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Talk:Nonstandard analysis

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?
Your article states:

"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 $$[*f]$$ 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
 * 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
 * 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 .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
 * $$   A_k = \{k \in *\mathbb{N}: k \geq n\} $$

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).

 * 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)

 * It should say "finite". I've made the change for future clarity. --Jordan Mitchell Barrett (talk) 06:06, 15 February 2020 (UTC)

Name change - removal of hyphen
I suggest that the title of this article should be Nonstandard analysis rather than Non-standard analysis. It seems that since about the 90s, the unhyphenated form has been about twice as common (see this ngram). Certainly, in every context I have seen, the hyphen has been omitted. I would say non-standard is the complement of standard, while nonstandard refers to the use of ultrapowers (not quite the same thing). So, I believe we should remove the hyphen as per WP:COMMONNAME. — Preceding unsigned comment added by Jordan Mitchell Barrett (talk • contribs) 07:21, 11 February 2020 (UTC)

 * ✅ A quick Google Scholar search supports this. Although Google Scholar suggests usage of "nonstandard analysis" and "non-standard analysis" is roughly equal, it does show "nonstandard" is much more likely than "non-standard" to refer to ultrapowers. "Nonstandard" also appears to be used by the large published journals, e.g. Notices of the AMS.  (talk) 21:05, 11 February 2020 (UTC)
 * ✅ this seems to be the standard for recent books on the topic. —David Eppstein (talk) 06:05, 14 February 2020 (UTC)
 * ✅ The above arguments have convinced me. XOR'easter (talk) 14:00, 14 February 2020 (UTC)

 * I'll go ahead with the change then. --Jordan Mitchell Barrett (talk) 06:01, 15 February 2020 (UTC)

 * Need a bit of help performing the move - I've done the main page and fixed up that page's content, but still need to do this talk page, everything in this category, etc...

Naïve Infinitesimal Analysis
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)

Image of Abraham Robinson
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)

Generalized functions
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)

title: Talk:Nonstandard analysis


text:
Ernst Curtius

Ernst Curtius (2 September 1814 – 11 July 1896) was a German archaeologist, historian and museum director.

Biography
He was born in Lübeck. On completing his university studies he was chosen by C. A. Brandis to accompany him on a journey to Greece for the prosecution of archaeological researches. Curtius then became Karl Otfried Müller's companion in his exploration of the Peloponnese, and on Müller's death in 1840 he returned to Germany. In 1844 he became an extraordinary professor at the University of Berlin, and in the same year he was appointed tutor to Prince Frederick William (afterwards the Emperor Frederick III), a post which he held until 1850.

After holding a professorship at Göttingen and undertaking a further journey to Greece in 1862, Curtius was appointed (in 1863) ordinary professor at Berlin. In 1874 he was sent to Athens by the German government and there concluded an agreement by which the excavations at Olympia were entrusted exclusively to Germany. Curtius was elected an International Honorary Member of the American Academy of Arts and Sciences in 1876. In 1891 Curtius was elected a member of the American Antiquarian Society. He was elected an International Member of the American Philosophical Society in 1895. Curtius died in Berlin on 11 July 1896.

Excavation in Olympia
On 10 June 1852 Ernst Curtius delivered his famous oration on Olympia at the Singakademie in Berlin in the presence of royal family, which marked the first step towards excavations of Olympia, and gave great impetus for the historical archaeological works in Greece. Following the death of Eduard Gerhard in 1867, Curtius succeeded him as professor of classical archaeology at Berlin. At the same time, Curtius served as a director of the Altes Museum and Antiquarium. The Turko-Russian War (1877–78) delayed the process of reaching an agreement between Greek and German governments for undertaking excavation at Olympia.

Curtius's imperial connections helped him to convince the government to nationalize the German Archaeological Institute in 1874, and open a branch in Athens. In 1874, the Greek government granted the exclusive right to the German Archaeological Institute to excavate at Olympia. A landmark agreement in the history of archaeology was signed by Curtius on 25 April 1874 at Athens, which mandated the Germans to leave all finds in Greece. A special museum was built on the site for this purpose.

In 1875 Curtius led large-scale expeditions of archeologists to systematically unearth Olympia. The first excavation was begun at Olympic in Elis, the original Olympic site, where for a thousand years, the Olympic Games had been held. Within six years, they cleared the Olympic stadium, with its runner's starting blocks and judges’ seat. The excavators also uncovered temple of Zeus’s and Hera.

The findings were published in scientific journals in voluminous throughout Europe between 1890 and 1897. Similarly, the German archaeologists were able to bring to life the knowledge of Olympic with paintings of the Olympic site, and the procedures of the Olympic Games. These helped the western world to understand the importance of reviving the Olympic Games.

Works
His best-known work is his History of Greece (1857–1867). It presented in an attractive style what were then the latest results of scholarly research, but it was criticized as wanting in erudition. It is now superseded. His other writings are chiefly archaeological. The most important are:
 * Die Akropolis von Athen (1844)
 * Naxos (1846)
 * Peloponnesos, eine historisch-geographische Beschreibung der Halbinsel (1851)
 * Olympia (1852)
 * Die Ionier vor der ionischen Wanderung (1855)
 * Attische Studien (1862–1865)
 * Ephesos (1874)
 * Die Ausgrabungen zu Olympia (1877, etc.)
 * Olympia und Umgegend (edited by Curtius and F. Adler, 1882)
 * Olympia. Die Ergebnisse der von dem deutschen Reich veranstalteten Ausgrabung (with F. Adler, 1890–1898)
 * Die Stadtgeschichte von Athen (1891)
 * Gesammelte Abhandlungen (1894)

His collected speeches and lectures were published under the title of Altertum und Gegenwart (5th ed., 1903 foll.), to which a third volume was added under the title of Unter drei Kaisern (2nd ed., 1895).

Family
His brother, Georg Curtius, was a noted philologist.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Ernst Curtius


text:
User talk:Radius

Re: Parsec.
Okay I misplaced the decimal point, somehow, but surely a number with 15 significant figures doesn't deserve to be marked as an approximation! (And the Astronomical unit seems to be known to only 12 digits, so the parsec should be the same.) wwoods 21:12, 15 Mar 2004 (UTC)

 * Based on what you say, I would not object to reducing the number of digits to 12. I do not feel strongly either way about the "~".
 * Radius 08:42, 18 Mar 2004 (UTC)

 * Actually, when I do the math I get 3.085 677 581 31×1016, using the 12-digit value for AU given in astronomical unit and 30-some digits of pi. The value used for AU further down in the article is different.
 * Radius 09:31, 18 Mar 2004 (UTC)

Re:Sally Gerber
Dear Ray: Hi1 Thanks for your message on my talk page. Im flattered when someone reads an article I originated. As far as Sally Gerber, you have a point. When I began the article, the Gerber website said it was Sally Gerber. A couple days ago, (since I had mailed Sally through the Gerber company asking for an autograph), I received a letter from the Gerber company stating they think the baby was Clark Gable! or Humphrey Bogart or just a baby from the street that they liked. Not even they are sure who the baby is.

We need to clean up the article. Leave it, because Sally was, actually, a real person, but point out all the other possibilities as well.

Thank you and God bless you! Your friend, "Antonio Crocodile Swamp Martin"

WikiProject Contract bridge
Hi. You might be interested in participating in new WikiProject Contract bridge. Regards, Duja 10:17, 28 June 2006 (UTC)

Drury convention
Hello, and welcome to Wikipedia! We welcome and appreciate your contributions, such as Drury convention, but we regretfully cannot accept original research. Please find and add a reliable citation to your recent edit so we can verify your work. Uncited information may be removed at any time. Thanks for your efforts, and happy editing! Burntsauce 20:03, 24 April 2007 (UTC)
 * I have added four references to the article. Radius 23:46, 29 April 2007 (UTC)

ArbCom elections are now open!
Hi, You appear to be eligible to vote in the current Arbitration Committee election. The Arbitration Committee is the panel of editors responsible for conducting the Wikipedia arbitration process. It has the authority to enact binding solutions for disputes between editors, primarily related to serious behavioural issues that the community has been unable to resolve. This includes the ability to impose site bans, topic bans, editing restrictions, and other measures needed to maintain our editing environment. The arbitration policy describes the Committee's roles and responsibilities in greater detail. If you wish to participate, you are welcome to review the candidates' statements and submit your choices on the voting page. For the Election committee, MediaWiki message delivery (talk) 22:13, 30 November 2015 (UTC)

title: User talk:Radius


text:
Talk:Forbes 500

Copyright Infringement
Pakaran. thought that the list of Forbes 500 companies (originally here, now at List of Forbes 500 might be copyrighted.

But since the list was created by taking certain facts (not copyrightable) ordering them by size (obvious ordering) and giving each company a rank for each factor (a simple fact from its placement in the list). Then, for each company, they summed the rank in each factor (again, a fact), ordered the result by size, took the top 500, and then putting them in alphabetical order (obvious ordering). The result is not copyrightable under Feist v. Rural Telephone Service.

There are some theories that Forbes might have something here that's copyrightable (because they chose what factors to weigh and how to define them), but this seems unlikely, and Forbes is also unlikely to mind much.

I'd like to see this contain the merged alphabetic list of all 824 in the individual top 500s, since that's really a better indication of the companies which we should cover first and this list already doesn't give the actual ranks. That happens to be more assuredly safe on the copyright side as well, so it would be good all around, IMO. Jamesday 04:21, 16 Feb 2004 (UTC)

I double-checked this using the list available by clicking on rank here. The list is no more than the list formed by adding the ranks in the four public domain lists and sorting by lowest total score. Since that's the first way that I thought of doing it myself, I don't think it passes the creativity test. Jamesday 04:48, 16 Feb 2004

(In any event, it should be cleaned up, and we should incorporate (if we can) some of the other lists (e.g. all the US dollar billionaires in the world, etc).  Pakaran. 04:54, 16 Feb 2004 (UTC)

List
Why not have the top 100 in each category - could be something this simple:

 * 1) Microsoft
 * 2) General Electric
 * 3) ExxonMobil
 * 4) Wal-Mart Stores
 * 5) Pfizer

BTW Forbes doesn't give the actual number for anything except employees. Ford, which ranked fifth in employees, doesn't make top 500 in profits (they may have taken a loss, or most of those employees may be in sales).

Note that in my list above, I pipe-linked those for which our articles use different names. Pakaran. 00:30, 25 Feb 2004 (UTC)

Still published?
"The Forbes 500 is an annual listing of the top 500 American companies produced by Forbes Magazine"
 * I don't think it's being published anymore. The 2003 list for 2002 was the last one it seems.--Jerryseinfeld 01:57, 6 Jan 2005 (UTC)

I received my copy in the mail this past April (2005). — Preceding unsigned comment added by 143.166.226.18 (talk • contribs) 21:00, 25 July 2005 (UTC)

 * The list has not been published in over a decade, and the online site (https://www.forbes.com/500s/) is not available so I removed it from the article but used an archived link instead now. At the moment, the article lacks references and possibly WP:Notability too. Shalom11111 (talk) 05:30, 26 August 2018 (UTC)

title: Talk:Forbes 500


text:
Mark Tobey

Mark George Tobey (December 11, 1890 – April 24, 1976) was an American painter. His densely structured compositions, inspired by Asian calligraphy, resemble Abstract expressionism, although the motives for his compositions differ philosophically from most Abstract Expressionist painters. His work was widely recognized throughout the United States and Europe. Along with Guy Anderson, Kenneth Callahan, Morris Graves, and William Cumming, Tobey was a founder of the Northwest School. Senior in age and experience, he had a strong influence on the others; friend and mentor, Tobey shared their interest in philosophy and Eastern religions. Similar to others of the Northwest School, Tobey was mostly self-taught after early studies at the Art Institute of Chicago. In 1921, Tobey founded the art department at The Cornish School in Seattle, Washington.

Tobey was an incessant traveler, visiting Mexico, Europe, Palestine, Israel, Turkey, Lebanon, China and Japan. After converting to the Baháʼí Faith, it became an important part of his life. Whether Tobey's all-over paintings, marked by oriental brushwork and calligraphic strokes, were an influencer on Jackson Pollock's drip paintings has been left unanswered. Born in Centerville, Wisconsin, Tobey lived in the Seattle, Washington area for most of his life before moving to Basel, Switzerland in the early 1960s with his companion, Pehr Hallsten; Tobey died there in 1976.

Early years
Tobey was the youngest of four children in a Congregationalist family. His parents were George Tobey, a carpenter and house builder, and Emma Cleveland Tobey. The father carved animals from stone and sometimes drew animals for young Mark to cut out with scissors. In 1893, the family settled in Chicago. He studied at the Art Institute of Chicago from 1906 to 1908, but, like others of the Northwest School, was mostly self-taught. In 1911, he moved to New York City where he worked as a fashion illustrator for McCall's. His first one-man show was held at Knoedler & Company in lower Manhattan, in 1917. The following year, Tobey came in contact with New York portrait artist and Baháʼí Juliet Thompson—an associate of Khalil Gibran—and posed for her. During the session, Tobey read some Baháʼí literature and accepted an invitation to Green Acre where he converted to the Baháʼí Faith. His conversion led him to explore the representation of the spiritual in art. In the following years, Tobey delved into works of Arabian literature and teachings of East Asian philosophy.

Early years
Tobey's arrival in Seattle in 1921 was in part an effort for a new start following a marriage and quick divorce. When his ex-wife found Tobey's address, she sent him a box of his clothes topped with a copy of Rudyard Kipling's The Light That Failed. In the following year, Tobey met Teng Kuei, a Chinese painter and student at the University of Washington, who introduced Tobey to Eastern penmanship, beginning Tobey's exploration of Chinese calligraphy. The beginning of his lifelong travels occurred in 1925 when he left for Europe, settling in Paris where Tobey met Gertrude Stein. He spent a winter at Châteaudun, and also traveled to Barcelona and Greece. In Constantinople, Beirut and Haifa, he studied Arab and Persian writing.

Upon returning to Seattle in 1927, Tobey shared a studio in a house near the Cornish School (with which he was intermittently associated) with the teenage artist, Robert Bruce Inverarity, who was 20 years younger. Inspired by Inverarity's high-school project, Tobey developed interest in three-dimensional form and carved some 100 pieces of soap sculpture. The next year, Tobey co-founded the Free and Creative Art School in Seattle with Edgar Ames, and in autumn, he taught an advanced art course at Emily Carr's Victoria studio.

In 1929, he participated in a show that marked a change in his life: a solo exhibition at Romany Marie's Cafe Gallery in New York. Alfred H. Barr, Jr., then a curator at the Museum of Modern Art (MoMA), saw the show and selected several pictures from it for inclusion in MoMA's 1930 exhibition: Painting and Sculpture by Living Americans. In 1931, Tobey became a resident artist of the Elmhurst Progressive School while teaching at Dartington Hall in Devon and painting frescoes for the school. He became a close friend of Bernard Leach, who was also on the faculty. Introduced by Tobey to the Baháʼí Faith, Leach became a convert. During his stay in Devon, Tobey found time to travel to Mexico (1931), Europe, and Palestine (1932). In 1934, Tobey and Leach traveled together through France and Italy, then sailed from Naples to Hong Kong and Shanghai, where they parted company. Leach went on to Japan, while Tobey remained in Shanghai visiting his old friend, Teng Kuei, before departing for Japan. Japanese authorities confiscated and destroyed an edition of 31 drawings on wet paper that Tobey had brought with him from England to be published in Japan. No explanation for their destruction has been noted; possibly they considered his sketches of nude men pornographic. In early summer, he studied Hai-Ku poetry and calligraphy at a Zen monastery outside Kyoto before returning to Seattle in autumn.

Tobey's first solo exhibition at the Seattle Art Museum occurred in 1935; he also traveled to New York, Washington, D.C., Alberta, Canada, as well as Haifa for a Baháʼí pilgrimage. Sometime in November or December, while working at night at Dartington Hall and listening to the horses breathe in the field outside his window, he painted a series of three paintings, Broadway, Welcome Hero, and Broadway Norm, in the style that would become known as "white writing" (an interlacing of fine white lines).

Mid-career
Tobey expected to return to teaching in England in 1938, but the mounting tensions of war building in Europe kept him in the US. Instead, he began to work on the Federal Art Project, under the supervision of Inverarity. In June 1939, when Tobey attended a Baháʼí summer program and overstayed his allotted vacation time, Inverarity dropped him from the Works Progress Administration (WPA) project. Tobey met the Swedish scholar, Pehr Hallsten (died 1965, Basel), in Ballard in 1939 and they became companions, living together from 1940. By 1942, Tobey's process of abstractionism was accompanied by a new calligraphic experiment. Marian Willard of the Willard Gallery in New York had seen some of Tobey's WPA paintings and gave him a show in 1944, which was considered to be a major success. In 1945, he gave a solo exhibition at the Portland Art Museum in Oregon, and the Arts Club of Chicago held solo shows of his work in 1940 and 1946. He studied the piano and the theory of music with Lockrem Johnson, and, when Johnson was away, with Wesley Wehr, who was introduced to Tobey in 1949 by their pianist friend Berthe Poncy Jacobson. Wehr, an undergraduate at the time, happily accepted the opportunity to serve as a stand-in music composition tutor for Tobey and over time became friends with him and his circle of artists, becoming a painter himself, as well as a chronicler of the group.

Tobey showed at New York's Whitney Museum in 1951. He also spent three months as guest critic of graduate students' work at Yale University on the invitation of Josef Albers, and had his first retrospective show at the California Palace of the Legion of Honor in San Francisco. In 1952, the film "Tobey, Mark: Artist" debuted in the Venice and Edinburgh film festivals. Acknowledging "academic responsibility," Hallsten enrolled in graduate school at the University of Washington's department of Scandinavian languages and literature in the early 1950s and, after receiving his master's degree, Tobey began referring to him by the honorific, Professor.

On September 28, 1953, Life magazine published an article on Tobey, Guy Anderson, Kenneth Callahan, and Morris Graves entitled, "Mystic Painters of the Northwest," which placed them in the national limelight. The four were considered founders of the Northwest School. He held a solo show at the Galerie Jeanne Bucher in Paris in 1955, and traveled to Basel and Bern. He began his ink wash paintings two years later. In 1958, he became the second American, after James Abbott McNeill Whistler, to win the International Grand Prize at the Venice Biennale.

Later years
Tobey and Hallsten emigrated to Basel, Switzerland in the early 1960s. Tobey, who had been an incessant traveler in earlier years, concentrated on his art, while Hallsten felt restless and traveled through Europe, returning to Basel. In 1960, Tobey participated in the Association of Visual Artists Vienna Secession, and in the following year, he became the first American painter to exhibit at the Pavillon de Marsan in Paris. Solo exhibits occurred at MoMa in 1962, and at the Stedelijk Museum in 1966, the same year that he visited the Baháʼí World Center in Haifa. In 1967, he showed again at the Willard Gallery, and held a retrospective show at the Dallas Museum of Fine Arts the following year. Another major retrospective of the artist's work took place at the Smithsonian's National Collection of Fine Arts in 1974. Tobey died in Basel in 1976.

In 2017 (from 6 May to 17 September), an important retrospective exhibition of Tobey's mature work was mounted in Venice, Italy, sponsored by the Peggy Guggenheim Collection, organized by the Addison Gallery of American Art Philips Academy of Andover, Massachusetts, and curated by Debra Bricker Balken. The exhibition was able to draw crowds from the Venice Biennale, gaining international attention and spurring an international reassessment of Tobey's significance before traveling to the Addison Gallery of American Arts, Phillips Academy, Andover, Massachusetts and exhibited 4 November 2017 to 11 March 2018.

Style
Tobey is most noted for his late "white writing" style, where an overlay of white or light-colored calligraphic brush strokes is painted over an abstract field of muted color, which is itself composed of small, interwoven brush strokes. This method, in turn, gave rise to the type of "all-over" painting style made most famous by Jackson Pollock and later painters.

When unveiling his white writing work at the Willard Gallery, where a lot of the future Abstract Expressionists were then exhibiting, Tobey did not want to confuse people as he was based in Seattle with strong ties to Asian art. Willard hired Sidney Janis (who would in 1948 open his own gallery in New York, pulling the Abstract Expressionists Jackson Pollock, Willem de Kooning, Robert Motherwell, and Philip Guston with him) to write an essay for the Tobey work at Willard to clarify Tobey's position. This essay both acknowledged Tobey's orientation toward Asia and emphasized an important distinction between his white writing and the automatic writing of the Surrealists which would inspire many of the American Abstract Expressionists. To quote Janis:"Nine years ago in 1935, Mark Tobey evolved the technique of white writing, which has distinguished his work. This method, a fusion of the spirit of Chinese writing with morphic characters rooted in twentieth-century painting, derives from Tobey's intensely personalized vision. ... It is presumably different from the psychic automation in [that it] is essentially under conscious direction."Janis claimed that Tobey's style was "at odds with the latest iterations of modernism," because "Tobey's white writing was more studied and controlled, the outcome of prolonged deliberation." In this regard, Tobey's painting Threading Light (1942), which was in the Willard show of 1944, has been compared to Pollocks's Night Mist of 1945 and André Masson's Automatic Drawing of 1924, which is an example of work directly influenced by Surrealist automatic writing.

Influence
Tobey, the senior of the 'mystical painters', was an influence on Graves. Tobey studied piano and music theory with John Cage, and thereafter, it was Tobey who had an influence on Cage.

Elizabeth Bayley Willis showed Tobey's painting Bars and Flails to Jackson Pollock in 1944. Pollock studied the painting closely and then painted Blue Poles, a painting that made history when, in 1973, the Australian government bought it for $2 million. A Pollock biographer wrote: "...[Tobey's] dense web of white strokes, as elegant as Oriental calligraphy, impressed Jackson so much that in a letter to Louis Bunce he described Tobey, a West Coast artist, as an 'exception' to the rule that New York was 'the only real place in America where painting (in the real sense) can come thru." Pollock went to all of Tobey's Willard Gallery shows where Tobey presented small to medium-sized canvases, measuring approximately 33 x. After Pollock viewed them, he went back home and blew them up to 9 x, pouring paint onto the canvas instead of brushing it on. Pollock was never really concerned with diffused light, but he was very interested in Tobey's idea of covering the entire canvas with marks up to and including its edges, something not done previously in American art.

Helmi Juvonen, another Northwest School artist and diagnosed manic-depressive, was obsessed with Tobey, and suffered the delusion that she and Tobey were to be married, even though Tobey was gay.

Legacy
At least five of his works are in the permanent collections of the Museum of Northwest Art. Tobey's work can also be found in most major museums in the U.S. and internationally, including the Smithsonian American Art Museum, the Tate Gallery in London, the Museum of Modern Art in New York City, the Metropolitan Museum of Art, and the Whitney Museum of American Art. There have been at least four posthumous individual exhibitions of Tobey's work: National Gallery of Art, Washington, D.C., USA, 1984; Museum Folkwang, Essen, Germany, 1989; Galerie Beyeler, Basel, Switzerland, 1990; and Museo Nacional Centro de Arte Reina Sofía, Madrid, Spain, November 11, 1997 – January 12, 1998 where the exhibition brought together about 130 works from some 56 different collections, covering the years from 1924 to 1975. Two of Tobey's paintings are in Guggenheim collections. A number of his figurative and abstract works are held by the Dartington Hall Trust. Four of Tobey's signed lithographs hang in the reception hall in the Seat of the Universal House of Justice, the supreme governing institution of the Baháʼí Faith, as his work was inspired by Oriental influences and his involvement in the Baháʼí Faith.

Anatoma tobeyoides, a species of sea snail, is named in honor of Tobey.

Awards

 * 1968, "Commander, Arts and the Letters of the French Government"
 * 1961, won first prize, Carnegie International, Pittsburgh
 * 1959, became the first American since James McNeill Whistler to win the Painting Prize at the Venice Biennale
 * 1956, elected at the National Institute of Arts and Letters
 * 1956, Guggenheim International Award

title: Mark Tobey


text:
Rostra

The Rostra (Rostri) was a large platform built in the city of Rome that stood during the republican and imperial periods. Speakers would stand on the rostra and face the north side of the Comitium towards the senate house and deliver orations to those assembled in between. It is often referred to as a suggestus or tribunal, the first form of which dates back to the Roman Kingdom, the Vulcanal.

It derives its name from the six rostra (plural of rostrum, a warship's ram) which were captured following the victory which ended the Latin War in the Battle of Antium in 338 BC and mounted to its side. Originally, the term meant a single structure located within the Comitium space near the Roman Forum and usually associated with the Senate Curia. It began to be referred to as the Rostra Vetera ("Elder Rostra") in the imperial age to distinguish it from other later platforms designed for similar purposes which took the name "Rostra" along with its builder's name or the person it honored.

History
Magistrates, politicians, advocates and other orators spoke to the assembled people of Rome from this highly honored, and elevated spot. Consecrated by the Augurs as a templum, the original Rostra was built as early as the 6th century BC. This Rostra was replaced and enlarged a number of times but remained in the same site for centuries.

In 338 BC the Rostra got its name when, following the defeat of Antium by the consul Gaius Maenius, the Antiate fleet was confiscated by Rome, of which the prows (literally rostra in Latin) of six ships were set upon the Rostra. Maenius paid for it out of his share of war booty. He also erected a victory column, the Columna Maenia, close to the Rostra.

Julius Caesar rearranged the Comitium and Forum spaces and repositioned the Senate Curia at the end of the republican period. He moved the Rostra out of the Comitium. This took away the commanding position the curia had held within the whole of the forum, having advanced extremely close to the Rostra during its last restoration. Augustus, his grand-nephew and first Roman emperor, finished what Caesar had begun, as well as expanded on it. This "New Rostra" became known as the Rostra Augusti. What remains in the excavated forum today, next to the Arch of Septimius Severus, has endured several restorations and alterations throughout its historical use. While a few different honorary names are attributed to those restorations, scholars, archeologists and the government of Italy recognise this platform as the "Rostra Vetera" encased inside the "Rostra Augusti".

The term rostrum, referring to a podium for a speaker is directly derived from the use of the term "Rostra". One stands in front of a Rostrum and one stands upon the Rostra. While, eventually, there were many rostra within the city of Rome and its republic and empire, then, as now, "Rostra" alone refers to a specific structure. Before the Forum Romanum, the Comitium was the first designated spot for all political and judicial activity and the earliest place of public assembly in the city. A succession of earlier shrines and altars is mentioned in early Roman writings as the first suggestum. It consisted of a shrine to the god Vulcan, that had two separate altars built at different periods. This early Etruscan mundus altar originally sat in front of a temple that would later be converted into the Curia Hostilia.

During the late Republic the rostra was used as a place to display the heads of defeated political enemies. Gaius Marius and consul Lucius Cornelius Cinna captured Rome in 87 BC and placed the head of the defeated consul, Gnaeus Octavius, on the Rostra. The practice was continued by Sulla and Mark Antony, who ordered that Cicero's hands and head be displayed on Caesar's Rostra after the orator's execution as part of the Proscription of 43 BC.

Caesar spoke from the Rostra in 67 BC in a successful effort to pass, over the opposition of the Senate, a bill proposed by the tribune Aulus Gabinius (the lex Gabinia) creating an extraordinary command for Pompey to eliminate piracy in the Mediterranean. Brutus and Cassius spoke from the Rostra to an unenthusiastic crowd in the Forum after the assassination of Caesar in 44 BC. Millar comments that during the late Republic, when violence became a regular feature of public meetings, physical control and occupation of the Rostra became a crucial political objective.

Tribal assemblies and tribunals
Until about 145 BC, the Comitium was the site for tribal assemblies (comitia tributa) at which important decisions were taken, magistrates were elected and criminal prosecutions were presented and resolved by tribal voting. Before an assembly, the convening magistrate, acting as augur, had to take the auspices in the inaugurated area (templum) on the Rostra from which he was to conduct the proceedings. If the omens were favorable and no other magistrate announced unfavorable omens, the magistrate summoned other magistrates and senators and directed a herald to summon the people. Heralds did so from the Rostra and from the City walls. During an assembly, magistrates, senators and private citizens spoke on pending legislation or for or against candidates for office. Before bills were presented for voting, a herald read them to the crowd from the Rostra. At the culmination of the process, the tribes were each called up to the templum on the Rostra to deliver their votes. After about 145 BC, the voting population of Rome grew too large for the Comitium, and tribal assemblies were then held at the opposite end of the Forum around the Temple of Castor, the steps of which served as an informal Rostra. The Rostra was also used for meetings of courts. In Republican Rome, criminal prosecutions took place in the Forum either before a tribal assembly with a magistrate prosecuting (a procedure specified in the Twelve Tables and the normal mode of prosecution in the middle Republic) or in a jury-court (quaestio de repetundis) established by statute and presided over by a magistrate with a jury (after 70 BC) of about 50-75 jurors. For trials held in the Comitium, the Rostra served as the tribunal upon which the magistrate sat in his curule chair with a small number of attendants. "This was enough in itself to establish a court, though it was supplemented by benches (subsellia) for the jurors, the parties to the case, and their supporters." The circle of onlookers (corona) either stood or sat on nearby steps.

The original structure was built during the middle years of the Roman Republic in approximately 500 BC It subsequently became known as the "Rostra" after the end of the Latin War in 338 BC when it was adorned by Gaius Maenius with naval rams (rostra) of ships captured at Antium as war trophies.

The Rostra was located on the south side of the Comitium opposite the Curia Hostilia (the original Senate house), overlooking both the Comitium and the Roman Forum. In addition to the prows of captured ships, the Rostra bore a sundial and, at various times, statues of such important political figures as Camillus, Sulla and Pompey. Private citizens also erected a number of honorary columns and monuments on the Rostra and throughout the forum. At one point, the Senate threatened to have them removed if the donors did not do so themselves.

Rostra Vetera
In form, the original Rostra may have been a simple raised platform made of wood, similar to the Roman tribunal. The Rostra had a curved form, possibly along the outer south rim of an amphitheatre. The structure was described by Christian Charles Josias Bunsen, based on his examination of two Roman coins depicting the Rostra, as "a circular building, raised on arches, with a stand or platform on the top bordered by a parapet; the access to it being by two flights of steps, one on each side. It fronted towards the Comitium, but later speakers often faced in the opposite direction to address larger audiences in the Forum. The Rostra Vetera's form has been in all the main points preserved in the ambones, or circular pulpits, of the most ancient churches, which also had two flights of steps leading up to them, one on the east side, by which the preacher ascended, and another on the west side, for his descent. Specimens of these old churches are still to be seen at Rome in the churches of San Clemente al Laterano and San Lorenzo fuori le Mura.

As part of his reconstruction of the Roman Forum in 44 BC, Julius Caesar is believed to have moved the republican Rostra Vetera. This Rostra, referred to as the "Rostra Nova" or "Caesarian Rostra", reused and incorporated nearly all of the original Rostra. Located on the southwest side of the new Julian Forum (Forum Iulium), the new Rostra was no longer subordinated to his new Senate House, the Curia Julia (still standing); Caesar had it placed on the central axis of the Forum, facing toward the open space. Left uncompleted at Caesar's death, Augustus finished and extended the new Rostra into a rectangle at the front, the dimensions being 28.8 m long, 10 m broad and 3.4 m above the level of the forum pavement. Traces of this Rostra can be seen today.

At the opposite end of the open Forum, the Temple of Caesar, completed by Augustus in 29 BC, included another Rostra at the front of its elevated base, facing the Caesarian Rostra. This Rostra was decorated with the rams from the Battle of Actium. John E. Stambaugh, professor of classics at Williams College, described the new arrangement as "a reflection of contemporary taste and the relentless Augustan desire for order."

In contemporary news
In November 2008 heavy rain damaged the concrete covering that has been protecting the Vulcanal and its monuments located in the Imperial comitium space since the 1950s. This includes the stele accorded the name of "The Black Rock" or Lapis Niger. The marble and cement covering is a mix of the original black marble, said to have been used to cover the site by Sulla, and modern cement used to create the covering and keeping the marble in place.

Professor Angelo Bottini, Superintendent of Archeology in Rome, stated that an awning or tent covering is in place to protect the ancient relics until the covering is repaired, giving tourists of this millennium a look at the original suggestum for the first time in 50 years.

Other known Rostra
In 29 BC Augustus ordered the construction of another Rostra in front of the Temple of Caesar, at the opposite end of the Forum Iulium from Caesar's Rostra. This was used as a tribunal and was adorned with the prows of galleys captured at the Battle of Actium.

title: Rostra


text:
Georg Curtius

Georg Curtius (April 16, 1820 – August 12, 1885) was a German philologist and distinguished comparativist.

Biography
Curtius was born in Lübeck, and was the brother of the historian and archeologist Ernst Curtius. After an education at Bonn and Berlin, he was for three years schoolmaster in Dresden, until (in 1845) he returned to Berlin University as a privatdocent. In 1849 he was placed in charge of the Philological Seminary at Prague, and two years later was appointed professor of classical philology in Prague University. In 1854, he moved from Prague to a similar appointment at Kiel, and again in 1862 from Kiel to Leipzig. He was teaching lndo-European and the historical grammar of the classical languages at Leipzig. His contributions were focused "to bridge the gulf between classical philology and Aryan linguistics." As a professor he constantly attempted " to bring Classical Philology and the Science of  Language  into  closer  relation   with  each  other." This is clearly reflected in the works of his pupils, and of his own.

His philological theories exercised a widespread influence. The more important of his publications are: The last two works were translated into English by Augustus Samuel Wilkins and Edwin Bourdieu England.
 * Die Sprachvergleichung in ihrem Verhältniss zur classischen Philologie (1845; Eng. trans. by FH Trithen, 1851)
 * Sprachvergleichende Beiträge zur griechischen und lateinischen Grammatik (1846)
 * Grundzüge der griechischen Etymologie (1858–1862, 5th ed. 1879)
 * Das Verbum der griechischen Sprache (1873).

From 1878 until his death Curtius was general editor of the Leipziger Studien zur classischen Philologie. His Griechische Schulgrammatik, first published in 1852, passed through more than twenty editions and was edited in English. In his last work, Zur Kritik der neuesten Sprachforschung (1885), he attacked the views of the emerging Neogrammarian school of philology.

Curtius died in Hermsdorf am Kynast, aged 65, and was succeeded at Leipzig by his student Karl Brugmann. The Opuscula of Georg Curtius were edited after his death by Ernst Windisch (Kleine Schriften von E. C., 1886–1887). He was posthumously elected to the American Philosophical Society in 1886.

title: Georg Curtius


text:
User:Astronautics~enwiki/lottery

Wikimoney Lottery
This is the first lottery for Wikimoney. Each slot costs ψ4.

If you wish to enter, sign any slot below and transfer the funds to my Wikimoney account. There is one first-place prize of ψ20, one second-place prize of ψ10 and one third-place prize of ψ5.
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title: User:Astronautics~enwiki/lottery


text:
Christian August Brandis

Christian August Brandis (13 February 1790 – 21 July 1867) was a German philologist and historian of philosophy.

Biography
Brandis was born at Hildesheim, and was the son of the physician Joachim Dietrich Brandis. His father moved to Copenhagen in 1810, where he became personal physician to Queen Marie.

He was educated at Kiel University. In 1812 he graduated at Copenhagen, with a thesis Commentationes Eleaticae (a collection of fragments from Xenophanes, Parmenides and Melissus). For a time he studied at Göttingen, and in 1815 presented as his inaugural dissertation at Berlin his essay Von dem Begriff der Geschichte der Philosophie.

In 1816 he refused an extraordinary professorship at Heidelberg in order to accompany BG Niebuhr to Italy as secretary to the Prussian embassy. Subsequently, he assisted Immanuel Bekker in the preparation of his edition of Aristotle. In 1821 he became professor of philosophy in the newly founded University of Bonn, and in 1823 published his Aristotelius et Theophrasti Metaphysica. With Böckh and Niebuhr he edited the Rheinisches Museum, to which he contributed important articles on Socrates (1827, 1829). From 1836 to 1839 he was tutor to the young King Otto of Greece. Brandis was elected a member of the American Antiquarian Society in 1839.

His great work, the Handbuch der Geschichte der Griechisch-Römischen Philosophie (1835-1866; republished in a smaller and more systematic form, Geschichte der Entwicklungen der griechischen Philosophie und ihrer Nachwirkungen im römischen Reiche, 1862-1866), is characterized by sound criticism.

Brandis died on 21 July 1867 in Bonn, Germany.

He was the father of the forestry academic and administrator Sir Dietrich Brandis.

title: Christian August Brandis