Talk:Calculus/Archive 8

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Calculus in other languages: wrong German wikipedia association

The link to the German wikipedia article "Kalkül" is wrong. Both words may stem from the same origin "calculation, ..." but the real translation for calculus in German is Infinitesimalrechnung or "Analysis" or "Integral- und Differentialrechnung". The currently linked "Kalkül" (through the "language bar") has a meaning in the area of "logic". 80.219.208.33 (talk) 14:05, 22 January 2011 (UTC)

I second this opinion. Tried to change link to point to https://de.wikipedia.org/wiki/Analysis , fails because it is linked to https://www.wikidata.org/wiki/Q7754 already. "Kalkül" is definitely not the German analogue. BjornVDM (talk) 13:52, 24 October 2013 (UTC)

Merge with infinitesimal calculus

I would like to merge infinitesimal calculus into calculus. Right now, the infinitesimal calculus article is mostly about the historical origins of the subject—it's about calculus when it was done with infinitesimals—and there's a small mention of non-standard analysis. All the historical material is already covered in much greater detail at history of calculus. This edit of mine I think effectively merges all of the content of the infinitesimal calculus article here (in addition to some reorganization and copyediting). The only things that don't currently appear here are the bibliographic reference to the book by Baron and the infinitesimal navbox.

I don't see what could possibly go at infinitesimal calculus that doesn't belong either here or at history of calculus. Any objections to turning infinitesimal calculus into a redirect? Ozob (talk) 12:15, 21 May 2011 (UTC)

I lean against; the present text is spread out over parts of the calculus and non-standard analysis articles, and it seems to serve readers better to have it in one place. I appreciate your work in merging the information into the calculus article, but I don't think removing the duplication serves readers. — Arthur Rubin (talk) 14:00, 21 May 2011 (UTC)

Oppose. Infinitesimal calculus is currently linked by mostly historical pages,such as biographies of Fermat, Wallis, Newton, Leibniz, Bernoulli, l'Hopital, etc. We should include information relevant for readers coming from those pages. Tkuvho (talk) 18:28, 21 May 2011 (UTC)

Is infinitesimal calculus actually a separate subject from modern calculus? I agree that the philosophical foundations are vastly different, but there are ways in which they are the same. For instance, they are applied to the same physical processes, and they produce the same functions as outputs of the derivative and indefinite integral operators. I don't see modern calculus as an essentially different topic from historical calculus. I feel like that having separate articles for these is like having separate articles for, say, functions pre-set theory and functions post-set theory. Or geometry pre-Descartes and geometry post-Descartes. Ozob (talk) 21:51, 21 May 2011 (UTC)

Infinitesimals are different from functions and geometry, in that you had functions before and after Cantor, and geometry before and after Descartes. Meanwhile, you are not going to find too many infinitesimals in Georg Cantor. Check out the article: he thought they were an "abomination" and the "cholera bacillus of mathematics". He didn't feel that way about pre-set theoretic functions. Tkuvho (talk) 22:22, 21 May 2011 (UTC)

You had calculus before and after Cauchy, Weierstrass, Riemann, et al. Cantor's views on functions seem like a good parallel here to Weierstrass's views on derivatives: Weierstrass had completely different foundations for derivatives from Newton and Leibniz, but he still recognized their notions as derivatives, just as Cantor recognized his predecessors' functions as functions. Ozob (talk) 23:23, 21 May 2011 (UTC)

But we are not talking about derivatives here, we are talking about infinitesimals. Incidentally, Leibniz was mostly working with differentials, not derivatives; while Newton turned more and more away from infinitesimals as he got older, so we are not talking about Newton's approach to infinitesimal calculus. Again, the point is that the basic objects in calculus based on infinitesimals are different from the basic objects of the epsilontic approach. That's different from formalizing functions relative to a set-theoretic foundation. Tkuvho (talk) 23:37, 21 May 2011 (UTC)

As I see it, the basic objects in calculus are derivatives and integrals, and this has not changed since the subject was invented. There are many ways of defining derivatives and integrals; there are infinitesimals, Weierstrassian epsilons and deltas, Robinsonian non-standard analysis, Kähler differentials, synthetic differential geometry, and probably more that I'm forgetting. But regardless of the exact definition, we still call the resulting operations the derivative and the integral. There's still a product rule, a chain rule, a fundamental theorem of calculus, and so on; the different foundations give the same results. It seems to me that having two articles will confuse readers (who might think that different facts are true of the derivative and integral of "calculus" as compared to the derivative and integral of "infinitesimal calculus") and causes needless duplication of effort. Ozob (talk) 02:29, 22 May 2011 (UTC)

I agree with the thrust of your argument when applied to the concept of epsilontic limit, but not when applied to infinitesimals. As you mentioned, derivative and integral are still derivative and integral before and after the technical tool of epsilontic limits is introduced. On the other hand, infinitesimals have a significance of their own. You cannot define a delta function, as Cauchy did, if you don't have infinitesimals. No amount of epsilontics will help you here unless you go out of an Archimedean system. Tkuvho (talk) 03:29, 22 May 2011 (UTC)

I am not proposing to merge either limits or infinitesimals here. I'm proposing to merge the page "infinitesimal calculus", which—as I see it—is about derivatives and integrals. Ozob (talk) 12:23, 22 May 2011 (UTC)

I hear you, but I think the range of possible meanings of "infinitesimal calculus" is broader than the standard epsilontic calculus. It includes both the historical infinitesimal calculus, with issues ranging beyond mere mathematical applications such as derivatives and integrals, to modern infinitesimal approaches to the calculus. If you check the history of the page, you will see that there were hardly any hits before links from historical pages were added. People come here generally because they are interested in history, and perhaps the meaning of the term. In modern usage, the term "infinitesimal calculus" has turned into a bit of a dead metaphor, but it has other uses as well. Tkuvho (talk) 12:40, 22 May 2011 (UTC)

You say that "infinitesimal calculus" can have meaning "beyond mere mathematical applications such as derivatives and integrals". Perhaps this is the heart of our disagreement. I see no meaning to the term beyond derivatives and integrals. Can you name for me something that should be on the "infinitesimal calculus" page that is not derivatives, integrals, non-standard analysis, or history? Specifically, something that is not a mere mathematical application? Ozob (talk) 13:16, 22 May 2011 (UTC)

Hmm. Perhaps we should start a new thread. Ozob, as research mathematicians, we are naturally suspicious of history. After all, all that matters is the results. Nonetheless, the field of history exists, and hard as I find it to believe, some people are even more passionate about it than about mathematics (hope the platonists among us aren't listening). There is any number of historical issues that can be clarified in a predominantly historical page, that need not necessarilyl be submerged by a focus on the mathematics. By your logic, infinitesimal should also be redirected to calculus. After all, it's just a technical tool just like limits. Why should there be a separate page for infinitesimals when all they are is a way of expressing derivatives and integrals? Where are the constraints in your argument? Tkuvho (talk) 13:29, 22 May 2011 (UTC)

No, as I said above, I do not want to merge infinitesimals into this page. Infinitesimals have other uses than derivatives and integrals and are a notable topic on their own. But I am still convinced that infinitesimal calculus is limited to derivatives, integrals, and history, and these topics are already adequately covered by calculus and history of calculus. I still have the same question: Can you name for me something that belongs on the infinitesimal calculus page that is not derivatives, integrals, or history? Ozob (talk) 17:22, 22 May 2011 (UTC)

Certainly the multiple uses of the term "infinitesimal calculus". Tkuvho (talk) 18:12, 22 May 2011 (UTC)

What uses are there that do not refer to derivatives, integrals, or history? Ozob (talk) 19:37, 22 May 2011 (UTC)

I generally favor this idea. Infinitesimal calculus is usually just another name for the calculus, disambiguating it from, say, the propositional calculus, or any of several other things with calculus in the name. It's the same subject, whether you do it with infinitesimals or with epsilons and deltas. It's plausible that there could be an article specifically on applications of infinitesimals to the calculus. but infinitesimal calculus is the wrong name for such an article. --Trovatore (talk) 20:18, 21 May 2011 (UTC)

I don't think you are sufficiently familiar with the historical literature on the subject. Authors such as Baron use the term in the historical sense. Tkuvho (talk) 20:21, 21 May 2011 (UTC)

The problem is that searches and links are quite likely to be intending simply the calculus. --Trovatore (talk) 20:26, 21 May 2011 (UTC)

No problem, the page provides a link both to standard calculus and true infinitesimal calculus. Tkuvho (talk) 20:33, 21 May 2011 (UTC)

There could be a hatnote at the top of calculus, saying

"infinitesimal calculus" redirects here. For infinitesmial methods in the calculus, see infinitesimal methods in the calculus.

, and that would address the needs of people looking for it in your sense. --Trovatore (talk) 20:41, 21 May 2011 (UTC)

Another solution would be to provide a hat at infinitesimal calculus, with a note that people looking for epsilontic methods in the calculus should go to calculus. Tkuvho (talk) 20:45, 21 May 2011 (UTC)

Calculus is just one subject, no matter what methods you use. In my estimation, the term infinitesimal calculus is mostly understood to mean just the calculus, and therefore should redirect there. --Trovatore (talk) 21:34, 21 May 2011 (UTC)

Merge Per Trovatore's comment above. Thenub314 (talk) 19:53, 22 May 2011 (UTC)

I am agree to merge the 2. Infinitesimal Calculus is just a way of proving the calculus that we know.Barticles (talk) 19:35, 21 June 2011 (UTC)

I do not agree completely as the majority of people who will search the word 'calculus' on wikipedia will probably not have enough knowledge of the topic to beable to comprehend the more subtle area's of this subject, the reason i think this is because calculus is introduced at A-level and if it had been around then I would have used wikipedia to research it. Potentially, this could just cause confusion amoung some viewers. — Preceding unsigned comment added by 193.82.129.38 (talk) 12:13, 1 September 2011 (UTC)

Agreed. The article "Infinitesimal calculus " should be merged with this article since that article creates a lot of confusion. — Preceding unsigned comment added by 220.255.2.143 (talk) 06:37, 25 November 2011 (UTC)

What do you find confusing about that article? Tkuvho (talk) 13:03, 27 November 2011 (UTC)

I agree with this suggestion."Infinitesimal calculus " is a part of calculus. It also creates confusion for the readers.Please make the article "Infinitesimal calculus " as a part of the article "Calculus." — Preceding unsigned comment added by 59.93.2.228 (talk) 09:42, 8 May 2012 (UTC)

Oresme and Riemann Integral?

Why is Oresme, the first discoverer of the Riemann integral, not mentioned here? — Preceding unsigned comment added by 62.194.6.46 (talk) 14:21, 28 May 2011 (UTC)

If you can find a reliable source that says he independently discovered the Riemann integral, then you are welcome to mention him in the article. Ozob (talk) 14:23, 29 May 2011 (UTC)

Addition of an external link

I added an external link that was deleted by Favonian. The reason was that it was a kind of advertisement, promotion etc. The link in question is http://www.scimsacademy.com/courses/SampleLessons.aspx#CalculusIntro. Yes I own the website and it is commercial. But the link to the material is a genuine, no-nonsense introduction to calculus; which I believe will be useful to any student (say at the freshman level). Many existing links in the article are definitely less useful than this one - e.g. Weisstein, Eric W. "Second Fundamental Theorem of Calculus." From MathWorld—A Wolfram Web Resource, which is just a statement of the theorem. If anything remotely commercial or promotional must be eliminated from Wikipedia; then any book reference should also be removed - a book like Stewart or Thomas Calculus (yes they are standard & good books) promotes those authors and publishers & they cost a good deal of money to buy. The material I have put is free to read, & gives a quick (yet fairly rigorous) introduction in maybe a few hours; as opposed to several weeks required to read many books on the subject. From a usefulness point of view, I think it should be accepted. I also believe, I should be allowed to add links to my introduction on Vectors and Mechanics (on the appropriate pages) because they are also useful in the same way. Let me know whether I can re-submit the link. Sivsub123 (talk) 15:57, 8 June 2011 (UTC)

Leibniz and Newton are usually both credited with the invention of calculus.

On my first day of college math the prof asked for a show of hands as to whether the Calculus was invented or discovered. He said the correct answer was discovered. I agree. The article doesn't. — Preceding unsigned comment added by 67.188.241.125 (talk) 19:19, 9 November 2011 (UTC)

Please sign your talk page messages with four tildes (~~~~)? Thanks.

Perhaps your prof tends more to Platonism than the contributors who created this article. It's an old philosophical question: are mathematical concepts invented or are they discovered? I think that nowadays most scholars prefer the invention view. Platonism has become a bit old-fashioned. - DVdm (talk) 19:43, 9 November 2011 (UTC)

"Perhaps your prof tends more to Platonism than the contributors who created this article." It should be unbiased, stating both opinions. — Preceding unsigned comment added by Brad7777 (talk • contribs)

Please sign your talk page messages with four tildes (~~~~)? Thanks.

It depends on whether the other point of view is sufficiently notable to explicitly state it. There is an interesting read (and policy) about that in wp:UNDUE. I don't think that it would be a good idea to replace multiple occurrences of phrases like "X invented concept Y" with phrases like "X invented — or, according to the Platonist view, discovered — concept Y," except of course in an article or in a specific section about the difference between inventing and discovering mathematical concepts. - DVdm (talk) 18:34, 27 November 2011 (UTC)

If there were a genuine Platonist/formalist issue here, then of course the Platonist view should be represented. It is not undue at all and I completely disagree that Platonism "has become old-fashioned".

But I don't really think there is. In a Platonistic sense, Hamlet, Prince of Denmark (at least, the sequence of characters corresponding to some given edition of it) has always existed, long before Shakespeare ever wrote it down, but we don't insist on saying that Shakespeare "discovered" it. A more neutral word might possibly be chosen if convenient, but there is no need to belabor the point. --Trovatore (talk) 23:06, 27 February 2012 (UTC)

Actully Calculus was invented in India. Claiming newton and liebniz discovered calculus is like claiming Columbus was the first to discover America. http://www.cbc.ca/news/technology/story/2007/08/14/calculus070814.html 72.53.146.220 (talk) 05:51, 19 September 2012 (UTC)

Changes to make the material more accessible

By the end of the first paragraph, I was confused as to what calculus is. The branch of mathematics part I get.

For those of us who are not mathematicians and are wanting to understand the concept of calculus with some of the basics, this article is not very satisfying.

After doing reading about in various places, watching videos on MIT OpenCourseWare, and other bits and pieces, there really does not seem to be a place that will give me a good concept of calculus, what it is used for, why was it invented, what need does it satisfy. I keep finding videos that show people doing equations and drawing simple 2D graphs of functions, always emphasizing how and not why or what.

My concurrent conception of calculus is something like the following.

Calculus is a type of mathematical language that is used to analyze how an algebraic function changes. Algebraic functions describe how a set of values are transformed into results. For instance to calculate the volume of a room the algebraic function would be area = width x length x height. The algebraic function is static in that the width, the length, and the height are static though the values may differ each time the function is used. For instance if the algebraic function is used to calculate the amount of water that can be put into several rooms in a house, the algebraic function is the same and the values used will differ depending on the dimensions of each room. Calculus functions describe how an algebraic function's output changes and what is changing. For instance for the algebraic function for the area of a room, the function is linear in that there are three variables which can be assigned values and the values of the function are constants and the result of the function is a constant. Calculus would describe this function as being constant.

Now if we consider a different type of room, say the trash compactor in an Empirical Star Destroyer, and much like Han, Luke, Chewie, and Princess Leia we have fallen inside and are very interested in whether we can survive by standing on the pile of trash as the walls close in. Our algebraic function of area = width x length x height is not changing. However we have a new variable, time, and we describe the volume of the room as being a particular width, length, and height at a particular time. So our original algebraic function is now modified so that the variable of width is no longer a value. Rather it is now a function that generates a value based on the time during the compaction operation. If the compactor is empty then as the walls move in and the width of the floor becomes smaller, the height of the room stays the same since we are standing on the floor which is not changing. However if there is trash in the compactor then as the walls move in and the trash is shoved together, the height of the trash grows (same amount of trash in smaller floor space) and so as we stand on the top of the trash, the distance between our heads and the top of the room grows smaller.

So we now have an algebraic function, area = width x length x height, that rather than having constants for the width and the height now have functions that generate the values for the width and the height. These functions have a parameter, time, that is changing. The compactor is a deterministic system. When the button is pushed to start compaction, the various hydraulics and other systems operate in a particular sequence so that, barring equipment breakage, if the compactor start button is pressed every morning of each day over the course of a year, it always operates the same with the walls closing in at the same rate. So even though the parameter time is changing during the operation of the compactor resulting in changes to the width, it is changing in a constant manner and the width changes in a constant manner and this also means that the height is changing in a constant manner. So there is no real calculus here since everything is constant for a particular compactor scenario.

However the amount of trash in the compactor may vary depending on activities within the Star Destroyer such as whether Spring Cleaning has started or the Emperor is visiting. This is where the true variability of the system comes into play and this decides whether we survive or not. If we are lucky and picked a day for trash compactor entry when there is little trash then we have lots of headroom. If we are unlucky and picked a day after lots of cleaning up and tossing of trash, headroom may become a tight fit leading to a tight squeeze.

And this is where the calculus part comes in, the function that specifies the beginning level of trash and how this beginning level affects the algebraic function.

Or have I got this all wrong? — Preceding unsigned comment added by 204.116.60.138 (talk) 18:03, 15 January 2012 (UTC)

I think you have got this wrong.

I don't know what you mean by "algebraic function". Calculus is done using certain types of functions, but these functions are usually called differentiable, not algebraic.

The historical impetus for the development of calculus was basic physics. If x(t) and v(t) are the position and velocity of a ball at time t, then the derivative of x(t) is v(t) and the integral of v(t) is x(t). You might find it easier to learn some basic physics before studying calculus. Ozob (talk) 19:09, 15 January 2012 (UTC)