# Talk:Non-standard analysis

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## older discussion

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)

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)

The author for the books at http://www.serve.com/herrmann/books.htm lloks rather questionable to me.

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)

This really ought not to be the redirect from Nonstandard Model. I think it would be nice to have an article on nonstandard arithmetic as well. Maybe I'll add something in the next few days, but not enough time right now.

Well have you read Herrmann's take on what is true in actual reality can't be known by science. He says that the existing laws of physics need not have applied in the past and thus we can't infer the age of the Universe. This has tremendous theistic implications. He apparantly proved this in 70pages of dense mathematics known as Ultra-logics a derivative of Non-Standard Analysis. Any comments on this?

## What are you saying?

"Some authors maintain that use of infinitesimals is more intuitive and more easily grasped by students than the so-called "epsilon-delta" approach to analytic concepts."

Epsilon and delta are infinitesimals by definition. An infinitesimal is a Cauchy sequence in which the limit of the terms is zero. All e-d proofs imply that epsilon and delta tend to zero. This is what an infinitesimal means.

This article together with other articles (real analysis, 0.999...) in your math pages needs serious revision. 70.120.182.243 12:14, 8 April 2007 (UTC)

Absolutely ... wrong. Epsilon and delta are numbers. If done correctly in one of the formulations of non-standard analysis, most epsilon-delta statements and proofs can be translated into statements about infinitesimals "epsilon" and "delta", but it's not the same. — Arthur Rubin | (talk) 13:53, 8 April 2007 (UTC)
According to your article on infinitesimals - "An infinitesimal is a number that is smaller in absolute value than any positive real number. A number e is an infinitesimal if and only if for every integer n, |ne| is less than 1, no matter how large n is. In that case, 1/e is larger in absolute value than any positive real number." According to the definition of a limit, I can make epsilon as small as I wish. You say epsilon is a number - in what sense is it not an infinitesimal? Is an infinitesimal "not a number"? The Wiki article states it is. 70.120.182.243 13:06, 9 April 2007 (UTC)
An infinitesimal is a fixed number. In context, epsilon is not a fixed number. — Arthur Rubin | (talk) 13:18, 9 April 2007 (UTC)
How interesting. A fixed number? What is the definition of a fixed number? 70.120.182.243 13:28, 9 April 2007 (UTC)
Actually, I misread your quote. "Epsilon" is not a number in that context. It's a variable. It represents numbers, but it isn't one itself. Although the section #In mathematics needs some more work, you may get the idea. — Arthur Rubin | (talk) 14:05, 9 April 2007 (UTC)
A variable represents numbers in this context, therefore it can only be a number. The fact that it can be different numbers makes it variable. But I still don't see how this changes anything. Are you saying an infinitesimal is not a variable? If yes, then I can agree. However, an infinitesimal is a number and if epsilon can take on infinitely many values close to but greater than zero, then by implication it must itself be an infinitesimal at some or other point close to zero. Well, by the requirement of continuity, epsilon must also be an infinitesimal for otherwise the definition of a limit is untrue. 70.120.182.243 20:24, 9 April 2007 (UTC)
Usually epsilon is an arbitrary positive real number. An infinitesimal is a positive hyperreal, that is smaller than any positive real. In action this means that with epsilons you pick epsilon small enough, and with infinitesimals they are automatically small enough. Or something like that. Basically the notion simplifies a lot of proofs and sometimes gives a nice intuition, after a lot of preparation of course.
Let's see: If an infinitesimal is a positive 'hyperreal' (no such thing exists, but let us assume it does) that is smaller than any positive real, then what is the 'smallest positive real'? 70.120.182.243 00:06, 20 April 2007 (UTC)

Would 70.120.182.243 please note that if he or she wishes to learn really really basic mathematical reasoning, then disussions of this kind in this kind of forum are really not a very efficient way to do that. Michael Hardy 20:38, 20 April 2007 (UTC)

Hmmm, that would be me. Really, really basic reasoning? Well, if it's so basic, then how is it that a PHd (Arthur Rubin) is having such a hard time explaining? Perhaps you can explain this basic reasoning Mr. Hardy? 70.120.182.243 01:56, 21 April 2007 (UTC)

I'm not having a hard time explaining (although I often do with elementary concepts.) You're having a hard time understanding. — Arthur Rubin | (talk) 08:32, 21 April 2007 (UTC)
It's not that you are having a hard time explaining, it's just that you are not explaining anything because you don't know what you are talking about. Mr. Hardy appears to know less given that he thinks these concepts are elementary. Way I see it is that Wikipedia has articles about ill-defined or undefined concepts such as infinitesimals. 70.120.182.243 16:14, 21 April 2007 (UTC)

Look: For any positive number ε, there exists a positive integer N, such that for all integers n > N, we have the following inequality: 1/n < ε. The values of εthat that statement contemplates are positive real numbers that are not infinitesimal. An understanding of that statement is an example of what I'm calling "elementary".

As far as existence of infintesimals goes, it's quite easy to exhibit examples that are constructive and in no way philosophically controversial. For example, the set of rational functions with real coefficients, ordered in a certain way that is easy to define. Again, fairly elementary. Undergraduate-level. Michael Hardy 01:31, 22 April 2007 (UTC)

The statement you made: "For any positive number ε, there exists a positive integer N, such that for all integers n > N, we have the following inequality: 1/n < ε" is elementary but what does it say about an infinitesimal? You are making a statement about what is not an infinitesimal. So what is an infinitesimal then? I have comments on your second paragraph too but I am more interested in keeping this discussion simple and to the point. At this time your second paragraph is irrelevant. 70.120.182.243 16:50, 22 April 2007 (UTC)

## Question in regards to Applicability of Godel's Theorem

Sometimes, when I read contents such as the below on the internet, it occurs to me that they are just random jabberings. However, is there any 'truth' to the below quote from http://www.physicsforums.com/archive/index.php/t-115523.html "Hurkyl 03-28-2006, 08:10 PM all other rigorously logical systems powerful enough to be of interest Actually, Gödel's incompleteness theorems do not apply to several rigorously logical systems of interest.

For example, the theory of real closed fields is very interesting -- but it's not powerful enough to define the word "integer".

On the other hand, the theories constructed for the purposes of nonstandard analysis are powerful enough to talk about the integers... but they are far too `powerful' for Gödel's theorems to apply. "

If so, why does the part in bold hold?

MrASingh

THE DEFINITION OF ${\displaystyle [*f]}$ is missing.

## problem with user 70.120.182.243

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)

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)
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)
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)
For once I agree with you. --Ian Maxwell (talk) 17:57, 1 December 2009 (UTC)

## "dominated by a standard integer"?

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)

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)

## Clean up references and links?

• 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)
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)

Are you talking about the reference list that is included below? If so, all these are notable references published by reputable publishers.

• Sergio Albeverio, Jans Erik Fenstad, Raphael Hoegh-Krohn, Tom Lindstrøm: Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press 1986.
• P. Halmos, Invariant subspaces for Polynomially Compact Operators, Pacific Journal of Mathematics, 16:3 (1966) 433-437.
• T. Kamae: A simple proof of the ergodic theorem using nonstandard analysis, Israel Journal of Mathematics vol. 42, Number 4, 1982.
• H. Jerome Keisler: An Infinitesimal Approach to Stochastic Analysis, vol. 297 of Memoirs of the American Mathematical Society, 1984.
• H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
• Edward Nelson: Internal Set Theory: A New Approach to Nonstandard Analysis, Bulletin of the American Mathematical Society, Vol. 83, Number 6, November 1977. A chapter on Internal Set Theory is available at http://www.math.princeton.edu/~nelson/books/1.pdf
• Edward Nelson: Radically Elementary Probability Theory, Princeton University Press, 1987, available as a pdf at http://www.math.princeton.edu/~nelson/books/rept.pdf
• Robinson, Abraham (1996). Non-standard analysis (Revised edition ed.). Princeton University Press. ISBN 0-691-04490-2.
• Allen Bernstein and Abraham Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific Journal of Mathematics 16:3 (1966) 421-431
• Curt Schmieden and Detlef Laugwitz: Eine Erweiterung der Infinitesimalrechnung, Mathematische Zeitschrift 69 (1958), 1-39
• L. van den Dries and A. J. Wilkie: Gromov's Theorem on Groups of Polynomial Growth and Elementary Logic, Journal of Algebra, Vol 89, 1984.
• Robert, Alain: Nonstandard analysis, ISBN 0-471-91703-6

--CSTAR (talk) 18:15, 21 January 2008 (UTC)

Oops, my bad. You re referring to the external links section. I agree with your assessment.--CSTAR (talk) 18:18, 21 January 2008 (UTC)

## ZFC

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)

P.S. There is a bit of an overlap between the current versions of "criticisms" and "refutation of criticisms", which may require editing. I think Robinson's defense could be presented in a more ample fashion than what currently appears at the end of "criticisms". I must say that with all due respect to a fields medalist I do not understand Connes objection very well. Results such as the existence of maximal ideals routinely use the full AC in standard mathematics. We (standard mathematicians) are thoroughly accustomed to such a state of affairs. Nobody doubts the Banach-Tarsky paradox. Why should one be more embarrassed of the non-constructive nature of the hyperreals than those other "standard" results? Of course, from the perspective of the constructivist school, all such results are invalid. But does Connes consider himself a constructivist? Katzmik (talk) 09:50, 31 August 2008 (UTC)

## Refutation of criticism

I concur with User:CSTAR; non-editorial comment is contained within the last sentence of the preceding section. — Arthur Rubin (talk) 10:26, 31 August 2008 (UTC)

It seems to me it is worth mentioning ZFC for a reader familiar with set theory and/or mathematical logic, but not yet familiar with Robinson's theory. Similarly, shouldn't the point about maximal ideals be mentioned explicitly? If you happen to be an expert in Robinson's theory I would have to defer to your opinion, but other than that it would seem to be a matter of elementary politeness to explain your repeated reverts. Katzmik (talk) 10:43, 31 August 2008 (UTC)

There are two points here that you could briefly mention as an addendum to the last sentence of the criticisms section.
• IST is a conservative extension of ZFC (This is shown in Edward Nelson's AMS 1977 paper in an appendix written by William Powell)
• Model theoretic non-standard analysis (for example based on superstructures, which is now fairly commonly used) does not need any new set-theoretic axioms.
Everything else you added is editorial comment, IMHO.
--CSTAR (talk) 15:59, 31 August 2008 (UTC)
Great, I copied your remarks. They are a little technical, though, please elaborate when you get a chance. It is with trepidation that I mention that I also made some changes at non-standard calculus, hope they pass the muster. Katzmik (talk) 08:37, 1 September 2008 (UTC)

## Common misconceptions

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)

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)
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)
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)
I don't think the "complaints" are Connes' alone.--CSTAR (talk) 15:56, 1 September 2008 (UTC)

## Tomita-Takesaki theory

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)

## Connes criticism

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)

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)
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)
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 [1].Likebox (talk) 21:43, 15 December 2008 (UTC)

Do we know for certain he that he meant his comment to be critical? Mathsci raises that issue that perhaps this is not the case, do we know any other documentation of his criticisms? On his blog he speaks of falling in love with NSA, and trying to be non-polemic. I think that if we can find a source other then Connes himself we would be in less danger of mis-interpreting his comments. Thenub314 (talk) 11:00, 18 December 2008 (UTC)

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)

Terry tao is an amazing clear expositor, and his blog is written for a general (mathematical?) audience. Here is the jist (but I really can't do it justice), by choosing a unbounded hyper integer you can split the non-standard interval (0,1) into infinitely many intervals you may build a set *E consisting of every other interval. Then you may Build a set E consisting of "every other" real number. This set E has Lebesgue density 1/2, contradicting the fact that it is either the Lebesgue density is either 0 or 1 a.e. Hence E is not measurable.

Now, a would like to digress on a comment about the danger of quoting authors about their views. This blog entry contains the quote which reads like a criticism:

The main drawbacks to use of non-standard notation (apart from the fact that it tends to scare away some of your audience) is that a certain amount of notational setup is required at the beginning, and that the bounds one obtains at the end are rather ineffective (though, of course, one can always, after painful effort, translate a non-standard argument back into a messy but quantitative standard argument if one desires).

This should be contrasted with the beginning of the paragraph where he states (and this to me sounds pretty positive)

I hope I have shown that non-standard analysis is not a totally “alien” piece of mathematics, and that it is basically only “one ultrafilter away” from standard analysis. Once one selects an ultrafilter, it is actually relatively easy to swap back and forth from the standard universe and the non-standard one (or to doubly non-standard universes, etc.).

Finally we should keep in mind that the article is about using NSA for "epsilon management" and "...here we shall be focusing on the reverse procedure, in which one harnesses the power of infinitary mathematics - in particular, ultrafilters and nonstandard analysis - to facilitate the proof of finitary statements."

For this reason I am rather concerned that Connes criticism is not a criticism. Particularly since the quote begins "The answer given by nonstandard analysis, a so-called nonstandard real, is equally deceiving." He is clearly in the middle of comparing two things. What is to say he has negative feelings about either, he may just be warning the reader that both are confusing. Sadly I do not have the article to read, but it almost doesn't matter. We should really find a secondary source that comments on his criticism. If we cannot we really should take it out of the article. Thenub314 (talk) 15:27, 19 December 2008 (UTC)

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)

## Connes criticism bis

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) P.S. Does anybody know who put those remarks there in the first place? Apparently they have been there for a long, long time. Katzmik (talk) 18:05, 20 December 2008 (UTC)

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)

Reply: I inserted the material originally. There were various remarks by Connes in an essay and several articles (not just the ones listed) that I and various other individuals I have spoken with had interpreted as being disparaging about NSA. Apparently, though, this interpretation is quite wrong.--CSTAR (talk) 18:14, 20 December 2008 (UTC)

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)
As far as the Bourbaki seminar, I am planning to get to it. Katzmik (talk) 09:08, 21 December 2008 (UTC)

## Quotes in Intro

This article's introductory paragraphs include extensive quotes. The same comment applies to a number of other articles in the now sprawling and rapidly expanding empire of non-standard analysis articles. Is this inclusion of quotes helpful? --CSTAR (talk) 18:43, 21 December 2008 (UTC)

I don't think there is an expanding empire. A few months ago I created articles such as hyperinteger, standard part function, etc. I have not created any new NSA articles recently. The article Criticism of non-standard analysis is, as you know, a renamed version of an old article. Following mathsci's suggestion, I created Influence of non-standard analysis and thought that you support this particular direction. Now merging the latter does not make any sense to me at all, as they simply deal with different subjects. We can discuss this further if you like.
As far as the quotes go, I tried to make general remarks in this direction, but you correctly pointed out at the time that they tended to be "editorializing". Therefore I replaced them by quotes from Robinson himself. I have the impression they do a good job of giving a general idea of what is going on. Obviously this can be discussed. Incidentally, if you get a chance I would appreciate your expert input concerning a recent addition to internal set. Katzmik (talk) 18:51, 21 December 2008 (UTC)
Would you object if I trimmed the quotes down a bit and moved them into the historical section?--CSTAR (talk) 20:19, 21 December 2008 (UTC)
No problem--I wish I knew as much about the topic as you do. Katzmik (talk) 13:28, 22 December 2008 (UTC)
As to the motivation for my "empire" comment, it's my inability to keep up with all the discussions in various talk and AfD pages.--CSTAR (talk) 20:21, 21 December 2008 (UTC)

## Merger proposal

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)

Maybe the article is not "gentle enough", but the notion of hyperreal number is a special thing. Nonstandard analysis is more general, as described in the article. Therefore I do not support the proposed merger. Peta 77.104.243.33 (talk) 20:23, 11 March 2009 (UTC)

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)

There is also a vast amount of overlap between Non-standard analysis and Non-standard calculus. In theory, it might make sense to have one article that would do via NSA the kind of elementary calculus covered in a lower-division calc course, and another that would tackle the topics generally found in an upper-division analysis course (compact sets, proofs of things like the intermediate value theorem that are never proved in freshman calc). But in reality, there is no such division between the articles. We have a situation where there are essentially three different articles on NSA, with tons of overlap and no clear criteria for what goes where. I'm not saying that merging all three articles is the only right choice. However, leaving them in their current duplicative, ill-defined state is also clearly not the right choice. Non-standard calculus is also a horribly disorganized mess right now. It's a random jumble of topics picked for no apparent reason and in no particular order.--76.167.77.165 (talk) 19:10, 15 March 2009 (UTC)

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)

Why can the merger not be subsumed under the surreal numbers? (John Marks)

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

NSC and Hyperreal numbers are essentially special applications of NSA (which is much more general and more technical). I think perhaps NSC and hyperreals should be merged, but probably not also merged with NSA (although the latter should definitely link to this special application). —Preceding unsigned comment added by 125.27.110.115 (talk) 13:09, 23 May 2009 (UTC)

I don't think that it is appropriate to merge Hyperreal number and Non-standard analysis, because they are really quite different ideas. The hyperreal numbers are a specific algebraic structure, so the Hyperreal number article should describe the properties of this particular structure. Non-standard analysis is a field of mathematics which involves the use of structures with infinitesimals and infinitely large elements (of which one such structure is the hyperreal numbers). Merging these articles would be akin to merging the articles Real number and Real analysis, which isn't a good idea. There is enough to say about hyperreal numbers and non-standard analysis separately that they don't need to be contained in the same article. —Babcockd (talk) 19:10, 13 July 2009 (UTC)

I disagree that these articles should be merged. I came here to express precisely the same argument as Babcockd, that merging these articles makes as much sense as merging the corresponding standard analysis articles. By the way, I don't think that "the current content of the articles needs improving" is justification for moving the articles around. Quietbritishjim (talk) 10:50, 5 October 2009 (UTC)

## Error in formula?

The formula

${\displaystyle A_{k}=\{k\in *\mathbb {N} :k\geq n\}}$

makes no sense to me. Perhaps it should be

${\displaystyle A_{n}=\{k\in *\mathbb {N} :k\geq n\}}$

Bo Jacoby (talk) 09:40, 6 June 2009 (UTC).

Fixed, thanks. — Arthur Rubin (talk) 16:32, 6 June 2009 (UTC)

## Tao article

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)

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)

## Leibniz Caption

We credit Leibniz in the caption as being the discoverer of infinitesimal calculus, but it was my understanding that both Newton and Leibniz used infinitesimal techniques. Which matches more in line with what is written Infinitesimal Calculus. I am going to change this and if anyone disagrees we can change it back and discuss. Thenub314 (talk) 17:20, 22 May 2011 (UTC)

It would be better to modify what we write at infinitesimal calculus, because even Pourciau admits that Newton wasn't so thrilled about infinitesimals, particularly in his later years. The current form of the caption is not sourced. I have doubts about its accuracy. Tkuvho (talk) 18:08, 22 May 2011 (UTC)

The current form of the quote was based on the first quote by Robinson listed in the page. To quote the quote: "G. W. Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latter." Thenub314 (talk) 19:15, 23 May 2011 (UTC)

That's what Robinson wrote, but it is not what you wrote. You wrote: "idealized numbers containing infinitesimals be introduced". Tkuvho (talk) 19:36, 23 May 2011 (UTC)

You feel this says something different then the quote above. How would you improve it? Thenub314 (talk) 23:35, 23 May 2011 (UTC)
By leaving it as it was. Leibniz was doing infinitesimal calculus in the true sense of the word, and Robinson created an appropriate mathematical foundation for certain procedures in Leibniz that were only heuristic, such as law of continuity which became the transfer principle. The main point is infinitesimal calculus, and that's all the caption needs to mention. Tkuvho (talk) 01:36, 24 May 2011 (UTC)
Leaving it as it was and leaving it as it is are options that make neither of us happy. Perhaps we can find some new alternative. We could correct the wording of mine, I am still not sure in what way my summary is misleading. Or perhaps we could find a new caption to put underneath, or simply just have his name as we do for Robinson. Thenub314 (talk) 06:20, 24 May 2011 (UTC)
I don't remember anymore what makes you unhappy about the old caption. What's wrong with saying that Leibniz created infinitesimal calculus? You worry about the priority dispute? That seems to have been resolved without passing through wiki arbitration :) Tkuvho (talk) 14:45, 24 May 2011 (UTC)
Yes I felt it was misleading to label him the "discoverer of Infinitesimal Calculus", the corresponding comment at Infinitesimal Calculus lists both Newton and Liebniz, which is closer to what I have read in history text. Putting that aside, I also thought it might be better to include a caption more related to the text of the article. Thenub314 (talk) 14:58, 24 May 2011 (UTC)
How about "co-discoverer of infinitesimal calculus"? The problem with saying "idealized numbers containing infinitesimals" is that it attributes perhaps too much foresight to Leibniz, and at any rate Robinson did not say this. Tkuvho (talk) 15:40, 24 May 2011 (UTC)

It is true that Newton and Leibniz both independently invented the Calculus (Newton was first but Leibniz published his ideas first). However, they had quite different ways to justify the new mathematics. Newton, being a physicist, used concepts from physics like velocity while Leibniz, being a logician and philosopher, used only math and logic. He also pondered a lot about if infinitesimal numbers could have an ontological existence. He never found a solution to this question and in the end he doubted if it would ever be possible to define these numbers in a logical way. Newton had a purely pragmatic relation to this subject and he never thought that he had discovered any new type of numbers. If Newton would live today he would not even care about Robinson's discovery, while Leibniz would be delighted and read all about it. iNic (talk) 14:46, 25 May 2011 (UTC)

So yes, it is correct to say that Leibniz and Newton both discovered the Calculus, but only Leibniz realized this could be the first step towards the discovery of infinitesimal numbers. However, it is to say too much to say that Leibniz argued that idealized numbers containing infinitesimals should be introduced. He never was really sure about their existence as numbers. iNic (talk) 14:46, 25 May 2011 (UTC)

Thanks. Your input at http://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics#Jaggedalia.2C_part_2 would be similarly appreciated. Tkuvho (talk) 15:31, 25 May 2011 (UTC)
This is not quite what books on math history texts say on the matter. (See for example The History of mathematics: An introduction by Burton pages approx 373-377). Mathematicians of the time had long been using infinitesimal quantities. And Newton certainly used them in developing Calculus. Thenub314 (talk) 15:38, 25 May 2011 (UTC)

Sure, but use them as a tool and believe in them are two different things. Use of infintesimal reasoning as a tool can be traced back to Aristotle. But clearly he didn't believe in any infinitesimal numbers, as ancient Greek mathematicians only granted finite quantities existence. So it sure was a gradual process. My point is that Leibniz was the first one really considering the possibility that such numbers could exist. This is why Robinson singles out Leibniz and not say Fermat or Aristotle as the true historical predecessor to his discovery. Newton didn't call them infinitesimals but fluxions, and he invented a mathematical notation that didn't at all support the idea of infinitesimal numbers. On the contrary Leibniz notation did and his notation quickly became the standard notation. He also developed his philosophy of monads heavily inspired by his thoughts about infinitesimals. iNic (talk) 16:29, 25 May 2011 (UTC)

I agree Leibniz deserves to be pointed out and that he should be recongnized as the proponent of infinitesimal quantities. Robinson is correct to signal him out. But as I understand to Newton fluxions and infinitesmals were not precisely the same. According to Burton it was "In the De Methodis Fluxionum [that] Newton abandoned his use of infinitesimals in favor of fluxions." Though he points out that in the modern prespective the two ideas "were not essentially different". But he certainly used both, as well as later justifying calculus using prime ultimate ratios as in Principia.
To give a slightely longer quote:

The difficulties in understanding Newton’s creation were due in part to the change of approach in each of his three works on the calculus. Infinitesimals were emphasized in the De Analysi but abandoned in favor of a theory of fluxions in De Methodis Fluxionum; and still later, that theory came to rest on prime and ultimate ratios in the De Quadratura Curvarum. The De Quadratura Curvarum, the last-written of Newton’s trio but the first published, was the climax of his effort to establish the calculus on sound foundations.

Thenub314 (talk) 17:45, 25 May 2011 (UTC)

Yes you are right, Newton changed his way of expressing himself over time. The only reason he talked about infinitesimals in his earliest writing was because that was the standard way to talk about these mathematical techniques at the time, not because he really wanted to. His main focus was to find a sound and consistent foundation for his new mathematics and new physics. He couldn't care less about the ontological status of infinitesimal numbers. That discussion didn't interest him. He was very happy to get rid of the infinitesimals, first by focusing not on the infinitesimals themselves but their "ratios," which are the physical quantities that did interest him. He made these instead of infinitesimals the basic concept of his calculus and called them fluxions of different orders. So fluxions are not infinitesimals. This is the whole point. Next step he took was to talk about prime and ultimate ratios of ordinary numbers, much in the same spirit as what Weierstrass later did. It's not a controversial interpretation to say that his struggles of finding a good foundation for his mathematics was the main reason why he hesitated to publish his new mathematics for such a long time. Leibniz published his corresponding ideas much faster. iNic (talk) 23:56, 25 May 2011 (UTC)

## invariant subspace

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)

## Too technical?

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)

I tried to provide a simple definition. Tkuvho (talk) 15:06, 3 December 2012 (UTC)

## "Bounded" quantities?

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)