Talk:Infinitesimal calculus

Infinitesimal calculus is an area of mathematics pioneered by Gottfried Leibniz based on the concept of infinitesimals, as opposed to the calculus of Isaac Newton, which is based upon the concept of the limit.

Is it really true that Isaac Newton had the concept of limit? I tend to think of that as a 19th-century innovation. Michael Hardy 20:58, 7 Sep 2004 (UTC)

Would like to second that last statement. The defintion given for Infinitesimal includes Newton as using infinitesimals and the idea of the limit wasnt introduced until the 19th century by Karl Weierstrass. Would be nice to keep definitions noncontradictory.

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)

Article needs cleaning up for readability

Reading this article is like wading through treacle. Please tidy up the language used to make it accessible to more people. Calculus is not a complicated subject if it is taught well. The text in this article may be accurate but if it fails to pass on understanding it is not doing the job an encyclopedia should do.

Best wishes, — Preceding unsigned comment added by 92.20.167.150 (talk) 21:49, 22 December 2012 (UTC)

Your comment is so common that we (WikiProject MATH) have a templated response for it; suffice it to say that the "infinitesimal calculus" is so complicated that, although created in the 1660s, mathematicians didn't get it right until 1960. — Arthur Rubin (talk) 06:47, 23 December 2012 (UTC)

Nonsense. That is purely your opinion. Not every mathematician accepts non-standard analysis. 197.64.207.71 (talk) 08:16, 25 June 2013 (UTC)

I agree with Arthur. Which aspects of the page does the IP find inaccessible? Is it the lede or the later sections? Tkuvho (talk) 08:22, 24 December 2012 (UTC)

15000 views

This was viewed 15000 times yesterday. Any idea why? xkcd does not seem to be it. Tkuvho (talk) 15:23, 16 April 2013 (UTC)

Misleading lead

The lead says:

Infinitesimal calculus is the part of mathematics concerned with finding tangent lines to curves, areas under curves, minima and maxima, and other geometric and analytic problems

This is incorrect. The "part of mathematics concerned with finding tangent lines to curves, areas under curves, minima and maxima, and other geometric and analytic problems" is called calculus.

Infinitesimal calculus is just one approach to calculus.

--Redaktor (talk) 14:00, 7 January 2014 (UTC)

@Redaktor, you may wish to consult this text by Dieudonné which uses the term "infinitesimal calculus" in the sense of "the calculus". Tkuvho (talk) 12:58, 8 January 2014 (UTC)

See also this review. Tkuvho (talk) 12:59, 8 January 2014 (UTC)