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Hi reimundo,
I think it's generally OK to do whatever you like if it's not adversely affecting someone else, and under that rubrick I think everyone should feel welcome to distribute their upvotes as they see fit, even following quite peculiar schemes! (In particular, I think that the more upvotes are handed out, the better the site will be, even if people are deciding how to upvote in strange ways.)
Another way of saying this is that I'm happiest when someone decides to use up their 30 vote quota every day, and how they decide to allocate those is second-order.
But an individual of course needs a rule to decide what to upvote, and I think a very good rule of thumb is simply to ask "Would I like to see more of this sort of question here?" As we know, opinions vary on this question, but the median and distribution of these opinions across the user population seem to be pretty fortunate.
best, Scott
Shevek: could you please explain me what precisely is asked in the question under discussion?
quid: As with many MO questions, if you don't understand, it might be out of your area of expertise. Noam Elkies and Andreas Blass both gave very insightful comments; I believe I understand what is asked and I could give an answer if the question is reopened. In my mind, this is a rather deep question on the decidability of equality between natural numbers, and there are several (well documented) mathematical perspectives on the subject...
Reimundo: Personally I have nothing against your vote. I would not upvote a question I consider as unclear, after all the description is 'useful and clear' for an upvote. But I understand what you mean and to me this is a minor detail. And, if I may say so, I consider it as unnecessary that Igor Rivin implictly complained about the upvote. Sorry, for ignoring your request, but I need to reply to François.
François: First, right when I saw this question, I was essentially sure something like this would happen. This happens so often with this type of question.
It is true that I am not an expert on this, but I am and always was aware that this could be a subtle question. And I commented the following, and the spirit of the very first comment of Igor Rivin, which I upvoted before voting to close, is the same. [Indeed, I wrote my own 'first closing comment' but deleted it when I saw a similar one; only when I saw the first vote to reopen I decided that perhaps Igor was too brief with his objection, so I elaborated.]
My comment:
I think it would be good if before the question is considered for reopening somebody makes precise in the question what 'constructed' and alike should mean in this question. At the moment the usage seems to be vague/informal (which is my issue with the question)
Now Andreas Blass's comment after this reads, replying to Darij.
@darij: There are several constructively inequivalent notions of "finite". One requires exact knowledge of the number as you said. Another only requires a surjection (not bijection) from a set of known finite size. The two definitions would disagree about whether Noam Elkies's example is known to be finite-dimensional.
Hmm, it seems to use 'finite' without explanation is indeed vague in that context. Now, one could argue that the question only makes sense for one of these two notions, and so 'obviously' this one is meant.
True, I am no expert, but I know this is vague. Perhaps the questioner is an expert and was just a bit informal in the formulation, yet could also be even less expert than I am. Might it not be good to know this to write the answer at the right level for the person asking it?
quid, this is an issue that touches foundations more than other areas of mathematics because of its peculiar status at the bottom of it all. Foundations questions arise in all sorts of contexts and they are (rightly) almost always strangely formulated. Logicians everywhere regularly get foundational questions from colleagues and others, and they are nearly always formulated in this form: I've been dealing with this kind of situation in my work, but I've been in doubt, could you tell me why I shouldn't be worried about this? (The word 'why' is frequently replaced by 'that', but let's not worry about such linguistic quibbles.) This question is not much different. In fact, it is very precise in asking about a specific mathematical topic (dimension of vector spaces). Perhaps this is misleading, in that the problem is probably not really about the existence of the dimension of a vector spaces in general, but rather about a specific vector space of unknown dimension. However, the specific circumstances are irrelevant. The basic question remains: why does the fact that every finite dimensional vector space has a dimension allow us to compare that dimension with another natural number (or even the dimension of another vector space, to remain in the same type of data)? This is a hard question, and there is probably no better way to ask it for a "regular" mathematician.
François, you say typically the regular mathematicians you mention give some motivation. "I've been dealing with this kind of situation in my work..." In the present question there is no motivation. So then let us wait for that? Also, why the dimension of vector spaces? Couldn't one maybe start with that question for the cardinality of finite sets? If I were to ask such a question, I might say something on this. Even if I am not an expert.
Let me say it very directly.
This question is not well written. As it lacks motivation and context, and for several other reasons.
Quite likely it is asked in a naive spirit.
In particular, any "regular" mathemtician that thinks about this problem for a while will come up with a counter example to a naive interpretation of this question. Therefore, if I were the questioner, I would at least give an example showing what type of examples I do not want, assuling I want something else.
In my opinion, however you interpret this, it is either a 'sloppy' or a very naive question. Neither is to be encourage.
quid, I also don't think its a good question, I thought you were just asking if the question was understandable at all.
This discussion made me remember Borges's ornithological argument for the existence of God...
In case people don't remember Borges's argumentum orthinologicum...
“Cierro los ojos y veo una bandada de pájaros. La visión dura un segundo o acaso menos; no sé cuántos pájaros vi.
¿Era definido o indefinido su número? El problema involucra el de la existencia de Dios.
Si Dios existe, el número es definido, porque Dios sabe cuántos pájaros vi.
Si Dios no existe, el número es indefinido, porque nadie pudo llevar la cuenta.
En tal caso, vi menos de diez pájaros (digamos) y más de uno, pero no vi nueve, ocho, siete, seis, cinco, cuatro, tres o dos.
Vi un número entre diez y uno, que no es nueve, ocho, siete, seis, cinco, etcétera.
Ese número entero es inconcebible; ergo, Dios existe.”
And here's an English translation (by Mildred Boyer):
I close my eyes and see a flock of birds. The vision lasts a second or perhaps less; I don’t know how many birds I saw. Were they a definite or an indefinite number? This problem involves the question of the existence of God. If God exists, the number is definite, because how many birds I saw is known to God. If God does not exist, the number is indefinite, because nobody was able to take count. In this case, I saw fewer than ten birds (let’s say) and more than one; but I did not see nine, eight, seven, six, five, four, three, or two birds. I saw a number between ten and one, but not nine, eight, seven, six, five, etc. That number, as a whole number, is inconceivable; ergo, God exists.
I don't understand Borges' argument. God doesn't need to exist, just someone who was reading your mind when you imagined those birds. It seems he has only proved that Professor X exists.
«It seems he has only proved that Professor X exists» is one of the funniest things I've read here :)
voloch, A parody?! I don't believe it. Next you'll tell me Swift didn't really want to eat Irish babies. Mariano, glad to entertain!
@HJRW (and Mildred Boyer): Good translation! Couldn't have done it better myself! Cheers.
@voloch: A quick google search reveals numerous earnest discussions of the validity of Borges' argument. Apparently Borges' own delivery was also too deadpan for many.
voloch, No worries! On the internet, everyone is deadpan.
Following Theo's request, I made a first attempt at reformulating the question. Any suggestions?
If MathOverflow and its community were A) disposed toward discussing issues in philosophy of foundations of mathematics, or B) willing to answer in Wikipedia style (like those of Joel Hamkins, but with less brevity and more breadth), or C) likely to answer basic undergraduate questions in the subject, or one of a few other reasons which I won't list here, then I would say the question should be opened.
The question is basically a vector-space version of an old chestnut in philosophy of foundations: when is something supposed to be treated as existing? I would use the traditional version about e + the truth value in C of some proposition like Fermat's last theorem, suitably stated; that particular example has lost its punch, but I have faith in the reader coming up with their own version.
I do not see this question as adding anything interesting. If it were stated as a reference request for where an example first appeared, that would be suitable for MathOverflow, but I think either the question should stay closed, or there instead should be a formal reversal on community standards on at least one of A,B, or C listed above.
Gerhard "Ask Me About System Design" Paseman, 2011.10.15
@François: Thank you for trying to reformulate the question. At present, my mathematical philosophy says that the answer to the question at the end is "yes," and that there's not much more interesting to say.
If an expert in foundations makes a compelling case that as written, the question is of, say, graduate-student level, then I will happily vote to reopen it.
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