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Frivolous theorem of arithmetic on Wikipedia

by Icebreaker
Tags: arithmetic, frivolous, solved, theorem, wikipedia
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arildno
#19
Apr10-05, 06:16 AM
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Perhaps Wikipedia needs a section of "The most false theorems" as well:
Here's mine:
The primes are closed under multiplication..
dextercioby
#20
Apr10-05, 09:03 AM
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Zurtex,[itex] 1+1=2 [/itex] is an equality,not an equation...


Daniel.
master_coda
#21
Apr10-05, 11:37 AM
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Quote Quote by Zurtex
Oh and please do prove for me rigoursly that 1+1=2, but that's an equation not really a theorem.
1+1=2 is much more of a theorem that the frivolous one being discussed. It's a provably true mathematical statement. "Almost all natural numbers are very, very, very large" is not a mathematical statement, and so it certainly isn't a real theorem.

But the real problem with the page is not that the theorem is useless, it's that the page is. The page provides no useful information at all; it doesn't explain why the theorem is true or what its significance is. If you already know the theorem then the page doesn't tell you anything new or interesting. And if you don't know the theorem then you aren't going to learn anything from the useless remarks on the page.
Icebreaker
#22
Apr10-05, 01:19 PM
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The page can be updated, modified; information can be added. Why don't you go add something to the page instead of deleting it altogether?
shmoe
#23
Apr10-05, 02:06 PM
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Quote Quote by master_coda
"Almost all natural numbers are very, very, very large" is not a mathematical statement, and so it certainly isn't a real theorem.
I've always looked at this frivilous theorem as an amusing (but crude and imprecise) summary of "For every real number M almost all natural numbers are larger than M". 'Almost all' being defined in the usual asymptotic sense-if P(x) is the set of naturals less than x that are larger than M, then P(x) is asymptotic to x as x goes to infinity. I admit I'm easily amused though.

Quote Quote by master_coda
But the real problem with the page is not that the theorem is useless, it's that the page is. The page provides no useful information at all; it doesn't explain why the theorem is true or what its significance is.
This is definitely true. A more precise statement, especially how we would try to quantify 'very large' and 'almost all', would be ideal in the currently non-existant body of the article as well as an explanation of practical implications. I'm not sure if there's any interesting folklore behind this, but if there is it would be a nice addition as well.
Hurkyl
#24
Apr10-05, 02:20 PM
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There's an interesting interpretation of this theorem in the context of a nonstandard model:

Most natural numbers are external -- here, that means they're bigger than any natural number we can "get" to. (i.e. as long as we stay in the word of natural numbers, doing anything imaginable with internal numbers will never produce an external number)
Zurtex
#25
Apr10-05, 03:21 PM
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Quote Quote by dextercioby
Zurtex,[itex] 1+1=2 [/itex] is an equality,not an equation...


Daniel.
Very true, my miswording sorry.
master_coda
#26
Apr10-05, 03:31 PM
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Quote Quote by Icebreaker
The page can be updated, modified; information can be added. Why don't you go add something to the page instead of deleting it altogether?
Because I don't care about this theorem enough to spend time trying to produce a worthwhile article about it. Of course, I haven't deleted it either; I just think it should be deleted, since the current article is nothing but noise. It should be up to the people who actually think the theorem is interesting to produce good content describing it.


Quote Quote by shmoe
This is definitely true. A more precise statement, especially how we would try to quantify 'very large' and 'almost all', would be ideal in the currently non-existant body of the article as well as an explanation of practical implications. I'm not sure if there's any interesting folklore behind this, but if there is it would be a nice addition as well.
I don't think there's anything wrong with the fact that this isn't a "real theorem" (although a more mathematical restatement of it would be ideal). I just thought it was strange that someone would tell me that 1+1=2 isn't a theorem in defence of something else which is very clearly not a theorem.
Zurtex
#27
Apr10-05, 04:34 PM
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Quote Quote by master_coda
1+1=2 is much more of a theorem that the frivolous one being discussed. It's a provably true mathematical statement. "Almost all natural numbers are very, very, very large" is not a mathematical statement, and so it certainly isn't a real theorem.

But the real problem with the page is not that the theorem is useless, it's that the page is. The page provides no useful information at all; it doesn't explain why the theorem is true or what its significance is. If you already know the theorem then the page doesn't tell you anything new or interesting. And if you don't know the theorem then you aren't going to learn anything from the useless remarks on the page.
I didn't realise there was scale of theromness, please tell us how this scale works so I know how to order my theorems from now on in their level theoremality. I would certainly like to know which theorems are more of theorems than your average theorem.

I found the equivalent MathWorld page quite enlightening when I came across it.
master_coda
#28
Apr10-05, 06:01 PM
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Quote Quote by Zurtex
I found the equivalent MathWorld page quite enlightening when I came across it.
What new insight into mathematics (or anything else) did you get from finding this theorem?
Icebreaker
#29
Apr10-05, 06:18 PM
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Are you implying that the theorem is not useful?
Hurkyl
#30
Apr10-05, 06:20 PM
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I put this theorem into the class of things everybody knows, but few actually realize until they have it pointed out to them. There are lots of such facts... for instance, once I saw the example of a nonzero function whose taylor series was zero, it took me several years before it dawned on me that most infinitely differentiable functions are not analytic. But, if at any point, someone had stated that fact to me, my reaction would've been "Oh right, that's obvious!"
master_coda
#31
Apr10-05, 06:33 PM
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Quote Quote by Icebreaker
Are you implying that the theorem is not useful?
No, I think the page is not useful. It could probably be made useful by adding more content. But all it has right now is filler. For example, the "one of the more interesting theorems of mathematics" comment could be replaced by text that actually explains why the theorem is supposed to be interesting.
Zurtex
#32
Apr11-05, 04:50 PM
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Yeah yeah I know, some times I take things too seriously, I've edited the front page, please edit anything I've wrote if you can think of something better.
Icebreaker
#33
Apr14-05, 12:33 PM
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I remember it being proven by someone "jokingly" a while back on one of the threads. That proof is much more simple, iirc.


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