# Talk:The Analyst

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Berkeley's "ghosts of departed quantities" is usually interpreted as describing infinitesimals, his "evanescent quantities." It seems to me that the "ghosts" actually describe their "velocities" or ratios that remain after the quantities have vanished. The ratio dx/dt remains like the smile of the vanishing Cheshire cat when ∆x and ∆t have vanished. The "ghosts" are Newton's "ultimate ratios," the limits of those ratios that would not be rigorously defined until Weierstrass's δ-ε formulation. Alan R. Fisher (talk) 22:07, 6 January 2008 (UTC)

I disagree with this analysis of the expression Ghosts of departed quantities. See my analysis there. Katzmik (talk) 08:52, 26 October 2008 (UTC)

## Merger proposal

I propose that Ghosts of departed quantities should be merged into The Analyst, as the quote from The Analyst that is the subject of the former page is already covered in this page; we do not need to duplicate essentially the same material in two places. Gandalf61 (talk) 13:49, 25 October 2008 (UTC)

I would like to reiterate the comment I have made elsewhere regarding mergers. I feel that a generally recognizable term (such as the title of this page) deserves a page of its own. It is far more recognizable, by the way, than the title of Bishop Berkeley's book. Similarly, moving the Bishop Berkeley page to George Berkeley is a mistake. It might satisfy a pedant of wiki regulations, but it does not correspond to common usage. Furthermore, the material on this page is not covered at the Analyst page. I am referring to the specific explanation of the paradox. Katzmik (talk) 08:50, 26 October 2008 (UTC)
Ghosts of departed quantities can redirect here, so it will still be available as a search term. The resolution of Berkeley's complaint through the limits approach of Cauchy, Riemann and Weierstrass is decribed here. The alternative approach via non-standard analysis is mentioned here. There is no point duplicating the same material in two different places. Gandalf61 (talk) 10:07, 26 October 2008 (UTC)
I understand your concern, but I feel you have not addressed the points I have made in favor of retaining the separate page. I would like to that I don't happen to think The Analyst is particularly well-written or researched. Katzmik (talk) 11:12, 26 October 2008 (UTC)

(ec) I support the proposal. Currently there is barely enough material here for one decent article. In the unlikely event that the merged article grows significantly it can always be split again. Katzmik, I am not familiar with the comments you have made elsewhere, so quite possibly I just don't understand your point if you didn't repeat everything here. But I don't see how it is common usage for encyclopedias to have separate, tiny, articles on somewhat famous books and somewhat famous quotations taken from them. I also don't understand where the "wiki pedantry" comes in. (Are you under the impression that this is about removal of information? There is no reason not to discuss the "ghosts" in much more detail in the present article.) If anything, keeping articles separate for formal reasons strikes me as "pedantic", but I only mention this to illustrate that "pedantry" is sometimes in the eye of the observer.

In my opinion it's a question of quality of writing. Virtually all interested readers of one article will click the link to the other one. (Provided that they notice it, and that they are not using media that make it impossible. E.g. I sometimes print an article to read it later without a computer. Once I am on a train it's too late to print any companion article. But this is a minor point.) As a result the readers move around between two tiny articles that cover basically the same topic using two vastly different approaches. That looks very unprofessional to me, although I find it hard to describe what, exactly, is the problem. --Hans Adler (talk) 11:29, 26 October 2008 (UTC)

The article The Analyst is more historical in nature, while Ghosts of departed quantities addresses the mathematical paradox rightfully criticized by Berkeley. That there is a need for such an explanation is proved by the present talk page. Namely, the first comment on this page shows that it is easy to misunderstand Berkeley's comment. Note that more people have visited Ghosts of departed quantities than The Analyst, which tends to indicate that the former is the better known term. When I was speaking of the points that I had made already, I was referring to the point that a recognizable concept should have its own page (my favorite example being computational formula for the variance). The ghost quote is certainly recognizable. The truth is that The Analyst is less so: a casual reader might think the term refers to Larry Zalcman :) Katzmik (talk) 11:41, 26 October 2008 (UTC)

Ah, you want a page that covers the paradox as something that's still relevant for mathematics teaching today? That makes some sense. I am not sure what you mean by "the present talk page". The talk page on which we are now doesn't have such discussions, and Talk:Ghosts of departed quantities doesn't even exist yet. By the way, now that I have looked at the history of Ghosts of departed quantities I think I understand your situation a bit better; it's a bit like an AfD immediately after the creation of an article. Be assured that I wasn't aware of this. I looked strictly at the two articles in their present state, and had never seen them before. --Hans Adler (talk) 14:12, 26 October 2008 (UTC)
I was referring to the comment by Fisher at the top of the page. Actually, I looked over Berkeley's article since I made the comment above, and I am not as sure as I was that Fisher is wrong. At any rate, what he writes certainly goes contrary to the received wisdom on Berkeley's famous witticism. Katzmik (talk) 14:15, 26 October 2008 (UTC)
Katzmik, you refer to "points" you have made. But the article Ghosts of departed quantities makes no points, since it doesn't bother to explain the premises on which any paradox can be inferred.
Specifically, at a certain point, delta-x is referred to as an infinitesimal, but (as of this writing) no groundwork whatsoever has been laid in that article for why it should be considered an infinitesimal.
Rather, the article Ghosts of departed quantities -- if it is to be retained in Wikipedia at all -- should simply be an exposition of Berkeley's objection to calculus as it existed when he wrote it. It should not be an opportunity for you to promote your personal agenda regarding infinitesimals.Daqu (talk) 21:40, 19 March 2009 (UTC)

I support the merging proposal. The expression "ghosts of departed quantities" might deserve a section of its own at this page, but that's all. I don't think that this expression is of such importance, neither historical or otherwise, that it deserves a wikipedia page of its own. In addition, to explain what the expression is all about from scratch, which is what we need to do if the expression has a page of its own, becomes a huge task. But placed in the correct context on this page the expression is a very simple thing to explain. iNic (talk) 02:46, 29 May 2011 (UTC)

Shall we vote?

• I vote yes. iNic (talk) 02:46, 29 May 2011 (UTC)
• I !vote yes, the phrase is notable enough to deserve a redirect but there is not really enough to make an article out of it. Thenub314 (talk) 03:48, 29 May 2011 (UTC)
• Against. We have trouble enough agreeing on a limited issue which is the meaning of the expression "ghosts of departed quantities". Merging this to the Analyst will merely transfer the content dispute to another page. The expression is well-established and deserves a separate page. It would be helpful to establish its meaning, which is at variance with Thenub's interpertation of it as referring to derivatives. Tkuvho (talk) 05:16, 29 May 2011 (UTC)
• support: I've only just come across this merger proposal but with both articles short and all relevant material in Ghosts of departed quantities already here there's no point having two articles. Readers and editors will be much better served by having essentially duplicate material in one place. This is especially true of contentious material: otherwise we're in danger of having two articles saying quite different things on the same topic.--JohnBlackburnewordsdeeds 12:06, 29 May 2011 (UTC)

Conclusion: merge

As only one editor want to keep the Ghost-article as a separate article I conclude that we should merge these articles. The only reason he has for keeping the Ghost-article as a separate article is that there are disagreements on what the Ghost-article should contain. As I see it that is a good argument for deleting the article rather than keeping it. No positive or objective argument for keeping the article has been provided. iNic (talk) 10:51, 1 June 2011 (UTC)

Given the current dispute content at Ghosts of departed quantities, merging it would merely transfer the content dispute to another page. I don't think a merge is appropriate in this situation. Tkuvho (talk) 14:00, 1 June 2011 (UTC)
As I noted above this is if anything good reason to merge: having all the information in one place where more editors can look at it is more likely to achieve consensus. Having two articles is likely only to prolong and fragment any discussion and possibly result in two articles which say quite different things on the same topic.--JohnBlackburnewordsdeeds 15:39, 1 June 2011 (UTC)

OK, so as long as no consensus is reached on what a badly defined article should contain, it can't be deleted? Let's say we never reach a consensus just because it's a badly defined article from the start, does that mean that Wikipedia have to keep it forever? In that case I think the rules for deleting an article at Wikipedia is in urgent need of some revision! In my view all badly defined articles where no one can explain why it's there in the first place should be deleted immediately. Please remember, Wikipedia is an encyclopedia, not a blog. iNic (talk) 15:31, 1 June 2011 (UTC)

I believe a merge would be best because a major part of the argument is over whether there is only one criticism in the whole of the Analyst whict is expressed in the Analyst or a number. Putting the argument here would allow better documenting of what the criticism/criticisms is/are rather than being constrained to talk about that single phrase. Dmcq (talk) 14:09, 3 June 2011 (UTC)
I have merged, as that seemed to be the consensus. I invite anyone still interested to take a look at what I have done and make sure I haven't fouled it up entirely. Thenub314 (talk) 19:35, 16 June 2011 (UTC)
Bravo! Thanks a lot. I think your summary is right on spot. This is how much attention this quote deserves. iNic (talk) 01:41, 18 June 2011 (UTC)
Thanks.  :) Thenub314 (talk) 02:10, 18 June 2011 (UTC)

## Claim of Berkeley having been 'essentially correct'

Prof. Grabiner offers this statement without supporting argument. The word essential is misleading because she refers (presumably) to the essence of Berkeley's argument, not the essence of calculus. I believe her point is that the mathematical practices of the time were not rigorous by Euclidean standards, although even this is debatable and she make no attempt to convince us of a claim made in passing. To say that the bishop was essentially correct would mean that calculus is essentially flawed. I agree that it is possible to interpret Berkeley as only criticizing the subject as practiced and not in essence. Moreover, I personally agree that this is his main point. I will point out, however, that he often refers to the calculus as 'impossible to understand', not poorly understood. His argument that the infinitesimal method was inherently contradictory has been refuted by so-called non-standard analysis. Therefore, in this crucial point, his 'correctness' is debatable at best. —Preceding unsigned comment added by 138.16.100.49 (talk) 01:53, 4 February 2009 (UTC)

## Limits in Newton

An allusion to limits in Newton has recently been deleted. Newton seems to have had a kinetic notion of limit (though not an epsilontic one). I would like to have it restored, perhaps with a clarification along these lines. Tkuvho (talk) 11:15, 4 May 2011 (UTC)

[1]? If you want the text back, the obvious way would be to provide a reference for it William M. Connolley (talk) 11:22, 4 May 2011 (UTC)
Your text was "But only beginning around 1830, first in the hands of Augustin Cauchy, later in those of Bernhard Riemann and Karl Weierstrass, were the derivative and integral redefined using a rigorously defined new concept, that of limit, exploited already by Newton". I am absolutely certain that Newton did not exploit or use the concept of the limit as developed and understood by Cauchy, Riemann and Weierstrass. In fact, exactly the opposite is true - it is Newton's fluxions that Berkeley calls "ghosts of departed quantities", and this description partly motivated the subsequent reform of calculus by Cauchy et al. If you actually meant to say something else about Newton, then state what you meant more clearly, and, as said above, provide a source for your claims. Gandalf61 (talk) 11:46, 4 May 2011 (UTC)
Wow, someone still remembers I exist! That is amazing... So where are we. Right, I have been called into the middle of some argument so I will do my best to play both sides of the field and be fair, because I haven't been around for a long long time.
OK on the one hand, some people do feel that Newton had a fairly complex notion of limit, see for example the paper I reference in this edit or more directly [2]. On the other hand this paper states specifically he was the first to produce an epsilon argument. So this paper could be used to support Tkuvho's statement that Newton used limits in the article, but not so much his comment on this talk page.
Background: At the time I dug out that paper I was involved in an argument with Katzmik about whether limits played a critical role in calculus. As I recall he felt that Newton and others for two hundred years were operating for 200 years or so infinitesimals, and so limits did not play a critical role. Reading through some of the secondary literature my impression is that Newton's ideas were heavily based in physics and so notions of quantities approaching each other were frequently invoked where has Leibniz's ideas were more directly algebraic in nature, dealing with infinitely small quantities.
Now for the other side of the coin. The third source literature is clear that it was neither Newton or Liebniz who provided the notion of limit to calculus. For example D. Burton's text on the history of mathematics states that: "The first prominent mathematician to suggest that the theory of limits was crucial in calculus was Jean d'Alembert... ", and you can find references that say most anything if you try hard enough. For example, I pointed out in Talk:Limit_of_a_function#History_section a reference that cites Wallis with the first limit proof somewhat before Newton. I suppose my point is that, interpreting the primary sources is difficult and frequently contradictory, and I am not to worried if the reference to Newton is either put back or removed.
My real problem with the current article is the sentence starting "And finally..." finally? That seems a bit dramatic somehow. Thenub314 (talk) 20:50, 4 May 2011 (UTC)
Maybe we can elaborate on what Pourciau said in his text. Incidentally, Jesseph in his 2005 paper used the adjective "kinematic" to describe Newton's approach. Other scholars use the term "kinetic". I don't have access to Pourciau. Does he document several instances or just one? It is not surprising that other historians prefer to deny any limits in Newton. This fits better with their ideology that Weierstrass invented analysis. Tkuvho (talk) 05:03, 5 May 2011 (UTC)
What's the basis for the claim that Newton did not use his limit as a foundation for calculus? Tkuvho (talk) 17:58, 5 May 2011 (UTC)
Well, there are many references to quote, but citing Burton's book. He is very clear about the fact that d'Alembert was the first person to suggest using limits as a foundation for calculus. The date of the article he cites was after Newtons death. So if d'Alembert was the first person to do so, and d'Alembert did so after Newton was dead, it stands to reason it was not what Newton was doing.
Pourciau's paper examines a new translation from latin to english of Newtons's Principia published in 1999. He cites various passages from this translation of Principa to support the idea that that Newton clearly understood limits and examines the proofs of three propositions whose proofs (after decoding them into modern language from greek geometry) that seem to be limit type proofs. But a few lemmas or propositions is far from the foundation of the subject. Thenub314 (talk) 22:35, 5 May 2011 (UTC)
The deduction you propose from Burton's book is WP:OR. Really, I expected better from you. It is much simpler to assume that Burton simply was not aware of Newton's clear understanding of limits. What exactly did d'Alembert say that made such an impression on Burton, and where did he say it? Tkuvho (talk) 04:58, 6 May 2011 (UTC)
What deduction do you mean? If Burton states that d@alembert was the first, after Newtons death, then there is (rather obviously) no OR involved. You look to be defending your point rather tendentiously William M. Connolley (talk) 07:15, 6 May 2011 (UTC)
Is it Burton who says "after Newton's death"? Then he doesn't mean that Newton didn't use limits, does he? He means that after Newton, the first one, etc., was d'Alembert. Does someone have the exact quotation from Burton? I do not think it is tendentious to think that Burton may not have been aware of the full extent to which Newton had a clear understanding of limits. After all, Pourciau's article appeared after Burton's book. Tkuvho (talk) 08:13, 6 May 2011 (UTC)

(outdent) Since you asked for Burton's specific quote it follows, but understand that quotes are a very bad way to understand sources, at this point in the text we have been discussing Newton's life and work and the invention of calculus for 30 pages, and will continue doing so for another 15. The exact quote from Burton is:

The first prominent mathematician to suggest that the theory of limits was fundamental in calculus was Jean d'Alembert (1717-1783). D'Alembert wrote most of the mathematical articles in that cardinal document of the Enlightenment, the Encyclopédie (28 volumes, 1751-1772) and in an article entitled "Différential" (volume 4, 1754) said 'the differentiation of equations consists of simply in finding the limits of the ratio of the finite differences of two variables in the equation.' In other words, he cam to the expression of the derivative as the limit of a quotient of increments, or as we write it,

${\displaystyle {\frac {dy}{dx}}=\lim _{\Delta x\to 0}{\frac {\Delta y}{\Delta x}}.}$

Unfortunately, d'Alembert's elaboration of the limit concept itself lacked precision. Therefore, a conscientious mathematician of the 1700's would have been no more satisfied with this definition than with currently available interpretations of the derivative.

Now I accept that Pourciau's article appears after Burton. I have two points to make about that, first Pourciau is primary research in the field of the history of mathematics, while Burton is a textbook. In terms of verifiability, that makes Burton a better source. It could be that someone publish an article next week challenging Pourciau's results, etc.

My second point is that my first point is moot. Regardless of how authoritative we see Pourciau's article, it simply doesn't say that Newton based his notion of calculus on limits. It states he clearly understood them, and he used them to prove 3 of his propositions. Not that he used them to define the notions of derivative or integral, etc. The point of the article was that Newton was the first person to give a real limit argument. Which is a radical statement in and of itself and contradicts many more well established sources. Pourciau is fairly consistent with Burton, who gives many examples of when he uses limit type arguments and many examples of when he used infinitesimal arguments. Over all I think what we have now is consistent with both sources. Thenub314 (talk) 19:22, 6 May 2011 (UTC)

What I find curious is the role of d'Alembert in this. Suppose he used the word "ultimate ratio" in place of "limit" in the citation above. Would this still be considered as visionary? Is he using the term "limit" as an intuitive concept, or as a technical concept? If he is using it as an intuitive concept, what is his visionary role and how is it different from Newton's ultimate ratios? If he is using it as a technical term, then who did the technical development that he is relying on? It certainly wasn't d'Alembert as far as I know. Tkuvho (talk) 04:46, 8 May 2011 (UTC)
"d'Alembert proposed that calculus be based on the concept of limit ... he identified dy/dx as the limit of a quotient of finite terms ... Missing was a clear definition of "limit" and the subsequent derivation of basic calculus theorems from it. In the end, d'Alembert did little more than suggest the way out of trouble." William Dunham The Calculus Gallery Chapter 5. Gandalf61 (talk) 08:58, 8 May 2011 (UTC)
What exactly was the "little" that he did do, propose to rename "ultimate ratio" by "limit"? Offer a lucky guess as to what the future mathematical concept will be called? Tkuvho (talk) 12:22, 8 May 2011 (UTC)
"Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity.... [Newton 1999, 442; Newton 1946, 39]." Tkuvho (talk) 16:20, 8 May 2011 (UTC)

(outdent) Perhaps we can all agree to say something akin to as the following quote from Edwards:

The first step towards resolving Berekely's difficulties by explicitly defining the derivative as a limit of quotients of increments, in the manner suggested, but not stated with sufficient clarity by Newton, was taken by Jean d'Alembert (1717-1783).

Regardless of how we might personally feel about d'Alembert's contribution, the history texts all seem to mention him has putting forward the first substantial step. Perhaps, if you don't recognize he made a significant mathematical contribution at perhaps he can be thought of as the best student of Newton, there were many other attempts to place calculus on a rigorous footing after this article was published, but he was the first to turn attention back to Newtons original approach and tried to make sense of it. Thenub314 (talk) 17:54, 8 May 2011 (UTC)

I am perfectly willing to leave the article as is, but I am just wondering about the nature of d'Alembert's contribution. In what way did he "try to make sense of newton's original approach" other than re-stating that the ultimate ratios are limits? I imagine his mathematical reputation needs no defenders, including his proof of the fundamental theorem of algebra (which however was erroneous, as pointed out by Gauss). Do you have any idea why traditional historians are so fond of him, to the extent of presenting him as a limit visionary, whereas he seems to have gone no further than Newton? Tkuvho (talk) 18:10, 8 May 2011 (UTC)
Well I will give you my opinion, but it will be just that, my opinion. Berkely called into question how calculus was being done, rather correctly by the understanding of the day. There was significant confusion about how to rigorously justify calculus. The mathematicians of the time tried many things but were not very close to figuring it out. Newton himself described things in the language of fluxions in some works, or used infinitesimal quantities, or by ultimate ratios as in principia.
So now there are several competing notions of how to rigorously justify calculus, and d'Alembert was the first to explicitly write that it was the notion of limit that was the way to go, and he was correct even if he couldn't complete the program. Perhaps subsequent French readers of this article may have been strongly influenced by this idea, I am not sure. It seems conceivable to me that Cauchy was introduced to this idea of doing things rigorously via limits by d'Alembert's article. Though I haven't tried to verify this was the case. If so, it would be pretty reasonable to single this article out as a significant step. Thenub314 (talk) 19:33, 8 May 2011 (UTC)
Thanks for your comment. How many approaches did Newton have exactly? You mention "fluxions" but I have the impression that "fluxion" is merely his term for the derivative. This would leave us with two approaches: (1) infinitesimal, and (2) ultimate ratios, which as Newton himself said, is the same as "limits". Shall we say that there are basically two approaches then? Tkuvho (talk) 04:55, 9 May 2011 (UTC)
Yes, mathematically speaking two approaches. Three sets of language though. Newton apparently explicitly tried to take all references to the word fluxion out of Principia (he apparently missed only one reference in the final manuscript). So for what ever reasons he didn't consider the terminologies to be of equal rigor. It was at that time considered to be the case that Euclidean Geometry was the pinnacle of mathematical rigor and at least some of the responses to the Analyst were to try to recast calculus in terms of Euclidean Geometry. Thenub314 (talk) 17:44, 11 May 2011 (UTC)
Most scholars agree I think that Newton distanced himself from the infinitesimal thread, thinking of it as a continental malady and more precisely unrigorous. As far as d'Alembert is concerned, I noticed that Pourciau is referring to a beginning of an epsilontic argument in d'Alembert. He is not very explicit but he seems to imply that this is mentioned in Grabiner's book. If you have access to the book, could you please check this out? If d'Alembert really has this type of argument, perhaps I can begin to fathom the reason for the credit he gets. Tkuvho (talk) 18:13, 11 May 2011 (UTC)
Ok I have checked out what she had to say about d'Alembert's work. It seems he was the first (him and Lagrange) to really bring inequalities into the mix. He "proved" various limits existed using inequalities. For example she mentions that before him mathematicians would examine the first for terms of a series to decide its convergence, he showed this was flawed by considering (1+200/199)½ expanded as a series and showing that the first few terms decreased but the ratio of successive terms was greater than 1 for all n>300, and so the series could not converge. He emphasized that limits should not involve appeals physical motion, and practiced what he preached. But he failed, somewhat crucially, to define limits in terms of inequalities. As an amusing side remark she mentions he used absolute values, but had no notation for them, and would simply use a phrase to describe it (something along the lines of "up to an abstraction of the sign", but I don't have the book in front of me just now.) Thenub314 (talk) 03:14, 2 June 2011 (UTC)
Thanks, that's very helpful. Do you recall what kind of limit he calculated using inequalities? Tkuvho (talk) 12:04, 2 June 2011 (UTC)
Well proved limits might be a bit strong, which is why I put it in quotes. Grabiner says: "In computing the slope of the tangent to the parabola ${\displaystyle y^{2}=ax}$, d'Alembert found that the slope of the secant was equal to ${\displaystyle a/(2y+z)}$. D'Alembert said, ' As we can take z as small as desired, we can make the ratio ${\displaystyle a/(2y+z)}$ approach the ratio ${\displaystyle a/2y}$ as closely as desired." Thus, he concluded, ${\displaystyle a/2y}$ is the limit of the ratio ${\displaystyle a/(2y+z)}$ and therefore is equal to the slope of the tangent." He also applies term by term comparisons with the geometric series when investigating ${\displaystyle (1+\mu )^{m}}$ and used this to derive an error term. Grabiner says about this "D'Alembert had given a completely worked-out example of how the partial sums of a series could be proved to differ as closely as desired from some fixed value." Thenub314 (talk) 21:02, 2 June 2011 (UTC)

## Summary of Sherry.

The summary of Sherry's paper is very misleading. It summarizes only one part of the paper, there are other criticisms he discusses. We need to improve this, but I am refraining from a simple undo. Thenub314 (talk) 05:13, 2 June 2011 (UTC)

Sherry and Jesseph are among the leading experts on Berkeley. They both say there were two criticisms: the logical one and the metaphysical one. Authors writing general history of math books don't always look at the details the way the experts do. It is pretty clear that in case of conflict between Jesseph/Sherry as opposed to a fat history book spanning numerous centuries, the former are more reliable. Tkuvho (talk) 12:06, 2 June 2011 (UTC)
I said nothing about "fat history book[s]". I said our summary of his paper was misleading. I have tried to improve it. Thenub314 (talk) 21:59, 2 June 2011 (UTC)

## Odd use of /harv/ ref style

Most of the references here would be more readable as inline citations; with the added benefit that then you autogenerate links back from the reflist to where those refs are cited in the article. Worth converting? I haven't looked into how and where harvard citation style are used elsewhere; nor time right now, so I'm making a note here. – SJ + 20:46, 2 November 2013 (UTC)

That would be fine. I am not much of an expert on references so I leave this up to you. Tkuvho (talk) 13:15, 4 November 2013 (UTC)