Talk:Criticism of non-standard analysis

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Removal of "scandal" assertion; synthesis[edit]

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)

Please note that the question of the inclusion of this topic, which can certainly be debated, should not be debated in terms of the current content of the article. That is very much a separate issue. The point you raise can be looked into further, but I believe that some people did believe that. Charles Matthews (talk) 13:32, 18 October 2009 (UTC)
It seems that what you are saying is that Solomon Feferman in http://math.stanford.edu/~feferman/papers/relationships.pdf is misquoting Bishop. That is, when on p. 3 of the Feferman writes
Bishop criticized both non-constructive classical mathematics and intuitionism. He called non-constructive mathematics a "scandal", particularly because of its "deficiency in numerical meaning".
then Feferman is mistaken on what Bishop's view was. Charles Matthews (talk) 13:39, 18 October 2009 (UTC)
Thank you for helping to clear this up. It appears that Feferman is responsible for the "scandal" mistake, assuming that there is no other original source for it than the above quote by Bishop. Bishop said Brouwer had exposed a scandal, he didn't say mathematics WAS a scandal. Feferman also made the odd mistake of saying Bishop "died of leukemia"; in reality Bishop died of pancreatic cancer, he never had leukemia. These two mistakes suggest that any assertions derived from Feferman's paper should be double-checked for accuracy. From a quick glance, Feferman's paper appears to express approval of Bishop's philosophy. Presumably Feferman's misrepresentation was unintentional, just a little careless. Introduction of the "scandal" assertion into this Wikipedia article, on the other hand, was evidently motivated by a desire to portray Bishop as an eccentric with a bad attitude towards mathematics in general, in order to discredit his criticism of non-standard analysis. I say "evidently" on the basis of reading not only the current article and its talk page, but also their histories. I don't understand what you mean, "should not be debated in terms of the current content of the article". The whole article suffers from the disease of synthesis, this was just one small example. Fix all the falsehoods and the article may still be fundamentally incoherent. What is it about? Bishop's criticism of non-standard analysis in an undergraduate calculus textbook, largely confined to a single book review? Connes' alleged criticism of something else, currently disputed? Halmos' alleged criticism of something else, also currently disputed? Anything more anyone would care to throw in as long as it's controversial and permits a variety of mathematicians to be misquoted? 66.245.43.17 (talk) 18:39, 18 October 2009 (UTC)
Well, without the text that Feferman is quoting from in front of me, I think I'd like to suspend judgement on the nature of his supposed "mistake". It seems to me that in the quote (introduced on 8 January 2009 verbatim, by the way) it is unlikely that the word "scandal" was not in the passage referred to. And if you look at the next sentence I doubt that that inclusion was intended to misrepresent Bishop's views. But, in any case, your speculations on people being "evidently motivated" don't get us very far at all. The rules of the game here are that content can be included if verifiable: the Feferman quote is verifiable and later editing into indirect speech may have been a mistake, but I don't see that the initial inclusion was a mistake at all. And you fail to take the point I made about the topic. We decide whether "Criticism of non-standard analysis" is an encyclopedic topic in one way, by deletion process; we decide whether aspects of the current content should stay or not, on whether the article is neutral or not, or otherwise contravenes content policy, in another fashion. Charles Matthews (talk) 21:25, 18 October 2009 (UTC)
Please look again at the quotation I provided from page ix of Bishop 1967. The word "scandal" is in it; I guess you hadn't seen that when you wrote it's unlikely the word was not in the passage. The word is there, and Feferman's mistake is clear (unless there's another passage in which Bishop really did say non-constructive mathematics WAS a scandal). I agree with your call for adherence to Wikipedia principles and procedures, and I wish you all the best of luck turning this into a decent article if you think it has potential value. If you prefer to ignore the motivations of other editors, good for you. Thank you for your effort to be positive. 66.245.43.17 (talk) 23:11, 18 October 2009 (UTC)
To get to the point about the existence of the article: the respectable argument against such a criticism article comes under Wikipedia:Content forking, and the notion of POV fork. This guideline is what should be read with your objections to the idea of bringing various critical views together. That deals with the principle: it happens that it is very unusual even to envisage a "criticism of X" where X is a mathematical topic, rather than, say, Wal-mart. What is going on here is somewhat unusual in that "non-standard analysis" is both (i) a branch of model theory, and (ii) a proposed way of (re)considering some very old material in calculus and its pedagogy. The article could of course make that clearer. It could in fact make many things clearer. I would have some sympathy with the view that this is a POV fork (and so better merged into the NSA article); and also has not managed to clarify the "meta" issue which is the debate about the aspect (ii) of NSA. In summary, it might be possible to have sensible article on (ii) which was not in the form of a POV fork of the article on (i); this is not yet that article; and I wonder if the nature of the literature on NSA will allow such an article to thrive. Charles Matthews (talk) 07:55, 19 October 2009 (UTC)
There seem to be two points here. First, there is the quote from Feferman that
"Bishop criticized both non-constructive classical mathematics and intuitionism. He called non-constructive mathematics a "scandal", particularly because of its "deficiency in numerical meaning"."
66.245.43.17 has claimed this is an erroneous reading of Bishop, who wrote
"Our motivation is the well-known scandal, exposed by Brouwer (and others) in great detail, that classical mathematics is deficient in numerical meaning.""
I think Feferman's reading is fine. I do not think it is realistic to argue that Bishop is only saying that "the deficiency in numerical meaning" is a scandal, but that classical mathematics itself is not. The entire point of Bishop's book, after all, is to present a significant chunk of constructive mathematics in an effort to convince people that the idealistic aspects of classical mathematics are unnecessary. As Bishop says,
"This book is a piece of constructivist propaganda, designed to show that there does exist a satisfactory alternative" to classical mathematics.
Viewing the entire book in context, I think that Feferman's statement is perfectly within the bounds of reasonable scholarly interpretation of Bishop's words.
Second, whether this entire article is some form of original synthesis. I am more sympathetic to 66.245.43.17 about this. The article has three broad sections at present: Bishop, Connes, and Halmos. But these are criticisms of different things.
  • Bishop was criticizing the use of non-standard analysis to teach calculus.
  • Connes was advocating his own foundational system, and so he was just describing why infinitesimal calculus does not meet his needs. This is highlighted by his choice of commutativity of infinitesimals as a problem in the theory.
  • Halmos was criticizing nonstandard analysis as a matter of taste, arguing that it is an idiosyncratic way of writing up results.
None of these criticisms really relates to the others, so the article has the appearance of a collection of every sort of "criticism" that has a appeared in print. I think that, in general, it is better to integrate all these "criticisms" into the main article, in appropriate places. — Carl (CBM · talk) 13:39, 19 October 2009 (UTC)
Unless someone proposes to reintroduce the "scandal" assertion, maybe this is a moot point, but please consider an analogy. Suppose Smith says, "Woodward and Bernstein exposed a scandal in the US government, known as Watergate." Then someone paraphrases: "Smith believed that the US government was a scandal." Would that be perfectly within the bounds of scholarly interpretation? I don't think so. There is a meaningful distinction between saying Watergate is a scandal, and saying the government is a scandal. 66.245.43.17 (talk) 16:46, 19 October 2009 (UTC)
I think you are not considering the context. Things would be somewhat different if Smith's book was entirely about reforming the US Government, and Smith was well known for his viewpoint that the US government was fatally flawed and in need of reform, and the quote was not about a specific incident (Watergate in the US government) but about a perceived-to-be endemic problem (lack of accountability of the president under the US constitution). So, as I was saying, I think it takes a lot of effort to argue that Bishop is only saying that "the deficiency in numerical meaning" is a scandal, but not criticizing classical mathematics as scandalous at the same time. — Carl (CBM · talk) 17:18, 19 October 2009 (UTC)
I'm aware of the context. The burden of proof is nevertheless on whoever would claim Bishop called classical mathematics a scandal. Your argument is synthesis. Just because you want to reform something, that doesn't mean you think it's "scandalous" or "fatally flawed". Those are your words. Bishop said what he said. He didn't say what he didn't say. 66.245.43.17 (talk) 17:39, 19 October 2009 (UTC)
I'm not sure what you're saying; I didn't call anything a scandal. Since Feferman has already published his paper, he doesn't have to prove anything, and Wikipedia would consider his comments reliable by default. So, if you want to convince other people that Feferman's argument is incorrect, the burden of proof is on you. After looking through the prologue to Constructive Analysis again, I can only say you have not convinced me that Feferman's comment is incorrect in this situation. (I am a classical mathematician, by the way, but I don't think it affects my judgment here.) — Carl (CBM · talk) 19:16, 19 October 2009 (UTC)
It's a moot point as the article now stands, so I won't waste any more of our time on it unless the disputed assertion finds its way back into the article. 66.245.43.17 (talk) 20:41, 19 October 2009 (UTC)

Feferman's published comment[edit]

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 --

In Foundations of Constructive Analysis (1967, page ix), Bishop wrote:
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.
According to logician Solomon Feferman, Bishop criticized both non-constructive classical mathematics and intuitionism on constructive grounds:
What [Bishop] simply meant was that if you say something exists you ought to be able to produce it, and if you say there is a function which does something on the natural numbers then you ought to be able to produce a machine which calculates it out at each number. [1]

-- 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:

In Foundations of Constructive Analysis (1967, page ix), Bishop wrote:
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.
Bishop's comments have been reflected in secondary literature, as when logician Solomon Feferman wrote:
Bishop criticized both non-constructive classical mathematics and intuitionism. He called non-constructive mathematics "a scandal", particularly because of its "deficiency in numerical meaning".
According to Solomon,
What [Bishop] simply meant was that if you say something exists you ought to be able to produce it, and if you say there is a function which does something on the natural numbers then you ought to be able to produce a machine which calculates it out at each number. [2]

Thanks! 66.245.43.17 (talk) 05:19, 3 November 2009 (UTC)

It should be kept in mind that Feferman may have been misquoted in the previous section. Feferman did not quote Bishop as saying that classical mathematics was a scandal. Rather, Feferman quoted Bishop as saying that non-constructive mathematics was a scandal. Tkuvho (talk) 15:43, 3 November 2009 (UTC)
How about simply removing the second sentence ("He called ...") from the Feferman quote? 66.245.43.17 (talk) 00:19, 11 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)

If you feel that Feferman's published comments are misleading you are free to publish a rebuttal, but doing so in the framework of wiki is WP:OR. Tkuvho (talk) 09:58, 14 December 2009 (UTC)
It's not clear how removal of a sentence could be considered original research or "publishing rebuttals". You reverted it again, with the edit summary "Penjari withes to quarrel with [name of distinguished mathematician]". In the future please don't put my name into your edit summary, especially not in a false accusation of conflict with a distinguished mathematician. (The accusation is slanderous, although I'm not saying you intended it that way.) Anyone (and I have no reason to think the distinguished mathematician would be an exception) might agree that the direct quotation of Bishop here is a more appropriate representation of Bishop's own philosophy. Let's stick to the point. Why is the paraphrase in this context "significant", and not redundant? Pendjari (talk) 13:55, 15 December 2009 (UTC)
Feferman is a distinguished logician and foremost expert in Bishop's field. His opinion of Bishop's attitude toward classical mathematics in general and non-standard analysis in particular is the most pertinent one. You clearly disagree with Feferman and your opinion should be respected. Please report on published material that reflects your opinion. Tkuvho (talk) 15:56, 15 December 2009 (UTC)

Questionable edits[edit]

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)

You're mistaken; that sentence was not recently added to the page. The only recent changes to that sentence are deletion of the the word "particularly" and change of the word "discussed" to "mentioned"; otherwise it appears to be unchanged since January 2009. You've reverted several changes that I put a lot of work into, especially the organization, and you haven't explained why you find them questionable or controversial. I'm happy to discuss the changes. The word "particularly" wasn't clear so I deleted it. The word "mentioned" seems more accurate if the essay only contained a few sentences concerning non-standard analysis. The organizational change is essentially a matter of putting things under headings where they belong, in logical or chronological order. Since there was already a section "Responses", put all the responses in that section. There was already a section header for the review, but none for the essay; those are the two pieces of criticism, so have a section header for each, and add the years to clarify ("1975 essay" and "1977 review"). The only remaining material that doesn't clearly belong under any of those three headings is related to constructivism, so I made a fourth section heading "Relation to constructivism". The section "Responses" mentions constructivism repeatedly; for the readers' comprehension, it makes some sense to put the constructivism section before the responses section, but it could go after the responses. Of the four sections, the constructivism section seems most problematic in terms of avoiding bias, synthesis, original research, undue weight, etc. Currently there may be a problem with the implication that constructivism and non-standard analysis are incompatible. How about adding the following to the end of that section:

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

See my comment above concerning Feferman. His published comments are appropriate as an opening, to set the stage for other comments. Constructive n.s.a. is a fascinating topic. What is relevant at this page is Robinson's n.s.a., and Bishop's criticism of it. Tkuvho (talk) 16:01, 15 December 2009 (UTC)
Thanks for compiling the material on constructive non-standard analysis. The new page should probably be expanded. Tkuvho (talk) 13:40, 16 December 2009 (UTC)
You obstruct me, you make false accusations against me, you exploit me, and then you thank me. Amazing! You had demanded prior discussion of any proposed contribution, so I posted here the material I compiled following the lead of Thenub314. Instead of discussing it, you took it and made your own use of it in creating a separate page, as though it were your own contribution rather than mine and Thenub314's. This is what I mean by "exploit". You're not welcome. You still haven't justified your claim that my changes, which you reverted, were controversial. This is what I mean by "obstruct". Pendjari (talk) 19:04, 18 December 2009 (UTC)

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)

Thanks. Pendjari's contributions to this page are welcome, of course modulo Charles' note. The controversial change I was referring to above is the deletion of Feferman's comment. I added a comment at the talk page of constructive non-standard analysis to the effect that the material is due to Pendjari. Frankly, I don't see anything controversial about the creation of that page. Tkuvho (talk) 20:10, 19 December 2009 (UTC)

Superstructure approach and ZFC[edit]

The current form of the introduction is a bit misleading. Tkuvho (talk) 07:13, 7 April 2010 (UTC)

How? It does not need any new set-theoretic axioms beyond those of ZFC - that's the truth. ––虞海 (Yú Hǎi) 14:29, 12 October 2011 (UTC)
Certain forms of the superstructure approach exploit large cardinals which are not part of the standard package of ZFC. Which form of the superstructure approach are you familiar with? Tkuvho (talk) 22:46, 15 October 2011 (UTC)

Archiving[edit]

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)

I object. Tkuvho (talk) 13:42, 25 July 2010 (UTC)
Why?--Oneiros (talk) 13:58, 25 July 2010 (UTC)
I archived the older discussions, and the page size is reasonable now. I prefer infrequent manual archiving for most article talk pages, because it's easy to miss comments if they are archived too quickly. Automatic archiving is only needed for pages that have nearly steady streams of comments. The discussion here is much more sporadic, so it just needs someone to do it once a year. It had been put off too long, though. — Carl (CBM · talk) 14:01, 25 July 2010 (UTC)

Robinson[edit]

The article mentions the name of "Robinson" quite a few times without explaining who he was. GregorB (talk) 20:38, 11 December 2010 (UTC)

Well, the first use of his name is "Robinson's non-standard analysis...", which correctly implies that he's the guy who developed it, which a quick hop to the non-standard analysis page will confirm. Hga (talk) 15:27, 12 December 2010 (UTC)

Bishops Criticism[edit]

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 (talkcontribs)

The previous comment was added by user Thenub314. In response, these comments are part of a long-established version of the article. They are relevant because the background of Bishop's attack is his philosophy of mathematics, and if it were not for his approach to the philosophy of mathematics, he would not have attacked NSA and Keisler's remarkable book. Please avoid any further one-sided massive deletions of material agreed upon by a number of editors. Tkuvho (talk) 04:12, 24 May 2011 (UTC)
Ok, I will be a little patient, but if memory serves this is not the most highly trafficked page so it may be that you and I are the only two give some input, it would be better to try to compromise. My feeling is the current quotes do a fairly poor job of conveying any meaning about what is going on and we would be much better to linking to the articles on constructive mathematics where one can find a more in depth explanation. Also there is quite a bit of repetition. The section cites the same paper of S. Feferman three. The first two times refer to the deficiency in numerical meaning. The last quote from Feferman is somehow the most clearly related, but it too falls short and leaves the impression that Bishop envisioned computability theory, which he openly criticized for being non-constructive. The quote from Bishop's book, again brings in that he was interested in numerical meaning, which has been well established already by Feferman's article. The sad part is that none of it even suggests that constructive analysis is at odds with NSA. And it some sense it couldn't be fully convincing because as I pointed out here a long time ago there are people studying constructive non-standard analysis (glad to see it made it to its own page.) So I am really left asking myself how does any of these quotes help? Thenub314 (talk) 05:45, 24 May 2011 (UTC)

Connes section[edit]

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)

Do you feel that saying that nonstandard analysis is "inadequate" cannot be described as a "Critique" thereof? Tkuvho (talk) 17:55, 24 May 2011 (UTC)
Depends on how and why "inadequate" was being used. Consider the statement "The 20 dollar bill in my wallet is an inadequate amount of cash to get me through the weekend because of (a list of expenses)." It is not a critique of 20 dollar bills. But more importantly there is a distinction between critique and criticism. A critique is (in my own words) a critical analysis of some work, which may be positive or negative. A criticism is a negative review, commentary, etc. As the article is about criticisms of NSA, even if you felt the Connes was offering a critique, he certainly isn't offering criticism.
More specifically, this is standard practice when introducing a new tool in mathematics (in my own experience). You describe why it is the problem forced you to deviate from the tools at hand. I frequently hear it stated that "Tool X" is inadequate to solve "Problem Y" because..., it is fairly standard parlance. But it is never considered a criticism of "Tool X", usually the statement concerns the nature of "Problem Y" more then anything else. Thenub314 (talk) 06:09, 25 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

Which definitions of infinitesimals as ur-objects are you referring to? Tkuvho (talk) 13:31, 24 November 2011 (UTC)
An infinitesimal displacement in a generalized manifold, if that's notable. Collin237 166.147.104.143 (talk) 02:56, 10 February 2012 (UTC)

Reposting of Connes Section[edit]

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 seem to have been reverted without discussion here. I did see the added references, as I said above they do not say anything relevant to this article. Thenub314 (talk) 19:22, 26 November 2011 (UTC)
The article you mentioned deals with Connes' criticism of nsa in section 6.5. Tkuvho (talk) 19:52, 26 November 2011 (UTC)
I missed the foot note on my first reading, fair enough. Thenub314 (talk) 20:35, 26 November 2011 (UTC)
What's our policy on POV? Is it whatever Thenub314 disagrees with? Tkuvho (talk) 19:29, 27 November 2011 (UTC)
No, I hope nothing ever comes down to my opinion in this way, but there were a lot of unsubstantiated claims in section that were POV is for example "His major work" is an immediate red flag. The section just needed a bit of clean up. Thenub314 (talk) 05:35, 28 November 2011 (UTC)
The phrase "In the view of M. Katz and K. Katz Connes' comments are critical of non-standard analysis" is a bit bizarre. It is generally acknowledged that Connes is a critic of nsa. What do you expect him to say to establish him unequivocally as such a critic: "I became aware of an absolutely major flaw in non-standard analysis, an irremediable defect" or something? Tkuvho (talk) 14:02, 28 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)

Are you referring to Sullivan's AMM article? I am not sure how your comments relate to the discussion in this section. NSA is clearly not mainstream. Tkuvho (talk) 14:54, 28 November 2011 (UTC)
Apparently I edited the wrong talk page. — Carl (CBM · talk) 15:54, 28 November 2011 (UTC)
I responded at Elementary Calculus. Tkuvho (talk) 16:00, 28 November 2011 (UTC)

Article should either be renamed to "Critics of non-standard analysis" or be heavily rewritten[edit]

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)

Too much space[edit]

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)