Talk:Ugly duckling theorem
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Yes, this is pretty much a copy of the page on Everything2 with the same name. I can do that: I wrote that one. Probably needs some formalizing and formatting. Clsn 18:15, 9 May 2006 (UTC)
wtf?
Blueaster 08:16, 24 October 2006 (UTC)
References and wrong content
[edit]Please add some print references to this article, not just a link to another page you have written.
- Link added to Watanabe's book, or at least a scan of the relevant part of it, which is what I could find. Maybe my statement of it is original, but the research isn't.
The sentence you added again is clearly wrong; there must be additional assumptions on the theorem that says that the set of attributes representing each object has a 50/50 chance of containing any particular attribute. CMummert 03:51, 9 January 2007 (UTC)
the German version
[edit]I find a quite detailed article on this topic in the German wikipedia: de:Ugly-Duckling-Theorem, but I don't understand German. So it would be great if someone could translate that one into English. Took 02:45, 5 March 2007 (UTC)
- It would be nice if someone could either translate that one or fix it, as the mathematics is wrong. — Arthur Rubin (talk) 22:19, 3 May 2008 (UTC)
Dead link
[edit]http://www.igm.hokudai.ac.jp/crg/download_files/watanabe.pdf —Preceding unsigned comment added by 212.198.66.165 (talk) 20:12, 25 September 2010 (UTC)
"if they are only distinguished by their names"
[edit]This statement is blatantly false and ruins the entire purpose of the ugly duckling theorem. It stands to reason, I believe, that this should be changed to "if each can be distinguished by a unique name" or something similar. Regardless of how many extra distinguishing features there are, the total number of possible classes will not change if the names are unique, which is the entire point, if I understand this theorem correctly.
I will let someone with a better understanding of the theorem rephrase that statement. — Preceding unsigned comment added by 213.125.208.5 (talk) 12:48, 28 June 2013 (UTC)
Capitalisation
[edit]The article uses "Ugly duckling theorem", "Ugly Duckling theorem", and "Ugly Duckling Theorem". This should be unified. 1234qwer1234qwer4 (talk) 22:49, 9 June 2020 (UTC)
Done I changed all occurrences to "Ugly duckling theorem", as in the article title. However, the title of Anderson's tale is still capitalized as "The Ugly Duckling"; I hope this is ok. - Jochen Burghardt (talk) 07:11, 10 June 2020 (UTC)
- Thanks. Yeah, I thought it would make sense to write "Duckling" with a capital letter, but a web search returned mostly occurrences with all letters capitalised (probably in headings) or with only the first of them (I think). There is still the possibility to write "ugly" with a lower-case u though, as the article title doesn't tell how to capitalise the first word. 1234qwer1234qwer4 (talk) 08:27, 10 June 2020 (UTC)
- @Jochen Burghardt: Apparently, you didn't, but I fixed the rest now. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 12:56, 24 December 2020 (UTC)
- @1234qwer1234qwer4: I made an attempt (on 10 Jun), but apparently missed some occurrences. Thanks for fixing that! - Jochen Burghardt (talk) 16:16, 24 December 2020 (UTC)
- @Jochen Burghardt: The choice to only capitalize "Ugly" does not make any sense. Either the theorem uses "Ugly Duckling" as a proper name or both "ugly" and "duckling" should be lower case. In general, Wikipedia prefers to avoid unnecessary capitalization, per MOS:CAPS and more specifically MOS:SCIMATH. Lester Mobley (talk) 03:34, 28 April 2021 (UTC)
- @Lester Mobley: I guess you are aware that the name refers to Anderson's fairy tale "The ugly duckling". I am not sure how this is to be handled per MOS:CAPS. If you are, I won't object to your suggestions. When I reverted Trappist the monk's recent edit, I just wanted to hint at this discussion. In case any of you decides to change the capitalization, please do it consistently throughout the whole article, the I'll be fine with it. - Jochen Burghardt (talk) 12:08, 28 April 2021 (UTC)
- For the record, I have edited this article exactly once and at this writing, have not been reverted. My edit had nothing to do with this uppercase-lower-case squabble about which I have no opinion. Please leave me out of it.
- —Trappist the monk (talk) 12:46, 28 April 2021 (UTC)
- Oops, sorry, the edit I meant was by JayBeeEll. - Jochen Burghardt (talk) 18:06, 28 April 2021 (UTC)
- Although I am not that well-versed in the manual of style, I think that in line with WP's general preference for lower case and the naming of theorems on Wikipedia (for example, as JBL mentioned below, "hairy ball"), I would pick "the ugly duckling theorem"; in this capitalization, the theorem can be referring to the ugly duckling from the story, rather than the title of the story. Since you are fine with a consistent change of capitalization and JBL supports lower case, I will go ahead and make it all lower case. Sorry if I created a "squabble". Lester Mobley (talk) 19:41, 28 April 2021 (UTC)
- I agree with Lester Mobley: the current situation is very weird, and "the ugly duckling theorem" is consistent with both how other theorems are called on Wikipedia ("the hairy ball theorem" etc.) and with common sense. (Currently capitalization is not standard in the article.) --JBL (talk) 18:15, 28 April 2021 (UTC)
- @Lester Mobley: I guess you are aware that the name refers to Anderson's fairy tale "The ugly duckling". I am not sure how this is to be handled per MOS:CAPS. If you are, I won't object to your suggestions. When I reverted Trappist the monk's recent edit, I just wanted to hint at this discussion. In case any of you decides to change the capitalization, please do it consistently throughout the whole article, the I'll be fine with it. - Jochen Burghardt (talk) 12:08, 28 April 2021 (UTC)
Countable set of objects to which the Ugly duckling theorem applies
[edit]This discussion has been closed. Please do not modify it. |
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The following discussion has been closed. Please do not modify it. |
That is true that subtraction is undefined for limit ordinals. It is moreover true that addition and multiplication of ordinals is non-commutative. Therefore why would one introduce ordinals, let alone limit ordinals to the Ugly duckling theorem? The author of this profound theorem clearly considered natural numbers: "We start with a certain number of predicates" (cf. ER p. 6). The term certain number is clearly non-negative integer. "An object satisfies or negates each starting predicate, therefore it corresponds to an atom [atomic predicate]." (cf. ER p. 6). Then he shows that "the number of predicates of rank that includes given two atoms (...) is independent of which atoms are (...) given" (cf. ER p. 7), which conclusion is the Theorem of the Ugly Duckling. Derivation of this theorem in terms of predicates and their ranks allowed Watanabe to additionally consider implicational constraints among the predicates, the existence of which reduces the number of the predicates. But the same conclusion can be reached in a simpler way if (as in this article) one considers all the possible ways of making sets out of the objects. For three objects it gives the list illustrated below the picture with 2 swans and the duckling which clearly shows that any two birds share 4 classes out of 8 available. That remains true for any objects. There is a close connection of such an articulation of this theorem with Cantor's diagonal argument, but it nevertheless remains valid for any countably infinite set of objects in the universe.Guswen (talk) 16:13, 15 April 2021 (UTC)
Yes. is a natural number and that is why it is countably infinite, as the set of natural numbers is countably infinite (). There is just one version of this theorem, clearly for countably infinite (any countably finite is but a subset of the countably infinite ). Similar theorems for countably infinite numbers are plenty. Take, for example this paper about Laplacians. Fact, here is taken to denote the dimensionality of a continuous Euclidean space but I do not see a reason to exclude that this could not be countably infinite. Take the combinatorial proof of the Boltzmann’s H-theorem. It is not based on tangible physics. Just a mathematical model of absolutely elastic molecules (non-dissipative mathematical points) "colliding" with each other. Take any theorem that employs/uses, at least in part/is based on a concept of "a certain number". "Certain numbers" are simply countably infinite. They do not even remotely resemble a fallen Christmass Tree of ordinals. Guswen (talk) 21:34, 15 April 2021 (UTC)
The definition of the successor establishes countable infiniteness () of natural numbers . There is no finite limit as . Therefore, indeed, each natural number is finite but they are, as a set, countably infinite. In my opinion this theorem applies both to any finite , as well as to itself, even though the latter is formally undefined (or rather unknown).Guswen (talk) 10:40, 16 April 2021 (UTC)
Well, I think the definition of a natural number (or ) is precisely (rigorously) defined by Peano axioms of arithmetic. Namely it is any number greater than or equal to zero, such that differs from the preceding and/or ("or" applies solely to 0) succeeding number by one (successor of zero). And that is all that is required for the proof of the Ugly duckling theorem. Guswen (talk) 08:22, 18 April 2021 (UTC) I have just found a citation to support countably infinite number of objects in this theorem. Obviously the definition of a natural number in the context of this theorem should be modified to "any number greater than zero, such that it differs from the preceding and the succeeding number by one". This theorem makes no sense for "zero objects".Guswen (talk) 20:53, 18 April 2021 (UTC)
Well, it doesn't matter much if a publication is sloppily written, as long as the information it provides is verifiable. Again, UDT applies to natural numbers that commute under addition ( [objects]), not to ordinals that do not commute under addition. Natural numbers are established by Peano axioms and the axiom of the successor function establishes their countable infiniteness (). The definition of a countable set, the elements of which can always be counted (even if the counting may never finish), pertains to objects to which the UDT applies. Any set of objects can be classified in ways and any objects from this set share classes. If makes you uneasy, kindly note that one could define natural numbers using a predecessor (instead of successor) function such that is false (that is, there is no predecessor of 0). Woodward proofs concern "all Boolean functions with combinations of inputs (...). There are possible Boolean functions" (cf. V.A). This is the same as in the proof of the UDT that I have just quoted. If you disagree with these arguments kindly consider putting our dispute under some arbitration.Guswen (talk) 08:21, 27 April 2021 (UTC)
References
Thank you. I do not see a problem with recursion (cf. e.g. factorial recursive formula) for the predecessor function . However, natural numbers are not closed under subtraction and that, indeed, can render such a definition problematic or impossible. Guswen (talk) 14:29, 27 April 2021 (UTC) As we received no feedback whatsoever to our dispute at Wikipedia talk:WikiProject Mathematics after two weeks since it has been posted, it stands clear to me that the UDT applies to countable sets (not necessarily infinite) of objects. Commutativity under addition is a prerequisite to count the objects (2+3=3+2=5 [objects]).Guswen (talk) 07:39, 10 May 2021 (UTC)
@Guswen and JayBeeEll: I have available both Watanabe's 1969 book, and his 1965 book chapter (in French, cited in the 1969 book). As far as I could see, he didn't explicitly discuss finiteness or infiniteness of m (his variable name for the number of "object types") in any of them. However, in the 1969 book, there are some strong indicators that he tacitly assumed m to be finite: in the preparatory section 7.5. referred to from the proof, he uses a "matrix" (also called "table") of size m×n (p.363 top), a "permutation" of its rows (p.367 mid), and the binary logarithm "log2" of m (p.371 mid). In the proof itself (sect. 7.6), he uses subtraction and binomial coefficients involving m (p.377 mid, cf. the Wikipedia article). None of this makes sense for infinite m. - Jochen Burghardt (talk) 18:16, 24 May 2021 (UTC)
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