Talk:Criticism of nonstandard analysis/Archive 1

Archive 1

Misc

I am not familiar with Gillies' book. Is he quoting Bishop there? Where else do the additional comments by Bishop appear? Katzmik (talk) 13:05, 4 September 2008 (UTC)

Importance of this

Why is this important? What influence did this have on mathematics at the time, or since? Was it more notable than other controversies at the time? Why? The article leaves out too much essential context. -- Dominus (talk) 15:08, 8 September 2008 (UTC)

Hi, I would like to answer your questions one at a time if you don't mind.

>Why is this important?

I tried to summarize what I feel is the importance of the controversy in the body of the article. Namely, for those who see mathematical truth as unquestionable, it is startling to witness so much debate about what actually constitutes the said truth.

What is missing here is a demonstration that anyone but you thinks that this particular "controversy" was of historical importance. Negative book reviews are published all the time. Scientists often argue with each other about philosophical issues. What is especially significant about this particular argument? Are there any third-party sources that refer to it? Do histories of mathematics identify this as a watershed moment?

>What influence did this have on mathematics at the time, or since?

I think the influence was enormous. The dismissive comments that standard mathematicians allow themselves about what could be, to quote Keisler, one of the crowning achiements of twentieth century mathematics (see, for instance, the talk page of non-standard analysis), are partly due to an intemperate standard set by Bishop.

That might be interesting, but that article does not make it clear, and at present it appears to be your personal opinion.

>Was it more notable than other controversies at the time? Why?

I am not sure which controversies you are referring to. In my mind, this controversy is a re-incarnation of the classical controversies of meta-mathematics, such as the one over Cantor, Brouwer-Hilbert controversy, etc.

>The article leaves out too much essential context.

Please add whatever seems appropriate. Katzmik (talk) 11:37, 9 September 2008 (UTC)

It is still not clear to me that the subject is worth this much discussion in the first place. —Dominus 17:18, 12 December 2008 (UTC)

Notability?

Why is this a controversy? Reading through it all it seems to say is that Bishop wrote a very negative book review, and Keisler defended his work with a pamphlet. Did this episode involve anyone beyond these two people? Thenub314 (talk) 08:51, 31 October 2008 (UTC)

The current name was chosen by Matthew Charles. By consulting the history you will notice that the original name was different. Katzmik (talk) 15:48, 27 November 2008 (UTC)

Bishop "vs" Keisler is no better. Why do you think this kind of synthesis, based on one book review, is of any value for this encyclopedia? Mathsci (talk) 22:03, 27 November 2008 (UTC)

Note that the page registered 65 visits on the first of the month alone (I was not one of the visitors). Apparently I am not the only person involved. What is your opinion? Katzmik (talk) 08:54, 30 November 2008 (UTC)

That you might have got up too early this morning :) "Number of page visits" has never before been used as a means of judging the worth of a wikipedia article. On the other hand it might be a suitable way of assessing the popularity of the gossip column of an online tabloid journal. @+ Mathsci (talk) 09:15, 30 November 2008 (UTC)

BTW I just noticed Matthew Charles :-) Scimath, 09:21, 30 November 2008 (UTC)

Third reply: that the visits were by me in view of the remarks added by me on that day. Mathsci (talk) 09:24, 30 November 2008 (UTC)

Thanks for your edits. You will notice that there are at least five other editors who have contributed to this page, indicating that the page is of some value for this encyclopedia. Katzmik (talk) 13:35, 1 December 2008 (UTC)

The number of editors is not a measure of encyclopedic value. I, for one, feel this article should be nominated for deletion. It is not clear there is any controversy here to speak about. What makes article about something more than a negative book review? Thenub314 (talk) 19:22, 10 December 2008 (UTC)

I agree that the page should be deleted. Mathsci (talk) 07:01, 11 December 2008 (UTC)

Digging through the literature on mathematics education, an experiment was conducted in the US and at an English university to teach calculus using Abraham Robinson's infinitesimals. This article has taken a point of view, not reflected in the literature, that it sparked a controversy. The article of David Tall shows that the article might be WP:SYN. Keisler happened to be a logician who was part of the experiment. The article itself seems extremely unbalanced and determined to make a point. No secondary sources have been used. I have no idea why this article was created - an article about an adverse book review? Will somebody now write a WP article on Baker's scathing review of Lang's Elliptic Curves: Diophantine Analysis ... [1]? I could imagine an article on "Teaching of non-standard analysis", since this has been much discussed by educationalists. Mathsci (talk) 21:50, 1 November 2008 (UTC)

The page from the article by Artigue that I have added to the references gives a concise history of the teaching of non-standard analysis to undergraduates. As she writes, "However, it is necessary to emphasize the weak impact of non-standard analysis on contemporary education. The small number of reported instances of this approach are often accompanied with passionate advocacy, but this rarely rises above the level of personal conviction." Mathsci (talk) 22:18, 1 November 2008 (UTC)

broader lead?

The current lead seems somewhat reductionist. The original version of the lead placed this particular disagreement in a broader context. It would be helpful if the editors involved could work out an agreed-upon version of the lead, rather than proceeding by a not-necessarily-convergent sequence of reverts and countereverts :) I was hoping to mention something about the general context of the tension between the "opposing" mathematical sensibilities of constructivism versus the mainstream of modern mathematics. Clearly this particular disagreement inscribes in the long line going back at least to Cantor, Dedekind, Kronecker, Brouwer, Hilbert... Katzmik (talk) 14:08, 2 December 2008 (UTC)

I just noticed this section, sorry I didn't reply sooner, I was away. Now, I disagree completely that this controversy was about opposing mathematical sensibilities. The only time Bishop in his writing discusses non-standard analysis is in relation to the teaching of calculus. I do not feel that non-standard analysis is particularly some sort of polar extreme to Bishop's constructivism. His issue was about the notion of existence, and at odds with many subjects of mathematics (for example topology he specifically mentions as a very non-constructive subject). We could argue about whether or not this review was part of the larger picture of Constructivism vs Formalism, etc. But I know of no reference saying so, so to include it in the article constitutes OR in my opinion (or perhaps synthesis). Thenub314 (talk) 15:40, 12 December 2008 (UTC)

edits aimed at deletion

I see that there are two editors in favor of deletion. They are of course free to nominate the article for deletion. Meanwhile, I feel that their current edits are detrimental to the article. Since they have taken a hostile attitude toward it, wouldn't it be more appropriate to refrain from editing it? Sufficiently many hostile edits will degrade the article to a situation where a deletion will be a self-fulfilling prophecy :) Katzmik (talk) 08:11, 11 December 2008 (UTC)

P.S. If you do insist on editing an article you dislike, it would be preferable to have a discussion in this space first, as I suggested above without receiving any input. Katzmik (talk) 08:14, 11 December 2008 (UTC)

I take it my edits are the main ones your addressing. Which of the edits do you disagree with and why? Thenub314 (talk) 08:24, 11 December 2008 (UTC)

I would be perfectly willing to engage in a discussion of this, inspite of your stated intention of having the page deleted (which I obviously disagree with). However, as I mentioned already, the changes should be reverted first. Of course the flags can be left in if you feel they are justified. Katzmik (talk) 08:36, 11 December 2008 (UTC)

Your reverts have been restored by Mathsci. I do feel the article is not particularly notable. But I stand by my edits as improving the article. So I recall my edits here, so We can discuss them individually.

1. I removed the sentance "Meta-mathematically speaking, Bishop's constructivism lies at the opposite extreme to Robinson's non-standard analysis, in the spectrum of mathematical sensibility." Which to my understanding is not true, philosophically speaking Bishop's constructivism is different point of view on the notion of mathematical existence. It is equally at odds with many parts of mathematics, there is nothing particular about Robinson's non-standard calculus that makes it the other extreme. If there is a source for this comment then I will concede the point, but I feel the statement is not true.

2. I asked for a citation for the statement: "The philosophical origins of the disagreement go back to Cantor, Dedekind, Kronecker, Brouwer, Hilbert, and more recently Robinson and Halmos". As I understand this controversy from reading the articles we list it is about the teaching of calculus and not about Bishop's constructivism vs. Formalism.

3. I deleted the statement "Note that Keisler presents a detailed 3-page construction of the hyperreals starting on page 911 of the textbook." This is a comment about the Online addition of his book, which is based on the second edition. Bishop was reviewing the first addition of the book. I could say more here, but I feel that is enough of a reason to delete this comment.

4. Lastly I deleted a quote from Bishop's review, because in the sentences just before the content of the quote. Further the relate how Bishop meant the comment to apply to Keisler's book, so the quote seems redundant.

I look forward to your comments. I tried to do my best to explain these in the edit summaries, which is why I made them 4 separate edits. Thenub314 (talk) 09:56, 11 December 2008 (UTC)

Why do you write "an article you dislike"? This is a very poorly written unencyclopedic article which seems to be your personal opinion. Wikipedia is not a blog. There seems to be no need to have Katzmik's interpretation of a book review. Mathsci (talk) 08:19, 11 December 2008 (UTC)

epsilon, delta

I have limited time this week to answer your legitimate queries and hope to get back to them more fully next week. At this time I would like to comment on the proposed deletion of Bishop's reference to the epsilon, delta definition of limit. The whole heart of the controversy centers on what Keisler calls Robinson's solution of a 300 year old problem, namely justification of infinitesimals conjectured by Leibniz. Robinson's solution obviates the need for epsilon, delta definitions. Now it is irrelevant at the moment whether or not we think the standard part function definition is better or worse than the epsilon delta definition. The point is that this issue is at the heart of the controversy. Katzmik (talk) 15:53, 11 December 2008 (UTC)

I removed the prod because, although the arguments for deletion may have merits, it is certainly controversial. So WP:AfD is the place to take this. Martin 15:59, 11 December 2008 (UTC)

How about discussing the article before we take it anywhere? Katzmik (talk) 16:04, 11 December 2008 (UTC)

P.S. I would like to make a request that if the article is to be challenged at AfD that such a move should be postponed for three days, as I will not have a chance to comment before sunday. It seems reasonable that an editor who created a page should have a chance to comment before the deliberations are almost complete. Katzmik (talk) 16:15, 11 December 2008 (UTC)

Keep?

If deletion is proposed, I'd lean toward keeping this article for several reasons:

• In the community of mathematicians there seems to be some folklore surrounding this episode;
• Errett Bishop was a major figure among those taking a certain minority position in philosophical foundations of mathematics. He would be a notable mathematician even if that were not the case, but he wrote a detailed book showing how to rewrite the principles of mathematical analysis from the constructivist point of view.
• Jerome Keisler may be the only person to bring Abraham Robinson's non-standard analysis down to the freshman level, and his textbook doing that is what this is about. I think it is deplorable that infinitesimals were banished from the freshman calculus curriculum. I don't necessarily think that means one should follow Robinson's approach with most undergraduates learning calculus for the first time, since I don't think logical rigor fits very well with non-math majors (e.g., I wonder why the mean value theorem should be included at all, since its actual role can only remain quite obscure given the way the first-year calculus course is usually done).

Michael Hardy (talk) 20:08, 11 December 2008 (UTC)

Your first bullet is certainly the most interesting. As a mathematician I was unaware of any folklore around this episode. I went out of my way to ask a few of my colleagues before suggesting deletion, the result was no one had heard of it. I suppose in some sense this is really my question? Was this a negative book review or did this extend in some way beyond Bishop's article?

I don't think I could argue with your second and third bullet (well I might have a few words to say about MVT, but that is off topic.) But I think the articles Errett Bishop and Non-standard calculus discuss Bishop and Keisler discuss this matter, and thereby give it enough weight. I feel that this article is about simply a scathing book review and nothing more. For the record, I read the secondary sources we list that I could get a hold of, and searched for others to try to improve the article before I decided it should be deleted. As a side comment on this subject: I cannot, for the life of me, find any evidence that the pamphlet Keisler published was a direct response to Bishop's review. Thenub314 (talk) 12:31, 12 December 2008 (UTC)

I haven't done an extensive (or even not-particularly-extensive) survey about the "folklore", but that has been my impression. I'll see if I can find out something more specific. (I could phrase this by saying I don't yet have enough material for a section titled "Bishop–Keisler controversy in popular culture", but that seems a bit much so I won't say that.) Michael Hardy (talk) 20:06, 12 December 2008 (UTC)

Another mention of this affair

Chihara, Charles (2007), The Burgess-Rosen critique of nominalistic reconstructions, vol. 15, pp. 54–78 {{citation}}: Unknown parameter |publication= ignored (help) mentions this affair and the negative book review. But I have trouble finding in its discussion anything meriting the description of "controversy".

This is very weak evidence, of course. But if there truly was a controversy, I wish someone would produce some record of it. —Dominus 17:28, 12 December 2008 (UTC)

Another mention of the review: Schubring, Gert (2005), Conflicts Between Generalization, Rigor, and Intuition, Springer, p. 163, ISBN 9780387228365 {{citation}}: Unknown parameter |subtitle= ignored (help), on page 163. —Dominus 05:14, 15 December 2008 (UTC)

Ahem, it has been in the references in the mainspace article since November 1st, added by me (I might have made an error in the page number). Mathsci (talk) 08:26, 15 December 2008 (UTC)

So it is. Thanks. —Dominus 15:14, 15 December 2008 (UTC)

Why I do not support the merge with nonstandard calculus (from the WikiProject mathematics discussion page)

OK. I don’t understand why a perfectly good article should not deserve its own page. I am certainly not an expert on non-standard calculus but knowing a bit about it, I can say that quite a few mathematicians work on it and appreciate it (I certainly appreciate it). An article discussing a controversy behind whether it is appropriate to include this in the teaching of calculus is certainly very significant. In fact, as mentioned in the article, this controversy also involved other famous mathematicians in the past. Therefore, in my opinion, the article should be kept.

If at all the article is merged (which I am not in support of):

• All references should be added to the article into which it is merged

• Every important point in the article should be included in the article into which it is merged

Topology Expert (talk) 21:51, 12 December 2008 (UTC)

The reason I suggested this is because this particular episode really is about just one reference, and it is a pedagogical debate on nonstandard calculus. I think that anyone who takes an interest in teaching nonstandard calculus will be interested in this debate, and anyone interested in this debate will be interested in nonstandard calculus, so that the two topics can be talked about together. Then you can talk about the general historical issues, like Robinson's approach, Cavalieri and Leibniz, Bishop Berkeley, etc, in the main article, and you don't have to repeat them here.Likebox (talk) 22:39, 12 December 2008 (UTC)

As to technical points about how to do the merge--- I deleted a few sentences which were summarizing Bishop's position, because I thought they were unclear. I also deleted the irrelevant cross references to other foundational debates, because they are tangents. Finally, I removed some quotes where I didn't see the point. If you're unhappy about that, you can reinsert the material that was removed, and people can talk about it. I didn't think I omitted any important points, but I might be wrong. I thought I kept all the references, I placed them right after the original references for nonstandard calculus (although I think that there are too many references here--- Bishop's article and Keistler's response were the only two needed for the body of the text).Likebox (talk) 22:44, 12 December 2008 (UTC)

I support putting good coverage of the teaching (and thus this exchange) in the article on Non-standard calculus. I don't feel this exchange should have its own article for various reasons.

I wanted to comment directly on one of aspect of this discussion. The lead mentions that this episode that this exchange included other famous mathematicians and places it as part of the constructive vs. formalism debate. My opinion is that this is not a part of that debate. As a constructivist, Bishop had reason to dislike almost any text on calculus. He particularly disliked this text, and wrote a bad review of it. He does not fault it for failing to use non-constructivist logic in his review. He instead attacked the book on more direct grounds (he felt it failed to give intuition about infinitesimals are, etc.) It maybe that the only reason he disliked the book was on constructivist, but that would be me guessing about his motivation, and purely synthesis. If there is not a reference that puts this book review as a part of constructivist philosophy, we should not infer it, and we should not appeal to Hilbert, Brouwer, Dedekin, etc. to establish the importance of the subject. Also this should not be confused to be about NSA as a subject. Instead exchange was about the teaching of NSA at the freshman level, and Bishop was careful to make comments to explain his views did not apply to NSA in general, that he only meant them to apply to the teaching of elementary calculus. Thenub314 (talk) 09:49, 13 December 2008 (UTC)

(edit conflict) I support the merge. Some arguments for it:

1) As Likebox mentions above, the entire subject matter of the Bishop/Kreisler exchange is non-standard calculus, so it is not limiting in any way to describe this exchange in the article on nonstandard calculus. Conversely, many readers of the article on non-standard calculus will find this exchange to be interesting and directly relevant.

2) Following Likebox again, if the B/K exchange is placed in an article on non-standard calculus, the issues and subject matter can be made clear without duplication of content.

3) I don't believe that having a stand-alone article on a subject means that that subject is more important than a subject which appears in a more expansive article, at least not in any meaningful sense. However, if an article appears outside of its wider context, it is harder for casual editors to appreciate its notability, and thus it seems more likely that the article will be a candidate for deletion. In this case, nonstandard calculus seems like a relatively "safe haven" for the material on the B/K exchange, which I myself do find to be of interest.

Plclark (talk) 22:53, 12 December 2008 (UTC)

Please vote at WikiProject mathematics. That way, we can gather everyones' opinions. Topology Expert (talk) 22:55, 12 December 2008 (UTC)

As others have already explained, wikipedia is not a democracy: decisions are not made based upon a plurality of votes. Moreover, the appropriate place to discuss changes to a single article is here, on the article's talk page. Plclark (talk) 05:19, 13 December 2008 (UTC)

I just want to chip in with an explanation: The reason voting is not the greatest idea is because you are polling a very biased sample, the people editing this article. Some of them start out very passionate one way or another. Everything is fluid, and subject to constant change, but usually if you find a really good solution, people come to agreement after a bit of honest discussion, and then if you held a vote, it would be 80/20, so the vote is superfluous. The number of people who would care enough to vote on nearly any matter is not going to be more than about 5. So consensus is best. Consensus minus 1 (the entrenched fanatic) is also OK. But a close vote on a proposal just means that the proposal is sucky.Likebox (talk) 05:47, 13 December 2008 (UTC)

I would hope, as already suggested here some while back, that a brief account of the "teaching of NSA" could be included as a subsection of the article on NSA. It could include a reference to this review. There is no need to explain constructive mathematics in any detail. As said before, this article seems to be an essay about a negative review. Its whole conception ignores context and existing literature on mathematics education, which makes it clear that NSA has had a very weak impact on undergraduate education. At present this information does not appear in the article (I added three sources in the references a while back). Mathsci (talk) 07:21, 13 December 2008 (UTC)

epsilon, delta bis

There are several threads here but I was hoping we could focus a bit on the article itself for a moment. Bishop's mentioned the epsilon, delta definition of limit explicitly in his review. It seems to me that this issue is a bit of a focus of the disgreement (if people don't like "controversy" perhaps we can move this to "Constructivist criticism of NSA"?). There is clearly a sharp disagreement between the protagonists regarding whether the hyperreal definition is an improvement or, on the contrary, a debasement of the standard one. This disagreement is obviously not merely about teaching calculus. If this were the case, Bishop would have stated simply that this is a lousy textbook, and be done with it. Furthermore, the Bulletin would have never asked him to review it in the first place (note that the textbook has never been reviewed by Math Reviews). Clearly, there are fundamental issues at stake here, involving fundamental disagreement of what "theorem" means, and of what "proof" means. I think this contoversy is more significant mathematically than the old Newton-Leibniz priority story, which is after all mostly about personalities and history, rather than mathematics.

In short, I am in favor of mentioning Bishop's mention of epsilon, delta explicitly, and would like to get some input from other editors on this issue. Katzmik (talk) 20:10, 13 December 2008 (UTC)

I think it should be enough to edit the last paragraph to make clear he intended his comments about common sense to extend to the epsilon-delta definition of limit. We don't need to explicitly quote him, we already do a lot of that. I think that this is a disagreement about teaching, Bishop explicitly says in another article that the issue is about the teaching of Calculus and not NSA (he claims to not know enough about it to comment). Thenub314 (talk) 14:31, 14 December 2008 (UTC)

Voting

I propose the following ‘voting guidelines’:

Voting Guidelines:

Vote 1 if you agree with this merge and 0 if you don’t. Every vote must be supported with a one sentence justification. The main contributors to the article may not vote (this is because it is obvious what they are going to vote for as can be concluded from the above discussion).

Voting (currently, 'not merging' is more favored, 1:0):

• 0. Non-standard analysis is an important topic of research in mathematics and an article on a discussion as to whether it should be included in the teaching of mathematics is definitely a necessity. Topology Expert (talk) 21:05, 12 December 2008 (UTC)

I am confused--- you say a discussion on pedagogical inclusion is "definitely a necessity", and then you voted "0", but doesn't 0 mean that you don't like the merge? I don't know if this needs a formal vote--- People will probably all agree, but just in case...Likebox (talk) 22:35, 12 December 2008 (UTC)

I said that an article such as this is important and therefore deserves its own page. It should not have to be compressed into a small section on another page. Topology Expert (talk) 22:49, 12 December 2008 (UTC)

Please follow the guidelines. Since you are the one who wanted the merge, you are not allowed to vote. Topology Expert (talk) 22:51, 12 December 2008 (UTC)

• 1. This debate is not notable enough to stand alone, because it is a description of exactly one reference and a reply to this reference. Similar discussions on pedagogical issues are included in mass in special relativity, where the Einstein/Okun relativistic mass detractors have their say, against the Tolman relativistic mass camp.Likebox (talk) 22:35, 12 December 2008 (UTC)

Stop voting. Wikipedia:Polling is not a substitute for discussion. Go to Talk:Bishop–Keisler controversy and figure it out there. Ozob (talk) 23:35, 12 December 2008 (UTC)

I agree. In the article, I believe all the non-Keisler references were added by me. They include academic articles that discuss the teaching of NSA in universities: reputable educationalists confirm that this was an experiment that did not work and the few people that advocate it do so with a disproportionate passion. My own view is that the NSA article should have a section on the teaching of NSA which might or might not mention Bishop's book review. The best place to discuss what should be done is probably an AfD, not the talk page of the article. Mathsci (talk) 00:34, 13 December 2008 (UTC)

I am fed up of pointless guidelines.

Please read (or reread) WP:NOTDEMOCRACY. To quote from the page: "This page documents an official English Wikipedia policy, a widely accepted standard that should normally be followed by all editors." Being "fed up" is not a good reason to ignore core policy, especially when others do not wish to do so. Plclark (talk) 20:14, 13 December 2008 (UTC)

Who wants to follow them anyway?

In case you actually don't know: at least, the people who made the policy and those that have reminded you about it in this instance: myself, Ozob, Mathsci. But more importantly, anyone who participates in a community tacitly agrees to abide by the rules. If there is a rule you do not like, the honorable choices are to work with the community to change the rule, to get the community to agree that the rule does not apply to your particular situation, or to withdraw from the community. For more information on this important idea, see Trial of Socrates and Crito. Plclark (talk) 20:14, 13 December 2008 (UTC)

Anyhow, this is a discussion: you are supposed to give a one sentence justification with your vote. If you are really worried about whether this should be on the talk page or not, I can copy it there once voting is over. Anyway, as we vote, we are discussing our justifications.

Continue voting:

Rather, please abide by the rules and the wishes of those present to abide by the rules, and discuss the matter either on the article talk page (as I have) or on an AfD. Plclark (talk) 20:14, 13 December 2008 (UTC)

A series of one-sentence justification is not a discussion. Rather, it seems designed to avoid thoughtful discussion and to sabotage the possibility of useful discussion leading to consensus. I believe this particular Wikipedia policy is very wise, and that you are doing everyone a disservice by ignoring it, for reasons that have already been explained. I reject your unilateral attempt to set the rules of the discussion. —Dominus 22:44, 13 December 2008 (UTC)

• 0. Non-standard analysis is an important topic of research in mathematics and an article on a discussion as to whether it should be included in the teaching of mathematics is definitely a necessity. Topology Expert (talk) 21:05, 12 December 2008 (UTC)

But the article is not about how NSA should be taught. There is not clear evidence there was a dicussion, just a book negative book review. The article claims that Keisler responded to the review, but none of the references I can find do. As was points out, a section about teaching would be a good to the NSA article, but I feel that is not what this article is about. Thenub314 (talk) 16:19, 13 December 2008 (UTC)

• 0. I don't agree with the merge as I think such a discussion is premature. If you consult the talk page of the article, you may notice that it is an article in progress. Once the text stabilizes, it may be possible to have a meaningful discussion of the desirability of a merge. Katzmik (talk) 20:19, 13 December 2008 (UTC)

You introduced an unexplained and unsourced parenthetical allusion to the invariant subspace problem in the lede. Wikipedians can puzzle about why you did that, but it is hardly progress. The general statements you are making at the moment suggest that you are now ready to present your arguments on an AfD page if somebody decides to start one. Mathsci (talk) 21:53, 13 December 2008 (UTC)

A tale of two definitions

I feel this section that has just been added exhibits a definite point of view. Thenub314 (talk) 12:30, 15 December 2008 (UTC)

You are right, "rhetorical fleurish" (I am still not sure about the spelling) is a bit too sarcastic. Can you suggest another wording? Katzmik (talk) 12:44, 15 December 2008 (UTC)

Well, it is really the whole section I have a problem with, not just those words. How is it your interpreting this quote, because it is much different then I interpret it. And what does the statement about equivalence of the definitions of limit have to do with anything? Thenub314 (talk) 16:39, 15 December 2008 (UTC)

The epsilon, delta definition of limit is a bit of an Apple of discord here, or should we call it a lighting bolt? Litmus test? (Likebox likes "touchstone".) How one relates to this issue pretty much determines one's position in this debate. In particular, Bishop's comment is revealing of his particular position. Katzmik (talk) 16:56, 15 December 2008 (UTC)

Why is this the litmus test for the issue (here I am thinking of the issue as assessing the quality of Keisler's book)? Bishop criticizes many things about the book, and this is just one of them. Even if it is the litmus test, what does the statement about equivalences of definition have to do with anything? Bishop is talking about which definition is common sense and expressing his opinion that the epsilon delta definition is. I think you mean something else by "this debate", could you clarify? Thenub314 (talk) 18:40, 15 December 2008 (UTC)

I have removed the latest section written by Katzmik because it is completely unencyclopedic. It is very poorly written and lacks secondary sources. It reads like a blog. It fails on WP:NPOV, WP:RS, WP:V, WP:OR, WP:SYNTH. Katzmik should urgently search for secondary sources if he wishes to add this kind of content to wikipedia. Wikipedia is not about what Katzmik thinks (that's what blogs are for), but what secondary sources have to say. Mathsci (talk) 22:18, 15 December 2008

Your comment is puzzling. Your edit on the page, of 22:10, 15 December 2008, to which you presumably refer, removed a chunk cited from the review. Charles Matthews (talk) 16:39, 16 December 2008 (UTC)

Controversy goes back to Cantor?

I find this sentence doubtful:

"The philosophical origins of the disagreement go back to Cantor, Dedekind, Kronecker, Brouwer, Hilbert, and more recently Robinson and Halmos (controversy over the proof of the Invariant subspace conjecture)."

It is true that the general debate about foundations of mathematics goes back at least to the early 20th century (much further than that, really). But that general debate is not related to criticism of nonstandard analysis. Similarly, in my reading about Brouwer, I have never seen it mentioned that he had a criticism of nonstandard analysis. I don't think there is really a strong relation between the debate about foundations in the 1920s and the debate about nonstandard analysis (which is not strongly related to foundations) in the later part of the century. — Carl (CBM · talk) 14:28, 16 December 2008 (UTC)

That's an interesting criticism. I was taking it for granted that there is a foundational component involved, since after all Bishop is a constructivist. I don't think you need to read about Brouwer to be able to assert that he never criticized NSA, as the latter originated in the 60's (some trace its origins to work in the 50's, see the main article). I am no expert in the history of math, and if there is a consensus that the debates are unrelated, the lead paragraph would certainly need to be revised. Katzmik (talk) 14:36, 16 December 2008 (UTC)

I already moved the sentence here from the lede. — Carl (CBM · talk) 14:38, 16 December 2008 (UTC)

A tale of two definitions

What is the section "A tale of two definitions" about? I don't know, and it doesn't say. — Carl (CBM · talk) 14:29, 16 December 2008 (UTC)

The original version of the article contained a direct quote from Bishop concerning the epsilon, delta definition of limit. We have been having a field day reverting each other's edits on this one. My position is that an editor who nominates an article for deletion should refrain from... well, deletions, until the issue is resolved at AfD. Katzmik (talk) 14:39, 16 December 2008 (UTC)

I didn't nominate the article for deletion. But I also have no idea what point is being made in this section. There appears to be no topic sentence, no indication of the purpose for including the quote, and no explanation why "it may be helpful" to look at the two definitions. What conclusion do we want the reader to see there, and why can't we come out and say it? — Carl (CBM · talk) 14:43, 16 December 2008 (UTC)

These are legitimate questions. We don't "want" the reader to reach conclusions, rather he is invited to compare the definitions favored respectively by the protagonists of the incident/controversy. I tried to explain this above in terms of the "litmus test". If this is not convincing, the section can certainly be removed. I do find it revealing of Bishop's position on this issue. It also shows that his claims elsewhere of holding "no opinion" on NSA may be disingenuous. Katzmik (talk) 15:04, 16 December 2008 (UTC)

When I write a paragraph, I have a particular point in mind that I want to convey to the reader. That's the cornerstone of sound technical writing - each piece has a purpose, and that purpose is clearly conveyed, so there is no doubt what was meant. In this case, even with a lot of mathematics background, I can't figure out what moral I am supposed to draw from that section. So I do favor removing it, unless it can be improved to come out and say something. — Carl (CBM · talk) 15:07, 16 December 2008 (UTC)

Moved to talk page

In a final rhetorical fleurish, Bishop writes:

"Although it seems to be futile, I always tell my calculus students that mathematics is not esoteric: It is common sense. (Even the notorious (ε, δ)-definition of limit is common sense, and moreover it is central to the important practical problems of approximation and estimation.)"

For the benefit of a reader unfamiliar with the technical aspects of infinitesimal calculus, it may be helpful to recall the following pair of equivalent definitions of what it means for the limit of f(x) as x tends to 0, to be 0:

• standard definition: for every ε>0 there exists a δ>0 such that if 0<|x|<δ then |f(x)|<ε;
• non-standard definition: if x≠0 is infinitesimal then f(x) is infinitesimal, as well.

Bishop's opposition to the latter definition stems from his constructivist approach to mathematics. Such an approach excludes the Axiom of choice from the mathematical toolkit. The constructivist approach in particular jettisons non-standard analysis, as the axiom is used in a fundamental way in all non-standard constructions.

I'm moving this here so we can mull over it. Katzmik, I can see why other people may be concerned with "original research" in your writing. In his review, Bishop never even mentions the "non-standard definition" of continuity.

In the text you added, you first claim that Bishop "opposes" the nonstandard definition - which is not supported by Bishop's actual review - and then go on to ascribe a motivation why Bishop opposes it.

It's true that Bishop was a constructivist, but it appears to me that in his review he was comparing Keisler's book with the standard mathematical theory of calculus, not comparing it with a constructivist version. Bishop never even mentions the axiom of choice in his review.

This article is not about the debate on constructive mathematics, it is about the reception of Keisler's book on nonstandard calculus. I think you are mixing the two issues in a way that involves more speculation on motivations than we can include in our articles. — Carl (CBM · talk) 15:40, 16 December 2008 (UTC)

I disagree. A lot of Bishop's criticisms would apply to standard calculus as well as to non-standard calculus. Does he allow "completed infinity" in his approach? If not, the standard assumption in calculus textbooks that you can talk about R is distinctly non-Bishoplike. In fact, he did rewrite standard analysis, replacing it by his constructivist approach. Furthermore, his position on foundations is well known--there is no secret how he feels about AC. Secondary sources discuss the fact that, since his criticisms apply to standard mathematics as well, it was in fact absurd to get him to review Keisler's book. This is my understanding of his position, on which my edits were based. Katzmik (talk) 15:48, 16 December 2008 (UTC)

I am very familiar with Bishop's work in constructive analysis. However, when I read the actual review, it appears to me that he wrote it in a moderate tone. His criticism of Keisler's book is the book neglects geometric intuition in favor of a set of non-intuitive axioms. And it is clear that Bishop's criticism extends to new math. But Bishop's review does not discuss constructivity at all, or the axiom of choice.

Regarding secondary sources, I love them. However, in this sort of article, they need to be included from the beginning, rather than added on. However, my guess is that many of the secondary sources are really just passing mentions, and so we will have to take them with a grain of salt. Do you have a particular one in mind? — Carl (CBM · talk) 15:56, 16 December 2008 (UTC)

(Rather than copying your edits, I will make a comment on the spot--if it bothers you please let me know) Bishop's opposition to non-standard definition of limit is not OR, it is obvious. Now it is a fine thing to give an editor the benefit of the doubt, but somehow I don't think this is what the two editors in question had in mind. Do you seriously mean to challenge this as OR? Katzmik (talk) 15:55, 16 December 2008 (UTC)

I moved this down so I can reply to it without breaking up my comment. I disagree that it's "obvious" that Bishop's review attempted to address the particular definition of continuity in nonstandard analysis. On the other hand, if we want to start with Bishop's program of constructive analysis, there are much more concrete things that he would object to than this definition. For example, the theorems of classical analysis that are not true constructively. But I don't think the point of this article is to discuss Bishop's program. It would be good, I think, to give a short amount of background on Bishop to inform the reader about the context of the review. — Carl (CBM · talk) 16:01, 16 December 2008 (UTC)

Also, I think that "the axiom [of choice] is used in a fundamental way in all non-standard constructions" misses the mark. AC is used to construct a model of the hyperreals within ZFC. But if the hyperreals are developed axiomatically not inside ZFC, where is the application of the axiom of choice? I don't think that just axiomatically applying a transfer principle counts as a "fundamental" application of AC. — Carl (CBM · talk) 16:04, 16 December 2008 (UTC)

I don't have a copy of the first edition of Keisler's book. The edition I have has an Epilogue constructing the hyperreals (a first approximation, of course). Did the first edition lack the appendix? Even if that's the case, Bishop was certainly aware of the fact that NSA can be constructed in ZFC. I think it is very misleading to emphasize the axiomatic approach of NSC, as if one had to add axioms hitherto unknown to get the infinitesimals as a prize for our daring. Surely Bishop was aware of the fact that when he attacked the supposedly consistent system of axioms, it is ZFC he was talking about. Please respond to my OR comment above. Katzmik (talk) 16:13, 16 December 2008 (UTC)

I did respond to the OR comment above. I don't think it's clear at all that Bishop was referring to ZFC, after looking at the actual language of the review. In Keisler's book, it looks like Keisler does present a basically axiomatic approach, along with a proof of consistency. And the axiomatic approach is also crucial when one wants to look at weaker fragments of nonstandard analysis, e.g. [2] Simply because the axiom of choice is used in proving consistency, does not mean that every result from the first-order theory of nonstandard analysis is somehow "fundamentally" about AC. — Carl (CBM · talk) 16:20, 16 December 2008 (UTC)

Perhaps discussing AC is going too far afield. On the other hand, Bishop does explicitly criticize infinitesimals in his review. If he is critical of infinitesimals, I don't see how it would be OR to say that he is critical of the infinitesimal definition of a limit. Would you oppose the inclusion of the direct epsilon, delta quote with a minimum of explanation concerning the two definitions? Katzmik (talk) 17:01, 16 December 2008 (UTC)

That was the problem before - that there was a quote and the two definitions, but no apparent reason why they were being presented. I think it is slightly misleading, in a particular sense, to say that Bishop "was opposed" to the infinitesimal definition of limits, if what he was actually was opposed to was the use of infinitesimals in the first place, or the axiomatic use of them without the simultaneous development of good intuition about what they are, geometrically.

I think the real solution would be to actually refer to some secondary source, if we want to make a claim about what Bishop was thinking when he wrote the review. — Carl (CBM · talk) 19:27, 16 December 2008 (UTC)

And I am still not comfortable with the unsourced claim that the review was related to Bishop's constructivist opinions, since Bishop apparently avoiding mentioning constructivism entirely when writing the review. If you would like to reference someone else who says in print that the review is related to Bishop's constructivism, and attribute that opinion to them, I can go along with that. But when I read the review, it is not plain to me that Bishop's opinion there would be out of place if written by any other (non-constructivist) mathematician at the time. — Carl (CBM · talk) 19:31, 16 December 2008 (UTC)

And, point blank: what is the point of including the two definitions? What information is being conveyed to the reader through including them? — Carl (CBM · talk) 19:36, 16 December 2008 (UTC)

unmetamathematical

Hi Carl, Thanks for your edits. The term metamathematics is frequently used in a sense close to "philosophy of mathematics", see the page in question. I thought that it may be helpful to point out in the lead paragraph that the controversy concerns deeper issues than merely truth of a particular theorem or proof. Perhaps "philosophical" would be better? Katzmik (talk) 14:44, 16 December 2008 (UTC)

These days, metamathematics is used to mean "the use of mathematical methods for the study of mathematics itself" - essentially, mathematical logic. I'll work on clarifying the article on metamathematics about that. I don't think any contemporary mathematical logician would recognize the discussion between Bishop and Keisler as being about "metamathematics".

The question whether the discussion is "philosophy of mathematics" is a different question. In the broadest possible sense, it might be, but that would stretch the word "philosophy" to be essentially meaningless. Really Bishop's review was just about the pedagogical advantages or disadvantages of using nonstandard analysis to study calculus. And I don't think that nonstandard analysis really has any affect on the big issues in philosophy of mathematics, which include "what are mathematical object", "how do we learn about mathematical objects", "what is the true nature of mathematical proof", etc. So I don't see that there is philosophical content to the debate.

The clearest explanation in the lede would be that the discussion between Bishop and Keisler was about the possible pedagogical benefits of teaching calculus using nonstandard methods. — Carl (CBM · talk) 14:52, 16 December 2008 (UTC)

Ostensibly it was about the teaching of calculus, but clearly there are deep philosophical differences between them about what it means to do mathematics. Bishop does not just disagree with teaching calculus using infinitesimals. His remarks about obfuscation apply to NSA as well. As far as the question 'what is a mathematical object" is concerned, you would be surprised how many comments I have gotten from fellow editors to the effect that infinitesimals are not "real" the way the standard real numbers are real. I don't know how the professional philosophers of mathematics define their discipline today, but these are certainly issues that there are wide disagreements on. Katzmik (talk) 14:59, 16 December 2008 (UTC)

Well, it's certainly true that infinitesimals are not part of the "real" line, and so no wonder people find them less "real". But my point is that the debate between Bishop and Keisler did not appear to extend into general philosophical or metamathematical discussions. You say the debate was "ostensibly" about the teaching of calculus - what evidence is there that is wasn't simply about the teaching of calculus? — Carl (CBM · talk) 15:05, 16 December 2008 (UTC)

I would cite the following evidence: (a) Bishop himself uses very broad language in criticizing Keisler, which suggests a sharp philosophical (if I may be allowed to use the term in its lay meaning) disagreement that goes beyond the classroom; (b) Halmos reportedly deliberately chose Bishop for the job, knowing well the likely outcome (whether or not Halmos was chief editor at the time can easily be checked; furthermore, senior colleagues who have told me about this twenty years ago may have personal recollections of the matter; if it is important to prove this through witness accounts, I may be able to do it); (c) Michael has documented a number of references in secondary literature that seem consistent with my position; (d) Bishop clearly refers to ZFC when he criticizes "the supposedly consistent system of axioms" in his review (I think my direct quotation may have been replaced by a weak paraphrase by the two editors in question). Since when is ZFC taught in elementary calculus? Katzmik (talk) 15:13, 16 December 2008 (UTC)

As far as the reality of the infinitesimals, an astute editor recently argued, based on quantum mechanics, that even the complex numbers (phases, etc) are more real than infinitesimals. I don't think this attitude can be explained merely in terms of habit. Katzmik (talk) 15:15, 16 December 2008 (UTC)

I have just looked through Bishop's review again - he does not ever explicitly mention ZFC. The phrase "supposedly consistent set of axioms" on p. 207 refers to the axioms Keisler uses to develop infinitesimal calculus. See three paragraphs higher on p. 207 of the review. It looks to me that Keisler revised the second edition to address this criticism - for example I tried to find an axiom V* but could not.

I don't see how Halmos picking someone can influence whether the debate is philosophical. Halmos, like most mathematicians, didn't see any particular benefit in nonstandard analysis, and so it's no wonder he would look for a negative review. That may be questionable from the viewpoint of professional ethics (or not, depending on your viewpoint), but it's only related to "philosophy" in a vague, lay sense.

The last paragraph of the review appears to me to be written to support the argument about pedagogy that students should learn more than just abstract technique, they should learn the geometric ideas behind the techniques. I believe that Bishop's main complaint is that the nonstandard presentation removes the geometric aspect of calculus, leaving only a set of nonintuitive axioms. — Carl (CBM · talk) 15:30, 16 December 2008 (UTC)

New Lede

The lede has currently been changed to:

A review by Errett Bishop of a textbook on nonstandard calculus by H. Jerome Keisler served as the touchstone for a debate among mathematicians in the 20th century concerning the role of nonstandard analysis in mathematics.

I don't believe this at all and seems to have a strong POV to me. Which mathematicians debated? What is the evidence that Bishop's review sparked the debate? Can someone explain to me why this is about the role of nonstandard analysis in mathematics and not about the teaching of calculus? Thenub314 (talk) 21:29, 16 December 2008 (UTC)

Please feel free to rewrite it. I was thinking already of the merge to "criticism of nonstandard analysis". — Carl (CBM · talk) 21:47, 16 December 2008 (UTC)

Connes

Is there a secondary source supporting the claim that Connes is a critic. I am now afraid that the quote may have taken this quote out of context, and I am moving the section here until we can improve it. Thenub314 (talk) 10:49, 18 December 2008 (UTC)

Connes' critique

Despite the elegance and appeal of some aspects of non-standard analysis, there is a great deal of skepticism in the mathematical community about whether this machinery really adds anything that cannot just as easily be achieved by standard methods. One noted critic of non-standard analysis is the Fields Medalist Alain Connes, as evidenced by the following quote:

The answer given by nonstandard analysis, a so-called nonstandard real, is equally deceiving. From every nonstandard real number one can construct canonically a subset of the interval [0, 1], which is not Lebesgue measurable. No such set can be exhibited (Stern, 1985). This implies that not a single nonstandard real number can actually be exhibited.
– A. Connes "Noncommutative Geometry and Space-Time", Huggett et al., The Geometric Universe, p. 55

The point of Connes' criticism is that nonstandard hyperreals are as fictitious as non-measurable sets. These sets can be shown to exist, assuming the axiom of choice of set theory, but are not constructible. Non-measurable sets are usually considered pathological.

In his now famous book Non Commutative Geometry, Connes offers an alternative approach to infinitesimals based on ideals of compact operators on a Hilbert space. In this treatment, the Dixmier trace plays a central role, but its definition is itself dependent on the choice of a free ultrafilter on the natural numbers, which is certainly nonconstructive. Moreover, Robinson notes on page 48 of the 1966 edition of his book that his theory does not require the axiom of choice but can also be based upon the ultrafilter lemma. Robinson infinitesimals can also be obtained using a free ultrafilter over the natural numbers.

These criticisms notwithstanding, however, there is absolutely no controversy about the mathematical validity of the approach and the results of non-standard analysis. In particular, the following two points should be kept in mind:

• IST is a conservative extension of ZFC. This is shown in Edward Nelson's AMS 1977 paper in an appendix written by William Powell;

• Model theoretic non-standard analysis, for example based on superstructures, which is now a commonly used approach, does not need any new set-theoretic axioms beyond those of ZFC.

Bishop's criticism

As I mentioned before, Bishop's criticism is about the teaching of NSA in calculus classes, and not a criticism of NSA. Does if have a place on the page as currently titled? Thenub314 (talk) 16:02, 18 December 2008 (UTC)

It is mentioned explicitly in secondary sources that Michael cited that Bishop's criticism stems from his foundational position on the axiom of choice (this is not my opinion but rather the secondary source). Thus Bishop's criticism fits well within the context of a foundational controversy. Katzmik (talk) 18:45, 18 December 2008 (UTC)

Now, correct me if I am confused. As I remember the article your referring to it was criticizing Bishop's review of Keisler's text. The comment was that Bishop criticized Keisler for not show that there is a model which satisfies the axioms, and this criticism only made sense from the point of view of Bishop's constructivism. But this was specifically a criticism of Keisler's text. (It was known a model did exist.) No other reference tries to put Bishop's constructivism into the picture. Even the Tall reference, which talks about the same point in Bishop's criticism, says Bishop is criticizing Keisler's text and makes no mention of constructivism. Specifically he says (emphasis added by me)

"Meanwhile, Bishop [1] fiercely criticised Keisler’s text for adopting an axiomatic approach when it is not clear to the reader that a system exists which satisfies the given axioms."

And Bishop explicitly states in his "Crisis in Mathematics" article that his only interest in non-standard analysis is the attempts to bring it into elementary calculus courses. I really don't understand how this is a foundational issue.

All of that being said, I don't mind including something about it, if we rework it. There doesn't need to be a blow by blow account of the review. Thenub314 (talk) 09:07, 19 December 2008 (UTC)

I was not referring to Davis' article, but to the last reference found by Hardy (see AfD). You already mentioned several times that Bishop said what he said, and I already replied that I think it is being disingenuous. At any rate, being a primary rather than secondary source, his claims of innocence are irrelevant. Katzmik (talk) 09:16, 19 December 2008 (UTC)

Here is the choice quote I had in mind: "Natural and Formal Infinities", by David Tall: Criticism of the use of the axiom of choice in the non-standard approach however, draws extreme criticism from those such as Bishop (1977) who insisted on explicit construction of concepts in the intuitionist tradition[.....]Bishop, E., 1977: "Review of ‘Elementary Calculus’ by H. J. Keisler", Bulletin of the American Mathematical Society, 83, 2, 205–208." It is very difficult to believe that there will not be further secondary sources connecting Bishop's constructivism and rejection of AC, to his negative attitude toward NSC; John says Halmos did it for precisely this reason, and committed himself in print. Katzmik (talk) 09:22, 19 December 2008 (UTC)

Tall is again talking about the teaching of Elementary calculus using NSA! That is exactly my point, if you want to take this quote and keep it in context then "Criticism of the use of the axiom of choice in the non-standard approach [to teaching calculus] however, draws extreme criticism from those such as Bishop (1977) who insisted on explicit construction of concepts in the intuitionist tradition." Thenub314 (talk) 11:14, 19 December 2008 (UTC)

Halmos' "scepticism"?

Where is there any sign of skepticism in the quote from Halmos. He said the use of non-standard analysis was a matter of personal preference, not of necessity. All that means is that the theorem can be proved by other methods than non-standard analysis. Typically a theorem can be proved by various different methods. The fact that Halmos recognized that some of those methods in that particular case do not require non-standard analysis is nota case of skepticism about the validity of non-standard analysis. I'd be surprised if the most ardent advocates of the use of non-standard analysis disagree with Halmos' comment. Michael Hardy (talk) 03:46, 19 December 2008 (UTC)

I agree with this comment (and have been having similar thoughts myself for several days now). It seems clear that Halmos' sentence refers to his own work on the problem at hand (invariant subspaces for polynomially compact operators) which was based on the original work but managed to remove the appeal to NSA. Beyond disagreeing that this quotation connotes skepticism (or scepticism...), I honestly can't figure out exactly what it is that Halmos is claimed to be skeptical of.

To my mind, this entire article has a strange point of view in that it is focused on criticism and controversy. Again I am not even sure quite what these terms mean when applied to mathematical ideas. Without further clarification it seems that the most likely interpretation of "criticism" is that so-and-so mathematician has concerns about the validity of NSA. Since NSA has (in at least one of its common formulations) exactly the same foundations as the rest of modern mathematics -- the ZFC axiomatization of set theory -- attributing such "criticisms" to famous mathematicians has the effect of simply making them look foolish. (Similarly, giving lots of attention to the opinions of Bishop on NSA seems very strange to me, since Bishop was one of the small minority of research mathematicians who truly did not accept ZFC -- or even conventional logic, such as the law of the excluded middle -- as being a correct foundation. Of course he is going to be critical of NSA...)

Another purely mathematical way of construing the "critique" is that NSA is only capable of proving things which are also possible to prove using conventional methods. This at least is an opinion that I think might be reasonably attributed to a majority of working analysts (as evidenced, for instance, that only a small minority of papers on classical analysis make any use of nonstandard methods). But, even assuming this is true, to me this seems more like a statement of preference rather than a "criticism" or "controversy". I think that in most fields of modern mathematics there are more than enough tools that could be fruitfully applied to the same circle of problems, to the extent that most people choose to become more adept in the use of certain tools than others. So what?

Overall I would like to see more content about nonstandard analysis (and related areas of nonstandard mathematics) than claims of the form "so-and-so didn't/doesn't like it", even if true. Plclark (talk) 04:44, 19 December 2008 (UTC)

Please note that the current proposal at the AfD for this article is to change the title of the article to "Impact of nonstandard analysis" (or slight variants) with a considerably broader scope. Mathsci (talk) 07:46, 19 December 2008 (UTC)

Please see John Z's comment at the AfD. The Halmos quote indicates to me his scepticism that NSA can actually produce significant results, and is therefore merely a question of personal preference. Now it is true I am reading a lot into this short quote, but after all this is not the only thing Halmos said. It is generally known that Halmos had such a sceptical attitude, and according to John Z, one should be able to find longer quotes in this direction. Katzmik (talk) 08:46, 19 December 2008 (UTC)

I saw John Z's comments, he was not exactly sure if there were secondary sources out there, and I haven't been able to find any (and I have been looking). I placed an interesting example on another talk page about more or less the same topic. In this edit I give an example about two quotes you could take from Terry Tao's blog that make him look like a critic of Nonstandard analysis or a supporter of non-standard analysis. If we are going to say someone is a critic, we really need a source (other then the supposed critic) saying "Person xyz is a critic of non-standard analysis." Until we have one we shouldn't advertise to the world via Wikipedia that they are. Thenub314 (talk) 15:53, 19 December 2008 (UTC)

This doesn't seem to be criticism, just merely the statement that it is possible to prove this result with or without non-standard analysis. I have realized at an older paper that the use of the phrase meta-mathematical to describe non-standard analysis was not derogatory. He is doesn't directly express any skepticism that non-standard analysis "can actually produce significant results." And it seems OR to portray his comment is a setting that suggests this is his intended meaning. We need to find a secondary reference for Halmos's criticism, and until that time we should remove this comment. Thenub314 (talk) 07:49, 13 January 2009 (UTC)

I am confident that we should be able to find suitable material in his mathography, in light of John Z's comment. If you could leaf through it to see what's there, I would much appreciate it. For the time being I think it would be sufficient to place a "source" flag at the section. Katzmik (talk) 13:16, 13 January 2009 (UTC)

Unfortunately my library doesn't have it. So I cannot look through it. Thenub314 (talk) 15:25, 13 January 2009 (UTC)

Controversy and mathematical ideas

Plclark wrote: "To my mind, this entire article has a strange point of view in that it is focused on criticism and controversy. Again I am not even sure quite what these terms mean when applied to mathematical ideas."

The same is true of Controversy over Cantor's theory. Michael Hardy (talk) 19:28, 19 December 2008 (UTC)

Perhaps what Michael is saying is that, like Cantor's theory, Robinson's theory has all the trappings of a scientific revolution a la Thomas Kuhn, including a period of incomprehension, scepticism, and denial (exemplified by Errett Bishop's intemperate attack in the guise of a textboook review), a period of adjustment of traditional paradigms so as to take in the new phenomena (Halmos' feverish race to produce an alternative standard proof of the invariant subspace conjecture before Robinson's gets published), gradual acceptance and success of the new paradigms (see Google Scholar figures). Katzmik (talk) 08:56, 21 December 2008 (UTC)

[comment removed by the author.] 66.245.43.165 (talk) 00:09, 15 December 2009 (UTC)

[To the user who persists in unremoving my comment: stop. It's abusive. Don't disregard what an administrator wrote on your user talk page.] 66.245.43.165 (talk) 19:39, 16 December 2009 (UTC)

Bishop's funeral may have been well-attended, but do any of the attendees agree with him? The current version of the page cites only authors who disagreed with Bishop's criticism. Tkuvho (talk) 09:14, 14 December 2009 (UTC)

The above comment is a response to user 66.245.43.17, who wrote: "When he died at age 54, many famous mathematicians traveled far to attend his memorial service, even logicians. I'm sorry if he hurt your feelings, or Keisler's. His criticism was motivated primarily by sympathy for students who are subjected to bad textbooks, as the review makes perfectly clear, if read in its entirety." —Preceding unsigned comment added by Tkuvho (talk • contribs) 13:24, 17 December 2009 (UTC)

attempts to derail this article

There was wide consensus on the AfD for retaining this page under the current title "Criticism of non-standard analysis". Attempts to move all of its contents to a different page constitute disdain for a community-wide decision, and should be refrained from. Katzmik (talk) 08:59, 21 December 2008 (UTC)

By the end of the discussion the consensus seemed to be that would be better titled "Impact of Non-standard analysis." I don't think that there was any "disdain for a community-wide decision" or any attempts to derail this article. First R.e.b. was not part of the AfD discussion, and may not have been aware that there was any consensus as the debate was closed as a "complete trainwreck" (which wouldn't encourage me to read through it.) I removed the sections on Connes and Halmos because of the possibility of misinterpreting them. If they were outspoken critics of NSA then surely someone would have commented on their critiques in print, and we should refer to those. I did a few google and jstor searches, but I didn't find anything yet. Thenub314 (talk) 09:39, 21 December 2008 (UTC)

If you go through the discussion carefully, and tote up the number of people in favor of retaining CBM's title "Criticism of non-standard analysis", you will notice that there is a majority in favor. Connes and Halmos expressed themselves in print. It could be that what they said is ambiguous, and this should be duly noted. I think it is very odd to delete the comments because someone a quarter century later thinks they are ambiguous, though. Katzmik (talk) 09:43, 21 December 2008 (UTC)

latest deletions an error

The notes section is empty because thenub unfortunately deleted the secondary reference by Gillies that Matthews put in. Instead of deleting the "notes" section, please restore the important reference.

Furthermore, CSTAR wishes to be cautious at this stage about the interpretation of Connes' work that was part of the section on criticisms at non-standard analysis; I don't think he meant there is anything wrong with the quote from Connes itself. That Connes is on record as criticizing NSA is beyond dispute, in my opinion. Instead of reaching for the delete button, please make yourself useful and try to find some sources. Katzmik (talk) 16:35, 21 December 2008 (UTC)

The fact that Connes' criticism is mathematically incorrect needs to be noted; if we can't find references for the fact, we, at least, need to tag Connes as potentially not being a reliable source. — Arthur Rubin (talk) 17:27, 21 December 2008 (UTC)

The statement

From every nonstandard real number one can construct canonically a subset of the interval [0, 1], which is not Lebesgue measurable.

is clearly false. To begin with, depending on the model, there may be more non-standard reals than subsets of [0, 1]. — Arthur Rubin (talk) 17:33, 21 December 2008 (UTC)

Here is a sense in which Connes' comment could be technically correct: the notion of an infinitesimal makes sense only once an ultrafilter, say UF, has been chosen. Thus when one refers to an infinitesimal I, strictly speaking one is referring to a pair (UF, I). Now applying the forgetful functor

$(UF,I)\mapsto I$

we do obtain a canonical way of associating a non-measurable set to an infinitesimal. This may be a little silly, but if my argument above for associating a non-measurable set with an UF is correct, then technically speaking Connes would be right. Of course, I have no idea whether this is what he meant. Some comments above seemed to suggest otherwise. Can we really pin him down for an error here? Katzmik (talk) 17:38, 21 December 2008 (UTC)

I suppose an (free) ultrafilter on N is a non-measurable set. We don't really know that Connes is requiring the concept of infinitesimal to use an ultrafilter, or even whether Kiesler uses ultrafilters. The compactness theorem and relatives (upward Löwenheim–Skolem theorem) gives a non-ultrafilter approach for any specific transfer result in non-standard analysis. — Arthur Rubin (talk) 19:45, 21 December 2008 (UTC)

Just a small clarification: Keisler does construct the hyperreals using an ultrafilter in his Epilogue starting on page 911 thereby bringing down the twin towers of ε and δ (a small joke, to avoid any misunderstandings) Katzmik (talk) 13:08, 22 December 2008 (UTC)

Various comments

Ok, various comments. If we quote him on a page that says he criticized, then we are implying he did. The quote we give makes it appear as if he doesn't accept the existence of Lebesgue measurable sets! He starts the quote by saying non-standard analysis gives an answer that is equally deceiving. Answer the following questions if your so sure he is criticizing.

What question is he answering?

What is the other deceiving answer, (that he must also be criticizing)?

I comment above I have been looking for resources. I have reading websites about him, read his blog, search jstor, search google and visited my universities library. I haven't found anything saying he was critical. Have you been searching? If you have then your search seems to be going as well as mine.

I have made my argument at the talk page on non-standard analysis that we must be very careful interpreting comments, as sometimes it can be misleading. Should I add a section on Tao's criticism?

Lastly, I can't answer much on mathematical correctness. There is the technical sense he could be using the word canonical and I am not sure what that would mean in this category. Tao's construction seems rather canonical, but this is probably just informal usage. Beyond that, Tao does (as discussed at the non-standard analysis page) give a very nice proof that if you have infinitesimals to work with you can build a non-measurable set from them.

Two last comments. One is that this is a famous individual. It is really much to ask that we find appropriate secondary sources Before we make claims about their point of view, and possibly cast a negative shadow on them?

And really, he has a fields medal, I can understand pointing out mathematical errors if they exist. But it kind of looks silly to say That fields medalist is an unreliable source on the subject of mathematics. At least I would be very careful before I was the one who stood up and said it. Thenub314 (talk) 20:10, 21 December 2008 (UTC)

I know, from personal experience, that even Nobel Prize winners, such as Feynmann, make mistakes in areas outside their particular expertise. Feynmann, at least, was willing to admit his errors when he strayed. Feynmann frequented the Math Department lounge at CalTech during the time I was a graduate student there, asking bizarre mathematical (or apparently mathematical) questions. I usually found them interesting, but sometimes meaningless or with an incorrect premise.

As for Wikipedia's including Connes' comments, we would need to verify

1. Exactly what he said,
2. Whether it was published as a peer-reviewed article, and
3. Whether he (or possibly the editor, if it's an article collection and the editor actually reviewed the articles) can be considered an expert in the field (non-standard analysis, rather than Connes' stated field of operator theory).

If neither (2) nor (3) is true, the quote shouldn't appear. — Arthur Rubin (talk) 20:24, 21 December 2008 (UTC)

I disagree entirely. Connes stature is sufficient that were he to criticize some field of mathematics in some reliably published source (not necessarily a peer-reviewed journal), it is certainly worth mentioning in a criticism type article. We might as well say Bill Clinton's quotes about some economic policy shouldn't be included because he is not an economist. Obviously that's not a suitable attitude to take. In fact, I'm not sure Connes "specialty" can be so neatly tied up in a package...he has, after all, his own competing definition of infinitesimal in his noncommutative geometry (indeed, I find this Connes interview where he describes his early background in NSA [3]). Now the question really is, did Connes actually criticize NSA in the way portrayed? The fact that the quotes start off as "Equally deceiving is..." suggests he is "criticizing" a plethora of things. We have no context for why he is doing so. For all we know, he's presenting his own better solution to these issues (alluded to by Mathsci in the AFD), and pointing out deficiencies (or so he thinks) in other approaches. (The Connes interview I found indicates the deficiency is related to use in physics: "...so it seemed utterly doomed to failure to try to use non-standard analysis to do physics.")

There's also a related issue of whether Connes is a "noted critic of NSA". Frankly, this is not the first thing I think off when I hear Connes and NSA in the same breath. If he is such a noted critic, I would like to see some evidence he is. To be a "noted critic" of something is to be far more than someone who has critiqued possible deficiencies on a topic on occasion. The person must be fairly frequently referred to as a critic of the topic. AFAIK, there are no reliably sourced such mentions of Connes as an NSA critic. So there may be a synthesis problem with just using his quotes. As an illustrative example, consider the claim that Serre is a noted critic of the proof of the classification of finite simple groups. Now this is probably a rather strong claim to make, but I can actually find an AMS Notices interview where he is asked about the classification theorem for finite simple groups, and he answers, "For years I have been arguing with group theorists who claimed [the theorem]...had been proved." Even this evidence, for Serre being a noted critic of the proof, can be viewed as tenuous, but it's undeniably stronger than the evidence being provided for the claim about Connes (and Serre's words leave no doubt as to what he thinks, at least). --C S (talk) 22:00, 21 December 2008 (UTC)

It is a dangerous idea that one has to be an acknowledged expert on a particular idea or subfield in order to render an opinion on it that is worth reading. By that train of thought, scientists should not be able to write critical articles on intelligent design, because that is "outside their particular expertise". (Any implicit comparison here between ID and NSA is certainly not my intent!) Or, by your logic, Feynmann's testimony on issues of aerospace engineering and the effect of low temperature on structural integrity of matierals should not count as a reliable source...This is just silly. I think it is a poor bet that Connes is simply mistaken in the quoted passage. Rather, the quoted passage in and of itself is rather obscure, whereas it seems to me that in order to include a critical quote, one needs to first understand its meaning. (If the quoted passage were truly notable and there was disagreement on its meaning, secondary source interpretations would be appropriate.) But I don't get any information from the quote except that NSA is -- in some sense that the quote itself does not make at all precise -- nonconstructive. But (i) we knew that already, or should have; and (ii) why is this a problem? As I said before, we need more information (isn't that the purpose of an encyclopedia, to inform?) and less he-said / she-(= Michele Artigue, so far) said.

Can someone supply a link to the entire article of Connes? Plclark (talk) 00:04, 22 December 2008 (UTC)

(unindent) Here it is on google books. Mathsci (talk) 04:12, 22 December 2008 (UTC)

He is not criticizing non-standard analysis, nor is he criticizing probability. He just seems to be justifying why he needs something different technique for his problem. Do other people agree with this interpretation of this section of his article? Thenub314 (talk) 08:59, 22 December 2008 (UTC)

1. The relevant field for ID is evolutionary biology. The relevant field for NSA is — well, I don't know, but it's not compact operator theory.
2. Feynmann didn't claim to be an expert in the field of structural integrity; and (IIRC) he agreed that his investigation wouldn't have proved anything until verified by structural engineers.
3. When Connes says something which appears false to someone (me) familar with the field (in this case, mathematical logic and metamathematics), his specific expertise has to be indicated. We can't quote me, but we also cannot imply that his statements are mathematically accurate unless it meets the peer-reviewed standard.
4. What Connes seems to be saying is that his infinitesimal formulism has fewer paradoxes than "standard" NSA. It still has paradoxes, in the case of non-commuting "infinitesimals".

I'm late for a dental appointment. See you all later today. — Arthur Rubin (talk) 15:24, 22 December 2008 (UTC)

Robinson did apply NSA to compact operators (see the Academic Press book of Hoegh-Krohn, Albeverio, et al). Also Connes has published papers on NSA; I don't think he can be dismissed in this way. Connes' work (with Henry Moscovici) applies to local index theorems: there are no paradoxes, but their results - related as they are to the Novikov conjecture - are quite deep. Connes is here making a parenthetic side-remark, as a quick reading of his paper shows. On the other hand his side-remark doesn't seem notable enough to be included in this encyclopedia. Mathsci (talk) 15:48, 22 December 2008 (UTC)

I do not find Arthur Rubin's points above to be worthy of debate, except: a cursory check of the literature reveals the following two papers of Connes on NSA.

MR0286645 (44 #3854) Connes, Alain Détermination de modèles minimaux en analyse non standard et application. (French) C. R. Acad. Sci. Paris Sér. A-B 271 1970 A969--A971

MR0286646 (44 #3855) Connes, Alain Ultrapuissances et applications dans le cadre de l'analyse non standard. (French) 1970 Séminaire Choquet: 1969/70, Initiation à l'Analyse Fasc. 1, Exp. 8, 25 pp. Secrétariat mathématique, Paris

By actually looking at his papers, one can see clear relationships between his work on NSA and on operator algebras. If I were Alain Connes, I would not be pleased to find statements like "it may be as an unedited essay by Connes, who is NOT an expert in non-standard analysis" on this webpage. I believe that at least the spirit of WP:BLP applies here: one should be more careful when talking about living people. Plclark (talk) 20:28, 22 December 2008 (UTC)

With a bit of insider information, I was able to locate an additional reference: Brisure de symetrie spontanee etc. from Bourbaki 95/96 and J Geom Phys '97. A quote from math reviews: "Connes has developed in this noncommutative setting the right notion of infinitesimal calculus: a real variable is given by a self-adjoint operator, an infinitesimal is given by a compact operator, teh integral of an infinitesimal, etc." I would not be surprised if some opinions as regards NSA are contained in the article, as well. Katzmik (talk) 20:32, 22 December 2008 (UTC)

It has pointed out to me that there is a map from (positive) infinite hyperintegers to ultrafilters on N, which correspond to non-measurable subsets of the reals. I wouldn't call the map "canonical", but I wouldn't argue.

It also appears that his definition of the (modified) trace requires the Hahn-Banach Theorem, which also demonstrates the existance of objects, where we cannot identify a single one. (At least in the translation pointed to by some of the helpful editors here.) He goes on to prove that which one he uses doesn't matter, exactly as is true of ultrafilters....

I see his criticism as hypocritical, whether or not notable. — Arthur Rubin (talk) 02:15, 26 December 2008 (UTC)

It is still not clear to me he is criticizing. From the article I have read he is simply justifying why he is introducing a new tool. He does never makes any claims that non-measurable sets do not exist, simply they are difficult to get a grasp on. I really think we should take this section out of the article. Thenub314 (talk) 19:48, 26 December 2008 (UTC)

Gillies' Revolutions in mathematics

This book was added as a footnote for Bishop's 1975 remarks about NSA. I couldn't find any mention of a Keisler-Bishop controversy on page 76 of the book. This was a reference to an article by Joseph Dauben, an expert in HPS and biographer of Abraham Robinson, charting "crises in mathematics" from Pythagoras onwards. Mathsci (talk) 04:04, 22 December 2008 (UTC)

Just a quick note: I remember a New Yorker cartoon from the cold war era, picturing a US general attempting to buy a greeting card. The caption reads: "I would like something that reflects the spirit of the season, without neglecting the need for a credible deterrent". I believe many participants in this discussion have reason to take a bit of a pause until after the holidays, could we please adjourn for a little while? I certainly don't have time for any bibliographic searches right now. Katzmik (talk) 13:14, 22 December 2008 (UTC)

I think you're right, Katzmik, we all have wiki-fatigue. Today has been spent santon-hunting, tasting some of the thirteen desserts and working out how to smuggle foie gras into California (actually it is legal). Mathsci (talk) 15:56, 22 December 2008 (UTC)

Brisure de Connes

In his article "Brisure etc" published in refereed journal J. Geom Phys ('97), Connes details his claim above. Namely, on page 211 he writes in elegant French: "the answer, namely a non-standard real, provided by non-standard analysis, is equally disappointing: every non-standard real canonically determines a Lebesgue non-measurable subset of the unit interval [0,1], so that it is impossible [Ste] to exhibit a single such [non-standard number]". Here [Ste] refers to a '85 article by J. Stern. Connes proceeds to assert that "the formalism that we propose will give a substantial and calculable reply to this question". I suppose if we can document the claim that his substantial and calculable reply in reality depends on AC, we will be able to claim that in fact a Fields medalist has made an error. Katzmik (talk) 20:52, 22 December 2008 (UTC)

Why would showing he used AC imply that he made an error? Also, Why are we looking for errors in his work, isn't that a bit beyond the scope of the an encyclopedia? Thenub314 (talk) 22:57, 26 December 2008 (UTC)

Connes does reference Dixmier's '66 paper on traces, but so far I have not found the place in the text where he uses it. Katzmik (talk) 21:02, 22 December 2008 (UTC)

On page 214, Connes outlines certain technical difficulties in using his version of infinitesimals, and then writes: "These two problems are resolved by the trace of Dixmier [Dx] i.e. by the following analysis of the logarithmic divergence of partial traces, etc." It seems as though Connes is using Dixmier's technique in an essential way. Having said this, I do not understand a word of what is going on here, but I have a feeling CSTAR's comment, recently deleted as a result of criticism by mathsci, were correct all along. Katzmik (talk) 21:07, 22 December 2008 (UTC)

traduttore, tradittore

I notice that earlier on this page, Connes' "decevante" was translated as "deceiving". In fact, the French word means "disappointing". Katzmik (talk) 09:18, 23 December 2008 (UTC)

Ce n' était pas moi qui en a fait la traduction.--CSTAR (talk) 21:27, 23 December 2008 (UTC)

Is this article a translation? The book describes it as a lecture and describes Olga Kravchenko as a note taker. Thenub314 (talk) 23:09, 26 December 2008 (UTC)

I originally inserted the material in the non-standard analysis article tel quel from Kravchenko's notes in English. I was never aware that a French original existed. I wonder where Katzmik saw this French version? This uncertainty about the source (for example, did (a) the note taker work from lectures given in French or (b) work from a from draft written in French or (c) were the notes translated by someone other than Kravchenko from her French notes or (d) the lecture itself was in English) seems to provide one more reason to delete this material attributed to Connes entirely. --CSTAR (talk) 04:25, 27 December 2008 (UTC)

I provided detailed references below. The same remark by Connes has appeared in numerous publications and at least two languages. Katzmik (talk) 20:35, 27 December 2008 (UTC)

I am also in favor of removing this section. But if we keep it we should quote another source, in this source there is no evidence a translation occurred, so it doesn't make sense to change the text because we feel it was mistranslated. Thenub314 (talk) 22:58, 27 December 2008 (UTC)

Robinson-Dixmier-Connes

The Dixmier trace (Dixmier's article dates from '66) was exploited by Connes to give a uniform explanation of the pseudodifferential residue of Guillemin and Wodjicki. The basic idea of his theory is to represent an infinitesimal by a compact operator. The exotic traces constructed by Dixmier have the property of being positive on a compact operator. In his article "Brisure de symetrie", Connes claims his solution to be "substantial and calculable", unlike what he claims to be the non-exhibitable nature of the Robinson's infinitesimals. Meanwhile, Dixmier's construction from '66 involves an ultrafilter on the integers.

To elaborate, note that in his '95 article "Noncommutative geometry and reality", Connes gives a detailed account of the role of the Dixmier trace in his theory. On page 6207, Connes states as the goal of section II, to develop a calculus of infinitesimals based on operators in Hilbert space. He proceeds to "explain why the formalism of nonstandard analysis is inadequate". Connes points out the following three aspects of Robinson's hyperreals:

(1) a nonstandard hyperreal "cannot be exhibited" (the reason given being its relation to non-measurable sets);

(2) "the practical use of such a notion is limited to computations in which teh final result is independent of the exact value of the above infinitesimal. This is the way nonstandard analysis and ultraproducts are used [...]".

(3) the hyperreals are commutative.

Connes proceeds to establish a dictionary relating classical and quantum notions. The last quantum item in his dictionary is the Dixmier trace, corresponding to "integrals of infinitesimals of order 1". On pages 6210-6211, Connes presents two technical difficulties with the theory, and states (page 6211): "Both of these problems are resolved by the Dixmier trace". On page 6212, the Dixmier trace Tr_\omega is defined as any limit point of suitable functionals, where "the choice of the limit point is encoded by the index \omega".

On page 6213, Connes states that "for measurable operators T, the value of Tr_\omega(T) is independent of \omega and this common value is the appropriate integral of T in the new calculus".

Such independence would seem to relativize the impact of the objection (2) raised above, affirming precisely a similar independence. Furthermore, to the extent that Dixmier's construction depends on the existence of an ultrafilter, it would appear that the foundational status of both theories is similar (namely, both rely on the axiom of choice), which tends to relativize the objection (1).

Thus the remaining objection (3), namely the non-commutativity of the hyperreals, is the strongest of the three. The remarkable coincidence of dates: Robinson ('66) and Dixmier ('66) suggests that cross-fertilisation of ideas may have taken place. Dixmier's traces are, of course, acting on operators, thus squarely within a non-commutative framework.

On pages 303-308 of his book "Noncommutative geometry", Connes presents a detailed construction of the Dixmier trace. On page 305, he chooses a positive linear form on the vector space of bounded continuous functions on R^*_+, such that L(1)=1, and which is zero on the subspace of functions vanishing at infinity. The construction of the trace is eventually proved to be independent of such a choice. Does the choice of L rely on the Hahn-Banach theorem?

Noncommutative geometry is independent of NSA. The above is pure synthesis. Whoever wrote it (it is unsigned) does not seem to have understood Connes' quest to find an alternative expression for the cyclic cocycle defined by an unbounded Fredholm module as a local formula or residue. At this stage the theory of noncommutative geometry is not well written on wikipedia. Arguing that the above text merits inclusion in a wikipedia article in these circumstances is simply perverse and seems to verge on wilful disruption. Sentence one is fine (it is just a copy and paste from what I have written elsewhere, so of course I'm flattered). Sentence two and three are just wrong, etc, etc. I have no idea why we are being subjected to this regurgitation of completely undigested mathematics. Mathsci (talk) 21:14, 23 December 2008 (UTC)

Well, I was kind of hoping you would help me digest it better. Katzmik (talk) 14:18, 24 December 2008 (UTC)

Section title

Summary of the facts ... Doesn't that sound more like a legal briefing?--CSTAR (talk) 16:59, 23 December 2008 (UTC)

Your edit is fine. Katzmik (talk) 18:11, 23 December 2008 (UTC)

Axiom of choice

The recently added material to the section on Connes comments seems like synthesis to me. It seems to only exist to make the comment that Connes work depends on the axiom of choice. And... ? Who said his work didn't? Why does the axiom of choice have anything to do with Connes comments?

I disagree (to quote a comment above) that "... if we can document the claim that his substantial and calculable reply in reality depends on AC, we will be able to claim that in fact a Fields medalist has made an error."

As was pointed out above he was making a side remark. It was meant to give a brief suggestion thinking of infinitesimals from a probabilistic or non-standard view did not solve the problem and so he is introducing a new notion of infinitesimal.

It is perhaps notable he has been making these remarks for on the order of 15 years. If he was criticizing, why hasn't someone commented (in print) by now? If he had made an error and his results are not calculable why hasn't this been pointed out? Thenub314 (talk) 19:58, 1 January 2009 (UTC)

I think you are making an error of confusing politics and mathematics. I don't know whether or not anybody mentioned his criticisms in print, and why they did or did not. The mathematical facts are: (1) he did make a criticism of NSA, and repeatedly; (2) he made such criticisms in a specific fashion that invites analysis; (3) he proposed the solution involving Dixmier traces; (4) Dixmier traces rely on ultrafilters. If you disagree with any of these facts, we can have a serious discussion. Katzmik (talk) 09:12, 4 January 2009 (UTC)

P.S. I don't think we need to commit ourselves one way or the other, as to whether "a fields medalist has made a logical error". I don't think we should make any such dramatic announcements. However, we should present the facts as best we can. Katzmik (talk) 09:13, 4 January 2009 (UTC)

I certainly don't agree with (1) he often has explained why he is introducing a different notion of infinitesimal, but he was criticizing neither probability nor Non standard analysis. As far as (2) we may certainly analyze his comments if you like, but this section is still being written from a very biased point of view. Even if you don't explicitly say he made an error, the presentation of these "facts" seems like an attempt to suggest to the reader that he did. Yes, he used Diximer traces, yes the general construction involves ultrafilters. What does that have to do with this article? As an aside I disagree that 1) and 2) are not mathematical facts. Thenub314 (talk) 10:53, 4 January 2009 (UTC)

Analyze his comments

Thenub wrote above: "As far as (2) we may certainly analyze his comments if you like, but this section is still being written from a very biased point of view" I listed three concrete criticisms of his in the main article. To the extent that the title of this page is "criticism of non-standard analysis", it would seem legitimate to include criticisms expressed by Connes, explicitly, and in refereed publications. Katzmik (talk) 14:03, 4 January 2009 (UTC)

I think you can explain why a particular mathematical tool doesn't solve a problem without criticizing the tool. Frequently mathematicians make comments like "One can not use [insert some theory] because [insert some reason]" this is not criticism, it is motivation for introducing a new concept. Why is Connes comments criticism and not motivation? Thenub314 (talk) 14:53, 4 January 2009 (UTC)

Well, Connes proceeds to "explain why the formalism of nonstandard analysis is inadequate" (and most likely he is correct). Now if you prefer the term inadequate perhaps we can either write a philological analysis justifying lumping "inadequate" and "criticism" together, or alternatively, rename the page "inadequacy of non-standard analysis". But frankly our time would be spent more productively understanding more of the mathematics involved. Katzmik (talk) 15:30, 4 January 2009 (UTC)

Like I said above, mathematicians often describe why one tool will not work, particularly if they are introducing another. If I read an article that introduces a new notion of infinitesimal, then I would expect some discussion about what is different from the other common notions, and a justification of why there needs to be another. I don't think it makes sense to have an article about "inadequacy", I just don't think these comments deserve inclusion in this article.

Let's talk about the mathematics.

I still would like to know why the insistence on pointing out that the Dixmier trace involves an ultrafilter (and hence the axiom of choice). The implication of the comments above would be that if this were true then Connes made an error. What error does it imply? Thenub314 (talk) 10:04, 5 January 2009 (UTC)

I don't think Connes made an error. Katzmik (talk) 13:05, 5 January 2009 (UTC)

OK, then what purpose does pointing out that Dixmeir trace relies on the axiom of choice serve in this article? Thenub314 (talk) 13:28, 5 January 2009 (UTC)

Bishop's review

The section about the review begins with explaining he quotes Keisler's book, and then we give the quotes. What does this have to do with his criticism. Why are we mentioning this aspect of the review? Thenub314 (talk) 10:06, 5 January 2009 (UTC)

My personal impression is the following. I emphasize the fact that this is my impression, and therefore I am not proposing that this should be included in the article. Having made that clear, my personal impression is that Bishop is so scandalized by what he feels are Keisler's grandiose claims (300 year old problem and the like), that he does not even bother to comment on this. Bishop's review is not particularly well written. He does quote Keisler to provide background for his review. It would be difficult to understand Bishop's remarks on infinitesimals without such background. Katzmik (talk) 13:05, 5 January 2009 (UTC)

I am a bit confused, he does comment on these two quotes. Specifically he says: "No evidence of these claims is given in Keisler's book, but students will not notice that." And again later (after several more quotes) he says "... If not, and his statements are true, the evidence should be somewhere in the book." But you had removed the statement that the points of these quotes was to criticize Keisler for not providing evidence to support them. (Over all, I am in favor of removing them, the section is too much of a "blow by blow" account of an already short review.) Thenub314 (talk) 13:38, 5 January 2009 (UTC)

I think it would be better to quote Bishop rather than synthetically summarizing him. Katzmik (talk) 15:44, 5 January 2009 (UTC)

Material deleted

"In his essay cast in the form of an imaginary dialog between Brouwer and Hilbert,(Bishop 1975), Bishop anchors his foundational stance in a variety of mathematical constructivism. The reviewer for Math Reviews calls into question the coherence of Bishop's particular foundational stance. Be that as it may, Bishop's opposition to infinitesimals, expressed in a scathing review (Bishop '77) of Keisler's textbook, was to be expected (and in fact was anticipated by editor Halmos)." I would appreciate an explanation for Thenub's deletion. Do you feel that all of this material is "synthesis"? Katzmik (talk) 15:42, 5 January 2009 (UTC)

Well, mentioning the easy was cast as an imaginary dialog is not relevant. Nor particularly is the coherence of his foundational stance. And the assertion that Halmos anticipated Bishop's scathing review is not substantiated. Over all the paragraph seemed to be attempting to take Bishop "down a notch" without discussing his criticism. I didn't quite see how to improve it, so I removed it. Thenub314 (talk) 16:20, 5 January 2009 (UTC)

well, I think you should put it back. If Bishop is to be allowed to criticize others, I don't see why others can't criticize Bishop. The fact that mathscinet puts into question his foundational stance is relevant on several counts. The fact that Bishop himself describes his article as an imaginary conversation between the two "titans" is significant, as it places his essay squarely in the context of a foundational controversy, which is the proper perspective to view his comments on NSA, as well. Katzmik (talk) 14:54, 6 January 2009 (UTC)

Well, I have various problems with it. For example, the reviewer's last sentence is:"Constructivism is surely a coherent alternative; whether it is the only such, however, is a matter that waits on further investigation." He doesn't seem to be suggesting Bishop's foundational stance is not coherent. There is not evidence I have seen that Halmos anticipated anything. Bishop did not anywhere give direct opposition to infinitesimals. Thenub314 (talk) 15:56, 6 January 2009 (UTC)

Concerning the comment "If Bishop is to be allowed to criticize others, I don't see why others can't criticize Bishop." Others are certainly allowed to, but not on a wikipedia article. If other people have, in some verifiable form, we can certainly include that (probably in the article on Bishop). But we cannot bring our own criticisms of Bishop into the writing of Wikipedia articles. Thenub314 (talk) 16:25, 6 January 2009 (UTC)

further deletions

The following material was deleted by user:thenub:

In his essay (Bishop 1975), Bishop anchors his foundational stance in a variety of mathematical constructivism. Thus, on pp. 507-508 he writes:

To my mind, it is a major defect of our profession that we refuse to distinguish [...] between integers that are computable and those that are not [...] the distinction between computable and non-computable, or constructive and non-constructive is the source of the most famous disputes in the philosophy of mathematics...

On page 511, Bishop defines a principle (LPO) as the statement that it is possible to search "a sequence of integers to see whether they all vanish", and goes on to characterize the dependence on the LPO as a procedure both Brouwer and Bishop himself reject. Given that a typical construction of Robinson's infinitesimals (see Keisler, p. 911) certainly relies on LPO, Bishop's opposition to infinitesimals, expressed in a scathing review (Bishop '77) of Keisler's textbook, was to be expected (and in fact was anticipated by editor Halmos).

The material was carefully documented and relevant to the present page. Katzmik (talk) 14:05, 6 January 2009 (UTC)

Perhaps, it was. Nonetheless it is synthesis. Which source talks about LPO and Bishop's views about infinitesimals? The paragraph above is drawing new conclusions, not presenting facts detailed in other sources. Thenub314 (talk) 14:42, 6 January 2009 (UTC)

Bishop's views are clearly stated in a paragraph on page 513-514, which I have just added to the page. Katzmik (talk) 14:57, 6 January 2009 (UTC)

I have read these pages. The do not discuss Robinson's construction of the infinitesimals, he doesn't state any explicit opposition to infinitesimals. His review only criticizes Keisler's book. And no support is given to the statement that Halmos anticipated his reaction. I still think the above paragraph is drawing conclusions.

Constructivism

It seems unhelpful, really, just to cut my mention of constructivism in the lead section. After all, Bishop is arguing from a constructivist point of view: this is the locus of his beef with NSA or NSC. Constructivism is very largely not compatible with classical analysis, from the point of view of the general reader. It seems to me a quibble to cut out this mention, when the point of having that in the lead section is to locate where these criticisms lie. By all means amplify that point or qualify it, but the idea of a lead section is to summarise what goes on in the rest of the article, and taking out the context of one main issue seems retrograde. Charles Matthews (talk) 16:11, 6 January 2009 (UTC)

Constructivism in the style of Bishop is explicitly consistent with classical mathematics. He makes clear he is only interested in NSA or NSC in terms of calculus teaching, and hence is not specifically objecting to it because it is not constructive. I do not see any explicit conflict between constructivism and NSA in this article. I see Bishop fiercely criticizing the idea we should teach this in calculus classes. Thenub314 (talk) 18:21, 6 January 2009 (UTC)

Well, I think that's wrong about Bishop. Bishop criticized both non-constructive classical mathematics and intuitionism. He called non-constructive mathematics "a scandal", particularly because of its "deficiency in numerical meaning". What he simply meant was that if you say something exists you ought to be able to produce it, and if you say there is a function which does something on the natural numbers then you ought to be able to produce a machine which calculates it out at each number.[4] In other words, he rejected classical mathematics on constructive grounds. It's not that relevant that he worked within the language of classical analysis. Charles Matthews (talk) 19:54, 6 January 2009 (UTC)

The quote is one hundred precent correct. I had previously made had a comment in the article that pointed out he rejected all classical mathematics including NSA (it has since been taken out). I have a problem if the article tries to state he had a specific problem with NSA. As far as I can find references for he did not, he did have a specific problem with its use in teaching calculus. And he generally lumped it together with all classical mathematics. Unlike intuitionism, all theorems proved under Bishops system are classically true. I interpreted your sentence in the lead to suggest otherwise, which is why I removed that part. Thenub314 (talk) 20:09, 6 January 2009 (UTC)

When I quoted Bishop concerning his principle LPO, I did not mean to say that he has any more specific problem with NSA as compared to other ZFC material such as standard mathematics (I think this may be a better term than classical mathematics, as the latter is less specific). Still, it is clear as day that you cannot do NSA without LPO, and that's why I quoted Bishop concerning LPO. This material has been deleted. I think we should either reinstate this material, or add the quotation from Feferman cited above, so that our discussion of Bishop's constructivism would be more specific. Katzmik (talk) 10:52, 7 January 2009 (UTC)

It is clear as day (to us) that the standard definitions of NSA require LPO. The subject of "Constructive nonstandard analysis" exists, so there is clearly some compromises that can be made so NSA can be done constructively. But all of this is beside the point. Bishop did not try to connect LPO to NSA, and none of the references do either, so we should not make the connection and explain it to the reader, that would be synthesis. Thenub314 (talk) 11:08, 7 January 2009 (UTC)

None reference is incorrect, as Charles pointed out. If you read Feferman's comment carefully, you will notice that he is merely spelling out in words what Bishop called LPO. Katzmik (talk) 11:13, 7 January 2009 (UTC)

Well it is inappropriate for us to say he rejected Keisler's book and NSA because of LPO. None of the reference said that. Feferman's reference doesn't claim this. Feferman's reference doesn't mention NSA, Bishops review of Keisler's book, or reference the "Crisis in math" paper. We can point out he rejected classical mathematics including NSA, I had done that at some point. Let me try to explain what I have a problem with. Consider the sentence "Given that a typical construction of Robinson's infinitesimals (see Keisler, p. 911) certainly relies on LPO, Bishop's opposition to infinitesimals [...] was to be expected." It is

• a) Not clear Bishop new the first thing about the construction of Robinson's infinitesimals, and if he did it was most likely not the version given in Keisler's book.
• b) Not clear Bishop had any opposition to infinitesimals. He was careful to qualify his remarks so that he was only speaking about their use in the classroom.

This sentence and takes two facts, Bishop's rejection of LPO, a typical construction in NSA, and combines them to come up with the idea Bishop would have opposed infinitesimals. Perhaps it would have been true, but I have not read anywhere that he did. This is why I feel it is synthesis.

Tall wrote explicitly that the non-standard approach was severely criticized by E.Bishop. This is why I feel you are splitting hairs here. Katzmik (talk) 15:30, 7 January 2009 (UTC)

The papers by Tall only bring up Bishop in reference to using non-standard analysis as an educational tool. I don't think that I am splitting hairs. Thenub314 (talk) 15:56, 7 January 2009 (UTC)

Tall explicitly mentions the axiom of choice as the culprit as far as Bishop is concerned. We see therefore that according to secondary literature, Bishop's objections are foundational in nature. I would appreciate it if other editors interested in the field expressed their opinion as to the synthesis/OR allegations being pursued here. Katzmik (talk) 17:12, 7 January 2009 (UTC)

The subject of "Constructive nonstandard analysis" exists, so there is clearly some compromises that can be made so NSA can be done constructively

The comment reproduced in the subject/headline was made by Thenub. I think there is a misunderstanding here. ISF (internal set theory) of Nelson has sometimes been described as a constructive version of NSA. I invite editors more knowledgeable than myself to comment, but it seems to me that such a description is a bit tongue-in-cheek. All it means is that Nelson minimizes the foundational material exploited in the theory, limiting it to ZFC without certain axioms that may or may not have been used by Robinson (I don't have enough technical knowledge of the subject to be more precise here). At any rate, Nelson's theory is certainly taking place in ZFC not ZF. Moreover, even ZF is taking place in a Cantorian paradise of set theory which is totally unacceptable to Bishop as it goes far beyond his LPO. In his '75 article he describes it as tasting the forbidden fruit. The idea that there is anything controversial about the claim that Bishop opposes ANY version of NSA on foundational grounds is mere obstructionism. Katzmik (talk) 16:31, 8 January 2009 (UTC)

IN what sense does anyone claim Internal Set Theory is constructive? Since IST is a conservative extension of ZFC in whatever sense IST is constructive, ZFC is as well. --CSTAR (talk) 17:58, 8 January 2009 (UTC)

I was completely serious in my comment. Two random examples of papers in this direction would be:

• E. PALMGREN, A constructive approach to nonstandard analysis, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 297-325.
• I. MOERDIJK, A model for intuitionistic nonstandard arithmetic, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 37-51.

I can only claim a vauge understanding gleaned from some survey articles I have looked at, but I stand by the statement that is the subject of this section. No humor was intended at all. Thenub314 (talk) 20:35, 8 January 2009 (UTC)

Palmgren's paper does indeed propose some form of nonstandard analysis, with a transfer principle even, but it is definitely not IST. Moreover, Keisler's approach to NSA in his calculus book and his work on stochastic processes is definitely not along these constructivist lines. To make this claim is hardly OR. Whether or not Bishop would object to the constructive versions of NSA we will never know, short of engineering his resurrection. But again to suggest that Bishop was dead set against NSA as developed by Robinson hardly seems synthesis.--CSTAR (talk) 21:10, 8 January 2009 (UTC)

I agree it would not be OR it say that Keisler's approach to NSA was not constructive. And I never meant to imply otherwise. I do feel like it is synthesis to say Bishop was dead set against NSA because (quoting from the synthesis section) the sources cited do not explicitly reach the same conclusion. I agree it seems very plausible he was, but I am unable to verify it. Trying to verify is how I stumbled upon the fact the people do NSA within Bishop's framework. Thenub314 (talk) 22:58, 8 January 2009 (UTC)

I tend to agree with Thenub314 that the claim that Bishop opposed infinitesimals is synthesis and in fact could be completely wrong. For example, the transcendental extension Q[t] is an ordered field and 1/t is a positive infinitesimal in it: this example is more relevant than the constructive nonstandard analysis of Palmgren and Moerdijk, since it is well known to any student that studied van der Waerden.

However I don't think the right claim is that Bishop opposed infinitesimals. The relevant claim is that Bishop rejected Robinson's nonconstructive approach to NSA. OK I don't have the relevant Bishop quote "This stuff sucks" but I would be surprised if some reliable and verifiable claim that he had entertained this view doesn't turn up.--CSTAR (talk) 02:54, 9 January 2009 (UTC)

I am not against pointing out Bishop rejected Robinson's nonconstructive approach to NSA. I would like to put this into some context that makes it clear he rejected all nonconstructive mathematics. If we do this well, I do not think we need a "smoking gun." Previously I had put in the sentence the statement "The view of Errett Bishop, most of mathematics (including non-standard analysis) is done by "formal finesse."

I think that in veiw of Feferman's quote I think that we can strengthen this a lot more to something like: "In the view of Errett Bishop non-constructive mathematics, which includes Robinson's approach to nonstandard analysis, was a scandal that was deficient in numerical meaning (cite Feferman)." How do people feel about this statement? Thenub314 (talk) 07:44, 9 January 2009 (UTC)

CSTAR wrote above: "IN what sense does anyone claim Internal Set Theory is constructive? Since IST is a conservative extension of ZFC in whatever sense IST is constructive, ZFC is as well. --CSTAR (talk) 17:58, 8 January 2009 (UTC)" To reply, I was merely quoting Arthur Rubin, who wrote to this effect a few months ago. Katzmik (talk) 11:39, 11 January 2009 (UTC)

I understood you were referring to someone else's comment, but I am surprised Arthur Rubin said that unless he meant something else. Also, the proposal above by Thenub314 seems reasonable to me. --CSTAR (talk) 18:08, 11 January 2009 (UTC)

Why no mention of (topological) completeness?

A significant benefit of the standard reals is that they form a complete ordered field, in fact the only such. The Fundamental theorem of algebra and the Jordan curve theorem are just two of many theorems that depend on completeness of the real line and the complex plane for most if not all of their proofs. The nonstandard real line is perforated throughout with gaps through which a curve might pass without hitting any point of the line, e.g. between the infinitesimals and the positive standard reals, compromising theorems depending on continuity and connectedness. Yet I have never seen this raised as a potential weakness of NSA. Why does this not bother people?

In "A Nonstandard Proof of the Fundamental Theorem of Algebra," AMM 112:8 705-712 (Oct. 2005), George Leibman offers an NSA proof of FTA. From a technical standpoint it is competently executed, but from a pedagogical standpoint it is a prime example of mathematics made difficult. He starts with the nonstandard complex numbers as a field (for the sake of the arithmetic), but then turns off most of the nonstandardness by retracting all the finite nonstandard complex numbers to their standard values (for the sake of defining roots of polynomials) and retracting the infinite complex numbers to a circle at infinity that remains connected to the finite part. This is accomplished via a topology manufactured to that end. The only substantive difference from a standard proof is that the circle at infinity permits speaking of the value of a polynomial p(z) actually at infinity instead of as |z| tends to infinity. All the values computed by the field operations are nonstandard, but for the purpose of comparing them for equality they are treated as standard (except for the circle at infinity). However the artificial topology making them standard enters into the argument in a way that requires the reader to understand in detail how the topology succeeds in turning off the nonstandardness, which while not rocket science is nevertheless tedious to follow unless one simply ignores it and assumes the finite part is standard to begin with.

The upshot is an essentially standard proof that would be easy to follow were it not cluttered up with the arcane machinery of a nonstandard topology whose sole purpose to make the nonstandard complex plane work like the standard plane except for the circle at infinity, while contributing negligibly to the key ideas of the proof. If the price of NSA is to greatly obscure what should have been straightforward proofs of theorems that depend on the topological completeness of the real line and the complex plane, then at least for those theorems it seems far preferable, from the standpoint of pedagogy, taste, and common sense, to formulate the real line and the complex plane standardly. NSA may have benefits elsewhere, but treating it as a panacea is wishful thinking. --Vaughan Pratt (talk) 11:13, 5 March 2009 (UTC)

If you know of a good secondary source that criticizes non-standard reals for being incomplete, then we should include this criticism. While I understand your point, incorporating it into the article at this point (I believe), we have been attempting to keep the article to criticisms that have appeared in print somewhere. Probably this criticism has, but I don't off hand know where to find it.

Agreed. It would be great if this objection could be found in print somewhere. If not then maybe a little note somewhere with a title like "Yet another nail in the nonstandard-analysis coffin" might be in order. --Vaughan Pratt (talk) 21:23, 6 March 2009 (UTC)

I have am a stickler for references. If it is not in print in a reliable source, it shouldn't be on Wikipedia. When I have first started editing I didn't realize this, but I have since had to delete facts I knew to be true, because they are not recorded in print anywhere. It makes it very difficult to record on Wikipedia some of the rich and interesting mathematical about the personalities behind the theorems. Like, why did Langlands choose his thesis advisor? (and what did his thesis have to do with the functional analysis texts produced by Dunford and Schwartz?) Thenub314 (talk) 10:46, 7 March 2009 (UTC)

It's easy to find a reference that shows the hyperreals are not complete. What you're not likely to find is a reference that shows that this is widely considered to be a problem with the hyperreals. There are plenty of valid reasons to object to the hyperreals (e.g., the strongly nonconstructive nature of most treatments), but this just isn't one of them.--76.167.77.165 (talk) 21:01, 14 March 2009 (UTC)

On a minor technical point it is not correct to say that the reals are the only complete ordered field, if you take complete to mean that Cauchy sequences converge. There is a construction in Gelbaum and Olmstead's "Counter examples in analysis" of another such field. This doesn't influence your point though. —Preceding unsigned comment added by Thenub314 (talk • contribs) 08:06, 6 March 2009

Yes, except that Gelbaum and Olmsted write, following Definition III in Chapter I, "Any two complete ordered fields F and F' are isomorphic." Maybe Atlantic states like NY and Scotland define things differently, but in Pacific states like California (Gelbaum founded the math dept. at UC Irvine) and Australia, "complete" and "Cauchy-complete" are distinct concepts in the context of non-Archimedean ordered fields. G & O are careful to state their example as being for Cauchy-complete ordered fields. "Complete" means that every nonempty set bounded from above has a sup, and in any non-Archimedean ordered field, whether or not Cauchy-complete, the infinitesimals (x^-1, x^-2, … in the usual ordering of the G&O example) have no sup while the positive rationals have no inf, as obtains in both NSA and the G&O example. --Vaughan Pratt (talk) 21:23, 6 March 2009 (UTC)

Fair enough. But I don't think there is any geographic aspect to comment. I am not from Scotland, so can't say how things here are usually defined (nor have I mathematically visited NY). When you used the phrase "completeness of the real line and the complex plane" I assumed you meant Cauchy-completeness. It was not clear the term completeness was to mean two different notions for the two different objects.

It is a pedagogical point that when people say "complete" the almost invariably mean Cauchy completeness. (As I assume you mean when speaking of C.) This tends to make people forget that you need to take a different notion of completeness if you want have the statement "R is the unique complete ordered field." I meant my comment as a friendly "Did you know..." and not as any lack of respect for a more accomplished mathematician then myself. Thenub314 (talk) 10:46, 7 March 2009 (UTC)

I think you're a little off base in complaining that the hyperreals aren't complete. When you look at definitions and theorems in an NSA context, some are of the type that you clearly want to generalize from the reals to the hyperreals, but others are not. For example, a compact set B on the real line is most naturally defined, in an NSA context, as one whose hyperreal version *B has the property that any x in *B has a well defined standard part st x, and st x is in B. This is a definition that's talking about the properties of the reals from the "outside," so it's clearly not one that you would even *want* to generalize to the hyperreals. Completeness is the same way. It's also worth noting that the lack of completeness doesn't show up in typical applications. For example, consider the question of whether the graph of y=x^2 ever cuts the graph of y=2. In the rationals, the answer is no. In the reals the answer is yes. That's a clearcut, important difference between the reals and the rationals. What about the hyperreals? Well, actually the two graphs do cut each other in the hyperreal plane, and that's not a coincidence; the fact that the two equations have a simultaneous solution is a statement to which the transfer principle applies. Here's another way of putting it; if the hyperreals didn't have *any* properties that were different from the properties of the reals, there would be no reason for studying them. Re the proof of the fundamental theorem of algebra, there's a boring technical part, which is the part that establishes that a polynomial can't have a zero that's at an asymptote or a point that's missing from the domain or something; and then there's the actual meat of the proof. The technical part seems to be what you're referring to, but it's going to be kind of boring and technical in any approach.--76.167.77.165 (talk) 01:43, 14 March 2009 (UTC)