Talk:Standard part function

Old thread
This is one of two articles that link to internal object. If that article is ever created, one of those links will become grossly absurd. Which one, remains to be determined. All of this, of course, would be anticipated by anyone with any sense linking to that strange phrase. Michael Hardy (talk) 12:09, 4 September 2008 (UTC)


 * Whew, it took me a while to figure out what you meant. I say, let's beat the psychologists to it :) Katzmik (talk) 12:16, 4 September 2008 (UTC)


 * Perhaps internal set (alternate usage) would be better here, although we'd have to write that article. (As an aside, that article suggests internal set theory should be a disambiguation.)  Of course, internal object would be necessary in non-standard category theory.  — Arthur Rubin  (talk) 13:42, 4 September 2008 (UTC)


 * One could describe the standard part function as a non-internal object but not as a non-internal set without causing confusion. This was my only reason for preferring "object".  Perhaps there is a better word. Katzmik (talk) 13:46, 4 September 2008 (UTC)


 * Perhaps internal function ? — Arthur Rubin  (talk) 13:52, 4 September 2008 (UTC)


 * Good idea. The current version of internal set is unfortunately very brief.  Could one reproduce some of the material at hyperreals?  By the way, I find the constructivist claim hard to believe.  Can non-standard analysis really be constructivist?  Nelson should have explained this to Bishop :) Katzmik (talk) 13:57, 4 September 2008 (UTC)

Maybe a lot of these terms ought to get redirected to just one article that deals with all of them. Michael Hardy (talk) 14:03, 4 September 2008 (UTC)


 * Mikhail Gromov always taught me that there should be one idea per article. Perhaps the same applies to wikipages :) Katzmik (talk) 14:29, 4 September 2008 (UTC)

Fond of paradox?
I don't understand this revert. Are you fond of paradox? William M. Connolley (talk) 11:33, 4 May 2011 (UTC)


 * Hi William! I appreciate your interest in these pages.  Perhaps we can start on a more amicable foot.  As I explained in the edit summary, the key word here is "infinitesimal".  As you are probably aware, I am perfectly happy with the epsilon, delta definition.  On the other hand, such a definition does not resolve the paradox of Leibniz's infinitesimal definion.  Rather, it replaces it by a different definition without infinitesimals.  At the moment it is irrelevant whether that's better or worse.  What is relevant is that the paradox of the infinitesimal definition was not resolved until Robinson.  This does not at all mean that there was not a consistent definition before Robinson, only that the infinitesimal definition was justified by him and not by Weierstrass.  Tkuvho (talk) 11:37, 4 May 2011 (UTC)
 * We are probably close to agreement, it is a matter of wording. As it says elsewhere, In early calculus the use of infinitesimal quantities was thought unrigorous and this is correct: they were unrigourous. Thus rather than L's "definition" being paradoxical, it is merely informal, incomplete, unrigourous: call it what you will. The std d-e version does away with the problem entirely. The non-std way formalises infinitesimals (as I understand it from these pages; I'm not familiar with that way) William M. Connolley (talk) 11:51, 4 May 2011 (UTC)


 * The "paradox" though is Berkeley's criticism: how can dx be at the same time zero and nonzero? This is the puzzle that's answered by the standard part, which is a mathematical implementation of adequality.  The issue discussed here is only tangentially related to epsilon, delta.  Tkuvho (talk) 13:19, 4 May 2011 (UTC)


 * To summarize: the person who is fond of paradoxes is Berkeley, not me :) Tkuvho (talk) 13:51, 4 May 2011 (UTC)

Some recent edits
An earlier edit by Connolley insisted that Weierstrassian epsilontics was a resolution of the paradox of Leibniz's infinitesimal definition of the derivative. The most recent edits appear to concede that Robinson was the one to have resolved the paradox, after all. Why is Connolley still pursuing the opposite agenda at ghosts of departed quantities? Tkuvho (talk) 22:09, 21 May 2011 (UTC)


 * In response to your edit summary: I will endeavor to improve my spelling, please also try to improve your history. Tkuvho (talk) 22:38, 21 May 2011 (UTC)


 * Per WP:TALKNEW ("Never use headings to attack other users") I have changed the section title. Hans Adler 01:06, 22 May 2011 (UTC)


 * OK, thanks. Though I was criticizing the edits, not the editor.  Tkuvho (talk) 01:43, 22 May 2011 (UTC)

Not internal
An editor asked for a clarification of the fact that the standard part function is not internal. There are several ways of explaining this. Perhaps the simplest is that its domain D, which is the collection of limited (i.e. finite) hyperreals, is not an internal set. Namely, since D is bounded (by any infinite hypernatural, for instance), D would have to have a least upper bound if D were internal. And it doesn't. I certainly appreciate your interest. Please try to listen to reason and understand that your historical claims at ghosts of departed quantities are utterly flawed. Tkuvho (talk) 22:33, 21 May 2011 (UTC)

Non-standard real numbers
The expression "non-standard real numbers" is not appropriate. In Edward Nelson's internal set theory, all numbers are real but the background set theory is modified by the introduction of a "standard" predicate, in such a way that we have non-standard real numbers. However, here we are not dealing with Nelson's approach, but rather with Robinson's. I am a great admirer of Nelson's approach but it needs to be specified that that's what one is dealing with, if one talks of non-standard real numbers. Otherwise, it is preferable to leave "finite hyperreals" of the previous version of the page. Tkuvho (talk) 18:05, 22 May 2011 (UTC)


 * Not sure quite what you mean by that. In the phrase provides a method of of translating calculations in the non-standard real numbers back to the real numbers the use of "non-standard" seems entirely appropriate William M. Connolley (talk) 18:25, 22 May 2011 (UTC)


 * The phrase non-standard reals is, if you'll pardon the pun, standard. Thenub314 (talk) 19:22, 22 May 2011 (UTC)
 * (Invited by Tkuvko through Wikipedia E-mail). reals:hyperreals::standard reals:non-standard reals?  I'm afraid I agree with Thenub314; the term "non-standard reals" is standard.  However, "finite non-standard reals" is clunky, and, if needed, "finite hyperreals" would be better.  — Arthur Rubin  (talk) 02:31, 23 May 2011 (UTC)


 * I am often afraid when I have to agree with that thenub314 bloke, he's a bit of a git. Just kidding :).  More seriously, I did intentionally leave the word finite out of the lead sentence.  I wanted to give the correct idea about what the function is and what it is used for without getting bogged down in technicalities. Leaving that for later in the article.  Looking at the first sentence, do you feel it should explicitly say finite? Thenub314 (talk) 02:56, 23 May 2011 (UTC)


 * What is your source for the claim that the term "non-standard real" is standard? Every element of R is standard.  Elements of *R that are not in R are non-standard.  The expression "non-standard real" is a contradiction in terms from the point of view of Robinson's theory. Tkuvho (talk) 04:11, 23 May 2011 (UTC)


 * I am a bit surprised your unfamiliar with the expression but here goes... I will give a few different examples of its use in the world. First there are several text books use the phrase "non-standard real numbers" such as An Introduction to Nonstandard Real Analysis by Hurd and Loeb.  Next people often refer to them this way in general discourse, Terry Tao for example refers to them this way in his blog  (republished in Structure and randomness: pages from year one of a mathematical blog by T. Tao).  Also it is use this way in research articles, see for example Nonstandard  Measure Theory-Hausdorff Measure by Frank Wattenberg, Proceedings of the American Mathematical Society, Vol. 65, No. 2 (Aug., 1977) (pp. 326-331) for a nice example where it is done on page 1.  Finally, and perhaps most weakly, other wikipedia articles also use the phrase see for example Hyperreal number.  Now this last example I grant is not really a citable resource but I just wanted to point out that I am not the unique person accustomed to referring to them in this way. Thenub314 (talk) 19:02, 23 May 2011 (UTC)


 * "Nonstandard real analysis" should be parsed "nonstandard (real analysis)", namely a nonstandard approach to real analysis. Real analysis is a field dealing with certain phenomena such as derivatives, integrals, etc.  Nonstandard approach to this field is an approach using the hyperreals.  I am not too familiar with Hurd and Loeb's book but I assume they are not using Nelson's approach.  I will check the other references when I have time and patience. Tkuvho (talk) 19:18, 23 May 2011 (UTC)
 * I didn't mean the title, I meant the interior of the book. Say on page 25 following definition 6.1: "Since *R is called the set of nonstandard real numbers..." Thenub314 (talk) 22:33, 23 May 2011 (UTC)
 * I can't find Tao's article anymore. The link you provided seems to lead only to a discussion of his article, not the article itself.  Perhaps this is because it got published and the publisher requested that it be removed from the blog.  At any rate, you can hardly be seriously suggesting that loose blog talk should define normative usage.  Tkuvho (talk) 19:23, 23 May 2011 (UTC)
 * Odd, when I click it it takes me right to the article. In and of it self, a blog entry would not be so convincing.  But as part of a larger picture it is certainly not so silly as it sounds.  Keep in mind this particular blog entry was published by the AMS and it is authored by a fields medalist. Thenub314 (talk) 22:33, 23 May 2011 (UTC)
 * Wattenberg has an odd mannerism of avoiding the "hyper" prefix at all costs, e.g. *finite, *integer instead of hyperfinite, hyperinteger. No wonder he says "nonstandard real number".  I guess he couldn't bring himself to write "nonstandard *real number".  You would never catch Goldblatt doing that. Tkuvho (talk) 20:06, 23 May 2011 (UTC)

(outdent) I could spend my day typing in more references and posting them here but it is not really a fruitful use of my time. I feel compelled to quote an article I read whose subject doesn't quite apply, but the comment itself is applicable. "It would seem that expecting people to be convinced by the facts flies in the face of, you know, the facts." That is, scientifically speaking presenting people with evidence is more or less a fruitless way to try to convince them. I tried to give not a long list of examples but cross-section of examples from different places. But in the end, if I continue to list examples, you'll just find a problem with each of them in turn has you have done above. So in effect there is no way I can see for me to convince you on this point. Thenub314 (talk) 22:33, 23 May 2011 (UTC)

confusion about internal sets
The newly added phrase "Some authors develop the theory of non-standard analysis using Internal Set Theory. In this setting the standard part function "st" is not defined by an internal set" is confused. Internal sets are used in Robinson's approach. In this approach, the standard part function is not internal. Nelson's approach internal set theory develops a different viewpoint. Here the standard part is not defined by a "set" at all. Namely, what were called "internal sets" in Robinson's approach, become "sets" in Nelson's approach. Note that this is not that unusual: even in traditional set theory such as Bernays set theory, you have entities that are not sets but rather only "classes". These are not really similar to Nelson's approach, though. Tkuvho (talk) 18:45, 22 May 2011 (UTC)
 * Reverted. The courses I took in NSA did not use this terminology, but now that you point it out I see what you mean.  Thenub314 (talk) 19:24, 22 May 2011 (UTC)

adequality and non-standard analysis
It is a pity the reference to adequality has been deleted. To respond to the edit summary, Stillwell in his book cited at adequality mentions explicitly that Fermat was ahead of his time, and this approach was only implemented with the advent of non-standard analysis. Tkuvho (talk) 18:48, 22 May 2011 (UTC)


 * Thanks for the reference. I looked it over but it makes no mention of the standard part function as being the implementation of adequality. Thenub314 (talk) 22:13, 22 May 2011 (UTC)


 * He mentions it as being implemented in non-standard analysis, and that's enough. Furthermore, some recent articles make the same point. Tkuvho (talk) 04:13, 23 May 2011 (UTC)


 * Actually he only mentions that the idea was revived, which certainly allows for the idea to change, as opposed to being implemented which means the idea was put into action more or less as is. But putting that differences aside he in know ways mentions how or where in non-standard analysis this idea came up or whom it influenced.  Certainly he mentions no connection to the standard part function.  So it in general doesn't seem enough to me.  Thenub314 (talk) 19:09, 23 May 2011 (UTC)


 * Another source is the paper where you got the Sherry reference. Tkuvho (talk) 19:30, 23 May 2011 (UTC)

Derivative
In the derivative section $$f$$ is assumed to be a real function. How can we then plug in hyperreal values? --Jobu0101 (talk) 22:08, 26 October 2019 (UTC)