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.
]]>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
]]>]]>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.
]]>“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.”
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.
]]>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?
]]>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
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