Talk:Nonstandard analysis

message: Should you really be redirecting from nonstandard model?
I was looking for a page on nonstandard models of first order versions of Peano's Postulates.
Re Connes' criticism, is this really a criticism of the hyperreals, or of one particular construction of the hyperreals? The construction using an ultrafilter is not very "constructive," since an ultrafilter can't be constructed. But there's another, more recent construction by Kanovei et al. that doesn't require an ultrafilter. I'd be grateful for a pointer to any further info on Connes' criticism, both because I'd like to understand it, and because I think this point in the article is not clear as currently written. --Bcrowell 17:52, 5 Mar 2005 (UTC)
replies: text: The point in the article is not very clear, because Connes is not very clear himself.  The arguments he gives against NSA (in his now classic book Noncommutative Geometry) seem to suggest his criticism lies with its nonconstructive aspects.  On the other hand, the Dixmier trace which plays an important role in his non-commutative theory of infinitesimals is non constructive, using a generalized limit in its definition. Fundamentally, Connes started his career in nonstandard analysis futzing around with logic, didn't get far with it, discovered Tomita-Takesaki, and has had a very low esteem of NSA ever since. This low regard is quite widespread among analysts, and is somewhat undeserved, but you do have to ask the question: "what has nonstandard analysis really done?" Not much, unfortunately. The one really unfortunate result of all this is that some very respected analysts (such as Edward Nelson) are now very unfairly regarded as somewhat marginal because they have used NSA or published articles in NSA. CSTAR 18:14, 5 Mar 2005 (UTC)

text: Hi, 84.132.219.203, could you explain why you say that? He's a professor at the U.S. Naval Academy. I'm actually a physicist, myself, not a mathematician. Although I was initially a little put off by the title of his book "Einstein Corrected," it doesn't have any of the hallmarks of crank literature. I haven't sat down and read the whole thing, just browsed through it.--Bcrowell 14:37, 5 May 2005 (UTC)
thread_title: older discussion

message: I have been trying to improve an article about infinitesimals but user 70.120.182.243 is up to the same kind of trouble he is doing here: a refusal to understand that nonstandard analysis is rigorous. I really think his sort of remarks should not appear as they are no help whatsoever at improving the articles. Can we do anything about it? Odonovanr 09:27, 8 May 2007 (UTC)
replies: text: Let's see: Anytime you disagree with what anyone else has to say, you simply remove their comments using a lame excuse such as the preceding one? I am all for finding a new way to teach calculus. I do not support real analysis either - it is flawed in my opinion. However, nonstandard analyis is a bunch of baloney. Unlike real analysis, the concept on which it is built (the infinitesimal) is ill-defined. This article and the article on infinitesimals should be deleted or revised so it is clear to anyone who happens to read these articles, the subject matter is pure research and should be treated as such. 24.167.4.177 14:23, 8 May 2007 (UTC)
replies: text: I agree with Odonovanr, after talking with one or the other anon on Talk:Infinitesimal, and his objections are mathematically unsound.  — Arthur Rubin |  (talk) 22:58, 9 May 2007 (UTC)
replies: text: I am the same anon in both the infinitesimal and non-standard analysis discussions. It seems to me that anyone who reads your comments will see who is 'mathematically unsound'. 70.120.182.243 14:10, 10 May 2007 (UTC)
replies: text: For once I agree with you. --Ian Maxwell (talk) 17:57, 1 December 2009 (UTC)
thread_title:  problem with user 70.120.182.243 

message: Under  First consequences, the article states that "r is limited or bounded if and only if its absolute value is dominated by a standard integer" (my emphasis), yet nowhere else is domination explained or even mentioned.  Will someone elaborate on this or link to an explanation?  Le coq d'or 07:57, 23 July 2007 (UTC)
replies: text: Explained in the article now as "dominated by (less than)".  I'll see if I can find an article about "dominated" which explains it.  — Arthur Rubin |  (talk) 13:52, 23 July 2007 (UTC)
thread_title:  "dominated by a standard integer"? 

message: * There are 12 references here. Perhaps they can be reorganized so that someone with no knowledge of the subject knows where they should start reading?
* I had a quick look at the external links, and I would recommend that somebody who is more qualified than me do the same.
** Keisler should be worth keeping, of course.
** I am very skeptical about the first and third. See the edit to this talk page from 27 August 2006. (And I have just read another, published, paper from the same author that doesn't make me very optimistic.)
** The fourth throws a strange light on the last two. --Hans Adler (talk) 19:49, 23 November 2007 (UTC)
replies: text: No expert has responded and looked at the external links. The link to Keisler's book was also in the References section. All the others seemed to be non-notable and possibly of questionable accuracy. It also looks as if all the others were added by their authors without taking WP:COI into account. Therefore I have now removed the external links section completely. --Hans Adler (talk) 18:07, 21 January 2008 (UTC)
replies: text:  Oops, my bad. You re referring to the external links section.  I agree with your assessment.--CSTAR (talk) 18:18, 21 January 2008 (UTC)
thread_title:  Clean up references and links? 

message: my edit at non-standard analysis was reverted with the justification that the content of the section is contained in the last sentence of the previous paragraph.  The sentence you are referring to mentions vaguely that there is no argument about the mathematical validity of non-standard analysis.  I don't think this is sufficiently precise.  Namely, even a system containing additional axioms could also be mathematically valid, so long as nobody has found an internal contradiction in such a system.  The specific point that non-standard analysis is "conservative" in the sense that it does not go beyond ZFC deserves to be mentioned explicitly.  If you disagree please raise the issue at WP math rather than using deletions.  For the time being I will revert my edits. Katzmik (talk) 08:11, 31 August 2008 (UTC)
thread_title:  ZFC 

message: What the typical wikipedia reader needs to know is not whether Robinson writes in a '66 edition that non-standard analysis can be based on an ultrafilter lemma rather than the axiom of choice.  Rather, we need to disabuse the typical reader of the misconception, based on decades if not centuries of "public opinion", that infinitesimals are "not rigorous".  It suffices to look through this very page to see the influence of such an attitude.  Thus, the reader needs to be told clearly what, for instance, Keisler says in his book, namely that the theory is "conservative" in the sense of not going outside the standard ZFC.  The current section on "criticisms" is written with an expert in mind, not a general reader.  Similarly, the fact that standard mathematics routinely relies on the axiom of choice needs to be mentioned explicitly, rather than implicitly as in the current version.  I don't claim to be the world's authority on non-standard analysis but these are likely to be the questions on the mind of a typical reader and they need to be addressed, if non-standard analysis is to be presented in a convincing fashion to more than a circle of experts.  Any comment? Katzmik (talk) 13:15, 31 August 2008 (UTC)
replies: text: Maybe you can think of a more neutral term that "criticisms" as a section heading. If I come up with something I'll post it here.--CSTAR (talk) 02:46, 1 September 2008 (UTC)
replies: text:  I would change it to "Connes' critique" if I thought I stood a chance of not being reverted :)  At any rate, how about including a mention of those two points in the lead of the article?  Namely, that (using Keisler's language) non-standard analysis is conservative, and that standard mathematics routinely uses AC whether or not this is acknowledged.  Do you agree with my phrasing when I say that non-standard analysis is based on the same ZFC that the rest of standard math is, or is this too naive from a logical viewpoint?  Katzmik (talk) 08:31, 1 September 2008 (UTC)

text: Sorry, I see that you responded to those two points above (I did not notice it before).  I will read it now. Katzmik (talk) 08:32, 1 September 2008 (UTC)
replies: text:  I don't think the "complaints" are Connes' alone.--CSTAR (talk) 15:56, 1 September 2008 (UTC)
thread_title: Common misconceptions

message: At the top of the page there appears an intriguing comment by CSTAR: "Fundamentally, Connes started his career in nonstandard analysis futzing around with logic, didn't get far with it, discovered Tomita-Takesaki, and has had a very low esteem of NSA ever since."  Could you elaborate on this?  Is there a connection between T-T and nonstandard analysis, or are you merely saying that he found an alternative method and therefore abandoned NSA?  Katzmik (talk) 13:38, 2 September 2008 (UTC)
thread_title:  [[Tomita-Takesaki theory]] 

message: Connes criticism seems unfounded to me. He says that given an infinitesimal e you can construct a nonmeasurable set in a canonical way. He doesn't say how to do this, so I have to guess. I am pretty sure he means construct a set with measure e. That's the interval 0 to e. This set has the property that it has infinitesimal measure, so it is "nonmeasurable" in the sense that the standard part of its measure is zero. But it is perfectly measurable in the hyperreal sense, because its measure is e.  It should be impossible to prove something like "there exists a non-measurable set" in the sense of AC, and this is the type of nonmeasurable set which was shown to be non-constructible by forcing, because infinitesimals are not that sophisticated logically. They shouldn't change the theorems, only the conceptual framework for the proof. Maybe Connes means something else, or maybe I'm wrong here. I don't know. But given that he's vague, is there a second source which explains in more detail what he is talking about?Likebox (talk) 16:29, 15 December 2008 (UTC)
replies: text: I was wondering if perhaps he means something else by this.  Constructing *R involves in particular choosing an ultrafilter on the set of sequences containing only 0s and 1s, half of which end up being infinitesimals.  Thus an infinitesimal would typically correspond to a kind of a "random" sequence of 0s and 1s.  The set of positions occupied by 0s will typically be a non-measurable in a suitable sense.  However, I have no idea how to do this in a canonical way, and maybe this approach is a red herring.  Perhaps CSTAR and Arthur can help?  Katzmik (talk) 17:15, 15 December 2008 (UTC)
replies: text: At the moment, I don't see it, but the ultrafilter construction may result in a non-measurable set.  However, the rational power series ring R(((ε))) describes the reals with an additional infinitesimal, which doesn't seem to require sufficient choice to construct a non-measurable set.  — Arthur Rubin  (talk) 19:12, 15 December 2008 (UTC)
replies: text: Yes he meant something totally different, relating to a choice construction of a nonstandard analysis model. I always thought about it syntactically, I never heard of this set theoretic formulations, so in short, I was totally wrong. Sorry. There's a discussion of nonmeasurable sets from NSA by Terrance Tao online .Likebox (talk) 21:43, 15 December 2008 (UTC)

text: I am absolutely thrilled at Mathsci's recent addition of a flag at this subsection, which will hopefully lead to a clarification of these issues.  I look forward to email clarifications he promised at the current AfD.  Incidentally, as far as non-measurable sets are concerned, perhaps the simplest way of interpreting it is in terms of the ultrafilter itself, on sequences of zeros and ones.  If we interpret the sequence as a binary expansion of a real in [0,1], then the ultrafilter produces a subset of [0,1] which picks one out of every pair of "complementary" expansions, which is certainly unmeasurable.  I have not looked at Tao yet, if someone did could they please provide a summary?  I generally find that editors at wikipedia are frequently clearer than published articles, as they are not writing for "posterity" but rather to explain a point.  Katzmik (talk) 08:59, 19 December 2008 (UTC)

text: My own opinion now is that all this material related to Connes' criticism  of NSA should be removed immediately until a suitable correct and verifiable formulation can be found. Indeed there may be some misconception about what Connes' actually believes on the matter, and *gasp* I don't think we should make the level of worldwide ignorance worse by having a Wikipedia article misrepresent his thinking. However, it is important that eventually something replace the deleted text. Whatever that replacement is, if correct and verifiable, is IMHO encyclopedic material. Please comment. If no one strenuously objects, I'll remove the material myself.--CSTAR (talk) 20:07, 19 December 2008 (UTC)
thread_title:  Connes criticism 

message: I just noticed CSTAR's remarks now, and have removed the Connes material from Criticism of nonstandard analysis, as well, except for the last few lines written, as I recall, by CSTAR.  If would be helpful to discuss Connes' criticism in an appropriate forum.  In the meantime, it does seem that he expressed himself in print on this subject.  Would it be imprudent to reproduce some of his remarks verbatim?  Katzmik (talk) 18:04, 20 December 2008 (UTC)
replies: text: I think it would.  As the remarks I produced by Tao are verbatim.  As was the original quote in the section on Connes Critique. Thenub314 (talk) 18:09, 20 December 2008 (UTC)

text: At any rate it would not hurt to quote Connes directly.  There is no doubt that he did express criticism of NSA in print.  I recently heard from one of his closest colleagues (closer than mathsci, I am quite certain) that he had expressed such criticism in a Bourbaki seminar in '96.  For now I merely added the direct quote as a separate section in Criticism of non-standard analysis.  As there was indeed a lot of such criticism, perhaps we should retain the latter article in its current form until more material can be added.  There have been attempts to undermine the decision at AfD and I would appreciate assitance in restoring the article to the title and form decided at AfD. Katzmik (talk) 09:08, 21 December 2008 (UTC)
replies: text: As far as the Bourbaki seminar, I am planning to get to it.  Katzmik (talk) 09:08, 21 December 2008 (UTC)
thread_title: Connes criticism bis

message: I think this article ought to be merged with hyperreal number. Hyperreal number is more detailed and readable. Non-standard analysis has more on formal logic. As it stands, non-standard analysis is not nearly as accessible to the general reader as it could be, because it lacks any gentle introduction to the basics.--76.167.77.165 (talk) 16:42, 8 March 2009 (UTC)
replies: text: Just because one is more general than another, I don't think that necessarily means they need to be separate articles. I think hyperreal number should be part of the nonstandard analysis article.--76.167.77.165 (talk) 23:36, 11 March 2009 (UTC)

text: I agree that Non-standard calculus is a mess, and that parts of Hyperreal number should go into Non-standard analysis. And as someone new to NSA I have to say that Hyperreal number is much more readable than the others. --Riyaah (talk) 12:25, 10 May 2009 (UTC)

text: I don't think so.  — Arthur Rubin  (talk) 06:52, 14 May 2009 (UTC)
thread_title: Merger proposal

message: The formula 
:    A_k = \{k \in *\mathbb{N}: k \geq n\} 
makes no sense to me. Perhaps it should be
:    A_n = \{k \in *\mathbb{N}: k \geq n\} 
Bo Jacoby (talk) 09:40, 6 June 2009 (UTC)
replies: text: Fixed, thanks.  — Arthur Rubin  (talk) 16:32, 6 June 2009 (UTC)
thread_title:  Error in formula? 

message: I can't understand much of this, but it looks pretty good:
http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/ 
66.127.52.47 (talk) 01:52, 16 March 2010 (UTC)
replies: text: Goldblatt's book presents the same material in a clearer way (no fault of Tao's, who was not trying to write a textbook). Tkuvho (talk) 10:33, 16 March 2010 (UTC)
thread_title:  Tao article 

message: I added a summary of a proof of the invariant subspace theorem.  I would appreciate if someone could go over it to check for correctness.  Tkuvho (talk) 08:21, 3 May 2012 (UTC)
thread_title:  invariant subspace 

message: I was looking on the page for a definition of the system, but I can't find that from looking at section headers (or with some very quick skimming). It seems that definitions start at "Logical framework" and/or "Internal Sets". That seems an awful organization; the text should support a reader wanting to skip "Approaches to non-standard analysis" (which is too complicated to get), the "Invariant subspace problem", "Other applications" and "Critique" to try to get at the basic definitions. At a minimum, there should be a section named "Definitions" containing "Logical framework",  "Internal Sets" and so on as subsections, with a reasonable section introduction.
--Blaisorblade (talk) 13:21, 3 December 2012 (UTC)
replies: text: I tried to provide a simple definition.  Tkuvho (talk) 15:06, 3 December 2012 (UTC)
thread_title:  Too technical? 

message: The  Pedagogical section mentions both infinitesimal and bounded quantities, but I do not see any indication of what bounded quantities are. This is opaque to a newcomer like me. 2602:306:CD0E:D410:CC97:6D73:35CC:91C0 (talk) 19:12, 2 February 2016 (UTC)
replies: text: It should say "finite". I've made the change for future clarity. --Jordan Mitchell Barrett (talk) 06:06, 15 February 2020 (UTC)
thread_title:  "Bounded" quantities?  

message: The Revision as of 20:56, 18 July 2020 adds a paragraph about Anggha Nugraha's so called Naïve Infinitesimal Analysis where the author reinvents formal Laurent series. This does not seem to be relevant to the topic since it is not an approach to nonstandard analysis. So I think it should be removed. I also find it displeasing that the term "technical nonsense" be used with respect to other approaches of non-standard analysis.
Also, non-prominent approaches to Nonstandard analysis should not be mentioned in the same way as IST or Robinson's approach, but rather be adressed, at most, as a list of links. Vincent Bagayoko (talk) 08:23, 19 July 2020 (UTC)
thread_title:  Naïve Infinitesimal Analysis 

message: Can someone please post the image of Abraham Robinson for obvious reasons? : https://mathshistory.st-andrews.ac.uk/Biographies/Robinson/Robinson.jpeg  — Preceding unsigned comment added by 108.41.98.105 (talk) 09:32, 14 May 2023 (UTC)
thread_title:  Image of Abraham Robinson 

message: I wondered if there was a nonstandard analysis approach to generalized functions.  A quick web search shows that the answer is yes.  The first two hits are arxiv preprints and I will try to look at them.  But I don't feel qualified to try to expand the article.  I hope someone more knowledgeable than I am could add something about this.  NSA for the purpose of making "dy/dx" meaningful is cute, but making the Dirac delta function meaningful seems to go a step better. 2601:644:8501:AAF0:0:0:0:86EA (talk) 20:45, 5 October 2023 (UTC)
thread_title:  Generalized functions