Talk:Mathematics/Archive 9

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Custodian of the realm of numbers

It is proposed that mathematics re-establishes itself as custodian of the realm of numbers. There is universal evidence that numbers are being misused in every scientific discipline, usually though ignorance of the limitations of real numbers when used in calculations founded on measurements and estimates. Mathematicians should insist that scientists remember that significant digits inevitably curtail the significance of their results. For example, if this surveillance were in place we would not see absurditiies such as those in Terry Quinn's 2003 paper in Metrologia, (p105) giving the wavelength of various radiations to a precision of 14 or 15 significant figures. In 1.1 for example lamda is given as 236 540 853. 54975 femtometres. The trailing digits after the decimal point imply that this measurement was accurate to one ten-thousanth part of the diameter of an electron. The standard metre bar was only ever measured to seven signifcant figures; later the metre was established by convention with nine significant figures. Where did the additional five significant figures in Quinn's results come from? Geologician 10:28, 30 June 2006 (UTC)

What has this got to do with the article??? JPD (talk) 10:33, 30 June 2006 (UTC)

If mathematics' responsibilities as Custodian of the realm of numbers are spelled out in the article it will be less likely to shirk them, as is apparently the case. Geologician 12:08, 30 June 2006 (UTC)

Your question is irrelevant to this article, but here is the answer. The metre was not "established by convention with nine significant figures". It was defined exactly in terms of the time standard, which in turn is defined exactly in terms of a particular physical process. The accuracy of the wavelength measurements reflects the accuracy of measuring radiation frequency in terms of the time standard. McKay 12:56, 30 June 2006 (UTC)

Mckay's comment returns us rather neatly to exactly the point I have been making in the previous section. Karl Popper said "no empirical hypothesis, proposition, or theory can be considered scientific if it does not admit the possibility of a contrary case." As McKay so helpfully points out, here at the very heart of physics is clear evidence that metrology cannot be regarded as a science. This is because it is impossible to calibrate any future measurement against a standard that has merely been defined. When a universally respected yardstick such as the standard meter rod existed, an investigator could check his laser measuring device against a physical length. As this has apparently been abandoned the wavelength of light has itself become the standard and does not admit the possibility of a contrary case. Wavelength is itself dependent upon relativistic effects. Therefore there is no future possibility of determining whether there is any actual drift in the velocity of light with time. In short no electromagnetic method of distance measurement is subject to calibration against a physical distance that exists on Earth. Can nobody else smell a rat? Geologician 22:36, 30 June 2006 (UTC)

I am not sure exactly what you think "mathematics" is, if you say it "promotes itself as the Queen of the Sciences" and "shirks its responsibilities", and I am even less able to see how changing this article is likely to have any significant effect on how it acts. Please stick to issues relevant to the article. JPD (talk) 13:07, 30 June 2006 (UTC)

Perhaps a perusal of WP:SOAP#Wikipedia is not a soapbox would be in order. Stephen B Streater 15:29, 30 June 2006 (UTC)

Okay. Wikipedia was not made for opinion, it was made for fact. Fine. I have been pointing to a series of facts that appear to support the need for a paragraph in the Mathematics Article that outlines the realm of responsibility of mathematics in the real world. Of course there will always be mathematicians who prefer to conjure with abstractions in an airy-fairy world, and good luck to them. Perhaps they might avail of an equal opportunity to explain in Wiki why the rest of us owe them a living. Geologician 23:02, 30 June 2006 (UTC)

Mathematics is a subject, not a person or organisation. A subject cannot have responsibility any more than a word can. Stephen B Streater 23:09, 30 June 2006 (UTC)

I much prefer to avoid semantic discussions, but a discipline's name usually provides clues to its intended realm of responsibility. For example, the Wiki on Bioinformatics begins with a nice clear description of its relevance to and place in the real world. The start of the Mathematics article talks about concepts and abstractions without ever mentioning numbers, which are its vital link to the real world. Geologician 09:42, 1 July 2006 (UTC)

Mathematics is an area of knowledge which gives power without responsibility. There is no "intended realm of responsibility" intrinsic in Mathematics. Numbers are only one link between Mathematics and the real world. On the other hand, many parts of Mathematics have no link to the real world. Stephen B Streater 18:03, 1 July 2006 (UTC)

Of course Math has a realm of responsibility. Check out The Mathematical Atlas If you are a dyed-in-the-wool semanticist you can call it the mathematical landscape. You will find 'Significant Figures' tucked away obscurely in Area 62 (Statistics) rather than upfront where it belongs. Geologician 11:23, 2 July 2006 (UTC)

I might have misunderstood what you mean by responsibility. What do you mean by responsibility? Stephen B Streater 15:37, 2 July 2006 (UTC)

Exactly the same as Concise Oxford Dictionary (var2). Geologician 17:42, 2 July 2006 (UTC)

Are you saying that Mathematics has some sort of duty? Stephen B Streater 18:22, 2 July 2006 (UTC)

I refuse to be drawn further into semantics. Read with more care what I have already written. Geologician 20:47, 2 July 2006 (UTC)

Maybe the article should also state that mathematicians must be guardians against the abuse of very large numbers, like disgustingly rich people can only be that rich because their bank accounts are specified with very large numbers. If we spell this out in the article, we will witness a new dawn of humanity. --Lambiam^Talk 22:38, 30 June 2006 (UTC)

We should be taking out cut somewhere. If only we were lawyers ;-) Stephen B Streater 22:43, 30 June 2006 (UTC)

I think there are some mathematicians who do have this sense of duty. Its maybe more common in statistics where missuse of numbers can have a large effect on policy. I've seen talks by statisticians devoted to how statistics are missused and ways to combat this problem. See for example A Mathematician Reads the Newspaper. --Salix alba (talk) 21:04, 2 July 2006 (UTC)

Yes, I agree, many mathematicians have private senses of duty. I have of sense of duty concerning every good act I do. However I don't really see what this discussion has to do with this article. Paul August ☎ 21:25, 2 July 2006 (UTC)

The introduction needs to emphasise the importance of number as the foundation in reality of all mathematics. Even crows can count. Thus when other disciplines err when they intrude into the realm of mathematics then mathematicians should not stand idly by. For example when folks whose background was in the discipline of religious studies began revising the tenets of evolution, others whose background was in the realm of paleontology were quick to point out the relevant fallacies. Likewise when physicists make serious errors in their assumptions about what thay can do with numbers, mathematicians are bound to defend their territory with a rejoinder. In the case of metrology (mentioned above), this apparently hasn't happenend. Geologician 08:59, 3 July 2006 (UTC)

I agree with this too, but I will look back at what Geologician has written in more detail as he suggests. Stephen B Streater 21:46, 2 July 2006 (UTC)

You and Geologician seem to be using the term "responsibility" in two different senses, and thus have been talking past each other. Geologician is using the term to mean the set of concepts covered by the subject of mathematics. This is the same sense in which we use the word when we say, "The division's area of responsibility stretched from the edge of the river to the town of Metz." It is a way of determining what the boundaries of some subject are. The Mathematical Subject Classification System (MSC) is a good example of this. You have taken it in the sense of a "duty" as in, "We all have a responsibility to feed our pet aardvarks." capitalist 02:36, 3 July 2006 (UTC)

Perhaps we could use the word scope to avoid ambiguity. Stephen B Streater 06:34, 3 July 2006 (UTC)

Geologician is using "responsibility" in all sorts of ways and using a dislike of "semantics" as an excuse for not being clear about what he is saying. Of course the intro should emphasise the scope of mathematics in relation to the "real world", which it already does. Quantity is actually a better description of what we are talking about than number, and space, change and structure are just as important. The possible responsibility of mathematicians to correct people's misuse of numbers is another matter, and is completely irrelevant to the article. JPD (talk) 10:47, 3 July 2006 (UTC)

Agreed, this sums up exactly how I feel. If I can, taking the Quantity is actually a better description of what we are talking about than number sentence a bit further, wouldn't number in fact just be one way of representing a quantity? Hence, quantity being the better word to use here. Cheers --darkliight^[πalk] 12:34, 3 July 2006 (UTC)

Preference for the word 'quantity' is simply a semanticist's way of avoiding the issue. The Concise Oxford Dictionary again: quantity: 1 the property of things that is measurable. 2 the size or extent or weight or amount or number. etc. etc. Thus the fundamental meme is number itself, without which quantity becomes just another of the abstract notions so beloved of the pure mathematician. To reiterate: number is the fundamental link between mathematics and the real world. Say "one, two or three" to the barest vestige of human intelligence and they will know what you are alluding to. Say "small quantity" and they will be completely baffled. Geologician 13:52, 3 July 2006 (UTC)

Tie a knot in one rope and two knots in another rope, put two stones together and another stone by itself or scratch two marks in a stick and one scratch on another part of the stick. Show these to the barest vestige of human intelligence and they will know what you are alluding to. Draw a picture of a '2' and '1', or say 'two and one' and they will be completely baffled. A sense of quantity should not rely on a good understanding of a language. Numbers are a way of representing quantity, not the other way around. If it is the other way around, then please show me the number that is represented by the area of a unit circle. Cheers --darkliight^[πalk] 14:16, 3 July 2006 (UTC)

Although a geometer could probably prove a few basic straight line propositions without recourse to number, for mathematics to progress beyond the abacus level Darklight depicts, representational numbers were found necessary by even the earliest civilisations. To demote them now to a position inferior to abstract notions of quantity is just semantic balderdash. Geologician 15:19, 3 July 2006 (UTC)

Ok. --darkliight^[πalk] 16:08, 3 July 2006 (UTC)

Darklight is quite right. I would like to add that using the word "quantity" is not avoiding the issue, but actually addressing it. If mathematics were simply about number then it would be perfectly happy with the sort of statements that Geologician is objecting to. It is precisely because scientists are dealing with quantities rather than simply numbers that error margins need to be taken into account. JPD (talk) 14:43, 3 July 2006 (UTC)

Not good enough. It is the scientists' approach to dealing with error margins that is at the root of my concern. An error margin is simply an excuse for exceeding the accuracy of the scientist's measuring device, which of course is eqivalent to trying to get something for nothing. Emphasising quantity is merely a way of fudging this issue. Geologician 15:19, 3 July 2006 (UTC)

No one is demoting numbers to an inferior position, but simply stating that quantity, etc are the fundamental concept that are being studied, not the abstraction which is numbers. You say that some scientists abuse error margins and numbers, which is undoubtably true, but has nothing to do with this article. The article says that mathematics includes the study of quantity, which includes not only abstract numbers, but issues of accuracy. What more is there to say? The lead of an article on mathematics is definitely not the place to deal with your concern about scientists. JPD (talk) 15:50, 3 July 2006 (UTC)

I shall not enter (probably futile) discussion of whether quantity or number is the most abstract concept. I should have thought the answer obvious but will abide by the consensus of any practical mathematicians who read this section (Always assuming that such a species exists). Geologician 17:08, 3 July 2006 (UTC)

I think a discussion of the various uses (or misuses) of mathematics might be appropriate lower in the article in the "connection with reality" section, or even in a new section of its own, but not prominently in the lead. I'm sure there could be a lively section on the approach of physicists vs. pure mathematicians.

But I don't quite see how one can discuss writing an article without paying attention to the meaning and use of words - semantics. John (Jwy) 16:27, 3 July 2006 (UTC)

I prefer to be judged by the grammatical correctness and logic of what I write, rather than hair-splitting on the meaning of words. Geologician 17:08, 3 July 2006 (UTC)

At risk of hairsplitting myself, then its the hair-splitting you are concerned about, not the use of "semantics." Grammatical correctness and logic require clear understanding of meanings (ask Humpty Dumpty). I understand this can be overdone, but it has to be done! John (Jwy) 17:23, 3 July 2006 (UTC)

John apparently misunderstood the words that mathematician Charles Lutwidge Dodgson had Humpty Dumpty say, which were: 'When I use a word,' Humpty Dumpty said, in a rather scornful tone,' it means just what I choose it to mean, neither more nor less.' I'm in complete agreement with HD, which is why I avoid semantic discussions. Geologician 17:42, 3 July 2006 (UTC)

An encyclopaedia is more than a collection of syntactically correct sentences. The semantics is the important part of the content. Stephen B Streater 18:07, 3 July 2006 (UTC)

I believe Dodgson is intending to point out that meaning IS important by using an extreme case. Word meanings are not a direct connection between the word and reality. We make that association and have to agree to that connection. As a mathematician, Dodgson must have known the importance of precise definitions of the terms used. I'm not saying we need to hair-split, but we do need to make sure we pay enough attention to the meanings of the words. I love this topic, so I will probably fade out as I could get yacking about it well beyond its usefulness here. I probably already have. John (Jwy) 18:27, 3 July 2006 (UTC)

Numeral addition

Webster's definition of 'numeral' confirms that it is perfectly appropriate in the place it has been put.

numeral adj 1. Expressing, denoting or representing number. 2. Of or pertaining to number. n. 1 A word expressing a number. 2. A figure or character , or group of either , used to express a number. OED says much the same. Geologician 16:38, 4 July 2006 (UTC)

Giving us a definition of "numeral" does not at all explain why it should be in that sentence. It was reverted because it was redundant and didn't read well because it was too specific, not because it didn't somehow make sense. JPD (talk) 16:49, 4 July 2006 (UTC)

Yep, this addition did not read well. Agree to keep it out. Oleg Alexandrov (talk) 16:54, 4 July 2006 (UTC)

That my addition of a single innocuous word to the math intro has generated such an instant and puerile response caused me to wonder, quietly, just where on a spectrum that ranges from 'genuine concern' through 'dog-in-the-manger' to 'extreme prejudice' this criticism might lie. However, taking a cautiously benevolent view of the motives of my interlocutors, I'll attempt, briefly, to respond to their worries. Every introductory text to the history of mathematics I ever read mentions the use of symbols to represent numbers as a significant advance in human communication. To ignore this fact, or include it under a portmanteau word like 'abstraction' seems tantamount to obscurantism, which is deliberate obfuscation to emphasise the impenetrability of one's own speciality subject. To introduce mathematics without mentioning this significant stage in its development is at best an oversight and at worst deliberately perverse. Geologician 07:44, 5 July 2006 (UTC)

You mean something like the beginning of the history section of this article? The introduction isn't the place to mention all the details. As for your concerns about how quickly this was reverted, I draw your attention to the long discussions concerning the introduction above. At lot of effort has been made to ensure it is accurate and readable. If you think it needs to be changed to more explicity mention numbers or numerals (in case readers don't understand what abstractions from counting, etc. are!), it would be simple courtesy to bring it up on this page first, and to take some effort to provide a well written alternative. JPD (talk) 10:18, 5 July 2006 (UTC)

The introduction is the place to introduce fundamentals. Numerals are fundamental to mathematics, not mere details. As JPD requested in my user/talk I have read the whole sentence, parsed it and pared it, and I remain completely at a loss to understand what is wrong with: It evolved, through the use of numerals, abstraction and logical reasoning, from counting, calculation, measurement and the study of the shapes and motions of physical objects. What is it about the addition of the word numerals followed by a comma, that JPD finds so alienating? Regarding his point about abstraction and following the Wiki link, we find that Abstraction is the process of reducing the information content of a concept, typically in order to retain only information which is relevant for a particular purpose. If one reduces the information content from counting, you are left with nothing, nada, zilch, not numerals, which are the accepted conventional way to express the results of a count. Geologician 11:57, 5 July 2006 (UTC)

The word "numerals" is out of place in that sentence. Paul August ☎ 12:34, 5 July 2006 (UTC)

I agree with JPD and Paul. McKay 13:07, 5 July 2006 (UTC)

Mathematics is presumably based on logic. Logic requires reasons. None of the triumvirate of JPD, Paul and McKay seem capable of explaining exactly why the word 'numerals' is out of place in the sentence. Encylcopaedias are not written by edict. Geologician 13:13, 5 July 2006 (UTC)

The fact that the three of us and Oleg think that it doesn't read well suggests that there is something funny about it in a subjective sense at the very least. The main reason is that it is quite strange to include "numerals" in a list together with "abstraction" and "logical reasoning". As for your comments concerning abstraction, I think you have missed the point. Perhaps it would be clearer if the intro linked to abstraction (mathematics) rather than the more general article. If you are still confused about numbers/numerals as an abstraction, try following the link to number. JPD (talk) 13:28, 5 July 2006 (UTC)

'Something funny' is not really an explanation for abrupt rejection. Your link for 'abstraction' led to my quote above and I attempted to link my 'numeral' to your 'number' site but this was apparently unacceptable. Geologician 14:40, 5 July 2006 (UTC)

The fact that a sentence does not read well is a very good reason to revert to an adequate prior version. I have explained why it does not read well, and Paul below continues the discussion on whether any mention of numbers/numerals is necessary - a separate issue. I am not sure why you say the links are mine, but I don't get the impression that you understood what I was saying about them. Number makes it quite clear that numbers are an abstraction from counting and measurement. JPD (talk) 15:21, 5 July 2006 (UTC)

My reasons: abstraction and logical reasoning are methods, the use of which were fundamental to the development of mathematics. The use of numerals to represent numbers was a convenient, (but fundamentally inessential) tool, which made that development easier, for particular branches of mathematics (e.g. Euclidean geometry can be developed quite easily without using numerals). In my opinion, numerals are appropriately addressed in the second section of the article. Paul August ☎ 13:44, 5 July 2006 (UTC)

Finally someone is trying to address the nub of the matter. But, as I pointed out in the Webster definition above, a numeral is a figure or character, or group of either, used to express a number and this idea has been around for at least 4000 years. So numerals have always played an integral and essential role in every aspect of arithmetic, algebra and geometry. How then can they not be fundamental? It is incorrect to say that Euclidian geometry can be developed without using numerals. It is perhaps a liitle-known fact, but the very names of geometric figures, triangle, quadrilateral, pentagon etc., came from the words for the numbers three, four and five. So presumably even the earliest geometers considered numbers essential when trying to describe what they were doing. Geologician 14:40, 5 July 2006 (UTC)

Yes numbers (not numerals) are fundamental in development of mathematiccs, hence the mention of "quantiy" and "counting" in the lead. Paul August ☎ 15:10, 5 July 2006 (UTC)

(edit conflict)The fact that numerals have been around for a long time does not at all show that they have always played a role in every aspect of mathematics. Avoiding semantics concerning "numerals", even the use of numbers in names of geometric figures does not mean that numbers are needed for all geometry. Numbers obviously have been very significant in the history of mathematics, but are not quite as significant as you are suggesting. In my opinion, they are rightly emphasised in the history section, and rightly included in a more general description in the intro. JPD (talk) 15:21, 5 July 2006 (UTC)

Good. Now we're getting somewhere, however quantity is fine but counting is a gerund form that immediately begs the question "How do I count without numerals?" (Which of course include the words we count with.) Since no-one else has advanced the foundations of higher mathematics as an argument against the role of numerals, I shall address the matter myself in a new section below. Geologician 16:24, 5 July 2006 (UTC)

We haven't got anywhere. The whole point right from the start was that mentioning numbers, let alone numberals, specifically, is not necessary, because things like quantity and counting are mentioned. The very fact that the word counting immediately brings to mind numbers is one reason why a specific mention is not necessary. Since noone is advancing the foundations of higher mathematics as an argument, you probably don't need to address the matter at all, unless you are simply looking for an argument. JPD (talk) 16:49, 5 July 2006 (UTC)

Of course we are getting somewhere. We are plumbing the actual depth of thought of would-be custodians of a section of Wiki encyclopaedia and recording it for posterity. Responding to JPD's last entry, It is a trivial matter to count things without recourse to numbers: just move beads from one side to another of an abacus. Thus the gerund 'counting' is an inadequate substitute for the concept of number. 'Quantity' is far too vague a word to convey the concept of number. Geologician 06:24, 6 July 2006 (UTC)

Recording the same thing over and over again is hardly getting somewhere. If the words we count with are numerals (which is a quesiton of semantics, so I'm not going to bother disputing it), then so are collections of beads. At the very least, they are representations of numbers. Of course "counting" isn't a substitute for number. None of "quantity", "counting" and "measurement" are the same as "number". They are all more basic concepts which have led to the use of numbers. Numbers are covered in the intro by the meaning of whole sentences, not individual words. The only question as I see it, is whether this coverage is adequate or not. I personally am not completely against a more specific reference to numbers. I just don't find it necessary, and insist that if it is included, it should be well written. JPD (talk) 12:03, 6 July 2006 (UTC)

It is not disputed that quantity, counting and measurement can sometimes be regarded as more fundamental attributes than number, but we are considering here the subject of mathematics not 'quantity surveying'. Mathematics apparently is unwilling to sully its delicate hands with the grim details of how the numbers are obtained, or even (given their dimensions), what they represent. They could be apples and oranges and the prices thereof, or they could be the orbit parmeters of Mercury. However, without numbers, obtained invariably by someone else, Mathematics cannot analyse the data. Therefore as far as Mathematicians are concerned the numbers are as fundamental as it gets. This seems to put paid to the question of whether mention of numbers is necessary. Regarding whether existing coverage is adequate or not, it seems it cannot be considered adequate if it points to matters that are the province of other disciplines, yet omits to mention a factor that forms the very foundation of its own discipline. Geologician 13:29, 6 July 2006 (UTC)

Your repeated personification of mathematics is nonsense, so I will ignore it and your strange ideas about what mathematicians are apparently concerned with. The foundations of mathematics are spelled out, as are its basic methods, without going in to the province of other disciplines. The basic methods of abstraction and reasoning are mentioned. The earliest abstraction, which is the basis and tool for many other things, is implied by these but not explicitly mentioned. If you can't see that, and insist on an explicit reference, perhaps you could suggest a more appropriate way of including it. JPD (talk) 14:42, 6 July 2006 (UTC)

Several people have patiently explained to you several times why you are beating a dead horse. All of us understand perfectly well the distinction between "number" and "numeral" and know when it is appropriate to make that distinction and when it is not. For a more complete explanation, see pedant. Rick Norwood 14:01, 5 July 2006 (UTC)

Rick appears to have completely missed the point of this discussion. Geologician 14:40, 5 July 2006 (UTC)

New intro suggestion

I moved the discussion into a new subsection - it was getting too cramped --darkliight^[πalk] 16:58, 6 July 2006 (UTC)

Okay I would replace the introduction with what follows, which avoids as far as possible the use of unfamiliar terms that require explanation elsewhere. (People who actually read introductions to encyclopaedia sections want to know in simple terms what a subject is about, not to be sent elsewhere to learn the basic concepts). Mathematics is the language of numbers, harnessed to explain in simple terms complex observations made by others. Practical mathematics analyses the underlying structure of things in the natural world by seeking patterns that can be described with numbers, such as musical scales, or by familiarity with certain basic shapes, such as the pyramids of Egypt or the path followed by the planet Mercury. Pure mathematics is the study of all aspects of the abstract logic demanded by the use of numbers. Mathematical principles are used in many fields, including science, engineering, medicine and economics. Most advances in mathematics have been made by mathematicians working in these mainstream fields, relating their work to actual needs. (For example, finding ways to predict the flooding of the Nile or breaking enemy codes in wartime). The word "mathematics" is often abbreviated math in the U.S. and Canada and maths in Britain, Ireland and many Commonwealth countries. Geologician 16:39, 6 July 2006 (UTC)

I disagree with this as an introduction to mathematics. I really don't know where to start with this either, in fact, the only parts I can agree with are the parts you have borrowed from the existing introduction. Briefly, some problems I have with this are, the fact that earlier you gave examples of mathematics that does not require numbers, so how can you justify defining mathematics as the language of numbers? My understanding of mathematics is not summed up by pattern matching - what does area have to do with patterns? Numbers do not demand logic, sometimes we use particluar numbers to represent a particular logic. I would argue that most advances in mathematics are not made in other fields. The current introduction does an excellent job of introducing mathematics (it would have to given how much work has gone into it), but regardless of this fact, I'm still happy to admit that it is not, and no intro will ever be, perfect and that some work still needs to be done - your suggestion does not help this cause though, imho. --darkliight^[πalk] 17:19, 6 July 2006 (UTC)

Its a commonplace phrase to describe mathematics as the language of numbers. Just google the two bits in bold here and see. I am happy to see improvements in this draft so long as it remains instantly intelligible to a ten-year-old kid, without wiki links. An Intro cannot be expected to cover everything. It is bait to capture the full attention of folk with a genuine sense of curiosity about the subject. There is plenty of scope in the main article to expand on the subtleties and complexities. Perhaps you meant to say that that most advances in mathematics in the past 50 years were not stimulated by the necessities of other fields, but even this is disputable. Geologician 20:25, 6 July 2006 (UTC)

For the record, I did a search of the phrase "language of numbers" and got 25k hits, none of which seemed to be included in a definition of mathematics. A search for "purple monkey dishwasher" yielded 45k hits for comparison. The search for a definition of mathematics was interesting though. Cheers, --darkliight^[πalk] 21:30, 6 July 2006 (UTC)

Mathematics covers many more abstract ideas than numbers. I think an over-emphasis on numbers misses the wide ranges covered by the subject. Stephen B Streater 20:29, 6 July 2006 (UTC)

I must say that I'm not to happy with the current introduction, it is very abstract referring to concepts which we take for granted but for the lay audience its a bit puzzling, not easy reading. I've been running the first sentence past a few people in the last day or so and also asking then what does mathematics mean to you, to recap the current first line is:-

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.

• First respondent 14 year old girl. Whats mathematics: BODMAS i.e. the rules of elementary algebra, response to first line: yeah right?
• Second respondents 16 year old girl. Whats mathematics: numbers. Her response to first line was interesting, at first did not get it but after a minutes thought she said something like Quantity, ah yes that's numbers. Structure, that's how things are organised, space, whats that? Me: 3D. Her oh yes. Change? Me: that how things change through time. Her: yep.

So we have a first sentence which take a minutes thought to understand, and even then requires help from a mathematician. Is this good for an encyclopedia article? Do we want the stop the reader dead in his tracks at the very start? Maybe we should try to draw then in gently, add a few more concrete examples which have relevance for the reader.

The simple English mathematics article goes a bit down this line, explaining each of the four terms:

Mathematics (math or maths for short) is the study of quantity (how many things there are), structure (how things are organized), space (where things are) and how these three things change.

I would do something like

Mathematics is the study of quantity (for example numbers), structure: how the different quantities relate to each other and the patterns which occur, space: the shape of objects, how they are arranged in 2, 3 and higher dimensions; and change: how the quantities change through time

Far from perfect, but it provides a little more context for the reader and they could hopefully be able to read it one go with out having to stop at each word (well it would be if it wasn't for my bad grammar).

I have certainly heard some people, Ian Stewart for one, refer to mathematics as the study of patterns. The patterns in the natural numbers give the primes, the patterns in crystals giving rise to space groups. Many of important theorems are about some pattern observed in a particular branch.

Anyway its a fun game to play, run the first line past someone you know and se e what thay make of it? --Salix alba (talk) 20:46, 6 July 2006 (UTC)

I'll test it out. Looking at it again, Mathematics is apparently four things. Is there a unifying theme behind them? Stephen B Streater 20:51, 6 July 2006 (UTC)

(And before anyone says it, numbers is too specific for what I'm after, which is more of an encompassing idea). Stephen B Streater 20:52, 6 July 2006 (UTC)

In reply to Salix alba, how does asking a 14 year old to define mathematics help us? Would it be acceptable to ask a 14 year old to recite an encyclopedic biography of Queen Elizabeth II on the spot, without letting them do any research or collaborate with other people? I don't think so, I mean don't get me wrong, the girl would know who I was talking, but ... would she come up with something so precise as the opening sentence of the queen's page? The purpose of a wikilink is to provide additional insight into a topic if the user is a little unsure, in essence, when you coaxed the 16 year girl a little she understood and agreed with the definition given - you were the wikilink. Cheers, --darkliight^[πalk] 21:14, 6 July 2006 (UTC)

It is a difficult task: I don't think I knew what mathematics was until a college Advanced Calculus class (and I might have had further revelations of my ignorance had I gone even further!), so trying to explain to someone who doesn't have that background is a daunting task. But I don't think we need to simplify it so much that we describe it inadequately. The current intro is quite good, as I see it. The suggested one above leans too heavily on the applied side of mathematics. For example, it was the work of David Hilbert and those around him that gave us the theoretic grounding of relativity and quantum theory. Their work was not originally focussed in that direction, but suited the subject fine. Mathematics is so much more than numbers. I wish there were some way we could include some indication of set theory, topology and the like into the introduction. That's REAL math! John (Jwy) 03:22, 7 July 2006 (UTC)

The current intro probably has room for improvement, without changing it to Geologician's essay on numbers, applied mathematics and mathematics as a person. However, I have to question some of the implied aims in the above discussion. Wikipedia is not aimed at ten year olds. Not even the introductions need to be instantly intelligible to them. In my opinion, the level of language used shouldn't be any lower than could be expected from something such as a secondary school textbook. Secondly, as darkliight has referred to, an encyclopedia aims to inform, not to tell the reader something they already knew. What a 14 yr old or 16 yr old thinks mathematics is is irrelevant. What matters is whether the intro informs them and/or encourages them to read more. Lastly, I think running only the first sentence by anyone is a mistake. Apart from when reading mathematics or other technical writing, or analysing a text, it is not normal to read each word or sentence by itself. We can be quite happy with not completely understanding something immediately if the context helps to understand it. We need to judge the first sentence together with the rest of the paragraph. As for Stephen's question, I think it is possibly the use of abstraction ("patterns") that is the unifying theme, more than anything about quantity, structure, etc. in particular. JPD (talk) 10:11, 7 July 2006 (UTC)

While I do not like the "quantity, structure, etc." definition, it corresponds roughly to dictionary definitions, and is echoed throughout this article and others. If it is changed, then many other changes will need to be made downstream.

I seems clear to me that mathematics is knowledge gained by deductive reasoning, that science is knowledge gained by inductive reasoning, and that truth is a correspondence between symbols and reality. But you will not find any of these opinions in Wikipedia. They are origianl research. Rick Norwood 15:08, 7 July 2006 (UTC)

Even Higher Mathematics is founded on Numerals

Higher maths, like arithmetic and geometry is founded explicitly on arithmetical principles, which themselves require numerals to demonstrate. The following quotation is definitive: " In the expression of algebraical principles we proceed analytically: at the outset we do not lay down new names and new ideas, but we begin from our knowledge of abstract Arithmetic; we prove certain laws of operation which are capable of verification in every particular case, and the general theory of these operations constitutes the science of Algebra. Hence it is usual to make a distinction between Arithmetical Algebra and Symbolical Algebra, and to make a distinction between them. In the former we define our symbols in a sense arithmetically intelligible, and thence deduce fundamental laws of operation; in the latter we assume the laws of Arithmetical Algebra to be true in all cases, whatever the nature of the symbols may be, and so find that they may obey these laws. Thus gradually we transcend the limits of ordinary Arithmetic, new results spring up, new language has to be employed, and interpretations given to symbols which were not contemplated in the original definitions. At the same time, from the way in which the general laws of Algebra are established, we are assured of the permanence and universality, even when applied to quantities not arithmetically intelligible." QED Geologician 16:24, 5 July 2006 (UTC)

Since I completely miss the point, maybe you can explain to me why a long quote that does not even mention the word "numeral" says something definitive about the importance of numerals. Rick Norwood 23:30, 5 July 2006 (UTC)

Sure. The entire warehouse of mathematics collapses in a pile of dust, leaving perhaps a couple of orthogonal straight lines and a few boring expressions in set theory, if we remove all the proofs that depend ultimately on proofs that depend on abstract arithmetic that cannot be expressed without recourse to numerals. QED Geologician 05:59, 6 July 2006 (UTC)

No. And certinly the paragraph you quote doesn't suggest that. The laws of mathematics are independent of numeration systems, and the real numbers, for example, can be defined and their properties deduced without any reference to a numeration system. I love the wonderful utility of the Indo-Arabic numerals as much as anybody, but numeration systems are important but not fundamental to mathematics. I think you would be hard pressed to find any quote by a mathematician that said otherwise. On the other hand, you find set theory boring, so we obviously have different taste as to what is interesting and what is not. Back up your taste with a citation. Rick Norwood 13:48, 6 July 2006 (UTC)

History of Notation

In the section on notation and language, I think it would be pertinent to mention the role of geometry in historical math. Yes, proofs were "written out in words", but often with accompanying geometric diagrams. This is the style in which Newton first wrote his works. However, I'm not entirely sure if this falls under the start of "modern notation" or not. Even so, the role of geometric style proofs seems to have played a large enough part in the development of mathematical notation to be mentioned here. --Archmagusrm 15:11, 12 July 2006 (UTC)

Second and third paragraphs

Continuing, I had a stab at getting something together for the second and third paragraphs. This is really rough and most of it is borrowed from the History of mathematics article and the current second paragraph, which were both quite good. The previous discussion resulted in a brilliant opening paragraph so hopefully we get something similar going here.

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement and the study of the shapes and motions of physical objects. Mathematicians explore these and related concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

Historically, almost all cultures made informal use of basic mathematics. Further development of these basic ideas began with the most ancient mathematical texts originating in ancient Egypt, Mesopotamia and ancient India. From this point on, the development continued in short bursts until the Renaissance period of the 16th century where new mathematical developments, interacting with new scientific discoveries, were made at an ever increasing pace that continues to the present day.

Today, mathematics is used throughout the world in many fields, including science, engineering, medicine and economics. These fields inspire and make use of new discoveries in mathematics, often dubbed applied mathematics, and have in some cases led to the development of entirely new disciplines. New mathematics is also created for its own sake, without any particular application in view, a common practice in the branch known as pure mathematics.

So, per WP:LEAD we should briefly summarize the most important points covered in an article in such a way that it could stand on its own as a concise version of the article and I think something like this will bring us in line with that. The only section not really addressed here being the Common misconceptions section, but I think we can safetly leave that out of the intro. I tried to keep the paragraphs of similar length and made sure not to be specific about any one discovery or person. Problems with what is above that I can see are

• Being specific about where mathematical development began. I think it would be better to leave out anything specific, but it did seem important so I left the mention there for now. I think the biggest problem with leaving specific place names in there is it will be a source of conflict between different points of view.
• Overuse of the word development in the second paragraph.
• I made a sloppy mention of applied mathematics and I think what I actually added is incorrect, or at least innacurate, but I couldn't work out something that worked better. I think it's important to offer a brief distinction between applied and pure mathematics for a few reasons though. 1. We have an entire section devoted to it 2. We already discuss pure mathematics in the lead and 3. I think it offers a bit of insight into the motivation behind mathematical development without going into details.

No doubt there are other problems too, so I guess let's see where this takes us. Cheers, darkliight^[πalk] 07:30, 15 July 2006 (UTC)

This is looking good. Here is a variation of paragraph 2 with fewer developments, and more verifiability eg changed exponentinal growth which is uncited and not obvious how to measure, and almost all softened, and began would be hard to verify too:

Historically, informal use of basic mathematics was widespread. Refinements of these basic ideas are visible in the most ancient mathematical texts originating in ancient Egypt, Mesopotamia and ancient India. From this point on, the development continued in short bursts until the Renaissance period of the 16th century where mathematical innovations interacted with new scientific discoveries leading to an acceleration in understanding that continues to the present day. Stephen B Streater 08:14, 15 July 2006 (UTC)

Hi again. Looks good, but the applied math section is confusing because it looks like the phrase "often dubbed applied mathematics" refers to the new discoveries in mathematics instead of to the use of math in these other fields as I think you intended. How about this?

"Today, mathematics is used throughout the world in many fields including science, engineering, medicine and economics. The application of math to such fields, often dubbed applied mathematics, inspires and makes use of new discoveries in math and has in some cases led to the development of entirely new disciplines. In contrast, pure mathematics is that branch of math which is done for its own sake, without any particular application in view."

capitalist 03:23, 16 July 2006 (UTC)

"Today, mathematics is used throughout the world in many fields including science, engineering, medicine and economics. The application of math to such fields, often dubbed applied mathematics, inspires and makes use of new discoveries in math and has in some cases led to the development of entirely new disciplines. In contrast, pure mathematics is that branch of math which is done for its own sake, without any particular practical application in view."

How about adding practical in? People might think that Art is an application, even if it has no practical use. Stephen B Streater 06:41, 16 July 2006 (UTC)

Sounds good to me, though I think the word particular is superfluous now. Cheers, darkliight^[πalk] 10:53, 16 July 2006 (UTC)

The text proposed as second paragraph is very good. However, I don't quite like linking "Historically" to history of mathematics. I didn't expect this link, and I think it can safely be left out. Furthermore, the use of the past tense and the word "historically" in "Historically, informal use of basic mathematics was widespread." implies that maths is no longer widespread. So, I'd prefer to replace that sentence by something like "Knowledge and use of basic mathematics is widespread, both in the past and the present".

Personally, I think that the most important development in the history of mathematics is the increased rigour of the Greeks. But we need to keep it short, so I don't mind if that's left out.

I need to think a bit more about the third paragraph. My first impression is that the proposed text is not as brilliant as the rest. -- Jitse Niesen (talk) 13:01, 16 July 2006 (UTC)

I do think the second paragraph must mention the Greeks. In the proposed third paragraph, the use of "math" instead of "mathematics" seemed, well, unencyclopedic. Also, I would drop "In contrast", since it seems to suggest that pure mathematics does not inspire or make use of new discoveries. Rick Norwood 13:35, 16 July 2006 (UTC)

I don't like using "math" that much either, but I was trying to reduce the repeated use of the word "mathematics" in the paragraph which sounded odd. capitalist 02:24, 17 July 2006 (UTC)

Repeated use of "math" sounds more odd, at least to those of us for whom it doesn't sound like a real word ;-) JPD (talk) 12:06, 17 July 2006 (UTC)

LOL! Well yes, then that might be a problem. How about this:

"Today, mathematics is used throughout the world in many fields including science, engineering, medicine and economics. The application of mathematics to such fields, often dubbed applied mathematics, inspires and makes use of new mathematical discoveries and has in some cases led to the development of entirely new disciplines. In comparison, pure mathematics is done for its own sake, without any practical application in view."

This includes the previous edits above and takes into account Rick's concern about "in contrast" as well. capitalist 02:43, 18 July 2006 (UTC)

Second paragraph with some improvements suggested above: Knowledge and use of basic mathematics is widespread, as it has been over history. Refinements of basic ideas are visible in the most ancient mathematical texts originating in ancient Egypt, Mesopotamia and ancient India, with increased rigour introduced by the ancient Greeks. From this point on, the development continued in short bursts until the Renaissance period of the 16th century where mathematical innovations interacted with new scientific discoveries leading to an acceleration in understanding that continues to the present day. Stephen B Streater 12:43, 19 July 2006 (UTC)

I think we're getting there. Two more suggestions. I would drop the word "most" from "most ancient mathematical texts". There aren't enough of them to talk about "most". Also, how about "In comparison, pure mathematics...in view, though in many cases applications are found." Rick Norwood 12:51, 19 July 2006 (UTC)

I think the second paragraph is close enough to edit in place now. I'll look at the third as well, as this is also developing nicely. Stephen B Streater 19:03, 19 July 2006 (UTC)

I agree with Rick's point about applications being found later as well, but the last sentence is sounding run-on and forced to me now. The passive voice sounds weak too. How about this:

"In comparison, mathematicians also engage in pure mathematics for its own sake without having any practical application in mind, although others may discover such applications later."

capitalist 02:37, 20 July 2006 (UTC)

I like the new second paragraph, but I have a few minor comments:

• I think the first sentence should be throughout history instead of over history.
• Should it be noted that the increased rigour introduced by the greeks came later than the development of basic ideas?
• I'm not sure about the last sentence. I like that it implies alot happen around the 16th century, but, and this may just be me reading too much into the wording, I think it's in danger of giving the impression that this was the last 'big thing' in mathematics. I think a little more emphasis should be put on continues to the present day, either by expanding on it or rewording it. I haven't got any suggestions for this bit yet, but I'll think about it tommorow. Cheers, darkliight^[πalk] 11:28, 20 July 2006 (UTC)

Throughout is clearly better. I'll think about the other ideas too. Stephen B Streater 11:42, 20 July 2006 (UTC)

Does anyone mind if I add the word later here: with increased rigour later introduced by the ancient Greeks? Also, the very last sentence, I think the word such should be removed, ie. "although others may discover such applications later" -> "although others may discover applications later". Is it possible to reword it slightly to give the idea that this does indeed happen? Eg. "although others often discover applications later."? I think often is too strong, but on the other hand I think may is too weak. Pedant, I know, but any ideas? :) Cheers, darkliight^[πalk] 12:06, 29 August 2006 (UTC)

Pictures

I've had a go at adding some pictures. I make no claim that they are the best for the purpose, so feel free to replace them if you can do better, but I thought it was strange that something so eminently visual as mathematics was so lacking in illustrations. Soo 14:03, 16 July 2006 (UTC)

Yes I like the new images and the captions which explain in nice simple language some aspects of mathmatics. You might also be interested in Wikipedia:WikiProject Mathematics/Graphics where there is a small degree of cordination in mathematical graphics. --Salix alba (talk) 17:38, 16 July 2006 (UTC)

The images rock! The captions are really helpful too. capitalist 02:27, 17 July 2006 (UTC)

Well that was a better reaction than I expected :) Soo 06:00, 17 July 2006 (UTC)

Instructions for creating beautiful mathematical visualisations

Is there a way for providing something in the way of instructions regarding how it is that the above referred to visualisations were created? The link to Wikipedia:WikiProject Mathematics/Graphics

This question (as always) has been implicitly answered. However, the codes, etc... and explicity instructions for generating large amounts of information have not yet been collated into one place. --Nukemason 10:58, 08 August 2006 (UTC)

Etymology

Since the "to do" list for this page includes adding inline references, I'm puzzled as to why my version (with highly authoritative reference) was reverted to the previous version (with no reference) (diff). I've checked the Greek and there's no plural/singular mistake that I can spot (μαθηματικα τεχνα is neuter plural). As for diacritics, I believe they are a recent invention to aid pronunciation of Classical Greek, so omitting them from an etymology is no big deal. I might be wrong about that, and if editors feel that diacritics are important then by all means restore them, but there's no need to revert the whole edit over it. Soo 05:59, 17 July 2006 (UTC)

I should say that I see that a lot of work and heated discussion has gone into this article, so I understand editors being reactionary. I'm new to this article and edit with WP:BOLD in mind, so I apologise in advance for when I inevitably tread on toes. For this reason at least, I'm leaving the lead paragraphs well alone for the time being! I suspect that, as a computer science undergraduate with a more-than-passing interest in maths, I am among the least qualified of the regular editors of this article, so please do revert me if I get things wrong. In this instance, though, I'm just going with the Oxford English Dictionary, which seems about as reliable a source as one could hope for. I would like to see this article featured, and I have no doubt that Wikipedia has the brains to make that happen. Soo 06:27, 17 July 2006 (UTC)

Hi Soo. I can certainly understand if you are bit put out by my reverting your recent work on the etymology. Too much good work gets deleted carelessly, I know. I'm sure you're not the only one to feel aggrieved. But now let's look at the two versions in question. First, what I have restored and tweaked a little:

The word "mathematics" (Greek: μαθηματικά) comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματική τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art.

The apparent plural form in English, like the more usual French plural form les mathématiques, goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical". Despite the form and etymology, the word, like the names of arts and sciences in general, is used as a singular mass noun in English today.

And what you had, before that reversion:

The word "mathematics" comes ultimately from the Classical Greek μαθηματικα τεχνα ("mathematical arts"), from μανθανειν ("to learn"). Latin inherited the phase as ars mathematica, and this in turn passed into Old French as the adjective mathematique ("mathematical"). "Mathematics" was adopted as a noun in English, with the earliest written reference dating from the late 16th century. Originally it was considered a plural noun but despite the form and etymology, the word is usually treated as a singular mass noun in modern English.

(So that I can get this done efficiently, I'll omit some diacriticals.) I had commented that there was a problem in your version with singulars and plurals; you say you checked, and found no problem. Let me now point out that τεχνα is not a neuter plural, as your phrase μαθηματικα τεχνα would require it to be. In fact, τεχνα is not a normal form at all. In the standard version of Greek that is appealed to in these matters, the noun in question is τεχνη (feminine). Its plural is τεχναι. So "mathematical arts" (plural), if it were to exist and were to use some form of τεχνη, would need to be μαθηματικαι τεχναι. But in fact the phrase that is used is μαθηματική τέχνη (singular): "mathematical art". (OED to which you appeal gives ἡ μαθηματική: supplying ἡ, the definite article, and with τέχνη understood.) As for your referencing, it is a good idea! I propose to restore it, because OED is an obvious source for this etymology, in either version. Not that I always agree with OED on such matters. The exact routes by these neuter-plural names for sciences come into modern languages are circuitous, and disputed.

On the matter of diacriticals, the transcription should, I think, represent them if they are present in the Greek. OED gives them, and so does just about every other source. They are in fact useful. One sort of diacritical in such transcriptions is the macron over e and o, to mark eta and omega respectively. I think they are unquestionably needed.

I hope that helps! I'll now go and restore your reference to OED. Noetica 01:39, 20 July 2006 (UTC)

Hmm, my Greek is a little rusty. You're right, though. Glad to see this section of the article reach FA standards. Soo 10:14, 20 July 2006 (UTC)

"Fields in mathematics" vs "Major themes in mathematics"

These two sections really ought to be merged since they deal with essentially the same thing, albeit in a different format. I really like the graphical boxes but the lists of articles are not useful for someone unfamiliar with topics and are likely to be unpopular with FAC. I suggest a hybrid approach, which can mainly be achieved by just reorganising the two sections into one. I'm going away for a few days now, so I'll leave this suggestion here and see what the other editors around here make of it. If the idea proves popular then I'll act on it when I get back. Soo 08:12, 18 July 2006 (UTC)

The merger looks like a lot of hard work, but I agree that it would be a good thing. Rick Norwood 12:48, 19 July 2006 (UTC)

I've had a go. The layout may still need a bit of work, but I'll wait for the text to stabilise before fiddling with that. I defer the Applied mathematics section to someone more knowledgable than me - it needs to be transformed into prose, and ideally a box like that of other subsections would be added, but I don't have the expertise to do that. Anyone want to take it on? Soo 17:42, 21 July 2006 (UTC)

Messy sections

The "Fields of mathematics" section is a bit messy. For example, the "Structure" sections contains illustrations for arithmetic and number theory, but these are both covered in the text under "Quantity". I also don't see how vector calculus belong under "Structure". Fredrik Johansson 20:24, 21 July 2006 (UTC)

You're right, I didn't really review the contents of the boxes, although I do like the idea. Go ahead and add/remove/rearrange as you see fit. Leave the illustrations blank if you can't find anything suitable, there are plenty of talented people around here who can sort out such things. Soo 23:52, 21 July 2006 (UTC)

Where does arithmetic belong?

An anonymous user removed the topic of "arithmetic" from the structure section because they said that arithmetic is not structural. I only have a very rudimentary understanding of what a "group" is in group theory, but I thought that arithmetic was structural in the senses that:

it deals with binary operators

it deals with properties of those operators like associativity, commutativity, etc.

we can make categorical statements like "The integers under addition are a group but the integers under division are not a group."

Anyway, I don't know enough about the issue to determine if arithmetic is "structural", so can someone more knowledgeable answer this. Also, if arithmetic doesn't belong under structure, where DOES it belong? Thanks. capitalist 02:52, 25 July 2006 (UTC)

This classification scheme is somewhat arbitrary. Arithmetic might reasonable be said to be in either the number or structure, sections or both. Paul August ☎ 03:05, 25 July 2006 (UTC)

I'm not an anonymous user. I moved it because, as Fredrik Johansson mentions above, it's discussed in "quantity" so it makes no sense to have it elsewhere. There isn't any more room in the Quantity box, and those things are not intended to be exhaustive anyway. Soo 09:19, 25 July 2006 (UTC)

OOPS! I thought there was just an IP address there yesterday when I looked at the edit history. Anyway, now I see the inconsistencies in the section and the reason for the moves. I just noticed that Number Theory is in the same boat. It's discussed in the "Quantity" text but illustrated under the "Structure" section. capitalist 03:01, 26 July 2006 (UTC)

I think it makes more sense for number theory to be in the structure section. Paul August ☎ 03:45, 26 July 2006 (UTC)

Personally I think the Quantity/Structure/Shape/Change classificastion is rather arbitary, and possibly bordering on original reserch. Do we have any sources which choose to classify mathematical topics in this fashion? Could it potentially be misleading for readers who first read this page thinking that this is a standard classification and then go to college where things follow a more standard Algebra/Analysis/Geometry system. It also seems to be at odds with the section title Fields of mathematics a field will typically study two or more of these, take topology, it starts with a space, calculates quantities (elements of the the various homotopy groups etc) examines the structure the groups themselves and the relations between the groups (commutative diagrams etc) and has some aspect of change - homotopy equivilence, bordism theory.

While I'm on a rant, are we mixing up mathematical objects and mathematical fields? Integers and groups are mathematical objects which are studied in Number theory and Group theory. And finally how come prime numbers don't get a mention? --Salix alba (talk) 07:30, 26 July 2006 (UTC)

I might join in slightly. As I mentioned earlier, dealing with uncertainty is a major aspect from where I work, so the four areas mentioned are a simplification. The debate centred around the origins of Mathematics and these were the broadest categories which covered most areas. The introduction is not designed to be exhaustive, but to lead in to the rest of the article.The alternative risks ending up with too much detail early on in the article, listing hundreds of subjects which the non-mathematician will be unfamilar with. Stephen B Streater 08:24, 26 July 2006 (UTC)

I think Salix has a point or three. The problems he mentions are made worse perhaps, by the merge of the old "Major themes in mathematics" into the "Fields of mathematics" section. Should we reconsider this merge? Also note the "themes" section had the following disclaimer: The following list of themes and links gives just one possible view.

Regarding Quantity/Structure/Shape/Change versus Algebra/Analysis/Geometry, I don't think these are meant to be essentially different. Rather, as pointed out in the article they are meant to correspond. Quoting from the first paragraph of the "Fields" section: These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry, and analysis).

Paul August ☎ 14:58, 31 July 2006 (UTC)

The dichotomy between Fields and Major themes still seems rather artificial to me. I do think it needs to be made clearer that our divisions are fairly arbitrary, as any division system must be, and that it is not in any way canonical. But to call it original research seems to be over-stating the point. Any article must contain original material (e.g. original prose, pictures, heading heirarchy), and the idea that the various disciplines are related is hardly research. Soo 17:00, 31 July 2006 (UTC)