Talk:All horses are the same color

Old talk
I moved this:
 * A similar paradox is a proof (by Raymond Smullyan) that all horses have thirteen legs. First take all your horses and paint them red. Now look at the horses. If all the horses have thirteen legs, then we can stop. But what if one or more of the horses don't have thirteen legs? Well, that would be a horse of a different colour! However, we've painted all the horses red, so it would actually be a horse of the same colour. Hence, we have a contradiction, so by reductio ad absurdum, all horses have thirteen legs.

I don't see why a horse that doesn't have 13 legs would have a different color. It would still be red, like all the horses. AxelBoldt 20:02 22 Jun 2003 (UTC)


 * It's a phrase: "horse of a different colour". Martin 21:34 22 Jun 2003 (UTC)


 * What does that phrase mean? AxelBoldt 20:55 24 Jun 2003 (UTC)


 * I would like to know too. In any case it is a word play, not a logical paradox, and doesn't belong here. -- Arvindn 06:58 25 Jun 2003 (UTC)


 * To answer Axel, "horse of a different colour" is a (chiefly American? British?) slang phrase that means roughly "one that does not fit the expected pattern". So this example is cute but has nothing to do with math or logic.  Incidentally, I have usually heard this logical paradox referred to as "the billiard ball paradox", not the "horse paradox", proving that all billiard balls are the same colour. -- Revolver


 * Smullyan's argument has everything to do with logic: the ontological argument is a logical argument for the existence of God that many believe is flawed due to semantic issues, just as Smullyan's "proof" is flawed. To be sure, it has nothing to do with formal logic, or with math. Martin 20:46 26 Jun 2003 (UTC)


 * Agreed, Smullyan's argument is an example of a proof that is flawed based on semantics, so yes it is related to logic. My formal bias shines through.  I wouldn't say that it is "similar" to the original horse paradox, though, because the horse paradox fallacy is based on a technical (i.e. purely mathematical) mistake in applying induction, namely the assumption that two sets have nonempty intersection, when in fact they do not.  This is not a semantic mistake.  It's a very interesting example to consider if you're discussing paradoxes in general.  I'm just not sure if it's useful here, because it's a different type of logical paradox (a paradox of a different colour...sorry :-/), and because the root of the paradox lies is a fairly culture-specific slang expression that may not be familiar to a lot of readers outside North America or England.  With regard to the name, I didn't mean to suggest it should be changed or anything, but I have heard "billiard ball paradox" so often, that there should probably be an entry for this pointing to the horse paradox entry.  (I haven't tried to make any of these pointers yet.) Revolver


 * Maybe if there was a wordplay paradox article, or something? I see your point, though. I wouldn't really mind if it was moved elsewhere - as long as there's a link from here to there :) Martin 16:22 28 Jun 2003 (UTC)


 * Good grief, I looked up "billiard ball paradox" on google, and apparently this name is shared with a paradox in theoretical physics about time travel. Too many paradoxes, not enough names! Revolver


 * OK, I guess I'm highly biased towards formal logic too. In my mind "logical paradox" is implicitly "formal logical paradox". BTW, the paradox page implies that too:


 * The identification of a paradox based on seemingly simple and reasonable concepts has often led to significant advances in science, philosophy and mathematics.


 * Look at this one which is a more obvious word-play: "A penny is better than nothing. Nothing is better than eternal bliss. Therefore a penny is better than eternal bliss." Would you consider that a logical paradox? (Just to see how far apart our viewpoints are :-) -- Arvindn 04:05 27 Jun 2003 (UTC)


 * Yeah, I think I would (the version I heard related to peanut butter sandwiches). Quite an important one, imo, because it shows why you probably shouldn't reify concepts like "nothing" and "existence". But hey, I'm an amateur. Martin 16:22 28 Jun 2003 (UTC)

I'm happy with leaving Smullyan's story in since it is amusing, but I also wouldn't call it a paradox, and I don't think it has anything to do with logic or semantics. If it did, it could be formulated in any language, but it only works in English. AxelBoldt 02:41 28 Jun 2003 (UTC)

I'd like to point out that Smullyan's story, apart from the language-dependent "flaw" mentioned above, also has a logical fallacy. Just if one or more horses don't have 13 legs, we can't say they don't belong to the group. In fact (in our world) they form the whole group (since all horses have 4 legs). We can only deduce that a horse that doesn't have 13 legs is of a different color (doesn't belong to the group) if we also know there is also a horse with 13 legs. -- Paddu 18:50 29 Jun 2003 (UTC)


 * I think that should be added. AxelBoldt 23:02 29 Jun 2003 (UTC)

I always thought that "the horse paradox" referred to the story I added. It was what I thought of when I first saw the article title.

I know another "paradox" which is actually a word play and is somewhat related to horses. Don't know, maybe it sounds quite silly in English, but in Russian it's a bit confusing and funny.


 * Let's prove that unicorns exist. We mean true unicorns here, not rhinoceroses or something else, we mean a mythical creature that resembles a horse with a single straight horn on its forehead.


 * To prove that set something exists, we can for instance, prove that some specific subclass of it is non-empty. To prove that rectangles exist, it suffices to prove that squares exist.


 * So, let's prove that there is a subclass of unicorns that is non-empty, i.e. that exists. For determinancy, let's prove that existing unicorns exist.  First, consider that existing unicorns don't exist.  But that's a contradiction---how can something that exists (existing something) not exist?  Therefore, we've come to conclusion that existing unicorns exist.


 * Now, if some (existing) unicorns exist, this means that unicorns in general exist. Statement proven.

Don't take this seriously ;)


 * Paul Pogonyshev 23:01, 16 Jul 2004 (UTC)

11 horse paradox
The puzzle needs either a wriggle wording (I have inserted the words "in his stable") or expanding in some way, to explain how the Lawyers horse "counts". E.G. the lawyer (or in some verions a folk-tale hero type) says "I will solve your problem. I'll lend you my horse....")

Is this an entry for bad jokes or serious conundrums?

Removal of other paradoxes
In the VfD discussion Votes for deletion/Horse paradox, I count 4 votes for keeping only the first section, 2 votes for keeping the whole article, 6 keep/cleanup votes that do not specify what to do, and 5 delete/merge votes. Therefore, I deleted the other "paradoxes". -- Jitse Niesen 01:19, 8 Jun 2005 (UTC)

Related Paradoxes
I discuss this more to bring light to something... What do you think about the relationship between this paradox and Paradox of the heap? They are related, in that induction is used. But they aren't quite the same. This paradox is about using induction to prove the validity of something for which you haven't proved the base case. The latter paradox is more about human speech, and the difficulty of defining or using logic to define what would seem to be something quite simple to define.

Granted, the current (latter) article could be improved somewhat (how I'm not sure, but some sourcing would be nice). I just wonder what if any mention could be stated in this article in relation to the other article, or if anything needs to be said at all.

Root4(one) 03:29, 22 March 2007 (UTC)

I've gotta say, the picture of that horse is so useless in this article. Why on earth is it there? 128.135.230.204 19:18, 9 April 2007 (UTC)


 * It's certainly not useless. I find it akin to some visualization/mnemonic tricks used to remember word definitions for tests like the SAT.  I mean, even though horses actually have nothing to do with the paradox, that is the paradox's name, so why not have a picture of a monochromatic horse? Root4(one) 01:06, 10 April 2007 (UTC)

Polya?
I've seen this attributed to George Polya. Is that right? If so, the article should say so. Michael Hardy 21:08, 8 May 2007 (UTC)

Say, assume, induce_________makes no sense.
Is there something missing about being one color? I see nothing to indicate that. The article has as follows: ''Now assume the truth of the statement for all sets of at most n horses. ... By the induction assumption, all horses in this set are the same color.''

If you just assume anything, and it's not true, is that a paradox?

Saying a horse with whatever is a different color is not necessarily true either. There is no establishing these facts. —The preceding unsigned comment was added by 68.180.38.41 (talk • contribs).


 * No, you're just missing everything in the article completely. It says at the outset that this is about mathematical induction.  If you don't understand what that is (and it certainly appears that you don't), then you won't have any idea what's being said here until you find out what mathematical induction is.  There is something called an "induction hypothesis".  To find out what that is, read mathematical induction. Michael Hardy 23:33, 24 May 2007 (UTC)

That did not offer any explanation. I see no reason to go to insults & not try to elaborate on the principle at hand. Please keep the maturity level to offering more help/info. Thank you though for the reference to the induction page. I will read that to educate myself further.68.180.38.41 01:28, 11 June 2007 (UTC)


 * I did not insult anyone. I said in order to understand this argument, it is necessary to understand what mathematical induction is, and I linked to the article on it. Michael Hardy 19:07, 4 October 2007 (UTC)


 * Sorry M[rs]. IP-address felt insulted, but I think Michael Hardy was helpful enough and polite. Linking is much better than explaining here. PJTraill (talk) 22:13, 7 January 2008 (UTC)

Original version
Which book of Polya first introduces this paradox?

I've found the paradox as an exercise in his book "Mathematics and Plausible Reasoning (Volume I)" chapter VII exercise 17. But there is no mention to horses:

"Are any n numbers equal? You would say, No. Yet we can try to prove the contrary by mathematical induction. It may be more attractive however, to prove the assertion "Any n girls have eyes of the same color"."

Rend 03:49, 6 October 2007 (UTC)

not mathematical induction?
I'm not a logician but it seems to me there is another problem with the line of reasoning that hasn't been mentioned. Even ignoring that the case for 2 horses fails, this is not mathematical induction anyway.

Compare with this: If we had a proof (P) that all horses are white we could prove that one horse is white. My horse is white. Therefore the proof P is valid. Therefore all horses are white.

This is obviously flawed because my horse being white can occur even if not all horses are white. Similarly, we could argue that in the OP's 4 horse proof, having all horses in a group of 1 horse be the same colour can occur for reasons other than all horses being the same colour (as, indeed, it can).

For this to be mathematical induction we would have an argument along the lines of:

I have a proof (P) that shows for any size group of horses all of one colour, a group of one more horse will also be all the same colour. A group of one horse is a group of all one colour, therefore all groups are the same colour.

This is a correct form of argument but of course, since we don't have a proof P, it isn't true.

Thus, as well as the explanation due to 'special cases', there is also an explanation due to a flawed form of argument that only resembles mathematical induction. Fontwell (talk) 14:43, 10 September 2008 (UTC)


 * I don't understand. This is in the form of mathematical induction, and the only flaw is the 2 case. Of course, it's much better written at Mathematical induction. Starting from an example of 5 horses seems very strange for anyone familiar with induction, and those not familiar with it will hardly be helped by it anyway. Can't we just copy the section from the MI article? -- Jao (talk) 15:19, 7 December 2008 (UTC)


 * Why don't we replace it with a redirect to Mathematical induction?--Prosfilaes (talk) 21:35, 12 August 2010 (UTC)


 * I agree with Jao. This may be the "form" of mathematical induction, but it has nothing to do with mathematics, because, (if I may attempt to explain things that are over my head) adding an unknown element to a set gives an unknown result and can't be approached inductively, and frankly I think it's one of the biggest horses*** articles in all of wikipedia.  72.177.123.145 (talk) 03:34, 15 May 2013 (UTC) Eric


 * You're right, they are over your head. If out of a set of items, you can take any two of them and they have the same color, then all of them have the same color. That's the essential point of the mathematic induction being done here, that that case is important.--Prosfilaes (talk) 12:05, 27 July 2013 (UTC)

WP:ENGVAR
"Color" and "colour" are currently mixed throughout the article. -- Jao (talk) 15:19, 7 December 2008 (UTC)

Hmm?
I don't see a paradox, what I see is a sad attempt to convince someone that a lunchbox is a stapler by removing real facts and making false statements upon obviously false statements. The article says that the groups of one have the same color, okay, now combine them back into the group of 5, oh look! a black horse, that's not brown! This is not a paradox!

Also, this is exactly like saying that all keys on a keyboard are the same, this is untrue, because you can't prove that q,w,e,r,t,y etc. are all 'q' (especially by grouping them into groups of one) because once combined into the group that originally brought up the question, you once again have "qwerty", not "qqqqqq"!

A paradox consists of two statements that seem true but contradict each other because one impossible fact was assumed or let possible. Paradox's are usually pointed out to prove something impossible, like the grandfather paradox: it involves killing, conception, and time travel, but something in the equation makes it a paradox because both sides ultimately contradict eachother, one part of this must be false or impossible. Killing someone is possible, being born is possible, therefore time travel must be impossible, and the fact that we never have accomplished--or even come close to--such a feat further proves that. The same thing can be said about uncyclopedia's 1=2 article, in which the proof is that 10 = 1, and 20 = 1, therefore 10 = 20, simplifying the equation by removing the powers (10 --> 1; 20 --> 2) ends in 1 = 2, but the false statment allowed true for this to seem true is that you can remove powers without changing the value of a number. This is wikipedia, not uncyclopedia, so don't make articles that seem like they came from uncyclopedia. This horse article is somthing I would expect from uncyclopedia: completely lacking of any and all logic. This article is 1 = 2, and somebody might even make an article based off of this equivalent to everything = cake.

Another really stupid paradox is the drinker's paradox, there is no contradiction what-so-ever, nothing but positives, maybe if the statement was "if one person is drinking, nobody is drinking" then it would make sense because this one person is also included, and therefore it can't be said that no one is drinking, because one person is. But this is not the case, in fact, the article says that when one person drinks, everyone drinks. Let's bring up the paradox formula again: Drinking is possible, Coincidence is possible, there is nothing here that is impossible, there is also no contradiction, but positive upon positive. The drinker article should really be removed because it makes no sense.

Unless someone can explain these articles without resorting to false and circular logic that ignores all common sense (like the two articles), AND I myself admit that you have a good point and that I agree (you can't just say something stupid or give a long mathematical equation and then proclaim that you are right, because you could be making it up), my argument stands undisputedly correct. Prove me wrong 168.103.126.103 (talk) 18:05, 8 December 2011 (UTC)


 * It's notable. It is not in Wikipedia's rules that we have to convince you of anything.--Prosfilaes (talk) 09:37, 9 December 2011 (UTC)
 * But it is in the rules to at least be coherent, and this "paradox" is not coherent if it implies that you can magically make all the keys on the keyboard "Q" just by saying "Q in a group of 1 is the same." 168.103.126.103 (talk) 19:17, 13 December 2011 (UTC)
 * It's simple incorrect induction. It's not magic.--Prosfilaes (talk) 14:17, 14 December 2011 (UTC)

n=2 case is not the one that fails (notational confusion)
The section "Explanation" says


 * The argument above makes the implicit assumption that the two subsets of horses to which the induction assumption is applied have a common element. This is not true when n = 2, that is, when the original set only contains 2 horses.


 * Let the two horses be horse A and horse B. When horse A is removed, it is true that the remaining horses in the set are the same color (only horse B remains). If horse B is removed instead, this leaves a different set containing only horse A, which may or may not be the same color as horse B.

But this passage assumes wrongly that n is the size of the "original" set, that is, the one before removal. But in the problem statement the size of the original set is n+1, not n. From the first sentence of the problem statement: Let us consider a group consisting of n+1 horses.

So the second quoted paragraph deals with the case n+1=2 hence n=1. So I'm going to change "n=2" to "n=1" in the first quoted paragraph. Duoduoduo (talk) 23:17, 8 March 2013 (UTC)

Proof by induction is often falsely presented as "First you prove the proposition to be true for n=1. Then under the assumption that it is true for n you prove it to be true for n+1. Then you are done." It need not be "true for n=1" but "true for a smallest value of n". The example with the horses comes in handy as an example where it can't be "n=1" since the smallest GROUP of horses is a pair of horses. And the proof by induction fails at the Base case: Two horses (who need not be the same color). (And it need not be n+1 either: "prove that this and that holds true for any odd number >7." 1) Prove it to be true for n=9. 2) Under the assumption that it is true for n prove it to be true for n+2.) — Preceding unsigned comment added by 83.223.9.100 (talk) 10:52, 15 March 2017 (UTC)

unexpected hanging paradox
I added a link to the Unexpected hanging paradox, because it looks to me like another case of incorrect application of mathematical induction: after proving the judge's proposition false for the index case (Friday) the prisoner assumes the judge's proposition is true to prove that it is false for the previous day and so on through the entire week.--Wikimedes (talk) 23:00, 25 July 2013 (UTC)

Could this be an alternative explanation?
Let H be the set of all horses. We set out to prove by induction that, for all n in N, if there are exactly n elements in H, then all horses are the same colour.

The base case is trivial since a single horse will always have the same colour as itself.

The induction hypothesis can be stated: If there are exactly n elements in H, then all horses will be the same colour.

Then we suppose that there are exactly n+1 elements in H. The problem is that the induction hypothesis is cannot be applied in this case. It can only be apply in the case that there are exactly n elements in H. It does not follow from the induction hypothesis that all sets of n horses are the same color. --Danchristensen (talk) 14:50, 1 June 2015 (UTC)
 * Your understanding of induction is incorrect. And no, I will not explain induction to you. Please take classes or read a book (or wikipedia :-). If you want to imp[orve thie article, please look for information in reliable sources which can be added here. Your own ideas cannot be used in wikipedia. Staszek Lem (talk) 19:00, 1 June 2015 (UTC)

P.S. I've noticed that in several other places you have already been told that wikipedia is not a forum. Please stop distracting wikipedians from working on the wikipedia's task: building encyclopedia based on reliable sources. 20:00, 1 June 2015 (UTC)

Explanation should simply reference the well-ordering principle
The reason that this is a fallacious "proof" is simply because it ignores the ordering requirement of mathematical induction. In order to apply mathematical induction, you must be able to assign a fixed ordering to the elements of the set under study. See https://en.wikipedia.org/wiki/Mathematical_induction#Equivalence_with_the_well-ordering_principle

As written, the inductive hypothesis "n horses are the same color" does not satisfy this requirement. A valid inductive hypothesis would instead say "the first n horses are the same color". Under this hypothesis, it would be impossible to assert that "any subset of n horses are the same color" because the hypothesis explicitly references the first n horses.

--James Monroe (talk) 21:31, 11 August 2018 (UTC)
 * Do you have a reference for this reasoning? Because what you write makes no sense. Staszek Lem (talk) 20:02, 13 August 2018 (UTC)
 * "n horses are the same color" is not the rigorous, precise statement of the inductive hypothesis, which is rather "any subset of the set of horses with cardinality n is such that there exists one color that is the color of every horse in that subset". The induction is on the power set of the set of horses, not the set of horses itself, so ordering within each particular subset is irrelevant, as shown by Michael's reformulation of the inductive step section.--Jasper Deng (talk) 00:58, 12 July 2019 (UTC)

. The numbers are indeed ordered. And the induction step is in fact correct, except when n = 1. The fact that it doesn't work for getting from 1 to 2 is the only flaw in the argument. Michael Hardy (talk) 00:09, 12 July 2019 (UTC)
 * You don't ever seem to understand that our job is emphatically not to parrot what the sources say. There is inherent rephrasing that must occur in this situation.--Jasper Deng (talk) 00:48, 12 July 2019 (UTC)
 * It was not a simple "rephrasing": this was a fundamental change (on this foundation-level of mathematics). Don't assume people who disagree with you are idiots. You don't seem to understand that theorems are to be stated verbatim, and it is not "parroting the sources", but mathematical precision. Hardy's reformulation changed the fundamental math of the original statement. Staszek Lem (talk)
 * You are blatantly and completely incorrect in both regards. There isn’t even a theorem being proven here!—Jasper Deng (talk) 17:33, 12 July 2019 (UTC)

The real issues
While I basically agree with D.Lazard and Jasper Deng that the actual mathematical content seems sufficiently sourced and the battle over a missing footnote or supposed OR seems largely misplaced, there are however some more subtle issues with the sources upon a closer inspection.
 * a) A minor but annoying point is that the article gives several books (Polya, Martin and the one containing Cohen's article) without page numbers.
 * b) The article claims the paradox is due to Polya and gives Polya's book as a source. That is actually really problematic case of OR. What is needed here to source the claim is a publication by a third party attributing the the paradox to Polya rather than just giving Polya's "original" publication.
 * c) The content of Polya's publication is actually somewhat misrepresented. Not with regard to the actual mathematical content but the framing. Polya writes about it in his book at page 120 and calls the problem "Are any n numbers equal?" and rephrases it in term of girls and the color of their eyes. There is however no mentioning of horses.

While it is true that some publications attribute the paradox/example to Polya (for instance, ,), I'm right now not really clear whether Polya ever phrased the paradox in terms of horses and their color. In fact i'm not even convinced that it is due to Polya at all. All I know for sure at this point is, that some publications attribute the horse paradox to Polya (without giving Details or references) and that Polya describes the paradox without horses in his book from 1954. So imho we need to formulate the attribution part in the lead more carefully or need to consult additional sources. I didn't have access to Martin or Cohen though may they shed some more light on the issue (I#m skeptical though).

Btw some sources that could be used for the explanation section:, ,

--Kmhkmh (talk) 04:06, 8 July 2019 (UTC)
 * I admittedly did overlook the history of this paradox, not being that much of a math historian myself.--Jasper Deng (talk) 04:28, 8 July 2019 (UTC)

Using the string "Are any n numbers equal?" I have immediately found Polya's text and update the article accordingly, unlike y'all page owners.

By the way, an someone add an illustration, similar to that of given by Polya. It does help understanding the "proof":

Staszek Lem (talk) 20:49, 9 July 2019 (UTC)

A vaguely remember a pop math book from my childhood which picked on Polya's formulation "Any n girls have eyes of the same color": it was pointed out that the base step is already false due to the possibility of  heterochromia. Staszek Lem (talk) 20:59, 9 July 2019 (UTC)

This is the link to a PDF file of VanDrunen (search for the word "girls" there), but I am not sure whether we can use it in the article due to the unclear copyright status of the pdf there. Staszek Lem (talk) 21:36, 9 July 2019 (UTC)
 * The copyright status of the pdf doesn't really matter far as its use as source is concerned. What matters is the reputation of the author and that is is reliably publoshed somewhere. This seems to be case, the author and its book can be found here. Assuming the pdf is not legal it can't of course be linked, but the original publication of which it is a copy can be cited.
 * While this looks like good source for the article and points to Polya's publication as the origin for his description, it does however say anything about the horse story. So we're still more or less stuck with what we've known before. That is, we know what Polya published in 1954, but it is unclear from where the framing in terms of horses originates and it still seems a bit uncertain whether Polya's publication was really the first one on that subject.
 * A graphic illustrating the critical non-empty intersection would imho be helpful.
 * --Kmhkmh (talk) 22:31, 9 July 2019 (UTC)
 * This 2003 self-published book says that it is a joke found on the internet. Staszek Lem (talk) 23:17, 9 July 2019 (UTC)
 * This is a 1965 book called "The Worm Re-turns: The Best from the Worm Runner's Digest" meaning there exists an earlier text in a Worm Runner's Digest issue, which is pretty close to Polya's time frame. Staszek Lem (talk) 23:18, 9 July 2019 (UTC)
 * Getting closer: From here: " Pejorative Calculus (Joel Cohen, On The Nature Of Mathematical Proofs, The Worm-Runner's Digest, Vol. III, No.3,  December 1961), where Lemma I (all horses are the same color) is  credited to Professor Lee M. Sonneborn, then at U. of Kansas. " - i.e.,. the very same  Cohen cited in our article. Staszek Lem (talk) 23:26, 9 July 2019 (UTC)
 * and it is a hilarious read. Staszek Lem (talk) 23:34, 9 July 2019 (UTC)
 * Ok I think we know definitely need a copy of Cohen's original article. If nobody here has access I'm trying to request it via REREQ.--Kmhkmh (talk) 23:42, 9 July 2019 (UTC)
 * Well the whole thing (with and without horses) is certainly math folklore for a long time (see also this article from 1986 unfortunately without a specific reference to its origin or Polya). In German speaking countries I've also see also at least dating back to the 80s, interestingly there is is often phrased with cats rather than horses (alluding to the idiom "all cats are grey in the dark"). The according article in the ProofWiki gives additional background on horse thing and cites a source from 1964. I'm not sure how reliable that is though and how much of the content was actually taken from the given source.--Kmhkmh (talk) 23:35, 9 July 2019 (UTC)
 * P.S. I was right to distrust the sourcing on Wikiproof. The source from 1964 is accessibke via Google books, it has the basic faulty induction, but nothing on the horses. In fact it proves any n objects are equal.--Kmhkmh (talk) 23:55, 9 July 2019 (UTC)
 * Cohen (1961) beats them all so far. Staszek Lem (talk) 23:39, 9 July 2019 (UTC)

There is one thing I don't quite get. Was Cohen's article just published in The Worm's Digest and Opus is part of the satire or was the first(?) publication slightly earlier in journal/magazine called Opus.--Kmhkmh (talk) 09:32, 10 July 2019 (UTC)
 * I only get this in snippet view, but here it is in The Worm Runner's Digest in Google Books: . It was apparently collected in a 1965 book "The Worm Re-turns: The Best from the Worm Runner's Digest". I don't know about the Opus part. —David Eppstein (talk) 09:45, 10 July 2019 (UTC)
 * Opus is mentioned here, where a shortened version of On the nature of mathematical prrofs was republished in 1973, it does however not mention the Worm Runner's Digest explicitly for this article. I also saw elsewhere that Cohen had published in Worm's Runner's digest, but a different article. Also Opus was given in the original reference in this WP article (Cohen, Joel E. (1961), "On the nature of mathematical proof", Opus. Reprinted in A Random Walk in Science (R. L. Weber, ed.), Crane, Russak & Co., 1973.). It would be great if somebody could get a copy of the original Worm Runner's digest article (or at least a copy from the compilation book you've linked. Or the publication in Opus if there ever was one (and such a journal or magazine).--Kmhkmh (talk) 13:35, 10 July 2019 (UTC)

Statement of paradox
I grudgingly may agree that in proofs wuikipedians can be allowed some slack, as long as their reasoning may be verified. However the statement of the paradox must follow the source as close as possible, because the only verification of the formulation of the problem is a published source. One may reword the formulation a bit, but one cannot change it in its essence. The published formulations say "first horse" and "last horse", not "some horse" and "some other horse". Hint for smartasses: in order to work with the latter formulation, one has no invoke the axiom of choice, i.e., one has to prove that "some other horse" exists. Dont say it is trivial: axiom of choice was a shaker for set theory. Staszek Lem (talk) 15:57, 12 July 2019 (UTC)
 * P.S. If you think that the initial formulation was somehow "imprecise" and you may write up  "a better one", please keep in mind that sometimes paradoxes are intentionally obfuscated to make it more difficult. And if someone finds another solution by picking of the phasing, kudos for them. For example, I remember someone picked on Polya's original formulation with girls by noticing that the statement "one girl has eyes of one color" is wrong. Unfortunately I cannot find the ref now, but this would be a nice addition to the article. Staszek Lem (talk) 16:30, 12 July 2019 (UTC)
 * Your reference to the axiom of choice here shows you’re not qualified to be assessing the mathematical details of the paradox (we do not need the axiom of choice here: the argument applies for all members of the power set of the set of horses with at least two distinct members simply by definition of cardinality).—Jasper Deng (talk) 17:36, 12 July 2019 (UTC)
 * OK I suppose flunked the set theory. Still, the replacement text "(not identical to the one first removed)" raises the red flag. What is "identical"? Wikipedia says "Two things are identical if they are the same" . Now, if we make a set of the two things in question, what is the cardinality of the set? Since we are discussing "two things" then the cardinality must be 2, right? Staszek Lem (talk) 21:11, 12 July 2019 (UTC)
 * Yes, but it is not appropriate to force the English language to comply with set theory. Half of all English words don't comply with its rules.--Jasper Deng (talk) 21:13, 12 July 2019 (UTC)

Explanation section
The "Explanation" section refers to "the horses in the middle." However, neither the "Inductive step" nor anywhere in the attempted proof there is any mention of middle.

A concise and clear explanation of the erroneous use of the inductive step is given in [| Mathematical induction, Example of error in the induction step] --UriGeva (talk) 10:17, 5 November 2023 (UTC)


 * Suggestion: Revise the Explanation section by replacing the existing text with something like the following:
 * The inductive step implicitly assumes that every set of n+1 horses can be split into at least 2 subsets of n horses, such that these two subsets have a nonempty intersection; that is, the intersection of the two sets contains at least 1 horse.
 * The fault with this argument is due to the fact that for n+1 = 2, only 2 subsets are possible and each subset contains a single horse: one subset contains the first horse and the other subset contains the second horse. Therefore, their intersection is empty (having no horses), which contradicts the assumption underlying the inductive step.UriGeva (talk) 23:27, 7 November 2023 (UTC)
 * The current explanation is actually wrong. It states "The argument above makes the implicit assumption that the set of horses has the size at least 3." Someone may infer that to correct the faulty argument just reset the base case of the induction from 1 to 3. Obviously starting the induction at 3 should also fail. The implicit assumption is more general: As described above, it requires that, given n+1 horses, the intersection subset of every pair of n horses subsets, is not empty. Once n = 2 fails, the case for n = 3 horses is no longer possible. This invalidates the option to start the induction with (set the base case to) n = 3.UriGeva (talk) 02:41, 11 November 2023 (UTC)

References section
In Reference #5, the link "the original" is broken. It should be updated with the following currently valid URL: https://math.hmc.edu/funfacts/all-horses-are-the-same-color/ UriGeva (talk) 23:47, 7 November 2023 (UTC)


 * I updated the link.UriGeva (talk) UriGeva (talk) 02:47, 11 November 2023 (UTC)