore 17:43, 3 October 2005 (UTC)
Goldbach's conjecture is a good example. It is still a conjecture, and I believe there is a million dollar prize offered for a proof. You may think mathematicians should accept authority, the word of God, or an overwhelmingly large number of examples, but the fact is, we don't. If we did, there would have been no fuss at all when Wiles proved Fermat's Last Theorem -- there were millions of individual cases checked. Once again, I suggest the OED be taken as definitive. Rick Norwood 20:34, 3 October 2005 (UTC)
The prize offered for proof doesn't demonstrate that anyone seriously doubts its truth; there are other reasons for desiring proofs, even when you're convinced of a proposition's truth. By the way the heuristic argument for Goldbach is not simply a matter of checking individual cases. That would not be especially convincing.
I didn't say mathematicians should accept "authority", though I suggested one particular case where one might. That particular authority is notoriously loath to publish; there's a piece floating around somewhere on why He couldn't get tenure....
What it comes down to is that mathematicians simply do not all agree with you about what constitutes mathematics. It is in fact a philosophically contentious point, one that the introduction ought not to trivialize or present as settled. --Trovatore 20:44, 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)

Thanks for explaining your reason for reverting, Vb. I actually haven't said my view about the definition of math. I don't think my view on that belongs on Wikipedia. I agree with you that a mere list of things is not a definition. (Contrary to the current opening, I think it is not true that math is "often defined as" that list; that list is no definition.) My concern is to make the intro introduce the topic: not by defining it precisely, but by broadly outlining its scope and mentioning some of the most noteworthy (for most people) facts about math. A broad sketch of the main topics of math is how most other sources characterize math, and seems pretty NPOV to me. What is an even clearer way to indicate that the opening line is not a definition?
It would be great to have a section on the various definitions of mathematics that people have proposed. Then the opening could refer to that. How would you like to write it?
Ben Kovitz 15:03, 4 October 2005 (UTC)
Could you try reading and applying Wikipedia:Lead section, rather than spinning theories of your own? Having the intro changed here every 30 minutes achieves exactly what? Charles Matthews 17:05, 4 October 2005 (UTC)

## false

"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)

## Beauty in mathematics

for recreation—discovering and cataloging patterns for their beauty, without regard for practical application

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)

Recreation (compare with leisure, both in terms of definition and categorisation - it makes no sense) seems to be something done for enjoyment, and that seems to be the correct term here. Brian Jason Drake 05:08, 16 October 2005 (UTC)
Yes, well, with that attitude, all of politics, music, art, literature, history, science and computer programming are "recreation". But that would be a silly over-broad use of the term. Professional math, like professional sports, is not exactly "recreation" even though what football players do in the stadium vaguely resembles what you might do at the beach. linas 19:01, 16 October 2005 (UTC)
I think "recreational math" is maybe better defined as math done without regard for application even within mathematics. For example, stuff that works only in base 10. I've modified the passage accordingly; see what you think. --Trovatore 19:27, 16 October 2005 (UTC)
I'm going to take a shot at that passage. It seems clear to me that there is a difference between pure mathematics and recreational mathematics. Euclid did not write The Elements because he thought it would be of use to carpenters, but neither did he write The Elements just for fun. Is the Mona Lisa "recrational art"? Rick Norwood 23:10, 16 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)

## Definition of "mathematics"

I have rewritten the much-disputed intro sentence as follows:

Mathematics is the field of study which investigates the exact formal relations between numbers and other abstract entities, and the methods of deduction, reasoning and proof based on the relationships between these objects. Applied mathematics is the study of spatial and quantitative relations in practical situations.

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:

[Mathematics is] That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations.

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)

Well, of the basic three things, analysis, algebra, geometry, it slants towards analysis; as one would expect in 1913, given the history. Charles Matthews 10:58, 19 October 2005 (UTC)

## Culture

I don't like the sentences:

Mathematics plays a role in every culture, and mathematical concepts are widely thought to be universal across all cultures. But few cultures have contributed new ideas to mathematics.

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 you. No point in provoking the "mathematicians are sexist and racist" crowd. Whether the passage is descriptively accurate or not (a point on which I'm undecided), it just doesn't add enough value to be worth its inflammatory potential right at the top of the article. --Trovatore 06:47, 19 October 2005 (UTC)
It's obviously unverifiable. The author has checked with all the people in the Amazon rainforest? Charles Matthews 08:46, 19 October 2005 (UTC)
While I will grant the possibility of rainforest mathematicians, whatever mathematics they may have done has not been "contributed" to the worldwide body of knowledge.
This concept has been in and out of the introduction many times. Everyone agrees that this article is important, but everyone has strong feelings about what it is most important to say. The modifying "only a few cultures" was added to offset the earlier "Mathematics plays a role in every culture."