Talk:Interesting number paradox

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[edit] Analysis

(Copied from 1729 number discussion page - the comment seems more pertinent here)

a) Should the statement that a number is interesting because it is the smallest uninteresting number be treated as an axiom? Acceptance or rejection of the axiom would be a matter of individual choice. Xenoglossophobe

Agreed, so I disagree with the artcile's statement that
there would be a smallest number with no interesting properties (for instance, 38 could be a candidate). This in itself would be an interesting property of the number, so it would no longer be dull.
—msh210 18:20, 18 Nov 2004 (UTC)

b) As the application of this putative axiom to the smallest uninteresting number leads to its removal from the list of uninteresting numbers and places the next lowest uninteresting number in a similar position that also requires its removal (and so on, ad infinitum), should this be regarded as proof by mathematical induction rather than proof by contradiction? Xenoglossophobe

I wouldn't call it an axiom, it's more a definition. It could be made into a proof by induction, but it is currently expressed as proof by contradiction. "Say we wish to prove proposition p. The procedure is to show that assuming 'not p' (i.e. that p is false) leads to a logical contradiction. Thus p cannot be false, and must therefore be true." P is "there is no smallest uninteresting number." We assume not p (there is a smallest uninteresting number), and show the contradiction. Therefore p. A proof by induction would work completely differently. anthony (this comment is a work in progress and may change without prior notice) 21:46, 29 Mar 2004 (UTC)
The suggestion is made based on the following understanding of the prinicple of mathematical induction: induction requires a) the demonstration of some property of the first item in a list (which, I agree, has has been claimed in this particular case using contradiction) and b) the demonstration of a chain of implication in which the presence of this property in a general item in the list implies the presence of the same property in the next item, such that the implication extends to every member in the list, even one of infinite length. Xenoglossophobe
All cases have been shown by contradiction. If there is no smallest uninteresting number, then there must not be any uninteresting number. I'd say that step is obvious. I guess you could form an inductive proof for it, though. anthony (this comment is a work in progress and may change without prior notice) 22:20, 29 Mar 2004 (UTC)

[edit] Kudo

Most hilarious Wikipedia article I've ever read. Bravi, tutti! Ventura 23:46, 2004 Oct 14 (UTC)

[edit] Resolvable?

Removed:

This paradox is, however, resolvable. It may be noted that in this example 38 is the infimum for the set of dull numbers: while not dull itself, there are no dull numbers less than 38.

This is not a resolution of paradox. It is a treachery, done by substituting the definition. What kind of logical fallacy is this? Mikkalai 23:56, 26 Oct 2004 (UTC)


[edit] Cleanup notes

(text moved here from Wikipedia:Cleanup/October)

  • Interesting number paradox a mathematician has to rewrite this article away from its sensationalist, rather than encyclopedic style. Some statements are plain false if under scrutiny (I've already removed the most evident one). Mikkalai 00:03, 27 Oct 2004 (UTC)

Added {{cleanup}} tag, but IANAM. --Jim Henry 20:11, 15 Feb 2005 (UTC)

[edit] Why is this page so serious?

This page is written as if the Interesting number paradox is a serious mathematical concept which actually proves something. It's meant to be fun. People reading this artcile will get the impression that there are no uniteresting numbers and that this has been proven mathematically!--Heathcliff 23:35, 17 May 2005 (UTC)

It isn't a paradox! It's a proof that there are no dull elements of a well-ordered set.--SurrealWarrior 02:37, 29 July 2005 (UTC)

Yes, well... it is a "paradox" in an intuitive sense that "we all know that numbers can't all be interesting," but we cannot find the smallest dull one -- perhaps, SurrealWarrior, it's a proof that the natural numbers are not a well ordered set? (ha ha, just joshing). Anyway, the tone is a bit too stony-faced: an encyclopedia article should let the reader in on the joke. I think Martin Gardner came up with the idea, and the great man often wrote tongue-in-cheek about perfectly serious mathematics. This paradox is semi-serious. Since the idea of "interesting" is so very subjective, this will never be really serious; and yet the self-referential nature of the paradox follows in the footsteps of many paradoxes that rely on naive set theory. And to paraphrase Martin Gardner, this last sentence of my comment is totally uninteresting, so I say don't even bother to read it. :-) --LandruBek 05:26, 24 April 2006 (UTC)

[edit] Encyclopedias are not joke books

Note: added tone template--Keerllston 11:43, 8 December 2007 (UTC)

[edit] Regarding "human knowledge is countable and finite"

What could possibly be the justification for such a strong claim? I propose it be removed until justified. 17:09, 10 October 2005 (UTC)

Under normal assumtions about the way the Universe works humans cannot have infinite amounts of knowledge. You cannot learn an infinite amount of things, because that would generate an infinite amount of heat and require an infinite amount of energy. Tbjablin 22:31, 14 November 2005 (UTC)
I feel you presume too much, largely through fuzzy definitions. If "human knowledge" is meant to be interpreted as some abstract list of facts determined to date then it is of course countable but I believe this is discretizing/modelling what we actually mean by "human knowledge". I would concede this latter thing is naturally bounded but strictly uncountably infinite.
If human knowledge does not mean the union of the set of things known by each human, I don't know what else that term could mean. Also, even if human knowledge were infinite, I think it still would be countable. The set of all strings in any alphabet is coutable. Tbjablin 12:27, 19 December 2005 (UTC)

[edit] Surreal numbers

Similarly, I removed the mention of surreal numbers. I don't see how they could have anything to do with the interesting number paradox. There are sets smaller than the surreals that are not well-ordered. Isomorphic 22:48, 13 November 2005 (UTC)

[edit] 197.3341 is a singularly bad example of an uninteresting real number

because it is one of the very few real numbers that have been used throughout history to illustrate the concept of an uninteresting real number. Dmharvey 03:17, 12 March 2006 (UTC)

I think the problem here is that any number given as an example of an uninteresting number would in itself be interesting, by virtue of being singled out as an example of an uninteresting number. When I read 197.3341 as an example of an uninteresting number, I immediately assumed it was the author's intent that this, in itself, be a paradox. BGreeNZ 04:28, 11 April 2006 (UTC)

[edit] Smallest uninteresting numbers in Wikipedia

As of 30-May-2006:

  • The smallest integer in Wikipedia without its own page was 201
  • The smallest integer about which there was no comment apart from its factors was 237
  • The smallest integer not mentioned was 1004
Uh...203.184.25.49 08:32, 31 January 2007 (UTC)

[edit] Does not prove that all numbers are interesting

Ok, suppose 38 is the smallest uninteresting number, but by virtue of being the smallest uninteresting number, it is therefore interesting. Fine. And then 39 is the smallest uninteresting number, so it becomes interesting as well... um... what about 38? If 39 is the smallest uninteresting number, 38 no longer is, and so 38 is no longer interesting. But then it becomes the smallest uninteresting number again, so 39 goes out and 38 comes back in. You can't have two smallest uninteresting numbers, so only one number can be made interesting for this reason.

So there's a paradox, yes, but this paradox stops the chain dead in its tracks. 38 and 39 will keep switching places, but this roadblock prevents the logic from being extended to 40, 41, 42, etc. So there is a major problem with trying to say that this proves that all numbers are interesting. Argyrios 17:36, 4 June 2006 (UTC)

I think it would be quite interesting if both 38 and 39 were the smallest uninteresting number. Dmharvey 02:18, 5 June 2006 (UTC)
lol. Sure, "if," but the point is they can't both be. There can only be one "smallest uninteresting number," so you can't say that every uninteresting number is actually interesting by virtue of being the smallest. Or, you know what, let's just say for the sake of argument that you can have multiple "smallest uninteresting numbers." Now there is a different problem: You can't just forget about why you made the previous numbers interesting. If you have this whole smorgasbord of numbers that are supposedly all interesting because they are all the smallest uninteresting number, then it's no longer particularly special to be the smallest uninteresting number, is it?
I tend to regard the smallest uninteresting number as an honorary degree which didn't involve passing any exams. MaxEnt 21:02, 15 June 2006 (UTC)
If 38 and 39 are in some kind of quantum state of uncertainty over being the smallest uninteresting number (like Schrödinger's cat is both living and dead at once), then perhaps that would make 40 interesting as the lowest definitively uninteresting number not part of this fog of uncertainty... but making it so brings it into the fog itself, and then you can extend this by induction upward. *Dan T.* 14:21, 12 May 2007 (UTC)

Flaw who says the smallest uninteresting number is interesting on that basis?--Keerllston 11:45, 8 December 2007 (UTC)

[edit] Origin

I'm trying to track down the source of this paradox, but I haven't had any luck yet. Chris Caldwell[1] traces it back to a letter of G. G. Berry (0000206 in the Russell Archives of McMaster University) and calls it Berry's paradox: "You will often find Berry's paradox stated as 'every integer is interesting.'". I think Berry's paradox is usually understood to be the proof about the smallest number definable in less than X words/letters/keystrokes, though, and from a quick Google search it appears the letter is about this paradox.

The article suggests that Gardner may be the origin of this paradox. Any thoughts? CRGreathouse (t | c) 23:29, 16 September 2006 (UTC)

I read it in a Swedish book at least 20 years ago. I don't know it if was a Swedish original or an translation. The paradox was that there are not an least interesting integer. The book was about popular mathematics and if I remember correctly that story was about 2 mathematicians that made a cab ride and the number of the cab was dull. I just found that number, its 1729 as mentioned in this discussion page http://en.wikipedia.org/wiki/1729_%28number%29 . I guess that the paradox was written in association with that story. —The preceding unsigned comment was added by 81.231.32.208 (talk) 15:44, 21 April 2007 (UTC).

[edit] Dull numbers

Is the last paragraph about dull numbers WP:OR? I haven't heard about them before, and the reasoning about uninteresting properties is rather contrived. You might certainly argue that unknown properties are uninteresting rather than interesting - besides the previous paragraph attempted to formalize the "interesting"-predicate as a finite list of properties, which contradicts the assertion that there should be an infinite number of interesting properties.

Unless someone objects, I will remove the paragraph in a few days. Rasmus (talk) 12:42, 9 January 2007 (UTC)

No comments, so I removed the paragraph. Anyone interested can read it here. Rasmus (talk) 13:02, 12 January 2007 (UTC)

Hey, I made a completely logical, true, and relevant edit and it was removed because it wasn't sourced. Can I please put it back in? --74.134.8.244 02:34, 28 May 2007 (UTC)

That depends. If you have a source for it, yes. If not, the best thing to do would be to mention it here on the Talk page so someone can find a source. CRGreathouse (t | c) 05:51, 28 May 2007 (UTC)