I'm fairly certain that this answer http://mathoverflow.net/questions/40082/why-do-we-teach-calculus-students-the-derivative-as-a-limit/53042#53042 was posted over the weekend and received several down-votes; it seems that Misha has simply deleted the downvoted version and reposted an identical answer. Am I right about this, and if so, what's the appropriate thing to do?

A little heads up: Misha has been known to get into long wrangles about this sort of thing (as signaled by his referring to "foes"). It seems to be almost like a religious thing with him. My advice would be not to engage him in argument at all.

And now the answer has been posted for a third time. It seems like a situation in which the moderators will want to contact the user.

To be clear: I was not suggesting that the moderators' involvement should be disciplinary. I just meant that we seem to have entered a periodic cycle of downvoting, locking/deletion and reposting, and that this is a waste of people's time (including the responder). So some intervention seems helpful.

Let me also say that I am sincerely interested in the ideas expressed in the responder's post, although I don't yet understand them very well and feel that, perhaps, they are not being expressed very clearly. In any case, in that it is a program for doing calculus without limits, it is 100% on target as an answer to the question.

[@Yemon: I have no idea what you're talking about. But that's okay with me if it's okay with you. :)]

@Pete: I warned you! But I should have said, "don't engage him at all, unless you like getting into long drawn-out interchanges that go nowhere," because this is the (predictable) course it would take. (Misha has a mathematics PhD, from Brown University if I remember correctly, and I think he may have written a few papers. But as I say, this business to do away with limits is something of a religious cause with him, and he is ready to attack all comers-on. It's strange, and a little sad.)

@nielkj: actually, my impression of this thread is that he didn't do that: the posts were deleted by a moderator (see Qiaochu's response above). So in so far as the originally stated topic of this discussion is concerned, that much has already been settled.

I haven't really followed the discussion very closely, but I thought I might point out that it is not at all impossible for one person to be totally brilliant and to behave like a complete bastard at the same time. In fact, it seems to be disturbingly common. Perhaps being brilliant and misunderstood can push a person towards the nasty end of the behaviour spectrum, which certainly is unlikely to improve the “misunderstood” part.

@Qiaochu: yes, it's the same guy.

@nielkj: I personally am not "worried" whether Misha has a PhD, but since he seems not to be very capable of having a civil conversation about his chosen topic, I thought it might be good to counterbalance any impression that he is a <i>complete flake</i> with the observation that he does have a substantial mathematical background. I don't see that registering that data point is in any way elitist -- it is consonant with what you wrote in the last sentence of your first comment here.

The interactions I had with him on Armstrong's blog did not, however, suggest that he was in any way a deep thinker; what I saw was a great deal of sloppiness. Are you seriously comparing him to Gromov?

@Harry: there are many literary critics, and I wouldn't paint them all with the same tarbrush. Perhaps you are thinking of a certain type or school of literary critic?

It sounded to me that nielkj was suggesting that the mathematicians there were just about as fawning in their attentions to said critic as the paparazzi, in other words that we mathematician-types are equally taken in by superficial trappings of reputation. (It's perhaps a little ironic to be talking about reputation here however!)

@nielkj: we are mostly in agreement then.

However, Misha/Michael, whatever his employment status, doesn't do himself any good carrying on as he does. He is really, as Pete found out, insulting to his interlocutors.

Seconding what Hailong has said; and seconding Andy's admonition to Harry. (I'm resisting the temptation to argue the rights and wrongs of Harry's/Hardy's claims, because that's not really the issue; the issue is that it just derails this thread.)

Harry, since you seem to hold documentary film in high regard, let me quote from the documentary "Man on Fire", in which Christopher Walken has this to say: "A man can be. an artist. at anything. Food, whatever. Creasy's art is literary criticism. And death. He's about to paint his masterpiece." Presumably your blanket denunciation of literary critics does not include Creasy? Because he will come for you.

"Promotion of obscurantism" -- you know, some people say that about higher topos theorists, too.

Yeah, but those people don't know what they're talking about. First of all, everything is as precisely defined, and the underlying ideas are easy to grok. As usual, the devil is in the details (the underlying ideas are far simpler than the underlying ideas in algebraic geometry, for instance (of course not that I'm implying that algebraic geometers are obscurantists either (quite the opposite, in fact))).

If that still isn't convincing, I suggest that such people take a look at Lurie's DAG I, where he applies the general theory to show that the fundamental category of a stable $\infty$-category (the definition of this is extremely elegant) is a triangulated category, where the shift autoequivalence (whatever you call it) (resp. its inverse) is induced by the natural suspension functor (resp. loop functor).

The answer and its comment thread have degenerated into something as discussion-y as things come---predictably...

Maybe it is time it were moved to a blog or email or...?

Now, is a public discussion of controversial issues appropriate for MO?

I'll say "no". Not because I don't think that these discussions are important, but actually because I think that they are important enough to have somewhere where they can actually be easily followed and participated in than stuffed in the comments on a question that most people won't look at.

Discussions and MO just don't fit well together!

I rescinded one of my downvotes because, as jbl says, the answer has now at least been fleshed out, and the ratio of detail to "denunciation" has improved.

I'm also inclined to agree with Andrew, but have to admit that if this discussion were taking place on ML's own blog then I would be even less likely to read it. The fact I do occasionally check the discussion going on in that comment thread is more to do with the calibre of the "defenders of the orthodoxy" than with it belonging on MO.

@nielkj: He has definitely been combative with me. All I was doing was pressing him for details, while continually making clear that I had no "skepticism" towards his ideas whatsoever and only wanted to understand them better. He did not answer any of my questions. He did send me an unsolicited private email in which he gave me some further links -- which I appreciate -- but the rest of the email was rather insulting. In particular, he seems to have decided that I don't know much analysis based upon the fact that I cannot supply a proof that the Weierstrass Approximation Theorem allows one to extend the derivative from the class of polynomials to all continuously differentiable functions. But come on -- the WAT says no such thing. Is this a major concern of mine with his work? Not at all -- it seems to simply be an offhand remark that a mathematician made that turned out not to be true. If you follow me on this website, you'll find plenty of instances of that sort of thing coming from me. But his reaction was frustrating: it discouraged me from further engaging him.

On the other hand, let me say that I did spend some time reading some of the materials on his webpage. His short "textbook" had the most details -- including some details that were not supplied on my requests here -- and I think there is definitely some interesting mathematical content there. What he has written is incomplete by any standard -- there are, for instance, whole blank sections in the text -- and it is way underexplained from the pedagogical side (I.e., given that we agree that most freshman calculus students actually get very little out of the formal definitions of limits and derivatives, is there any reason to believe that replacing one technical foundation they ignore with another technical foundation they will mostly ignore is going to make any significant difference?). But as a mathematician I definitely enjoyed parts of his work and acquired insight from them. When I have the time I would like to take some notes on it and think harder about how or why to incorporate some of these ideas into my teaching (probably not at the calculus level, but that's okay).

I feel the need for some concluding paragraph summing up my take on the situation, but I guess it's pretty clear that I find it rather strange and have very mixed feelings about it. That will have to do for now.