Talk:Calculus

Misuse of sources

This article has been edited by a user who is known to have misused sources to unduly promote certain views (see WP:Jagged 85 cleanup). Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent.

Please help by viewing the entry for this article shown at the cleanup page, and check the edits to ensure that any claims are valid, and that any references do in fact verify what is claimed. Tobby72 (talk) 21:42, 24 August 2010 (UTC)

This goes all the way back to 2007, and there are a large number of edits to check. I checked the last one, and it seems to accurately reflect a reliable source. Rick Norwood (talk) 13:10, 25 August 2010 (UTC)

Here is a typical misuse of sources: In this edit Jagged introduced the text which now reads "the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus" with this source. But the source merely mentions that Egyptians knew a certain formula, and it speculates (in a student exercise) that dividing a volume into small blocks might have led to the formula. The statement in the article should be removed. I sampled Jagged's edits and they appear to be connected with Calculus#Ancient and Calculus#Medieval where Jagged introduced the text on Ibn al-Haytham ("was the first to derive the formula") and much more. To fix the mess, we need to read the sources with a skeptical eye while checking each claim in the Ancient/Medieval sections (there is no need to check each of Jagged's edits). I would remove anything that looks somewhat dubious in those sections (if really unsure, the removed text could be copied to here). In particular, when Jagged used the word "first", it is usually wrong (or fails verification). Johnuniq (talk) 00:49, 26 August 2010 (UTC)

Everything under the Medieval section appears to check out. If anyone would like a copy of the unlinked sources, contact me. Anyone that finds other discrepancies should post them here. Hexagon70 (talk) 00:29, 10 October 2010 (UTC)

In Our Time

The BBC programme In Our Time presented by Melvyn Bragg has an episode which may be about this subject (if not moving this note to the appropriate talk page earns cookies). You can add it to "External links" by pasting * {{In Our Time|Calculus|b00mrfwq}}. Rich Farmbrough, 03:01, 16 September 2010 (UTC).

Maria Gaetana Agnesi

I just reverted this edit by Lorynote which added:

File:Maria Gaetana Agnesi.jpg

Maria Gaetana Agnesi' was an Italian linguist, mathematician, and philosopher. Agnesi is credited with writing the first book discussing both differential and integral calculus. The plane curve, known as versiera, is also called the "Witch of Agnesi" [ref]Agnesi Witch Agne Scot, Agnesi.

My edit summary mention "unsourced". I see that is not correct since the refs do offer some support for the statement (since the refs were on the second sentence I jumped to the conclusion that the only support was Maria Gaetana Agnesi). However, the material definitely should not be in the "Significance" section, and there needs to be consideration of whether the material is WP:DUE (did the book describe as in a text book, or did it develop?). Johnuniq (talk) 00:29, 4 December 2010 (UTC)

The fact is that she is a pioneer, a major figure for the history of maths and calculus. It answer the question "who wrote th first book on calculus?". I see this as a fundamental information. Lorynote (talk) 10:35, 4 December 2010 (UTC)

From About.com: Women History, Agnesi; And: Belle vue college, Agnesi. Lorynote (talk) 11:00, 4 December 2010 (UTC)

This source says that "her two volume textbook was the first comprehensive textbook on the calculus after L'Hopital's earlier book", so L'Hopital would be more fundamental, yet is not mentioned in the article. The other source says that "it was one of the first and most complete works on finite and infinitesimal analysis.". So there is no support for "Agnesi is credited with writing the first book...", and neither here, whereas this source is "Condensed from "The Pioneering Women Mathematicians" by G.J. Tee in The Mathematical Intelligencer", so not really a reliabe source. The sources do mention the Witch of Agnesi, but that is more on-topic in analytic geometry, not in calculus. DVdm (talk) 11:10, 4 December 2010 (UTC)

Ok, so one can say she wrote first comprehensive book. Lorynote (talk) 12:07, 4 December 2010 (UTC)

No, we can not say that. She wrote the first comprehensive book after L'Hopital's earlier book, and there is no reason to mention that, since L'Hopital is not mentioned in the article either. DVdm (talk) 12:49, 4 December 2010 (UTC)

Although there are two sources to support that she was the pioneer and only one to support the after someone else we can keep the second; mentioning she wrote one of the first books. Besides, the source clearly cites his book and her compreheensive book. The fact L´Hopital is not mentioned here it´s not relevant; once his name is on WP anyway and the link will not be red. Lorynote (talk) 13:15, 4 December 2010 (UTC)

Also note that the first source says in its disclaimer on http://jwilson.coe.uga.edu/ : "The content and opinions expressed on this Web page do not necessarily reflect the views or nor they endorsed by the University of Georgia or the University System of Georgia.". This is someone's personal web site, so It cannot be taken as a wp:reliable source either.

I think it would be okay to use the second source to write this short statement in the article:

One of the first and most complete works on finite and infinitesimal analysis was written In 1748 by Maria Gaetana Agnesi.

No details about or picture of Agnesi are needed here. Readers who click the link get all they want. What do the other contributors think about this? DVdm (talk) 13:42, 4 December 2010 (UTC)

You can guess my opinion; I vote for a picture and more details about her (just the very short 'she was italian, mathematician, philosopher'). My very honest opinion is that she has a brilliant work and she was recognized as such. Lorynote (talk) 14:13, 4 December 2010 (UTC)

I agree it needs to be short, no bio needed (no-one else has one), but a mention needs to be there. The Agnes Scott College source is adequate - there are several mathematical sources on the Witch of Agnesi curve - I think as well as your sentence, one sentence on the Witch (which is very well known and has its own article, and there's no contention that it was studied by and named after Agnesi) would be adequate. People can read both articles for more info, but if we don't put a mention in, people won't know to look. This is always the challenge of an article like this.

One of the first and most complete works on finite and infinitesimal analysis was written in 1748 by Maria Gaetana Agnesi. The Witch of Agnesi curve is named after Agnesi, who wrote about it in 1748 in her book Istituzioni Analitiche. (this citation taken from the Witch article)

Picture I'm in favour of. We have Newton and Leibzitz - the fathers of calculus (that sounds odd I know). Agnesi I agree is probably not more notable as a mathematician (as opposed to a female mathematician) than several others who advanced one area of calculus or other, but I think the addition of her picture would possibly attract the interest of more readers, because she is a woman. It's a bit tokenistic I know, but pictures should mho pique the interest, not just decorate the page. Elen of the Roads (talk) 16:27, 4 December 2010 (UTC)

Elen, I would sign next to your words as if they were mine. Lorynote (talk) 16:34, 4 December 2010 (UTC)

I have no problem with Agnesi being mentioned, but shouldn't there be a sentence about l'Hôpital as well, since he actually produced the first textbook (plus l'Hôpital's rule of course)? Favonian (talk) 16:43, 4 December 2010 (UTC)

I´m for L´Hôpital as well! Sure, he´s welcome! Lorynote (talk) 17:15, 4 December 2010 (UTC)

@Favonian, could do. That entire modern section is pretty unsourced at the moment, except for the rather random closing sentence about calculus being taught in schools (I'm sure that's wandered in from somewhere else) so I'm not averse to anything with sources. That last paragraph could be a little enlarged without any ill effects as well I feel.Elen of the Roads (talk) 17:26, 4 December 2010 (UTC)

But let's make sure we have a proper source for l'Hôpital's first textbook -- note that the Wilson source cannot be taken as a wp:RS (see disclaimer, cited above). DVdm (talk) 17:30, 4 December 2010 (UTC)

It appears that L'Hospital's claim is somewhat dubious [1] - his book was actually written by Bernoulli. Havent seen anything yet - article on the book is sourced to a book I don't have Elen of the Roads (talk) 18:32, 4 December 2010 (UTC)

FYI, there might be no further comments from Lorynote again on this. - DVdm (talk) 22:55, 4 December 2010 (UTC)

So she did turn out to be Jackiestud. Figures. Still, I'd rather AGF a bit than bite the heads off newbies (I'll leave that to Darwinbish). I would still support putting my two penn'orth above about Agnesi into the article. Your thoughts?Elen of the Roads (talk) 02:21, 5 December 2010 (UTC)

Did you notice that there is currently something in the article? I would prefer someone with knowledge of the area to comment on the DUEness of the wording, and the sentence on the curve should be removed since it is not relevant here, and is in the linked article. Until we encounter someone with relevant knowledge (so they can find the refs), I am happy with the current image and wording if the curve sentence is removed. Johnuniq (talk) 03:23, 5 December 2010 (UTC)

I've removed the section entirely, since it's not clear how notable Agnesi and her book are. If this is in fact a notable event in the history of calculus then it could come back. Ozob (talk) 04:26, 5 December 2010 (UTC)

Agree with Johnuniq. The one-of-the-firstness of the book seems to be sufficiently sourced and (thus) notable. The curve is indeed off-topic. Don't care about the pic: afaiac it can stay. DVdm (talk) 10:22, 5 December 2010 (UTC)

@Johnuniq - wording and pic fine by me - I'll defer to others on the curve. I will however revert Ozob's deletionadd back DVdm's contribution. It was already agreed it was sufficiently notable to put in the article, but should be limited to one sentence. I've also put the picture back, per discussion above.Elen of the Roads (talk) 10:52, 5 December 2010 (UTC)

Looks good to me! Ozob (talk) 12:20, 5 December 2010 (UTC)

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)

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)

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)

Merge

The proposal to merge this with non-standard calculus doesn't make sense. Non-standard calculus did not emerge until the 1950s; this article is about something introduced by Leibniz in the 17th century. Michael Hardy 21:48, 12 Jun 2005 (UTC)

infinitesimals were put onto a rigorous basis by nonstandard calculus --MarSch 15:18, 16 Jun 2005 (UTC)

Everybody knows that, but it would be absurd to limit the topic to its rigorous formulation when the non-rigorous version played a far more prominent role in the history of the subject. Michael Hardy 21:55, 16 Jun 2005 (UTC)

Both infinitesimal calculus and non-standard calculus have changed considerably since 2005. Since calculus using Robinson's infinitesimals is a natural continuation of what was known historically as infinitesimal calculus, a case could be made for merging non-standard calculus into infinitesimal calculus, which is obviously the more appealing title. Tkuvho (talk) 14:18, 24 June 2010 (UTC)

I also find the merge to be a bit inappropriate. Perhaps a disambiguation is a better idea. Based on a Google books search, it seems that the term "infinitesimal calculus" is used both for calculus in the ordinary sense and the sense of non-standard calculus. Sławomir Biały (talk) 13:19, 2 July 2010 (UTC)

I agree with Michael Hardy and Sławomir Biały, there should be 2 different articles. To my knowledge infinitesimal calculus is usually not associated with non standard analysis as a field, but it is rather used for the "normal" calculus before the analysis rigor of the late 19th/early 20th century and today mostly used as a term for a somewhat less rigorous approach to analysis (like first calculus primers).--Kmhkmh (talk) 14:11, 2 July 2010 (UTC)

I agree also. -- Radagast3 (talk) 01:01, 3 July 2010 (UTC)

Reprising my comments from WT:WPM:

• In my experience infinitesimal calculus simply means the integral and differential calculus considered together; the infinitesimal calculus is as distinct from, say, the calculus of finite differences. The term has no particular implications for foundational approach, and is compatible with a limits-based exposition, even though the latter does not officially use infinitesimals per se.
• Therefore the proper target for infinitesimal calculus is simply calculus.
• Non-standard calculus should be merged into non-standard analysis, the more common term. --Trovatore (talk) 21:52, 4 July 2010 (UTC)

I second this proposal. Sławomir Biały (talk) 13:54, 5 July 2010 (UTC)

copied from a talkpage

Hi, there used to be a parenthetical remark following "standard calculus", clarifying that the standard approach was developed by Cauchy and Weierstrass. The parenthetical remark did not say that infinitesimal calculus was developed by them. What we have today is a pair of approaches that provide rigorous foundation to infinitesimal calculus as envisioned by Newton and Leibniz, namely the approach by Cauchy and Weierstrass on the one hand (called the standard approach), and the much more recent approach of Robinson (non-standard). It would be incorrect to suggest that standard calculus was developed by Newton and Leibniz, as the current version states. Katzmik (talk) 15:48, 8 January 2009 (UTC)

I can't say I agree with the assertion that Cauchy and Weierstrass provided a rigorous foundation to infinitesimal calculus. They didn't (though a number of maths textbooks and poorly-researched histories of mathematics claim otherwise). They avoided the problem of infinitesimals by using an alternative concept - the concept of a limit - around which they build a set of analytical techniques that are fundamentally distinct from those of the infinitesimal calculus. Read Imre Lakatos's paper on non-standard analysis if you want a reasonable account. If you're still not convinced, get back to me for more references. zazpot (talk) 01:31, 10 January 2009 (UTC)

Very interesting. Lakatos is one of my favorite authors. Somehow I was never aware of the fact that he had something to say about NSA, I will try to follow up your link. Meanwhile, historical issues aside, it is hard to argue with the fact that the approach of Cauchy and Weierstrass has been adopted in the teaching of calculus worldwide. Are you attempting to make a distinction between "calculus" and "infinitesimal calculus"? There again, numerous calculus courses around the world are named "infinitesimal calculus" even though they may not be taught using infinitesimals. I think using the term "infinitesimal calculus" to describe The Calculus is consistent with common usage. Moreover, originally, and for many years, the page infinitesimal calculus was only a redirect to calculus, so adding the NSA link should perhaps be viewed as an improvement from your point of view? Katzmik (talk) 11:25, 11 January 2009 (UTC)

P.S. I found no mention of NSA at the link you provided. Katzmik (talk) 11:27, 11 January 2009 (UTC)

Infinitesimal calculi rely upon the (assumption of the existence of and) use of infinitesimals: quantities small enough to be insignificant under some, but not all, circumstances. Both the ontology and epistemology of infinitesimals were disputed from the 17th to the 20th centuries; Abraham Robinson seems to have been the first to put infinitesimals on a footing equivalently sure with those of less disputed mathematical objects. In the intervening period, many mathematicians (e.g. Newton, Leibniz, Maclaurin) worked to improve the epistemology of infinitesimals, none very successfully. Consequently, an alternative way to achieve the results of "the infinitesimal calculus" (broadly, that consisting of the common parts of Newton's and Leibniz's work, as extended by their many followers of the same and subsequent generations) was sought. Notably, Cauchy made major strides in this direction, which Weierstrass consolidated. The calculus of Weierstrass cannot be called infinitesimal, because it avoids (and aims to avoid) the use of infinitesimals. During the period from Weierstrass's work to Robinson's, infinitesimals were widely considered to have been "consigned to the dustbin of history" (cf. Eric Temple Bell, IIRC). A "calculus" is, of course, merely any system of calculation rules, but during this period (Weierstrass to Robinson), "the calculus" came to refer to the Weierstrass calculus (which, as I've stated, obtained the results of the infinitesimal calculi of Newton and Leibniz but by different means). Before the introduction of the limit concept, however, "the calculus" referred to the infinitesimal calculi of Newton and Leibniz, et al. In practice, since mathematicians are rarely good historians, some confusion between the two meanings of "the calculus" persisted, and continue to persist, among them. You appear to be one such mathematician.

It is utterly false to assert, as you have, that Cauchy and Weierstrass put infinitesimal calculus on a rigorous footing. They did not. They created an alternative calculus by which they were able to obtain the same results.

If you want to distinguish clearly between the two pre-Robinson families of calculus, you may wish to refer to them as "the infinitesimal calculus" and "the calculus of limits", or suchlike, but do not conflate the two, or you will be being factually inaccurate.

IIRC, the title of the Lakatos paper was Cauchy and the continuum, pub. in 1970s. zazpot (talk) 16:42, 11 January 2009 (UTC)

I made a few changes to your recent edit, and thought you might like me to explain. As Katzmik points out, and I agree with, the term infinitesimal calculus generally refers to simply calculus (done with or without infinitely small quantities.) I think most historians do not make a distinction between the calculus developed by Newton and Leibniz, and the later work of Cauchy and Weierstrass. The ideas in Lakatos's paper are interesting and perhaps merit inclusion elsewhere, perhaps in History of Calculus, but mostly seem to confuse the disambiguation page. Lastly, I took out the comments about limit being a new idea, because the concept of the limit of a function pre-dates calculus, and is present in Newton's work on the subject (perhaps in Leibniz as well, but I haven't research this as carefully). Thenub314 (talk) 12:05, 12 January 2009 (UTC)

Just to make my position clear: I entirely agree with Zazpot's remark that what seems to be the generally accepted usage of the term "infinitesimal calculus" as a synonym for "differential and integral calculus", is in many ways a misnomer. On the other hand, as Thenub correctly points out, we cannot take it upon ourselves to correct accepted usage. Katzmik (talk) 12:35, 12 January 2009 (UTC)

Thanks to both of you for clarifying your positions here, and for working, as I have, to make improvements (esp.: Katzmik, thanks for spotting my typo with the century!). I think the infinitesimal calculus page as it stands now is in much better shape than it used to be, and do not plan to edit it further myself for the time being.

I agree with Thenub314 about Lakatos's paper possibly meriting inclusion elsewhere, in the sense that I feel a general rationalisation of the contents of "infinitesimal" calculus-related topics is probably overdue on Wikipedia. But at the moment, I don't have the spare time to undertake such a large-scale operation.

I do feel it's erroneous for mathematicians or historians to refer to the calculus that came out of Cauchy & Weierstrass as "infinitesimal calculus", unless this is clarified by saying that this is just a common or colloquial name for it; as you can see, I'm concerned to rebut any suggestion that "infinitesimal calculus" is a proper or descriptive name for it. I spent many months unpicking these subtleties over the course of my undergraduate degree (in History and Philosophy of Science, if you want to know), and wrote about them in my final year dissertation, which I have wished over the last few days I had to hand. One of the many things I learned during the course of that enterprise was that there are very few satisfactory histories of mathematics. Most seem to have been written by retired mathematicians with much bluster but no historiography, and they are best at revealing the fashions that were present in mathematics when their authors were peaking in their careers, rather than at telling the history of the discipline and its protagonists. As for Thenub314's suggestion that, "most historians [of mathematics] do not make a distinction between the calculus developed by Newton and Leibniz, and the later work of Cauchy and Weierstrass", I would say it's not quite true, but it's far from being as false as I'd like it to be. The decent histories of mathematics are truly few and far between; but it would be a mistake to use the bad ones as our guides. zazpot (talk) 00:09, 13 January 2009 (UTC)

Charles Sanders Peirce

In mathematical logic, Charles Sanders Peirce had interesting writings about infinitesimals. Secondary literature includes the mathematics historians Joseph W. Dauben, Carolyn Eisele, John L. Bell; c.f. the wider discussions of Peirce's mathematics (and mathematical logic) by the mathematical logicians by Hilary Putnam (e.g. in Peirce's "Reasoning and the Logic of Things") and Jaakko Hintikka (e.g. in "Rule of Reason"). See also:

• Peirce on Infinitesimals |Author(s): P. T. Sagal | Source: Transactions of the Charles S. Peirce Society, Vol. 14, No. 2 (Spring, 1978), pp. 132-135 | Published by: Indiana University Press
• The Genesis of the Peircean Continuum |Moore, Matthew E. Transactions of the Charles S. Peirce Society: A Quarterly Journal in American Philosophy, Volume 43, Number 3, Summer 2007, pp. 425-469 (Article) | DOI: 10.1353/csp.2007.0037

Thanks, Kiefer.Wolfowitz (talk) 15:17, 4 July 2010 (UTC)

disambiguation

Trovatore and tkuvho have come to a tentative agreement that IF it can be verified that the term "infinitesimal calculus" can be documented to be used in a historical sense to describe the calculus using infinitesimals prior to Weierstrassian reform, then this page can be turned into a disambiguation page with perhaps three items: (a) calculus; (b) history of calculus; (c) infinitesimal calculus in the sense of Henle's textbook. Tkuvho (talk) 11:07, 9 July 2010 (UTC)

Encyclopaedia Britannica

The 9th edition of the EB has a 68 page article about "Infinitesimal Calculus".WFPM (talk) 20:03, 15 March 2011 (UTC)

Do you have any more details about this? When did it come out? what's in the article that's not in the earlier editions? Is there an electronic link? Tkuvho (talk) 03:12, 16 March 2011 (UTC)

I don't know about the details. It's just in my copy of the 9th EB (R. S. Peale Company, Chicago, 1892. The Infinitesimal Calculus article was managed by Benjamin Wilson, F. R. S., Professor od Mathematics, Trinity College, Dublin. Encyclopardia Britannica Volume 13 (INF - KAN) Pages5 through 72.WFPM (talk) 11:58, 16 March 2011 (UTC)