The Interesting Number Paradox

http://en.wikipedia.org/wiki/Interesting_number_paradox

by chaosmachine • 17 years ago

65 9 17 years ago

en.wikipedia.org

spuz• 17 years ago

Does the same apply to HN submissions? ;)

serhei• 17 years ago

If it did, you would be forced to classify spam and old Valleywag articles as "interesting".

mixmax• 17 years ago

The theory also applies to trees, ropes and pipes.

roundsquare• 17 years ago

The set in question needs some ordering (at least for the proof to easily cross over).

Though, I suppose we can use "number of molecules" and almost always get away with it.

DanielBMarkham• 17 years ago

This and the hanging prisoner paradox in the afternoon.

Must be epistemology night on HN :)

michael_nielsen• 17 years ago

The Wikipedia Paradox: what's the most notable subject that's not notable enough for Wikipedia?

Now, one could reasonably argue that merely being the answer to this question would not, by itself, be enough to make a subject notable. However, the act of arguing this point, if seriously engaged in by enough people, would, in fact, make the question (and the answer) notable, and thus the answer to the question would deserve inclusion in Wikipedia.

Inductively, you can use this argument to show every subject is notable enough for inclusion in Wikipedia. Take that, deletionists!

(Inspired by a lunchtime conversation with a group of physicists and philosophers.)

mkyc• 17 years ago

Were you among the physicists? :) Notability is not an intrinsic property of articles: it's like weight, not like mass. Your argument fails at the "if seriously engaged in by enough people" point, since it's simply false (by definition) that at time T1 the first uninteresting thing is sufficiently popular, at T1. It's also false that people will a) notice and b) care enough to make it sufficiently popular even at time T2.

Same goes for numbers. The fallacy is in confusing intrinsic and extrinsic properties.

mayanks• 17 years ago

sounds like a catch 22 kinda situation to me. Or for that matter Heisenberg uncertainty principle.

Dove• 17 years ago

Persuasive enough, mathematically. Practically, though, people wouldn't allow the definition to recurse like that.

Consider a hypothetical person filling in descriptions for all the interesting numbers. "Smallest prime", "Smallest odd prime", "Smallest square" . . . He would get to the smallest uninteresting number and write, "Smallest uninteresting number." Then when he got to the next one, he'd start to write that, realize he already had one, and ponder for a bit.

The divergence occurs here.

If he was a mathematician, he would shout, "Aha! There are no uninteresting numbers!" If particularly diligent, he would go on to write "Was the smallest uninteresting number until that made it interesting" on all the other numbers.

If he was a normal person, he'd criticize the definition. Yes, he does think the smallest uninteresting number is interesting, but he doesn't think the next one is. So the technical definition shouldn't be recursive, and that keeps the initial annotation from being so self-defeating. He'd then go back and write, "Smallest number which would otherwise be uninteresting" on the first one, and leave the rest alone.

I side with the normal person. A definition which allows so many numbers with the same description to be considered "interesting" doesn't actually correspond with what I think interesting means. I think it is reasonable to require that the definition of interesting be stable regardless of what order you evaluate the integers in, so a definition that allows for this sort of nonsense should be modified until it doesn't.

The Interesting Number Paradox

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