Talk:Nonstandard calculus

message: If someone named Robinson made "one of the major mathematical advances of the twentieth century" then perhaps one of the editors of this article would be kind enough to say who is this "Robinson" character and what was the "three hundred year old problem".  —Preceding unsigned comment added by 160.33.66.118 (talk) 01:29, 7 November 2008 (UTC)
thread_title: Old thread

message: The current version of the article is a little disappointing as it plunges into transfer operators and internal functions almost as soon as it starts.  Somehow a novice is looking for something more elementary than a sequel to non-standard analysis.  If I were more competent in the area I would write an introduction to non-standard calculus.  Anyone interested? Katzmik (talk) 11:08, 28 August 2008 (UTC)
thread_title: Disappointing

message: The proof of the IVT currently in the article seems to rely on the claim that the positive hyperintegers are well-ordered, which claim it justifies by the transfer principle. But the principle does not apply here (as it applies only to first-order formulae, and well-orderedness is second-order) and the claim is obviously wrong: if M is an infinite positive hyperinteger, then \{M-n|n \in \mathbb {N} \} is a nonempty set of positive hyperintegers without a least element. Can the proof be rescued? Algebraist 11:13, 2 September 2008 (UTC)
replies: text: Thanks very much for your correction.  I think the answer to your question is the remark that the set you define is not internal.  I thought transfer should apply to arbitrary order.  Hopefully CSTAR or Arthur Rubin are watching and will straighten us out. Katzmik (talk) 11:43, 2 September 2008 (UTC)
replies: text: I think the proof is correct; the (positive) hyperintegers are not well-ordered, as Algebraist quite properly noticed, but any non-empty internal set of positive hyperintegers has a first element.  (I can prove this in the ultrafilter representation, but I'm not sure of the details of the transfer principle in Robinsons's.)  But it's been a while since I've done this.  — Arthur Rubin  (talk) 13:11, 2 September 2008 (UTC)
replies: text:  It is correct that any non-empty internal subset of the non-negative integers has a least element. This follows by transfer. For the IST approach see Nelson's 1977 paper. For the model-theoretic approach see Albeverio, Fendstad, Hoegh-Krohn, Lindstrom.--CSTAR (talk) 15:34, 2 September 2008 (UTC)
replies: text: This comment should certainly appear somewhere, how about hyperinteger?  Katzmik (talk) 15:37, 2 September 2008 (UTC)

text: Thanks, I am very relieved.  Please improve the article by adding whatever clarifications may be appropriate.  Katzmik (talk) 13:15, 2 September 2008 (UTC)

text: Another matter: the article currently makes much of the shortness of this proof of the IVT compared to standard proofs. And indeed it is much shorter than the standard proof by interval bisection. But it's just as short as (and very similar to) the standard proof directly from the least-upper-bound axiom. Algebraist 14:28, 2 September 2008 (UTC)
replies: text: Looks like you are right.  Let's find a standard theorem that requires nested intervals for its proof, and include it in this page.  The advantages of NSA should be clearer then. Katzmik (talk) 14:34, 2 September 2008 (UTC)
replies: text: I seem to recall a theorem involving Hilbert spaces in which the non-standard proof was much shorter than the standard proof.  I can't recall what it was, though.  — Arthur Rubin  (talk) 14:49, 2 September 2008 (UTC)
replies: text:  Re: Your question I can't recall what it was.  You may be thinking of the spectral Theorem for arbitrary self-adjoint operators.  The non-standard proof is very elegant and short. --CSTAR (talk) 15:37, 2 September 2008 (UTC)

text: Not really calculus, though.... — Arthur Rubin  (talk) 14:53, 2 September 2008 (UTC)
thread_title:  Error in proof of intermediate value theorem 

message: To answer your question about the extension of the Dirichlet function, I will apply the extension principle which states that every real function f has a hyperreal extension f* (the professionals seem to prefer the notation *f which proves conclusively that I am not a hyperprofessional).  In particular, there exists a hyperreal extension of the Dirichlet function.  As you know, the axiom of choice is used in all these constructions, therefore certain things cannot be made very contstructive, which is why Bishop checked out of this particular club (as well as Connes, apparently; see non-standard analysis and the interesting discussion at talk page there). Katzmik (talk) 15:48, 3 September 2008 (UTC)
replies: text: As far as duct tape, it may be helpful to illustrate the above as follows.  Let us show that the natural extension of the Dirichlet function is not continuous at π.  Consider the continued fraction approximation an of π.  Now let the index n be an infinite hyperinteger.  Then the hyperrational  an is infinitely close to π.  Thus the natural extension of the Dirichlet function takes different values at these two infinitely close points, and your head is whole :)  Hope this helps Katzmik (talk) 12:11, 4 September 2008 (UTC)
replies: text: I think there's an even easier method. If you want to prove that this function is discontinuous in standard analysis, you would probably make use of the fact that between any two rationals there's an irrational. By the transfer principle, the same statement is valid for the hyperreals. So let r be a rational, and let another rational be given by r'=r+1/n, where n is an infinite hyperinteger. Between r and r' there is an irrational hyperreal number, so discontinuity follows directly from the definition.--76.167.77.165 (talk) 23:26, 15 March 2009 (UTC)
thread_title:  Dirichlet function bis 

message: Hi Arthur,
Thanks for your edits.  I see that you removed the "infinity" category at the bottom of the page.  I thought Robinson's infinitesimals were a hands-on way of working with infinity.  Why do you feel such inclusion is inappropriate? Katzmik (talk) 15:40, 17 September 2008 (UTC)
replies: text: I'm not sure what the intent of :Category:Infinity is exactly, but I didn't think it fit.  I still don't, but perhaps it could be in a logical subcategory.  — Arthur Rubin  (talk) 17:47, 17 September 2008 (UTC)
replies: text: Infinitesimal is in the category, and probably should not be;  as a subcat might satisfy both of you. Septentrionalis PMAnderson 22:14, 17 September 2008 (UTC)
thread_title:  [[infinity]] category 

message: The discussion of non-uniform continuity ended, in an earlier version, with the following comment: 
:By way of comparison, in the standard approach one must show that there exists an ε such that for every δ there exist x and y such that |x − y| < δ while |ƒ(x) − ƒ(y)|>ε.  This type of calculation is generally considered more difficult due to its quantifier complexity, see comment in Kevin Houston's book.
This comment has been removed by another editor, who has not responded to my request for a clarification.  I would like to open a discussion as to the appropriateness of comparing the non-standard and the standard arguments in such a fashion.  Katzmik (talk) 13:21, 30 October 2008 (UTC)
replies: text: I'd give Thenub a day or so to respond. I think the removed text was OK, although we really should include the page numbers for the Houston reference. I am curious what argument Fitzpatrick makes, but I don't have the book. Thenub, can you sketch it here? — Carl (CBM · talk) 13:30, 30 October 2008 (UTC)
replies: text: A proof is given at Uniform continuity#Properties, which might be the same one. Katzmik (talk) 14:14, 30 October 2008 (UTC)
replies: text: In a way, this is a powerful challenge to the non-standard approach.  After all, hyperreals are defined in terms of sequences, with a LOT of additional material from logic, whereas here one just uses sequences.  Once one develops an intuition for the hyperreals and is comfortable with infinitesimals, hyperintegers, etc, the "points" of R* seem more elementary than "sequences" of R.  However, this will not convince someone who has not yet developed such an intuition... I hope there is a better answer, will keep thinking.  Katzmik (talk) 15:03, 30 October 2008 (UTC)

text:  I am not attempting to issue a challenge.  Presumably, one needs to study non-standard calculus before they get further in non-standard analysis. And what question are you trying to answer? Thenub314 (talk) 12:50, 31 October 2008 (UTC)

text: Thenub, thanks. That all makes sense. I had been blinded by the formal definition and didn't think about the sequential alternative definition. — Carl (CBM · talk) 13:00, 31 October 2008 (UTC)
replies: text: The sequential definition does involve quantification over sequences, which puts it outside of first-order logic.  I think its effectiveness in teaching undergraduates requires proof. Katzmik (talk) 11:25, 27 November 2008 (UTC)
replies: text:  I am not sure what sort of proof would be appropriate.  All I can say is that I took the example from an undergraduate text used at a university where I taught. Thenub314 (talk) 12:59, 16 February 2009 (UTC)
thread_title:  Quantifier complexity 

message: I am a bit confused by phrase:
:While the thrust of Robinson's approach is that one can dispense with the limit-theoretic approach using multiple quantifiers,...
I think of the thrust of an argument to be it's most powerful part.  And I think of Robinson's approach as, his approach to defining infinitesimals.  So at this point I am a bit confused, because I don't think of Robinson's work has having much to do with multiple quantifiers.  I think what is meant here is:
:"The main advantage to teaching infinitesimal calculus at in introductory level is that one can dispense with the limit-theoretic approach using multiple quantifiers,..."
But I am not sure, so I thought I should ask before I changed anything. Thenub314 (talk) 14:45, 31 October 2008 (UTC)
thread_title:  Confusing phrase. 

message: We claim in limit of a sequence in the non-standard setting is quantifier-free.  But in the definition we have the phrase "for every hyperinteger n". Am I missing something? Thenub314 (talk) 14:51, 31 October 2008 (UTC)
replies: text: Good point.  — Arthur Rubin  (talk) 18:09, 31 October 2008 (UTC)
replies: text:  Looks like it has been taken care of by CSTAR.  Thanks! Thenub314 (talk) 08:06, 1 November 2008 (UTC)
thread_title:  Limit of a sequence. 

message: In the basic theorems section we talk about Cauchy integrals and Cauchy sums.  I am unfamiliar with these phrases and it seems a lot like we mean Riemann integral and Riemann sum. Thenub314 (talk) 14:57, 31 October 2008 (UTC)
thread_title:  Cauchy Integral? 

message: In the current version of the page, the second section is entitled "Robinson's argument".  The section is rather technical.  Moreover, the construction presented here is only one possible construction of the hyperreals.  For example, Keisler uses a different construction.  All such constructions are excellent, but I wonder if it would be appropriate to present material that's more basic at this stage.  A reader confronted with infinite lists of axioms may not be motivated to continue reading.  I assume part of our purpose here is to serve the larger public.  I think detailed logical constructions are just as inappropriate at an early stage of non-standard calculus as equivalence classes of Cauchy sequences would be at an early stage of calculus. Katzmik (talk) 14:46, 11 January 2009 (UTC)
replies: text: I tend to disagree.  It was the description first given in my undergraduate course on the subject.  The course was desgined for non-math majors and was geared for many people who have not had calculus.  The idea is intuitively simplier, if technically more difficult, then many other treatments of the subject.  Thenub314 (talk) 15:31, 11 January 2009 (UTC)
replies: text: That's interesting.  Let's see what other editors think.  Incidentally, I did not mean to imply that this construction is worse than any of the others; I explicitly stated they are all excellent.  I had the feeling any details of a construction at this point are somewhat premature.  Was this description given in the first week of classes in your undergraduate course?? Katzmik (talk) 15:39, 11 January 2009 (UTC)
replies: text: See Dacu's comments below for an illustration of what sort of misconception I have in mind.  Katzmik (talk) 19:10, 14 January 2009 (UTC)
thread_title: subsection "Robinson's argument"

message: It seems to me the new lead section is overusing quotes a bit.  I think it should be possible to paraphrase Courant. Thenub314 (talk) 16:05, 11 January 2009 (UTC)
replies: text: I have relegated Courant to a footnote. Paraphrase may be a better solution. Clearly, too, another source is necessary in order to complete the paragraph adequately. --P64 (talk) 18:50, 21 April 2011 (UTC)
thread_title:  Lead is over using quotations 

message: Is "pointwise cluster" standard terminology? Thenub314 (talk) 19:05, 14 January 2009 (UTC)
replies: text: I don't know about pointwise but cluster has the advantage over the other term in that it is self-explanatory (I was not familiar with the other word until you introduced me to it, whether in mathematics or outside).  Katzmik (talk) 19:09, 14 January 2009 (UTC)

text: I've haven't heard this term before and my copy of Keisler's book doesn't mention it in the index. Maybe Katzmik meant monad.  Also what's the difference between  \approx, \cong ? Finally, it seems to me there are some quantifiers missing in the latest edit on uniform continuity (e.g., what's y quantified over?).  I think the definition was right before. --CSTAR (talk) 19:14, 14 January 2009 (UTC)
replies: text: I mentioned below that Keisler routinely leaves out the quantifiers in this type of condition, when the domain is clear from context.  Perhaps leaving out the domain of y is a bit artificial, but it makes the pointwise property much clearer, but you are the best judge.  As far as clusters are concerned, they have been used informally though not by Keisler.  Katzmik (talk) 19:23, 14 January 2009 (UTC)

text: I was also puzzled by the two different symbols used for the "infinitely close" equivalence relation, namely, the \approx in the introduction and the \simeq in the discussion of uniform continuity. If they are distinct, \simeq should be defined or explained. If they are not, \simeq should be replaced by \approx.Akvilas (talk) 16:50, 14 February 2009 (UTC)
replies: text: Keisler uses \approx, so I've changed all the \simeqs to that. Algebraist 16:58, 14 February 2009 (UTC)
thread_title:  Terminology 

message: Keisler would unhesitatingly give the definition in terms of a single quantifier as I did.  I think taking about a pair of hyperreals obscures the "pointwise" nature of the definition.  There were a lot of scandalized comments at the talk page of uniform continuity until this sank in, and they were presumably by the experts, whereas the article is supposed to be generally accessible.  Katzmik (talk) 19:13, 14 January 2009 (UTC)
replies: text: Here y cannot be interpreted as running over the standard reals (which is a source of concern for the latest edit), since for example if x is infinite the condition x\approx y will never be satisfied.  Katzmik (talk) 19:27, 14 January 2009 (UTC)
thread_title:  pair of hyperreals 

message: Two sections above may be relevant, Quantifier complexity and Intermediate value theorem. In particular someone has previously asked what is quantifier complexity of a proof.
Does quantifier complexity refer simply to the number of nested quantifiers? perhaps the maximum in a clause, maximum in a statement, maximum in a proof, maximum in textbook, whatever is said to be complex? Whatever the answer, it will be useful to provide a wikilink for "quantifier complexity" rather than simply "quantifier". --P64 (talk) 18:12, 21 April 2011 (UTC)
replies: text: There are actually two issues here, both the number of quantifiers and also alternations of quantifiers.  The latter are thought to be particularly confusing to students. Tkuvho (talk) 07:57, 22 April 2011 (UTC)
thread_title:  Quantifier complexity again 

message: Berkeley talked about lots of errors but not about the error term that I could see. The error term may be an infinitesimal but not necessarily if there isn't a derivative. I don't believe the business about Berkeley referring to the error term should be added to this article. Dmcq (talk) 14:42, 30 May 2011 (UTC)
replies: text: The "error term" is just the standard mathematical expression for whichever term clutters up the formula you want to obtain at the end.  If you find references to "error" objectionable, we could go with different terminology: Berkeley criticized dropping the terms at the end of the calculation that were previously assumed nonzero." Tkuvho (talk) 14:46, 30 May 2011 (UTC)
replies: text: He did talk a bit about the remainder and discarding ab when multiplying (A+a) by (B+b) but the main comment about that was "Nor will it avail to say that ab is a Quantity exceeding small: Since we are told that in rebus mathematicis errores quam minimi non sunt contemnendi". Dmcq (talk) 15:50, 30 May 2011 (UTC)
replies: text: It's the same criticism.  The term dt cannot be dropped even though it is exceedingly small.  Tkuvho (talk) 16:13, 30 May 2011 (UTC)

text: I agree with Dmcq, the business about error terms does not belong in this article.  I particularly object to using the phrase "ghost of departed quantities" in relation to error terms.  Since at that particular part of the critique he definitely was not discussing error terms. Thenub314 (talk) 17:16, 30 May 2011 (UTC)
replies: text: If he was not criticizing the disappearance of the higher order terms, what was he criticizing?  As I mentioned, I was using the expression "error term" in its modern mathematical sense, equivalent to higher order terms.  Tkuvho (talk) 18:50, 30 May 2011 (UTC)
replies: text: In the ghosts of departed quantities part he was criticizing treating two values of the x axis as different when determining fluxions and then treating them the same in the result. He said either they're different or they're the same. The ghost of a departed quantity was the infinitesimals used in the calculation of the fluxions. They had to be different to make the division meaningful and yet they had to be the same to be the tangent at a point. Dmcq (talk) 19:31, 30 May 2011 (UTC)

text: Berkeley could be referenced but it should be to the right place, the bit about the remainder and compensating errors and the quote from Newton is the bit which is relevant to taking the standard part. The evanescent bit is relevant to where Δy is divided by Δx and they become infinitesimals. Dmcq (talk) 19:37, 30 May 2011 (UTC)
replies: text: Whether Δx is an infinitesimal or a vanishing increment is a separate issue.  Today we know that both approaches can be implemented mathematically to our satisfaction.  In either case, we have to get rid of the Δx at the end.  The Δx is the difference between x and z mentioned by Berkeley earlier in the paragraph.  The paradox of x being at the same time equal to and not equal to z is the same paradox as Δx being zero and nonzero at the same time. Tkuvho (talk) 05:12, 31 May 2011 (UTC)
replies: text: The main thing is that Berkeley was identifying two different problems in these two sections, not one. Dmcq (talk) 07:46, 31 May 2011 (UTC)
replies: text: OK.  What are the two problems?  Incidentally, we are talking about a single section, namley section XXXV. Tkuvho (talk) 09:32, 31 May 2011 (UTC)
replies: text: XXXV only deals with Δx not being zero when determining fluxions but having to be zero if it applies to a single point, i.e. Δx being an evanescent increment. It does not deal with the remainder term. Dmcq (talk) 17:30, 31 May 2011 (UTC)
replies: text: Paragraph XXXXV specifically talks about dropping the higher order term at the end of the calculation, in the following terms: Divide now zzz  xxx by z  x and the Quotient will be zz +zx+xx: and, supposing that z and x are equal, this same Quotient will be 3xx which in that case is the Ordinate, which therefore may be thus obtained independently of Fluxions and In�nitesimals. But herein is a direct Fallacy: for in the first place, it is supposed that the Abscisses z and x are unequal, without such supposition no one step could have been made; and in the second place, it is supposed they are equal; which is a manifest Inconsistency.  Note that he uses the terms "fallacy" and "inconsistency", which are quite a bit stronger than paradox. Tkuvho (talk) 17:53, 31 May 2011 (UTC)
replies: text: And where does it refer to an error term or a remainder in this? What it refers to is the error of supposing they are different and then supposing they are the same. If he wanted to refer to an error term or remainder he would have and does so explicitly in other sections. Dmcq (talk) 20:32, 31 May 2011 (UTC)
replies: text: We can certainly avoid using the terms "error" or "remainder" if it bothers you.  However, the mathematical point is identical: x and z are "the same" if and only if their difference is zero!  It seems that we are quibbling about technicalities. Tkuvho (talk) 03:07, 3 June 2011 (UTC)
thread_title:  Error term 

message: This thread is getting unwieldy, so I am starting a new one.  I don't really follow what you are saying.  Apparently Berkeley is making the same mathematical point here, which is what he sees as the logical inconsistency of the procedure used in the calculus, where delta-x=z-x is nonzero at the start of the calculation, but is set to zero at the end.  What do you see as too different criticisms here, either philosophically or mathematically?  It seems that we are splitting hairs here.  Do you have a source that there are two separate criticisms of the logic of the dy/dx calculation?  Tkuvho (talk) 03:42, 1 June 2011 (UTC)
replies: text: The criticism is the same in that he is complaining about starting with one hypothesis, getting a result and then changing the hypothesis. That does not mean that when he was referring to the ghosts of departed quantities he was referring to the error term or remainder in his discussion about compensating errors. If you look in Sherlock Holmes in Babylon: and other tales of mathematical history By Marlow Anderson by Robin Wilson you'll find he discusses fluxions as proportional to nascent increments in one place and he discusses the derivation of the increment of a rectangle AB in another paragraph and then he talks about the last and most fundamental of Berkeley's criticisms about changing the hypothesis afterwards. There is not just one criticism, there is a few different ones. By the way he discusses also that some mathematicians around then talked about infinitesimals as fluxions which might explain a confusion I saw about the evanescent increments was fluxions, it is pretty clear Berkeley was referring to an actual velocity which was Newtons ultimate ratio. Dmcq (talk) 20:44, 1 June 2011 (UTC)

text: Anderson and Wilson are editors of this collection.  Which essay are you referring to exactly and by whom?  Both Jesseph and Sherry identify what they call the logical criticism and the metaphysical criticism.  They are among the leading experts on Berkeley.  They describe the logical criticism as dx being nonzero at the start and zero at the finish.  Tkuvho (talk) 13:19, 2 June 2011 (UTC)
replies: text:  I haven't read Jesseph in his entirety yet, but he definitely discusses other logical criticisms by Berkeley then just assuming something is non-zero and later assuming it is zero.  For example, he starts by discussing Berkeley's logical criticism of the product rule, in which he says that Berkeley correctly points out that Newton is taking the wrong difference quotient.  (The difference quotient Newton takes is akin to [f(x+h)g(x+h)-f(x-h)g(x-h)]/(2h))  There is more then just one logical criticism contained in The Analyst.  So I would like to offer the following correction: "They describe the [a] logical criticism ..." Thenub314 (talk) 22:47, 2 June 2011 (UTC)
thread_title: Remainder term

message: There is no need to mention "standard real number", as every real number is standard.  In Nelson's approach, you have standard and non-standard real numbers.  The notion of "non-standard real numbers" is a colloquial short-cut and is always in the plural. Tkuvho (talk) 03:04, 3 June 2011 (UTC)
replies: text: We have been through this before I have given you examples of papers where people use phrases like "for a non-standard real number x". Taking it out further is just you enforcing your POV that it is a nonsensical statement.  Thenub314 (talk) 03:59, 3 June 2011 (UTC)

text: In Nelson's framework, you can have a nonstandard real number.  This page is not working with Nelson's framework.  I have already analyzed the references you provided and showed them to be unconvincing.  This, Tao always uses the phrase in the plural, as I mentioned above.  It may be appropriate in a blog page.  Tkuvho (talk) 05:05, 3 June 2011 (UTC)
replies: text: I agree with Thenub314 on this issue.  In Nelson's framework, we have non-standard real numbers.  In other frameworks, "non-standard real number" is often shorthand for a member of the "non-standard real numbers" which is not a (standard) real number, which comes to the same thing.  — Arthur Rubin  (talk) 06:59, 3 June 2011 (UTC)
replies: text: The plural is used as an informal shorthand, as you say.  I think the singular is rarely used.  I suggest we follow a standard textbook such as Goldblatt, rather than informal shorthands on blog pages.  Tkuvho (talk) 07:35, 3 June 2011 (UTC)
replies: text: A standard textbook on non-standard analysis?  As I went to school before non-standard analysis became a subject for standard textbooks, I can't confirm textbooks, but the term is used in expository papers on non-standard analysis, such as in the American Mathematical Monthly.  — Arthur Rubin  (talk) 17:07, 3 June 2011 (UTC)
thread_title:  every real number is standard 

message: Looking at the non-standard analysis article, we are tending to put the stars on different sides of the function.  Is there any feelings that we should standardize, and if so to which.  My personal feeling is that consistency is good if there is consensus for it, but I have no feelings one way or the other about which way is preferred. Thenub314 (talk) 17:30, 7 June 2011 (UTC)
thread_title:  *f vs f* 

message: I have issues with the new last paragraph in the motivation section.  It is hard not to see this as a direct response to edits at Elementary Calculus and (ε, δ)-definition of limit, in which I included a reference which says something to the effect that "the hope that non-standard calculus could be done without ε-δ methods can not be realized in full."  Now I understand Tkuvho disputes my reading of the resource, but adding unsourced statements to the contrary only serves to escalate tensions. Thenub314 (talk) 21:47, 18 December 2011 (UTC)
replies: text: I tried to source it in the book. Tkuvho (talk) 12:49, 19 December 2011 (UTC)
thread_title:  Last Paragraph in Motivation Section 

message: The following insightful comment by Madison and Keith Stroyan was recently deleted from the page:
:Infinitesimals provide a method for calculating limits ... (Epsilon and delta methods require the answer in advance). 
Tkuvho (talk) 13:58, 2 January 2012 (UTC)
thread_title:  madison and stroyan