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  1.  
    • CommentAuthorRon Maimon
    • CommentTimeApr 29th 2011
     
    I gave a question two days ago, about differentiating with respect to pi. It was closed as "unanswerable". I still
    don't know if that was because the answer was "obviously no" or "obviously yes", or a mix of readers holding these
    two opinions.

    The people who closed it suggested I come here and ask about reopening.

    Here is the link:
    http://mathoverflow.net/questions/63127/is-this-joke-unfunny-closed

    The question is whether there is a consistent set of rules within a model of the reals (think ZF--- it doesn't
    seem to me to matter how strong the system is) to differentiate with respect to a sufficiently generic
    transcendental constant symbols (I chose pi, although that might not be sufficiently generic), so that finite
    additivity/chain rule/product rule hold, the derivative of any rational multiple of pi with respect to pi is that
    rational number, and the answer is zero for real numbers whose definition is independent of pi.

    In the question, I gave one way of explicitly doing this for the restricted class of elementary functions
    applied to "pi" and "e". If you make sure that $exp(i\pi)$ is constant, you find the derivative of e w.r.t
    pi, and then all the normal trigonometric functions get a derivative.

    I was wondering if the model theory literature understood this. It is asked to make rigorous the following
    classical statement, essentially dating back to Archimedes:

    "The volume of a sphere depends on pi, but the volume of the intersection of two cylinders does not."

    One way of asking this in a less ideosyncratic way is as a question about automorphisms of models of ZF.
    Is pi rigid under all of these(no way)? Is there a continuous parameter of such automorphisms that translate pi(yes)?
    Are there constants whose change is linear in the change in pi (yes)? I believe the answer is yes, I was wondering
    who had done this.

    Perhaps the answer is "nobody did this.", or "it is impossible" But I will accept that too, so long as there is some
    indication of what the appropriate literature is, if there is any. I googled automorphisms of models of the reals,
    but I didn't find anything related.
  2.  

    I voted to reopen. This seems like a sensible question, even if I think the answer is just "pi is rigid."

  3.  

    I also think the question is fine. Note though that titling it "Is this joke unfunny?" is not really helping readers see that it is a serious math question.

    • CommentAuthoralias
    • CommentTimeApr 29th 2011 edited
     
    ..
    • CommentAuthorAnixx
    • CommentTimeApr 29th 2011
     
    This question has only two answers: either pi is interpreted as constant, then the derivative is infinite or it is interpreted as a variable, in this case there should be partial derivative. In any case this question does not belong to MO. This is a question about understanding differentiation. In one of the comments the asker said that there are no definite rules of differentiation which points that he does not have understanding of elementary calculus.
    • CommentAuthorEmerton
    • CommentTimeApr 29th 2011
     

    Dear Ron,

    I voted to reopen the question. But I would also suggest that you change the title of the question, and edit the body of the question so that the information/motivation described in your post here is present in the actual question. (Note that you can edit the question even though it is closed, and the closure of a question is in fact an invitation to edit the question into a better form which you can then argue should be reopened.)

    Regards,

    Matthew

    • CommentAuthorAnixx
    • CommentTimeApr 29th 2011 edited
     
    ..
    • CommentAuthorMariano
    • CommentTimeApr 29th 2011 edited
     

    The question «does automorphisms of models of ZF fix \pi?» is interesting, and that is how I understood it when I first read it at the time of its posting. The title could be changed, and the cuteness toned down, surely. Andreas and, in a comment, Joel have actually answered this question (it is really nice to have such a nice bunch of logicians around!)

    On the other hand, I think that the question if we can actually differentiate with respect to \pi, on the other hand, is rather off topic and should be asked in some other forum---its answer should be well-known to people doing research at the math level: one differentiates functions with respect to its arguments. Anixx's answer, which again is showing his/her great ability to gather downvotes, is answering this reading of the question, and, I think, is as off-topic as the reading of the question it answers.

  9.  
    Mariano: I honestly don't know how such a question would be interesting. Andreas's answer addresses the problem. Joel's suggestion leads to the to the tables-vs.-beer-mugs issue, expressed in technical terms. The point is: There are no *internal* automorphisms, but the question is about an internal property of pi. And *external* automorphisms, if any, would not change any internal properties.
    • CommentAuthorMariano
    • CommentTimeApr 29th 2011 edited
     

    Andrés:

    At some point one realises that one cannot really tell i from -i, that there is a choice involved in talking about "i". It is natural to wonder what other choices there are. The answer may well be uniteresting, of course.

    (I did have in mind external automorphisms)

    (Do models exist of ZF with interesting automorphisms? Do they do anything interesting?)

  11.  
    Mariano:

    That's a good question, actually. The answer would be a clear yes if instead of ZF we were looking at PA.

    I can say a couple of things that are a bit technical.

    For models of ZF, there are methods to create models with automorphisms; for example, by ensuring that the ordinals of the model are non-rigid, we can obtain models of V=L (or, more generally, of V=HOD) with automorphisms (an automorphism of the ordinals extends to an automorphism of V if V=HOD, since every set is definable from ordinals), and we can tweak these constructions in technically interesting ways.

    As of whether they have interesting *applications*, I confess I do not know. They do, for the model theory of ZF, but I'm drawing a blank on applications to ZF proper. The expert in these matters is Ali Enayat, I think. A recent result of his is that if T is a completion of an appropriate extension of ZF+V=HOD by large cardinals (call it ZF'), then there is a model M of T with an elementary extension N such that there is an automorphism j of N with M as its class of fixed points. Moreover, the theory ZF' is in a sense weakest with this property.

    I would think there ought to be applications of these ideas to determinacy at the low levels of the projective hierarchy, but I would have to think about this more carefully.

    For ZFC, the "right" notion is internal: Rather than automorphisms, we look at elementary embeddings. These maps are ubiquitous in set theory, both for the theory of large cardinals, and for the development of what we call fine structure. (Still, a result of Kunen is that there are no internal elementary embeddings j:V --> V, in the presence of choice.)
    • CommentAuthorRon Maimon
    • CommentTimeApr 30th 2011
     
    I don't understand why this question is generating so much hatred. Somebody said "I don't have enough middle fingers for this question", then it was closed as "spam". I genuinely want to know the answer, I have absolutely no stake in it either way, I am just curious about how much freedom you have in models of reals, and I wanted some literature pointers into this field, and there are people here who are experts. Andreas killed the obvious idea, but I was hoping to go think about the answer, see if it resolves the question once and for all, and if so, accept it as the final word. I am sorry the question is generating annoying responses which are not adressing the issues, but I wasn't expecting it to be nebulous--- I genuinely believed this sort of thing had been discussed somewhere before. I don't necessarily want it reopened, but I am not sure that I am happy with the idea that I am responsible for spam.
  13.  

    I think your question would have been taken much more seriously if you had left out all mention of jokes and the extensive formal manipulations with pi. If you had just asked the mathematical question about models of the reals (possibly with a brief example using pi) or even "is there a way to interpret the notion of dependence on the value of pi", you might have received a warmer reception. As you can see, some people seized on your motivating example as an opportunity to display their prowess with formal manipulations, and perhaps a more focused question would have made it more clear that such answers were not sought.

    I think the answer to your question may depend a lot on context. If you are given a polynomial in pi in a setting where you are only doing polynomial manipulations, you can treat pi as you would any abstract transcendental, and derivatives make sense. If your formula occurs in a setting where you are (implicitly or explicitly) using periods or analytic properties of the real line, then the question of taking derivatives becomes more delicate (and outside my expertise). My bold guess is that you should consider working in a system that is substantially weaker than ZF.

    • CommentAuthorRon Maimon
    • CommentTimeApr 30th 2011
     
    The question is serious, and while going through all the rigamarole on this forum, I found the answer (I wrote it on the page). This is not spam, it is not unanswerable, and I am unhappy that people were so hostile. The question is motivated by the observation that a randomly chosen real number has no special properties. It will never have the property that E(x)=F(x) for two different elementary functions E and F, so it should be possible to fiddle with x in a model and all the values of model-functions of x to keep everything ok. I was wondering how far you could go for non-generic constants, in particular, could you use a very special number, like pi, in place of a generic number like "x". The answer is no for pi, although I am not sure about other constants. Is there any way that this question can be identified as non-spam? I don't like being thought of as a spam-monger.
  15.  
    Ron. Let me _stress_ that the question being "closed as spam" most definitely does _not_ mean that the question was spam. This is a "feature" of the software we are using here. When one votes to close a question one is _forced_ to choose one of a small set of "reasons" and in many cases none of these reasons apply, so people often just choose a random one and then the system gets a multitude of different "reasons", none of which are accurate, and just picks the last one. In your case I think the question was "closed because it was generating a lot of nonsense which kept making it be bumped to the front page". Please do not think that anyone is accusing you of writing spam; this is just a defect of the system (which we cannot, unfortunately, change, because we cannot edit the software itself).

    Let me also absolutely encourage you to start again with another question, if you still feel there are things that need to be said. Give that other question up for dead; the internet just sometimes doesn't work very well and there was a lot of nonsense being posted on the original question that had absolutely nothing to do with your question. Although in most cases people whose questions are closed are encouraged not to start another thread about the same question, perhaps in this case there is a case for an exception to be made, because the question wasn't closed because it was intrinsically poor/inappropriate for this site, but because of other people's comments. Just as Scott said, if you ask again and make your title some precise technical statement about models of the real numbers then the people who want to divide by zero will not even read the thread, but the logicians will, and you will find a much more pleasant and welcoming environment here.

    As I think someone else implicitly said, I think that in retrospect was that your original choice of title for the question was something that happened to work strongly against the question. This was perhaps not something that could have been guessed at the time, but another feature of this site is that titles can be, and often are, very long and technical. On the other hand, pithy titles clearly attract a lot more attention, not all of it wanted on this occasion. In some sense the first version of the title was a disaster and the second still wasn't precise enough to get rid of the "trolls" in the sense that it admitted many interpretations that were very far from what was intended.

    So don't give up on us, but try again with a long and technical and precise title that can admit no possible "high school math" interpretation, and see what happens. You might be surprised.
    • CommentAuthorgilkalai
    • CommentTimeMay 1st 2011
     
    Chosing to close the question as "spam" in the second round after it was closed an opened is inappropriate. (Of course, not all the people who voted to close it chose "spam," "span" won by plurality.) In any case, I propose that the moderators will change the wording.
  17.  
    But I don't think the moderators cannot change the wording because the wording is part of the program and the moderators cannot change the program. The reasons for votes to close have time and again been criticized---see other threads here---but there is nothing the community or the moderators can do about this. "spam" might have only won because there was a tie and "spam" was mentioned last.
  18.  
    Kevin Buzzard, it seems to me you misunderstood Gil Kalai's suggestion (at least I understand it differently, and what I understand should be doable, which in turn suggests to me that this is his suggestion).

    A concern of Ron Mainon is that the question was closed as 'spam,' and this is also prominently visible on the question.
    Now, one could keep the question closed but change the reason 'spam' for which it got closed to the milder reasons 'not a real question' (used the first time) or perhaps even better 'too localized.' And, this (at least I strongly believe so) could be done by a moderator. I do not know whether directly, but at least indirectly by reopening it and then reclosing it with this different reason, it should be possible. (If this were done the visible reason would be a milder one, and this would also reassure the questioner that it is not really considered as 'spam'.)

    So, I believe Gil Kalai's suggestion is to change the reason for closure for this particular question to another one from the fixed list.
  19.  
    I understood and accepted the "spam" business, and I have no further complaints. I might suggest (if this is not impossible) that you have a category "not of general interest", as opposed to "closed" which will prevent questions which annoy people from coming up prominently on the front page.

    I also don't like the idea that there are different tiers of mathematics, with some being "high school mathematics" and others being "advanced mathematics". I asked this question because I was genuinely confused about it, and I am still genuinely confused about it, and I still don't know if there is a contradiction even just from high school mathematics (that would be great, but I didn't give one). I will not reopen the question, because I don't think I am going to get anywhere with it here. The theorem about automorphisms of models of reals quoted by Andreas is sufficient as an entry point into the model theory literature, and I will (if I have time) read about model theory, and see if there is a less open ended way to phrase the question that isn't immediately wrong (like the automorphisms business). Until then, I will leave it alone.
    • CommentAuthorMariano
    • CommentTimeMay 1st 2011
     

    Ron said:

    I also don't like the idea that there are different tiers of mathematics, with some being "high school mathematics" and others being "advanced mathematics".

    While you may not like it, it is a fact of life.

    There is a difference between high school mathematics, and research mathematics. The difference shows up in the subjects the two involve, in the point of view each takes, in the emphases each puts on different things, and in the group of people which deal with each, among other things. Commingling everything under the title of "Math" will not help either subset, and will rather introduce problems.

  21.  
    Certainly a moderator could change the reason for closure by reopening it themselves and immediately reclosing it with a new reason.
  22.  

    While this is technically possible, I don't think it's appropriate. While the original list of people who voted to close would remain visible in the history, most people won't know this (or how to find it), and would only see a moderator unilaterally closing the question.

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