# Talk:Ghosts of departed quantities

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In relation to the merge proposal: as things stand, not too much would be lost by placing the current content here into The Analyst. What needs to be said, though, is that Berkeley was not the only such critic of calculus. Other early critics were Detleff Clüwer, and Bernhard Nieuwentijt. The latter, anyway, seems to have argued the same point in the 1690s (see s:1911_Encyclopædia_Britannica/Infinitesimal_Calculus/History_2). Therefore it seems a shame not to take the chance to broaden the scope here, to include some more of the history of the debate. Charles Matthews (talk) 14:38, 30 October 2008 (UTC)

Very interesting. I was not familiar with these other criticisms, it would be helpful if someone could contribute a few paragraphs. According to your date, their criticism preceded Berkeley's? Katzmik (talk) 14:47, 30 October 2008 (UTC)
Note that the discussion on the merger has become active again here: Talk:The Analyst#Merger proposal --JohnBlackburnewordsdeeds 12:09, 29 May 2011 (UTC)

## Delete merge suggestion?

The merge suggestion listed at the top of the article does not seem to be going anywhere. If nobody objects, I will delete it. Katzmik (talk) 20:08, 29 November 2008 (UTC)

I am now deleting the merge proposal. There is a separate proposal to rename the page "Early challenges to infinitesimal calculus", which is still current, pending addition of new material. Katzmik (talk) 07:55, 8 December 2008 (UTC)

The text "To consider an example, the function y = x2 might be regarded in a pedagogically simplified fashion as being "differentiated in calculus" by forming the quotient ${\displaystyle {\frac {\Delta y}{\Delta x}}}$ of the y-increment, usually denoted Δy, over the x-increment, usually denoted Δx. The resulting expression simplifies algebraically to ${\displaystyle 2x+\Delta x.\,}$" is perfectly unproblematic. There is no need to add "can be considered pedagogically", which merely obfuscates a simple point. Tkuvho (talk) 16:04, 3 May 2011 (UTC)

Did you mean to call my text "perfectly unproblematic"? I don't quite agree with you; it is awkward. However, it is better than what was there before, the function y = x2 is differentiated in calculus by forming the quotient, which was inaccurate. If you consider the former text "perfectly unproblematic" then I disagree with you William M. Connolley (talk) 16:31, 3 May 2011 (UTC)
I think the "pedagogically simplified" doesn't help here. I've changed the original to read
"To consider an example, the function y = x2 is differentiated in calculus by first forming the quotient ${\displaystyle {\frac {\Delta y}{\Delta x}}}$ of the y-increment, usually denoted Δy, over the x-increment, usually denoted Δx.
I inserted the word "first" to make clear that this isn't quite all you do. But the definition of the derivative is defined in terms of a difference quotient. There is nothing "pedagogically simplified" about it. Of course, the next step is we'd like to take a limit as ${\displaystyle \Delta x\to 0}$, but that's something at odds with the point being made. Sławomir Biały (talk) 21:04, 3 May 2011 (UTC)
No, this is all wrong. Derivative (mathematics) gives you the answer; its the value of a limit, if that limit exists, etc etc. Its not a sequential process with something you do first then something you do second William M. Connolley (talk) 21:15, 3 May 2011 (UTC)
You seem to be splitting hairs here. (And indeed one does form the difference quotient first.) Rather than edit-warring, you might consider trying to improve the explanation, rather than adding content that definitely obfuscates the point being made. I have tried yet another wording. Sławomir Biały (talk) 21:22, 3 May 2011 (UTC)
Oh, spare me. "rather than edit warring" is rich from someone who reverted out my text. All you've done now is hide the vagueness / wrongness by using the "difference quotient" which is itself a very poor article William M. Connolley (talk) 07:27, 4 May 2011 (UTC)
The overall text is much better now. Thank you. Sławomir Biały (talk) 10:58, 4 May 2011 (UTC)
Thanks for your input, Sławik. Tkuvho (talk) 11:09, 4 May 2011 (UTC)
Glad we can agree. I've struck my somewhat intemperate comment William M. Connolley (talk) 11:21, 4 May 2011 (UTC)
I'm glad as well (and have responded in kind). I had hoped that our relationship could become more collaborative rather than adversarial, as you are a much more experienced editor than I am. My hope has been restored! Cheers, Sławomir Biały (talk) 11:36, 4 May 2011 (UTC)

## Stick to Islam, you are more knowledgeable in that area

Or so says Tkuvho [1]. That doesn't seem acceptable to me. Nor do I think Tkuvho's edit is an improvement, which is why I reverted it William M. Connolley (talk) 16:08, 20 May 2011 (UTC)

On the contrary, I appreciate your work on ensuring objectivity in Islamic articles. As far as infinitesimal calculus is concerned, you have admitted yourself on the talkpage that you are not fully knowledgeable about the area. You should therefore stick to areas you are knowledgeable in. Tkuvho (talk) 20:16, 21 May 2011 (UTC)
Your comments make little sense; if it comes to expertise, I have a degree in maths, but this kind of silly credentialism is of little value. As to the substance: your version Thus, the infinitesimal quantity Δx is assumed nonzero at the stage of calculating the quotient, and yet appears to be assumed zero in the last phase of the calculation when Δx is discarded does not correspond to how the derivative is actually defined, nowadays William M. Connolley (talk) 21:09, 21 May 2011 (UTC)
I have just restored the version WMC restored previously as it much better reflects mainstream thinking on calculus. In particular the 'paradox' of calculus is resolved by the modern rigorous approach of limits. There may be other ways of approaching it but they are generally ignored in modern calculus. The links used, to non-standard calculus and non-standard analysis suggest this too. And yes, this is not Citizendium, we do not ask for credentials here and we comment on the edits, not our perceptions of other editors' areas of expertise (which are often wrong)--JohnBlackburnewordsdeeds 21:26, 21 May 2011 (UTC)

## unsourced whiggish history

Probably best to draw a veil over this intemperate section

Far from being mainstream thinking about calculus, the ideas you are pushing are unsourced whiggish history. Nobody in the 19th century thought of epsilontics as being a resolution of the paradox pointed out by Berkeley. On the contrary, they all accepted what Berkeley claimed to be a proof of the inconsistency of infinitesimals. The paradox of the infinitesimal definition of derivative was resolved only in the 20th century. Connolley implicitly acknowledged this fact through his latest edits at standard part function. Tkuvho (talk) 22:27, 21 May 2011 (UTC)

Yes, indeed, the "paradox" was only resolved well after Newton / Leibniz. No-one is claiming otherwise. As the current article says this is really only a reflection of the informal nature of the "definition". When Berkley wrote his criticism, the notion of derivative had not been formalised; it was not until 1830, beginning with the work of Augustin Cauchy, that the standard resolution of the apparent paradox was provided, as a rigorous definition of "derivative", via the (ε, δ)-definition of limit. But if this article talks about derivative (unqualified) then it must refer to derivative, as currently understood. If you want to talk about derivative (as understood at the time) then it would need to be very clearly labelled as such, to avoid confusion. Also, you can't spell my name; I've taken the liberty of correcting you William M. Connolley (talk) 22:33, 21 May 2011 (UTC)
William, this page is not about the definition of the derivative. It is about Berkeley's criticism of the infinitesimal definition of the derivative. Who in his right might is going to disagree that a consistent definition of derivative was developed in the 1870s? But that's not the subject of this page. Tkuvho (talk) 22:36, 21 May 2011 (UTC)
I just referred you to User:William M. Connolley/For me/The naming of cats. Be kind enough to read it, to avoid faux pas William M. Connolley (talk) 22:45, 21 May 2011 (UTC)
I will read your expository writing when I have more confidence in its integrity, William. Tkuvho (talk) 23:39, 21 May 2011 (UTC)

## Content disagreement

Let's try again. Leibniz gave an infinitesimal definition of the basic objects of the calculus. Berkeley argued that infinitesimals are incoherent, inconsistent, and fraught with paradox. After various attempts to find solid ground over the course of a century and a half, Weierstrass and his students abandoned the infinitesimal definitions. The reason they rejected them is because they largely accepted the gist of Berkeley's criticism. Instead, they grounded analysis in a real continuum without infinitesimals, based on epsilon, delta definitions. This is what is generally considered the "mainstream" approach today. A century later, Robinson stepped in and claimed to have been able finally to justify Leibniz's infinitesimal definitions.

These are the facts as I understand them. If you see the facts differently, please present your position. If you do accept the facts as I presented them, then the current version of the page is inaccurate.

I particularly object to the use of quotation marks around "paradox". Historians don't use such quotation marks even for veridical paradoxes, see drinker's paradox. There is absolutely no need to use the subjunctive tense in describing the paradox "as if" it were one but in reality not. Tkuvho (talk) 17:57, 22 May 2011 (UTC)

What you are missing is that L's "defn" wasn't a defn in the currently accepted sense; in that it was vague, and relied on intuition to skip over the gaps. Hence the quotation marks around paradox: there is no paradox (of course, there cannot be). What there is, is a lack of clarity, which leads to the appearence of paradox. It is like saying "light is both a wave anda particle; this is a paradox". It isn't. All that has happened is that you have pushed english too far William M. Connolley (talk) 18:20, 22 May 2011 (UTC)
Cantor, who called infinitesimals an "abomination" and a "cholera bacillus of mathematics", would turn in his grave if he heard you describing him as resolving the paradox of the infinitesimal definitions. Tkuvho (talk) 04:28, 23 May 2011 (UTC)
But since I didn't say that, three isn't a problem William M. Connolley (talk) 07:29, 24 May 2011 (UTC)
That's very different from the version you wrote in this revision which puts non-standard calculus first suggesting it is how modern mathematics resolves the 'paradox'. The quotes are needed as it's not a paradox, at least not anymore. See e.g. this news item yesterday. Not a real tiger so it's a 'tiger sighting'.--JohnBlackburnewordsdeeds 18:40, 22 May 2011 (UTC)
You seem to have misread the version of the page you cited: this. It mentions infinitesimal calculus, not non-standard calculus. If the derivative is defined as the infinitemal ratio dy/dx on the nose, this is indeed paradoxical, as it calls for dx to be zero and nonzero at the same time. Tkuvho (talk) 04:28, 23 May 2011 (UTC)
It looks as though you want this page to be about something different from what everyone else wants it to be about; and you're doing a rather poor job of explaining what you want William M. Connolley (talk) 07:29, 24 May 2011 (UTC)

(outdent) I noticed a lot of traffic on my watch list at this article and so I peaked in to see what was going on. Sadly there are some factual errors with the article. This particular phrase was not aimed at infinitesimal quantities, which Bishop did fault for being zero and not zero as convenient, but that wasn't the issue with he was addressing when he penned "ghosts of departed quantities". He was discussing limits that were of the form (in modern terms) of 0/0 (meaning something like ${\displaystyle {\tfrac {\sin x}{x}}}$ as ${\displaystyle x\to 0}$. The explanation I recieved for the term in my math history course years ago was "As x→0 the numerator and denominator go to zero, in other words they have departed, but a non-zero limit is somehow left over. And so it must be the ghost of a departed quantity. Boyer also points out that his criticism were seen as "fair, and well take" and that he he wasn't questioning the validity of the results just the justifications. Thenub314 (talk) 07:35, 25 May 2011 (UTC)

I will try to fix the explanation later, if someone doesn't beat me to it first. Now to bed. Thenub314 (talk) 07:41, 25 May 2011 (UTC)
I have done my best, after working on the article for a good part of the day I am left wondering why this has its own page. Couldn't we condense the information here and merge into the Analyst article? There is probably lots of cleanup/spelling/etc issues. I am a rather messy editor, so I encourage everyone to look it over. Thenub314 (talk) 23:01, 25 May 2011 (UTC)
The idea that the phrase "ghosts of departed quantities" refers to fluxions is your own fabrication. If you re-read the quote from Berkeley cited in the article, you will notice that the phrase refers to evanescent increments of fluxions, not fluxions themselves which are velocities. Tkuvho (talk) 04:18, 26 May 2011 (UTC)
Well, I am not quite sure how to interpret the above paragraph. It either means I botched the second sentence, which is a fair assessment. So I tried to fix it. You might possibly mean that evanescent increments of fluxions refers to infinitesimal increments of fluxions, in which case you think I botched the whole thing. To which I have a few comments:
• Regardless of how you or I might read the quote, the sources I cited are very clear that the quote is refering to defining fluxions via a limit.
• By my reading the term evanescent increments is a reference to how Newton describes ultimate rations in the Scholium of Section 1, Book 1 of the Principia. (see near markers 54/55 of the linked page.
• More correctly is speaks of "Velocities of increments" and not "increments of velocities". We agree that fluxions are velocities. But I disagree that the passage refers to increments of fluxions.
Thenub314 (talk) 17:17, 26 May 2011 (UTC)
Error, again. When Berkeley speaks of "Velocities of evanescent Increments", he is referring to velocities computed by using evanescent increments, the latter being infinitesimals. The evanescent increments mentioned in the context of fluxions are infinitesimals. The same increments are the ghosts. The sources you are using are flawed and will never be accepted. You are basically wasting your time developing this version. An editor at WPM suggested relying on the presentation in The Mathematical Experience, and I think that's a good idea. It is not very constructive to delete the page that was around for years, and replace it by your own version, over vocal opposition from one of the main contributors to the page. Tkuvho (talk) 17:49, 26 May 2011 (UTC)

(outdent) If you feel I have made an error in the second two bullets, your welcome to your opinion. I was simply offering mine.

What do you mean the sources I am using are flawed and will never be accepted? Seriously, you really think that? I am not sure how to respond to that. I should perhaps point out that Boyer is a famous mathematics historian, his textbook is fairly standard. Edwards book is likewise a respected Springer text. If it is the community consensus that for some reason these books are flawed and cannot be sourced, I would have to live with it. Though we would have to clean up all the other wikipedia pages that reference one of these two. (Which is approximately 90-100 pages at a glance.) But I suspect this version will, in some form remain a part of the page.

I never suggesting deleting anything, nor have I. I simply came by a more or less unsourced page where arguments were occurring over content. I decided to check a reference, the reference disagreed with the page, entirely. So naturally I rewrote the page to improve it. I couldn't, in good conscience leave the page as it was by the time I realized you had objections (which you very notably did not raise here or with myself) because it was a mess. I wasn't going to add back unsourced explanations that contradicted the sources I added. Thenub314 (talk) 18:56, 26 May 2011 (UTC)

You have pretty much deleted the original page, and replaced it by your own understanding of the subject. I am not sure I understand the source of your animosity toward Robinson. It is a commonly heard claim that Robinson resolved the logical paradox of the infinitesimal definition of the derivative pointed out by Berkeley. If you think such sourced claims are inaccurate, you can cite counterclaims, but deleting other people's work is inelegant. Tkuvho (talk) 15:23, 29 May 2011 (UTC)
Perhaps we use the word delete differently, everything the page was or ever has been is in the page history and nothing has been deleted. I did certainly re-write the page, and I remain unapologetic about the matter. If there is disagreement between a source and an unsourced statement, you removed the unsourced statement, that much is simple. Now I would like to know. Why do you think I hold any animosity toward Robinson? He is a great mathematician who will be remembered long long after I have been completely forgotten to the pages of history. Nonstandard analysis will live on and flourish, and that is great. Let me be very clear that I have no issues with him or his mathematics. Thenub314 (talk) 03:56, 30 May 2011 (UTC)

## Are ghosts infinitesimal?

The expression "ghosts of departed quantities" has been traditionally interpreted by historians as referring to Leibniz's infinitesimals, or equivalently to Newton's evanescent increments, etc. This is the intepretation found in Grabiner's essay that gave us the epsilon, for example. User Thenub insists on his personal interpretation of the phrase as referring to something other than infinitesimals. Of the four editors who have been active here, he is the only one to pursue such a novelty. Tkuvho (talk) 03:49, 27 May 2011 (UTC)

I may have been the first to put this in the page here, but I also the first to cite a mathematics history text. To be honest until I picked up a book and started reading a few days ago I was under the same impression you were. I just don't try to assume I am better versed in mathematics history then the actual historians.
So if I understand you comments at WT:WPM and your comments here, Boyer is too old and out of date to be a reference and it isn't traditional? Not quite a contradiction, but quite an interesting pair.
Anyways, Grabiner never says the phrase infinitesimal, infinitely small, or anyting of the sort in the article you reference above. The quote with a little context for those reading along:

[Berkeley] ridiculed fluxions-" velocities of evanescent increments"-calling the evanescent increments "ghosts of departed quantities" [11]. Even more to the point, he correctly criticized a number of specific arguments from the writings of his mathematical contemporaries. For instance, he attacked the process of finding the fluxion (our derivative) by reviewing the steps of the process: if we consider y = x2, taking the ratio of the differences ((x + h)2 - x2)/h, then simplifying to 2x + h, then letting h vanish, we obtain 2x. But is h zero? If it is, we cannot meaningfully divide by it; if it is not zero, we have no right to throw it away. As Berkeley put it, the quantity we have called h "might have signified either an increment or nothing. But then, which of these soever you make it signify, you must argue consistently with such its signification" [12].

And, for what it is worth, Grabiner consistently uses the phrase "vanishing increments" when she discusses in her other papers. Thenub314 (talk) 05:30, 27 May 2011 (UTC)
What does it mean to "let the h vanish" in your opinion? Tkuvho (talk) 05:15, 27 May 2011 (UTC)
Take the limit as h → 0, of course. She is clearly writting to a modern audience so inclined to use modern language. And if its clear from the modern perspective that the phases "let ... vanish" or "as ... vanishes" refers to a limit. I guess you feel she could mean "h is infinitely small, so neglect it" Thenub314 (talk) 05:27, 27 May 2011 (UTC)
At any rate she is not referring claiming that the phrase "ghosts of departed quantities" refers to ultimate ratios, i.e. derivatives. Explaining this phrase as referring to derivatives is erroneous. Tkuvho (talk) 20:03, 28 May 2011 (UTC)
Perhaps you could enlighten me, what does it refer to, she clearly doesn't mention infinitesimals. Is it perhaps something else? Thenub314 (talk) 04:37, 29 May 2011 (UTC)
She is referring to things that are very small. Certainly not derivatives. Tkuvho (talk) 05:09, 29 May 2011 (UTC)
So we are agreed she is not referring to infinitesimal quantities? Thenub314 (talk) 05:58, 29 May 2011 (UTC)
Berkeley is referring to infinitesimals. He explicitly states that his criticism addresses both vanishing increments and infinitesimals, and mocks Newton's supporters for acting as if their theory is any different from its continental counterpart. You may disagree with Berkeley, but you can't change the criticism as he expressed it. Tkuvho (talk) 15:20, 29 May 2011 (UTC)
I have nothing to agree or disagree with Berkeley about. His book attempts to 1) attack all contemporary foundations of calculus, 2) attack possible rebuttals 3) explain why people arrive at the correct result, despite flawed reasoning. I think we agree about these three things. The question still remains as to what he is referring to in this passage. My own feelings on the matter aside, some references seem to indicate he is discussing defining fluxions by ultimate ratios, others simply refer to vanishing/evanescent quantities which is part of the definition of limit. I am not trying to change his criticism, I am just attempting to report what the secondary sources say about it. Thenub314 (talk) 03:43, 30 May 2011 (UTC)

## The Mathematical Experience

Since there was some discussion about this book I went and found the section where he discusses this particular phrase

Newton, unlike Leibniz, tried in his later writing to soften the "harshness" of the doctorine of infinitesimals by using physically suggestive language. "By the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place, when the motion ceases nor after; but at teh very instant when it arrives. .... And, in like manner, by ultimate ratio of evanescent quantities is to be understood the ratio of the quantities, not before they vanish, nor after, but that with which they vanish." When he proceeded to compute, however, he still had to justify dropping unwanted "negligible" terms from his computed answer. Newton's argument was to find first, as we have done, ds/dt = 32 + 16dt, and then to set the increment dt equal to zero, leaving 32 as the exact answer.

But, wrote Berkeley, "it should seem that this reasoning is not fair or conclusive." After all, dt is either equal to zero or not equal to zero. If dt is not zero, then 32 + 16dt is not teh same as 32. If dt is zero, then the increment in distance ds is alos zero, and the fraction ds/dt is not 32 + 16dt but a meaningless expression 0/0. "For when it is said, let increments vanish, i.e., let the increments be nothing, or let there be no increments, the former supposition that the increments were something, or that there were increments, is destroyed, and yet a consequence of that supposition, i.e., an expression got by virtue thereof, is retained. Which is a false way of reasoning." Berkeley charitably concluded: "What are these fluxions? The velocities of evanescent increments. And what aer these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?"

This reference sounds to me like it is also explaining this passage from Berkeley's book has discussing Newton's use of ultimate ratios, and not infinitesimals. The article from Dunham has a nice passage where he says, after discussing ghosts of departed quantities as vanishing ratios, that Berkeley was no kinder to the infinitesimal quantities of Leibniz, and that these quantities "where above him". But, I digress. Thenub314 (talk) 19:29, 27 May 2011 (UTC)

Now that you have quoted it, I suggest we base the page on it. Berkeley indeed thought that Newton's procedures were no sounder than Leibniz's, and repeatedly emphasized that the "vanishing increments" are no different from infinitesimals. The British mathematicians at the time thought their approach was more rigirous than Leibniz's infinitesimals, and Berkeley specifically mocked this claim. You wouldn't find Ian Stewart, for example, saying that "ghosts of departed quantities" refers to ultimate ratios/derivatives, nor Grabiner. I really wish you would discuss proposed changes before deleting a page that was here for years and replacing it with your understanding of the subject. Tkuvho (talk) 20:07, 28 May 2011 (UTC)

Berkeley criticized all attempts that existed at the time to make the new mathematics understandable on a philosophical level, be it infinitesimals, fluxions or ultimate ratios. He set out to show that no explanation that had been presented so far could do the trick. Remember that he wanted to show that Halley was wrong when he had said that religion rested on ungrounded faith while science didn't. Berkeley could not have achieved his goal if he only had criticized one or two of the attempts to make the new mathematics intelligible. He had to criticize them all, which he also did. That British and continental mathematicians took different stances in the priority dispute that followed between Leibniz and Newton is a completely different matter. iNic (talk) 02:01, 29 May 2011 (UTC)

I am not sure what you are getting at. Let's stick to the subject. What is the meaning of the phase "ghosts of departed quantities"? Tkuvho (talk) 05:08, 29 May 2011 (UTC)

I think I stick to the subject very very much. This single expression that you have made into a "subject" is just a satirical reformulation of one of Newtons attempts to make the new mathematics intelligible. So it is targeted at Newton's fluxion-heuristics. But The Analyst is in no way limited to criticizing only Newton's ideas. This shows why it's so silly to have an entire wikipedia page about this expression as if Berkeley's entire goal with his book could be condensed to this single expression. It can't. iNic (talk) 10:15, 29 May 2011 (UTC)

The length of this whole discussion shows that there are still misunderstandings about the nature of Berkeley's criticism. I can't see how it can be described as so trivial given that there has been a week-long argument about it among people supposedly knowledgeable about the subject. Tkuvho (talk) 15:19, 29 May 2011 (UTC)

OK can you please explain what different opinions there are about what Berkeley criticized? iNic (talk) 17:27, 29 May 2011 (UTC)

My opinion is that Berkeley criticized what he saw as paradoxical definitions of the calculus, where vanishing increments or infinitesimals would be assumed to be nonzero at the start of a calculation, but zero at the end thereof, and that's what he referred to by the phrase "ghosts of departed quantities". Thenub at first argued that the phrase does not refer to vanishing increments or infinitesimals, but rather to fluxions. Then he seemed to be arguing that it does not refer to infinitesimals but only to vanishing increments. I am not sure what his current position is. Note that one of the articles he quoted at adequality specifically uses the term paradox, without quotation marks, in discussing Berkeley's criticism. This whole discussion started when some editors challenged my use of the term, and carefully placed quotation marks around it. The term has been deleted altogether since. Tkuvho (talk) 17:45, 29 May 2011 (UTC)

OK I see, thank you. But I don't think your opinions are that far apart really. You are definitely right when you say that Berkeley criticized inherent contradiction in all presented methods of what would become known as infinitesimal calculus. However, the ghost quote is actually only mentioned by Berkeley when criticizing Newtons fluxions. But so what? Why is this fact so important? It's just an insignificant historical detail. I fail to see the importance of this particular quote from the book. I would appreciate if someone could explain this to me too. iNic (talk) 20:52, 29 May 2011 (UTC)

## Berkeley's ghosts are not derivatives

Let's try this again. In response to user Thenub's interpretation, note that Berkeley's "ghosts of departed quantities" are neither fluxions nor derivatives. Berkeley is criticizing Newton's vanishing increments, which Berkeley himself claims amount to the same thing as infinitesimals. One can certainly think of them as things "tending" to zero if one prefers, which is what Newton did. But at any rate, claiming they are derivatives is erroneous. Derivatives don't tend to zero. Therefore Thenub's interpretation is not acceptable. I suggest we go back to the original version from a month ago, and rework it so as to fit better with The Mathematical Experience quote. Tkuvho (talk) 05:05, 29 May 2011 (UTC)

Sure, Newtons fluxions are derivatives, time derivatives. And they do tend to zero. This is the whole point with fluxions. By introducing time Newton can start to talk about real physical quantities that tend to zero when explaining his mathematical results. Could it be that this missunderstanding lies at the heart of this lengthy debate between you and Thenub? iNic (talk) 10:22, 29 May 2011 (UTC)
Exactly. Newton's fluxions are time derivatives. What Berkeley described as "ghosts" were not fluxions but rather evanescent increments involved in Newton's definition of fluxions. In Leibniz's notation, dx is nonzero when you form the differential quotient, but dx is supposed to be zero at the end of the calculation. This was the gist of Berkeley's argument. Tkuvho (talk) 15:14, 29 May 2011 (UTC)
No. What Berkeley calls "ghosts" are definitely the fluxions of Newton. Nothing else. But it is just an accidental historical fact, not a necessary or even interesting fact. Berkeley's satirical pen was not targeted at fluxions alone and it just happened to be that this particular quote including ghosts was about Newton's fluxions. It could very well have been used when he criticized any of the other ideas, Leibniz infinitesimals for example. Or just simply omitted. History would not have been changed a bit by any of that. So I fail to see the meaningfulness discussing this at all. Can someone please explain why this "ghost" quote is so very important? iNic (talk) 20:08, 29 May 2011 (UTC)
To be honest I don't feel the ghost quote is so very important. As I have said before, I saw there was a dispute here, before weighing in with my opinion I picked up a reference and re-read about the subject. The page disagreed with the reference, so I rewrote it. Before Tkuvho discussed my edits with me he I was being accused of a whiggish rewriting of history at WT:WPM, compared me to some illicit character Jagged 85, labeled as a deletionist in a completely unrelated discussion. Since then the insults have continued constantly saying I was in error, that I was fabricating the whole idea, that I would turn Wikipedia into a laughing stock, etc, etc. At this I stand by my work, and continue the discussion mostly because I feel it is necessary to defending myself. Thenub314 (talk) 03:34, 30 May 2011 (UTC)
I think Berkeley has been misread there. He called a fluxion the velocity of an infinitesimal and was referring to the infinitesimal as the ghost of a departed quantity, not the fluxion. Here's the usual quote "And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?" Dmcq (talk) 11:45, 30 May 2011 (UTC)
No, he is not misread. The difference between infinitesimals, evanescent quantities, fluxions and derivatives are very subtle indeed, so it's very easy to confuse these concepts. Berkeley is definitely criticizing Newton's fluxions in the passage you quote. Newton tried to remove infinitesimals altogether from the new mathematics. Therefore he introduced his concept of fluxions, which was his first attempt to get rid of the infinitesimals. Berkeley's point was that Newton's explanation using fluxions was exactly as confusing as the old one based on infinitesimals. iNic (talk) 12:36, 30 May 2011 (UTC)
Notice the current version does not claim are either fluxions or derivatives are ghosts of departed quantities. It did at mistakenly have this as part of the second sentence. When you complained, I reworked that sentence, saying that your comment was a "fair assessment". There is now not much of a difference between the explanation offered here and the explanation offered by any of the references discussed thus far. Except of course clarity of exposition, and length of discussion. Thenub314 (talk) 07:02, 29 May 2011 (UTC)
Indeed, the current discussion is needlessly obscure, as it avoids at all cost talking about infinitesimals, which Berkeley explicitly stated were the object of his criticism, as well. Berkeley criticized the discarding of the error terms at the end of the calculation, contrary to Thenub's claim in a recent edit. Tkuvho (talk) 15:17, 29 May 2011 (UTC)

## Try again

We have a number of editors currently participating in the discussion, and I would like to summarize the various positions, in alphabetical order.

Editor Dmcq feels that Berkeley's "ghosts" refer to infinitesimals.

Editor iNic feels that they refer to Newton's fluxions.

Editor Thenub originally held that they refer to fluxions, and currently considers them to refer to vanishing increments.

Editor Tkuvho feels that they refer to infinitesimals and evanescent increments.

Is that a fair summary? Tkuvho (talk) 12:11, 30 May 2011 (UTC)

Well he referred to evanescent increments but I would ally myself with Tkuvho. The only difference between infinitesimals and vanishing increments is if you're talking about Leibniz's formulation or Newton's but his arguments applied to both. I definitely don't believe they refer to fluxions. Dmcq (talk) 12:22, 30 May 2011 (UTC)
I haven't participated for a few days but have been watching for times when I could contribute and this is an obvious one. I feel he's referring to nothing definite: he's not indicating he thinks they are this or that, but is indicating that it's unclear whether they are non-zero or zero or something in between ('neither quantities or quantities infinitely small, nor yet nothing.'). He is pointing out the contradiction or paradox in the vague wording and lack of a formal definition, using very unscientific phrasing ('ghosts of departed quantities') to emphasise the lack of rigour.--JohnBlackburnewordsdeeds 13:04, 30 May 2011 (UTC)

I think this is the first time ever I have engaged in a debate what a particular ghost really is. Personally I don't think that Berkeley believed in ghosts at all. He was a religious man and believed in a christian god, but he didn't believe in ghosts. So this debate what a ghost really is, ontologically, is a bit silly I think. Ghosts doesn't exist and this is why Berkeley is calling the Newtonian fluxions ghosts. It is evident from this quote: "What are these fluxions? [...] May we not call them ghosts of departed quantities?" His point is that fluxions are about as real as a ghosts, i.e., they don't exist. But of course, ontologically the ghost of Berkeley is equivalent to everything that doesn't exist. And as he didn't believe that infinitesimals existed either you can of course say that they are equivalent to ghosts as wel. But the fact remain, the only time when Berkeley talks about ghosts in his book is when he is talking about Newtons fluxions. Everyone that can read will see that. Sorry if I hurt anyones feelings. ;-) iNic (talk) 13:11, 30 May 2011 (UTC)

Ghost ontology and epistemology should be discussed at a different page, not here. The question is, which scientific concept was Berkeley criticizing. He could have called it the scalps of departed quantities as far as I care. Was he criticizing infinitesimals or was he criticizing derivatives? Your quotation above is misleading, as it leaves out an intermediate sentence from which it is clear that he is no longer talking about fluxions, but rather evanescent increments. Thenub seems to have realized this. Correct me if I am wrong. Tkuvho (talk) 13:16, 30 May 2011 (UTC)
The first omitted sentence above is just one of the expressions Newton used to describe his fluxion concept. The second omitted sentence is just the same rhetorical question Berkeley stated in the first question about what a fluxion is, but now with the word 'fluxion' replaced with Newton's supposedly explaining words 'evanescent increments.' So nothing essential is omitted when leaving those two sentences out. Berkley wasn't changing the subject of his interest that quickly. And he for sure didn't in this case. iNic (talk) 15:23, 30 May 2011 (UTC)
Velocities of evanescent quantities are not the same as evanescent quantities. The velocity of a plane is not a plane. Dmcq (talk) 15:35, 30 May 2011 (UTC)
Sorry I meant 'velocities of evanescent increments.' This is the expression Newton used. A typo by me. Fluxions are instant velocities in the direction of a coordinate axis. Newton didn't talk about 'evanescent quantities' separated from 'velocity' as something meaningful in itself. It was Berkley that did that here in an attempt to dissect Newton's concepts into its constituents. iNic (talk) 17:36, 30 May 2011 (UTC)
The ... is 'The Velocities of evanescent Increments? And what are these same evanescent Increments?'. Fluxions are compared to the velocities of evanescent increments, they are not compared to evanescent increments. Dmcq (talk) 14:12, 30 May 2011 (UTC)
Thanks, that's exactly right. Berkeley criticized what he felt was a logical error in assuming dx is nonzero at the start, and discarding it as zero at the end of the calculation. Thenub insists on deleting my completely noncontroversial statement to this effect at non-standard calculus. Tkuvho (talk) 14:37, 30 May 2011 (UTC)
You said Berkeley was talking about the error term. Berkeley wasn't talking about the error term at the end. He was talking about the use evanescent quantities in the first place before you even got that far. That the error term was an evanescent quantity didn't mean he was talking about the error term. Dmcq (talk) 15:10, 30 May 2011 (UTC)
I am not sure I followed everything you wrote. At any rate, let's forget about error terms. There were higher-order terms left at the end of the calculation, e.g. in the case of x^2, that need to be discarded to get the right formula for the derivative. Berkeley was criticizing the discarding of those higher-order terms. Does that correspond to your thinking? Tkuvho (talk) 15:17, 30 May 2011 (UTC)
No it doesn't. I do not believe he was criticizing that. If getting that far was okay then he would probably have been happy with that but his problem with it all was a long way before. He was criticizing getting the velocity of an evanescent quantity which you do by dividing the distance evanescent quantity by the time evanescent quantity. Dmcq (talk) 15:30, 30 May 2011 (UTC)
I am confused again. By the "time evanescent quantity", do you mean the vanishing increment of time, dt so to speak? Or do you mean the t-dot with respect to some other parameter? Tkuvho (talk) 15:34, 30 May 2011 (UTC)
I mean the dt. I mean that he was talking about a velocity and how you calculate velocity is by dividing distance by time. The evanescent quantity he referred to was the distance travelled when getting the velocity. The velocity itself had problems from how it was calculated. As to error terms he did talk a bit about the discarded remainder ab when multiplying (A+a) by (B+b) but the main comment about that was "Nor will it avail to say that ab is a Quantity exceeding small: Since we are told that in rebus mathematicis errores quam minimi non sunt contemnendi". Dmcq (talk) 15:58, 30 May 2011 (UTC)
By the way he was quoting Newton back at himself with that bit of Latin. Dmcq (talk) 16:04, 30 May 2011 (UTC)
Let's get back to basics. Let's say distance s depends on time t in terms of s=t^2. Then we take an evanescent increment dt, calculate ds as (t+dt)^2-t^2, form the prime ratio ds/dt. We get 2t+dt. To pass from this to the ultimate ratio 2t, the term dt has to be dropped. It's true it is very small, even evanescent. But that wasn't good enough for Berkeley. Either it's zero or it's not zero. You can't have both. If we can't agree on a summary of Berkeley's criticism, it is going to be hard to make progress. Tkuvho (talk) 16:09, 30 May 2011 (UTC)
He talked about 'The Velocities of evanescent Increments' and that the evanescent quantities were the ghosts of departed quantities. He didn't refer to remainder or error terms in this. His argument here would have been just as valid for s=t which has no error term. Dmcq (talk) 17:51, 30 May 2011 (UTC)
OK, that's interesting. If he was not criticizing the disappearing dt, what was he criticizing exactly? And what was his doctrine of compensating errors supposed to correct? Tkuvho (talk) 18:42, 30 May 2011 (UTC)
The compensating errors bit was where he talked about that the tangent to a parabola calculated using a finite increment away from it wasn't the same as that using the geometric definition, but this error was compensated by ignoring the error even though ignoring the bit over is another error. The double error then gave a result which corresponded with the classical geometric tangent to the parabola. Dmcq (talk) 19:10, 30 May 2011 (UTC)

(outdent) I think my position was misinterpreted at the beginning, but given the volume of things written today it seems a bit late to chime in with that now. My opinion was that we should stick closely to whatever the secondary sources say. I agree that the changing from zero to non-zero was part of Berkeley's criticisms but not the part that ghosts is referring to. But I disagree with the whole business of reexamining Berkeley's work for ourselves and coming to our own conclusions. That is fine if we are writing a paper, but we are writing an wikipedia article, to the extent possible we should stick with the sources. Thenub314 (talk) 02:55, 31 May 2011 (UTC)

If the problem is secondary references, then they can be readily provided. Tkuvho (talk) 18:32, 31 May 2011 (UTC)
Which secondary source are you referring to? Dmcq (talk) 17:40, 31 May 2011 (UTC)
I had meant the sources I cited, that have been Tkuvho feels are flawed and that they will not be accepted. These happen to be the ones that that disagree with his opinion that this quote has to do with infinitesimals. For example Boyer. Thenub314 (talk) 21:06, 31 May 2011 (UTC)
Boyer seems to be quite explicitly referring to the problem of the calculation of the fluxions becoming 0/0, not to an error or remainder term. Dmcq (talk) 22:15, 31 May 2011 (UTC)
I agree, I had generally been removing remarks that referred to error/remainder terms as ghosts of departed quantities. Was your comment meant for Tkuvho, or am I missing something? Thenub314 (talk) 22:56, 31 May 2011 (UTC)
Why does the ratio allegedly become 0/0? Because we set delta-x equal to 0 at the end of the calculation. So Boyer's interpretation is not incompatible with what Berkeley says in XXXV. At any rate, even if we retain the Boyer references, I don't see why we need to rely on him exclusively. For instance, the Kleiner reference that Thenub introduced at another page, specifically criticizes Fermat for allegedly setting delta-x to zero at the end of the calculation, and states that Berkeley's criticism applies to him. Regardless of whether the criticism does or does not apply to Fermat, we see clearly that Kleiner holds that Berkeley is criticizing the business of setting delta-x to zero at the end of the calculation. Besides, since that's what Berkeley says himself, it is not surprising that Kleiner would interpret him this way. There are additional secondary sources that make this clear. Tkuvho (talk) 03:35, 1 June 2011 (UTC)
Which reference was that please? I thought there was more than one criticism in The Analyst. The one about starting off with non-zero getting a result and then supposing you started off with zero was his major one but I don't think that was what he was explicitly referring to in the bit about ghosts of departed quantities. Dmcq (talk) 21:10, 1 June 2011 (UTC)

(outdent) I think Tkuvho meant "The Role of Paradoxes in the Evolution of Mathematics" by I. Kleiner and N. Movshovitz-Hadar; The American Mathematical Monthly, Vol. 101, No. 10 (Dec., 1994), pp. 963-974,[2]. Where they discuss adequality on page 970, they use the phrase "ghosts of departed quantities" when describing what Fermat was doing, but note that what Berkeley was writing about something "in a somewhat different context". Thenub314 (talk) 23:51, 1 June 2011 (UTC)

Is this page about "a summary of Berkeley's criticism" in The Analyst or only about that damn quote about ghosts from The Analyst? Can someone please explain this to me? iNic (talk) 18:02, 30 May 2011 (UTC)

Do you wish to change the title? Or merge it into The Analyst and deal with some other matters there as well? Otherwise why do you think the title was chosen? Dmcq (talk) 18:17, 30 May 2011 (UTC)

This is not an answer to my question. This is instead three new questions. iNic (talk) 20:41, 30 May 2011 (UTC)

It is about that quote. Otherwise the title would be The Analyst. Dmcq (talk) 20:56, 30 May 2011 (UTC)

OK this is what I thought too for a while. But after having examining this quote in detail here other quotes from the same book are suddenly being discussed instead as if it was as relevant as the quote itself. And editor Tkuvho reveals that his mission is even broader: "If we can't agree on a summary of Berkeley's criticism, it is going to be hard to make progress." Eh? A summary of a whole book is in general something totally different from nailing the meaning of just one quote from a book. And if we have consensus that this is the real mission of the page why not just make a redirect to The Analyst page? There you should find the summary of Berkeley's criticism. I think that would be the best solution. Still no one has given any arguments why this spooky quote is of such an importance as to have it's own wikipedia page. Does every famous quote from the Bible, for example, have its own wikipedia page? I don't think so. So why must this particular quote haunt us? iNic (talk) 23:57, 30 May 2011 (UTC)

## Berkeley speaks

Here is what Berkeley had to say about the object of his criticism, a few lines before the "ghost" quote, in the context of two nearby points x and z: "But herein is a direct Fallacy: for in the first place, it is supposed that the Abscisses z and x are unequal, without such supposition no one step could have been made; and in the second place, it is supposed they are equal; which is a manifest Inconsistency, and amounts to the same thing that hath been before considered". Either x=z, or not, but you can't have both. This is the context of his "ghost" remark. Tkuvho (talk) 19:03, 30 May 2011 (UTC)

Yes I agree this is directly related to the ghosts remark. Whast are you trying to say about it? Dmcq (talk) 19:15, 30 May 2011 (UTC)
That x-z is nonzero at the start of the calculation, but at the end of the calculation is paradoxically treated as zero. that's the paradox Berkeley refers to by the term "ghosts of departed quantities". Tkuvho (talk) 02:29, 31 May 2011 (UTC)
I really shouldn't indulge in interpreting Berkeley, but for what it is worth, I would disagree that "ghosts of departed quantities" refers to this issue. Primarily because of the passage in between "The great Author of the Method of Fluxions felt this Difficulty, and therefore he gave in to those nice Abstractions and Geometrical Metaphysics, without which he saw nothing could be done..." That is, Newton noticed this difficulty in introduced some more abstract notion (which I understand to be ultimate ratios, but presumably others feel are infinitesimals) and then he leaves it to the reader to decide if Newton made a more satisfactory explanation, and proceeds to ask his questions about what Newtons abstraction — fluxions. I would like to understand your point of view a bit better, so I am a bit curious how you interpret one bit of the whole thing. If these vanishing increments, whose ratio is forms the fluxion, are infinitesimal quantities then why does Berkeley include the phrase "nor Quantities infinitely small" in his list of things that these vanishing quantities cannot be? Thenub314 (talk) 03:11, 31 May 2011 (UTC)
He is again alluding to the fact that Newton's supporters claim that Newton's approach is more rigorous than the continental approach, in that it does not use infinitesimals. In retrospect, we know that Newton's ultimate ratios can be justified without infinitesimals. But this is precisely what Berkeley was sceptical about. Tkuvho (talk) 04:46, 31 May 2011 (UTC)

## There is no such thing as infinitesimal.

It seems to me that the math gods of Wikipedia (fools that they are) are finally beginning to realize that there is no such thing as infinitesimal. If you want to define the number that succeeds zero on the "real" number line, just call it the incommensurable successor of zero because it is by no means infinitely small - zero is smaller. So is any negative "number". Abraham Robinson was a fool. 71.132.139.182 (talk) 19:34, 16 June 2011 (UTC)

It is not worth my time to argue. But your incorrect on many accounts. Particularly about Robinson. Thenub314 (talk) 19:37, 16 June 2011 (UTC)