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![]() | The contents of the Ghosts of departed quantities page were merged into The Analyst on 16 June 2011. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
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 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)
(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)
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?
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)
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)
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)
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)
(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,
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)
(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)
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)
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)
"Berkeley sought to take mathematics apart" - Did he? He certainly concentrated on taking apart the then foundations of calculus: "...the Object, Principles, and Inferences of the Modern Analysis". Of which, of course, Edmund Halley (he of the "offhand comment mocking" Berkeley's Alciphron) was a notable practitioner. See the first words of The Analyst: "Though I am a Stranger to your Person, yet I am not, Sir, a Stranger to the Reputation you have acquired, in that branch of Learning which hath been your peculiar Study;"
Perhaps this needs to be made clearer? CatNip48 ( talk) 12:57, 15 February 2024 (UTC)
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||||
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![]() | The contents of the Ghosts of departed quantities page were merged into The Analyst on 16 June 2011. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
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 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)
(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)
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?
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)
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)
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)
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)
(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,
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)
(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)
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)
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)
"Berkeley sought to take mathematics apart" - Did he? He certainly concentrated on taking apart the then foundations of calculus: "...the Object, Principles, and Inferences of the Modern Analysis". Of which, of course, Edmund Halley (he of the "offhand comment mocking" Berkeley's Alciphron) was a notable practitioner. See the first words of The Analyst: "Though I am a Stranger to your Person, yet I am not, Sir, a Stranger to the Reputation you have acquired, in that branch of Learning which hath been your peculiar Study;"
Perhaps this needs to be made clearer? CatNip48 ( talk) 12:57, 15 February 2024 (UTC)