Talk:Calculus

The history section

Shouldn't the history section have at least some content? I mean, more than the link? --31.45.79.44 (talk) 22:46, 27 February 2012 (UTC)

In the main text we read: "Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. (André Weil: Number theory. An approach through history. From Hammurapi to Legendre. Birkhauser Boston, Inc., Boston, MA, 1984, ISBN 0-8176-4565-9, p. 28.) " I think that this is misleading. Fermat borrowed from Bachet de Meziriac's Latin translation of Diophantus’ Arithmetika the noun adaequalitas and the verb adaequare where they have a meaning in a completely different context, namely solving a special class of arithmetical problems by the method of false position. Fermat, however, uses those words (and the word adaequabitur) in his method of determining maxima and minima of algebraic terms and of computing tangents of conic sections and other algebraic curves. Fermat’s method is purely algebraic. There is no "infinitesimal error term". Fermat uses two different Latin words for "equals": aequabitur and adaequabitur. He uses aequabitur when the equation describes the identity of two constants or is used to determine a solution or represents a universally valid formula. Example:

$x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})$.

And he uses the word adaequabitur ($\doteq$) when the equation describes a relation which is no valid formula. Example (Descartes' folium):

$x^{3}+y^{3}\doteq 3xy$.

Typical examples are:

$(x+e)(b-x-e)=bx-x^{2}-2ex+be-e^{2}$

and

$be\doteq 2ex+e^{2}$.

Equations which describe relations between two variables (x and y, x and e) were unknown to Vieta. So the 21-year-old Fermat felt compelled to introduce a new concept of equality and called it adaequalitas. Later, when he created his analytic geometry he abandoned this special terminology of his younger days. See my paper Barner, Klaus: Fermats <<adaequare>> - und kein Ende? Math. Semesterber. (2011) 58, 13-45, unfortunately written in German. 91.4.83.79 (talk) 17:28, 7 February 2013 (UTC) Klaus Barner (talk) 14:40, 5 February 2013 (UTC)

Very interesting. Is there perhaps a translation of your article online somewhere? Tkuvho (talk) 16:13, 6 February 2013 (UTC)

Unfortunately there is none. The reason is: my German is sophisticated and requires reading between the lines whereas my English ist rather weak. It is no artical about mathematics but about history of mathematics which requires a better command of English than I have. However I feel that I should produce a raw translation and ask a native speaker to improve it. Klaus Barner (talk) 19:09, 10 February 2013 (UTC)

https://www.google.com/imgres?imgurl=http://academic.pgcc.edu/~sjohnson/ancientcurvetext.jpg&imgrefurl=http://academic.pgcc.edu/~sjohnson/casestudy2.html&h=1012&w=1331&sz=324&tbnid=Wx15eqipFgydwM:&tbnh=90&tbnw=118&zoom=1&usg=__rwdu-Js_exE4O78S94CeAopgX1Q=&docid=H8UYTm3StwLcWM&sa=X&ei=yS5CUp0HrrbgA6TwgYAG&ved=0CDwQ9QEwAw

A text found at Saquara has a picture of a curve and under the curve the dimensions given in fingers to the right of the circle as reconstructed to the right. It was presumed the horizontal spacing was based on a royal cubit but the results if based on an ordinary cubit are different. It appears the value for PI being used is 3 '8 '64 '1024. which at 3.141601563 is slightly better than the Rhind value. The circumference of the circle is 1200 fingers and the diameter of the circle is 191 x 2 = 382 3 '8 '64 '1024 x 382 ~= 1200.0 The side of the square is 12 royal cubits and its area is 434 square feet. The area of the circle is 191^2 x 3.141601563. The algorithm suggests working with coordinates and numerical analysis to define a curve.

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
• 3
• 3 + 1/2y^3 is 3 '8, = 3.125
• 3 + 1/2y^3 + 1/2y^6 is 3 '8 '64,= 3.140625
• 3 + 1/2y^3 + 1/2y^6 + 1/2y^10 is 3 '8 '64 '1024 = 3.141601563

For purposes of comparison(3 '7 = 3.142857143)

From a published reference "Ancient Egyptian Construction and Architecture" Here it is as a Google Book. "Ancient Egyptian Construction and Architecture" See The referenced graphic Architects Curve What you see there was already referenced on the page as an artifact from Saqarra, but without the graphic. Its Fig. 53 p. 52. Figure 54 p. 53 is a scale drawing of it. The Graphic is in Wikipedia commons used in two other articles which have the text I copied to the talk page. "The algorithm suggests working with coordinates and numerical analysis to define a curve". Its similar to adequabiture and an exampl≤e of how Pascals triangle may be used to generate PI with a series formed by the third diagonal; 1,3,6,10.≤″24.93.139.28 (talk) 21:43, 2 June 2018 (UTC) https://upload.wikimedia.org/wikipedia/en/f/f3/Egyptian_circle.png 24.93.139.28 (talk) 21:53, 2 June 2018 (UTC)

Just for the records:

adequabitur is the verb form 3rd sg fut ind pass of the verb adequare. The construction of a noun the adequabiture is a mystification.

Purgy (talk) 07:01, 3 June 2018 (UTC)

In the intro to this section "Fermat uses two different Latin words for equals". I'm expecting he means not "equals" necessarily, but adequately equal or something to that effect. I couldn't find an etymology that goes back before "adequate" from early 17th century Latin where the noun means "equal" and the "verb made equal to". 24.93.139.28 (talk) 08:16, 6 June 2018 (UTC)

Balance and accuracy concerning controversy concerning origins?

Article says, "When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit." but I understand from the Wikipedia article on the controversy itself (to which this should link) that the controversy evolved many years after Leibniz published his results. Also, the slant of the story itself, as it is told here, seems to favor one side (Newton). — Preceding unsigned comment added by Rentstrike (talk • contribs) 18:25, 28 November 2012 (UTC)

Capitalization of Calculus

It is my understanding that the formal name for calculus is "The Calculus." While the "The" is generally dropped these days, isn't it still appropriate to capitalize the word Calculus?

"A calculus is a way of calculating, so mathematicians sometimes talk about the 'calculus of logic', the 'calculus of probability', and so on. But all are agreed there is really only one Calculus, pure and simple, and this is spelled with a capital C" (emphasis mine) (Crilly, Tony (2007). 50 Mathematical Ideas You Really Need to Know. London: Quercus Publishing Plc. p. 76, 208. ISBN 1-84724-147-6.)

Interestingly, the index of the book does not capitalize the word.

It would seem seems that we should at least mention the issue of capitalization in the article.

Billiam1185 (talk) 01:02, 4 March 2013 (UTC)

• Other common forms of calculus: There has been a general academic trend (20 years?) to not capitalize concepts, unless specifically discussing the "Ideals" of Plato ("Truth" or "Beauty"), and it has led to some confusion, such as any old "categorical imperative" versus "The Categorical Imperative" of Immanuel Kant. For types of calculus specializations, there is "differential calculus" versus "integral calculus" versus "tensor calculus" (similar to a type of algebra) verus "infinitesimal calculus". So, those are reasons to avoid capital "C" in "calculus". Does that make sense? -Wikid77 (talk) 20:01, 17 April 2013 (UTC)

Why is Leonhard Euler not mentioned ?

To be clear, most who add to the mathematical pages have probable forgotten more than I know about maths (I'm British, so I refuse to call it math). However it couldn't escape my notice that when Google put Leonhard Euler up in a Google doodle and specifically mentioned his historical significance to maths and his important work on infinitesimal calculus, that there is no mention of him whatsoever on the infinitesimal calculus page. Is Google incorrect in its highlighting the importance of Euler? Or is his work on infinitesimal calculus not as important as made out on his Wikipedia page? Either one or the other needs correction?

This is definitely an oversight. Euler should be mentioned both on calculus and infinitesimal calculus. Would you like to contribute a comment there? Tkuvho (talk) 15:05, 20 May 2013 (UTC)

There should be more technical information

Perhaps more examples on this matter. — Preceding unsigned comment added by Pedrovalle (talk • contribs) 13:11, 20 May 2013 (UTC)

The introduction should include when and who.

I think there should be something in the introduction about when it was invented and (gulp) who invented it. At the risk of being beaten over the head by all the history revisionists and refactorers out there, I think that should be the 17th century, Newton, and Leibniz. --Chetvorno^TALK 00:43, 27 July 2013 (UTC)

Quote hanging on Newton's edge

The Neumann quote box is hanging on Newton's image edge. Can someone fix it? Tried to, but got rolled-back. Formatting problems. --J. D. Redding 00:11, 28 July 2013 (UTC)

Really?

Come on, people, let's hold the line for general references. We don't have to give in to the inline-cite extremists, not here. For most aspects of the topic, all our refs are going to say the same thing, probably in almost the same words. Save the inline cites for the stuff that's a little particular, and don't make the reader work through a forest of little blue numbers. --Trovatore (talk) 19:14, 4 December 2013 (UTC)

My take on this... someone comes along and makes some subtle changes. Without inline refs it tends to be pretty hard—if not impossible—to check the sources, at least for me, having not significantly contributed to the article in the past. Inline refs make this much easier. - DVdm (talk) 19:22, 4 December 2013 (UTC)

I resist giving in to citation extremists on non-controversial facts, but having some citations are still useful. For instance, the large Principles section of this article has no citations. If a curious student wanted to learn more about principles of calculus, there are zero pointers to a good source or two on this. It's a flaw. I think the more relaxed citation guidelines in WP:SCICITE would be appropriate here: about one general ref per paragraph. --Mark viking (talk) 20:05, 4 December 2013 (UTC)

I am not a citation extremist. But this article doesn't even meet a one general reference per SECTION standard, much less one general ref per paragraph. Stigmatella aurantiaca (talk) 22:48, 4 December 2013 (UTC)

Orphaned references in Calculus

I check pages listed in Category:Pages with incorrect ref formatting to try to fix reference errors. One of the things I do is look for content for orphaned references in wikilinked articles. I have found content for some of Calculus's orphans, the problem is that I found more than one version. I can't determine which (if any) is correct for this article, so I am asking for a sentient editor to look it over and copy the correct ref content into this article.

Reference named "almeida":

• From Madhava of Sangamagrama: D F Almeida, J K John and A Zadorozhnyy (2001). "Keralese mathematics: its possible transmission to Europe and the consequential educational implications". Journal of Natural Geometry. 20 (1): 77–104.
• From Indian mathematics: Almeida, D. F.; John, J. K.; Zadorozhnyy, A. (2001), "Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications", Journal of Natural Geometry, 20: 77–104.
• From Kerala school of astronomy and mathematics: Almeida, D. F.; John, J. K.; Zadorozhnyy, A. (2001). "Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications". Journal of Natural Geometry. 20: 77–104.

I apologize if any of the above are effectively identical; I am just a simple computer program, so I can't determine whether minor differences are significant or not. AnomieBOT⚡ 21:02, 11 January 2014 (UTC)

Merger proposal

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

---

Request received to merge Infinitesimal calculus into Calculus. User:Unsigned request. Reason= unknown. Please discuss here. GenQuest ^{"Talk to Me"} 00:16, 30 March 2014 (UTC)

• Support, with qualifications. I think the main use of the term infinitesimal calculus is simply to mean the calculus; that is, the differential and integral calculus, as opposed to, say, the propositional calculus. So I think infinitesimal calculus should ultimately redirect to calculus. However, whether the content should be merged is a different question. Possibly the content should instead be moved to some title such as infinitesimal methods in the calculus, or else merged into nonstandard analysis, before pointing the redirect at calculus. --Trovatore (talk) 00:35, 30 March 2014 (UTC)
  • Update I wrote the above without really looking at the current content of infinitesimal calculus. As it stands, there's actually not that much about infinitesimal methods there, so I'm not sure how much there is to start another article with, or to merge to nonstandard analysis. Still, in principle, I stand by my remarks — for example, it could be (I haven't checked) that the article used to be more about infinitesimal methods, and in that case, that previous content could be used in the way I described. --Trovatore (talk) 00:48, 30 March 2014 (UTC)
• Support. There is not enough material specific to the use of infinitesimals in calculus to warrant an entire article on this topic. All the material presently in the infinitesimal calculus article should be in either the main calculus article or in some closely related article (such as on history, on derivatives or integrals specifically, on non-standard analysis, and so on). Ozob (talk) 02:01, 30 March 2014 (UTC)
  • Comment Even if there is enough material for such an article, I don't think infinitesimal calculus is the right name for it. Sorry to be picky about it when we're on the same "side", but I really think the point to stay focused on is, where should the search term infinitesimal calculus point? And in accordance with the "common name" principle, I think that term is more used for the integral and differential calculus (regardless of foundations) than it is for the use of infinitesimals in the foundations of calculus. --Trovatore (talk) 03:45, 30 March 2014 (UTC)
    • I think infinitesimal calculus should point to calculus. At this point I think that's what the term refers to; saying "infinitesimal calculus" distinguishes differential and integral calculus (considered together) from, say, propositional calculus. To me it also carries a hint of infinitesimal foundations; maybe they're Newtonian or Leibnizian instead of non-standard analysis, but regardless the term itself suggests that infinitesimals make an appearance in the theory somehow. Ozob (talk) 06:36, 30 March 2014 (UTC)
• Support. Theo (Talk) 10:00, 1 April 2014 (UTC)
• Support As far as I know as a math student with no background in history of math, calculus is essentially a shorthand for infinitesimal calculus. -- Taku (talk) 17:20, 1 April 2014 (UTC)
• Support I think infinitesimal calculus should be merged into calculus, with the redirect also pointing to calculus as the most common usage of the term. The infinitesimal calculus article is mostly redundant with calculus article, except for the "Non-standard calculus" and "Smooth infinitesimal analysis" sections--those could be usefully merged into the Calculus#Limits and infinitesimals section, which doesn't even mention the Non-standard calculus article. --Mark viking (talk) 17:38, 1 April 2014 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Madhava sources

I have undone this edit. It is based on 4 sources, all of which i.m.o. unreliable:

• source [1]: "Categories:Types, School Work, Essays & Theses"
• source [2]: wp:primary self published website by Luke Mastin. Who is Luke Mastin?
• source [3] gives the view of one author, George Gheverghese Joseph, "on a mission to reclaim :India’s pride of place in the world of mathematics." Seems not sufficiently notable to be mentioned here.
• source [4] is wp:circular: "Based on materials from: Wikipedia."

Comments? - DVdm (talk) 10:47, 22 February 2015 (UTC)

Comments:

Regarding the third source (the Telegraph article), please read beyond the first sentence. George Gheverghese Joseph is an "emeritus professor at Manchester University in the UK" who was "awarded a Royal Society Visiting Fellowship". You can read more about him and his arguments here. Why is Joseph "not sufficiently notable"? Exactly what are your criteria for notability? — Preceding unsigned comment added by 71.106.229.93 (talk) 15:45, 22 February 2015 (UTC)

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

Are there scholarly secondary sources who cite Joseph? See wp:secondary sources. - DVdm (talk) 15:49, 22 February 2015 (UTC)

Princeton University Press published Joseph's book "The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition)" wherein he makes the arguments cited, and two positive reviews from New Scientist and Times Literary Supplement appear on the PUP website: http://press.princeton.edu/titles/9308.html His scholarly credentials are bona fide: http://www.sagepub.com/authorDetails.nav?contribId=649701 ~~~~ — Preceding unsigned comment added by 71.106.229.93 (talk • contribs) 16:02, 22 February 2015 (UTC)

Just the four tildes, without the nowiki-tags please (~~~~). Thanks.

Yes, but that is a wp:primary source. We need other scholar sources who cite the book. Such sources would establish notability. - DVdm (talk) 16:19, 22 February 2015 (UTC)

From the Wikipedia article on Secondary source: "In scholarship, a secondary source is a document or recording that relates or discusses information originally presented elsewhere. A secondary source contrasts with a primary source, which is an original source of the information being discussed; a primary source can be a person with direct knowledge of a situation, or a document created by such a person. However, as discussed in detail in the section below on classification, deciding how to classify a source is not always an obvious decision. Secondary sources involve generalization, analysis, synthesis, interpretation, or evaluation of the original information. Primary and secondary are relative terms, and some sources may be classified as primary or secondary, depending on how it is used." Princeton University Press and the reviewers at New Scientist and Times Literary Supplement would seem to agree that Joseph's scholarly credentials are unassailable; and his book published in 2010 contains "generalization, analysis, synthesis, interpretation, or evaluation of the original information" pertaining to events of the fourteenth and fifteenth centuries.

Has the citing by other scholarly sources been verified for other works cited on this page, such as those of Katz and Zill? Have their works been published by Ivy League universities? Where might we find Wikipedia's rules for what sources are acceptable and which ones are not? Too many articles on Wikipedia are treated as the private fiefdoms of their self-appointed guardians of correctness, and I've seen excellent scholarship repeatedly censored by such individuals who elsewhere in the same given article fail to abide by the "rules" they invent to set the bar far higher for facts and/or interpretations that they dislike. 71.106.229.93 (talk) 16:55, 22 February 2015 (UTC)

Please indent your messages along wp:talk page formatting guidelines. Thanks.

Reviews of a book don't really count as secondary sources for particular statements made in that book. The statements you like to add to the article should be found in other articles or books—preferably not newpapers—, referring to the primary source in question. If other poor or unacceptable sourcing has taken place here, perhaps remedial action should be taken, but it certainly does not warrant addition of more such content—see wp:otherstuffexists. - DVdm (talk) 17:52, 22 February 2015 (UTC)

I asked you specific questions, viz.: (i) Exactly what are your criteria for notability? (ii) Has the citing by other scholarly sources been verified for other works cited on this page, such as those of Katz and Zill? (iii) Have their works been published by Ivy League universities (as Joseph's work has)? (iv) Where might we find Wikipedia's rules for what sources are acceptable and which ones are not? 71.106.229.93 (talk) 18:12, 22 February 2015 (UTC)

George Gheverghese Joseph's book makes arguments in the general direction you are interested in, namely examining Indian contribution to the development of the calculus, but I am not convinced by your claim that Joseph "makes the [specific] arguments cited" in your proposed revision. If you can source these and replace some of the dubious references you provided by page numbers in Joseph this may strengthen your case for inclusion. Tkuvho (talk) 08:42, 23 February 2015 (UTC)

Fair enough. My interest is not geographically specific but rather the acknowledgement of contributions by great minds. Joseph's book covers "the deep influence that the Egyptians and Babylonians had on the Greeks, the Arabs' major creative contributions, and the astounding range of successes of the great civilizations of India and China". I will look further and propose material in the future.71.106.229.93 (talk) 16:00, 24 February 2015 (UTC)

Note that the reliability of Joseph is put into question in a detailed review by Jens Høyrup here. Tkuvho (talk) 08:49, 23 February 2015 (UTC)

@71.106.229.93: Re (i) and (iv): see wp:verifiability and wp:consensus. Re (ii) and (iii) see wp:otherstuffexists. - DVdm (talk) 11:36, 23 February 2015 (UTC)

Your link contains basic bibliographic information on Joseph's book and nowhere puts into question his reliability. I am well aware of Wikipedia's articles on sources, etc. and they do not answer the questions that I asked. Please reply in complete sentences: (i) Exactly what are your criteria for notability? (ii) Has the citing by other scholarly sources been verified for other works cited on this page, such as those of Katz and Zill? (iii) Have their works been published by Ivy League universities (as Joseph's work has)? (iv) Where might we find Wikipedia's rules for what sources are acceptable and which ones are not? 71.106.229.93 (talk) 16:00, 24 February 2015 (UTC)

Again, see wp:verifiability and wp:consensus. In a complete sentence: whether the content is notable or not, and whether the sources are reliable or not, is ultimately decided here by consensus on the article talk page. - DVdm (talk) 16:37, 24 February 2015 (UTC)

User 71.106.229.93, your claim that the link to the mathscinet review contains only "basic bibliographic information on Joseph's book" is in error. The review lists numerous inaccuracies, distortions, and tendentiousness in Joseph's book. Tkuvho (talk) 18:32, 24 February 2015 (UTC)

You have linked to a page that only shows basic bibliographic information on Joseph's book and that requires special access including a 'MathSciNet license'. Is this review freely available elsewhere? 71.106.229.93 (talk) 15:26, 25 February 2015 (UTC)

Ancient History Section

I've removed this from the article's Ancient History section since it should be dealt with here.

{{clarify|post-text=(Why mention this if not relevant? <-Perhaps it is a first/early instance of volume and area calculation, which the paragraph suggests would be the evolutionary antecedent of modern calculus, thus introducing a historical and logical progression, which would seem to be the purpose of this section.)|date=August 2015}}

Should something be added to the article in way of clarifying this point? Bill Cherowitzo (talk) 04:37, 30 August 2015 (UTC)

dx + Δx or just dx?

A question. Mesmerate (talk) 04:32, 22 November 2015 (UTC)

It is dx+Δx. Mesmerate (talk) 04:36, 22 November 2015 (UTC)

The context for that part of the article is the historical development of calculus through infinitesimals. At that time, dx was used to refer to an infinitesimal change in x. Such a change in x is too small to be represented by a real number. The replacement of infinitesimals by limits of real-valued increments Δx happened over a century later. The revival of infinitesimals by Robinson happened several centuries later, and his concept of infinitesimals was philosophically different from those used by the pioneers of calculus. Ozob (talk) 04:49, 22 November 2015 (UTC)

Sir, the idea is you're taking the limit as it gets really close to zero, and as it gets closer, that is what it gets closer and closer to. As it gets closer and closer, infinitely, it can be considered infinitesimal. Mesmerate (talk) 04:55, 22 November 2015 (UTC)

You are using a post-Weierstrass interpretation of infinitesimals as limits. That is a historical anachronism. It is not at all how Newton and Leibniz saw what they were doing. Ozob (talk) 04:58, 22 November 2015 (UTC)

Please tell me HOW they saw what they were doing? Mesmerate (talk) 05:05, 22 November 2015 (UTC)

The change in x, or Δx, can be defined to include infintely small or infinitely large numbers in its list of possible numbers. It commonly is, and that is partly why i think it is a just fine canidate for being an infinitesimal. Mesmerate (talk) 05:08, 22 November 2015 (UTC)

So, you think infinitesimals cannot be defined using limits? Mesmerate (talk) 05:11, 22 November 2015 (UTC)

For the inventors of calculus, dx (in Leibniz's notation) represented a change in x that was positive, but smaller than any number. This sounds paradoxical because we want to know what kind of thing dx is (it is by definition not a real number). It was just as paradoxical at the time it was introduced; see the article for some history of objections to calculus. Yet calculus seemed to work, so people went about using it. There were still open foundational questions even as calculus developed into a practical discipline. These foundational questions eventually spurred the work of Cauchy, Weierstrass, Riemann, Lebesgue, and so on, who all came much later than Newton and Leibniz and who gave us the definitions of limits, derivatives, and integrals that we use today. It would be historically incorrect to say that the discoverers of calculus understood it in the same way we do. They didn't. (It's also worth noting that the idea that all of mathematics should be rigorously derived from a small set of axioms did not develop until the 19th century, and it really only triumphed with the rise of Bourbaki.) Ozob (talk) 05:17, 22 November 2015 (UTC)

We are talking about modern calculus, where Δx is allowed to approach zero, and where dx is defined as Δx as dx. Mesmerate (talk) 05:25, 22 November 2015 (UTC)

Sorry. Mesmerate (talk) 05:26, 22 November 2015 (UTC)

• where dx is defined as Δx Mesmerate (talk) 05:26, 22 November 2015 (UTC)

The part of the article you edited is specifically about the history of calculus. Modern foundations are discussed in the following paragraph (and in other appropriate articles). Ozob (talk) 05:31, 22 November 2015 (UTC)

No it isn't, check. Mesmerate (talk) 05:35, 22 November 2015 (UTC)

It's under "principals". Mesmerate (talk) 05:37, 22 November 2015 (UTC)

@Mesmerate: dx and Δx as used in this article are quite different concepts. Letting Δx → 0 is most definitely not the same as simply having an infinitesimal quantity dx. See the hyperreal numbers for a rigorous discussion of the distinction. Any particular Δx is a finite real number. The operation of letting it approach zero does not make it the same as an infinitesimal, which on its own, without taking a limit, is smaller than any positive real number, yet nonzero. Phrases like "As it gets closer and closer, infinitely, it can be considered infinitesimal" are not mathematically rigorous and do not suffice.--Jasper Deng (talk) 06:33, 22 November 2015 (UTC)

I still argue otherwise, I argue in modern calculus dx itself can be considered as the change in x as it approaches 0. Mesmerate (talk) 12:37, 22 November 2015 (UTC)

And I have already clarified that we are talking about modern calculus, "principals" is not a sub heading of "history". Mesmerate (talk) 18:43, 22 November 2015 (UTC)

The meaning of dx depends on the context and the foundations. In modern differentiation, it is just a notational convenience and has no intrinsic meaning. In modern integration, it is a notation for the variable of integration or for a measure. As a differential form, dx stands for the exterior derivative of the identity function x → x. In Robinson's theory of infinitesimals, dx is a hyperreal number.

What I think you are proposing is that dx is a notation for "the limit of Δx as Δx → 0." But this is not what you want: lim_{Δx → 0} Δx = 0 for trivial reasons, and you do not want dx = 0. To derive anything useful, you must take the limit of some function of Δx. For example, the derivative is the result of taking the limit as Δx → 0 of the difference quotient (f(x + Δx) - f(x)) / Δx. In the difference quotient, Δx is not infinitesimal; it is just a real number. Taking the limit detects infinitesimal behavior but does not actually require the use of infinitesimals.

If you'd like to learn more about these topics, I suggest Rudin, Principles of Mathematical Analysis. Ozob (talk) 21:21, 22 November 2015 (UTC)

Sorry, nevermind. I read in a text book that the (dy/dx) could be seperated by multyplying by "dy" and as such, it had a page where it said "Who are we to throw around these numbers?" and defined dx as the change in x. I also understand that something getting infinitely cl in all the mess. I admitzero by itself. I still think that delta x equals zero, i just added a small note and got tangled up in it all. I still think something approaching zero still works like an infinitesimal, except for that one case in which it just goes to zero. Sorry i'm a little put of tune in calculus, plus, please tell me if i am right in "It acts like an infinitesimal, except for that one case.". Mesmerate (talk) 23:44, 22 November 2015 (UTC)

Sorry, i mean you can seperate the (dy/dx) by multyplying by dx, not dy. Mesmerate (talk) 23:46, 22 November 2015 (UTC)

Sorry my sentence 2 sentences ago was messed up. Mesmerate (talk) 23:47, 22 November 2015 (UTC)

I meant to say i got caught in it all, and in all cases except for that case "approaching" acts like an infinitesimal. Mesmerate (talk) 23:48, 22 November 2015 (UTC)

Sorry, by "two sentences ago" i meant two comments ago. Mesmerate (talk) 23:50, 22 November 2015 (UTC)

Also, to clarify, i admit i was wrong. I will stop editing that part of the page, and i am sorry for disrupting wikipedia. Mesmerate (talk) 23:53, 22 November 2015 (UTC)

The idea i had in mind was "dx=the change in x", "the change in x can act infinitesimal in most cases", "the change in x is fine, smd should be mentioned." Mesmerate (talk) 23:57, 22 November 2015 (UTC)

• and Mesmerate (talk) 23:57, 22 November 2015 (UTC)

Still, i believe that dx in integrals stands for "the change in x" in integrals,as the definition of the integral uses it alot. infact, it used "the change in x" an infinite number of times. and in both case, dx is short hand for "the limit of this using delta x" Mesmerate (talk) 00:00, 23 November 2015 (UTC)

And as such i would like to start a new discussion. I would like to discuss the idea of mentioning that the meaning of "dx" is different across modern and old calculus. Mesmerate (talk) 00:01, 23 November 2015 (UTC)

Explanation of the change of the meaning to the notation "dx" from meaning infintesimal, to meaning shorthand for the change in x, whether or not x approaches 0.

Read description. Mesmerate (talk) 00:06, 23 November 2015 (UTC)

I believe it is important enough to be mentioned. Is there any consensus? Mesmerate (talk) 00:07, 23 November 2015 (UTC)

It's explained in Leibniz notation. I'd support including that but not more. Also, by the way, please indent your comments (your next comment below should have two colons (::) appended before it, just like this comment has one before it).--Jasper Deng (talk) 04:13, 23 November 2015 (UTC)

The title of this section is misleading. Whether or not x approaches zero has nothing to do with dx and never did. If, for example, we take the limit as x approaches 7, then dx is still the change in x. Rick Norwood (talk) 12:23, 23 November 2015 (UTC)

On one hand, that entire section is historical - there are no infinitesimals in the real line. I would remove both the δx and the dx from that sentence, which is really only trying to say what an infinitesimal is. The symbol dx is not an infinitesimal in modern treatments of calculus. — Carl (CBM · talk) 12:37, 23 November 2015 (UTC)

It's in "principals", not "history". If it is historical, that's a mistake. Mesmerate (talk) 22:53, 24 November 2015 (UTC)

So we should remove the mention of "dx" meaning infinitesimal all together and only say it's just the change in x? I'm fine with that. Mesmerate (talk) 22:56, 24 November 2015 (UTC)

The h in the limit definition of a differential

What does h stand for in:

$f'=m={\frac {f(a+h)-f(a)}{(a+h)-a}}={\frac {f(a+h)-f(a)}{h}}...?$

Why don't we explain it in Calculus#Differential_calculus? Warmest Regards, :)—_thecurran Speak your mind ^{my past} 13:54, 6 December 2015 (UTC)

It is explained in the article: "If h is a number close to zero..." - DVdm (talk) 15:03, 6 December 2015 (UTC)

I think thecurran means "why do we call it h, particularly, and not some other letter?". I don't know the answer to that. If it turns out there's a known and sourceable explanation, do I think we should put it in the article? Maybe. I'd probably have to hear the explanation first. --Trovatore (talk) 20:40, 6 December 2015 (UTC)

When a fellow student asked why h was taken for Planck's constant, our prof said, because all other letters were already taken by that time. Perhaps it's the same in math: a, b, c, d were reserved for constants, e for the exponentials, f and g for functions (and gunctions ), and h was free. I have never seen a comment about it. - DVdm (talk) 21:10, 6 December 2015 (UTC)

The letter 'h' represents a simple exhalation of air, lesser than any of the other vocalizations that shape vowels or define consonants. It is used for Planck's constant because it's the smallest possible articulation of a meaningful utterance. Willondon (talk) 22:36, 6 December 2015 (UTC)

A glottal stop doesn't involve any air at all, but nobody uses the symbol ʔ for a variable. There may be other reasons for that, though. Ozob (talk) 23:52, 6 December 2015 (UTC)

The h used to be a delta x, but at some point someone decided that a two letter symbol might confuse students, and replaced it with h. I think Thomas was the first, at least the first I saw. I think the explanation "h was free" is as good as any. Rick Norwood (talk) 12:24, 7 December 2015 (UTC)

I'm not quite sure what you mean by "used to". This article used to have a Δx? Or authors used to use it? I'm not going to bother to check the history to see about the article. If you mean authors, I imagine that there are authors who still do — as far as I'm aware, there was no meeting held to decide what to do.

My personal take on the question is that using Δx is a suboptimal choice (and to keep this on topic, we should not use Δx in the article) because it invites learners to read too much into it. Δx can be read "the change in x", and in the indicated formula, it's true that it's the change in x. But that isn't necessary for understanding the formula; the formula itself does not ascribe any intensionality to the Δ symbol, or at least no such intensional reading is necessary. Therefore it's better to use a meaningless variable name like h, precisely because it's meaningless, to avoid giving the impression that the meaning is of the essence in the formula. --Trovatore (talk) 22:24, 7 December 2015 (UTC)

Babylon Tablets Comment

While adding the note for babylonian astronomer clay tablets, some edits were reverted (and perhaps the reversion also reverted.) Assuming good faith, it is wise to allow for 10 minutes from initial edits before removing content and asking for more citations. The citations requested were being adding during the event. I for one, appreciate the watchful eyes keeping track of article improvement. Kyle^(talk) 20:03, 29 January 2016 (UTC)

I still think that these two sources [5] and [6] suffer from wp:recentism. These are non-mathematicians reporting about a non-mathematician's recent finding. A proper wp:secondary source would be one from a scholar i.e. a mathematician in a peer reviewed article. After all, this is an math article, not an archeology one. - DVdm (talk) 21:51, 29 January 2016 (UTC)

Even though I'm the person who entered the citation to the New York Times article, I don't have strong feelings about this recent development needing to appear in Wikipedia. I will go with the flow on this one. Isambard Kingdom (talk) 22:00, 29 January 2016 (UTC)

I agree with DVdm. These references are summaries of a single article containing one person's interpretation of some apparently related clay tablets. One needs only recall the checkered history of the interpretation of Plimpton 322 to see that this interpretation may not hold up under the scrutiny of several specialists. When that vetting is done the result will be in reliable secondary sources and we may freely cite these, but until then it should be treated as a viable theory and not as an established fact. Bill Cherowitzo (talk) 05:18, 30 January 2016 (UTC)

Ok, there seems to be no consensus to keep this addition, so I have removed it again ([7]). - DVdm (talk) 09:03, 30 January 2016 (UTC)

Calculus... "is the mathematical study of change"??

That sounds cute, but it's not correct, or at best it's only correct in some instances. While dx generally represents change in x, that doesn't mean calculus is "the study of change". O-m-g. Someone please rework that intro. 76.208.70.118 (talk) 02:04, 20 August 2016 (UTC)

That someone could be you! Go for it. WP:BOLD! Ozob (talk) 02:53, 20 August 2016 (UTC)

I apologize for abusing the reverting of an edit, just to continue the discussion.
May I submit my personal thoughts on this:

• First of all, I see calculus not as the study, but rather (as already mentioned in the article!) as a methodology to exploit the results of the respective studies, which belong to, say, (functional) analysis. I would not say this about geometry and algebra in a similar way. However, if pressed, I would say that water is wet as liquid nitrogen is, but not as wet as mercury is. I also admit that I know about linguistic constructs like pars pro toto, and related ones, but prefer to be highly discriminating in an encyclopedic context.
• In extremis, Calculus reminds me more of the Ricci calculus, which is per se not a study of multilinear algebra, but a viable path to deal with tensor algebra and analysis. Certainly, calculus contains more algorithmic parts, but is similarly away from a study (at least in a sense which is is not abused like for sociological "studies") of the underlying structures. I think calculus takes its importance from its outstanding applicability, already in quite early math education.
• Calculus also does not really deeply care on which bases it is applied. It works on the reals, on the extended reals with strictly defined infinitesimals, as well as in brownie-enriched physicists' world with just very small, very useful objects, also coined infinitesimal, and even on fairly discrete domains.
• I suggest to talk about local rates of change in favour of change, since the limit of a ratio of local changes is important in differential calculus, and not the change itself. The slope of a curve is exactly such a rate (or a ratio). I admit that the notion of relative rate of change is one more of a ratio and might frighten unwary beginners.
• Integral calculus then, is concerned with summing up these local rates of change over some given domain, directly relating the sum to the measured "amount" under a curve, or, more general, pertaining in the given domain.
• I think, the introduction of the antiderivative has to stay that vague and sloppy as it ever has been in my weak efforts of recon of education in calculus.
• To me, calculus is an important item in a syllabus, and it refers to successfully hammering without knowing details about what a hammer is.

Knowing too much about small details may be adverse to your mental health - see G. Cantor. Purgy (talk) 09:27, 28 August 2016 (UTC)

I've long been concerned that the phrase "rate of change" is more of a slogan than a rigorous description of anything. I'm not sure how I would define it, even in a nonrigorous and philosophical way, without implicitly making reference to derivatives. So I would be wary of making this phrase too prominent.

Also, "summing of local rates of change" translates to $\int f'(x)\,dx$, not to $\int f(x)\,dx$. Ozob (talk) 13:48, 28 August 2016 (UTC)

See also Calculus I

After having worked on the article Calculus I, user Samantha9798 added an entry in the See also section: [8]. It was immediately removed by user Mean as custard, without an edit summary: [9]. I restored it because I think it is a valid entry per WP:SEEALSO: [10]. It was removed again without an edit summary: [11]. Samantha9798 added it again: [12].

As it would be silly to start an edit-war over this, can we, in the spirit of WP:BRD, please have a least a minimal discussion here? - DVdm (talk) 11:53, 21 November 2016 (UTC)

• It is fine in its new location under "Other related topics". I only objected to it being right at the beginning of the "See also" section with no indication as to what the article is about. . . Mean as custard (talk) 12:55, 21 November 2016 (UTC)

Ok, thx. - DVdm (talk) 15:30, 21 November 2016 (UTC)

Sorry, I removed if before seeing this thread. Self reverted pending outcome of AFD. Meters (talk) 18:51, 21 November 2016 (UTC)

No problem. Seems unlikely to survive, so... see you later . - DVdm (talk) 19:13, 21 November 2016 (UTC)

Comparison to geometry and algebra in the lead

Recently, an editor removed the phrase, " in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations," and it was replaced, evidently all in good faith. I think this should be removed. I think an article about calculus should define and explain calculus in absolute terms, without comparing it to similar concepts at the same level, except when necessary, especially in the lead. Also, since calculus, geometry, and algebra are very general, it is arguable whether the existing definitions are general enough. — Anita5192 (talk) 17:02, 18 December 2016 (UTC)

• Delete: I agree to the above intentions to delete the comparision of "study of change", "study of shape", and "study of operations ..." under their respective names "calculus", "geometry", and "algebra" in the lede of an article about calculus. As said, these maybe comparable on a very specific level and in very special hindsight, only. Imho, the article does not gain anything by referring to the names of other mathematical topics. -Purgy (talk) 17:31, 18 December 2016 (UTC)

In the light of the overwhelming votes for keep, I want to point out that a statement ('A = B' like 'C = D') .and. ('A = B' like 'E = F') would not transport much information in the math sense. In my opinion, the hope that these comparisons would induce some illustration for the notion of calculus is futile. I think that an equal value would be achieved by a statement like "calculus is the study of change in the same way as other subjects in math are the study of other topics". Putting a warning next to a statement that it is vague, by adding additional vagueness, is to me alien to math.

As an aside, "Differential Geometry" flashed through my consciousness.

In case my above argumentation is not considered convincing, I want to mention my reservation to explicitly calling the calculus-change a "continuous" one, and putting algebra in the "discrete" environment, while agreeing to the apposition being too long and too technical.

How about

Calculus encompasses the introductory settings for the mathematical study of change, mostly demonstrated along functions of one real variable. replaced!

Just to throw in some alternative. -Purgy (talk) 11:23, 19 December 2016 (UTC)

Vague is fine with me, and it shows it isn't some precise definition. Your longer form just above is long and rambly and would just confuse. The extra words make it look like something more is meant, verbosity is not a reliable defense against misinterpretation. Dmcq (talk) 14:37, 19 December 2016 (UTC)

I really tried to mean more, and I try it again, shorter, less verbose, and even including continuity:

Calculus is an introduction to the mathematical study of change, mostly along continuous functions.

This is 16 words shorter than the current version, it offers new information: (1) the introductory level of calculus within the fullblown math setting, (2) the support of the studied change, (3) refers to D. Lazard's continuity, and is not loaden with the vague comparisions, which I would call rambly, because -as shown above- they contribute nothing to make the position of calculus within math topics more graspable. I cannot imagine that this sentence of 14 words could be confusing.

BTW, omitting the empty comparisions also saves the lede from concealing curved shapes and from connecting discrete and discontinuous to algebra. -Purgy (talk) 17:22, 19 December 2016 (UTC)

It isn't an introduction. It is a full topic in itself. I can't see the point of the 'mostly along'. Personally I'm not altogether happy with 'change' though I can see its point in distinguishing from topology and others here like it. I don't think of them as changing at all, change is a physical phenomenon. I would have thought calculus differed from topology in having a measure assigned though how one phrases that easily is another question. Dmcq (talk) 01:17, 20 December 2016 (UTC)

Would it be considered inappropriate to ask for a refutation of my "emptiness"-claim, or even better, for the content of the empty comparisons?

Would you prefer something along "relative local differences", or perhaps some algebraic approach along the Leibniz rule for the derivatives, instead of the study of change?

Could you imagine to help with improving the "along"?

I certainly agree that integration without referring to a measure is outdated, but OTOH, integration in the realm of calculus is mostly based on an approach via Riemann sums, dumping the continuity aspect at some very early stage, and not involving the Lebesque picture.

For the time being I'm mostly focused on improving the status quo, containing an empty non-description, and not on the future "best lede in town". -Purgy (talk) 14:07, 20 December 2016 (UTC)

I understand you think that calculus is best defined by only saying what is included without saying what it is distinguished from. That is not what others here think. I think saying what something isn't is a good way of outlining the extent of a mathematical topic and placing it in context. Dmcq (talk) 14:31, 20 December 2016 (UTC)

Just correcting your understanding of my intentions: while I do require definitions to positively declare the definiendum, I never intended to define the notion of calculus. It is fine with me too, to rather sketch this in the lede, with, roughly, a central point and a contrasting environment.

However, I see no sensible contrast of the portrayed notion to the notions listed by the "in the same way"-topics. In my perception the current wording boils down to the already mentioned "calculus is the study of change in the same way as other subjects in math are the study of other topics". The suggested epitetha do not improve this lack of manifest meaning, and my hint to "Differential Geometry", setting up even some connection, instead of a contrast, between (multi-variate) calculus and geometry has not been dealt with.

I do not perceive much cooperativeness in argueing about my objections, nor for discussing my proposals, but an overwhelming consensus, based on personal valuation and opinion, to rather insert some, imho, disconnected continuity into the current version, than pondering any change, as initiated by an IP and supported by Anita5192 and me. So for now, no additional suggestions from my side. I just hope, my use of italics does not annoy again. -Purgy (talk) 09:53, 21 December 2016 (UTC)

I thought I had discussed what was proposed directly and to the point. I don't agree with you. That is not the same as being uncooperative. I think the lead of this article should be at about the level where a young teenager who will soon start calculus would be able to read it okay - we should use what is common knowledge at that point and say what is new about this topic and distinguishes it from what they already know. Dmcq (talk) 11:01, 21 December 2016 (UTC)

It must be really embarrassing, what a blockhead I am. Having seen all this already in "numbers", and now here again: none of my 3 explicit questions answered, not even addressed, any concrete effort rejected without reasoning, just rejected on "existential non-qualification", and on "not agreeing". It won't bother you, but it does not bother me either. I'm just into constantly trying to improve Wikipedia, and be it one word for one word.

Claiming a disqualifying property, without giving a reason, is not the kind of reply I consider to show "cooperativeness". BTW, I agree with what you wrote about the level the lede should be in. -Purgy (talk) 11:00, 22 December 2016 (UTC)

• Keep I believe that one of the functions of the lead is to provide the context in which the article is to be viewed. In this case that context is, how does Calculus fit within the scope of mathematics? This can not really be addressed if you confine yourself to the intrinsic features of the subject. Also, I am uncomfortable with the bare phrase, "Calculus is the study of change", as this is too simplistic and stands a good chance of being taken too literally by the probable readers of this article. By placing the phrase in the context of similar phrases, the reader is being warned that the statement is very general and too vague to be taken as a definition. --Bill Cherowitzo (talk) 17:46, 18 December 2016 (UTC)

• Keep for necessary context in the lead, which can be as general as it is now. By the way, one of the best leads in town, i.m.o. - DVdm (talk) 19:35, 18 December 2016 (UTC)
• Keep + comment. IMO, every article, such this one, about an area of mathematics must give indications on what belongs to the area and what does not belongs to it. This cannot be done formally, as it cannot be any consensus on the exact limits of any scientific area. Instead, some indication must be given where calculus ends and where begin the other comparable areas. This is the reason of keeping the disputed sentence. (The end of this post and the resulting discussion are about another question, and have thus been moved in the next section.) D.Lazard (talk) 21:52, 18 December 2016 (UTC)
• Keep Putting it in context of simpler and better-known mathematical areas is a reasonable and useful part of the lede . Meters (talk) 22:05, 18 December 2016 (UTC)

Short description of mathematics areas in the lead

Start of the moved discussion:

However, I find the present formulation too vague for being useful for anybody. IMO, "study of change" should be replaced by "study of continuous change" (here "continuous" may be interpreted in its non-mathematical meaning, but if a reader interprets it in its mathematical meaning there is no problem). "Study of shape" is good. On the other hand, "study of operations ..." is too long and too technical. For algebra, I would prefer "study of discrete properties" or "study of discontinuous change and discrete properties". D.Lazard (talk) 21:52, 18 December 2016 (UTC)

I think of topology as the study of continuity and that discontinuity is fine in calculus though of course if something is continuous it makes life easier. And for a beginner they'd probably assume continuous anyway. And Lie groups are continuous and come under both algebra and topology but not really calculus. Dmcq (talk) 14:23, 19 December 2016 (UTC)

I have not written "calculus is the study of continuity" but "calculus is the study of continuous change". Topology is the study of continuity but not the study of change (at least at elementary level, before the study of homotopy). On the other hand, calculus is essentially the study of derivation and integration; derivation is the measure of the instantaneous rate of change of a continuously changing quantity, and (at calculus level) integration is the summation of a quantity that varies (almost always) continuously. The change of the content of a bag, in which apples are added or removed one after the other, does not belong to calculus. Therefore talking of "change" and not of "continuous change" for characterizing calculus is confusing for non-mathematicians. D.Lazard (talk) 15:04, 19 December 2016 (UTC)

That makes sense okay, yep the continuous is good. Thanks. Dmcq (talk) 01:09, 20 December 2016 (UTC)

End of the moved discussion.

It seems that there is a consensus (between only two editors) to change "study of change" into "study of continuous change" for qualifying calculus, and to keep "study of shape" for geometry. However, I am not fine with the present description of algebra nor with the alternatives that I have proposed in a previous post: many readers would not recognize algebra in such a description. Now, I suggest: Algebra is the study of generalizations of arithmetical operations. This is short, not technical (everybody should know of arithmetical operations), explains clearly that algebra is related to, but distinct from arithmetic and number theory, and, IMO, covers most aspects of algebra. Your opinion? D.Lazard (talk) 12:07, 21 December 2016 (UTC)

I was also unhappy with the short algebra description as the study of operations and think that your suggestion is a vast improvement. However, I am uncertain about the "calculus is the study of continuous change" statement. I have seen "calculus is the study of change" far too often in non-technical writing and introductory calculus texts. I fear that we are trying to change a popular misconception without adequate support in the literature. I don't have this issue with algebra since that topic does not have the same kind of track record (as a subject whose name, at least, is well known to the general public) as calculus does. --Bill Cherowitzo (talk) 19:54, 21 December 2016 (UTC)

May I, please, ask, what your specific reservations (beyond those referring to integration) to "study of change" wrt calculus are, and wherein you see the "popular misconception"? I think that "study of change" wrt differentiation is very appropriate and "better" than a more precise "local relative difference". As far as integration is concerned, I think that "summing up that change (in an interval)" equally well describes the integration, which is generally introduced either as antiderivative (perfectly corresponding to studying the "reversion of change"), or via Riemann sums, obviously summing "changes", weighted by a "duration", just rendering the continuity as contradictive. -Purgy (talk) 10:53, 22 December 2016 (UTC)

While I am fully convinced that beginners are thinking in a smooth environment, taking the "change" itself as contionuous, I do not see any gain adding a technical term, even when speculating on it being non-technically perceived. I do adhere to the thought that discrete changes are used in already in elementary natural sciences: Dirac-delta and Heavyside-step are used when their mathematical foundations are still very far away.

As for the algebra topic, I disagree on the to-day algebraic view on arithmetic being a "generalization". The notion of "inverses" is imho fully alien to elementary pedagogic, and the 4 basic arithmetic opereations with subtraction/division are not really an algebraic concept. May I throw into the debate "abstract mathematical structures" (for algebra), and "trigonometry about angles" (for an alternative topic, known to the intended clientel)?

I do not expect any of my suggestions to be realized, but I'm into trying to help to improve Wikipedia. -Purgy (talk) 10:53, 22 December 2016 (UTC)

I agree that the four arithmetic operations do not belong by themselves to arithmetic. This is why I proposed "generalizations of arithmetical operations". In fact, algebra began when one (in fact Viète) wrote x + y, that is when addition (and the other operations) were applied to symbols which were not numbers. Still now, algebra is mainly devoted to the study of algebraic structures, which, generally, are sets on which are defined some kinds of generalizations of the arithmetical operations. This is clear, as the symbols used for denoting operations in algebraic structures are almost always (except maybe in some "pedagogic" introductions) those of arithmetic operations. When I have suggested my formulation, I had algebraic structures in mind, but this term is too technical to be used here (moreover, it would induce some circular definition).

May I recall you that Wp is not a textbook with a specified audience, an that the intended "clientel" is not restricted to kids at school. D.Lazard (talk) 11:39, 22 December 2016 (UTC)

Doesn't the limit of a sequence properly belong to the study of the calculus? A sequence is neither uncountable nor continuous. — Anita5192 (talk) 17:59, 22 December 2016 (UTC)

You are right, but it is impossible to summarize a whole area in a single phrase. Moreover, one has to remember that "calculus" is, originally, an abbreviation of infinitesimal calculus, and that this means "computing with infinitesimal changes". I believe that "continuous change" is better understood by the layman than "infinitesimal change". Limits of sequences belong presently to calculus, but they have been introduced (for formalizing calculus) three centuries after the "invention" of calculus (see second paragraph of Calculus#Foundations). For these reasons, I think that there is no harm in omitting limits of sequences in a single-phrase definition of calculus. D.Lazard (talk) 18:46, 22 December 2016 (UTC)

@D.Lazard, you certainly do not need to remind me of Wikipedia not being a textbook, and I do not recall having written about "kids in school". I do recall however very well, how I was reminded of the "intended clientel" for some article. Furthermore, I do not see my original claim refuted, that the current view of algebra on arithmetic, disregarding all historical roots (Al Quarizmi(?)), were not a generalisation, but a total revamp of the view as presented in pre-calculus education. -Purgy (talk) 18:32, 22 December 2016 (UTC)

I wrote above about trying to make the lead accessible to young people about to start learning calculus. That is quite different from wanting the article to be like a textbook though. Wikipedia is an encyclopaedia not a coursebook. However we should aim to make the first half worthwhile to all reasonably likely readers. I'm not quite certain what you are saying about algebra. I think you are saying we should present a view of algebra as encountered by young children rather than saying something that is a fair approximation of what the subject is actually about. If the description given would not be accessible to young children about to start calculus I would agree it would present a real problem. However the description given is in simple terms and should be easily comprehensible, if they have a problem with it they can just skip over it or go to the article. Giving a description which is not a reasonably fair approximation would bowdlerize the article. It is a hard job balancing the needs of readers and the requirement to be an encyclopaedia. Doing that well is a mark of a good editor. Dmcq (talk) 22:46, 22 December 2016 (UTC)

* As can be seen from the indentation, my comment above is a reply to D.Lazard's reply, and does not immediately refer to your comments.

* As said, I do not need any reminder of Wikipedia being no coursebook, but an encyclopedia.

* I did not say anything about presenting a view of algebra for young children, but argued against the effect of D.Lazard's suggestion of "generalization" on the targeted clientel. Perhaps you must read my whole sermon.

* To my opinion the needs of the readers are not balanced well to the requirements of an encyclopedia in the proposed texts, starting with insisting on empty contrasts, and in sequence with "continuous" and "generalization".

Rest assured, I'm just trying to contribute to improve things I consider really suboptimal, I won't edit a word against the "good editors". -Purgy (talk) 19:32, 23 December 2016 (UTC)

It is better in general to address the topic rather than specific editors. I have some difficulty in understanding what you are saying or what your points are, that is why for instance I said " I'm not quite certain what you are saying about algebra. I think you are saying". If i just said "I don't understand" you would have no idea what impression I had got from what was said. I was hoping if I was wrong then knowing that would help you address the problem in getting your points across. For instance you said "I do not see my original claim refuted, that the current view of algebra on arithmetic, disregarding all historical roots (Al Quarizmi(?)), were not a generalisation, but a total revamp of the view as presented in pre-calculus education." As far as I can make out you think I did not address that point in what I wrote, is that correct? I think it needs also to be pointed out that a talk page is for discussing improvements to an article, it is not for asserting refuting or anything else particular peoples ideas. Dmcq (talk) 22:38, 23 December 2016 (UTC)

Until further notice or legal advice I won't stop

• to suggest proposals or post criticism I consider to be advantageous for Wikipedia
• to adress changes and proposals by their authors, if this appears advantagous to me
• to repudiate any unnecessary advice or instruction about any behaviour de rigeur or any purpose and target of Wikipedia
• to hope that people at the assumed level of education can overcome petrification, mislead solidarity, and any not-invented-here syndrome, allowing for improvements not written by themselves
• to consider refuting/asserting one side of opposing opinions to be the superior methods to decide about their prevalence, especially when compared to the methods of persevering on the status quo, of promoting a personal POV, or simply of executing a silly form of backslapping (my enemy's fiend is my friend) and therefore
• to consider talk pages as the prominent and appropriate place to put those refutations/assertions in the light of transparency.

For the time being I explicitly re-state my opinion on the 30(!) words under discussion and their improvement:

• The employed comparisions to geometry, ... are vacuous, and the suggested amendments of "continuity" (conveniant preliminary, generally present in the imagination of the uninitiated, but not intrinsically important to calculus) and "generalization" (rather a conceptualisation, e.g. reducing the arity is no generalisation, ...) are no improvement, but rather deteriorating, and that my second proposal above, even when being shorter (14 words!), encompasses new, relevant, and even characterizing information (introductory level, application to functions), and could be itself easily improved -with some good will of native speakers- to be better than the status quo.

Since continued refutation of incoherent and unnecessary advice on off topic agendae leads still more off topic, I herewith, at my discretion, stop commenting in this section. -Purgy (talk) 10:07, 26 December 2016 (UTC)

Claim of "undisputed"

In the line of the complete debate and especially of the last item in my last comment in the section above I feel urged to state that I definitely tried to utter a clear objection to D.Lazards suggestions, and that in no way I consider them to be "not clearly disputed", especially when not ignoring the whole debate above. I protest against the claim in the edit comments by D.Lazard per 25 January 2017.

I do not mind any further to be disregarded in this here setting. Purgy (talk) 09:56, 26 January 2017 (UTC)

Development of calculus

We discuss the contributions to calculus of the Indian mathematician Madhava of Sangamagrama and the Kerala school of astronomy and mathematics in this article in the Medieval section. A user is adding a claim that these two should be credited with the development of modern calculus [13] and a slightly modified [14]. This seems to be overstated, and three different editors have removed the material. The material in this article at Calculus#Medieval, and in the specific articles Madhava of Sangamagrama and Kerala school of astronomy and mathematics shows work on infinite series, but the attribution is uncertain as is the possible transmission of the techniques to Europe. Claiming that this work as the development of modern calculus does not seem justified. Meters (talk) 20:33, 3 March 2017 (UTC)

I agree with this assessment. Perhaps a reliable source is needed that shows that the development of these specific ideas happened in parallel or even before Newton and Leibniz's time. I note that the citation given, Mathematics in India by Kim Plofker, on the cited page (221), says

The crest-jewel of the Kerala school is generally considered to be the infinite series for trigonometric quantities discovered by its founder, Madhava. These are preserved in about a dozen verses explicitly attributed to Madhava in various works by later members of the school, and translated in the following sections. Many of the accompanying rationales and explanations of these results are probably also originally due to Madhava, even if not directly credited to him.

The fact that only many and not all, and that these results probably are due to Madhava is what makes the inclusion a bit iffy indeed, in my opinion. Penskins (talk) 20:48, 3 March 2017 (UTC)

So it seems like there are a few separate issues here that need to be treated separately:

1. To what extent did the results of the Kerala school influence modern calculus?
2. To what extent are the results of the Kerala school attributable to Madhava as an individual?
3. How much content that we would retrospectively recognize as "calculus" is there in the results of the Kerala school?

I think Penskins's remarks seem mainly addressed to question 2, which is an interesting question in itself, but not the one that people are likely to get excited about. Most of our readers have never heard of Madhava or the Kerala school, and just the fact that some of the Newton/Leibniz content was anticipated (question 3) is already surprising and worthy of note, even taking Madhava as a pars pro toto for the Kerala school (like Pythagoras for the Pythagoreans).

But the question that's likely to be really controversial is question 1. Remember the Columbus principle: It's not who discovers it first; it's who discovers it last. If the Kerala results anticipated some of modern calculus but then were limited to some dusty old books that no one knew how to find, then they're really just an interesting historical footnote. If Newton/Leibniz somehow knew of them, that's very different. So far I have not heard of any clear evidence that the Kerala results influenced modern calculus, and the claim that they did would need to be very well-referenced. --Trovatore (talk) 21:57, 3 March 2017 (UTC)

No Dutch translated version

There is no Dutch translated wiki to Calculus. Can someone please look into this? Because the subjects under calculus are available in Dutch, but not the main topic itself. Weird enough.

Would love to read it in my language. — Preceding unsigned comment added by 2001:1C01:2104:3000:9806:9E5B:420A:EEAD (talk) 16:21, 24 March 2017 (UTC)

Mathematics of the other beings

QUOTE: In the 14th century, Indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. Madhava of Sangamagrama and the Kerala school of astronomy and mathematics thereby stated components of calculus.END OF QUOTE.

It is only right to assume that when the animal world acquires the skill to use English, computers etc., they will also come up with similar astounding claims. — Preceding unsigned comment added by 137.97.8.238 (talk) 22:01, 2 May 2017 (UTC)

Did I address your concern in my edit? Maybe, you should be more explicit of your concerns, and try to implement this on your own. Purgy (talk) 05:59, 3 May 2017 (UTC)

I put Indian back in. There's no problem about describing people from the south of India in medieval times as Indian. Dmcq (talk) 17:47, 3 May 2017 (UTC)

Maybe, I'm definitely not sure, it's you (please, note the italics), who does not see the problem, which others try to articulate. Purgy (talk) 19:19, 3 May 2017 (UTC)

Perhaps you can articulate what you see as what they were saying and therefore you did something about? In short say why South Asian Subcontinent is better than Indian in that context. Dmcq (talk) 14:31, 4 May 2017 (UTC)

I followed the (wrong again) assumption that the intentions of the OP were well known, or even stereotyped. I'll revise my presupposition that it is sufficiently established that there are people from this region, who prefer not all of their possible ancestors being subsumed under "ancient indians", for whatever reason, and in a similar vein like others feel justified to enforce the singular they on all linguistic corpora.

I hope this is articulate enough. In no way I oppose to having been reverted, I just tried to make WP a safer space. ;] Purgy (talk) 06:35, 5 May 2017 (UTC)

Most of that subcontinent has been referred to as India from long before then. People looking up this article on the English Wikipedia cannot be assumed to be familiar with the constituent parts of India, it is not a Hindi or Indian encyclopaedia. The article does immediately afterwards refer to Kerala so they would know from that vaguely where Kerala is. Those are the reasons I think saying Indian is best. It is all very well trying to cater for people but Wikipedia's main purpose is not to be a social forum but to be an encyclopaedia. They haven't answered your query about what their concern actually is. What did you think the intentions of the OP were or what type of stereotype did you think I had? There just is not enough there to get any handle on anything that I can see. Dmcq (talk) 10:10, 5 May 2017 (UTC)

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Thank you, wikipedia, for this opening line

Thank you, wikipedia, for this opening line:

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, like on an abacus) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

— wp editors, Calculus, Isaac Newton

It clarifies so much for me. Actually, it says: "This is what math is". - DePiep (talk) 21:07, 1 February 2018 (UTC)

Well, this is the best "consensus" in WP could generate. You may want to look at the intense discussion about the lead here, here and here, too (scroll up ~20 lines, in case), to see the intense effort, just recently spent, to improve the lead, and you can also learn where to place your thanks more precisely. Purgy (talk) 08:22, 2 February 2018 (UTC)

Allow me to skip those readings. I'm so happy with the result as it is now. I didn't read "compromise", I read clarity. (I'm trying to understand QM these days. For example, this is the one, pre-wikipedia). -DePiep (talk) 00:18, 3 February 2018 (UTC)

No integral calculus page?

There is a nice article for differential calculus, which describes it as a branch of calculus and gives some history, but there is no corresponding page for integral calculus. There is only a redirect to integral. Why then is differential calculus not a redirect to derivative? 121.216.128.241 (talk) 10:59, 9 February 2018 (UTC)

Hi. Your best bet at finding editors who are interested and know enough about this subject is at Wikipedia:WikiProject Mathematics. Perhaps bring this up at their talk page (Wikipedia talk:WikiProject Mathematics). Bennv3771 (talk) 11:57, 9 February 2018 (UTC)

(edit conflict) "Integral calculus" is much less used than "differential calculus". This is probably the reason for not having Integral calculus. Also, having a separate article needs to have content that is not covered by the target of the redirect. It seems that this is not the case here. D.Lazard (talk) 12:39, 9 February 2018 (UTC)