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The origin of this assertion appears to be the following passage from Foundations of Constructive Analysis (Bishop, 1967):
"Our program is simple: To give numerical meaning to as much as possible of classical abstract analysis. Our motivation is the well-known scandal, exposed by Brouwer (and others) in great detail, that classical mathematics is deficient in numerical meaning." (Preface, page ix)
If so, it was a distortion to say anyone believed "non-constructive mathematics . . . WAS a scandal . . .".
Possibly this entire article is synthesis, and unsuitable for Wikipedia even if all the mistakes could be corrected. 66.245.43.17 ( talk) 17:36, 17 October 2009 (UTC)
As described in the previous section, there is a problem that Feferman's paraphrase of Bishop is inaccurate. Could we have a discussion of the pros and cons of the recent change from this version, which I prefer --
-- to this version, in which the paraphrase of the "scandal" statement is misleading in my opinion, and anyway superfluous when the primary source has already been quoted:
Thanks! 66.245.43.17 ( talk) 05:19, 3 November 2009 (UTC)
The paraphrase was removed since it was redundant at best and misleading at worst. The removal was reverted with the edit summary 'fact of "scandal" criticism being reflected in secondary source is significant'. The article by Feferman never mentions non-standard analysis. The original quote using the word "scandal", which Feferman paraphrased, concerned the history of mathematics before non-standard analysis existed. To provide background on Bishop's criticism of non-standard analysis, his general philosophy may be described, but a redundant or misleading paraphrase should not be included without a strong justification. If you insist on including the paraphrase, please provide an explanation of why it is "significant" in this context, and why it is neither redundant nor misleading. Pendjari ( talk) 20:21, 13 December 2009 (UTC)
The comment "Errett Bishop was concerned about the use of non-standard analysis in teaching calculus, as he mentioned in his essay "Crisis in mathematics" (Bishop 1975)" recently added to the page may be Bishop's spin on the story. Whether or not the page should adopt his view, as opposed to the view of Artigue, Dauben, Feferman, Komkov, Tall, and others, should be discussed in this space before any further controversial edits are attempted. Tkuvho ( talk) 12:50, 15 December 2009 (UTC)
Constructive non-standard analysis was developed subsequently (Moerdijk 1995, Palmgren 1998, Ruokolainen 2004).
Ieke Moerdijk, A model for intuitionistic nonstandard arithmetic, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 37-51. "Abstract: This paper provides an explicit description of a model for intuitionistic non-standard arithmetic, which can be formalized in a constructive metatheory without the axiom of choice." http://www.sciencedirect.com/science/journal/01680072
Erik Palmgren, Developments in Constructive Nonstandard Analysis, Bull. Symbolic Logic Volume 4, Number 3 (1998), 233-272. "Abstract: We develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. ..." http://projecteuclid.org/euclid.bsl/1182353577
Juha Ruokolainen 2004, Constructive Nonstandard Analysis Without Actual Infinity https://oa.doria.fi/bitstream/handle/10024/2865/construc.pdf
The paper by Ruokolainen says: "The possibility of constructivization of nonstandard analysis has been studied thoroughly by Palmgren (1997, 1998, 2001). The model of constructive nonstandard analysis studied there is an extension of Moerdijk’s (1995) model for constructive nonstandard arithmetic."
Pendjari ( talk) 15:54, 15 December 2009 (UTC) and also Surreal Numbers, which contain constructive infinitesimals, although they are neither intuitionist nor analytical. 198.228.198.52 ( talk) 22:10, 22 July 2011 (UTC) Collin237
I have added the standard {{ talk page}} header, as a reminder that discussion here should be carried on without personal animus. Charles Matthews ( talk) 10:38, 19 December 2009 (UTC)
The current form of the introduction is a bit misleading. Tkuvho ( talk) 07:13, 7 April 2010 (UTC)
This talk page is way too long (see WP:TPNO#When_to_condense_pages and it shows up in Wikipedia:Database reports/Long pages). Does anyone object to me setting up automatic archiving for this page using MiszaBot? Unless otherwise agreed, I would set it to archive threads that have been inactive for 30 days and keep ten threads.-- Oneiros ( talk) 13:41, 25 July 2010 (UTC)
The article mentions the name of "Robinson" quite a few times without explaining who he was. GregorB ( talk) 20:38, 11 December 2010 (UTC)
In this section, all of the material that follows Dauben's quote to the the start of the subsection "Bishop's review" is primarily about constructive mathematics. Which is unrelated to Criticism of NSA. The section opens with by pointing out Bishop's constructivist so I don't see the value of having several quotes about it later in the section. — Preceding unsigned comment added by thenub314 ( talk • contribs)
I think the section on Connes should be removed. There is simply not criticism in the article, we are completely taking his comments out of context. Notice that Connes is working in the area of Noncommutative geometry and for obvious reasons wishes to have a notion of infinitesimal that is not commutative. Given that the non-standard reals are commutative, he cannot use them. So he spends time justifying why he needs to go through the effort of producing a notion of infinitesimal when NSA already provides one. Consider the opening lead to the section "We shall develop in this section a calculus of infinitesimal real and complex variables based on operators in Hilbert space. Let us first explain why the formalism of nonstandard analysis is inadequate."
We have no secondary source material to suggest he is critical of NSA. We have simply quotes from a section that says: "This is why we need/want a new tool in this setting" and placed it along side more directly critical commentary. Thenub314 ( talk) 06:06, 24 May 2011 (UTC)
There are some definitions of infinitesimals that are considered (as I understand it) ur-objects, rather than numbers. In that case, constructibility is a non-issue, and non-commutativity can be simply imposed. (Although it might be claimed that such infinitesimals are not really infinitesimal.) Specifically I'm familiar with anti-commuting differential forms, although there may be other examples. 198.228.198.52 ( talk) 21:51, 22 July 2011 (UTC) Collin237
Connes article was not meant to be a criticism of non-standard analysis. After reading the apporiate sections of Katz and Katz, it was clear they were discussing how intuitionists should view Connes work and did not challange Connes comments, nor label them as a criticism of NSA. Since only new paragraph misrepresented the reference, I removed the section once more. Thenub314 ( talk) 04:21, 26 November 2011 (UTC)
I think that the reference to the AMM article is a good fit for the article. We have to keep NPOV in mind, and not paint nonstandard analysis as moremainstream than it actually is. Millions of students learn calculus each year, and all but a vanishing few learn it in a way without infinitesimals. We cannot expect to see a huge literature on it, so if something as prominent as the Monthly has an article, that ought to be mentioned here. — Carl (
CBM ·
talk) 14:50, 28 November 2011 (UTC)
This article focuses too much on the critics and the views that influence their criticisms (like Bishop's constructivist views) and too little on the actual criticisms. It's full of drama and who-said-what and who-responded-to-whom etc.
Why not have the article structured like an account of the actual criticisms (or at least mark it as requiring a rewrite), with headers like "Questions of rigor", "Incompatibility with constructivism", "Practical applications", "Pedagogical difficulties" etc.? The current content should be reduced to a History section of the article. Skl ( talk) 13:11, 21 February 2014 (UTC)
This is way too much space for basically one critique. This should be minor part of another article. The problem with the article is that if you read it through, you understand that there is no general and prolonged historical criticism that is justified. And, on the other hand, I did not see so far any article saying a criticism on this or that proof. That is truly nonsense. A proof can be complicated or something, but cannot be criticized as musical piece or painting. Non-standard analysis is a theory and anyone's attempt to criticize from the perspective of Facebook like/dislike principle is pure nonsense. Even worse is that this article is based on one book, which is by the way, an excellent book, I bought it since I was annoyed by this article, not understanding what it says actually.
Well, maybe that is not bad, in the end, I got a good book.
Aperisic ( talk) 08:52, 27 March 2014 (UTC)
As of now, the lead is a mess of quotations. While quotations are probably needed and can sometimes be a great way to illustrate and explain various things, leads are meant to present an overview of the article content and sum things up in a general way. I therefore suggest that the lead is rewritten and that the quotations are moved to the body of the article instead.
This topic can be viewed as in line with Skl's section above from 2014. RhinoMind ( talk) 16:17, 16 May 2015 (UTC)
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This is the
talk page for discussing improvements to the
Criticism of nonstandard analysis article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
Archives: Index, 1 |
This article was nominated for deletion on 14 December 2008 (UTC). The result of the discussion was Keep after confused discussion. |
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
The origin of this assertion appears to be the following passage from Foundations of Constructive Analysis (Bishop, 1967):
"Our program is simple: To give numerical meaning to as much as possible of classical abstract analysis. Our motivation is the well-known scandal, exposed by Brouwer (and others) in great detail, that classical mathematics is deficient in numerical meaning." (Preface, page ix)
If so, it was a distortion to say anyone believed "non-constructive mathematics . . . WAS a scandal . . .".
Possibly this entire article is synthesis, and unsuitable for Wikipedia even if all the mistakes could be corrected. 66.245.43.17 ( talk) 17:36, 17 October 2009 (UTC)
As described in the previous section, there is a problem that Feferman's paraphrase of Bishop is inaccurate. Could we have a discussion of the pros and cons of the recent change from this version, which I prefer --
-- to this version, in which the paraphrase of the "scandal" statement is misleading in my opinion, and anyway superfluous when the primary source has already been quoted:
Thanks! 66.245.43.17 ( talk) 05:19, 3 November 2009 (UTC)
The paraphrase was removed since it was redundant at best and misleading at worst. The removal was reverted with the edit summary 'fact of "scandal" criticism being reflected in secondary source is significant'. The article by Feferman never mentions non-standard analysis. The original quote using the word "scandal", which Feferman paraphrased, concerned the history of mathematics before non-standard analysis existed. To provide background on Bishop's criticism of non-standard analysis, his general philosophy may be described, but a redundant or misleading paraphrase should not be included without a strong justification. If you insist on including the paraphrase, please provide an explanation of why it is "significant" in this context, and why it is neither redundant nor misleading. Pendjari ( talk) 20:21, 13 December 2009 (UTC)
The comment "Errett Bishop was concerned about the use of non-standard analysis in teaching calculus, as he mentioned in his essay "Crisis in mathematics" (Bishop 1975)" recently added to the page may be Bishop's spin on the story. Whether or not the page should adopt his view, as opposed to the view of Artigue, Dauben, Feferman, Komkov, Tall, and others, should be discussed in this space before any further controversial edits are attempted. Tkuvho ( talk) 12:50, 15 December 2009 (UTC)
Constructive non-standard analysis was developed subsequently (Moerdijk 1995, Palmgren 1998, Ruokolainen 2004).
Ieke Moerdijk, A model for intuitionistic nonstandard arithmetic, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 37-51. "Abstract: This paper provides an explicit description of a model for intuitionistic non-standard arithmetic, which can be formalized in a constructive metatheory without the axiom of choice." http://www.sciencedirect.com/science/journal/01680072
Erik Palmgren, Developments in Constructive Nonstandard Analysis, Bull. Symbolic Logic Volume 4, Number 3 (1998), 233-272. "Abstract: We develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. ..." http://projecteuclid.org/euclid.bsl/1182353577
Juha Ruokolainen 2004, Constructive Nonstandard Analysis Without Actual Infinity https://oa.doria.fi/bitstream/handle/10024/2865/construc.pdf
The paper by Ruokolainen says: "The possibility of constructivization of nonstandard analysis has been studied thoroughly by Palmgren (1997, 1998, 2001). The model of constructive nonstandard analysis studied there is an extension of Moerdijk’s (1995) model for constructive nonstandard arithmetic."
Pendjari ( talk) 15:54, 15 December 2009 (UTC) and also Surreal Numbers, which contain constructive infinitesimals, although they are neither intuitionist nor analytical. 198.228.198.52 ( talk) 22:10, 22 July 2011 (UTC) Collin237
I have added the standard {{ talk page}} header, as a reminder that discussion here should be carried on without personal animus. Charles Matthews ( talk) 10:38, 19 December 2009 (UTC)
The current form of the introduction is a bit misleading. Tkuvho ( talk) 07:13, 7 April 2010 (UTC)
This talk page is way too long (see WP:TPNO#When_to_condense_pages and it shows up in Wikipedia:Database reports/Long pages). Does anyone object to me setting up automatic archiving for this page using MiszaBot? Unless otherwise agreed, I would set it to archive threads that have been inactive for 30 days and keep ten threads.-- Oneiros ( talk) 13:41, 25 July 2010 (UTC)
The article mentions the name of "Robinson" quite a few times without explaining who he was. GregorB ( talk) 20:38, 11 December 2010 (UTC)
In this section, all of the material that follows Dauben's quote to the the start of the subsection "Bishop's review" is primarily about constructive mathematics. Which is unrelated to Criticism of NSA. The section opens with by pointing out Bishop's constructivist so I don't see the value of having several quotes about it later in the section. — Preceding unsigned comment added by thenub314 ( talk • contribs)
I think the section on Connes should be removed. There is simply not criticism in the article, we are completely taking his comments out of context. Notice that Connes is working in the area of Noncommutative geometry and for obvious reasons wishes to have a notion of infinitesimal that is not commutative. Given that the non-standard reals are commutative, he cannot use them. So he spends time justifying why he needs to go through the effort of producing a notion of infinitesimal when NSA already provides one. Consider the opening lead to the section "We shall develop in this section a calculus of infinitesimal real and complex variables based on operators in Hilbert space. Let us first explain why the formalism of nonstandard analysis is inadequate."
We have no secondary source material to suggest he is critical of NSA. We have simply quotes from a section that says: "This is why we need/want a new tool in this setting" and placed it along side more directly critical commentary. Thenub314 ( talk) 06:06, 24 May 2011 (UTC)
There are some definitions of infinitesimals that are considered (as I understand it) ur-objects, rather than numbers. In that case, constructibility is a non-issue, and non-commutativity can be simply imposed. (Although it might be claimed that such infinitesimals are not really infinitesimal.) Specifically I'm familiar with anti-commuting differential forms, although there may be other examples. 198.228.198.52 ( talk) 21:51, 22 July 2011 (UTC) Collin237
Connes article was not meant to be a criticism of non-standard analysis. After reading the apporiate sections of Katz and Katz, it was clear they were discussing how intuitionists should view Connes work and did not challange Connes comments, nor label them as a criticism of NSA. Since only new paragraph misrepresented the reference, I removed the section once more. Thenub314 ( talk) 04:21, 26 November 2011 (UTC)
I think that the reference to the AMM article is a good fit for the article. We have to keep NPOV in mind, and not paint nonstandard analysis as moremainstream than it actually is. Millions of students learn calculus each year, and all but a vanishing few learn it in a way without infinitesimals. We cannot expect to see a huge literature on it, so if something as prominent as the Monthly has an article, that ought to be mentioned here. — Carl (
CBM ·
talk) 14:50, 28 November 2011 (UTC)
This article focuses too much on the critics and the views that influence their criticisms (like Bishop's constructivist views) and too little on the actual criticisms. It's full of drama and who-said-what and who-responded-to-whom etc.
Why not have the article structured like an account of the actual criticisms (or at least mark it as requiring a rewrite), with headers like "Questions of rigor", "Incompatibility with constructivism", "Practical applications", "Pedagogical difficulties" etc.? The current content should be reduced to a History section of the article. Skl ( talk) 13:11, 21 February 2014 (UTC)
This is way too much space for basically one critique. This should be minor part of another article. The problem with the article is that if you read it through, you understand that there is no general and prolonged historical criticism that is justified. And, on the other hand, I did not see so far any article saying a criticism on this or that proof. That is truly nonsense. A proof can be complicated or something, but cannot be criticized as musical piece or painting. Non-standard analysis is a theory and anyone's attempt to criticize from the perspective of Facebook like/dislike principle is pure nonsense. Even worse is that this article is based on one book, which is by the way, an excellent book, I bought it since I was annoyed by this article, not understanding what it says actually.
Well, maybe that is not bad, in the end, I got a good book.
Aperisic ( talk) 08:52, 27 March 2014 (UTC)
As of now, the lead is a mess of quotations. While quotations are probably needed and can sometimes be a great way to illustrate and explain various things, leads are meant to present an overview of the article content and sum things up in a general way. I therefore suggest that the lead is rewritten and that the quotations are moved to the body of the article instead.
This topic can be viewed as in line with Skl's section above from 2014. RhinoMind ( talk) 16:17, 16 May 2015 (UTC)
Hello fellow Wikipedians,
I have just modified one external link on Criticism of non-standard analysis. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
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source check}}
(last update: 18 January 2022).
Cheers.— InternetArchiveBot ( Report bug) 17:09, 14 August 2017 (UTC)