Talk:Calculus/Archive 3

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Obscure article Removed Status of Good Article

Unresolved

 – We are still working on restoring the article to GA status but many significant improvements have been made since this comment--Cronholm144 20:56, 10 May 2007 (UTC)

I came here in order to get a basic understanding of what calculus is and I have to say that after reading this article and trying to make sense of it, I don't know better. There isn't even a definition of what the word calsulus means! It is written in a very technical way that may be suitable for a mathematical encyclopedia but not a general one. What "which allows control over arbitrarily small and arbitrarily large numbers" means? What is an arbitrarily small number. You have to consider writing this article from the point of view of someone who has minimal knowledge in Math. There is a lot of unexplained and probably uneccessary jargon in the current form that does not really help.

There are many reasons to delist this article from WP:GA, one being that is does not follow the Well-Written criteria (point a, b, c and d) and another that there is an ongoing controversy regarding the historical origin of calculus. Once more, think that you are writing for a general audience. I look forward to the rewrite announced above ! Roger jg 06:02, 27 January 2007 (UTC)

Shortcomings of the present article

Resolved

 – the following issues are the some of the main points of contention currently,but these are expanded upon later in page--Cronholm144 07:59, 21 May 2007 (UTC)

Well, I don't think that a GA nomination is appropriate. Maybe I'm totally wrong, but let me explain my position.

Here are some specific things that I object to in the present version of the article:

• The derivative section starts off by talking about how hard it is. We should not be intimidating readers. Then it talks about notation. Only then does it talk about the derivative, and when it does, it jumps right into technicalities with no motivation.
• The integral section doesn't clearly explain the relationship between indefinite integrals and definite integrals. I concede that the precise relationship is the Fundamental Theorem of Calculus, which is explained in the next section. However, for a novice reader who's never seen anything about integrals ever in his whole life, the section is rather mysterious. Again, it jumps into technicalities and provides very little motivation.

My experience teaching calculus has taught me that it's important to move slowly and patiently and to provide copious motivation for everything. Students have a tendency to ask, "What is this good for?" and if the only answer I can give is, "Because it's on the final," then they stop caring. Not caring leads to not showing up to class, not showing up to class leads to failing, failing leads fear, fear leads to hate, hate leads to suffering. This article should not be responsible for turning students over to the Dark side, so the very least it needs to do is explain what's going on in very simple terms that anyone can follow. I figured that my ant thing, while somewhat weird, would be easier for the complete novice than the present approach. I don't mind if other people don't like it, but I do mind if the article stays as it is. The other alternatives that I see are:

• Use "I", "you", or "we". That's appropriate for more advanced articles, but I figured I would try to avoid it on this one, which is (comparatively) elementary. (I once showed a math book to a non-mathematical friend and she commented, "What is this, the royal we?")
• Consistently write declarative sentences. "The difference quotient is ..." "The integral is ..." I've never tried this for an entire article, but I don't think it would scan. It's brusque and forced, not gentle the way calculus ought to be.

I agree that my ant thing is not very encyclopedic. But it's probably good for pedagogy. (By the way, Rick Norwood, I challenge you to point out an incorrect use of "it's" and "its" in my text.) 141.211.120.93 23:42, 15 April 2007 (UTC)

Sorry, the "it's" was Minestrone Soup, not you. I actually liked the "ant" idea, but it was a big change to make without discussing it here first. Now that a lot of the discussion of the derivative has been moved to the differentiation article, maybe this article can replace the current discussion with the ant version. You would need to fix the picture, where one of the lines was wrong, but that is a minor point. What do others think? Rick Norwood 12:56, 16 April 2007 (UTC)

I have a lot of sympathy for your point of view, and agree with much of what you say (although others may not). This article should be really really accessible, and it is not right now, but this is an encyclopedia, not a textbook, so accessible is the word to bear in mind, rather than pedagogical. The article should not be intimidating, but also there is no hope to explain all of calculus to someone who has never met it before. Instead the diligent reader should be encouraged and given directions.

The encyclopedic tone means that first and second person subjects are rarely appropriate, and also many uses of the passive tense are not recommended (e.g. "the derivative can be found..." - by whom?). I think it is stimulating and challenging to find good subjects for sentences; the active tense, used well, can be so much more engaging. I encourage you to develop your ideas with this in mind. Also, making edits in stages often helps other editors to see what you have in mind, and your ideas are then more likely to be incorporated, rather than reverted. Geometry guy 00:12, 16 April 2007 (UTC)

PS. Just to cut to the chase regarding trival matters, I think the offending "it's" for "its" is here: Minestrone hold your head in shame ;-) Well, just for a moment, anyway. I know I've made this mistake before and I'm sure I will do it again :-). Geometry guy 01:10, 16 April 2007 (UTC)

I also feel for your point of view. However, as Geometry Guy mentioned, this is an encyclopedia article and to incorporate all the changes that you have suggested would undermine that express purpose. Certainly there is a place for your suggestions within this article, but perhaps the most appropriate place for the pedagogical tone you espouse can be found here wikibooks' calculus page

Cronholm144 06:03, 16 April 2007 (UTC)

There seems to be some initial support for the "ant" approach. Although I have reservations, I started to wonder how to write it in an encyclopedic tone. It is quite a challenge to be engaging without addressing the reader (e.g. with imperitives) or using reflexive verbs and other teleological language to describe the ant. I had a couple of thoughts. First, generally use the present tense, third person (e.g. "If instead the ant travels along a curve..." to convey the point of view of an abstract observer (the encyclopedia). The subjunctive may also be useful (e.g. "If an ant were to travel..."). Second, use subjects other than the ant, e.g., "The path of the ant bends upwards...". Other properties of the ant (the direction of travel etc.) are also good subjects. I would be happy to copyedit along these lines, but will be away for a few days, and so I wanted to leave a couple of suggestions.

I'm not sure that the ant idea will work so well for the "integration" part, because it involves the ant "knowing" the derivative, and I can't think of a way round this. But perhaps a different approach would be more refreshing anyway.

Another way out of this difficulty would be to cite the approach taken by a well-known textbook, and mix in a few quotes to liven up the text. Pity about the GA, but maybe this is also an opportunity. Geometry guy 17:05, 16 April 2007 (UTC)

PS. Mentioning Zeno's paradoxes in the introduction is a very nice idea. I also liked the "small-scale phenomena vs large-scale behaviour" idea, since it gets to the heart of what kinds of problems differentiation and integration (and differential equations) address.

I'm not thrilled with the "large scale"/"small scale" idea, since "large" and "small" are relative concepts. I'm also having second thoughts about the "ant" idea, because it skirts being too "original". I'm going to read some books, and think about this. Rick Norwood 18:50, 16 April 2007 (UTC)

I agree the "ant" idea may be a bit too original unless we can source it (or something similar). I see what you mean about "large scale"/"small scale". One could use relative terms such as "larger" and "smaller", but this doesn't quite capture the idea, since "small" really means "sufficiently small to model by the continuous or infinitesimal", whereas "large" often means something like "observable". I do think, though, that the question about what calculus "is" might come into sharper focus not by thinking about the mechanisms of limits, integration and differentiation, but instead about the spectacular success of differential equations (which involve differentiation to define/model and integration to solve/predict) in describing the natural world. Geometry guy 19:15, 16 April 2007 (UTC)

Well, what is calculus if not the relation of small-scale and large-scale behavior? The only other reasonable definition I can think of is "Poincaré duality and de Rham cohomology on Rⁿ", and not only is that inappropriate for an article like this one, it doesn't capture the feeling of the subject. You could talk about foundational concepts, like limits, but limits weren't introduced for their own sake, they were introduced to take derivatives and integrals, which are the real essence of the subject. This is what I was trying to capture when I said small-scale and large-scale behavior. Derivatives are local constructions which can be used to give global information like extrema; integrals are global (or at least not local) constructions which, by the Fundamental Theorem of Calculus, are really just the combination of a lot of local information. (The interpretation of integration as the cap product makes it clear that integrals really are global phenomena.) I was reluctant to say "local" and "global" in the article because those words are jargon, so I said small-scale and large-scale instead.

You could make a reasonable case that derivatives and integrals are mostly about time. After all, probably the first application that everyone sees is taking the derivative of a position function to determine velocity. And while this is good motivation, it's not really accurate because there are other useful things you can do with derivatives and integrals which have nothing to do with time. (Extrema, finding areas and volumes.) And you could talk about the utility of differential equations, but again I think that's a little too specific. I'm not sure what other kinds of motivation you could give.

Finally, I should mention that I'm not wedded to the ant idea. It was fun to write, though. 141.211.120.199 23:54, 16 April 2007 (UTC)

Good Article Review - Fail

Unresolved

 – These comments have spurred vigorous discussion and a resolution seems to be at hand.--Cronholm144 20:59, 10 May 2007 (UTC)

See What is a good article? for further details on the statements below.

GA review (see here for criteria)

1. It is reasonably well written.

   a (prose): b (MoS):

2. It is factually accurate and verifiable.

   a (references): b (citations to reliable sources): c (OR):

3. It is broad in its coverage.

   a (major aspects): b (focused):

4. It follows the neutral point of view policy.

   a (fair representation): b (all significant views):

5. It is stable.

   

6. It contains images, where possible, to illustrate the topic.

   a (tagged and captioned): b lack of images (does not in itself exclude GA): c (non-free images have fair use rationales):

7. Overall:

   a Pass/Fail:

Bear in mind that I am not well-versed in calculus and that I am therefore reviewing from an external perspective.

• 1b - What is calculus? The lead section does not actually state this - it merely says that it has 'applications in many fields'.
• 1c - Again, lead problem. Also, there are a few minor problems with jargon - arbitrarily large? Needs further explanation. However, I have not failed 1d, since the rest of the article is well-explained/linked with examples.
• 5 - The discussion above must be resolved before I can pass the article.

Some work is required on this article - but its basically simple explanations of what is already there and the resolution of the discussion above.

--ck lostsword|^queta!|_Suggestions? 09:52, 16 April 2007 (UTC)

Having read through the above discussion, it is clear that there are some major failings and discussion points in the article. Please renominate when these have been resolved - and perhaps read through my original review for some further suggestions. Thanks, and good luck ck lostsword|^queta!|_Suggestions? 14:16, 16 April 2007 (UTC)

Organisation

Unresolved

 – We are still working on resolution of these issues.--Cronholm144 21:43, 10 May 2007 (UTC)

In my quest to get this article back to GA status I have looked at numerous FA and GA articles for ideas about ensuring stability and precision within this article. As such I have added a to do list to the top of the discussion page and I would like to introduce the "responded" template to all the comments that have been addressed. I believe that a concerted effort for consensus on the few minor issues remaining will result in this article becoming GA within the next couple of weeks. After that, on to FA! I look forward to your input. Cronholm144 20:14, 21 April 2007 (UTC)

A great deal of work has already been done on the issue of clarity. If there are specific expository lumps remaining, please point them out. The issue of stability moved to the forefront when a new player appeared on the scene in the midst of the good article nomination. Things seem to have settled down, but one never knows what the future will bring.

I've looked through a number of widely used calculus texts, and was startled to find that none of the ones I've checked so far makes any attempt to "define" calculus. Usually they say a few words about "quantities that are changing continuously" and leave it at that. I'll continue to search. Rick Norwood 20:36, 21 April 2007 (UTC)

How about "Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which uses a common and specialized algebraic notation." Ironically enough this was once the first sentence of this very article. Cronholm144 22:42, 21 April 2007 (UTC)

I've already given it my best shot in the article. I doubt I can do any better than that. But thanks for asking. Rick Norwood 13:10, 25 April 2007 (UTC)

That's a true statement, but it's not a definition. You could say the same thing about probability, for example. If you interpret "analysis" in the non-technical sense, then you could also say the same thing about abstract algebra. Calculus is really about derivatives and integrals, and the first sentence should reflect that.

Here are some specific expository lumps which I object to:

• The derivative sections starts by talking about how hard it is. Phrases like "fundamentally more advanced" and "students must master mathematical notation" are inappropriate. (And I certainly wasn't a master of mathematical notation when I first studied calculus!)
• The derivative should be motivated in some way before the definition by difference quotients is discussed.
• The meaning of "instantaneous rate of change" should be explained. The phrase is jargon, not an informal description, despite what everybody seems to think.
• The integral section should begin with the motivating example.
• The definition of and notation for the indefinite integral should be together.
• The applications section needs more examples. It should mention differential equations.

And while I don't have any specific objections, for some reason I don't like the foundations section. But I can't put my finger on why, and I might be off base about that. 141.211.120.84 00:34, 22 April 2007 (UTC)

As regarding your objections,

• Calculus itself is an advanced topic and the article should make mention of it. Omitting useful and relevant information out of the fear that the reader will be frightened should never occur in a Encyclopedia. If you have a kinder way to relay that information regarding the advanced nature of the topic, please suggest it.
• Your comments about motivation I would like address similarly, this article is not trying to (and should not try to) teach a course on calculus. That is not to say that there should be no motivation, but the article makes mention of the uses of calculus in nearly every paragraph.
• I agree that instantaneous rate of change is jargon, so you comments about how to correct this problem would be welcome. The applications section is wikilinked to plenty of examples, and an article about calculus should not be bogged down with physics and engineering. I agree the section should link to differentials.
• I actually agree about the foundations section. It seems out of place. I think the foundations section can be merged into the article. Anyway I hope you understand what I am trying to say and look forward to your response.
• Oh I am aware that it is not a "definition" of calculus but I believe it is more clear than what we currently have. Further, like Rick, I have looked through 15 different books on single variable calculus and not one of them begins with a concise definition. Calculus certainly does center around the derivative and the integral but is not exclusively concentrated around them. Basically what I am saying is there isn't a really good definition out there so we will have to make do with what we have

Cronholm144 01:13, 22 April 2007 (UTC)

I'm going to take a stab at addressing some of the issues raised. To me, the "definition" of calculus is "applications of the concept of limits", but that wouldn't mean anything to someone who does not already know what calculus is, sigh. Rick Norwood 15:04, 22 April 2007 (UTC)

It may be helpful to distinguish between "calculus" and "analysis". I realise that one might consider these to be synonyms, but the categories of the maths WikiProject do make a distinction, in which analysis embraces continuity, sequences and series, but calculus does not (except for power series). Also, as you suggest, calculus suggests a more elementary "applications" context, whereas analysis emphasises rigorous formulations. Geometry guy 15:19, 22 April 2007 (UTC)

I would not object to calculus being called an advanced topic. But I do object to phrases like "fundamentally more advanced" and "master mathematical notation" which I feel are more intimidating than helpful. I'm glad that the latter is changed. I think I would like the word "fundamentally" removed.

I think any references to instantaneous rate of change should either be replaced with phrases like "behavior near a point" or they should occur after some description of derivatives and velocity. (A description which includes something like, "...for this reason, the derivative is sometimes called the instantaneous rate of change.")

As far as a definition, how about "Calculus relates small-scale phenomena with large-scale behavior"? I suppose that's not really a definition, either, but I think it makes the fundamental ideas pretty clear. 141.211.63.34 21:48, 23 April 2007 (UTC)

It is hard to tell what will be clear to the lay reader, but I strongly dislike the "small scale"/"large scale" idea -- it seems to me totally misleading, since most actual calculus can be done on any scale. I also like "fundamentally" -- there have been a number of studies (by the NSF, for example -- that show that calculus really is fundamentally harder for people to grasp than algebra, just as algebra is harder than arithmetic, because calculus involves what is called a "second order abstraction". I don't want to get into that here, but I do think a student beginning to study calculus should understand that calculus really is "hard", in some sense, and so even a good student should expect to have to struggle. It doesn't discourage people from learning chess to be told that chess is harder than checkers. I do agree with you that "instantaneous rate of change" is jargon to be avoided. Rick Norwood 13:00, 24 April 2007 (UTC)

There seem to be two essential points of disagreement here: how to define "calculus", and how to deal with the fact that it is a challenging subject. I do believe these gaps can be bridged by thinking carefully how the ideas are expressed. While I agree with Rick Norwood that "small scale"/"large scale" could be misleading, there is something in it. For me it is a way of saying that "calculus relates the infinitesimal to the finite" without using the taboo word "infinitesimal". In the real world, "infinitesimal" aka "small scale" means small enough that linear approximation is good enough. Can we find a way to express this more clearly? If Rick Norwood were forced to write a couple of sentences capturing this idea (despite his reservations), what would he write?

Similarly, while I agree with 141.211 that we should not intimidate the student, I share Rick Norwood's point of view that calculus is hard, and no one can expect to get it by reading an encyclopedia article: this article should inspire the interested reader to make the effort to learn calculus, rather than attempt to teach it. On the other hand, I'm not convinced that NSF studies should determine the way we express the difficulty of calculus in an encyclopedia article, so let me ask the same question of 141.211: if you were forced to include a paragraph on the difficulty of calculus (despite your reservations), what would you write? Just a suggested way forward... Geometry guy 16:50, 24 April 2007 (UTC)

Geometry Guy's suggestion is brilliant and I look forward to both Rick's and 141.211's response. I think we will soon reach a consensus on this article's most difficult issues. Good luck and do your best, Cronholm144 20:25, 24 April 2007 (UTC)

I'd write something like this: "Calculus is difficult because it requires a significant amount of abstraction. In more elementary subjects such as algebra, functions take only numbers as inputs and outputs. However, the main operations of calculus, differentiation and integration, take functions as inputs and can return numbers, functions, or sets of functions as outputs. For example, the derivative takes a function to another function, definite integration takes a function to a number, and indefinite integration takes a function to a set a functions."

I've made the assumption that it's the "function of functions" thing that's hard. That might not be the only thing, but my experience tells me it's at least one thing. 141.211.62.20 21:57, 24 April 2007 (UTC)

I agree that this is the main hard thing. You deleted a paragraph on this issue (which was originally based on this article) at Derivative. I agree the paragraph was intimidating. Can you replace it by something better along the lines of your suggestion? Geometry guy 22:07, 24 April 2007 (UTC)

That would be this edit. I felt like I kept all of the abstract facts in that paragraph. The article still states that the derivative is an operator, and it goes into a little more detail about what that means. What it used to do but doesn't do now is compare the derivative operator to functions of a real variable. That would be good to restore; I'll get around to it at some point. 141.211.62.20 23:13, 25 April 2007 (UTC)

OK, another comment. After thinking about my own experience learning and teaching calculus, I've realized that it's not just the "function of functions" thing that's hard. The other thing that's hard is the use of infinitesimals. Logically speaking, we don't use infinitesimals for calculus anymore, we use epsilons and deltas. But I remember thinking about calculus in terms of infinitesimals and, while I haven't conducted studies, I know I've seen my students try to reason with infinitesimals. It's probably a phase we all pass through.

It might be good to state things in terms of infinitesimals. It's a lot simpler than explaining difference quotients and Riemann integrals, for instance. Of course, it would require a statement to the effect of "The modern foundations of calculus have replaced infinitesimals by Cauchy's definition of a limit." Given that statement, I think it would be perfectly acceptable to argue with infinitesimals on the calculus page.

One other thing. User:Rick Norwood, you mentioned that there were NSF studies that showed that calculus was harder than algebra. It would be great if we could cite those on this page. 141.211.120.87 20:15, 29 April 2007 (UTC)

As with so much that I read, I remember reading it, but not where I read it. Sorry. Rick Norwood 13:36, 30 April 2007 (UTC)

I'm guessing that Theory of cognitive development might have some light to shed on the subject. Under that framework a case could be made that calculus represents a higher level of cognative development, indeed googling for cognitive development calculus gives some interesing results, notably David Tall's work[1]. --Salix alba (talk) 20:54, 30 April 2007 (UTC)

This article "Conceptual Knowledge in Introductory Calculus, Paul White; Michael Mitchelmore Journal for Research in Mathematics Education, Vol. 27, No. 1. (Jan., 1996), pp. 79-95. Stable URL: [2]" deals with the cycle of abstraction etc... If you don't have JSTOR I can send you the article by other means. I read some other interesting things in that journal. This might be dated, but the fail rates(Below a D) of Calculus in high school class were about 47% as of 1991 and the conceptual understanding of the basic concepts in calculus were extremely poor. I will look for more recent entires though. 17 May 2007, Cronholm144

That was an interesting article! No wonder students always object to word problems. Right now I feel like we shouldn't give them anything else. >:-)

You know, it would be nice if we had an article on calculus education. I see that we already have one on mathematics education, but it's mostly about K-12 education. In the meantime, does anyone feel like beginning a "Calculus education" section in the main article? The above-referenced article is a good place to start. 141.211.120.81 16:37, 19 May 2007 (UTC)

James F. Hurley; Uwe Koehn; Susan L. Ganter

   The American Mathematical Monthly, Vol. 106, No. 9. (Nov., 1999), pp. 800-811. 

   Stable URL: [3] 

This is the most recent article that I could find.--Cronholm144 21:34, 19 May 2007 (UTC)

Limits

Unresolved

 – Somewhat resolved, addressed further in very, very small--Cronholm144 21:01, 10 May 2007 (UTC)

I noticed Minestrone's comment with his last edit "Still, take some consideration maybe into having a more distinct section on limits within the derivative or differential areas" and in light of his words I would like to pose the following: The addition of a small intro to limits plus the definition given by Rick, which was that calculus is really just an application of the concept of limit. Instead of just having a limit link in both Integral and Derivative, I think the very first section after the lead should explain and introduce the concept, this would enable us to define calculus and simplify explanation later. Also with the addition of limit there should also be a mention of the limit definition of definite integral so as to illustrate the concept's appearance in both areas of calculus. Let me know if I am on track with this of not. Cronholm144 14:24, 5 May 2007 (UTC)

On second thought the limit definition of the definite integral may be a little much for the general calculus article, but... it seems like I am just arguing with myself, a bad habit to say the least. comments needed, lest I go mad ;) Cronholm144 02:01, 6 May 2007 (UTC)

The limit is, in my experience, the hardest concept for someone new to calculus to grasp. It is also the heart and soul of calculus, and one of the most beautiful concepts in mathematics. If you can find a way to get that across to the general reader, good for you. Rick Norwood 12:30, 6 May 2007 (UTC)

I agree with Cronholm the second ;) Furthermore, I do not agree that "calculus is really just an application of the concept of limit" or that the limit is "the heart and soul of calculus". It could be argued that the limit lies at the heart of analysis (including e.g. continuity, sequences and series, which are not calculus) but even there I think the emphasis is better placed on inequalities rather than limits. Limits, beautiful though they are, are a technical tool; they are one way to make many mathematical ideas, including the derivative and integral, precise. They are more the "liver of calculus" than its heart and soul!

The place to introduce limits is in the most elementary context (e.g. sequences for discrete limits and continuity for continuous limits) where there is nothing else going on. In differentiation and integration there are other things happening in the limiting process: the difference quotient and the Riemann sum. It is these I would emphasise, not the limit. At the heart of calculus I would place the following concepts:

• linear approximation
• rates of change
• tangency
• areas and volumes.

None of these are fundamentally about limits. Furthermore, limits are almost never used to compute derivatives and integrals: derivatives are usually computed symbolically (e.g. by a computer) and integrals are computed using the fundamental theorem. This was the Newton-Leibniz breakthrough which rendered obsolete the "method of exhaustion" (a limiting process) for computing areas and volumes, where progress had hardly been made since Archimedes. Emphasising limits in calculus is a historical step backwards! I hope I convince you ;) Geometry guy 13:30, 6 May 2007 (UTC)

I still think the limit is, mathematically, the tool that makes calculus work, even though higher level theorems allow us to take limits without realizing that we are really taking limits. In geometry we seldom have to use the Parallel Postulate, but without it, things fall apart, the center cannot hold. Rick Norwood 15:21, 6 May 2007 (UTC)

An interesting analogy! Whereas calculus had Newton and Leibniz, geometry had Descartes, Gauss, Klein,... These days, the "Euclidean plane" refers to R^2 with the notion of congruence given by the Euclidean group, not Euclid's axiomatic system of points and lines. From this point of view, one might as well say that "Dedekind cuts are at the heart of geometry", since without them (or some equivalent notion), R^2 would fall apart. I reply a little more seriously below. Geometry guy 20:52, 6 May 2007 (UTC)

Linear approximation/Euler's method is very useful, but I don't think it is at the heart of calculus. Rates of change I agree, but how does one go about finding those rates of change? Tangency, same question. Areas and volumes, well the integral as we have presented does not compute area, unless we believe in negative area and volume (I do, but that is besides the point). Rather, the definite integral acts a an accumulator of rates of change to yield total change over an interval, which can be either positive or negative. While calculus, because of the advances since Newton and Leibniz, can be computed without explicit use of the concept of limit, it underlies the entire process within both integration and differentiation. It is the concept that makes calculus possible. I agree that using limits to calculate an integral or derivative is an exhausting exercise, but it is where one begins when explaining the ideas behind why and how calculus works. Since this is an introductory article, I think the limit is precisely where we should begin. I am not suggesting that we throw the limit as applied to calculus at them just an overview of what the limit means so that when we do throw out the definition of derivative, it can be understood. I am rambling but I think that defining calculus as an application of the limit concept to functions is the best stab at really defining what calculus is thus far. DONE Cronholm144 18:16, 6 May 2007 (UTC)

Well almost... P.S. In many introductory calculus books the method of exhaustion is used to demonstrate the usefulness of integration as an area finder. I used it when I was younger to approximate Pi and really enjoyed it, I ended up using a limit to increase the number of polygons to infinity, suddenly I had a infant arc length integral. Although I did not know that at the time. The method of exhaustion becomes calculus once the limit is applied. at least for area integration.

I agree somewhat, although if we were to use method of exaustion, perhaps show an example of how it diverges towards calculus, as many introductory calculus books do. Thanks a bunch also, for taking into consideration my proposals! ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ ^{slurp me!} 18:34, 6 May 2007 (UTC)

I am conscious that I am talking too much and editing too little, which is against my general wiki-philosophy, but I'm working on the assumption that this discussion is useful, since we are not sure how to best to define "calculus" and the GA review suggests we should. My point of view is that this is the top level article for its category, and so we should take a top-down approach, motivated by what calculus is for, not a bottom-up approach motivated by the machinery which makes it work. Anyway, there isn't just one machinery: Newton and Leibniz managed fine without limits, and nonstandard analysis provides one of several ways to make this rigorous (call it a liver transplant, if you like ;). So limits do not necessarily underlie calculus. Indeed the derivative is essentially characterized by linearity and the Leibniz rule, independent of any particular definition (see e.g. Tangent vector#Definition via derivations and Differential algebra). When my computer computes the derivative of 2x^3+10 x, it is not using limits without realising it, it is using this characterization.

A few other comments...

• Linear approximation. I think Cronholm misunderstood what I meant: see Derivative#Definition via difference quotients.
• Tangency. You don't need limits to define tangents, as Greek geometry shows. For instance, away from inflection points, the tangent is the unique line which does not locally cross the curve. For curves arising in practise, we don't draw the tangent by computing limits.
• Rate of change. The speedometer on my car does not compute limits. Still, I am confident that it tells me how fast I am going. Maybe it is an approximation, but then so is the mathematical idealization of the real numbers.
• The integral as a change given by accumulating the rate of change. This comes close to the idea expressed by 141.211 that calculus relates small scale changes to large scale behaviour, which is an idea that I like and I think should be developed.

If on the other hand, this discussion is just bogging us down, then perhaps we should think more about what calculus is for, rather than what it is. The applications section is desperately inadequate at the moment. Perhaps that would help to clarify what calculus is. Geometry guy 20:52, 6 May 2007 (UTC)

Just as a slight follow up, if possible can some things be referenced? Not implicit referencing, as there is some kind of license for explanation here but we can still use references as a means of saying "we worked off this book". Similarly, can we all agree on a definite structure for the article and maintain it's continuity? It's all well and good having factual sections, but continuity would be a great thing to help those who haven't done calculus all their lives like us boring old farts (or maybe just me!) ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ ^{slurp me!} 18:38, 6 May 2007 (UTC)

Hmm... Well I agree that we need references for every section but the history. But how will we go about sourcing each particular section, can we just pick a book? I am not to familiar with the wiki policies regarding citation. I own about 8 text books on introductory calculus and I can pick one and cite places in the book that coincide with the article's information. Would this be appropriate? I mean we didn't really work directly out of a book, did we?(I didn't):0. I can use the following authors: Stewart, Ayres, Apostol, Smith, Euler, Snyder, Keisler, Kuttler. Stewart is the most popular college level book and I have all three of his editions, all of these books I also have in pdf form so if you want to see them I can send them to you. As for continuity, what would you propose exactly? I think the idea has promise but we are not writing a textbook so we have to be careful. Cronholm144 21:11, 6 May 2007 (UTC)

P.S. I am not sending these books to anyone but researchers for this article. No answer hungry students!;)

Geometry Guy I agree with you about the top-down approach, I was so desperate for a definition that I lost sight of the purpose of this article. Newton and Leibniz did it without limits, but the formulation of calculus spurred the formulation of a rigorous definition of limit because of its underlying importance to calculus, you cannot live without your liver for very long, unless you like jaundice. I agree that we are being wordy so off I go to search out real world applications!Cronholm144 21:18, 6 May 2007 (UTC)

Thanks, and good luck - you are making very useful contribibutions here! Anyway, I'm happy to see that the liver analogy appears to be working well, whether it be a limit liver, a nonstandard liver, or an algebraic one ;) Geometry guy 22:59, 6 May 2007 (UTC)

Do we still give a fig for Newton?

Unresolved

 – Continuation from limits, almost resolved--Cronholm144 21:03, 10 May 2007 (UTC)

The non-standard analysis boys had their fifteen minutes of fame, but I would guess that a high percentage of the current calculus texts today still begin with limits. Remember, we only report, we do not create, at least not here. Thomas set the template, back in the fifties. The biggest difference today is that now we expect students to learn analytic geometry in high school. Thomas, in the early editions of Calculus and Analytic Geometry, did not. (I taught Calculus III this semester, and what my students knew about Analytic Geometry could dance on the head of a pin, so Thomas was right.)

So, in fact, in practice, calculus goes: limit, derivative, applications, integral, applications, series. The only real disagreement is between early transcendentals and late transcendentals.

Which means this article should find some way of making the idea of a limit clear.

To me, it is all about control. If you can, by controlling the input, control the output, then the limit exists, and you get a useful answer. If you can't control the output, you may get an answer, but it won't be useful.

And I would say that, when my calculator tells me a derivative or integral, it is using some algorithm which we can implement with confidence because of theorems that go all the way back to the limit.

Did you know that in Newton's original manuscript he wrote that the derivative of a product was the product of the derivatives? Then he scratched it out and discovered what the truth was, huristically.

Rick Norwood 22:12, 6 May 2007 (UTC)

Yes, we report. This article is called "calculus". It is not called "the teaching of calculus", "the Thomas approach to calculus", "the nonstandard approach to calculus", "calculus with/without analytic geometry", "calculus III", or "the way current textbooks for students present calculus". It has been said it many times already, but it doesn't hurt to say it again: this is an encyclopedia, not a textbook. I am no fan of the nonstandard analysis approach (I find the necessity to take the "standard part" all the time ugly), but it exists and it works, and referring to "15 minutes of fame" is unnecessarily pejorative. We might have a more balanced view of calculus if Bishop Berkeley's criticisms had not come so early.

As I explained already, when a calculator computes derivatives (and hence integrals), it goes back to the characterization of the derivative as the unique linear operator up to scale which satisfies the Leibniz rule (the scale is easily fixed by x' = 1). I say this not to propose replacing the definition via difference quotients: I just want to illustrate the fact that any approach to calculus which dogmatically focuses on one point of view is misleading and unencyclopedic. Furthermore, no one has even attempted to answer my point that this is the wrong place to explain what a limit is. If limit of a sequence and limit of a function don't do it, then why not try to improve these articles, instead of attempting to do it here? Geometry guy 22:55, 6 May 2007 (UTC)

The use of limits in calculus is a historical anachronism. Here's how you get rid of them: Let ${\mathcal {O}}_{a}$ denote the ring of germs of functions at the point a. That is, it is the collection of all functions defined in a neighborhood of a modulo the equivalence relation f ~ g if and only if there exists a small neighborhood U of a such that f and g are equal once they are restricted to U. (In other words, ${\mathcal {O}}_{a}$ is a stalk of the structure sheaf of the real line.) ${\mathcal {O}}_{a}$ is a local ring, so it has a unique maximal ideal m consisting of function which vanish at a. Take f(x) - f(a); by f(x) I mean the function f, and by f(a) I mean the value of f at a. This is a new function which vanishes at a, i.e., lies in m. Look at its image in the module m/m². This is the derivative of f at a! You can check this easily for a polynomial: Write f(x) as f(a+h); expand; cancel the constant term; and then look at the terms containing exactly one copy of h (that is, the terms in m which don't get annihilated by m²).

All that's happened here is that I've applied the universal derivation to f. m/m² is the Zariski cotangent space at a, and the map that takes f to the image of f(x) - f(a) is the universal derivation d. More globally, one can work with Kähler differentials; the construction is similar, and it gives you the derivative (function) of f. Now, I've been slightly deceptive here in saying that you actually get the derivative, because what you really get is the differential form df. But df = f'(x)dx, so you're not actually that far off. Indefinite integration, of course, is just taking a preimage under d. Definite integration is just the cap product pairing in de Rham cohomology.

You could even argue that we should never use limits, because the construction above works in far greater generality (in any topos). (Correction: any ringed topos. 141.211.120.87 15:43, 9 May 2007 (UTC))

Rick Norwood, I don't mind if the calculus article has a brief discussion of limits, but it needs to be very brief. Its only purpose would be to introduce the limit so that we can use it when we discuss derivatives and integrals later in the article. From the perspective of calculus, limits are a technical tool, just like how the definition of a function as a graph is a technical tool to capture the intuition we have. Limits aren't necessary for calculus; you can replace them with the universal derivation or with nonstandard analysis. You get the same derivatives and integrals any way you do it. 141.211.120.92 16:32, 7 May 2007 (UTC)

There are many possible ways to do calculus, but some are more widely used than others. To say that limits are an anachronism is to claim that methods that do not depend on limits have effectively replaced the use of limits in calculus. I submit that this is clearly not the case.

I have not examined the programs that, say, the TI-95 uses to take derivatives. But I strongly suspect that it takes the derivative of, for example, x squared using, in effect, the power rule, a theorem that depends on limits, in the standard mathematical development of calculus. Rick Norwood 20:23, 8 May 2007 (UTC)

It is quite a challenge to take an intermediate point of view between the Zariski cotangent space and the TI-95 (both redlinks!), but I will do my best! The point made by 141.211 is essentially what I meant by the "characterization of the derivative as the unique linear operator up to scale which satisfies the Leibniz rule". However, to spell it out as 141.211 has done, one does need to be careful about the space of functions under consideration. In particular, continuous functions do not work (e.g., |x|^{1/2} vanishes at zero, but its square does not have zero derivative).

The idea that limits are a historical anachronism is one with which I partly agree: one of the reasons analysis has such specifically rooted foundations now was the knee-jerk response to Bishop Berkeley's attack on rationalism which I mentioned previously. Nevertheless, limits remain a widely used, and useful approach.

I have programmed symbolic differentiation. Yes, it typically uses the power rule, but to say this depends on limits is a bit like saying that the existence of infinitely many primes depends on the successor axiom in Peano arithmetic. Just as there are different ways to formalize arithmetic, so too there are different foundations for calculus. The key ingredient for the power rule is not the limit, but the binomial theorem.

Ironically, for Rick Norwood's argument, the case of x^2 is particularly transparent, since it is simply x times x. The derivative of this function is immediate from the Leibniz rule, which depends only on the characterization of the derivative, not on how it is defined. Geometry guy 21:44, 8 May 2007 (UTC)

I like 141's presentation of the modern way to approach calculus, it would be nice expanded in a seperate article. However it is likely to scare off 90% of our readers, germs modules wtf!

If I'm right the above can also be apllied to Jet (mathematics) (taylor series truncated at n-th term) so m is polynomials with no constant term, m² is polynomials with no constant and no linear terms so m/m² is just linear polynomials so the image of f-f(a) in m/m² just the linear aproximation to f which is another way to view the derivative. This might make it more understandable to mear mortals. --Salix alba (talk) 21:10, 8 May 2007 (UTC)

I agree, but I think 141.211 was just trying to make a point for the purposes of the talk page, rather than suggest e.g. that the cap product should be used to define the definite integral in this article! ;) Geometry guy 21:44, 8 May 2007 (UTC)

Actually don't, but it's personal preference.. i prefer texts the way they were around Newton's time as i can relate much easier to them due to their direct nature; when i read books today, they have about 35 different ways of explaining things with lovely little analogies of cats and dogs or rollercoasters, etc. I prefer good old physics-related stuff but i'm willing to do whatever to get this article to FA  :-) ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ ^{slurp me!} 21:55, 8 May 2007 (UTC)

Not intending to include this here. But there may be scope for an article modern aproaches to calculus. As we are an encyclopedia we should really document the history of the subject, indicating both the historical foundations and also more modern treatments. Certainly when you look at a subject like singularity theory where derivatives are studied in great detail it is the presentation in terms of jets, germs and modules where its actually possible to prove interesting results. --Salix alba (talk) 22:51, 8 May 2007 (UTC)

Let me reply to all the discussion above one point at a time.

Rick Norwood, I do not think that limits have ever had a place in the computational aspects of calculus proper. The derivative is the unique linear operator which satisfies the Leibniz rule and a normalization condition. One can use limits to construct such a linear operator, but, as I've sketched above, one does not have to. Similarly, the antiderivative is the inverse of the derivative, and definite integrals can be defined by evaluating antiderivatives (which is just taking the cap product in this context). This approach makes the Fundamental Theorem of Calculus a definition; then the real content of the FTC is that the Riemann integral is a construction of a definite integral, that is, it is an alternative way of constructing the cap product.

Oh, and another objection: You've said repeatedly that you believe that calculus is the application of the concept of the limit. I don't think this is correct; topological spaces are applications of the concept of the limit.

Salix alba, yes, everything you've written is correct. Let's restrict to the smooth case, since that's where I'm confident the following is correct: It's worth pointing out here that the stalk ${\mathcal {O}}_{a}$ is very nearly a graded ring with graded pieces mⁿ/mⁿ⁺¹, with the homogenous pieces of f corresponding to the various parts of the Taylor series expansion. A more precise description is that

${\mathcal {O}}_{a}=(\bigoplus _{n=0}^{\infty }{\mathfrak {m}}^{n}/{\mathfrak {m}}^{n+1})\oplus \bigcap _{n=0}^{\infty }{\mathfrak {m}}^{n}.$

(To see this, check that mⁿ consists of all functions which vanish n times at a and check the usual compatibility condition for a graded ring to see that the sum of the mⁿ/mⁿ⁺¹ is contained in ${\mathcal {O}}_{a}$; this is the stalk of the sheaf of analytic functions. Everything that's left must vanish to infinite order, hence it lies in the big intersection. Since the 0th graded piece is the field R, we must have a direct sum.)

Geometry guy, yes, I was just trying to make a point for the purposes of the talk page. To be honest, I hadn't thought all of it through very carefully, and I think (well, okay, I'm typing before I'm thinking again) that I've actually defined the exterior derivative in the sense of currents. This also answers your question regarding functions like f(x) = |x| which are not everywhere differentiable, because one can always take the distributional derivative.

Finally, we should all remember is that the point of this article is to be read. Put yourself in the position of a bright high-school student, say a 10th grader, who knows that some of the seniors are taking calculus but has no idea—none at all—what calculus is about. He's comfortable with algebra and he knows a little bit of geometry, but everything else is over his head. In particular:

• He won't know what a limit is
• He won't know what an infinitesimal is
• He won't know what a linear operator is

This article has to explain one of those concepts in enough detail that our reader can understand the rest. Historically, infinitesimals came first; I think that's because they're the ones that make the most intuitive sense. Linear operators came last, because they're the most technically demanding. The first two are appropriate topics for beginners, but the last one should be relegated to a parenthetical comment or a footnote.

What I think we should do is have a section entitled "Limits and infinitesimals" that says something like this:

Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". On a number line, these would be locations which are not zero, but which have zero distance from zero. No non-zero number is an infinitesimal, because its distance from zero is positive. Any multiple of an infinitesimal is still infinitely small, in other words, infinitesimals do not satisfy the Archimedean property. From this viewpoint, calculus is a collection of techniques for manipulating infinitesimals. This viewpoint fell out of favor in the 19th century because it is difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis, which provided solid foundations for the manipulation of infinitesimals.

In the 19th century, infinitesimals were replaced by limits. Limits describe the value of a function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but using ordinary numbers. From this viewpoint, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits are easy to put on rigorous foundations, and for this reason they are the standard approach to calculus.

What does everyone think of that? 141.211.120.87 15:43, 9 May 2007 (UTC)

Very good, I went ahead and added it to the article. The discussion page is getting long again and I think we might need another archive as soon as all of the issues mentioned in earlier sections of the page are addressed. (I think they have been for the most part.) Cronholm144 19:02, 9 May 2007 (UTC)

P.S. I fixed one redlink (TI-95), and the Zariski cotangent space is addressed in Zariski tangent space but I don't know how to do the cross link... I think I got it now.

Reference

Unresolved

 – references have yet to be added--Cronholm144 21:41, 10 May 2007 (UTC)

I think the question of references got drowned out in the recent discussion storm. What is the plan regarding adding references and citations to more than just the history section, or are they even needed? I offered earlier to provide all the refs I have and I think all of us here are fairly well armed in that department, its just a matter of consensus.Cronholm144 23:30, 6 May 2007 (UTC)

I think they are needed, again though as guidelines to explain the basics; if we include an explenation of differentiation, then i think it'd be best to add in a <ref> that has a note on how a particular author said something similar to how we did; that way we're able to have people pointed to areas of relevant bibliography (shock horror word!) to further their studies if neccesary. ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ ^{slurp me!} 06:56, 7 May 2007 (UTC)

Very, very, small.

Unresolved

 – Working towards resolution--Cronholm144 21:09, 10 May 2007 (UTC)

"Calculus is usually developed by manipulating very small quantities." seems to me a false statement. But I have expressed my opinion already several times. I'll wait and see what other mathematicians think. Rick Norwood 00:35, 10 May 2007 (UTC)

Well, just in case it wasn't fully clear, the "very small quantities" I was thinking about were the h in the difference quotient and the Δx in a Riemann sum. Feel free to clarify the article as you like. 141.211.120.87 17:21, 10 May 2007 (UTC)

I think it's a bit misleading, but I'm not sure how to explain it. Calculus isn't just the manipulation of small quantities - I think it's not small numbers that are important, but rather, the fact that these quantities vanish when we take the limits involved in both derivatives and integrals. Until they actually vanish, we're just talking about numerical approximations to derivatives and integrals. Cheeser1 17:34, 10 May 2007 (UTC)

Rick, how would you state it to better convey the concept? I am sure that no one will contest an edit along this vein.--Cronholm144 21:06, 10 May 2007 (UTC)

This is something I've thought and written about for many years, and my best shot at expressing the concept in writing used to be in the article, but has since been overwritten. The trouble with the current version is that it suggests that all limits are "very small", while in fact limits as x goes to infinity are equally important. In fact, limits don't necessarily have to have anything to do with "very small".

I'm leaving on a trip to China tomorrow, so I'll just put my version here, and if anyone likes it well enough to add it to the article, they should feel free to do so.

The concept of the limit was designed, by Cauchy and others, as one way to express precisely the concepts developed by Newton and Liebniz, and to answer the objections of Bishop Berkeley. A limit exists where control over the input allows control over the output. For example, the limit as x goes to 2 of ($x^{2}-4$) over ($x-2$) exists, because by keeping x close to 2, we can kep the output as close as we want to 4. On the other hand, we cannot take the limit as x goes to 2 of ($x-4$) over ($x-2$), because we cannot control the output when x is close to 2. This can be seen with the use of a calculator, plugging in numbers close to 2, such as 1.99 or 2.01. The precise definition of the limit, often called the "epsilon delta" definition, allowed mathematicians to prove theorems about derivatives, integrals, and infinite series. If the idea of the limit, or some equivalant idea, is not used, wrong answers, such as 1 + 2 + 4 + 8 + 16 + ... = -1, may occur.

Rick Norwood 12:26, 11 May 2007 (UTC)

I don't know if control is the proper term when talking about limits, perhaps we can expand on the concept of "closeness." .Iinstead of control maybe we could say that we can choose points as close as we like to the point being considered, assuming the limit exists. Of course much more eloquently than what I just wrote. A "rigorous" or "mathematical" definition of "closeness". maybe. I am not sure if we should mention epsilon delta per se though.--Cronholm144 01:47, 13 May 2007 (UTC)

The word that I would use is arbitrarily small, but maybe that word is too technical Tomgreeny 15:25, 2 June 2007 (UTC)

you are right, arbitrarily is typically the language used, but we are trying to make it universally understandable. However, the main proponent of the limit here is Rick, who happens to be on vacation, so this conversation has been put on hold for a while. As it stands now, the article deals with both the concept of limit and the concept of infinitesimals in a fairly approachable way. If you can word them in a more approachable manner, Go for it! --Cronholm144 16:20, 2 June 2007 (UTC)

GA ready?

Unresolved

Things have settled down somewhat here, I don't know if we have a pure consensus, but certainly the state of flux the article has been in recently has leveled off. Keeping this in mind, and the fact that we have "defined" calculus, I would like to point out that we have actually addressed the points raised at the GA Review. So, what do you think? I would hate to jump the gun on this one (like last time), but I think the article is definitely GA-nomination worthy now.--Cronholm144 08:06, 21 May 2007 (UTC)

Sorry, but i think the article still needs to be beautified and made more continuous. The information is there, it just needs to flow very smoothly from one section to the other -- at the moment it doesn't because there's definitions within the paragraph and it's somewhat choppy. I'll work on it shortly. ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ ^{slurp me!} 08:17, 21 May 2007 (UTC)

I strongly oppose submitting any article to the incompetent and illiterate criticism of GA. Why insult your good work? Septentrionalis PMAnderson 21:08, 28 May 2007 (UTC)

Well, in light of what has been going on recently, I think we might go straight to A once the article is better and the issue has cooled down,--Cronholm144 21:19, 28 May 2007 (UTC)

GA approval is a much less broken system than GA review, because it only requires the input of one editor, and there is a reasonable chance to get an experienced and thoughtful editor who is not obsessed with citations and will instead comment on improving the article overall. Still, on the road to A-class, we don't need to make such a big leap: I had another look at the article, and think at least a small promotion is long overdue. Geometry guy 21:59, 28 May 2007 (UTC)

I still think that it's got a while to go for A-Class, simply because some of the things there are not explained well enough, and seem to be there as residual information from former edits; things like the euler markup are fine, but also, the use of $\delta \lor d$ may explain things more clearly ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ ^{slurp me!} 00:12, 31 May 2007 (UTC)

Please, be sure to carefully copyedit the article before submitting it to any review! After looking at it for a few seconds I have already spotted a few punctuation problems (missing commas, etc), and there are probably more in there! Arcfrk 03:29, 31 May 2007 (UTC)

No worries, I don't plan on jumping the gun like I did last time :) It will be ready when we submit it.--Cronholm144 03:48, 31 May 2007 (UTC)

Imprecise Wording

A real question, however, is what is "complex and expansive problems" supposed to mean. We should avoid both adjectives anyway; "complex" could be confused with complex numbers; "expansive" with "expanding"; but what is meant here? Septentrionalis PMAnderson 20:38, 21 June 2007 (UTC)

I think what that sentence is trying to convey is that calculus is used in a wide variety of fields, and can solve a mathematically diverse set of problems. --Cheeser1 02:38, 22 June 2007 (UTC)

I gave the lead another try, removing the vague wording. Arcfrk 03:35, 22 June 2007 (UTC)

I assume by algebra you meant elementary algebra? It reads better now though. --Cheeser1 04:13, 22 June 2007 (UTC)

Good work everyone, it is good to see some fresh faces around here. :)--Cronholm¹⁴⁴ 04:56, 22 June 2007 (UTC)

Please be aware that everyone and his brother has taken a stab at rewriting the calculus article. Being dogmatic about the superiority of your version is not apt to lead to a stable article. Rick Norwood 15:13, 22 June 2007 (UTC)

You are right, of course, but do we want a stable article where no one quite understands what 'complex and expansive problems' means? Arcfrk 15:41, 22 June 2007 (UTC)

Power series

What do you all think of replacing the phrase "power series" in the first paragraph with the word "series", since arguably the Fourier series is at least as important as the Taylor series. Rick Norwood 15:15, 22 June 2007 (UTC)

Certainly the average reader won't notice the difference, but the change seems appropriate.--Cronholm¹⁴⁴ 15:25, 22 June 2007 (UTC)

Hearing no objection... Rick Norwood 16:13, 23 June 2007 (UTC)

Change it Rick, this is wikipedia after all :) --Cronholm¹⁴⁴ 16:20, 23 June 2007 (UTC)

Emending the lead

Some of the vague language has been removed from the lead, but I feel that it still needs a lot of improvement. The part that is especially objectionable to me is the statement that calculus builds on analytic geometry and mathematical analysis. Is this true?

Analytic geometry:

• Historically speaking, no, since the article itself makes clear that a lot of preliminary work on calculus predates Descartes and even Vieta; of course, the work of Fermat and Descartes motivated Newton and Leibniz, but there are better ways to express this heredity;
• In the way it is presently taught, no, since analytic geometry tends to be taught within calculus courses, or not at all;
• Methodologically, no, since the methods of calculus are not based on methods of analytic geometry.

Mathematical analysis:

• Historically speaking, not at all, it's the exact opposite of what has happened (first Newton, Leibniz, Euler, then Cauchy and Weierstrass);
• In the way it is presently taught, no, again, it's the reverse of the actual state of affairs, we first teach calculus (with or without proofs) to majority, then real analysis to selected few;
• Methodologically, yes, although I am not prepared to formulate the exact connection.

In addition, a somewhat more precise description of applications of calculus (going beyond the adjective 'widespread') would be in order. Finally, we may want to mention straight off what distinguishes calculus from more elementary mathematics (namely, emphasis on functions).

Below are the current lead (minus the first sentence) and my proposed amended version. Any constructive comments will be greatly appreciated. Arcfrk 21:14, 22 June 2007 (UTC)

Current version

Calculus has widespread applications in science and engineering and is used to solve a wide variety of problems for which algebra alone is insufficient. It builds on analytic geometry and mathematical analysis and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus.

Proposed version

Unlike algebra, which mostly deals with variables, calculus operates with mathematical functions. Analytic geometry is used to convert geometric and physical problems into the language of functions, leading to numerous important applications of calculus in science and engineering. A mathematically sophisticated underpinning for calculus is provided by mathematical analysis. Calculus is divided into two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus.

Discussion

For the record, I did not write the proposed version, nor did I add it to the article, regardless of what User:Rick Norwood would have you think [4]. --Cheeser1 21:52, 22 June 2007 (UTC)

I agree with Arcfrk's proposal. Geometry guy 22:09, 22 June 2007 (UTC)

I agree as well. Silly rabbit 22:15, 22 June 2007 (UTC)

This seems quite sensible. Leads should have actual content; and it will get us away from the curriculum/a silliness. Doubtless some people will find something else to revert-war about. Septentrionalis PMAnderson 00:03, 23 June 2007 (UTC)

Rick Norwood was likely not trying to insult you Cheeser, he just cares about the article and was reacting to the sudden change in content. Also, do you support the new revision?(I assume that you do but I just want to make sure)

Sept, Revert wars are very uncommon here, and hopefully now that all of the concerned parties are on the same page, any war-like impulses can be sublimated into positive contributions to the article.

The proposal seems reasonable to me.--Cronholm¹⁴⁴ 04:46, 23 June 2007 (UTC)

calculus builds on (that is, has as a prerequisite) analytic geometry. This informs the reader that, to learn calculus, one must first learn analytic geometry. It has nothing to do with history. Some colleges still teach analytic geometry and calculus out of the same textbook, but most, today, teach the analytic geometry in a separate course, called precalculus.

I agree that analysis does not belong here.

One can study functions extensively without calculus. What makes calculus different is the introduction of the limit or (if you want want to be historical) the introduction of the derivative and integral.

The "proposed version" above does not seem to me to improve on the current version. Also, Cheeser1, you do not help your argument by guessing what I do or do not "seem" to think. Rick Norwood 16:11, 23 June 2007 (UTC)

(On analytic geometry:) The two of you are saying slightly different things. Analytic geometry is (in educational practice) preliminary to calculus, but it is not logically necessary. Synthetic geometry will do fine, and it is possible to build calculus without any appeal to geometry at all. Septentrionalis PMAnderson 16:44, 23 June 2007 (UTC)

Just to clarify Rick, several other people did think that the proposed version improved on the current one, so your restoration of an earlier lead, to a version before even Septentrionalis's fix of 'complex and expansive problems', went against consensus. It is preferable to discuss first, then act. Of course, we all want this to be a well-written article, so let us not complicate our common task by unnecessary back-and-forth edits. Arcfrk 22:25, 23 June 2007 (UTC)

lede

Here is the current lede:

"Calculus (from Latin, "pebble" or "little stone") is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. Calculus has widespread applications in science and engineering and is used to solve complex and expansive problems for which algebra alone is insufficient. It builds on analytic geometry and mathematical analysis and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus."

On reading it over several times, I think it can be improved. We say where the word comes from, what larger subject it is part of (mathematics) and what its constituant parts are. So far so good. Then we seem to want to say something about its place in modern university education, about its applications, and about its structure. It seems to me that structure, then education, and finally applications is a better order to treat these topics. Here is my proposed rewrite:

Proposed rewrite

"Calculus (from Latin, "pebble" or "little stone") is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series. It includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus. In modern university education, prerequisites to calculus include algebra, trigonometry, and analytic geometry, which are often combined in a course titled precalculus. Most modern university students do not take calculus, but it is required for all majors in mathematics, any of the sciences, and engineering, and is often required for majors in business. Calculus has widespread applications in science and technology and is used to solve problems for which algebra alone is insufficient, especially problems that require optimum solutions."

Comments? Rick Norwood 16:31, 23 June 2007 (UTC)

Comments

• The educational aspects of Calculus get a bit too much attention, at the expense of the mathematical content.

• Can you, please, explain the meaning of used to solve problems for which algebra alone is insufficient? My interpretation, the one that you have disagreed with, was that algebra deals with variables and calculus deals with functions. The sentence as it stands is rather obscure.

• I think that more can and should be said about the subject of calculus. For example, that the derivative measures the rate of change, and the integral expresses 'accumulating quantities'. Some rather good ideas were contained in the earlier leads that survive in the history of the page.

• On the other hand, if we choose to postpone the discussion of the substance of calculus, then optimum solutions would be rather out of place, as it is neither the first, nor the only application of calculus.

Arcfrk 22:41, 23 June 2007 (UTC)

• Optimization does seem out of place in the lede.
• "The integral is an accumulator of rates that yields total change" is appropriate for the definite integral. Indefinite, I am not so sure. Saying something about the subject will be very tricky, because we are bound to say too much and too little.
• I would like to hear what other editors think of Arcfrk's interpretation of Cal vs. Al, (I am ambivalent, leaning no as possibly misleading. If we use those definitions we will have to emphasize the elementary aspect of algebra and calculus) as it seems that this is the crux of the current disagreement.

--Cronholm¹⁴⁴ 03:24, 24 June 2007 (UTC)

I'm sorry, Arcfrk, but the idea that "algebra deals with variables and calculus deals with functions" is very misleading, since obviously algebra also deals with functions (though not in the same way calculus does) and calculus also deals with variables.

Let me ask if there is any objection (modulo leaving out one or more topics) with the order: first structure, second education, third applications?

There seem to be the following points yet to be decided. They all take the form: Does X go in the lede or is it left to the body of the article. 1) Do we want to try to explain what a derivative and an integral are in the lede? 2) Do we want to discuss Calculus, the course, as well as calculus, the area of mathematics, in the lede? 3) Do we want to mention specific applications in the lede, and if so what?

I suggest we take these one at a time, starting with 1) Rick Norwood 13:56, 24 June 2007 (UTC)

(1) No; that's what the article's for. Septentrionalis PMAnderson 17:03, 24 June 2007 (UTC)

(2) Probably not; calculus exists without Calculus courses. Having taught them, I will add that this is probably just as well. Septentrionalis PMAnderson

(3) I'm torn about; "many applications" seems much more leadish. Septentrionalis PMAnderson 17:03, 24 June 2007 (UTC)

Are we basically happy with the body of the article yet? If we are and Good Article is our next goal, then determining what goes in the lead should be easy, because of the "no surprises" clauses in WP:LEAD, i.e., any substantial content in the article must be mentioned in the lead. Of course that does not mean such a lead is easy to write. However, this article is incredibly lucky to have so many of the best editors at WPM paying so much attention to it. (Indeed, even though this is one of the few articles permenantly on my watchlist, I rarely comment or contribute here because it is in such good hands.) The issue of what calculus is has now been well discussed, and so, if the body is also okay, I reckon that if any one of the editors here

• forgot about all previous versions of the lead,
• read through the body of the article again,
• and reminded themselves of the guidelines

they would write a fantastic lead! Geometry guy 17:56, 24 June 2007 (UTC)

Pyramid

I strongly object to the use of "method" in discussing the the Egyptians. In this context, it implies that they had some rudimentary form of calculus, for which there is no evidence whatsoever.

You don't need calculus to prove the volume of a pyramidal frustrum; all you need to do is subtract the volume of the missing pyramid from the volume of the whole pyramid, and the fact that the volume of a pramid is one third of the enclosing block is an exercise in solid geometry, which Euclid did quite well. To determine it, by trial and error, may well be easier; after all, the Egyptians did have extensive practical experience with packing blocks into pyramids.

Please remember what the Moscow papyrus actually says. To papaphrase slightly:

{Volume of pyramid frustrum] Base 4, top 2, height 6 Multiply 4 by 4? 16 Multiply 4 by 2? 8 multiply 2 by 2? 4 add? 28 Multiply by one third of 6? 56

"See, it is 56; your have found it correctly"

I do not recall the standard translation accurately enough to find it on the web, but it is widely available; Morris Kline's history of mathematics, for example. This is, I think, better described as "working out" the volume, than any of the more highfalutin phrases I and others have been using.

What is impressive to me here is the possibility they may have factored a³ minus b³ with only rudimentary algebra.

Regards. 17:33, 24 June 2007 (UTC)

I just changed it to method because the prose was ungainly. Why don't we say "determined by trial and error or solid geometry or whatever they actually did". I don't intend it to sound pretentious, but this is an encyclopedia--Cronholm¹⁴⁴ 17:53, 24 June 2007 (UTC)

We don't know how they discovered the calculation above; all we know is that they used it. That's what I'm trying to make clear. Septentrionalis PMAnderson 19:10, 24 June 2007 (UTC)

• "High-faluting" was primarily aimed at myself: I suggested "algorithm", for Pete's sake. Septentrionalis PMAnderson 19:19, 24 June 2007 (UTC)

(edit-conflict) Oh... that is troubling, I'm sorry I didn't read your comment more closely. Perhaps we could say that "the Egyptians found a rudimentary way to calculate the volume of a pyramid" or some such.--Cronholm¹⁴⁴ 19:21, 24 June 2007 (UTC)

P.S. Leave Pete out of this. what did he ever do to you? :)

Much better.--Cronholm¹⁴⁴ 19:23, 24 June 2007 (UTC)