Talk:Elementary Calculus: An Infinitesimal Approach

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Old thread[edit]

I object to including Bishop's criticism in this page, as it tells the reader strictly nothing about the book. Perhaps once there is already a considerable material here about the book itself, we can start thinking of criticisms, but as things stand, the page is not too inviting for someone who actually wants to find out about the book. Katzmik (talk) 08:46, 21 December 2008 (UTC)

Relative analysis[edit]

I am not sure "relative analysis" is sufficiently notable to be mentioned here. The authors clearly state in their introduction that the approach is not new, and list a number of references, the earliest of which (1977) is the paper by Nelson. Hrbacek himself published in 1978 and 1979. Tkuvho (talk) 14:43, 31 May 2011 (UTC)

Ok, I have taken it out. Thenub314 (talk) 01:24, 1 June 2011 (UTC)
Their criticism is aimed at technical foundations of the subject such as the ultrapower construction. Keisler does not envision teaching the ultrapower construction to freshmen, nor does Hrbacek mean this particular criticism to apply to Keisler's book, but rather to more advanced developments in non-standard analysis. If we include this criticism at all, we should make it clear that Hrbacek is not criticizing the infinitesimal approach per se, but rather offering an alternative infinitesimal approach. Tkuvho (talk) 03:51, 1 June 2011 (UTC)
Not at all. Their criticism is about the pedagogical difficulty inherent teaching which manipulations of infinitesimal quantities are allowable. Check the articles they cite, it is not about ultrapower constructions but teaching students to do nonstandard analysis "safely". Thenub314 (talk) 20:49, 1 June 2011 (UTC) PS. Safely is their word choice not mine, hence the quotes. Thenub314 (talk) 05:08, 2 June 2011 (UTC)
You misread their piece. They are criticizing difficulties of a technical nature, that they alluded to earlier in their article, namely the ultrapower construction, etc. Keisler obviously is not planning to teach the ultrapower construction in first year calculus. Tkuvho (talk) 05:10, 2 June 2011 (UTC)
Which piece are you referring to? We are talking about several papers now, and I want to be clear on which one I have misread. Thenub314 (talk) 05:17, 2 June 2011 (UTC)
Since you are citing the 2010 article, I will quote what they write:
The usual framework for nonstandard analysis is based on a suitable non-Archimedean extension ∗R of the field of real numbers R, often constructed as an ultrapower of R. Such an approach is satisfactory to research mathematicians, and a number of important results proved using nonstandard methods testify to the power of these ideas. The advent of nonstandard analysis also raised hopes that the teaching of calculus at the elementary level could be made easier by replacing the ε–δ arguments with simpler, physically intuitive, yet rigorous reasoning about infinitesimals. Attempts to do so using nonstandard analysis show that the concept of infinitesimal is easy for students to grasp; however, technical details inherent in the development of analysis in the Robinsonian framework (see, for example, [13, 19]) present some serious pedagogical difficulties (page 801 of the article you wish to quote).
Now the phrase technical details inherent in the development of analysis is not a phrase typically used to describe freshman calculus, which is the subject of Keisler's book. Note that they wrote "analysis", not "calculus". What they are referring to are the technical details of the ultrapower they mentioned a few lines earlier (see above). If you wish to mention other articles by these authors, you can try, but don't put words into their mouths. They said strictly nothing here about any alleged shortcomings of Keisler's approach. Tkuvho (talk) 11:57, 2 June 2011 (UTC)
Look, I am sorry to say but your mis-interpreting their paper. They specifically mean the technical details associated with Keisler's book (and other books). Fundamentally they are talking about pedagogical difficulties associated with the transfer principle and trying to describe which statements are transferable. The part that you failed to quote (you should always go to the end of a sentence or at least up an ellipse or something to let the reader know your cutting things off) is "(see [15, 8] for a discussion of these matters)". Those are two papers, one by Hrbacek, and the other by O’Donovan. They are much more clear about what they see to be the problem, I have read them both and one thing they definitely do not discuss is ultrapower constructions. It is clear (or should be) that such a thing would be left out of a book for freshman or high school students. What cannot be left out, and is discussed in Keisler is the idea of transfer, and on some level you always need to discuss this if you want to avoid the difficulties of analysis on general non-archemedian fields. Anyways I am taking out the business about ultrapower constructions because that is just not what they are discussing. Thenub314 (talk) 16:23, 2 June 2011 (UTC)
You are basing your edits on sources you have read but are not quoting. The paper you are quoting says strictly nothing about Keisler's book. Tkuvho (talk) 17:51, 2 June 2011 (UTC)
I am sorry, perhaps I misread. For the sake of anyone reading along could you tell me what reference [13] is in the passage you quote above? Thenub314 (talk) 18:44, 2 June 2011 (UTC)
The texts by both Hrbacek and O'Donovan are published in a conference proceedings "nonstandard mathematics 2004". It is in the nature of such proceedings that the papers are not always seriously refereed. AMM is certainly refereed seriously, but they say strictly nothing here about Keisler. Your current edits introduced a criticism that is too vague to be reliable, and should be deleted. Tkuvho (talk) 18:02, 2 June 2011 (UTC)
Correction: "should be [has been] deleted." In the interests of 3RR I am not adding anything even remotely critical back for the moment. Thenub314 (talk) 20:32, 2 June 2011 (UTC)
I am interested in Hrbacek's criticisms of the transfer principle that you mentioned above. I suggest you try it on the talk page. I am somewhat sceptical about the relevance of this at the freshman calculus level. After all, these systems are mathematically isomorphic, as Edward Nelson proved in 1977. How great an improvement can it be if you call infinitesimals "ultrasmall"? Tkuvho (talk) 02:51, 3 June 2011 (UTC)
I think it is slightly different then just calling infinitesimals "ultrasmall", the property of ultrasmallness is relative. So and might be ultrasmall compared to 1, but not ultrasamll compared to each other (and perhaps is ultrasmall compared to for epsilon that are ultrasmall compared to 1? I haven't tried to prove it but I wouldn't be surprised.) The two authors give a number of criticisms. Mis-intuitions about derivatives caused by the standard formula involving the standard part. Particular confusing functions that come up. Hbracek also writes that it is not possible to get away from and because certain statements do not transfer as you would hope (Such as a standard function continuous at standard points extends to a continuous at non-standard points). Apparently in the setting of stratified (or relative) analysis it is easier transfer is somewhat more general so you avoid this difficulty. O'Donovan writes (near the end of his intro) that the problems caused often cause people to think about teaching non-standard analysis without discussing the transfer principle, which basically has its own problems. Hence my summary about the transfer principle. Hope that helps. Thenub314 (talk) 04:41, 3 June 2011 (UTC)
I don't understand Hrbacek's criticism about a standard function continuous at standard points extends to a continuous at non-standard points. He gives the example of the sine function, saying that one would want it to be "continuous" at the non-standard points, as well. But this seems to miss the point of hyperreal definitions of continuity. Namely, unlike the sine function, we don't want the squaring function to be continuous at infinite points! The whole advantage of the hyperreal definition is that we can formulate the failure of the squaring function to be continuous, in terms of a failure of microcontinuity at a single infinite point. That's a feature of Robinson's approach, rather than a bug. The way it's presented in Hrbacek it's the opposite. Tkuvho (talk) 18:45, 29 November 2011 (UTC)
Do you have access to the paper, or do you want me to comment further? Thenub314 (talk) 19:05, 29 November 2011 (UTC)
I do have access to the paper. I still don't see how he can turn a feature into a bug :) Tkuvho (talk) 19:07, 29 November 2011 (UTC)
I think your misunderstanding. The squaring function is continuous at infinite points. This is the "basic and useful" fact of non-standard analysis discussed in the setting. Thenub314 (talk) 19:17, 29 November 2011 (UTC)
You are missing the point. The following property is called microcontinuity at x, following Gordon et al: if x' is infinitely close to x, then f(x') is infinitely close to f(x). Now the failure of f(x)=x^2 to be microcontinuous at an infinite point is precisely responsible for the failure of uniform continuity of f on R. In Nelson's approach, the squaring function is continuous at all points, but how useful is this? Tkuvho (talk) 19:22, 29 November 2011 (UTC)

Is the squaring function continuous?[edit]

I don't think I am missing the point, I understand what your saying perfectly well, and it is discussed very clearly in the paper. First, I would comment the paper is not saying that the squring function is continuous only in Nelson's approach, but in Robinson's as well. What you seem to be debating is that this is a "basic and useful" fact. Do I understand your last question correctly? Thenub314 (talk) 20:11, 29 November 2011 (UTC)

Let's see if we can agree on some basic facts: (1) with respect to the epsilontic definition (with epsilons varying over R*), the squaring function will be continuous at every point. This is true by the transfer principle. (2) the squaring function is not microcontinuous at an infinite point, in the sense defined above. This makes it possible to give a "pointwise" definition of uniform continuity, and also to have a "pointwise" test for the failure of uniform continuity: the real squaring function is not uniformly continuous because its natural extension fails to be microcontinuous at H. Here H is any infinite point. Are we together so far? Tkuvho (talk) 12:30, 30 November 2011 (UTC)

the nature of the allegations[edit]

To respond to Thenub's comment, Hrbacek may refer to [13] which is Keisler's textbook, but he says strictly nothing about the nature of the "pedagogical difficulties" that he alleges. Tkuvho (talk) 02:57, 3 June 2011 (UTC)

So if he allegdes there are pedagogical difficulties, why is it not OK to include in the article "Authors XYZ say there are pedagogical difficulties"? The sentence I in was clearly qualified so the reader immediately knew whose opinion it was. I am not detailing their opinion in the article, just stating they have this opinion.
Thenub, saying there are "pedagogical difficulties" without saying a word about what they are is not informative and is a smear. We have no need for smears. Tkuvho (talk) 05:08, 3 June 2011 (UTC)
Sure they are saying a word, they do better, they give references. You read the references, and they have references too. You can read lots about it if you desire. I have spent the past day doing that in order to satisfy you. The fact is it doesn't matter if they are informative, I was simply pointing out the fact that they said it. Heck the invented a whole program to get around Keisler's (and more generally Robinson's) approach and then told everyone about their program in the monthly along with where to get teaching materials. That alone would seem fairly clear they don't feel they can use Keisler's book for pedagogically reasons. It is fairly significant they also said so in print. Anyways why is it OK that we only present a one-sided opinion? We cite Sullivan's positive review, why is it so crazy to cite their negative appraisal? Thenub314 (talk) 05:26, 3 June 2011 (UTC)
We should elaborate on Sullivan, for instance "a field study showing that students who followed a non-standard approach developed better understanding of fundamental concepts of the calculus than the control group following a standard syllabus". I don't think we should include blanket criticisms of the "pedagogical difficulties" sort. As I mentioned, both references they cite have not been reliably published, and certainly not in anything remotely resembling an education outlet. In such a situation, it behooves us to use some common sense. While I am sceptical about O'Donovan's criticisms, I may be convinced that there is something to Hrbacek's. I suggest we discuss this here rather than edit warring. In general, when it comes to including negative information about living people or their work, we should err on the side of caution. Tkuvho (talk) 08:35, 3 June 2011 (UTC)

Hrbacek's criticism[edit]

What user Thenub wrote above about Hrbacek's criticism is interesting. I propose to discuss it here, try to arrive at a consensus, and then including it in the page. Tkuvho (talk) 05:11, 3 June 2011 (UTC)

OK, lets talk. First things first. Here are some of my thoughts.
  • The strength of non-standard analysis claims all the papers published in the book are peer reviewed, so we I feel it is in appropriate to deny them based on not being reliable enough. This doesn't seem to be just your average conference proceeding.
  • I am happy to expand on Sullivan (I even tried to do so a little myself), as long as we fair and discuss the fact not everyone is content with approach. We would like to keep our bias to a minimum.
  • I am not happy quoting the comment about students being happy with infinitesimal numbers. It is not about this book, and the quote should be in the Non-standard calculus page. Thenub314 (talk) 08:58, 3 June 2011 (UTC)
The recent paper by Hrbacek et al. does not deal with the alleged "pedagogical difficulties". Rather, it proposes an axiomatic approach to infinitesimal calculus. Citing this paper as a reference for a criticism of Keisler's book is therefore inappropriate. Tkuvho (talk) 15:04, 28 November 2011 (UTC)
In fact I am curious about what their pedagogical criticisms might be. As far as the "technical difficulties", as Thenub correctly pointed out this refers to the ultrapower construction. Keisler does not treat the ultrapower construction in the book but rather in the accompanying volume for the instructor. Teaching ultrapowers to calculus students would be as inappropriate as teaching them about equivalence classes of Cauchy sequences. Tkuvho (talk) 15:13, 28 November 2011 (UTC)

The paper seems very apropos to me. There are not going to be a lot of published references we can use. Indeed this is one of those areas where most of the published material is by those who advocate using infinitesimals, although though that position is very small within the overall mathematics community. So we need to avoid giving a false impression that infinitesimal methods are often used in teaching calculus. Millions of students learn calculus each year, and all but a vanishing few learn it in a way without infinitesimals. — Carl (CBM · talk) 15:52, 28 November 2011 (UTC)

Without seeking to give an incorrect impression that infinitesimal-based teaching is prevalent, we can stick to the point when it comes to a meaningful discussion of the book which is the subject of this page. Hrbacek's paper is very interesting but it just does not happen to be about Keisler's book, and the critical comments Hrbacek makes concerning "technical difficulties" refer to the ultrapower construction, not used by Keisler (except in an appendix on page 911). Tkuvho (talk) 15:56, 28 November 2011 (UTC)
I added a comment to clarify the situation with regard to the "prevalence" issue you mentioned. Tkuvho (talk) 15:58, 28 November 2011 (UTC)
Hrbacek writes as follows: while it is undoubtedly possible to do calculus by means of infinitesimals in the Robinsonian framework, it does not seem possible to do calculus only by means of infinitesimals in it. In particular, the promise to replace the (epsilon, delta)-� method by the use of infinitesimals cannot be carried out in full. This is eminently reasonable, though I have not yet analyzed his arguments in detail. In other words, he claims that some residual epsilon, delta arguments must remain even after we replace most definitions by infinitesimals. Citing such a concrete criticism would be more meaningful than vague claims about "pedagogical problems". Tkuvho (talk) 16:35, 28 November 2011 (UTC)
O'Donovan explains, in the paper the current article cites, that the problem is with the explanation of the star map and the transfer principle. Basically he seems to be saying your left with either sweeping the details of how to use these operations under the rug, or introduce the logical underpinnings in order that these can be used "safely" which is too much of a prerequisite. (Haven't I said this before?) The bit about an ultrapower construction was a mistake, I started by copying a pasting a sentence from an old version and reworking to include some additions you made, unfortunately I was a little careless and grabbed the wrong sentence. In his various papers he gives examples of the difficulties ones encounters. He considers questions that arise in teaching such as "What is wrong with the up-down function f(x)=2st(x)-x?" and "How does one calculate "f'(x+δ)?" where x is real and delta is infinitesimal. Thenub314 (talk) 19:38, 28 November 2011 (UTC)
As I already mentioned, Hrbacek's credentials are a lot more solid than O'Donovan. The latter seems to be neither a mathematician nor an education specialist. A citation out of Hrbacek would be a lot more convincing. To respond to the particular point you raised, O'Donovan seems to think explaining the standard part function is difficult in Robinson's approach. Is this supposed to be more easily explained in Nelson's approach? Note that with Robinson we can explain that "st" is not an internal function. Did you ever ponder what the corresponding explanation is in Nelson's framework? It is that "st" is not a function because it is not defined by a set. How's that for effective undergraduate teaching? Tkuvho (talk) 20:18, 28 November 2011 (UTC)
We do not pick and choose articles depending on your point of view of their credentials. These are published articles, published in the same place. Your point about Nelson's framework is meaningless here unless your claiming it somehow relates to Kiesler's book. Please explain why I cannot summarize what a published reference in a peer reviewed journal says about this book. Thenub314 (talk) 21:59, 28 November 2011 (UTC)
My comments about Nelson may be irrelevant. At any rate, what I was trying to say is that Hrbacek's axiomatic approach is similar to Nelson's (they developed them at the same time around 1977), and the axiomatic approach does not seem to bestow any particular advantage as far as the specific issues raised by Hrbacek and O'Donovan. Which comments would you like to include? Including the question "what's wrong with the upside-down function" is not going to be intelligible to the reader without further explanation, and at any rate the main point is that in calculus, we are interested in differentiating standard functions, which involves working with the natural hyperreal extensions of such functions. The standard part function does not happen to be the natural extension of any standard function, and this is the answer to O'Donovan's question. As far as evaluating the derivative f' at x+δ, the answer is just as clear: once we have the derivative f', it has its own natural extension also denoted f', and it's that extension that's being evaluated at x+δ. Let's not make a big ado about nothing. I would be in favor of including Hrbacek's criticism cited above. Tkuvho (talk) 13:20, 29 November 2011 (UTC)


I think it might be better to site an actual paper rather than User:CBM. I did a little googling and I fond at least one possibility, if it is acceptable to other people involved. It is an essay in the book "Book publishing I" By Rowland Lorimer, Shoichet, Jillian, John W. Maxwell. The essay titled "The Nature of Marketing in Higher Education Publishing" by Leslie J. Carson is an essay about increasing sales of Edwards and Penny in the Canadian market. The evaluate the top competitors (which this book is not among) and state that all other textbooks combined make up 2.5% of the market share. Where this number comes from I do not know. Does anyone else know of any other Market Share data for Calculus books?

The article seems to be reproduced here for anyone interested in looking at it. Thenub314 (talk) 23:15, 28 November 2011 (UTC)

I don't think this claim needs to be explicit in the article. I was simply pointing it out on the talk page as part of my opinion why we need to be sure to include what sources we have that point out the uncommonness of this approach. — Carl (CBM · talk) 23:50, 28 November 2011 (UTC)

Constructive approach[edit]

I changed the sentence "Constructivist Errett Bishop's review was critical of the book for adopting a non-constructive approach." because this was not what the reference said. There are two ways to interpret the phrase "non-constructive approach" one way being that it is not constructive from point of view of constructive philosophy, the second being that the mathematical objects defined are not constructed. What the reference explicitly says is that Bishop criticized the text for not constructing infinitesimals. If arguing from a constructivist point of view he also would have faulted the book for proving non constructive results such as the intermediate value theorem. Thenub314 (talk) 18:41, 29 November 2011 (UTC)

A more fundemental fact about Bishop's criticism is that it stems from his contructivism, and that it is part of his broadside criticism of classical mathematics as a whole. This was detailed by Martin Davis and others. Furthermore, the real numbers are not constructed in a typical calculus course, either. It is odd to hold this particular point against the book in a brief summary. In general terms, I object to an over-emphasis on Bishop's critique that you seem to be pushing here. The consensus among numerous commentators is that his critique is not coherent. Whether or not we agree with this, it should place a damper on overzealous Bishop admirers. Tkuvho (talk) 18:48, 29 November 2011 (UTC)
I myself am not a "Bishop admirer." I only mentioned Bishops review because for bettor or worse it was greatly discussed afterward. Perhaps as a compromise we can get rid of the sentence entirely, I will be bold and take it out for now, add it back if you don't like the idea. Thenub314 (talk) 19:34, 29 November 2011 (UTC)
I see that you eliminated it altogether. That's fine but the truth is that part of the controversy over the book was its foundational aspect. This tends to add to the importance of the book. Bishop's review reflects a clash between different philosophies of mathematics. It might not be a bad idea to mention this somehow. Tkuvho (talk) 19:50, 29 November 2011 (UTC)

The case of O'Donovan[edit]

I see you incorporated some criticisms by O'Donovan. I think there are two issues here: (1) are his criticisms coherent? and further more (2) if they are, are they answered in the framework they are proposing? Furthermore, please keep in mind that they are not proposing anything new. They explicitly state so on the second page of their AMM article, and proceed to cite sources from the 1970s including Nelson's article and Hrbacek's article. Historically speaking, Hrbacek got a bit of a raw deal to the extent that he developed an axiomatisation independently of Nelson, but Nelson typically gets all the credit. Consider, for instance, the case of Wikipedia. None of our articles on internal set theory refer to Hrbacek as far as I recall, and if they do at best in a footnote. This state of affairs should probably be corrected. Still, it does not justify certain excesses of language, particularly in O'Donovan. Now before I spend more time on this, I would like to find out whether you are interested in clarifying the situation with items (1) and (2) above. If you are going to pursue a line to the effect that "this is published material, therefore it is legitimate to cite it here", I am not sure it is worth my time. Tkuvho (talk) 13:02, 30 November 2011 (UTC)

I am happy to talk about it and try to keep an open mind to (1), but on my first few readings I regard his comments has coherent. As far as (2) is also interesting, but regardless of what I decided about this issue it wouldn't effect my decision about mentioning their comments in this article. They can correctly point out a difficulty without solving it, and they can try to solve with with or without being successful. But the difficulty is present either way. Thenub314 (talk) 19:36, 30 November 2011 (UTC)
OK, I interpret this as placing the importance upon the coherence of such a criticism rather than it being published in conference proceedings (which are frequently refereed in a sloppy way if at all). I assume this means that if we find a criticism to be incoherent, we will not insist on including it in the page. more later. Tkuvho (talk) 20:33, 30 November 2011 (UTC)
Exhibit A: O'Donovan explains that students do not understand when it is appropriate to use the standard part function st(·) and lead to the mistaken believing that ƒ'(x+δ)=ƒ'(x) for all infinitesimals δ. This does not seem to be fully grammatical, but apparently this means that the students' failure to understand the proper use of the standard part function leads them to misapply it in such a way as to conclude that f' is constant on the monad. Is this a coherent criticism? Keisler indicates early on that "st" is not on par with other functions because it is not a natural extension of a real function. "st" should not be used except in definitions of derivatives and integrals. The derivative f' is a real function in its own right. The way to evaluate it at a non-standard point such as x+δ, is to form its natural extension and apply it there. No big mystery here. Keisler is both a great pedagogue and a great mathematician. O'Donovan is neither as far as I am able to tell. His speculations are not supported by solid educational research. Note that Sullivan's study was a proper controlled experiment, meaning that half the groups were following a standard curriculum and the two approaches were compared at the end of the term. O'Donovan does not even claim he performed any studies of this sort. Why should anyone want to rely on his anecdotal speculations published in a basically unrefereed venue? The idea that students will walk away with a mistaken belief that the derivative is constant on a monad strains credulity. take f(x)=x^2. Its derivative is 2x. Why should anyone think this is constant on the monad? I can grant you that if O'Donovan did not properly explain this to his students by means of examples, they may be confused. But who is the lousy pedagogue here? Tkuvho (talk) 20:45, 30 November 2011 (UTC)
Well, first a couple of things, I am prone to grammatical mistakes. I'll try to improve it, and thanks for the feedback about that sentence. Firstly, there is no measure of pedagogical aptitude I am aware of, so I am unwilling to accept arguments Keisler is a good pedagogue or that that either O'Donovan is not. Secondly the editors of the volume state the papers are referred, and I have no particular reason to doubt them, and his paper in the monthly was certainly refereed.
But you make some good comments that I will try to discuss. Trying to keep an open mind I skimmed through Keisler's book again to look exactly at what was written. Looking at the definition of derivative on pages 44-45, in no way is it indicated that formula given is meant to apply only to standard x, and the sentence immediately preceding it says that we consider the derivative for any x. So I would find it reasonable for a student having read this to expect this formula to hold everywhere.
I agree that when faced with a specific calculation it students may readily see that for a specific function the derivative is not constant. But when considering second derivatives for general functions there is opportunity of confusion, as students do not have the explicit formula for the derivative to appeal to see the derivative obviously shouldn't be constant on monads. Glancing ahead, in Keisler's book when higher derivatives are discussed, when one would have an obvious reason to consider ƒ'(x+δ) there is no mention that this would be calculated differently then ƒ'(x), or that it is necessary to consider the extension, etc. So I find his argument fairly convincing that there is a pedagogical issue.
Finally, this is not his only published paper on education. And most papers on the subject are not controlled studies. Your question about why anyone should be concerned with his anecdotal evidence is reasonable, but the fact is that people are, and many papers in education are of this nature. The comments by the Katz & Katz paper are equally anecdotal. So I don't see as a reason to exclude his comments. Thenub314 (talk) 00:01, 1 December 2011 (UTC)
This is answered in the next section. Tkuvho (talk) 22:14, 3 December 2011 (UTC)


Well, first a couple of things, I am prone to grammatical mistakes. I'll try to improve it, and thanks for the feedback about that sentence. Firstly, there is no measure of pedagogical aptitude I am aware of, so I am unwilling to accept arguments Keisler is a good pedagogue or that that either O'Donovan is not.

The point is not that Keisler is a better pedagogue, but that NSA is not a magic wand that is supposed to make calculus easy as cake. It will never be easy as cake. The fact that O'Donovan had some students who were confused about the standard part function merely attests to the intrinsic difficulty of the subject. Had O'Donovan argued that his students understood epsilon-delta definitions better, that would be an interesting claim. However, he is not making such a claim, and I seriously doubt that anyone makes such a claim.

Secondly the editors of the volume state the papers are referred, and I have no particular reason to doubt them,

Every conference proceedings says that it has been refereed. Do me a favor, ask some colleagues around to see what they think about conference proceedings.

and his paper in the monthly was certainly refereed.

As I already mentioned, the paper says strictly nothing about the alleged "pedagogical difficulties", and is on an altogether different subject.

But you make some good comments that I will try to discuss. Trying to keep an open mind I skimmed through Keisler's book again to look exactly at what was written. Looking at the definition of derivative on pages 44-45, in no way is it indicated that formula given is meant to apply only to standard x, and the sentence immediately preceding it says that we consider the derivative for any x.

You are in error. The definition on page 45 clearly states that f is a real function. This means that its argument x is real. Keisler comments elsewhere that when the input is nonstandard then of course f(input) uses the natural extension of f, but the original x is real. He cannot repeat this comment on every page for the benefit of a reader who comes along looking for contradictions. Interpreting this definition as holding at non-standard x would be attributing an error to Keisler, which not even Bishop did. Here we come to what I referred to earlier as slander.

So I would find it reasonable for a student having read this to expect this formula to hold everywhere.

Incorrect, if he attended the lectures and/or read the book rather than page 45 thereof.

I agree that when faced with a specific calculation it students may readily see that for a specific function the derivative is not constant. But when considering second derivatives for general functions there is opportunity of confusion, as students do not have the explicit formula for the derivative to appeal to see the derivative obviously shouldn't be constant on monads. Glancing ahead, in Keisler's book when higher derivatives are discussed, when one would have an obvious reason to consider ƒ'(x+δ) there is no mention that this would be calculated differently then ƒ'(x), or that it is necessary to consider the extension, etc.

The reason is that f' was only defined at real points. Therefore f'(x+δ) can only be interpreted as the application of the natural extension of f' to x+δ.

So I find his argument fairly convincing that there is a pedagogical issue.

As I mentioned, in every calculus course there are going to be students who are confused. But attributing errors to Keisler is slander.

Finally, this is not his only published paper on education. And most papers on the subject are not controlled studies. Your question about why anyone should be concerned with his anecdotal evidence is reasonable, but the fact is that people are, and many papers in education are of this nature. The comments by the Katz & Katz paper are equally anecdotal. So I don't see as a reason to exclude his comments.

I do see a reason, and I mentioned it several times already. The reason is that when dealing with work of living persons we should be particularly careful to avoid critical information that is ill-founded. Unfounded praise is far less objectionable than unfounded slander. But if you prefer, you can delete both comments. Tkuvho (talk) 13:16, 1 December 2011 (UTC)
We will have to agree to disagree about conference proceedings. As commented above by other editors there is no slander, libel, etc. Also, there is no critical information about Keisler. It is important to distinguish the person from the book. So on this ground I see no need this needs to be removed before the discussion has been reached a conclusion. Thenub314 (talk) 15:51, 1 December 2011 (UTC)
The criticism of Keisler's book is inaccurate. There are no errors there. Your impression that there is an error is based on your failure to read the previous chapter on the extension principle, as I noted above. Tkuvho (talk) 16:12, 1 December 2011 (UTC)
Neither I nor O'Donovan are claiming their are errors. If you look at his papers as a whole, the real point is that to correctly understand and use these concepts, a fairly sophisticated level of mathematical depth is needed. One needs to be comfortable with the distinction between a real function, and its natural extension. You need to know how to define concepts such as the derivative at a nonstandard real, but this is never made sufficiently clear, and in order to make it sufficiently clear it is difficult for students. I did not say there was a mistake, even rereading the first chapter, his comments make sense. How can it be "inaccurate" if he states students were confused by this concept, as long as we accept he is being truthful? Now I was mistaken when I said he in no way pointed out that the definition was only intended for standard x, but my comments about the immediately preceding comment emphasizes that the following definition is for any x still holds. It is not surprising to hear that students were confused by this point. Thenub314 (talk) 16:41, 1 December 2011 (UTC)
I never said that there is a magic wand to learn all this. However, your claim about derivatives is incorrect. Evaluating derivative at a non-standard point is clearly explained in Keisler. O'Donovan's criticism here is incoherent. Tkuvho (talk) 16:45, 1 December 2011 (UTC)
In every calculus class there are students who are confused because the material is difficult. However, the specific claim about the derivative is unsound. Tkuvho (talk) 16:47, 1 December 2011 (UTC)
O'Donovan may be truthful but I suspect he did not explain the extension principle to his students, causing them to be confused. Come to think of it, there is a basic problem here: if he uses Hrbacek's relational framework in his teaching, how does he know that students are confused by Robinson's framework? I suspect his anecdotal claim is not even based on teaching but rather, in turn, on hearsay. Tkuvho (talk) 16:59, 1 December 2011 (UTC)
According to one website at least, O'Donovan taught using the Robinson approach for several years, apparently. He later switched to Hrbacek's because of the difficulties he discusses in the paper cited here. But to be honest your suspicions can never be smoothed away. We will never know how well he described things, if the real problem was one of his teaching etc. There will always be a reason for you to distrust what he says. I believe this is fundamentally what the wikipedia policy aims for "verifiability, not truth", because the truth of the matter is quite subjective. But he published in several reliable sources that he found difficulties with the Robinson approach, and so switched to Hrbeck's. Each time he does he cites this book.
The real question is how can we reach a compromise. I am not willing to take out O'Donovan's remarks entirely, your not content to let them stand as they are. We clearly disagree as to whether there is any truth to his words, and we don't seem to be making any progress. Here are some ideas, since the specific claims about the derivative I am willing to leave out that claim. Would that be an improvement? Thenub314 (talk) 19:49, 1 December 2011 (UTC)
I still find it odd that the page should devote so much space to O'Donovan's criticism whereas Davis's review is barely mentioned. Davis is a leading mathematician, whereas O'Donovan has no visible publication record either in math or in education. Tkuvho (talk) 09:13, 2 December 2011 (UTC)
I would be happy to reduce the length, from the beginning I was content with a comment that said they found pedagogical difficulties with this approach, and I am happy to return to a version that says little more then that. Thenub314 (talk) 19:40, 2 December 2011 (UTC)
I don't see that you have been able to identify any such "difficulties". I think your time would be spent much better including some ideas from Davis' review. Tkuvho (talk) 22:13, 3 December 2011 (UTC)
O'Donovan's stuff seems relevant to me for the topic of NSA's (and Keisler's book's) usefulness in teaching elementary calculus, since he's actually tried to do it. Davis is a logician and if he has stuff to say about the logical content of NSA, then fine, but I don't think NSA's logical status is a matter of much disagreement. Has Davis tried to actually teach calculus from Keisler's book? This description saying Davis gave the book a detailed positive review is completely inaccurate in my opinion; the old wording was better. The cite goes to a review by Davis of a completely different book, and a part of that review criticizes Bishop's earlier review of Keisler's book, but it's mostly a complaint about logicians getting dissed in general by other types of mathematicians. Tkuvho, just what are you trying to do? This isn't coming across well. (talk) 03:46, 4 December 2011 (UTC)
I agree with and on a similiar but slightly different note, I am also frustrated by that particular diff, in regards to the portion about Bishop. As I mentioned above repeatedly, and I have been attempting to compromise to find a version we could all live with. But putting back the sentence exactly simply ignores my point of view. Thenub314 (talk) 05:33, 4 December 2011 (UTC)
Actually the situation with Hrbacek is precisely the reverse of what the IP indicated. Davis and Hrbacek are both logicians, but Davis wrote a detailed review and a discussion of pedagogy concerning Keisler's book (by the way, it's not Davis's article that Thenub put into the page), whereas Hrbacek is precisely concerned with the foundations of nsa, or rather alternative foundations thereof. For this reason, a detailed discussion of Davis's article would be appropriate. As far as Bishop is concerned, the book triggered a foundational controversy, and it is important to point this out. Keisler's book did not find favor with a constructivist, but it did find favor with numerous classical mathematicians. Tkuvho (talk) 12:39, 4 December 2011 (UTC)

I see now, the article now cites a review by Davis and Hausner, that does talk about classroom experience with the book. The earlier citation was to Davis's review of a logic book by Monk, which mentioned Bishop's review of Keisler's book only in passing. I agree that the Davis/Hausner review's remarks should be expanded on. I'd want to see documentation for the assertion that "[a[s far as Bishop is concerned, the book triggered a foundational controversy" since in my reading, Bishop simply didn't like the book's axiomatic approach (Bishop's constructivism didn't seem to have much to do with that). I'm not aware of any foundational controversy about NSA. Nobody AFAIK doubts that it's correct (to the extent that classical epsilon-delta analysis is correct) and technically clever. There are only differing opinions about how useful it is. (talk) 02:33, 5 December 2011 (UTC)

The phrase [a[s far as Bishop is concerned, the book triggered a foundational controversy was not put in by me, I'll have to check the history. It would be more accurate to say that Bishop's critique was in itself foundationally motivated. He objected to the book because the book adopted a non-constructive approach inimical to a constructivist like Bishop. I don't think we need to put all this in, though. Let me know if I answered your question. Tkuvho (talk) 12:42, 5 December 2011 (UTC)
"foundational controversy" is from your earlier post here.[1] In reading Bishop's review I didn't especially get the impression that his criticism was foundationally motivated. I thought he just felt that the axioms in the book were hard to understand (presumably the ones related to hyperreals in the first edition; they seem to have been renumbered in the online second edition). I may try to re-read the review more closely to check this. But I don't see how to make much sense of Keisler's axioms without knowing more logic up front than could be expected of a calculus student. (talk) 11:14, 6 December 2011 (UTC)
The "foundational controversy" was triggered, not by the book, but by Bishop's review thereof. The fact that Bishop's review is foundationally motivated has been noted by a number of observers. In his review he alludes more than once that the problem with the book is its alleged "meaninglessness". He does not elaborate, but his earlier writings clearly state that such an alleged lack of "meaning" stems from an unbriddled application of the law of excluded middle. He explicitly found fault with classical mathematics as a whole in this regard, so his vitriolic (Dauben's adjective) review of Keisler's book is but a fragment of a total vision where Bishop predicted an imminent demise of classical mathematics. I can give the salient points if you are interested. Concerning your final point: there is certainly room for discussion here. There is no magic wand that will make calculus as easy as π but Keisler's preparatory chapter on infinitesimals is, in my view, more accessible than traditional treatments of limits with or without epsilontics. Tkuvho (talk) 11:57, 6 December 2011 (UTC)
I thought I would comment on the comment that Bishop's review was foundationally motivated has also been contested by other authors. It certainly has been noted by several authors and disputed by others, such as Stolzenberg. Thenub314 (talk) 18:35, 6 December 2011 (UTC)
The distinguished constructivist Stolzenberg does not appear either on this page or at criticism of non-standard analysis. I have a slight preference to incorporating whatever comments he might have made there rather than here, but we can discuss either option. Tkuvho (talk) 18:52, 6 December 2011 (UTC)
On this point we agree, it would be better to include Stolzenberg at criticism of non-standard analysis, similarly I think the comments about Bishop recently added by Katz & Katz would be better at criticism of non-standard analysis instead of here, since we would then be only including one point of view in this article. Thenub314 (talk) 20:26, 6 December 2011 (UTC)
Well, the constructivist context of Bishop's remarks was pointed out by virtually all of the commentators, I think including Stolzenberg himself. The reason constructivists dislike Robinson's infinitesimals is not because they used them in teaching, but because they fly in the face of their philosophy of mathematics. Tkuvho (talk) 12:05, 7 December 2011 (UTC)
I included Stolzenberg's comments, I am not opposed to placing both POV's instead in the article on criticism, but it doesn't seem appropriate to include just one side to me. Thenub314 (talk) 00:20, 8 December 2011 (UTC)
No problem, we can include him and his five dogmas, as well :) Tkuvho (talk) 13:54, 8 December 2011 (UTC)
I am confused, what is the point of the five dogmas remark. It relates neither to the book on this page, the review about the book. It is does not dispute Stolzenberg's remarks in any way. It seems to just be a off hand remark about how many times he uses the word dogma. What is the point exactly?
The point is to show the level of seriousness of Stolzenberg's review, and to indicate the extent to which he himself is "capable of the rational minded inquiry necessary to objectively review a textbook that is not constructive". Tkuvho (talk) 17:01, 8 December 2011 (UTC)
Does it say anything like this in Katz&Katz, or is this our inference? Thenub314 (talk) 17:07, 8 December 2011 (UTC)
The reader is intelligent enough to draw his or her own conclusions from an author's excessive use of colorful terminology such as "the spouting of dogma". I don't think we need to rub it in any further. Tkuvho (talk) 17:26, 8 December 2011 (UTC)
This quote is not related to the content of the book, the review, nor does it dispute Stolzenberg's comments. It seems to be a passing remark on his use of language and the reference, as far as I can tell, doesn't even suggest the use of language is inappropriate. This simply has nothing really to do with the article at hand.
(the previous comment is by Thenub) On the contrary, it is Stolzenberg's letter that is not related to the contents of the book, and the level of his diction is a good indication of his seriousness. Tkuvho (talk) 17:39, 8 December 2011 (UTC)
Both Katz & Katz and Stolzenberg are not directly related to the content of the book, which is why I suggested they both be moved to the page on criticism. You on the other hand felt that Katz & Katz belonged here as it relates to the review being discussed. Stolzenberg also relates to the review being discussed. Now, it is your opinion that this comment about how often he uses words whose root traces back to dogma indicates the seriousness of the response but this is not a verifiable fact. The paper which makes the comment, does so in a foot note, and in no way ties this foot note in with any argument that suggests an issue with his letter. So, excluding your own opinion from the analysis, how does this comment about is diction relate? Thenub314 (talk) 17:54, 8 December 2011 (UTC)
Stolzenberg's comments about "spouting of dogma" are as vitriolic as Bishop's review itself, and this should be obvious to any reader without having to hit him over the head. I don't think it is approapriate to include Stolzenberg without indicating the nature of his contribution to the debate. That contribution was: little content, much virtiol. Tkuvho (talk) 08:21, 9 December 2011 (UTC)
All of the comment above is purely your opinion, not supported by the published material. A comment about how often he uses a particular word with text to explain what it means or why we are mentioning it is simply unclear. Can we agree that all of this (Katz & Katz and Stolzenberg) include the comment about the number of times dogma is used would be better off at the criticism page? Or can you suggest an alternative? It really seems to be mainly you and I working on this article at the moment, and we really have to find some common ground. Thenub314 (talk) 17:51, 9 December 2011 (UTC)
I never received a reply here, but in the time intervening there have been several edits I strongly disagree with, and the article in general is moving away from a neutral tone. Thenub314 (talk) 23:34, 15 December 2011 (UTC)
Finding common ground sounds like a good idea. I am fine with removing Stolzenberg to the "criticism" page; you are the one who put him in, in the first place. I just think that if we do include him here, we should indicate to the reader the nature of his reaction, which was not pretty. That's clear from published material and is not "purely my opinion" as you write. Tkuvho (talk) 09:15, 16 December 2011 (UTC)

continuity at a non-standard point[edit]

One wonders whether continuity at a non-standard point is really a useful notion as far as elementary calculus is concerned. Note that the squaring function is "continuous" in this sense at an infinite point. This tends to go counter to the intuition. As a point of fine theory this is important, in order fully to understand why the transfer principle holds here. But is Hrbacek's criticism relevant to calculus teaching at this level? Tkuvho (talk) 16:37, 1 December 2011 (UTC)

Well I am not sure what to say. My area of research is not NSA, so when Hrbacek says this is a basic and useful notion I tend to accept it. Also I don't think he intended to give the idea that this was just a problem with continuity, but he mentions applies equally well to the other standard notions, such as the derivative. But, the papers seem to suggest that Hrbacek's observations come out of O'Donovan's difficulties teaching calculus. As such I believe the comments are relative to teaching calculus at this level. Thenub314 (talk) 19:46, 2 December 2011 (UTC)
Seeing that Keisler manages to define all the basic notions without epsilontics, your insistence on the inclusion of this criticism is puzzling. Tkuvho (talk) 22:12, 3 December 2011 (UTC)
Perhaps to you my behavior is puzzling. But your insistence at keeping out a reliably sourced point of view is equally puzzling to me. Thenub314 (talk) 05:36, 4 December 2011 (UTC)
Perhaps, but the notion of microcontinuity is a useful one and can be used to give an infinitesimal definition of the continuity of a function on an interval, as well as an infinitesimal definition of the uniform continuity of a function on an interval (using non-standard points). I don't think either of the authors you mentioned disagrees with this. Tkuvho (talk) 12:22, 4 December 2011 (UTC)

another review[edit]

Excellent work. I will try to add it to the article this evening. Thenub314 (talk)
Decided to do a little googling around in case others had been missed. Found another: Elementary Calculus. by H. Jerome Keisler by E. W. Madison and K. D. Stroyan; The American Mathematical Monthly, Vol. 84, No. 6 (Jun. - Jul., 1977), pp. 496-500.


The comments on Hrbacek are synthesis and are taken out of context. I don't think Hrbacek said what is attributed to him. Tkuvho (talk) 12:54, 18 December 2011 (UTC)

Here is a quote of the entire second paragraph from the first page of Hrbacek's article.
"It seems fair to say, however, that acceptance of “nonstandard” methods by the larger mathematical community lags far behind their successes. In particu- lar, the oft-expressed hope that infinitesimals would now replace the notorious ε-δ method in teaching calculus remains unrealized, in spite of notable efforts by Keisler [20], Stroyan [31], Benci and Di Nasso [4], and others. Sociological reasons — the inherent conservativity of the mathematical community, the lack of a concentrated effort at proselytizing — are often mentioned as an explana- tion. There is also the fact that “nonstandard” methods, at least in the form in which they are usually presented, require heavier reliance on formal logic than is customary in mathematics at large. While acknowledging much truth to all of the above, here I shall concentrate on another contributing difficulty. At the risk of an overstatement, it is this: while it is undoubtedly possible to do calculus by means of infinitesimals in the Robinsonian framework, it does not seem possible to do calculus only by means of infinitesimals in it. In particular, the promise to replace the ε-δ method by the use of infinitesimals cannot be carried out in full."
— Carl (CBM · talk) 13:26, 18 December 2011 (UTC)
Looking at the entire paper, I don't see any synthesis in the text here. Hrbacek directly argues that the "Robinsonian framework" is insufficient because it uses definitions that do not themselves transfer, so that ε-δ methods are required to prove things about continuity, derivatives, etc. at non-standard points. That is precisely the claim that is attributed to him in the article. — Carl (CBM · talk) 13:39, 18 December 2011 (UTC)
The relevant comment by Hrbacek seems to be this: while it is undoubtedly possible to do calculus by means of infinitesimals in the Robinsonian framework, it does not seem possible to do calculus only by means of infinitesimals in it. In particular, the promise to replace the ε-δ method by the use of infinitesimals cannot be carried out in full. Notice that in this particular passage, Hrbacek is talking specifically about Robinson's framework. I interpret it as referring to subtle issues involved in building non-standard analysis, at least in this particular sentence. I think we all agree that epsilon-delta remains an ingredient at the foundational level. However, as Keisler's book shows in great detail, the basic notions of calculus such as continuity, derivative, and integral can indeed be defined without epsilon, delta. Hrbacek's phrase is too vague to amount to a serious criticism of Keisler's book. When he does get into specifics, he discusses continuity at infinite points. To convince oneself that continuity of, say, x^2 persists at an infinite point, one indeed has to revert to epsilon, delta. However, one can question the relevance of this to teaching calculus. If anything, it is microcontinuity at such points that happens to be the useful notion. Rather than being a shortcoming, it is a feature, as it allows a single-variable definition of uniform continuity, as explained there. Tkuvho (talk) 13:48, 18 December 2011 (UTC)
I would like to add a comment to explain what I feel is going on with Hrbacek's criticism. The Hrbacek-Nelson framework for non-standard analysis is philosophically a very different way of looking at infinitesimals, and he feels it is the "right" one. Our page internal set theory describes some of their motivations; briefly, the idea is to find infinitesimals within the real number system. This approach has the advantage of not treating infinitesimals as anything "added on" to the standard package; rather, they have been there all along; we just haven't noticed them. This is done at the cost of a dramatic foundational shake-up (unary predicate, etc). In fact, I do find this approach very appealing. I think one has to understand what it is that's driving Hrbacek. He disagrees with Robinson at a deep foundational level. The relevance of this with regard to the specific question of Keisler's book is a very big question mark. It would be much better to discuss this at non-standard analysis or perhaps the criticism thereof. It's just silly to claim that you can't define continuity without epsilon delta when Keisler does just that in great detail in a text that has been used to teach thousands of students. Tkuvho (talk) 14:00, 18 December 2011 (UTC)
On one hand you are arguing that a statement which is a direct paraphrase of Hrbacek's argument is synthesis, and on the other you are referring to Hrbacek's motivations for making the comments. It seems to me that whether we have correctly quoted Hrbacek is separate from his motivations.
Separately, how does Keisler show that x2 is continuous at nonstandard points? That seems to be the thing that would determine whether Hrbacek's comments are relevant to Keisler's approach. — Carl (CBM · talk) 14:14, 18 December 2011 (UTC)
First of all, if both you and N are supporting the inclusion of this material then I am certainly going to stop arguing. Second, "continuity at a nonstandard point" is an ill-defined notion; you have to clarify which continuity you have in mind. Since the hyperreals are an elementary extension of the reals (in the strong sense, i.e. including all relations), everything true about f will be true about f*. The epsilon, delta definition of continuity is certainly an elementary formula, and therefore by Los's theorem a.k.a. transfer principle, it holds for f*, as well. What happens with microcontinuity of course depends on the function; sine will be microcontinuous but not x^2. As far as Keisler is concerned, he is a practical-minded fellow who is interested in teaching calculus rather than nonstandard analysis. Why should the students be bothered with continuity at nonstandard points? The point of the hyperreal approach is NOT to replace the reals by the hyperreals, but rather to use the hyperreals to understand the reals. We are interested in real functions, and we use the infinitesimal-enriched continuum to understand their properties. Hope this answers your question. Tkuvho (talk) 14:22, 18 December 2011 (UTC)
As far as Hrbacek is concerned, I mentioned his motivation to explain why he is being vague about what he is criticizing exactly. Note that quoting Hrbacek would be fine; he made a specific comment about continuity that I included in the article. However, N is insisting on including a much broader paraphrase of Hrbacek that goes beyond what he actually said. But be my guest and include it, I feel I have spent enough time on this already. Tkuvho (talk) 14:25, 18 December 2011 (UTC)
I don't have any real opinion about including it here, I just did a search for it to see what Hrbacek actually says, and it does seem to be accurately rephrased in the article. On the other hand I don't think Hrbacek was discussing this book directly, as much as discussing the Robinson framework in general. Compare O'Donovan's article which does directly give Keisler's book as an example. — Carl (CBM · talk) 18:25, 18 December 2011 (UTC)
Let me comment briefly that these comments ended up here because I tried to quote the a monthly article by Hrbacek, O'Donovan, and Lessmann which had a comment that said books using the non-standard approach had pedagogical difficulties, and cited this book as an example. Tkuvho seemed to feel this was unfair to the book because it gave no details. They pointed to Hrbacek's Paper and O'Donovan's paper for further details. But I do acknowledge Hrbacek only cites this book in the passage given above. I personally am happy and willing to go back to the version where we mention only the monthly paper, if acceptable to everyone else. Thenub314 (talk) 20:48, 18 December 2011 (UTC)
Let's include it then, but let's also make sure the reader understands the facts. Looking forward to a harmonious conclusion to this discussion. Tkuvho (talk) 13:13, 19 December 2011 (UTC)


It is currently claimed in this page that Bishop's review was "harshly critical" but this description is unsourced. I proposed to replace this by a description of his review as "vitriolic" sourced in an influential article (dedicated in part to that topic) by Joseph Dauben, a distinguished historian and head of many societies.

I also proposed to include a quote from the widely cited education study by K. Sullivan which is already discussed in the page (therefore there is no question of notability).

I similarly proposed to include a reference to a recent article by Vladimir Kanovei et al. specifically devoted to the topic of Bishop's review of Keisler' book, and therefore of obvious relevance to the page. Tkuvho (talk) 09:08, 19 January 2015 (UTC)

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